UNIVERSITY OF LJUBLJANA FACULTY OF MATHEMATICS AND PHYSICS DEPARTMENT OF PHYSICS Nejc Kosnik STANDARD MODEL AND SIGNATURES OF NEW PHYSICS IN WEAK AND RADIATIVE DECAYS OF HEAVY MESONS Doctoral thesis ADVISOR: prof. dr. Svjetlana Fajfer UNIVERZA V LJUBLJANI FAKULTETA ZA MATEMATIKO IN FIZIKO ODDELEK ZA FIZIKO Nejc Kosnik STANDARDNI MODEL IN SLEDI NOVE FIZIKE V ŠIBKIH IN RADIACIJSKIH RAZPADIH TEZKIH MEZONOV Doktorska disertacija MENTOR: prof. dr. Svjetlana Fajfer Abstract We study rare decays of heavy B and D mesons in the standard model and several new physics models. For the extremely rare process b ^ dds we use the experimental limit of B- ^ n-n-K + to constrain parameter space of model with Z' gauge boson and MSSM with broken Rp-parity. We predict upper bounds of several two-body hadronic B- decay branching fractions which turn out to be of the order 10-7 in the case of Z' model. For the c ^ u£+£where £ = e,p we study spectra and decay widths of D ^ Py and D(s) ^ n(K)£+£-. We model the long distance amplitude with the intermediate resonances using the Breit-Wigner shape. We find new bounds on the new physics parameters coming from the D+ ^ n+p+p- experimental upper bound and predict decay spectra and widths of the Ds ^ K£+£-. We find MSSM with broken Rp-parity and the model with singlet leptoquark in SM representation (3,1, -1/3) have best prospects to be probed in experimental searches. We use a combined heavy quark, large energy, and chiral symmetry of QCD and make a prediction of photon spectra of B ^ Kny in a kinematic region with hard photon and one soft meson. Precision of future experiments will allow for probing b ^ sy transition in this kinematical region. In experimental analyses of B- ^ D0£v decay there is a background of events with additional undetectable soft photon. We find that D0* contributes dominantly in this respect due to small mass splitting between D0* and D0. Future experiments should be able to detect photons of energy below 100 MeV in order to extract Vcb with precision of order 1%. Keywords: weak decays of heavy mesons, flavour changing neutral current, new physics searches, radiative decays PACS (2008): 13.20.-v, 13.20.He, 13.20.Fc, 13.25.Hw Povzetek V tezi raziščemo razpade težkih mezonov B in D v okviru standardnega modela in nekaterih modelov nove fizike. Za izjemno redek proces b ^ dds uporabimo obstoječo zgornjo eksperimentalno mejo razpada B- ^ n-n-K+, da omejimo parametre modelov nove fizike, kot sta model z dodatnim umeritvenim bozonom Z' in minimalni supersimetrični standardni model s krseno parnostjo Rp. Napovemo zgornje meje razvejitvenih razmerij za nekatere dvodelčne hadronske razpadne kanale mezona B-. Le-te so v primeru modela z bozonom Z' reda 10-7. Razisčemo spektre in razpadne sirine razpadov D ^ P7 in D(s) ^ n(K)£+£-, kjer je £ = e, ki temeljijo na pročesu c ^ u£+£-. Dolgosezne prispevke amplitude modeliramo z vmesnimi resonančnimi stanji in Breit-Wignerjevim nastavkom. Izhajajoč iz zgornje eksperimentalne meje razvejitvenega razmerja razpada D+ ^ , izpeljemo nove meje parametrov za minimalni supersimetrični model s krseno parnostjo Rp in model s skalarnim leptokvarkom v reprezentačiji (3,1, -1/3). S temi parametri napovemo tudi spektre in razpadne sirine razpadov Ds ^ K£+£-. Omenjena modela nove fizike imata tudi največ moznosti zaznave v eksperimentih. Za radiačijski razpad B ^ KnY v kinematičnem območju z visokoenergijskim fotonom in enim počasnim mezonom, z uporabo simetrije tezkih kvarkov, efektivne teorije visoke energije, in kiralne simetrije kvantne kromodinamike, napovemo fo-tonski spekter. V prihodnosti bodo eksperimenti lahko merili prehod b ^ sy v omenjenem kinematičnem območju. V eksperimentalnih analizah B- ^ D0£v ima pomembno vlogo dodatni foton, ki ga eksperiment ne zazna in tako prispeva k ozadju semileptonskega razpada. Ugotovimo, da je zaradi majhne razlike mas mezonov D0* in D dominantni mehanizem taksnega ozadja resonančni pročes preko vmesnega D0*. Spodnji rez na fotone nizkih energij bi moral biti nizje od 100 MeV, da bi lahko iz analize semileptonskega razpada ugotovili vrednost matričnega elementa Vcb z natančnostjo okrog 1%. Ključne besede: sibki razpadi tezkih mezonov, okus spreminjajoči nevtralni tokovi, iskanje nove fizike, radiačijski razpadi P ACS (2008): 13.20.-v, 13.20.He, 13.20.Fč, 13.25.Hw I acknowledge the financial support of the Slovenian Research Agency and throughout completion of this thesis. Parts of this work have been completed with the financing from FLAVIAnet network, whose support I also acknowledge. Above all, I am deeply grateful to my advisor prof. Svjetlana Fajfer for all the support, stimulating discussions, and patient guidance throughout my graduate study. It is to say that without her immense help this work would have never seen the light of day. My thanks go to Jernej Kamenik, Damir Becirevic, Ilja Doršner, Saša Prelovšek, and Tri-Nang Pham with whom I completed parts of this work and meanwhile learned a great deal about physics. For fruitful discussions I also thank Miha Ne-mevšek, Jure Drobnak, and Jure Zupan. Special thank goes to Damir Becirevic who allowed me to complete a part of this thesis at LPT Orsay. During the making of this work I have enjoyed a stimulating atmosphere at the Department of Theoretical Physics of Jozef Stefan Institute, where I've spent a good time in the company of my colleagues and friends. In this respect, I thank Matej Kanduš, Samir El Shawish, and Ana Hoševar for making the coffee and lunch breaks lively. Last but definitely not least, I would like to thank Alenka for her patience and support during the last few months. Contents 1 Introduction 17 1.1 Standard model and beyond....................... 17 1.2 Flavour physics.............................. 18 1.2.1 Flavour sector of the standard model.............. 18 1.2.2 Flavour changing neutral currents ............... 19 1.2.3 Heavy meson decays....................... 19 1.3 Goals and methods............................ 21 1.4 Structure of thesis............................ 21 2 Flavour violation at low energies 23 2.1 Flavour violation in the standard model................ 23 2.1.1 CKM matrix........................... 24 2.2 Operator product expansion in weak decays.............. 25 2.3 Low energy QCD............................. 28 2.3.1 Hadronic form factors...................... 28 2.3.2 Heavy quark symmetry ..................... 29 2.3.3 Chiral perturbation theory ................... 30 3 Rare process b ^ dds 35 3.1 Effective Hamiltonian .......................... 35 3.1.1 QCD corrections......................... 36 3.2 Inclusive decay width .......................... 37 3.2.1 Standard model.......................... 38 3.2.2 Minimal supersymmetric SM.................. 39 3.2.3 MSSM with broken Rp-parity.................. 40 3.2.4 Family nonuniversal Z' ..................... 42 3.3 Exclusive B- decay modes ....................... 43 3.3.1 Vacuum saturation of matrix elements............. 43 3.3.2 Hadronic amplitudes....................... 43 3.3.3 Hadronic decay widths...................... 45 3.4 Constraints on the short distance parameters............. 48 4 Neutral currents in charm and D ^ P£+£- 51 4.1 Charm decays and resonances...................... 52 4.2 New physics scenarios.......................... 53 4.2.1 Additional up-type quark singlet................ 53 4.2.2 Supersymmetry.......................... 53 4.2.3 Weak singlet leptoquark..........................................53 4.3 c ^ uy......................................................................54 4.4 D ^ P£+£- decays: short distance amplitudes ........................55 4.4.1 SM..................................................................55 4.4.2 Models with additional up-type quark singlet..................56 4.4.3 MSSM..............................................................57 4.4.4 Rp violating MSSM ..............................................58 4.4.5 Scalar leptoquark (3,1, -1/3)....................................59 4.4.6 D(s) ^ n (K) form factors........................................60 4.5 Long distance contributions in D ^ P£+£-............................61 4.5.1 D+ ^ n+£+£- ....................................................62 4.5.2 D+ ^ K+£-£+....................................................63 4.6 Decay spectra and widths of D(s) ^ n(K)£+£-........................64 4.6.1 D+ ^ n+£+£- ....................................................64 4.6.2 D+ ^ K+£+£-....................................................68 4.7 Summary ..................................................................70 5 Dalitz plot analysis of the B ^ KnY decays 75 5.1 Framework ................................................................76 5.2 Large energy limit of B ^ P form factor................................78 5.3 Hard photon spectra......................................................79 5.4 Summary ..................................................................81 6 Radiative background of B ^ D£v decays 83 6.1 Extraction of Vcb in B ^ D£v............................................84 6.2 Amplitude decomposition................................................85 6.2.1 Single particle poles in and ............................86 6.3 Hadronic parameters of B ^ D* and D* ^ Dy transitions ..........87 6.4 Irreducible background to B- ^ D°£v channel from D0* ^ D0y . . 88 6.5 Summary ..................................................................89 7 Concluding remarks 91 A Technical aside on b ^ dds process 93 A.1 Matching and renormalization of composite operators ................93 A.1.1 Mixing of effective operators in b ^ dds........................96 A.2 GIM mechanism in the b ^ dds..........................................97 A.3 Parameterization of B ^ n form factors................................97 A.3.1 B ^ n form factors ..............................................97 A.3.2 B ^ p form factors................................................97 A.3.3 K ^ n form factors ..............................................98 B Ward identity and amplitude expressions for B- ^ D0£z/y 99 B.1 Ward identities............................................................99 B.2 Invariant coefficients of the V^ and in B- ^ D°*£ž/ ^ D°£z7y . 100 B.2.1 Contribution to V®?.......................100 B.2.2 Contribution to .......................100 Bibliography 101 8 Razširjeni povzetek disertacije 111 8.1 Standardni model in njegove razsiritve.................111 8.2 Fizika teZkih kvarkovskih okusov....................111 8.2.1 Kvarkovski okusi v SM......................112 8.2.2 Razvoj produkta operatorjev v sibkih razpadih........113 8.3 Redek proces b ^ dds..........................113 8.3.1 Inkluzivni razpad.........................114 8.3.2 Ekskluzivni razpadi mezona B- ................115 8.4 Nevtralni tokovi carobnega kvarka in razpad D ^ P£+£- ......117 8.4.1 Razpad c ^ uy v MSSM ....................117 8.4.2 Kratkosezni prispevki k c ^ u£+£-...............118 8.4.3 Dolgosezni prispevki v D ^ P£+£-...............119 8.4.4 Primerjava resonancnih in kratkoseznih spektrov.......119 8.5 Analiza Dalitzovega diagrama za razpad B ^ Kn7..........121 8.5.1 Pristop z efektivnimi teorijami kvantne kromodinamike . . . 121 8.5.2 Fotonski spektri .........................122 8.6 Ozadje mehkih fotonov v razpadih B ^ D£v .............122 8.6.1 Enodelcna vmesna stanja....................123 8.6.2 Spekter mehkega fotona.....................124 8.7 Zakljucek.................................124 Notation Greek indičes v,... run over the spačetime čoordinates 0,1,2,3. Spačetime metrič is diagonal with elements -g00 = g11 = g22 = g33 = -1. Covariant derivative operator is dM = dX-. d'Alambertian d2 is defined as 3^3^. The four-dimensional Levi-Civita tensor is totally antisymmetrič and has e0123 = +1. Dirač gamma matričes 7^ satisfy the antičommutation relation 7^7^ + 7V7^ = 2g^v. We define 75 = i7°717273 = -i/4! 7^7V7^7^. Left- and right-handed proječtion operators are defined as PL,R = (1 T 75)/2. We use units where h = c = 1. H.č. denotes a Hermitian čonjugate of the prečeding expression. Chapter 1 Introduction The standard model of particle physics has been established as a satisfactory theory of smallest verifiable distances and time scales. It is a model with 19 free parameters which have been overconstrained by hundreds of measured observables collected during the last three decades. Still to date there has been no direct experimental evidence of an observable that would, including experimental and theoretical uncertainties, deviate more than 3 standard deviations from the standard model predictions. The only exception is the observation of neutrino mixing, which cannot be explained with massless neutrinos of the standard model. 1.1 Standard model and beyond Further indirect experimental evidence supporting incompleteness of the standard model (SM) originate from cosmological and astrophysical observations. Namely, SM lacks dark matter particle which should not be a baryon and must interact very weakly. However, there exists a possibility that axions of quantum chromodynamics could comprise the required dark matter. The observed abundance of matter with respect to antimatter in the present day universe requires baryogenesis with stronger breaking of charge-parity (CP) symmetry violation than is present in the SM. One aspect of the SM experimental tests has been to look for new energy thresholds where new degrees of freedom would become relevant. In this way, the gauge structure of SM was confirmed directly at the weak scale ~ 100 GeV at the Large Electron Positron collider (LEP), where the weak bosons and neutral currents were first discovered. The electroweak-breaking sector of the SM should be directly studied at Fermilab and the Large Hadron Collider (LHC) in the forthcoming years with typical energy of partonic reactions of the order 1 TeV. On the other hand, Yukawa sector of the SM exposes its richness in the quark flavor changing interactions, which are well suited for study in hadron and t lepton weak decays and are accessible at energies much below the weak scale. From theoretical perspective, hadron states are bound by a genuinely nonperturbative phenomenon of strong interactions — confinement — which is interesting in its own right, but in this case blurs the view of electroweak dynamics that drives the decay. Nonperturbative hadronic dynamics, whose quantitative treatment makes theoretical predictions rather uncertain, thus stands between a clean comparison of observables' measured values and predictions of SM or of a theory that would supersede SM at energies above the electroweak scale. Conceptual reasons have also been put forward for considering SM as merely an effective low-energy theory of its yet unknown ultraviolet completion. Certainly SM would fail to describe physical phenomena at the Planck scale (MP ~ 1019 GeV), at which gravity should be reconciled with principles of quantum mechanics. In principle, SM could hold up to almost Mp but in this case the fundamental scalar field in the SM — the Higgs doublet — which breaks the electroweak symmetry and provides masses for gauge bosons and fermions, seems unnaturally light. The hierarchy problem is the problem of instability of Higgs mass term, which is, due to being a coupling of relevant operator, naturally expected to be driven to the SM cutoff scale A. Namely, first order additive quantum corrections to the Higgs mass scale as A2 and setting A ~ MP requires excessive fine-tuning of the tree-level Higgs mass term. On the other hand, Higgs mass should not be too far from the electroweak scale in order to satisfy all experimental constraints coming in particular from the electroweak precision observables [1]. The common opinion about hierarchy problem is that there should be a scale close to or not far above 1 TeV, at which SM might loose its validity and is to be replaced by a more fundamental theory, usually termed new physics (NP) in the literature. Currently there are many viable possibilities about what the NP may be like, however none of them is clearly preferred over others owing to good agreement of SM predictions with experimental data. 1.2 Flavour physics Presently available experimental data from LEP, Fermilab experiments, B-factories, and other low energy experiments as well as astrophysical and cosmological observations impose important constraints on NP models. The experiments based on the colliding e+e- beams measured a whole plethora of quark flavor changing processes, driven by the tree-level charged weak currents as well as flavor changing neutral currents (FCNC) which lead to rare meson decays and neutral meson mixing. Violation of CP symmetry has been observed both in meson mixing amplitudes and directly in the decay amplitudes. Important contributions have also come from Fermilab experiments, most notably the discovery of Bs meson mixing. 1.2.1 Flavour sector of the standard model The number of quark generations N is a free parameter in the SM, and so are the quark masses and quark flavour mixing parameters. Anomaly cancellation condition [2] only requires that we have an equal number of quark and lepton generations, while the asymptotic freedom of quantum chromodynamics requires N < 8. It was pointed out by Kobayashi and Maskawa [3] before the third generation of quarks was discovered, that one needs at least N = 3 generations of quarks to accommodate one CP-violating phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix [3, 4]. Thus the observed phenomenon of CP violation is not a prediction of the SM but rather a consequence of observation of 3 generations. To put it differently, for observation of 3 generations SM predicts there should be exactly one real parameter desčribing all CP violating phenomena1. The magnitude of mixing is parameterized by 3 angles in the CKM matrix, whičh are known to be small, i.e. the CKM matrix is, to within A = 0.22 error, diagonal. The flavour čhanging transitions are hierarčhičally suppressed as A, A2, and A3 for the respečtive transitions between generations 1 o 2, 2 o 3, and 3 o 1. Sinče the angles are free parameters, SM čannot provide a reason for their smallness, and neither čan for huge disparity between quark masses whičh span over 5 orders of magnitude. Questioning the reason behind the measured flavour parameters is meaningful only in a suitable NP model. 1.2.2 Flavour changing neutral currents The Glashow-Iliopoulos-Maiani (GIM) mečhanism of SM suppresses FCNCs through sums over intermediate quark flavours whičh čontribute to the pročess at loop order. Presenče of FCNCs in the SM amplitude is rečognized by a fermion line running through the Feynman graph čonnečting two fermions of different generation but same čharge. GIM is direčtly related to unitarity of the CKM matrix and thus an inherent property of SM amplitudes. It is broken only by the nondegenerate quarks masses. NP models with new sourčes of flavor violation generally lačk a mečhanism analogous to GIM, thus leading to severe experimental čonstraints on flavour violation parameters of the NP model or potentially result in a člean signal of NP with small SM bačkground. Moreover, sinče FCNCs are loop-indučed they are sensitive to short distanče dynamičs that might play a role in quantum čorrečtions. One well-known example is the study of dependenče of the B ^ K*y dečay width upon the top quark mass [6]. However, all observables that involve quark flavour čhange čonfirm the CKM mixing mečhanism among the six quark flavours and is best illustrated by the unitarity triangle shown on Figure 1.1. Absenče of deviations at few perčent level requires a highly nontrivial flavour stručture of NP around the 1 TeV sčale, whereas sčale of a generič NP model is to be of the order 100 TeV or above. The above-mentioned flavour čonstraints are in tension with the hierarchy problem as they forče the NP sčale far above 1 TeV. The problem has been addressed in a model independent framework čalled minimal flavour violation [8, 9], where the flavour stručture of NP is exačtly aligned with the SM, i.e., the SM flavour group is broken only by the SM Yukawa čouplings. 1.2.3 Heavy meson decays Heavy B (D) meson čontains a heavy valenče b (c) quark and another light valenče anti-quark. Mass above 5 GeV allows the B meson to dečay weakly into numerous final states resulting in many observables where one čould find signatures of physičs beyond the SM. The CKM flavour mixing mečhanism has been thoroughly tested in K and B dečay observables [10], the most important of the latter we list in the following. Exčlusive and inčlusive čharm dečays čan be used to čonstrain VCb, while čharmless B dečays čonstrain Vub. Time dependent CP asymmetry in B ^ J/^KS 1 Another source of CP violation is the 0 term of quantum chromodynamics, which is however constrained by the electric dipole moment of the neutron (|dn| < 3 x 10-26 ecm) which implies 0 < 10-9 [5]. Figure 1.1: Unitarity of the CKM matrix is globally satisfied [7]. depends on the phase difference between the B°-B° mixing amplitude and the decay amplitude and provides a very precise measurement of sin 2/5 (see Figure 1.1). Decay widths of several B —>• DK decay channels and hadronic channels B —>• 7T7r,p7r,pp provide constraints on the respective angles 7 and a. The ratio of mass splittings determined in B° and Bs mixing is a measure of |Vtd/Vts\- Semileptonic and leptonic charm meson decays can probe magnitudes of \Vcd\ and j VCs I • Very recently also neutral charm meson mixing has been measured [11-15], offering various new possibilities in the up-type quarks FCNCs. However, the c o u FCNC process is dominated by cl, and s quarks in the loop (b quark contributions Cabibbo suppressed by factor A5) and light hadronic resonances dominate the decay widths. From the theoretical viewpoint, large masses of heavy quarks, compared to the QCD scale (m• q, the light quark is almost on the light cone where additional symmetries arise. 1.3 Goals and methods Considering the existing results from B-factories, with LHCb almost in the physics run, and recent approval of the Belle 2 experiment, precision of D and B meson decay observables will improve considerably. The two new experiments will hopefully find an evidence for NP or provide a good handle for discrimination between NP models. In this thesis we set out to analyse rare decays of B and D mesons in framework of SM and several NP models. We shall provide for considered decay channels a comparison between the SM prediction and experimental results (or bounds at 90% confidence level (CL) if the decay in question has not yet been discovered). Afterwards we study what would be the impact on observables of particular NP model. Among these we will consider minimal supersymmetric standard model with (MSSM) and without (RpMSSM) Rp-parity conservation, models with extra Z' neutral gauge boson, models with additional singlet up-type quark (EQS), and class of models with scalar weak singlet leptoquark (LQ). For meson decays dynamical heavy degrees of freedom like W bosons, t quark, and short distance NP dynamics are integrated out and absorbed in the short distance Wilson coefficients of operator product expansion [18, 19]. Where necessary we will account for the QCD corrections in the effective Lagrangian. For matrix elements between meson states we will either use form factors calculated by dispersive methods (QCD sum rules, light-cone sum rules), lattice QCD, or use form factors determined experimentally. Whenever feasible we shall utilize underlying symmetries of QCD which arise in specific kinematic circumstances. For heavy quarks interacting with light degrees of freedom this is the aforementioned heavy quark symmetry and in the case of energetic light quark the large energy effective theory, whereas low energy light degrees of freedom interact as dictated by chiral symmetry of QCD. 1.4 Structure of thesis In Chapter 2 we restate the mechanism of flavour violation in SM and outline the procedure of separating short distance from long distance dynamics when treating weak meson decays in SM. Relevant aspects of NP models will be described when needed, in situ. Chapter 3: Rare process b ^ dds. Quark transition b ^ dds and the corresponding hadronic decay channels would be too rare to be detected if SM was a complete theory. Currently, the best bound is obtained in B- ^ n-n-K+ channel. Experimental evidence would serve as indisputable proof of NP, but already existing bounds on branching ratios can be used to constrain some NP models and predict viability of searches in other hadronic modes. Chapter 4: Neutral currents in charm and D ^ P£+£-. Semileptonic decay channel D ^ n£+£- and Ds ^ K£+£- encompass the c ^ u£+£- process and are sensitive to FCNCs of up-type quarks. With existing bounds on parameters of NP models there is still some room for signals in the kinematical region with low and high invariant mass of the dilepton pair. Chapter 5: Dalitz plot analysis of B ^ KnY decay. We focus on two corners of Dalitz plot with one or other meson soft. We predict the photon spectra in the two regions in the SM. Chapter 6: Radiative background of B ^ D£za This exclusive decay channel is used to extract the magnitude of Vcb. We explore the impact on precision of Vcb measurement if the resonant D* meson decays into an unobserved photon and a D meson. In Chapter 7 we summarize the results and draw conclusions. Intermediate derivations and too long expressions are relegated to appendices whenever their presence would bother the main line of thought. Chapter 2 Flavour violation at low energies 2.1 Flavour violation in the standard model In this sečtion we will demonstrate how flavour violation arises in the standard model (SM). SM is a realization of relativistič quantum field theory čonsistent with Lorentz invarianče and čausality and has built in basič postulates of quantum me-čhaničs like positivity and unitarity [20-22]. Stručture of the SM interačtions is čompletely spečified with ločal gauge group G = SU(3)c x SU(2)W x U(1)Y and its fermionič representations: E(1, 2)-1/2 = , *r(1, 1)-1, ur(3, 1)2/3 = (uR uR uR) , dR(3,1)-1/3 = (dR dR dR) , where the numbers in the bračkets denote representations of the respečtive čolor, weak isospin, and weak hyperčharge groups. The above gauge stručture repeats itself in three generations of matter, whičh are distinčt only by the Yukawa čouplings in the SM Lagrangian (2.3). The Lagrangian density of SM is čommonly split into four terms L = Lkin + Lgauge + LYuk + LEWB- (2.2) These, in the order as written above, čorrespond to gauge-čovariant kinetič terms of fermions, kinetič terms of gauge bosons, and Yukawa čouplings between fermions and Higgs field. To break the elečtroweak symmetry and allow for masses of the gauge bosons and fermions, an additional sčalar field 0 — the Higgs sčalar - in the representation (1, 2)1/2 is introdučed [23-26]. The elečtroweak symmetry breaking potential, present in LEWB, then triggers the Higgs field to develop a spačetime-uniform vačuum expečtation value (VEV). Parameters responsible for fermion masses an flavour čhanging interačtions of quarks are čontained in LYuk = -Q0YddR - QiTVYuUR - £0YeeR + H.č., (2.3) where Yd>u>e are the 3 x 3 dimensionless Yukawa matrices in generation space. The Higgs field has develops a VEV w. = („/%) . (24) and mass terms for the fermions arise in the Yukawa potential Lyuk 9 - — dLYddR - — UlYuUR - -V= eLYeeR + H.c.. (2.5) In order to operate with conventional Feynman rules with flavour-diagonal propagators, we have to work in the mass-basis of fermion fields. For a general complex matrix Y two unitary matrices U and V can always be found such that Y = U^DV, where D is diagonal. We express the Yukawa matrices in terms of physical fermion masses and unitary rotations Yu = UL diag(mu, mc, mt) Ur, (2.6a) Yd = diag(md, ms, m6) Dr, (2.6b) Ye = EL diag(me, mr) Er , (2.6c) and absorb the rotations in fermion fields ul,r ^ UL Rul,r, dL,R ^ dl RdL , R, eL,R ^ ELReL,R in order to make the mass terms (2.5) diagonal. The flavour rotations connecting the weak- and mass-basis fields cancel out in the kinetic and neutral current terms of the SM Lagrangian because of their unitarity. This cancellation does not occur in the quark charged current coupled to W boson where the remnant physical parameter is the misalignment between rotations of up-type and down-type left-handed quarks1 Lkin 9 - — W+ U^Pl(UlDL)ijdj + H.c., (2.7) The CKM matrix V = UlDl thus contains all parameters of the SM flavour changing interactions. 2.1.1 CKM matrix The CKM matrix for N quark doublets is described by a total of N(N - 1)/2 Euler angles and N(N + 1)/2 phase factors. We may redefine the phases of each up- and down-type quark fields — 2N of them — and get eliminate 2N - 1 phases from the CKM matrix and end up with altogether (N - 1)2 = N(N- 1)/2 + (N - 2)(N - 1)/2 (2.8) Euler angles phases real parameters. Based on observed CP violation in K0-K0 mixing and the above-mentioned phase counting, Kobayashi and Maskawa [3] proposed existence of the 1We work here in the unitary gauge to avoid complications with flavour changing couplings of Goldstone modes. third quark generation before even charm quark was discovered. For the N = 3 SM, three angles #12, #23, #13, and phase 5 are conventionally taken as C12C13 V = I -S12C23 - Cl2S23Sl3e S12S23 - Ci2023Sl3elS iS iS S12C13 C12C23 - S12S23S13e -C12S23 - S12C23«13e iS iS (2.9) with Cij — Cos , Sij — sin #ij. Note that the CP phase is absent from CKM matrix when s13 = 0. Experiments have established the smallness of all mixing angles, meaning that CKM matrix is diagonal up to A corrections, a fact which we demonstrate below. Hierarchy of CKM elements becomes more lucid in the Wolfenstein parameterization [27] which is a power expansion in A — s12 along with redefined parameters [28, 29] s23 — AA2, s13e-iS — AA2(p - in): ( V= 11 2 A A 1 2 ^AA3(1 - p - in) -AA2 AA3(p - in)\ AA2 1 + O(A4). / (2.10) Working to A3 order we can take A = 0.226 ± 0.001, value of Cabibbo angle cos #c, while other parameters' values are [28] A = 0.81 ± 0.02, p = 0.14+0 q2, and n = 0.36 ± 0.02. Mutual orthogonality relations between columns ^k VkiVk*j = 5ij and rows k VikV*k = 5ij of V can be depicted as six unitarity triangles, of which the one corresponding to product between the first and the third column is commonly used to illustrate the unitarity triangle analyses (Figure 1.1). 2.2 Operator product expansion in weak decays Weak decays of heavy mesons are processes involving a typical kinematical energy scales of at most few GeV. In contrast, the dynamical degrees of freedom triggering a weak decay are the weak gauge bosons of a mass ~ 100 GeV or even heavier NP degrees of freedom. Full theory treatment becomes rather awkward due to large disparity of scales in the problem. Thus, to facilitate calculations of decay amplitudes and in particular their QCD renormalization effects one works instead with effective f -flavour theory, where f = 5, 4,... is the number of dynamical quark fields at the chosen renormalization scale p. To this end, we will integrate over the W boson field in the generating functional in the presence of external weak currents J±. We integrate over the weak gauge bosons degrees of freedom whereas the weak currents of fermions are here treated as external fields [29] Zw[J+] = J DW+DW- exp ij d4^Lw + ^(J+W+M + J-W-M) (2.11) Coupling constant of the SU(2) weak isospin is denoted g. The functional integral measure is defined as [30] DW ± = N (e) U dW± (x) dW±(x) dW2±(x) dW±(x), (2.12) where nx denotes the produčt over the points of infinitesimally disčretized spačetime with spačing e 2. 1 Lw = — - dvW+)(d^W-v - dvW+ mWW+W (2.13) J+ =Vij UiLY^djL + ViL7^£iL, J- = (J+)t. (2.14) We work in the unitary gauge of the elečtroweak interačtions, where the Goldstone sčalars are absorbed by the longitudinal čomponents of weak bosons. The kinetič terms čan be rewritten using per partes integration and dropping the surfače terms -dMW+d^W-v+dMWv+dv W-v + mW W+W -^Wv+ d2W-v - Wv+ dv dMW+ mW W+W =W+K (x,y)W-, (2.15) K(x, y) ^4(x - y) (dfy) + mW- Sfy)Sj (2.16) where the introdučed operator K is the inverse Feynman propagator of the W boson j d4y K«(x, y) iAav(y, z) = i^(4)(x - z) iA^v (x,y) = d4P i ^f) ip^(x-y) (2n)4 p2 - mW + ie Sčhematič stručture of the ačtion funčtional is then S [W +, W-] = Wx+Kxy Wy- + -g2(Jx+Wx+ + Jx-Wx-), and may be čompleted into a square with obvious substitutions W+ = Wx+ + -g= Jy-Ayx, W- = W- + -g^Axy Jy+. Expressed in terms of the new variables (2.20) the ačtion bečomes S[W + , W-] = W+Kxy Wy- - 9-J-Axy Jy+. xy" y 2 x ^-xyjy At this point, we čan integrate over the W fields in the funčtional integral Zw[J+] = J DW+DW- exp i J d4xd4y(W+(x)Kv(x,y)Wv-(y) - g^J"(x)A^v(x,y) J+ (y) (2.17) (2.18) (2.19) (2.20a) (2.20b) (2.21) (2.22) 2The normalization constant N(e) cancels out in the physical results. to find nonlocal interaction for the fermions Snoniocal = —d4xd4y J"(x)A^(x,y)J+(y). (2.23) Now follows the crucial step — expanding the nonlocal operator under the integral in terms of local operators, with consecutive highers orders carrying additional powers of 1/mW • Higher order terms will necessary involve derivatives to account for nonlocality, whereas the leading term is a point-like interaction Snoniocai = —f" / d4x d4y [J"(x)g^S(x - y) J+(y) + O(1/mW)] • (2.24) We read off the leading order term in the operator product expansion (OPE) [18, 19] to be 4G Lope =--iF J-(x)J+M(x) + O(1/m2W) x (higher D operators). (2.25) v2 Of particular importance for weak decays are dimension-5 and 6 operators. For definiteness we focus here on the dimension-6 composite operators Ldim-6 = -4GF E Ci(p)Qi(p), (2.26) i Qi = (Vix ri^i2 )(Vi3 ri^i4). (2.27) Here ii,i2,i3, and i4 denote species of particles, while ri, ri are matrices in the Dirac spinor space whose combination is scalar under homogeneous Lorentz transformation (barring discrete transformations like time-reversal and parity). Dimensionless Wilson coefficients Ci(p) depend on the renormalization scale p, couplings, and masses of heavy particles. On the other hand, composite operators' p-dependence is indirect through p-dependence of the renormalized fermion fields. We should emphasize that renormalization scale p (also called factorization scale) is an artificially introduced momentum scale, separating long distance dynamics (momentum scales below p) from short distance scales (scales above p) [29]. For a process in question we first choose a complete set of effective operators which will, together with their corresponding Wilson coefficients, reproduce the invariant amplitudes of the full theory below the matching scale A up to terms suppressed with additional powers of 1/A2. Calculating hadronic amplitude amounts to determining the Wilson coefficients and then evaluating matrix elements of the composite operators between hadronic asymptotic states A+f = -4GF E Ci(p) {f | Qi(p) | i) • (2.28) 2i Any possible hadronic states in asymptotic states |i), |f) require the matrix elements to be calculated in nonperturbative regime of QCD, which is notoriously difficult to solve. The only ab-initio technique is a discrete formulation of QCD on four dimensional lattice which employs direct evaluation of correlation functions using the Feynman path integral representation [31, 32] analytically continued to imaginary time. From the point of view of OPE, the important feature of matrix elements is their dependence on renormalization scale which is cancelled against ^-dependence of Wilson coefficients in the final expression for the amplitude. In practice however, the nonperturbative methods are usually performed at a single value of ^ = mhad, namely at typical momentum scale of hadronic process. Correspondingly also the Wilson coefficients' values should be known at scale ^had. In Appendix A we outline the procedure of composite operator renormalization as applied to the b ^ dds transition. Having separated the short distance dynamics by performing the perturbative OPE and QCD renormalization procedure, we now have at hand the necessary set of short distance Wilson coefficients, whereas the matrix elements in (2.28) must involve some nonperturbative QCD method. In parameterizing the matrix elements we are guided by the Lorentz covariance of quark bilinears which must be manifest in the resulting form factor decomposition3. We will first introduce the customary form factor decomposition for transitions of pseudoscalar or vector to vacuum (P, V ^ 0), pseudoscalar to pseudoscalar meson (P ^ P') and pseudoscalar to vector (P ^ V). Later on we will briefly delve into the underlying symmetries of QCD which can impose further constraints on the form factors. 2.3.1 Hadronic form factors Decay constants The standard decay constants of the pseudoscalar P and vector mesons V are defined where q'q are the flavours of P or V. Polarization vector of vector particle is denoted e, e2 = -1, e ■ p = 0. Transitions between pseudoscalars In semileptonic decays with one hadron in the final state we may encounter a quark bilinear inserted between the two meson states. We will use the standard form 3Also the parity transformation properties must be preserved by the form factor decomposition since we consider hadronic states as purely QCD bound states, not allowing for parity violation. 2.3 Low energy QCD (o I e'Y5q I P(p)> = i/p^, (ole'Y Vq|V (e,p)> = gv e^, (2.29a) (2.29b) factor parameterization [33, 34] between two pseudoscalar mesons =F+(q2) ((p + p'f — 2 2 ')M — mp — m,p, q^ q2 (2.30) 22 m2 m2 + Fo(q2) tp "tp/ q^, (P'(p') | q'^v(1 ± 75)q | P(p)} =is(q2) |_(p + p')^qv — q^(p + p')v ± ie^(p + p')aq^ (2.31) where q = p — p' is the momentum transfer. Pseudoscalar to vector transitions Matrix elements between a pseudoscalar and a vector meson are decomposed as customary (V(e,p') | qVq | P(p))=- ^ N II/ mp + mV \V (e,p')|q-'7^75q|P (p)> =ie 2mv Ao(q2) ^ q 2 + (mp + mv )Ai(q2)f — ^ A2(q2) (mp + mv) ((p + p')" — (2.32a) (2.32b) mp — mV M v —q^ q (P(p') | q'q^v(1 ± Y5)q | V(e,p)> =ieA — 2iTi(q2)ev^pApMpP (2.33a) ± T2(q2) ((mV — mp)gAv — qA(p + p')v) 2 ± Ts(q2)qA qv — q 22 m 2V - m 2p (p + p')v 2.3.2 Heavy quark symmetry Hadronic phenomena are effects determined by QCD in the nonperturbative regime, namely at scale Aqcd, where the perturbative expansion in powers of as cannot be used. Typical energy of quark and gluon degrees of freedom in a hadron are of order Aqcd. A heavy quark Q and accompanying light antiquark q comprise a heavy-light mesons, and since mg » Aqcd the heavy quark acts as almost static triplet colour charge interacting with dynamical light degrees of freedom [16, 17]. Accordingly, momentum of heavy quark is split into kinematic part owing to velocity of the parent hadron and a small residual fluctuation k ~ Aqcd pq = mgv + k. (2.34) On the level of QCD Lagrangian of heavy quarks (c, b) we can introduce new velocity dependent fields which have the momentum scales of order mQ factored out [35] hv (x) = eimQ 1^-J/ Q(x) (2.35) Hv (x) = eimQ v^x / Q(x), (2.36) where Q(x) is a quark field of QCD. Projectors (1 ±/)/2 project out the particle and antiparticle components of Dirac spinors, and Hv corresponds to small components which would vanish exactly in the limit mQ ^ to. Lagrangian of QCD expressed with new fields is [35] Lhqet = hviv ■ Dhv + hviD±--„ , 1--iD±hv, (2.37a) iv ■ D + 2mQ — ie D^ = D1 — v ■ D v1. We have used D as standard covariant derivative of QCD. In the exact heavy quark symmetry limit (HQS) only the first term survives. It is manifestly independent of quark mass and owing to the absence of Dirac gamma matrices interaction with gluons are independent of the heavy quark spin. Spin-independence implies that physical states of heavy-light mesons are grouped into doublets of according to spin of light degrees of freedom j* which is summed with the heavy quark spin jh into total spin of a hadron J = jh + j*. (2.38) The two degenerate doublet members' spin difference comes from ±1/2 contribution of heavy spin. The ground state j* = 1/2 doublet can be represented by a velocity-dependent field H„(v) = 11+^/ (/» — 75P„(v)) (2.39) with v the meson velocity and a the flavour of light antiquark. The factor (1 + /)/2 projects out the large quark components, fields P* and Pa annihilate vector and pseudoscalar mesons, whereas Dirac matrices and y5 are added to ensure that field Ha(v) transforms as a fermionic bilinear under the Lorentz group. Ha(v) are useful entities for incorporating the chiral symmetry of QCD, which couples to the light quark indices a. 2.3.3 Chiral perturbation theory Chiral perturbation theory (CHPT) is the effective theory of QCD valid below the chiral symmetry breaking scale Ax ~ 1 GeV. Perturbative expansion in as is meaningless in this regime and because of confinement also the quark degrees of freedom cannot be used. We consider Lagrangian of QCD in the approximation where we neglect masses of light quarks LQcd = + gsy Gi) ^ — 4G%, ^ = (u d s)T . (2.40) Here gs is the strong coupling constant, Aa are the standard Gell-Mann generators of SU(3)c fundamental representation, whereas G^ is the gluon field4. The above Lagrangian is invariant with respect to SU(3)L x SU(3)R x U(1)v x U(1)a global transformations. The U(1)v symmetry, acting as a phase rotation of all quarks simultaneously, is conserved even with quarks massive and its generator is the baryon number. The U(1)a is broken by the Abelian anomaly [36]. Remaining G = SU(3)l x SU(3)r is the group of chiral transformations that act independently on the left- and right-handed components of the quark fields ^ 9l,R^L,R. (2.41) The chiral group transformations (2.41) can equivalently be parameterized with the vector gv = gL+R and axial gA = gR-L transformations. The global symmetry of dynamics should be imprinted in degeneracy of physical states. The vector subgroup gv when acting on physical states spans the familiar octet of pseudoscalar mesons [37, 38], which are quite far from being degenerate because of questionable assumption of massless s quark. Once we consider only u and d as massless the gv becomes the isospin SU(2) symmetry which manifests itself in highly degenerate multiplets of hadrons (i.e. mass splittings in the nucleon doublet and pion triplet are tiny). Global axial symmetry gA, on the other hand, would, if conserved predict degeneracy also between multiplets of opposite parities. No such degeneracies are observed in the physical spectrum, which indicates that gA must be broken somehow. The central idea of CHPT lies in the assumption of axial symmetry breaking by the vacuum expectation value of the quark condensate (0 | ^ | 0) = 0 and in identifying the broken generators with Goldstone bosons [39, 40], which are the lightest pseudoscalar mesons of the spectrum. To construct the effective theory one has to include in the Lagrangian all terms with Goldstone fields that are symmetric under G [41] and devise power counting to be able to truncate the expansion. Higher dimension operators are suppressed by more powers of p/Ax [42], where p is a typical momentum of the process. Very convenient method to construct the terms in the effective Lagrangian is the CCWZ formalism [43]. The group element G is a product of axial and vector generators g = e^ a. (2.42) Group generators can be taken as Va = iAa/2 and Aa = VaY5 where Aa are the Gell-Mann matrices. G is broken into H = SU(3)v, and the Goldstone boson fields are represented by coordinates £a of the coset space G/H. We study the transformations of the broken subgroup elements u(£a) = e^aA°, defined by gu(£a) = U(£a )ev'a ya = U(£a )h(g,£a). (2.43) Transformation of u(£a) is found to be nonlinear U(£a) ^ U(£a) = gu(£a)h-1 (g,£a) = h(g, £a)u(£a)L(g)-1, (2.44) 4Gluon field strength is G= dMGa - 3VG" + gsf ahcG°;1Ghv, where fabc are SU(3) structure constants. where g £ G is arbitrary and L(g) denotes g with left and right components interchanged. Group element h belongs to the unbroken group H. A useful block of building invariants is S = u2 field which transforms as S(&) = u(Ca)2 ^ 9Ru((a)h-1(g,(a) h(g, £>(£«)£-1 = gRS(e„)g-1 (2.45) and can be parameterized as S = ei/2n/f. (2.46) We fix our parameterization of the Goldstone boson fields by specifying matrix n which contains the octet of pseudoscalar mesons n- - % + K° V K- K° - 1 (2.47) Parameter f with dimension of mass is introduced to cancel the pseudoscalar fields mass-dimension. The unique chiral invariant with two derivatives that reproduces the conventional normalization of kinetic terms is f2 ^XaV£t )ba. (2.48) To generalize the framework to include also explicit breaking of G, we find in the first order of light quark masses f2 L2 = 4 dMSab(d^St)ba + A°(rh abSab + (St )abm ba) • (2.49) Light quark mass-matrix is here m = diag(mu, ma, ms). The chiral noninvariant term accounting for the finite quark masses shares the same transformation properties under G as quark mass terms in the QCD Lagrangian [44, 45]. Expanding the leading chiral-order Lagrangian (2.49) leads to interactions of even number of pions whose vertices are accompanied by one power of external momentum p. In the low energy limit the leading order Lagrangian dominates as the terms with higher number of derivatives contain more powers of p. The power counting even works for loop integrals in the effective theory [41]. a Heavy meson CHPT In our calculation in Chapter 5 we shall encounter the amplitude for emission of soft Goldstone boson off a heavy meson line. One can combine both chiral symmetry of QCD and the heavy quark symmetry to construct the heavy meson chiral perturbation theory (HM%PT) (c.f. [46] and references therein). We use the velocity dependent heavy meson fields Ha(v) for the ground state negative parity-doublet (2.39). They transform under the unbroken light flavour group H as Ha ^ Hb(h 1)ba. (2.50) Fields Ha can be combined into chiral invariants together with the derivatives of Goldstone fields A = 2 (utdMu — ut) , (2.51) = |(utdMu + u^. (2.52) Leading order effective Lagrangian is LhmxPT = 05 = (d^bL)(jLsR). (3.3) We denote with 0i the chirally-flipped set of operators obtained from 0i by interchange L o R. Color contractions of the quark fields are indicated by repeated indices a and ft. For convenience we supplemented the operator basis introduced in [57] with additional scalar operators 04,5, 04,5, which are however redundant and can be reduced to 02,1; (O2,1 via Fierz transformation O4,5 = — 2 02,1, 04,5 = — 1 <02,1- (3.4) 3.1.1 QCD corrections Given an underlying full theory valid at short distances we perform matching to the effective theory (8.8) by equating the 4-quark 1-particle irreducible Green functions calculated in both theories. In this way we obtain the short-distance Wilson coefficients, as described in Sections 2.2 and A.1. In Appendix A also the anomalous dimension matrix of operators 01...3 are determined in leading order in as and leading logarithm approximation. For the 01 5 we get as Y = 2S 1 —3 0 0 0 0 —8 0 0 0 00200 0 0 0 —8 0 0 0 0 —3 1 (3.5) The operator 03 does not mix into its color-flipped counterpart (which has color contractions between the two currents), which is due to the two fields and Fierz rearrangement, equal to the original 03. Obviously, with (3.4) the anomalous dimensions for 01,2 and 05,4 are the same. We solve the renormalization group equations which govern evolution of the Wilson coefficients as the renormalization scale runs from A down to ^ ~ mb, where the hadronic matrix elements are calculated 1. These corrections might be substantial due to large separation between 1We will consistently denote A and ^ for the matching scale and the scale where hadronic matrix elements are calculated, namely ^ ~ mb. scales A > mW and u ~ mb. The resulting evolution matrix U (u, A) = Tg exp r dg' ytv) /S(A) č(g') (3.6) then relates the Wilson coefficients at the scale u to their values obtained at the matching scale Ci(u) = U (u, A)ij Cj (A). (3.7) Using the standard parametrization of the beta function (A.10) we obtain the sub-blocks of the evolution matrix from where we can directly read the Wilson coefficients at scale u The QCD corrections mix the operator O1 into O2 Ci(u) = as (u) _as(A)_ -1/^0 C1(A), (3.8a) as(u) 8/Ao as(u) 3 _as(A)_ _as(A)_ -1/Aox C1(A) + as(u) as (A) 8/^0 C2(A), (3.8b) while C3 is multiplicatively renormalized " as(u) 1-2/A° C3(u) = _as(A)_ C3(A). (3.9) We reproduce the mixing behaviour reported in [57]. To find mixing of C4 and C5 one is to substitute in expressions (3.8) C1 ^ C5 and C2 ^ C4. The beta-function coefficient ,0o = (33 - 2/)/3 depends on /, the number of dynamical quark flavours between scales u and A. Evolution matrix depends on whether the matching scale A is above mt and when this is the case we compose U from consecutive evolution matrices valid at / = 6 and / = 5 U (u, A)|a> mt = U(u mt)|f=5UA)|f=6- (3.10) Note that expressions (3.8),(3.9) assume that the number of dynamical quark flavours / is constant between u and A. Renormalization of the Wilson coefficients C1,...,5 of chirally-flipped operators is governed by the same set of equations as for C1,...,5. 3.2 Inclusive decay width In this section we will calculate inclusive decay width of b ^ dds in frameworks of SM, MSSM, RpMSSM, and the Z' model. Model with down-type singlet quark has been studied in [58]. First we will match the full theory onto the effective Hamiltonian (8.8) at scale A to find relevant Wilson coefficients C (we denote these without the scale argument). The RGE techniques, described in the last section, are then applied to obtain a set of Cj(u), Cj(u) in the given framework. Finally, we express the inclusive decay width in terms of relevant parameters of the underlying theory and its numerical value, if the parameters are already bounded from other observables. Figure 3.1: Box diagram generating b ^ dds in the SM. 3.2.1 Standard model In the SM process b ^ dds is mediated by box diagram 3.1 with two W bosons exchanged between the quark lines. The leading term of the order 1/mW in the OPE is local and thus independent of external momenta [59] so we set the external momenta to zero in calculating the box diagram of Figure 3.1. One should in principle also add nonlocal contributions of H^ff1^' and %ffS)(dc), which have been shown to be smaller than short distance contributions for decay B ^ K-K-n+ [54, 55] and we choose to neglect them as they cannot change the order of magnitude of our predictions. All the vertices are of the (V - A) ® (V - A) color-singlet type so only O3 is generated at scale A ~ mW. Note that we used the SM with dynamical t quark and integrated out both W and t at common scale mW. CSM G2f m 8n2 ^Ai~\j f (xi , xj) (3.11) i,j The sum runs over the charge 2/3 quark flavours (i,j = u,c,t) with CKM weights Ai = VibV*d and Aj = VjSV*d. Remaining dependence on masses of the quarks in the loop is contained in function f (xi,xj-) 2, where xi = m2u./mW f (x,y) = - 3xy + xy (x - 1)(y - 1) x - y 6x_3 1 - (x _ 1) 2 )ln x - [x ^ y] (3.12) The above f(x, y) suppresses contributions of light quarks in the loop (see however Section A.2). Intricate hierarchy of quark masses and CKM factors in (3.11) renders negligible all but two of the terms. Relative size of all 9 terms contributing to CfM, using the simple Cabibbo angle A = cos dc = 0.22 power-counting is given below. The two important ones are with i = t and j = c or t. Two top quarks contribute dominantly although the Cabibbo suppression is ~ A8. The subleading contribution comes from top and charm in the loop and its milder A4) Cabibbo suppression is compensated by small value of f (xt,xc) and is at par with the dominant contribution. With the abovegiven Wilson coefficient (3.11), adapted for the b ^ ssd decay, our numerical value of CfM agrees with the expressions reported in [50]. In analogy with K0-K0 and B0-B0 mixing processes we do not expect the QCD corrections to change significantly the decay rate [60]. Indeed, the RG evolution (3.9) only 2For inherent ambiguity in the choice of f (x,y) that is related to GIM cancellation, see Sec- tion A.2. i u c t u 4 10- 7 5 x 10-6 2 < 10-8 c 5 10- 6 5 X 10-2 1 10-3 t 9 10- 6 0.4 1 (3.13) Table 3.1: Hierarchy of contributions of virtual quarks in the SM box diagram for b —> dds . changes CfM by a negligible factor of 0.9. Using values for the parameters from the Particle Data Group [28], we get C3sm « 5.3 x 10-13 GeV-2. (3.14) To obtain the decay width, one has to consider two distinct cases, namely when: (i) the two final state d quarks have the same color and they cannot be distinguished, or else, (ii) when d quarks' colors are different and we can tell which one belongs to which of the currents in O3. In the end, these two cases have to be summed incoherently, i.e., on the level of decay width. The resulting inclusive decay rate, averaged and summed over spins and colors of quarks is rSM r incl. I^SM I2 „_5 |C3 | mb 384n3 (3.15) which amounts to B(b ^ dds)sM = (8 ± 2) x 10-14. This is far below the current and foreseeable experimental sensitivity and discovery of this decay mode would undoubtedly be a signal of NP. 3.2.2 Minimal supersymmetric SM In the supersymmetric extensions of the SM the b ^ dds process can be mediated by additional diagrams involving squarks and gluinos in the box [61]. The contribution depends on the quark-squark-gluino vertices which are flavour nonconserving in the mass-basis of fields [62]. A universal framework for dealing with these flavour violating interactions in the general low-scale supersymmetry is the mass insertion approximation (MIA) [63] where one chooses a mass-basis for squarks where the vertices with gluino are flavour diagonal, whereas the squark mass matrix and the squark propagator are flavour nondiagonal. One can expand the squark propagator in powers of a (matrix) parameter Sy = Ay/(mj.mj.), where Ay are the off- diagonal terms in the squark mass matrix and mt are the squark masses. Single t j insertion in the squark line is denoted by a cross (see Figure 3.2). We will use Wilson coefficients of the effective AS = 2 Hamiltonian derived in [61] and adapt them for the process b ^ dds. The left-handed squark contributions, (Sij-)LL, contribute to the O3 operator: Cmssm as 216m| (St^LL^LL 24x/e (x) + 66/6 (x) (3.16) = m2~/m~ is d' g and the loop functions /6 and /6 are [61] Here x = m|/m~ is the squared ratio of average squark mass to the mass of gluino „ , , 6(1 + 3x)ln x + x3 - 9x2 - 9x + 17 „ x /6(x) =-6^-' (3-17a) . 6x(1 + x) ln x — x3 — 9x2 + 9x + 1 /„_„, x /6(x) =-^-• (3-17b) The recent limits on [64-66] disallow significant contributions from the mixed and the right-handed squark mass insertion terms. Therefore, we only include the dominant left-left insertions given in the expression (3.16). We follow [65] and take x = m|/m~ = 1 and the corresponding values of |(5d3) LL (x = 1)| < 0.14 and K5di)LL (x = 1)| < 0.042 [61]. We take for the average mass of squarks mj = 500 GeV and for the strong coupling constant = 0.11, and find ^MSSM1 < 1.6 x 10-12 GeV-2. (3.18) Using then (3.15) and substituting for the CMSSM Wilson coefficient one finds that in the MSSM the branching fraction of b ^ dds inclusive decay is at most 7 x 10-13. bL <5ds )ll -x-- -x— )LL / \ dL bL dL SL dL Figure 3.2: The box diagrams in the MSSM with gluinos and down-type squark lines with 5ll mass insertions. 3.2.3 MSSM with broken Rp-parity One way to generalize the MSSM is to relax the implicit assumption of R-parity (Rp) conservation which prevents violation of baryon (B) and lepton (L) numbers. A complete review on the topic is available in [67]. Once the supersymmetric particles are assigned the same B and L numbers as their SM partners, the Rp of a given particle is given in terms of its spin (S), B, and L [68] Rp = (—1)2S ( —1)3B+L = {—1 ; susy • <-«9) which implies that all "normal" particles are Rp even and their superpartners Rp odd. Consequently, in the MSSM supersymmetric particles can only form in pairs and the lightest supersymmetric particle is stable. Another important virtue of Rp conservation is the absence of interaction terms which would violate B or L, and thus the proton is stable in MSSM. All the above features are not guaranteed in supersymmetric models with Rp violation (RpMSSM). The superpotential is supplemented by terms with trilinear couplings 1 1 WRp = 2 Aijk LiLj Ek + Aijk LiQj Dk + 2 Aijk UiCDcDk (3.20) of which A and A' violate L while A" violates B. Note that the proton cannot decay if at least one of the B, L is conserved [69]. In addition, A' coupling induces a flavour changing neutral current in the down-type quark sector mediated by a sneutrino exchange L = -Aijk ViLdkRdjL + Hx. (3.21) In this supersymmetric framework, the tree-level exchange of a sneutrino (Fig- bL dn bR dL dL SR dR Figure 3.3: Tree-level exchanges of sneutrino via trilinear couplings. ure 3.3) is expected to be the dominant contribution to b ^ dds transition and its OPE at common sneutrino mass scale A ~ m^ is CRp = CRp = 3 \' \'* V^ Ara31Ara12 ^ m- ' n=1 3 \' \'* V^ Ara21Ara13 ^ m? n=1 (3.22) (3.23) The renormalization group evolution of C4 p down to scale p ~ mb is according R to (3.8) with substitutions C1 ^ 0 and C2 ^ C4 p (with omission of squarks and gluinos contributions in the QCD beta function). The resulting multiplicative renormalization is contained in fQCD where C4/p (mb) = Iqcd (m^)C4/p 24/21 (3.24) fQCD (A) = as(mb) as(mt) as(mb) as (A) 24/23 24/23 as(mt) as(A) (3.25) A > mt A < mt In range of common sneutrino mass the fQCD assumes values from fQCD (200 GeV) « R/ 2 to fQCD(1 TeV) « 2.5 and we will fix fQCD = 2.2. Also C4 p is renormalized by fQCD. Inclusive b-quark decay width is then p r incl. m5fQ CD CrnO 2048n3 |cfp |2 + |Cfp (3.26) Present experimental bounds on the individual A' couplings contributing to the R ~ R Wilson coefficients C4 p and C4 p do not constrain this mode, and we extract the bounds on the relevant combination from exclusive decays in Section 3.4. 3.2.4 Family nonuniversal Z' In many extensions of the SM [70] an additional neutral gauge boson appears. Heavy neutral bosons are also present in grand unified theories, superstring theories and theories with large extra dimensions [71]. This induces contributions to the effective tree-level Hamiltonian from the operators O1,3 as well as O~1,3. Following [70, 71], the Wilson coefficients for the corresponding operators read at the interaction scale A — mZ/ Cf' = -4^2Gf yBd2L B^, Cf' = -4^2GF yB^ Bd3L, (3.27a) Cf = -4^2Gf yBd2L B^^, Cf = -4^2GF yB^ , (3.27b) where y = (g2/g1)2(p1 sin2 0 + p2 cos2 0) and p» = mW/m2 cos2 . Here g1, g2, m1 and m2 stand for the gauge couplings and masses of the Z and Z' bosons, respectively, while 0 is their mixing angle. Weinberg angle is denoted . Again, renormalization group running induces corrections and mixing between the operators. According to the RG evolution equations (3.8) the operator O1 ((~1) mixes into O2 (O2) and for typical mass of mf > 500 GeV their Wilson coefficients at the scale u — mb are expressed in terms of /qcd given in (3.25) Cf' (mb) = [/qcd (mz )]-1/8 Cf' (3.28a) Č2Z'(mb) = 3 /QCD(mZ/) - [/qcd(mZ/)]-1/8) Cf', (3.28b) whereas the value of C3(mb) is C3Z' (mb) = [/QCD (mz' )]-1/4 C3Z' - (3.29) In particular, for a Z' boson scale of mf — 500 GeV [70] one gets numerically /qcd (mZ/) — 2.3 and Cf' (mb) = 0.90Cf', Cf' (mb) = 0.81Cf', (3.30) C2Z' (mb) = 0.47C f. Again, the chirally flipped operators (~ 1,2,3 are renormalized in the same manner. The inclusive b ^ dds decay width is given in the closed form as 5 m5 3(/Q CD + 8/qcd)(|C f |2 + |Cf' |2) + 2/qcd(|C3z' I2 + Cz' |2) pZ' = mb incl. 768^3 3 v" -c^*-- - m i . . j- . - - qcd v,«, , u ~ , (3.31) In Section 3.4 we discuss bounds on Wilson coefficients Cf3 and Cf3 which might be estimated from the B- ^ n-n-K + decay rate. 3.3 Exclusive B decay modes We only consider the charged B meson decays driven by the b ^ dds transition, since the antiquark u is a spectator in this process and one should not worry about possible contributions of the SM penguins. In calculating decay widths of the B-meson decay channels driven by the b ^ dds transition, we will make use of form factors available in the literature. Alternative approach taken in [55] exploits the SU(3) flavour symmetry of light quarks to relate the amplitudes to the measured ones. 3.3.1 Vacuum saturation of matrix elements Since we aim for ~ 30% precision we can use the vacuum saturation (VSA, also called naive factorization) approximation assuming that vacuum state, inserted between the two currents, contributes dominantly with respect to other states [33, 72, 73]. The matrix elements in VSA are factorized and it is clear that long distance contributions between the two hadronic currents are neglected. Obtained matrix elements of single current operators are then decomposed into standard form factors (see Section 2.3.1). For B ^ n(p) transitions we use form factors calculated in the relativistic constituent quark model, with numerical input from the lattice QCD at large q2 [74]. For the Ds ^ D and K ^ n form factors we use results of [75, 76] where the approach combining heavy quark and chiral symmetries was used. 3.3.2 Hadronic amplitudes For three-body decays of B meson to pseudoscalars P, P1 and P2 we provide below the kinematically simplified expression for the factorized matrix element of the O3 operator, contributing in the frameworks of SM, MSSM, and Z' (P2(;P2)Pi(;Pi) I d>s I 0) (P(p) I dYMb I B-(Pb)) = (3.32) = (t - u)FfPl (s)FiPB(s) + (mpl - mp2 )(mB - mp) (s)FfB(s) - FPp2Pl(s)F0PB(s) We introduced Mandelstam variables s = (pB - p)2, t = (pB - p1)2, and u = (pB -p2)2. Crossing symmetry relates the transition 0 ^ P^1P2 to the P1 ^ P2, whose form factors are known. The above expression (3.32) also holds for contribution of operators O3, O1; and (O1 to the B ^ PP^1P2 amplitude, as only the vector parts of currents have the correct parity. Explicit parameterization of form factors F1 and F0 are shown in Appendix A.3. For the matrix elements of (pseudo)scalar operators and (O4 in the RpMSSM framework one can use equation of motion for the quark fields iDq = mqq, DM = - (3.33) s to express the operators as divergences of (axial) vector currents id^qY ) = ^ ' j, (3.34a) —j - mqi fcY5?; = - ). (3.34b) mqj + mq; Using (3.34) we write down matrix element expressed in terms of scalar form factor Fo(q2) (P2(P2)Pt(Pi) | ds | 0> (P(p) | db | B-(pb)> = (mPi - mP2)(mB - mP) fp2P1(s)fpb(s) (3 35) (m* - md)(ms - md) F (')F (S) (3.35) Again due to parity conservation, pseudoscalar components of operators 04 and 04 are irrelevant for the matrix element of B ^ PPiP2. In the Z' model we also encounter contributions of the operators 02 and 02, whose color structure is Fierz rearranged to 04 and 04 and then (3.35) applies as well. In two-body decays to vector meson V and pseudoscalar meson P in the final state we sum over the polarizations of V. The sum in our case reduces to El* (PV )pb|2 = ^, (3.36) where eV is the polarization vector of V and A is defined as A(x, y, z) = (x + y + z)2 — 4(xy + yz + zx). For decay to two vector mesons we use the helicity amplitudes formalism as described in [77]. Unpolarized decay width is expressed as sum of the three helicity widths lp1l 2 2 2 r 8nmR (|Ho|2 + |H+i|2 + |H_i|2) , (3.37) where p1 is momentum of the vector meson in B meson rest frame and helicity amplitudes are expressed as y A(mR, m2, m2) H± 1 = a ± 1-c, (3.38a) 2m1m2 Ho = -m2 - — - m2 a - A(mB'—f2) b. (3.38b) 2mim2 4m2m2 Vector meson masses are denoted m1,2, while definition of the constants a, b and c is in terms of general Lorentz decomposition of the amplitude Ha = ^(A)e2v(A) (ag"v + ——p£pB + —— e^Pi^) , (3.39) ^ \ mim2 B mim2 J where ei 2 and pi 2 are the respective vector mesons' polarizations and momenta. 3.3.3 Hadronic decay widths B- ^ n-n-K+ Hadronic matrix element entering in the amplitude for B- ^ n-n-K+ in SM (MSSM) is readily given by (3.32) after identifying P = n-, P1 = K+, P2 = n- and using appropriate form factors whose explicit form can be found in Appendix A.3. Expression (3.35) is used instead for /pMSSM, while for the Z' model the amplitude consists of both contributions (3.32) and (3.35). The two final state pions are indistinguishable and the crossed term with interchange u o s is to be present in the amplitude. After phase space integration the hadronic decay widths can be written in a compact form with only Wilson coefficients on the matching scale A left in symbolic form: T (MS)SM nnK (MS)SM nnK rZ' x nnK C CRp + CR cZ + CZ + + Re C3Z' + C3Z' ' x 2.0 x 10-3 GeV5, x 8.9 x 10-3 GeV5, x 3.5 x 10-3 GeV5 % 1.3 x 10-3 GeV5 Cf' + CZ') Cf + Cf x 3.2 x 10-3 GeV5. (3.40) (3.41) (3.42) B- ^ n-D-D+ In calculation of the B- ^ n-D-D+ decay rate again we use (3.32) and (3.35), now with P = n-, P1 = D+ and P2 = D-. Numerically this yields T (MS)SM ■KDDs (MS)SM -pi'-p 1 nDDs C. C4p + C74 r Z' nDDs cz + C( Z' + cZ' + CZ' x 8.7 x 10-9 GeV5, x 8.2 x 10-5 GeV5, x 1.6 x 10-5 GeV5 % 5.7 x 10-9 GeV5 + Re cf + CZ') CZ' + Cf' x 5.9 x 10-7 GeV5. (3.43) (3.44) (3.45) This mode turns out less favourable than B ^ n n K + due to phase space suppression. B- ->■ n-K0 In [57] this decay was addressed as the mode with the wrong kaon mode, being highly suppressed in the SM compared to the decay with K in the final state. The operators O1 3 and (91 3 that are present in SM, MSSM, and Z' model have the 2 ■i 2 2 * following contribution: (K0(pk) | d7(u75s | 0) (n-(pn) | d7^b | B-(ps)> = i(mB - mn)/k(mK). (3.46) In the RpMSSM and also in the Z' framework, operator O2 is first Fierz transformed to O4 and O4, which then result in (K0(pk) | d75s | 0) (n-(pn) | db | B-(ps)) imK(mB - mn) -/K(mK). (mb - md)(ms + ^^.......^ However, in RpMSSM and Z' models, the two chirally flipped contributions to the amplitude have opposite signs, resulting in a slightly different combination of Wilson coefficients compared to the B- ^ n-n-K + decay width rr-.........--2 r rf (MS)SM | (MS)SM C3 nK —p nK = flp /Op C4 - C4 | f' || nK = C f' - Cf'2 + -Re nZ' /O Z' C3 — C3 x 3.9 x 10-4 GeV5, (3.48) x 5.0 x 10-4 GeV5, (3.49) x 6.8 x 10-5 GeV5 2 x 2.6 x 10-4 GeV5 Cf - Cf') (Cf - Cf) *] x 2.7 x 10-4 GeV5. (3.50) B- ^ p-K0 Using the form factors parameterization (2.32) of the pseudoscalar to vector meson transition we derive the following two factorized matrix elements of axial-vector and pseudoscalar operators: (K 0(pk ) | doyfs | 0) (p-(ep,pp) | d7^75b | B -(pB)) = -2mp/KA0B(mK)e* ■ pb, (3.51) (K0(pk) | d75s | 0) (p-(ep,pp) | d75b | B-(ps)) 2mpmK /ka0b(mKk ' pb. (mb + md)(ms + m^)"7K"u (3.52) Finally, we sum over polarizations of the p meson using (3.36), and the unpolarized decay rates read r (MS)SM pK C (MS)SM -p 1 pK r f' pK C— + c4r Cf' + Cf + Cf' + Cf x 3.9 x 10-4 GeV5, x 5.0 x 10-4 GeV5, x 7.5 x 10-4 GeV5 2 2.6 10-4 GeV5 (3.53) (3.54) Re Cf + Cf ( Cf + C x 8.8 x 10-4 GeV5. (3.55) 2 2 p * B- ->• n-K -1^*0 Factorized matrix element here is a product of vector meson K0* creation amplitude (2.29b) and B- ^ n- transition amplitude. Operators with vector currents result in k*°(ek,pk) i dy^s i 0) (n-(pn) i dy^b i b-(pb)) = 2gk*fnb(mk*)ek • pb, * FnB (mi* V * (3.56) while the density operators O4 and O4 do not contribute as a result of (3.34) and (2.29b). Thus the RpMSSM model does not contribute to this channel in the naive factorization approximation. (MS)SM nK* is* rZ' r nK* C C Z' + c f + (MS)SM C3Z' + C3Z x 7.4 x 10-4 GeV5, : 5.9 x 10-4 GeV5 x 4.^ 10-4 GeV5 (3.57) + Re C f + C Z Z' CZ' + C73Z' x 1.1 x 10-3 GeV5. (3.58) 2 * B- ^ p-K*0 Like in the previous case, this channel does not receive factorizable contributions in the RpMSSM framework. In SM, MSSM, and Z' we calculate unpolarized hadronic amplitudes of the operators O 1 ,3 and O i,3 by utilizing the helicity amplitudes formalism. With form factor decomposition (2.29b), (2.32) we express the polarized amplitude as in (3.39) and identify constants a, b and c: a = -4(mB + mp)gK* ApB(mK*)(C - C), (3.59a) b = g k * A2B (mK* )(C - C), (3.59b) 2 mB + m p c = - i mK^mp gK* vpB(mK*)(C + C). (3.59c) 2 mB + mp K C and C are combinations of the Wilson coefficients present in a considered model. We have C = c3ms)sm Č = 0 (3.60) in the SM and MSSM and C = [fQCD (mf)]-1 /8Cf' + [fQcD (mf)]-i/4C3Z', (3.61) (5 = [fQCD (mf)]-1 /8Cf + [fQCD (mf)]- 1/4^3Z' (3.62) in the Z' model. Decay rates are then r (MS)SM pK * (MS)SM rZ' r pK * C cz ' + c z + + + + Re + Re C Z' - C Z' CaZ' + (7sZ' C3 _ C3 X 9.2 X 10_4 GeV5, X 2.8 X 10_5 GeV5 X 7.2 X 10_4 GeV5 X 2.3 X 10_5 GeV5 5.8 10_4 GeV5 CZ' + CZ' C3Z' + (73Z' cZ ' - C Z' cZ ' - c Z' (3.63) X 5.0 X 10_5 GeV5 x 1.3 x 10_3 GeV5. (3.64) 3.4 Constraints on the short distance parameters We have investigated the b ^ dds transition within the SM, MSSM without and with Rp terms and within a model with an extra Z' gauge boson. The SM contribution leads to extremely small branching ratio for this transition. Roughly one order of magnitude increase in the MSSM compared to the SM predictions is still too insignificant for any experimental search. The supersymmetry with Rp terms, however, might give significant contributions and a possibility to exclude down the parameter space even further. The Z' model exhibits its structure through interplay of different types of effective interactions and might also give opportunity to constrain its relevant parameters. In the b ^ dds decay a particular combination of the model parameters appear which can be constrained using the B_ ^ n_n_K+ decay mode. In our calculation we have relied on the naive factorization approximation, which is sufficient to obtain correct gross features of new physics effects. One might think that the nonfactorizable contributions might induce large additional uncertainties, but we do not expect them to change the order of magnitude of our predictions. Additional uncertainties might originate in the poor knowledge of the input parameters such as form factors. However, we do not expect these to exceed 30%. Using the most stringent experimental bound for the B(B_ ^ n_n_K+) < 9.5 X 10_7 [49] and normalizing the masses of sneutrinos to a common mass scale of 100 GeV we derive bounds on the JLMSSM terms given in (3.21) 2 V f100 Ge^2 (A' A'* + A' A'* ^ VAra3 1 An 1 2 + Xn2 1 An 1 3) n= 1 V m£>" J < 6.6 10 5 (3.65) Complementary bounds coming from measurements of K-K and B-B mixings have been established in [78] Re Im Re A /100 GeV V V mf„ ) ,n= 1 ' 100 GeV .«,= 1 £ .«,= 1 3 E 100 GeVy 1 \ ,n= 1 n An31An12 An31An12 An21An13 < 2.6 x 10-6, < 2.9 x 10-8, < 2.9 x 10-4. (3.66a) (3.66b) (3.66c) From (3.66a) and (3.66b) it becomes apparent that the A^A^ term is negligible in (3.65), and the bound becomes simpler E ^100 GeV Y n=1 J An21 Ari13 < 6.6 x 10 -5 (3.67) now being more restrictive than the bound (3.66c), obtained from B-B mixing alone. Assuming that new physics arises due to an extra Z' gauge boson we derive bounds on the parameters given in (3.27). We neglect interference between Wilson coefficients, namely the third term in (3.42). Experimental bound of this simplified expression now confines (jCf + Čf |Cf' + Cf'to lie within an ellipse with semiminor and semimajor axes as upper limits R^L r,dR , r>dR r,d L B12 b13 + B12 b13 RdL RdL i RdR RdR b12 b13 + b12 b13 < 3.2 x 10-4, < 5.2 x 10-4. (3.68a) (3.68b) Complementary bounds of the same couplings originate from neutral meson mass-splittings and CP violation in kaon system and have been derived [70] Re[(BdR,L )2] Re[(BdR,L )2] Im[(BdR'L )2] < 10 8 < 6 10 8 < 8 10 -11 (3.69a) (3.69b) (3.69c) The above bounds are stronger than our (3.68). Nevertheless, the bounds (3.65) and (3.68) are interesting since they offer an independent way of constraining the particular combination of the parameters, which are not constrained by the Bd-Bd, B°-B°, K°-K° oscillations, or by B- ^ K-K-n+ decay rate (c.f. [79]). Using these inputs we predict the branching ratios for the various possible two-body decay modes and the B- ^ n-decay. We apply the bound (3.65) on R expressions for hadronic decay widths r^lx from the previous section and find the experimental sensitivity to RpMSSM couplings of the given channel. The procedure is straightforward except for the B- ^ n-K° and B- ^ p-K*° decay channels. rpv n rpv* In those we have to assume as in [51, 52] that interference term CR C negligible, which leads to the approximation |CRPV - CRPV | ~ |CRPV + CRPV |. is 2 n In the case of the Z' model there are contributions from Wilson coefficients "1" (Cf' and Cf') and "3" (Cf' and Cf'). We have already neglected the interference terms between "1" and "3" in (3.42) to find bounds (3.68) and we assume that these terms are small for all considered decay modes. Using (3.68) we can now predict branching ratios for decay modes B- ^ n-D-D+, B- ^ p-K0, and B- ^ n-KThe remaining two decay widths B- ^ n-K0 and B- ^ p-K*0 can be approached after we neglect interference terms Cf C f * and Cf Cf *. The results are summarized in Table 3.2. Decay channel SM MSSM /pMSSM Z' B- n n K+ 1x 10- - 1 5 1x < 10" -14 constraint constraint B- 6x 10- -2 1 6x 10- 20 9x < 10-9 4x 10-9 B- n-K0 3x 10- 6 3x 10- 15 5x c 10-8 2x 10-7 B- p-K0 3x 10- 16 3x 10- 15 5x c 10-8 4x 10-7 B- n-K*0 5x 10- 16 5x 10- 15 — 5x 10-7 B- p-K*0 6x 10- 16 6x 10- 15 — 6x 10-7 Table 3.2: The branching ratios for the AS = —1 decays of the B- meson calculated within SM, MSSM, RpMSSM, and Z' models. The experimental upper bound for the B(B- ^ n-n-K+) < 9.5 x 10-7 has been used as an input parameter to fix the unknown combinations of the RpMSSM terms (IV column) and the model with an additional Z' boson (V column). The SM gives negligible contributions. The MSSM increases them by one order of magnitude, which is still insufficient for the current and foreseen experimental searches. Using constraints for the particular combination of the RpMSSM parameters present in the B- ^ n-n-K + decay we obtain the largest possible branching ratios for the two-body decays of B- ^ p-K0 and B- ^ n-K0, while for the B- ^ n-K*0 and B- ^ p-K*0 the RPV contribution is suppressed by the vanishing of factorizable contributions. However, these two decay channels are most likely to be observed in the model with an additional Z' boson, if we assume that interference terms are negligible. Since in the experimental measurements only Ks or Kl are detected and not K0 or K"0, it might be difficult to observe new physics in the B- ^ n-K0 decay mode. Namely, the branching ratio B(B- ^ n" -K0) = (23.1 ± 1.0) x 10-6 [80] is two orders of magnitude higher than our upper bound for the B(B- ^ n-K0) making the extraction of new physics from this decay mode almost impossible. Therefore, the two-body decay modes with K*0 in the final state seem to be better candidates for the experimental searches of new physics in the b ^ dds transitions. Chapter 4 Neutral currents in charm and D ^ P£+£- Charm mesons are the only low energy window into flavour changing currents (FC-NCs) involving up-type quarks. There are essentially two distinct up-type quark FCNC processes in the SM at low energy. One of them is the decay c ^ uy, which can be either on-shell or virtual, and other the neutral meson mixing cu o cu. This is to be contrasted with wealth of experimental information on FCNCs in the down-type quark sector. Top quark physics will bring in additional input but already a handful of charm observables is worth studying as new possibilities opened up with the recent measurement of D0-D0 oscillations. The mixing was reported by Belle, BaBar, and CDF collaborations [11-15]. Combining the measured quantities [80] resulted in determination of mass splitting between the two CP-eigenstates AmD as well as ArD x = ^Wfl =(0.98 ± 0.25) x 10-2, (4.1a) Td Ar y = ArD = (0.83 ± 0.16) x 10-2. (4.1b) 2rD These results immediately stimulated many studies (c.f. [81-86]). In light of the long distance dominated SM prediction for x and y, ranging from 10-5-10-2 [8790], the measured values of x and y are not in favour of NP effects. However, they give additional constraints on physics beyond the SM as observed in [82, 83]. On the other hand, also the study of rare D meson decays is not considered very informative in current searches of physics beyond the SM [91-98], as it is expected from B physics. Namely, most of the charm meson FCNC processes are dominated by virtual d and s quarks, signaling strong presence of long distance (LD) resonant contributions, which dominate over genuine short distance (SD) effects [91-99]. In light of new data on charm meson mixing we will study rare decays D ^ n£+£-and Ds ^ K£+£-, where £ = e, and provide updated constraints of Rp-violating MSSM and a model with scalar leptoquark. 4.1 Charm decays and resonances Conspiracy of CKM elements and quark masses makes FCNC charm decays very susceptible for presence of low energy QCD dynamics. Up to ~ A2 Cabibbo order, the third generation of quarks does not mix with the first two and in CP-conserving processes hadronic states of the first two generations saturate the decays widths. As for the genuine short distance contribution, the GIM mechanism and smallness of the down-type quark masses renders the radiative c ^ uy decay width strongly suppressed at the leading order in the SM [91, 95]. The QCD effects enhance it up to the order of 10_8 [100], however the overall decay width is saturated with the long-distance resonant contributions. We will show that when one includes into consideration the possible effects of MSSM with non-universal soft breaking terms on c ^ uy [101, 102], the enhancement relative to the SM value is a factor 10, still too small to give any observable effects in D ^ Vy decays (V is a light vector meson). The dominant long-distance (LD) contributions in the D ^ Vy decays give the branching ratios of the order B ~ 10_6 [91, 95], which makes the search for new physics effects impossible in D ^ Vy. Another possibility to search for the effects of new physics in the charm sector is offered in the studies of D ^ or D ^ exclusive decays which might be result of the c ^ transition [84, 92, 93, 96, 97, 99]. Within the SM inclusion of renormalization group improved QCD corrections of c ^ gave an additional significant suppression leading to the inclusive rates r(c ^ ue+e_)/rDo = 2.4 X 10_10 and r(c ^ )/rDo = 0.5 X 10_10 [103]. These transitions are largely driven by a virtual photon at low dilepton mass q2 = (p+ + p_)2, while the total rate for D ^ is saturated by the LD resonant contributions at dilepton invariant masses q2 = mp,m2 , [92, 97]. NP could possibly modify the dilepton invariant mass spectra below p or above 0 resonant peaks. In the q2 spectrum of D ^ decay there is a broad kinematical region of dilepton invariant mass above the 0 resonance which presents a unique possibility to study c ^ [97]. In order to compare effects of NP and the SM we have to estimate size of the resonant contributions. We will also extend our analysis on FCNC decays to the charm-strange mesons Ds ^ whose upper bounds are currently much weaker than for the corresponding D decays. There are intensive experimental efforts by CLEO [104, 105] experiment and Fermilab collaborations [106, 107] to improve the upper limits on the rates for D ^ X£+decays. Two events in the channel D+ ^ n+e+e_ with q2 close to m^ have already been observed by CLEO [104]. Currently the upper bounds are B(D+ ^ n+e+e_) < 7.4 X 10_6 [108], (4.2a) B(D+ ^ nW_) < 3.9 X 10_6 [109]. (4.2b) Other rare D meson decays are more difficult to access experimentally, but with the plans to make more experimental studies in rare charm decays at CLEO-c, Tevatron and charm physics sections of the forthcoming LHCb and Belle 2 experiments makes the study of rare D decays more attractive. 4.2 New physics scenarios 4.2.1 Additional up-type quark singlet Some models of new physics contain an extra up-type heavy quark singlet [110, 111] inducing FCNCs driven by Z boson exchange in the up-type quark sector [94, 112115], while the neutral current for the down-type quarks is the same as in SM. The most stringent bound on parameters of these models comes from the measured x of D0-D0 mixing as given in (4.1). In our calculation, we analyze how these bounds on the FCNC vertex cuZ affect the D ^ P£+£- decays. A particular version of the model with tree-level up-quark FCNC transitions is the Littlest Higgs model [116]. In this case the magnitude of the relevant c ^ uZ coupling is further constrained by the large scale / > O(1 TeV). The smallness implies that the effect of this particular model on c ^ u£+£- decay and relevant rare D decays is insignificant [94]. Similar effect can be produced in the model with an extra Z' gauge boson, which couples as c ^ uZ', and was shown to produce weaker constraints in D0 ^ £+£- than in D0-D0 mixing [117]. 4.2.2 Supersymmetry The leading contribution to c ^ uy in MSSM with conserved Rp-parity comes from one-loop diagram with gluino and squarks in the loop [92, 97, 102]. Using the new bound on the mass insertion parameters within MSSM [81, 82] from the D0-D0 oscillations constraints (4.1) and constraints from the MSSM vacuum neutrality we will argue there are no good prospects for using D ^ Vy as probe of MSSM. Same holds for the tree-level photon exchange which enhances the short distance c ^ u£+£- spectrum at small q2. Bounds on the mass insertion parameters make the abovementioned enhancement in D ^ P£+£- decay negligible [92, 93]. On the other hand, among popular models of NP the supersymmetric extension of the SM including the Rp-parity violation (RpMSSM) is still not constrained as other NP models. As noticed in [92, 103] one can test some combinations of the Rp-parity violating contributions in D+ ^ n+£+£- decays. We place new constraints on the relevant parameters and demonstrate the effects of RpMSSM in the D+ ^ K+£+£- decays which might be interesting for the future experimental studies. 4.2.3 Weak singlet leptoquark Leptoquark states are expected to exist in various extensions of SM. They were first introduced in the early grand unification theories (GUTs) in the seventies [1,2]. Scalar leptoquarks are expected to exist at TeV scale in extended technicolor models as well as in models of quark and lepton compositeness. Squarks in supersymmetric models with Rp violation may also have leptoquark-type Yukawa couplings. Usually, they are present due to some symmetry between leptons to quarks in the fundamental theory and consequently their interactions may trigger lepton and baryon number violation which might lead to proton decay. Recently, leptoquarks were revived in search of resolution of the so-called /Ds puzzle [118]. Namely, the measured decay widths of Ds ^ pv, tv have been with moderate significance > larger than the prediction in terms of Gf, Vcs, and fDs: r SM Ds^-rv G 22 F m; lVcs|2 fDs mDs 8n 1 mT mDs (4.3) It was first pointed out in [118] that scalar leptoquark exchange could, in contrast to other mechanisms such as s-channel charged Higgs exchange, add positively to the SM decay width of Ds ^ ^v. Several studies have been done [119-122] but in the meantime the fDs puzzle has lost its significance [123, 124] Either way, weak singlet scalar leptoquark can also contribute to the effective operators mediating c ^ In [119] this state was part of the 45-dimensional representation of the SU(5) group, which contains also other scalars states which might be light and contribute notably in low energy phenomena or top quark physics [125]. 2 2 4.3 C ^ UY Given the recent observation of D0-D0 mixing, we evaluate the possible effect of MSSM on c ^ uy, taking into account the D0-D0 mixing parameters (4.1). Since MSSM with universal soft-breaking terms is known to have negligible effect [102], we consider the case of non-universal soft-breaking terms. We take into account only the gluino exchange diagrams through (5U2)LR and (5U2)RL, since the remaining mass insertions cannot have sizable effect as shown in [101, 102]. The maximal value of (5f2) LR,RL insertions has been constrained by saturating parameter x with the gluino exchange mechanism in [82]. Their results corresponding to a value x = (0.79±0.34) x 10-2 are shown in second column of Table 4.1. Another constraint is obtained by requiring of minima of MSSM scalar potential not to break electric charge or colour and they are bounded from above (5 U2)LR,RL < \/3mc/mg [126], with values given in third column of Table 4.1. The second constraint is obviously stronger for mg > 350 GeV, while AmD gives more stringent constraint for lighter squarks. Using (5 U2)LR,RL < V3 mc/mg, mg = mg = 350 GeV, mc = 1.25 GeV and expressions from [102] we get the upper bound r(c ^ UY)/rDo < 8 x 10-7, (4.4) which is one order of magnitude larger than the SM prediction r(c ^ uY)/rD0 = 2.5 x 10-8 [100]. However, this possible MSSM enhancement by 1 order of magnitude mg = mg max |(5u2)lr,rl| max |(5 u2)lr,rl| from AmD from stability bound 350 GeV 0.007 0.006 500 GeV 0.01 0.004 1000 GeV 0.02 0.002 Table 4.1: Upper bounds on mass insertions |(5^2)LR,RL| from measured AmD and stability bound [126]. would not affect the rate of the D ^ V7 decays, which are completely dominated by LD contributions with 13 ~ 10-6 [91-93, 95, 99]. The only theoretically sound observable probing the c ^ uy remains the Bc ^ B^y decay, where LD contributions are strongly suppressed and thus comparable in size with SD contributions [127]. 4.4 D ^ P£+£ decays: short distance amplitudes 4.4.1 SM The c ^ £- transition is induced at one loop level in the SM. We will use the effective theory description where W boson, t and b quark degrees of freedom are already integrated out. Here we will follow the procedure taken in [103]. Starting at the weak scale ~ mW we have - 4Gf Leff = Ad E Ci(pw)(Qd - Qb) + As Ci(pw)(QS - Qb) i=1,2 i=1,2 with standard current-current operators (4.5) Q1 = (uLy^L )(qL Y^L), (4.6a) Q2 = (uly^^l) (qLY^CL), (4.6b) where we have denoted CKM mixing factors Aj = VCiVUj. The direct quark-lepton operators e2 Q9 = (ulymcl)(^V) (4.7a) e2 Q1° = (W(uly^cl) (^ymy5^) (4.7b) are not present in SM and neither are the standard QCD penguin operators Q3...6 (c.f. [128]). Operators Q9 and Q3...6 are not present as a consequence of CKM unitarity since at the matching scale mW we sum over all down-type quarks which are considered massless and GIM cancellation is exact.Operator Q1° is negligible in the SM and does not mix with other operators [103]. The electromagnetic dipole contribution em Q7 = 7enC2 cr) F^v (4.8) (4n)2 is further suppressed by a and thus neglected in (4.10). Matching of the Wilson coefficients at the weak scale is performed at NLO in QCD [128] n f \ 11 as (mw) /„ n x C1 (mw) = y—4^-, (4.9a) nf \ 1 11 as(mW) ,. nM C2(mw) = 1 - ---• (4.9b) 6 4n The scale dependent Wilson coefficients C1,2(p) are then run down to the b quark threshold using the 2 x 2 anomalous dimension matrix given in [128]. At pb the 5-flavour effective theory is matched onto the 4-flavour theory, which generates the penguin operators (see Appendix A of [103]). Evolution of the Wilson coefficients CdS and C3...6,9 down to hadronic scale is performed using the 7 x 7 anomalous dimension matrix (see [128], eqs. (6.25), (6.26) for Ci...6, and (8.11), (8.12) for C9). Finally, the effective Lagrangian at the charm scale is then [103] /•SM Leff 4Gf 72" Ad ^ Ci(^c)Qd + A^ Ci(^c)Q? - Afc Ci(^c)Q i= 1,2 i= 1,2 i=3,..., 1 0 (4.10) The term in (4.10) containing QCD penguins Q3...6, dipole operators Q7,8, and operators with (axial)vector lepton current Q9,10 are in SM rendered negligible due to strong Cabibbo suppression Ab ~ A5. The dominant contribution comes from the two loop diagrams on Figure 4.1 with Q2 insertion and additional virtual gluon [100]. The amplitude of the free quark decay c ^ in SM can be parameterized by u c Figure 4.1: Dominantly contributing diagrams to c ^ in SM. Box vertex denotes the Q2 insertion. the electromagnetic dipole operator Q7 (4.8) which is generated by the insertions of Q2 operator (Fig. 4.1). We take for the effective coefficient C7ff [100, 103] Cff = As(0.007 + 0.020i)(1 ± 0.2). (4.11) 4.4.2 Models with additional up-type quark singlet The class of models with an extra up-like quark singlet (EQS) induce FCNCs at tree level [94, 110] leqs g cos 3 - sin2 JEM). (4.12) c u c u c u c u c u Electromagnetic current is denoted Jem, the singlet quark T while the weak neutral current contains T * 3 T3 = Q iLY* QjL 1 "UjiY^Q jUjL - 1 diLY*diL- (4.13) There is mixing among up-type quarks present in (4.13) [116], where U1...4 and di...3 are the quark mass eigenstates. Using the convention where we assign the physical flavour rotations to up-type quark sector, the CKM matrix is generalized to 4 x 4 unitary matrix U = fVvd ^Us Vub Gu Vcd Vcs Vcb Gc Vtd Vts Vtb Gt vVTd VTs VTb gt/ (4.14) which causes tree-level FCNCs in the interaction term J^3 Z* in the up-type sector. The mixing matrix of the up-type quarks Q contains the elements of the last column of matrix T^ Q = (1 -|Gu|2 -GcGU -GtGU v-gtGU -GuG* 1 -|Gc|2 -GtG* -Gt G* -GuGT \ -GuG* -Gc G* -GcG: 1 -|Gt| -GtG* 1 - |Gt|2/ -GtGT (4.15) The unitarity of the extended CKM matrix then implies that off-diagonal elements of Q could be non-zero, e.g. Quc = -GuG* = VudVc*d + VusVS + = 0. The effects are encoded in Wilson coefficients C9 and C10. Relative to the negligible SM values, they are modified by the presence of an extra up-like quark: 4n VubVctčCg = — Quc(4 sin2 - 1), a * 4n VubVcb^C10 = -Quc - a (4.16a) (4.16b) The element Quc of the up-type quark mixing matrix is constrained by the measurements of D0-D0 mixing (4.1). Using expression Am^ = 2 x 10-7 |Quc|2 GeV derived in [116], we find (4.17) Quc < 2.8 x 10 4 4.4.3 MSSM The leading contribution to c ^ in MSSM with conserved Rp-parity comes from the gluino exchange diagram via virtual photon and could enhance c ^ at small q2. However, this enhancement is sizable only in vector decay channels, e.g. D ^ [92], whereas gauge invariance cancels the 1/q2 behaviour as the decay to D ^ P7 with on-shell photon is forbidden. To clarify this, consider the amplitude for D(p) ^ P (p') 7 *(q,e) with virtual photon - A(q2) [q2(p + p')* - (mD - )q*] e*, (4.18) which, as written is gauge invariant and vanishes in the limit q2 ^ 0. Scalar function A(q2) summarizes hadronic form factors and is analytic at q2 = 0. Replacing the photon polarization with propagator and coupling it to a lepton pair gives Au^p-r^Pii - A(q2) [q2(p + p'f - (mD - mp)q^] ^u(p_)7(Uv(p+) (4.19) - A(q2)u(p_)(/ + /Mp+), (4.20) where the term proportional to dropped out and the remaining term cancelled the 1/q2 enhancement. Consequently, penguin diagrams with gluino exchanges in MSSM can not produce interesting signatures in D ^ and we will not pursue them any further. 4.4.4 Rp violating MSSM In MSSM with broken Rp-parity (RpMSSM), the c ^ process is mediated by the tree-level exchange of down squarks [92] (see the discussion about aspects of RpMSSM in Section 3.2.3). The relevant trilinear interaction terms are contained in the superpotential [67] Aijk = KrsUrj DsR> (4-23) L/p 9 Aijkd*kR(liL)cUjL + H.c.. (4.21) Integrating out the squarks leads to the effective four-fermion interaction Leff = E ^(UL7MCL)(^iL7M^iL), (4.22) k=1 dkR Note that down squark is a leptoquark with SM quantum numbers (3,1, -1/3) whose effects in a more general setting will be studied in the next section. The CKM-rotated couplings between the L, Q, and D supermultiplets in the superpotential are denoted [92] ^ijk /^irsUrj1 where UL and DR matrices transform the up-type left-handed quarks and down-type right-handed squarks from mass to weak basis, respectively. In the effective theory framework (4.10), the tree-level squark exchanges contribute in Wilson coefficients C9,10 [103] = -VWC10 = =!JW£ (4.24) k=1 V dkR J where i = 1 (2) is relevant for the e+e_ (p+p_) mode. The A'12k and A'11k have been constrained from tests of charged current universality and neutrinoless double ,0-decay searches [103, 129] - m ~ A"k- < 0 021I50t!ev■ (4-25a) ^ < °'043100 GeV' (4-25b) Taking into account (4.25a) we determine the maximum value of 10 to be |Vc:V^C9e;10| < 4. (4.26) Value of £fc A22fcA21fc can be inferred from the experimental upper limit B(D+ ^ < 3.9 x 10-6 [109]. The latter experimental bound is almost saturated with LD amplitude. Analysis is undertaken where we include in the amplitude both long distance SM and short distance RpMSSM contributions in order to constrain the A'2xx/m(j couplings. 4.4.5 Scalar leptoquark (3,1, -1/3) Interactions of the scalar A particle in the (3,1,-1/3) representation of the SM gauge group with SM fermions are La = YLj QfiT2A*Lj + YR ur^r (4.27) Here we have restricted ourselves to the dimension-4 interactions. Doublets Q and L denote left-handed leptons and quarks in the mass basis, and we assigned the physical flavour rotations in the quark sector to down-type quarks Q* = (j) ■ (428) where V is the CKM matrix. yl,r are arbitrary matrices with i, j denoting generation indices, whereas = C0 T 1. Omitted colour indices are contracted between quark and A fields, both of which are the fundamental (anti)triplets of SU(3)c. Note that both YL and YR couple A to up-type quarks and charged leptons. Their contribution to the effective Lagrangian for the c ^ decay is V*vyc. = -™o = =0w mW Y,.^.*, (4.29a) 2a2 mA VubVcb5(A9 = v^aCu = ^^ mw YR.YR.*, (4.29b) 2a m a where C9,10 are Wilson coefficients of the chirality flipped operators Q 9 = (ur7mcr )(^v), (4.30a) QQ10 = (ury^cr) (4.30b) Comparison of (8.28) and (4.24) suggests that the RpMSSM model can be embedded into the LQ model if one sets Y^Y^/mA ^ ^k A.2kA.1k/mf . As the right-handed down-type squarks dkR have the quantum numbers of A they act as leptoquarks with exclusively left-handed couplings. In addition to Q9,10 and Q9,10, A exchange induces scalar (S) and tensor (T) operators LffT = ^ImA" [-(urcl) (^l) + c) (zr*l)] (4.31) + ^^ [-(ulcr) (4^r) + c) ^R)] . 2mA 1For charge conjugation matrix we take C = ij2,y0. Their contribution is small because new upper bounds [130] of leptonic branching fractions B(D0 ^ e+e-) < 7.9 x 10-8, (4.32a) B(D0 ^ p+p-) < 1.4 x 10-7, (4.32b) put strong constraints on the mixed helicity products of couplings Y^Y^1*, YL^YR^ Namely, saturating the above branching fractions with pure A exchange rA _ fDmDo |YL2iYRf + lYRY^I2 ( _ 256nm2 mA ' (4'33) where we use fD _ 206 MeV [131], mc _ 1.25 GeV, and me _ 0, to find bounds i y 21 y 11 | M. GLV)2 < x 10-4, (4'34a) Y22 Y12 L(R) R(L) < 1.6 x 10-4, (4.34b) (mA/100 GeV)2 which apply for the couplings to electrons (4.34a) and muons (4.34b). SM contribution is of the order 10-13 [92] and thus negligible with respect to the bounds (4.32). Clearly the products of the same helicity couplings YRYR or YLYL contribute to D0 ^ £+£- width with helicity suppression factor mf. Thus B(D0 ^ £+£-) is not a good probe of those coupling combinations. On the other side, in the D+ ^ n+£+£- all terms of (8.28) and (4.31) contribute. Differential decay width can be written in a quite compact way d2rD+^n+e+e- 1 dq2 dt (16nmD )3mA (IYl2^!2 + |YRYRf) F+ (q2)(q2 - mD)2 (4.35) 2 + (lY^Rf + lY^f) (s(q2)(mD - ml - q2 - 2t) + m^(q2)) q2 \ mc J We have set me _ 0 and introduced t _ (p' + p+)2. For matrix elements of scalar operators the QCD equations of motion (3.33) are used (3.34). Note that the decay spectrum is independent of possible phases contained in Y matrices. We will in the following set to zero products of type YL(R) YR(L) as they are too strongly constrained by leptonic decay branching fractions (4.34). 4.4.6 D(s) ^ n (K) form factors Evaluating the matrix elements of operators Q7,9,10 requires the knowledge of vector and tensor form factors of D —> P transition: (P(p') | U7"(1 - Y5)c | D(p))_F+(q2) 22 (p + p')" - mD ~ mp q" 22 m2 m2 (4.36) + Fb(q2)"tD 2"q", q2 (P(p') | (1 ± Y5)c | D(p)> _is(q2) [(p + p')"qv - q"(p + p')v ± (p + p')«q^ (4.37)" where P denotes n+ (K+) in the case of D+ (D+) decay. Momentum transfer q = p - p' equals momentum of the lepton pair. For the F+ form factor we use the double pole parameterization [132] of the analysis performed in the heavy meson chiral perturbation theory [133] F+ (q2) = F+(0) (1 - q2/mp*)(1 - aq2/mp*)' (4.38) with values F+(0) = 0.54 (0.62) for the D ^ n (Ds ^ K) transitions, whereas we fix a = 0.58 for both cases [133]. The latter parameter was determined also in the experimental analyses of D ^ P^v decays with the result a = 0.52(10) (c.f. [134] and references therein). Lattice simulation, on the other hand provided a = 0.44(4) [135]. We approximate the tensor form factor s(q2) by F+(q2)/mD which is valid in the limit of heavy c quark and zero recoil limit [136]. Finally, expression for the SD amplitude of c ^ decay is asd = -1—— C eff 1602 u(p-)/Y5v(p+) + (4.39) + C7ff 2n2 mD mc + r&n2) u(p-)/v(p+) F+(q2). Momenta p, p+ and p- belong to the decaying D (Ds) meson and the leptons in the final state, respectively. We neglect the lepton masses in our further study. 4.5 Long distance contributions in D ^ Pi+i The dominant short distance part of the SM amplitude is generated by the operator Q2 and the light quark loops accompanied by the virtual gluon. In addition to SD, poles in momentum transfer q2 may appear due to bound (quasi)stable states of QCD, whose properties are governed by nonperturbative QCD. The background they produce is crucial for isolating short distance physics in semileptonic decays D ^ Following procedure described in [94] we model the LD contributions with vector resonances V of appropriate quantum numbers. D meson first weakly decays to P and a neutral vector meson V, followed by decay of V ^ Weak nonleptonic decay is controlled by the Q1,2 operators of (4.10), which are, after Fierz transforming the operator with color non-singlet currents Q1 Lnonlep = —E V*? V* [a1U7*(1 - Y5)q 97* (1 - Y5)c q=d,s +a2U7*(1 - Y5)cqY*(1 - Y5)q] (4.40) The effective Wilson coefficients of naive factorization on the scale mc = 1.25 GeV are [73, 103] a1 = 1.26 ± 0.10, a2 = -0.49 ± 0.15. (4.41) The flavour structure of (4.40) allows intermediate resonance V to be either p, w or 0. Since branching fractions of separate stages in the cascade are well measured, we shall not use a factorization approximation (4.40,4.41) but will instead make use of currently available experimental data. Resonant decay spectrum must contain a pole due to intermediate resonance [137, 138] dr 1 /o2 dq2(D - KV - = n(02)(mV - + rVrv— («2)- (4.42) Here rD^Kv(q2) and rV^£+£- would be decay rates if mass of V were v^O2 and these rates are known experimentally at q2 = mV. Since the resonances V = p, w, 0 are relatively narrow (rv ^ mv) the narrow width (NW) approximation holds B [D — PV — ~ B [D — PV] x B [V — . (4.43) The Breit-Wigner resonant amplitude that reproduces the above behaviour (4.43) is then ald [d — KV — = q2 - m/+ imvrv u(p-)pv(p+). (4-44) In the NW approximation the av coefficient dependence on q2 is irrelevant and we assume av to be free parameters. We included phase 0v explicitly, so that individual av are positive. Equivalent description of the long distance amplitude was used in [92, 103] where they instead included it in the Cfff coefficient. 4.5.1 D+ — Right-hand side of (4.43) are measured experimental branching fractions (Table 4.2) which in turn fix the parameters av of (4.44). Decay mode D+ — n+w has not been decay channel D+ — n+p D+ — n+w D+ — n+0 B [10-3] 0.82 ± 0.15 < 0.34 5.53 ± 0.24 Table 4.2: Branching ratios of decays of D+ meson to the intermediate resonant states [28]. decay channel p — e+e w — e+e 0 — e+e B [10-5] 4.7 7.3 30 Table 4.3: Branching ratios of vector resonances decays to lepton pairs [28]. measured yet, but we can relate aw and its phase to the well measured contribution of the p resonance relying on the underlying nonleptonic weak Lagrangian (4.40) as in [94]. Relative phases and magnitudes of the resonances can be derived by considering the flavour structure of nonleptonic weak Lagrangian (4.40) and electromagnetic coupling of V resonance to photon. The flavour structure of the resonances then determines relative sizes and phases of resonant amplitudes. Detailed analysis has already been done in [94], where the relative phases of p and w contributions were found to be opposite in sign, while for the ratio of their magnitudes it was found aw/ap = 1/3. Also the phases of p and 0 are opposite. Thus we get for the final LD amplitude A LD BW(q2,p) - 1BW(q2, w) 3 - a^ BW(q2, 0) u(p_) /v(p+), (4.45) with individual contributions ap = (2.6 ± 0.2) x 10 9 and a^ = (4.0 ± 0.2) x 10 9, where uncertainties are estimated from the experimental ones. We have defined BW(q2, V) = (q2 - mV + imyry) -1 (4.46) 4.5.2 D+ — K+l-l+ In this case only the branching ratio of D+ —■ K+p is known (see Table 4.4). Contributions of p and w are related like in the case of D+ meson, namely aw/ap = decay channel D+ — K+p D+ - K+w D+ — K+0 B [10_3] 2.7 ± 0.5 — < 0.28 Table 4.4: Branching ratios of D+ meson to the intermediate resonant state [28]. 1/3 with relative minus sign between the two amplitudes. In the same way as for the D decays, we determine ap = 7.1 x 10-9. However, we cannot determine the a^ in the same manner due to unknown width of Ds —■ K0. Consequently, the total LD amplitude for resonant decay Ds — VK — is a sum of two terms: A LD = a« BW, (q2, p) - ^BW(q2, w) 3 u(p-) pv(p+) + A, LD 2 • (4.47) Last term of above amplitude can be calculated in the factorization approximation using the nonleptonic weak Lagrangian (4.40), which determines the width of Ds — K0. Both a1 and a2 parts of (4.40) can generate the flavour quantum numbers of 0 and K +. The a1 part connects initial D+ state to 0 through a charged current (sc)Vwhile the (us)V_a creates the K + out of vacuum. Neutral currents, namely the a2 part, act in the following way: D+ — K + and 0 — 0. Subsequent decay 0 — is measured (Table 4.3). The resulting 0 contribution to the LD amplitude in the factorization approximation is ALD 3 g2 cs q2(q2 - m2 + im^) x [aim2/xAo(mK) + a2g2/+(q2)] u(p_)pv(p+). where g2 is a 0 decay constant, defined as (0 | | 0(q, e)) = g2 = 0.233 GeV2. Value of g2 is determined from ^^e+e- r 2^e+ e- 4ng2 2 a2 27m0 (4.48) (4.49) (4.50) the value of which is taken from [28]. Transition D+ —■ 0 is parametrized by the form factor A0 (see Section 2.3.1) whose shape we take from [139]. a p 4.6 Decay spectra and widths of D(s) ^ n(K)t+t From the point of view of resonances it is by far most convenient to show spectra in variable q2, where one can isolate the resonance dominated region. Using the combined approach described in the previous section, where we account for the SD and resonant LD dynamics, we now show the impact of SD physics on decay spectra with respect to LD resonant background. Since the SD contribution of SM and MSSM is completely overshadowed by LD, we will only estimate the experimental prospects for discovering or constraining EQS and RpMSSM models. Current constraints on EQS model coming from the D0-D0 mixing already indicate only minor role in the total decay width. On the other hand, the contribution of RpMSSM model is still allowed by existing constraints to show up in the nonresonant part of the decay spectrum. When deriving the upper bounds for the underlying parameters of RpMSSM model we always vary free phases in the Lagrangian as to achieve the most conservative constraint. 4.6.1 D+->• n+t+t- decay experiment res. EQS res.+RpMSSM/LQ D+ ^ n+e+e- < 7.4 x 10-6 1.7 x 10-6 < 1.3 x 10-9 < 4.2 x 10-6 D+ ^ n+p+p- < 3.9 x 10-6 1.7 x 10-6 < 1.6 x 10-9 constraint on A' Table 4.5: Comparison of experimental branching fractions with predictions for branching fractions of D+ ^ n+decay. In the last three columns, separate predictions of resonant amplitude, short distance EQS amplitude, and the total amplitude in the RpMSSM or LQ case, are given. Branching fractions are listed in Table 4.5. Clearly, the EQS model is already too stringently constrained from D0-D0 mixing and measuring the D ^ cannot bring any further information with current experimental sensitivities (Figure 4.2). On the other hand, relevant couplings for the electron final states in the RpMSSM are already constrained (4.25a) and moderately increase the branching ratio almost to the upper experimental bound. Deviation from the LD amplitude is pronounced in the nonresonant region, either at q2 < mp or q2 > (Figure 4.3). However, the most promising mode is the channel with muons. The LD contribution (1.7 x 10-6) is at par with the experimental upper bound 3.9 x 10-6 and should be combined together with the SD part to derive constraints on the Wilson coefficients. The bound we obtained by saturating the experimental bound entirely with RpMSSM contributions was |VcbVubCg 10| < 18, whereas more stringent bound resulted after we included resonant amplitude in the analysis: im^J < 14. (4.51) The latter bound is the most conservative with respect to the unknown phase of A' couplings. Although the inclusion of the LD term does not make substantial difference in resulting bound, it should be included as the experiment will eventually measure a signal with branching fraction of the order 1.7 x 10-6. All the branching Figure 4.2: Distributions of the branching fractions in the EQS model for the decay channels D+ ^ e+e- (top) and D+ ^ (bottom). Blue line represents the combined resonant and EQS contribution, with EQS mixing matrix element Quc constrained from D0-D0 mixing (4.17). Red, green, and light blue lines show individual contributions of resonances, short distance EQS, and short distance SM, respectively. & 10 2B 101010101010- 0.5 1.0 1.5 q2 (GeV2) 2.0 2.5 3.0 Figure 4.3: Distributions of branching fractions in the RpMSSM model for the decay mode D+ — n+e+e-. Blue line represents total contributions of resonances and RpMSSM with parameters A'12fc/mj, A 'llfc/m(j constrained from charged current universality and neutrinoless double ft-decay (4.25a). Red, light blue, and green lines show separate contributions of resonances, SM short distance amplitude, and RpMSSM, respectively. 0 4 0 5 0 6 0 7 0 0 0 0 0-160 0.5 1.0 1.5 q2 (GeV2) 2.0 2.5 3.0 Figure 4.4: Distributions of the branching fractions in the RpMSSM model for the decay mode D+ ^ Blue line represents total contributions of reso- nances and RpMSSM with parameters A'22fc/mj, A21fc/mj adjusted as to saturate the experimental upper bound (4.2b). Red, light blue, and green lines show separate contributions of resonances, SM short distance, and RpMSSM contributions, respectively. 0 0 5 0 6 0 7 0 8 0 9 0 10 11 12 0 ratios within the jRpMSSM model with muons in the final state (Tables 4.5, 4.6) and their kinematical distributions (Fig. 4.4,4.7) are calculated using the bound (4.51). In the singlet leptoquark framework the couplings Y^R and YLR, with i denoting the lepton flavour, result in various terms, contributing to D+ ^ In the leading order, we only consider the resonant amplitude and its interference with the LQ amplitude, while dropping the LQ amplitude squared. The decay spectrum of intereference terms is l-pLQinterf. dl D^nl+l- dq2 1 3(8n)3mD [((m-D + mn)2 - q2)((mD - mt)2 - q2)]3/2 Y2iY + Y 2^Y T L ' R R (4.52) Xres.(q2) m A where a resonant shape was introduced Ares. — BW(q2, p) - ^BW(q2,w) 3 - a

tU o 0.02 0.01 2s G) -0.01 K (y22yl2* + YffYR*) /(mA/100 GeV) 0.00 0.01 2 Figure 4.5: Allowed region for couplings of the singlet leptoquark scenario, where, (i) only the interference term was considered (blue triangles), or, (ii) entire leptoquark contributions taken into account (red circles), for the semileptonic decays to electrons (above) and muons (below). (4.25a), or D+ ^ n+p+p- decay (4.51) and we use all those processes as input to decay process Ds ^ . The branching fractions contributions are summarized in Table 4.6. Again, the EQS model has negligible effect (Figure 4.6). /pMSSM has notable effect, especially in the p+p- mode, where it increases branching ratio by an order of magnitude (Figure 4.7). In this case, RpMSSM contributions overshadows the LD contribution throughout the phase space, except in the close vicinity of the resonant peaks. decay experiment res. EQS res. + RpMSSM/LQ D+ ^ K+e+e- < 1.6 x 10-3 6.0 x 10-7 < 3.4 x 10-10 < 1.9 x 10-6 D+ ^ K+p+p- < 2.6 x 10-5 6.0 x 10-7 < 3.8 x 10-10 < 1.8 x 10-6 Table 4.6: Comparison of experimental branching fractions with predictions for branching fractions of D+ ^ decay. In the last three columns, separate predictions for the resonant amplitude, short distance EQS amplitude, and total resonant amplitude including the RpMSSM or LQ contributions, are given. 4.7 Summary Recently measured D0-D0 mass difference constrains the value of tree-level flavour changing neutral coupling c ^ uZ, which is present in the models with an additional singlet up-type quark. We have studied the impact of this coupling on rare D+ ^ and D+ ^ Kdecays, where its effects are accompanied by the long distance contributions. Long distance contributions in D+ ^ Khave been assessed following the same phenomenologically inspired model as it has been done previously in the case of D+ ^ The constraint coming from D0-D0 mixing render the effects of additional singlet up-type quark to be too small to be seen in dilepton invariant mass spectra of either decay mode. In a previous study [94] forward-backward asymmetry in D0 ^ p0^+^- was considered and very small effect was found. New constraint reduces that asymmetry even more, making it insignificant for the experimental searches. Present constraints on mass insertions in MSSM with conserved Rp-parity still allow for increase of c ^ uy rate by one order of magnitude. For this reason MSSM could significantly increase c ^ rate at small lepton invariant mass q2. However, this MSSM enhancement is not drastic in D decays, since D ^ Vy and D ^ have large long distance contributions in the small q2 region, while D ^ rate is multiplied by factor of q2 owing to gauge invariance. The remaining possibility to search for new physics in rare D decays is offered by the MSSM models which contain Rp-parity violating terms or in a more general model with scalar weak singlet leptoquark. We have found new bounds on the combinations of these parameters in D+ ^ by including the long distance effects. Using current upper bound on the width of D+ ^ n+p+p- decay we derive Figure 4.6: Distributions of branching ratios in the EQS model for the decay modes D+ ^ K+e+e- (top) and D+ ^ K(bottom). Blue line represents the combined LD and SD contributions. Red line represents the resonant contributions, whereas green and light blue lines are the pure short distance spectra of EQS model and the SM. > CD a 0-4 0-5 0-6 0-7 0-8 0-9 0-10 0-11 0-12 0-13 0-14 0-15 0-16 0-170 0.5 1.0 1.5 q2 (GeV2) 2.0 2.5 Figure 4.7: Distributions of the branching ratios in the RpMSSM model for the decay modes D+ ^ K+e+e- (top) and D+ ^ K(bottom). Blue line represents combined LD and SD contributions. Red line represent the resonant contributions, whereas green and light blue lines are the pure SD spectra of the EQS model and the SM. 0 limits for the couplings of a weak singlet leptoquark model Y 21 Y11* JL(R)JL(R) 3 (mA/100 GeV)2 22 12* YL(R) YL(R) < 1.6 x 10 < 1.0 x 10-3. (mA/100 GeV)2 Second bound applies also for the Rp-violating MSSM, namely (4.56a) (4.56b) 3 \/ * EA22k A21k o -----—- < 1.0 x 10-3. (4.57) k=1 (mdkR/100 GeV)2 1 ' Since at Belle 2 there are plans to investigate rare D decays [140] we have used upper bounds (4.56b), (4.57), and calculated the dilepton invariant mass spectrum for the decay D+ ^ K+■£+This bound still gives small increase of the dilepton invariant mass distribution for the larger invariant dilepton mass, making it attractive for the planned experimental studies. Chapter 5 Dalitz plot analysis of the B ^ KnY decays An interesting proposal has been made by the authors of [141] on the possible effects of new physics in B ^ PiP2y decays. Namely, in these decays new physics might affect the polarization of outgoing photons. As it is known, in the SM photon emitted in b ^ sy is dominantly left-handed [142, 143]. Since most of experiments only have access to photon momentum and energy one has to rely on an indirect method of measuring photon polarization. Suitable observable is the time-dependent CP asymmetry of neutral B decays to a CP-eigenstate f and a photon: r(Bf) ^ fY) - r(Bf) ^ fY) = / sin(Amt) - / cos(Amt). (5.1) r(B(t) ^ fY)+r(B(t) ^ fY) /Y v ; /Y v ; v ; Mixing induced parameter S/Y has been studied in radiative decays of neutral B decays to K*y [142], B ^ PPy [141, 144], and also B ^ PVy [145], where P and V are a light pseudoscalar and vector meson. For three-body decay B0 ^ KSn0Y the authors in [144] used Soft Collinear Effective Theory (SCET) in the region with soft pion. They used the Breit-Wigner ansatz for the resonant channel via intermediate K*y and concluded that right-handed photons are mainly due to the resonance and related interference effects. In this chapter we focus on the decay width spectrum of B ^ K^y in kinematical region with the hard photon carrying energy of the order ~ mR/2 and one soft pseudoscalar whose energy is of the order Aqcd. Obviously, the remaining light meson is necessarily hard under these circumstances. These restrictions will allow us to simplify considerably evaluation of hadronic matrix elements. Emission of soft pseudoscalar off a heavy B meson line is driven by the leading order heavy meson chiral perturbation theory, whereas we use heavy quark symmetry and large energy effective theory for transition of heavy state transition to light energetic meson. We predict differential decay widths in these regions. This decay channel has been already observed by Belle and BaBar experiments [146-148], with the branching fractions [148] B(B0 ^ K0nY) = (7.1+2.1 ± 0.4) x 10-6, (5.2a) B(B+ ^ K+nY) = (7.7 ± 1.0 ± 0.4) x 10-6. (5.2b) Quoted errors are statistical and systematic, respectively. On the list of excited strange mesons, compiled by the Particle Data Group [28], one finds only two strange resonances with spin 2 and 3 which potentially contribute to the B0 ^ K°nY decays in the low to intermediate region of low K and n invariant mass . Their effects are not so important, as for the K|(1430), the product B(B ^ K2*(1430)y) x B(K2*(1430) ^ Kn) ~ 10-6 is one order of magnitude below branching fractions (5.2) and we can neglect it in the first approximation. Similar contribution from K|(1780) is 10-8 which is completely negligible. Smallness of resonant contributions has been confirmed by Belle experiment [146]. On the other hand, spectra of BaBar [148] show some excess of events in the 1.4 GeV < MKn < 1.8 GeV region, but due to large error bars one cannot draw any conclusion. Following this features we do not include any resonant contributions in our approach. 5.1 Framework The b ^ sy is induced by the AB = 1 effective Hamiltonian [128] ^2 Leff =--F VtSVb CiQi+C7Q7+^8^8 i=1 (5.3) The most important contribution in SM is due to electroweak penguin operator Q7 which couples tensor current between b and s quarks to the electromagnetic tensor e Q7 = 8^2 (1 + Y5)b + msš (5.6) e _ = 8^2mb ^Y(e, q) I 2dMAv | 0) x (K0(k) | (1 + Y5)b | B*(n, p)) = ^emb , where n and e are the respective polarizations of B* meson and the photon. For soft K0 (right-hand graph of Figure 5.2), the above manipulations are performed on flavor rotated states (B*,n) o (B*,K0). Virtuality of intermediate B* is zero up to 1/mb corrections, so use of the heavy quark spin symmetry is justified up to hard spectator effects [151]. In this picture, we assume heavy-quark interacts with light degrees of freedom solely through soft gluon exchanges and thus we use only upper-components field hv for the b-quark. This is similar to approaches in [151, 152]. 5.2 Large energy limit of B ^ P form factor In the following, we shall relate the B* ^ K0 tensor form-factors to the vector ones of B ^ K0. We repeat the standard form factors =2TfK(q2)ev^opB)MMo, (5.7a) =iT2BK(q2) [(M2 - mK)n - n ' q(PB + k)]v (5.7b) „2 + iTBK(q2)(n ■ q) q- q M2 - mK (Pb + k) (K0(k) | šyvb | B(pb))=FfK(q2) Pb + k - M2 m K (5.7c) + FBK (q2) M2 m2 q2 K qv, where M and mK are the B and K meson masses, respectively, and q = pB - k. Now we can use underlying heavy quark and large energy symmetries to constrain the number of independent form factors. Following [151], we express the matrix element between B and energetic K0 as Dirac-trace of their wave functions and the matrix r which is the Dirac structure of interaction (K 0(En_) <5n,rhv B(*)(n,Mv)^ = Tr [A(E)Mkr^B] . (5.8) E = M +2MK q is the energy of K and n_ is a light-cone vector almost parallel to the K momentum k = En_ + k', n2 = 0. (5.9) Residual momentum k' is of the order aqcd. sn is the effective large-energy field of the s quark with factored out dependence on large energy sn(x) = eiEn-xL^+ s(x), (5.10) v and = 2v — n_. Long distance physics is parameterized by function A(E), which does not depend on r, since Hamiltonians of heavy quark effective theory and large energy effective theory commute with quark spin operators. The most general parameterization of A(E) is then in terms of the four energy-dependent functions [151]: A(E) = ai(E) + 02(E)/ + as(E )/i_ + a4(E)n_/. (5.11) For wave functions of mesons we use Dirac structures transforming as fermionic bilinears under the Lorentz transformations XT Yn_n+ M 1+ / f / ; B = B*(n,Mv) (5.2) = ^"T"' = { (—75) ; B = B(Mv) • (5.12) Evaluating the traces on the right-hand side of (5.8), one can connect form factors with functions a1(E),..., a4(E) and find at q2 = 0 the symmetry relation TfK (0) = T2bk (0) = T3bk (0) = FfK (0). (5.13) Consequently, matrix element of Q7 for B * ^ KK0 transition = FfK(0) [2MeV^av^a (5.14) + iM2nv — in ■ q(Mv + k)v is proportional to FfK(0), the value of which has been determined with the light-cone sum rules approach [153] FfK(0) = 0.33 ± 0.04. (5.15) 5.3 Hard photon spectra The diagram on the left-hand side in Figure 5.2, representing the soft n region is then An sft = - .GfV«;VfOKm) = FfK(0)f (25* - S^) (Pa — V ■ p Va) V ■ p 2MevApavAfcp + iM2gav - i(Mv - k)a(Mv + k)v (5.16) where 0 = -15.4° is the ns - ni mixing angle [154] and f = 93 MeV is the pion decay constant. Wilson coefficient C7 on scale of the b quark is C7(p = 5 GeV) = -0.30 [128]. Electromagnetic gauge invariance is found to be valid in the limit of small En. The right-hand side diagram of Figure 5.2 with soft KK0 represents the amplitude of similar form Ak soft =.GfV«;V«fcC7(mfc)^FfK(0)gVW + an 0 (kg - V ■ fcVa) v • k 2MevApavApp + iM2gav - i(Mv - p)a(Mv + p)v (5.17) x In comparison to the soft n amplitude (5.16), the soft K amplitude (5.17) has interchanged momenta p o k and n8 — n1 mixing factors now originate from B**nY vertex, where we rely on flavor SU(3) symmetry to estimate form factor E^. To find the amplitude for n' in the final state, one only has to modify n8 — n1 mixing coefficients in the amplitudes (5.16), (5.17) and find for soft n' a;, soft = — iGF v:v*C7(m) em fBk (0)g (+ ^ (pCT — v ■ pVg) f2MevApCTvAkp + iM2gCTV — i(Mv — k)CT(Mv + k)v X v ■ p e * (5.18) Momentum of n' is here denoted by p. Amplitude for soft K and energetic n' is &' iG V*V C (m ) emb FBK(0) g cos g — ^2sin 6 AK soft = — iGF Vt6C7(m6) (0) f-- (kg — V ■ kVg) vk 2MevApCT vApp + iM 2ggv — i(Mv — p)g (Mv + p)v (5.19) ev. dB/dE, [GeV"1] dB/dEr [GeV"1] Ey [GeV] Ey [GeV] Figure 5.3: B° ^ K°nY spectra. Left: Photon spectrum in the region of En < 0.8 GeV (short dashes), En < 1.0 GeV (solid line), and En < 1.2 GeV (long dashes). Right: same for soft K, Ek < 0.8,1.0,1.2 GeV. 1. X 10 X 10 6. X 10 X10 2. X 10 Figure 5.4: B° ^ K°n'Y spectra. Left: Photon spectrum in the region of En < 1.1 GeV (short dashes), En < 1.2 GeV (solid line), and En < 1.3 GeV (long dashes). Right: same for soft K, EK < 0.8,1.0,1.2 GeV. 5.4 Summary We have investigated photon spectra of the B ^ n(n')KY decays in the region of Dalitz plot with energetic photon and one soft meson. We applied the combined heavy quark, large energy, and chiral symmetries. Use of this approach is justified owing to the fact that in the considered regions of Dalitz plot kinematical configuration allows simultaneous expansion in soft momentum, 1/mb, and 1/Ehard. In the approach we proposed only the nonresonant production is taken into account, neglecting in particular the resonant B ^ K*y ^ KnY decay. Since n and n' are isosinglets we do not expect any significant final state effects. Partial branching ratio integrated over both regions in Figure 5.1 with upper bound on soft meson energies set to 1.2 GeV accounts for about 10% of the B0 ^ K0nY branching ratio (5.2a). Thus, with increasing statistics, these two corners of phase space could be studied more thoroughly and bring in complementary information on the magnitude of C7. Chapter 6 Radiative background of B ^ D£v decays Many efforts have been devoted to experimentally check validity of the Cabibbo-Kobayashi-Maskawa (CKM) mechanism which predicts that all quark flavor observ-ables agree with the unitary CKM matrix which connects all CP -violating processes. If one is to confirm the CKM mixing then either measuring sides or the angles of the unitarity triangle the apex (p, n) should come out unique. The value of Vcb determines lengths of the two sides adjacent to the apex, among them also the side opposite to the angle ft which is precisely measured. CP-violating parameter eK of K-K mixing is one of the most important experimental inputs of the unitarity triangle analyses and is also extremely sensitive to the value of |VCb|. Full expression can be found in [155], here we isolate the dependence on CKM values and nonperturbative physics, which contribute largest uncertainties: |eKI « BknA2 A6. (6.1) Scale invariant bag-parameter of matrix element of AS = 2 matrix element between K0 and K0 states has been denoted BK, while n, A, and A are the CKM parameters in the Wolfenstein parameterization (2.10). Letting A = Vcb/A 2, we find |eKI« Bkn|Vcb|2 A2. (6.2) Cabibbo angle A = 0.226 ± 0.001 [28] is known with good precision. Bag parameter Bk has been computed on the lattice and in the last years unquenched results with relatively small errors have become available. Recent average of results amounts to [156] Bk = 0.731(7)(35), (6.3) where the numbers in brackets represent the respective statistical and systematic errors. A 5% percent error of the bag parameter is becoming less important than the error of |VCb|2, whose current values, determined by the inclusive and the exclusive methods disagree |Vcb| = (38.6 ± 1.2) x 10_3, exclusive B ^ Dčv and B ^ [157], (6.4a) |Vcb| = (41.5 ± 0.4 ± 0.6) x 10_3, inclusive B ^ X^v and B ^ X^y [80]. (6.4b) Values from exclusive analyses are consistently below the inclusive analyses. Precision of the inclusive method is better but in future flavor experiments one expects to approach precision of about 1% even for the exclusive method. 6.1 Extraction of Vcb in B ^ D£v The problem with extraction of the Vcb value using measured spectra and decay widths of a exclusive process are the ever pertaining hadronic dynamics. Effects and uncertainties of these are encoded in the form factors, which are to this day the main source of theoretical uncertainty. Spectrum of B ^ D£v semileptonic decay to a light lepton is conventionally expressed with G(w) form factor, where w = v ■ v' is the product of the two mesons' velocities dW-(B ^ D£v) = WmD(mB + mD)2(w2 - 1)3/2G2(w). (6.5) For the heavy ^ heavy transition B ^ D mediated by the (axial-)vector weak currents, starting point is the heavy quark symmetry (HQS) at the point where B and D velocities are equal and the light degrees of freedom dynamics is described by the Isgur-Wise [16] function. Then one has to invoke HQS breaking corrections in powers of as and especially Aqcd/mQ (Q = c, b) to account for the relatively light c-quark. The HQS supplemented by the breaking terms provides a reliable theoretical prediction for the form factors at the zero-recoil, a kinematical point with maximum momentum transfer squared between the heavy mesons tmax = (mB — mD)2. However, phase space vanishes at zero-recoil point and the decay spectrum is correspondingly small. Experimentalists measure the bulk of semileptonic events in the region away from the zero-recoil limit, where no theoretical predictions based on HQS are available. The usual procedure in this case is to fit the data to a physically viable shape of the form factor and extrapolate the measured spectrum to obtain the value of VCbG(w = 1). Commonly used parametrization is the one of Neubert, Lellouch, and Caprini [158] (CLN) which resides on the analytical properties of the current correlators and HQS. On the other hand, lattice QCD simulations have provided in the last years quenched values of the form factor G(w) at several points away from the zero-recoil thus establishing direct contact with experimental data [159161]. In future also unquenched results will be at hand for the heavy-to-heavy form factors, computed at several w [162]. In this chapter we will study the possible background of B ^ D£v decay due to radiative events, which are not recognized by the experiment. The control over systematics is very important if we are to reconcile the inclusive and exclusive values of Vcb. Furthermore, the exclusive decay B ^ D is determined by two hadronic form factors that are accessible to lattice QCD and have now been computed at several values of q2. In our case the dominant contribution to radiative events with soft photons will turn out to be the D°* resonance which decays to D0 and a photon, and is always very soft in the B meson rest frame. 6.2 Amplitude decomposition Radiative process b ^ cIvy is induced by the effective weak Lagrangian accompanied by the electromagnetic interaction. Leff =--FV^ Lqed = eB "(xJ (6.6a) (6.6b) We use B" for the electromagnetic field, whereas weak currents H", L" and electromagnetic current J" are defined as H cY"(1 - Y5)b = V" - A" J" = -+ 2/3uY"U - 1/3dY"d + 2/3cymc - 1/3KY"S - 1/3bY"b. Leading order S-matrix element for B(p) ^ D(p')^(p^)v(k)Y(e,q) is (6.7a) (6.7b) (6.7c) (divy | S | B) = ^divy - J d4x d4y T [Lff (x)Lqed(^)] B =(2n)4^4(Si Pi) eGF Vc6 "FT-' (D | Hv(0) | B) I d4y eiq'y (Iv | T [ J"(y)Lv(0)] | 0) + (Iv | Lv(0) | QW d4y eiq'y (D | T [J"(y)H(0)] | B) (6.8) T in the above expression denotes the time-ordering operator. The resulting invariant amplitude is A(B(p) ^ D(p'K(pi)K(k)Y(e, q)) = (6.9) eGF Vcb ^ F2 e*" u(pj) Fv (t) 2pi ■ q Y"(p + / + m) + V^v - A^v Yv(1 - Y5)v(k) First term in brackets is the amplitude for photon emission from the lepton leg, and is proportional to the vector form factors of B ^ D transition: Fv(t) = i (d(p') | ffvt)(0) | B(p)) , t = (p - p')2. (6.10) The last two terms in bracket of (6.9) correspond to photon coupled to the heavy line (Figure 6.1), and is given in terms of vector and axial hadronic correlators V"v d4y eiq'y A^v d4y eiq'y (d;(pn) | Vv(0) | B(p)) e-i(pn-p,)-y +8(-yo) (D(p') | Vv(0) | b;(p„)> (B;(pn) | Jf (0) | B(p)) e-i(p-pn)'y Sum runs over all beauty (B*n) and charm (D*n) flavoured states. Of the B** intermediate states, only B- introduces a pole in the physical region when photon is soft (EY ^ 0). This pole contribution precisely matches the V^. Amplitude for bremsstrahlung off the lepton leg is also singular at EY = 0. Consequently, the two amplitudes together are gauge invariant (as can be seen by comparing Ward identities (6.12) and the amplitude (6.9)) and they comprise the well-known soft divergence of quantum electrodynamics. According to Bloch-Nordsieck theorem [163] it cancels against one-loop virtual correction on the level of decay width. SD amplitude is divergence free at EY ^ 0, however, additional poles appear at finite Ey due to D^ states, and since the outgoing photon is assumed to be soft, only the lowest excited states should contribute dominantly. Here we will consider only the first excited state D* and argue in the end why contribution of higher states is not substantial. Close to the D* pole V.fVD takes the following form jUV i (D(p') | J | D*(p' + q)) (D*(p' + q) | Vy | B(p)) (p' + q)2 — mD* + im-D* Td* : (6.17) where we accounted for the finite width of the D* via the Breit-Wigner ansatz. AUV also has the same pole structure, only with axial current Av in place of Vv. 6.3 Hadronic parameters of B ^ D* and D* ^ Dy transitions D0* ^ d0y This decay is governed by the magnetic-dipole transition (DV)Y(fc, n) I D0*(p, e)> = egD0*D07eUva^n^vPaPj?, (6.18) whose value gDo*DoY = 2.0 ± 0.6 GeV-1 was computed on the lattice [164] along with the strong coupling constant gD*Dn = 20 ± 2. We combine the lattice results with the measured ratio [28] r(D0* ^ D0n0) Rn/7 ^ r(D0* ^ D0Y) = 1.74 ± 0.13 (6.19) to find a tighter constraint: 1.8 < gDo*DoY < 2.5 (see Figure 6.2). Knowledge of gDo*DoY allows us to predict the decay width of D0* meson from the measured branching fraction B(D0* ^ d0y) = 0.381 ± 0.029 [28]: 53 keV < rDo* < 108 keV. (6.20) This is much lower than the current experimental upper bound Tdo* < 2.1 MeV [28]. B- ^ D0* Vector and axial-vector form factors of B ^ D0* (see Section 2.3.1) have been computed using the quenched lattice simulation [165] at several values of w = (t — mB — mD* )/(2mBmD*). We perform the chi-squared fit on their stated values and errors using the CLN shapes of the form factors [158]. 1.4 1.6 1.8 2.0 2.2 2.4 2.6 gD*Dy [Ge T1] Figure 6.2: Blue region represents the lattice allowed values of gDo»Don and gDo»Do7, and red region is allowed by the measurement of Rn/Y. The resulting contribution of the D * resonance to structure dependent functions A1...4, V1...4 (6.14a) are expressed in terms of form factors VBD*, a^f^, (we use lowercase a for the form factors (2.32) to avoid ambiguity with functions A1...4 used in decomposition of ). Their explicit expressions are stated in the Section B.2. 6.4 Irreducible background to B ^ D°£v channel from D°* ^ D0Y The resonant decay chain B ^ D * ^v followed by D * ^ D7 forms an irreducible background to B ^ D^v if the experiment overlooks the final state soft photons whose energy is = 137 MeV in the center-of-mass frame of D *. Depending on the experimental ability to discern events accompanied by the photon from ordinary semileptonic events, a number of fake events are included in the sample of semileptonic events. The available experimental data [28] is sufficient to estimate the importance of B- ^ D0 *^v pollution in B- ^ D°^v. Using the narrow width approximation gives for the branching fraction of resonant B- ^ D°7^v a value of (2.2 ± 0.2)%, which is of the same size as non-radiative decay [28]. This clearly poses a serious problem since the major part of the photons are quite soft due to small mass splitting between D° * and D°. We show the photon spectrum, calculated using the framework defined in Sections 6.2 and 6.3, on Figure 6.3. The 137 MeV photon in the D° * rest frame is boosted to energies up to « 350 MeV in the B meson rest frame. Figure 6.4 shows the ratio of radiative events recognized as B- ^ D°^v to the total sum of radiative and semileptonic events. dT/čEj 6. x 10 -14 5. x 10 -14 4. x 10 -14 3. x 10 -14 2. x 10 -14 1.x10 -14 i i i ■ Ey [MeV\ 100 200 300 400 Figure 6.3: Spectrum of resonant B ^ D°*^v ^ D°y^v. 0.14 0.12 0.10 0.08 0.06 0.04 0.02 0.00 0 20 40 Figure 6.4: Fraction of misidentified events in the sample depending on the photon energy resolution of the analysis. The uncertainty corresponds to the range allowed for the rD. 6.5 Summary Precision of VCb determined by measuring an exclusive b ^ c decay channel is important in reconciling a discrepancy between inclusive and exclusive methods of Vcb extraction. Unitarity triangle analyses rely heavily on the CP -violating parameter of K-K mixing e^, and largest uncertainties are stemming from errors of the bag parameter Bk and the value of VCb. Advances in unquenched lattice calculations have provided a value of bag parameter with a good precision of ~ 5%. Thus, experimental value of |Vcb| is becoming a dominant source of errors in using eK as an experimental input to unitarity triangle analyses. Since B_ ^ D°^v channel is expected to be measured also at future Belle 2 experiment we have analysed the background radiative events B_ ^ D°^vy. We have found the dominant contribution due to D°* resonance, decaying into D°y. We have shown spectrum (Figure 6.3) and the fraction of fake events in the experimental sample (Figure 6.4). Due to small mass splitting 142 MeV between mDo and mDo*, all the photons are soft in the B meson rest frame. Higher excited charm states would necessarily produce more energetic photons, which experiment could detect, so those states have been of minor importance in our analysis. We have come to conclusion that for a required precision of Vcb of the one percent order, experiment should be sensitive to the photons of energy well below 100 MeV. Chapter 7 Concluding remarks The search for new physics in precision low-energy observables has been a long lasting aim of joint experimental and theoretical efforts. The physics programme of B-factories even exceeded the original goals and, together with input from other experiments, confirmed that a global fit of the standard model quark Yukawa sector shows no serious anomalies. The unitarity triangle analyses tested the Cabibbo-Kobayashi-Maskawa mixing mechanism at the ~ 10% level. The most precisely measured and theoretically unproblematic observables like CP -violation parameters in K-K mixing already push the scale of new physics up to ~ 100-1000 TeV. It has become clear that new physics contributes to the quark flavour observables on the level of few percent or even smaller. It is thus crucial to have the standard model theoretical predictions under good control. In first part of the thesis we have put emphasis on comparison between standard model and new physics signatures in rare decays. The standard model contributions in that context act as background of new physics signals. In Chapter 3 however, we studied a process b ^ dds, occurring at a negligibly slow rate in the standard model. This allowed us to consider contributions of the new physics scenarios alone. We found that certain hadronic decay channels shall play a role in constraining a model of supersymmetry with broken Rp-parity and a model with additional Z' gauge boson, once the experiment is able to probe their branching fractions on the 10-7 level. Chapter 4 dealt with semileptonic charm decays D ^ P£+£-, which are notorious for their poorly controlled long distance contributions. In this case we had to model the resonant background, whereas the short distance standard model contributions proved negligible. Same quark level transition c ^ u£+£- triggers the well constrained leptonic decays D ^ £+£-, which give useful additional information in this case. We showed that experiment is on the verge of discovery of D ^ n£+£-and currently puts severe bounds on the Rp-parity violating supersymmetry and a more general singlet leptoquark model. All the abovementioned decays will be searched for in future at LHCb and the Belle 2 experiments and will provide useful exclusion bounds in parameter spaces of new physics models. In the second part, namely in Chapters 5 and 6 we did not focus on any particular new physics model but instead devised methods of testing the standard model predictions themselves. For the B ^ KnY decay we predicted in the standard model photonic spectra for the decay channels with n and n' in final state. These spectra will be recorded at the Belle 2 and LHCb experiments and will probe the chiral nature as well as the scale of b ^ sy coupling. In Chapter 6 we exposed the pitfall present in the exclusive determination of Cabibbo-Kobayashi-Maskawa element |VCb|, a very important parameter in the unitarity triangle analyses. Namely, the discriminating power between B ^ d^z/y and B ^ D.£z/ decays depends crucially on the lower cut on photon energy. For the case of B- ^ D° we identified the dominantly contributing resonance D° and estimated the number of misidentified radiative events depending on the photon cut. Results of these two chapters may prove to be valuable in progressively stringent tests of the standard model. The quark flavour observables studied in the thesis constitute important aspect of new physics searches. The precision low-energy experiments, although affected by hadronic uncertainties, are best-suited to assess the new physics flavour properties. In forthcoming era of the Large Hadron Collider a fruitful interplay between the high pT experiments at hadron colliders and precision flavour physics will further test the standard model and hopefully expose the principles behind it. Appendix A Technical aside on b ^ dds process A.1 Matching and renormalization of composite operators We choose to regularize the loop integrals in the effective theory using the naive dimensional regularization (NDR) and renormalize them by minimal subtraction scheme [166, 167] (MS). Using a mass-independent renormalization scheme like MS results in a well defined power expansion in 1/mW of the amplitudes unlike mass-dependent schemes, where loop contributions of higher dimensional (dimension-8 and higher) composite operators is not suppressed. When we choose to include virtual QCD corrections we work in the modified minimal subtraction scheme [166, 167] (MS) to renormalize divergent loop integrals. However, in MS scheme also the QCD beta function is mass-independent and heavy particles do not decouple as the Appelquist-Carazzone theorem does not apply [168, 169]. This is why we have to decouple a heavy particle1 with mass M "by hand", integrating it out on the scale p ~ A [170]. At matching scale A we determine the Wilson coefficients, by imposing equality of 1-particle irreducible (1PI) Green functions calculated in the full theory and the 1PI Green functions (GF) calculated in the effective theory. Following [29] we demonstrate, how the matching and operator renormalization is calculated in LO in as and leading logarithm approximation. First we express GF in the effective theory as - 4G2 Aeff = ? Ci (A) (Qi(A)) = (QT (A)) C (A), (A.1) i where (Q^ denotes the vector of GFs calculated with composite operator insertions (Fig. A.1). Assuming our basis of operators is complete under QCD renormalization, GFs can be expressed as combinations of tree-level GFs (i.e. calculated without QCD correction), denoted S (Q(A)) = (1 + SrT) S (A.2) 1Here qualification "heavy" should be understood as heavy with respect to typical momentum scale of the problem, namely ^had C M. Figure A.1: 1PI Green function diagrams in the effective theory. Shaded squares represent currents in the effective 4-fermion vertex. The corresponding full theory GF is (Fig. A.2) (keeping only leading terms in lw): 1/mW Figure A.2: 1PI Green function diagrams in the full theory. Dots are the weak currents. - 4GF = ST (i(°) + ^ A(1)). (A.3) The above full amplitude as written (A.3) is finite after we have accounted for the 1/2 external leg renormalization by adding factor Zq for each external fermion. In the leading order in as the quark field renormalization constant is a CF , 2n „ N2 -14 . . Z = 1 - ^"T +O(a2)' CF = ^ = 3 (A-1) Also the effective theory result (A.3) is finite, but there a mere multiplicative factor does not suffice to render the GF finite. That should come as no surprise since the effective theory is not renormalizable as is the case with full theory. Especially the UV behaviour is different in the two theories. One should treat Wilson coefficients C as ordinary coupling constants and proceed with usual renormalization procedure where also Wilson coefficients get renormalized. Equivalently we can consider renormalizing the composite operators as Q (°) = ZQ. (A.5) This renormalization is chosen to be done in the MS scheme, subtracting only the 1/e poles2. Once we have adopted the same scheme for renormalization of QCD divergences in both theories and having included composite operator in (A.2), we read off the Wilson coefficients at the matching scale A from equality of (A.1) and (A.3) C (A) = a4 (°) + ^^(AW - rT AW). (A.6) 4n 2We calculate the momentum loop diagrams in D = 4 — 2e spacetime dimensions. Our final goal are the values C(-had) and thus we could in principle set A = -had in (A.6). However, if the -proportional terms in coefficients (A.6) contain large logarithms of scale separation lnA2/M2 they would invalidate perturbative treatment for A ^ M [29, 171]. Thus the matching calculation is valid only for A ~ M. The aforementioned large logarithms can be summed using the renormalization group improved theory, which amounts to performing a sequence of matching procedures with infinitesimal scale separations, allowing one to write the renormalization group equations (RGE) for a renormalized parameter. RG equations express the rate of change or "running" or "evolution" of a parameter as the renormalization scale - is being changed. Renormalized Wilson coefficients' running is determined by the anomalous dimension matrix 7 €7 = yt (g)C <">• where in the minimal subtraction scheme 7(g) is --dependent only indirectly through g(-). Same RG equation holds for the evolution matrix ^U^n-A) = yT (g)U (-, A), C (-) = U (-, A)C (A). (A.8) Since 7 is a function of g in MS scheme, it is more suitable to treat g as independent variable, namely dgU (-(g), A) = ^ U (-(g), A)), (A.9) where * is the standard beta-function of QCD df- = * 0 = -A + (A.io) *„ = (33 - 2f)/3. (A.11) f is the number of dynamical quark flavours at scale -. The solution to running as(-) = g2/(4n) in leading order in is ( ) =_as(mz)_= 4n (A 12) as(-) = 1 - *„asM ln m|/-2 = *„ ln -2/AQcd ' (A.12) Solution of (A.9) can be written iteratively resulting in U (-, A) = Tg exp dg' 7T(g') /S(A) *(g') (A.13) Operator Tg enforces products of matrices to be coupling-ordered, meaning that the matrix with largest g is placed leftmost, the one with second largest g is placed next to the leftmost and so on, if g (A) < g(-). If however, - > A, then the ordering must be reversed. The anomalous dimension matrix 7 is connected to the composite operator renormalization, in particular to coefficient Zi of the 1/e term of matrix Z (A.5) 7(g) = —2g2 ^. (A.14) A.1.1 Mixing of effective operators in b ^ dds Working in the leading logarithm approximation we can perform matching on scale A at order 0(a°). To determine the anomalous dimension matrix of the operators (8.8) we have to calculate the insertions of operators O1...3 into the 1PI Green function to first order in as. 0102 mixing We have to evaluate the loop diagrams on Figure A.1 with insertions of 01 and 02 K' > = (<>° bs ^ + K>„ S 7- -(n(°)> as 3 as3 ° ^"'M0^ ? where (...)° denotes the tree-level insertion of operator, i.e. without QCD corrections. Evidently, the operator 02 does not mix into 01. Insertions of the renor-malized operators 01,2 into the four point functions are finite and are expressed as ^ = Z2Z-1 ( (0(°)> I . (A.17) \o)r ^ \iof Inserting the quark field renormalizations (A.4) and Cf = 4/3 we find for the composite operator renormalization matrix Z=1+£ (-18)- (A-18) The anomalous dimension matrix is determined by the 1/e term in expansion of Z in powers of e (see (A.14)) Y=an (1 =3)- (A19) O3 renormalization The 03 insertion into graphs of Figure A.1 also generates at 0(as) an operator with the colour nonsinglet currents, which is not present in our basis. However, owing to the presence of two d quark fields we can Fierz transform the abovementioned operator back to 03. Thus, 03 only mixes into itself and will be multiplicatively renormalized l(°)> = (1 , as 2CF = 2\ (0(°) > ° °r) = 1 + )c. (A.20) After the quark field renormalization we find for the anomalous dimensions __ aa s 2 aa s ^. ~ \ Z = 1--, Y = —. (A.21) 4n e n A.2 GIM mechanism in the b ^ dds Note that in (3.11) the two independent sums over quark flavors imply exact GIM cancellation of any term in f (xi,xj-) that does not depend on both xi and Xj. Resulting freedom in choice of f (x, y) is f(x,y) ^ f(x,y) + u(x) + v(y), (A.22) with arbitrary functions u and v. In (3.12) we fixed u(x) and v(y) by imposing the constraint f(x, 0) = f(0, y) = 0 to suppress the contribution of light quarks and retain relatively simple form. We could also forbid contributions at chosen value of x and y. For example, t quark would not explicitly contribute in the box diagram if we redefined f (x,y) as f (x, y) ^ f (x, y) - f (xt, y) - f (x, xt) + f (xt, xt). (A.23) A.3 Parameterization of B ^ n form factors For the B- ^ n- and B- ^ p- transitions we use form factors calculated in the relativistic constituent quark model with numerical input from lattice QCD at high q2 [74]. A.3.1 B —> n form factors F nB (0) F?b(q2) = -- 2 1)[1( ) 2/ 2 ], FinB(0) = 0.29, -1 = 0.48, (A.24a) (1 - q2/mB*)[1 - ^1q2/mB*] F0nB(q2) =-o , 02 ()--r——4—, F0nB(0) = 0.29, -1 = 0.76, ^ = 0.28, 0 w ; 1 - a1 q2/mB» + ^2q4/mB^ 0 W (A.24b) A.3.2 B ^ p form factors VpB (q2) = ^-2, 2VpB (0)-^^, VpB (0) = 0.31, a1 =0.59, (A.25a) (1 - q2/mB»)[1 - CT1q2/mB*] ApB(q2)=(1 - qW-g1q2/mB] • A0B(0)=0-3». -1 =0-54. (A-25b> APB ^ )=1 - -1 q^/^ W/m.B. • APB (0>=0.26• =a73^2 = 0A0' B B (A.25c) APB(q2) = --2/AppB + 4 / 4 , APB(0) = 0.24, -1 = 1.40,-2 = 0.50. 1 - -1 q2/mB. + CT2q4/mB» B B (A.25d) The transition form factors between heavy mesons D_ ^ D have been calculated in the chiral Lagrangian approach in [75] Fi s (q2) = 0, (A.26a) FDDs (q2) = q2 (gn/4)/*(1430) VmDsmD (. 26b) F° (q ) = m2 m2 2 2 . . . (A.26b) mDs - mD q2 - mK(143°) + Vq2rK(143°) A.3.3 K ^ n form factors The same method has been used to obtain the K_ ^ form factors [76] FT* (q2) =-2gvK(892)gK *-, (A.27a) q2 - m*(892) + wq2r*(892)(q2) 2, 2gVK(892)gK* (1 - q2/mK(892)) F°n* (q2) = ; , + . ^ 7;2) (A.27b) q2 - m*(892) + wq2r*(892)(q2) q2 /*(1430)gSK(1430) + m* - q2 - m*(1430) + ^yq^r*(1430)(q2) ■ Here the decay widths of resonances K*(892) and K(1430) are taken to be energy dependent [76] / m2 \5/2 r*(892)(q2)4 ^Pj (A.28a) ( [q2 - (m* + mn)2][q2 - (m* - m,)2] \ 3/2 r V [m*(892) - (m* + mn)2][m*(892) - (m* - mn)2] / ^^ ' / m2 \3/2 r*(1430) (q2) 4 ^Pj (A.28b) „ ( [q2 - (m* + mn)2][q2 - (m* - m,)2] \ 1/2 r )21[^2 — (mm ^ — mm )2] I *(1430) Jm*(1430) - (m* + mn)2][m*(1430) - (m* - mn)2]_ Appendix B Ward identity and amplitude expressions for B- ^ D°£uy B.1 Ward identities To derive the Ward identity of electromagnetic gauge invariance for VMv, of Chapter 6 we write out explicitly the time-ordering (B.1) = —i | d4y (^eiq'y) (D | T [ J(y) Vv (0)] | B) = i J d4y eiq'y (D | e(y0) J(y)Vv(0) + 8(—y0)Vv(0)JM(y) | B> = ij d3y e-iq'y (D | [J^K (0)] | B). Using the canonical commutation relations shows [138, p. 444] that [J0(y),F(x)] = —qFF(x)^(y — x), where F is a local function of fields, present in the Lagrangian, and qF is the sum of charges of all the fields in F. So we have q^v = i J d3y e-iq'y(1/3 + 2/3)č(y) (D | Vv | B) = Fv(t). (B.2) B.2 Invariant coefficients of the and Apv in B ^ ^ D0£šY B.2.1 Contribution to V: SD juv d* _ 2#d*d7VBD (W2) mBEd VD = (mB + m-D*) ((p' + q)2 - m D* d* = -£d*d7VBD (W2)(mD + mD*) (mB + mD*) ((p' + q)2 - mD*) V2D = VD* = 0, V4D* = -2gD*D7 VBD* (W2) (mB + mD*) ((p' + q)2 - mD*) (B.3a) (B.3b) (B.3c) (B.3d) B.2.2 Contribution to A AD* "#D*D7 ((p' + q)2 - mD*) W2 2mD* aBD* (W2) - (mB + mD* )aBD* (W2) W2 + mB - mD* BD* + mB + mD B "tD* aBD* (W2) mD* - m D , (B.4a) a2 ad* gD*D7 ((p' + q)2 - mD*) W2 + 2mD*p ■ qaBD*(W2) - (mB + mD*)(p ■ q + W2)aBD*(W2) p ■ q (W2 + mB - mD*) bd*/w2x ad* a4 _-gD*D7_ ((p' + q)2 - mD*) W2 _ gp*p7_ ((p' + q)2 - mD*) W2 mB + mD* -2mD* aBD* (W2) + (mB + mD* )aBD* (W2) -aBD* (W2) (B.4b) BD^i t-2\ + W - mB + mD* aBD* (W2) mB + mD* 2mD* aBD* (W2) - (mB + mD* )aBD* (W2) (B.4c) BD^i/2\ + W2 + mB - mD* bd* (w2) mB + mD* (B.4d) 2 2 * Bibliography [1] H. Flacher et al., Eur. Phys. J. C60, 543 (2009), 0811.0009. [2] D. J. Gross and R. Jackiw, Phys. Rev. D6, 477 (1972). 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D 79, 017502 (2009) [arXiv:0810.4858 [hep-ph]] "On the Dalitz plot analysis of the B ^ k^y decays" S. Fajfer, T. N. Pham and N. Kosnik Phys. Rev. D 78, 074013 (2008) [arXiv:0806.2247 [hep-ph]] "Updated constraints on new physics in rare charm decays" S. Fajfer, N. Kosnik and S. Prelovsek Phys. Rev. D 76, 074010 (2007) [arXiv:0706.1133 [hep-ph]] "6 ^ dds transition and constraints on new physics in B- decays" S. Fajfer, J. F. Kamenik and N. Kosnik Phys. Rev. D 74, 034027 (2006) [arXiv:hep-ph/0605260] Poglavje 8 Razsirjeni povzetek disertacije 8.1 Standardni model in njegove razsiritve Standardni model (SM) je preverjena fizikalna teorija veljavna na najmanjsih dosegljivih dolžinskih skalah (10-15 m). Skladen je s principi posebne relativnosti in kvantne mehanike, ter za osnovne prostostne stopnje uporablja kvantna polja, katerih ekscitacije predstavljajo delce. Vsebuje 19 prostih parametrov, ki kvantitativno razložijo vse eksperimentalne opazljivke na omenjenih skalah v okviru teoreticnih in eksperimentalnih napak. Edina izjema so opazene nevtrinske oscilacije, ki jih SM ne napove, in nakazujejo na masivnost nevtrinov. Ker v naboru interakcij SM ni gravitacije, je le-ta veljaven najdlje do Planckove energijske skale (MP ~ 1019 GeV), kjer zagotovo odpove klasicen (nekvanten) opis gravitacije. V zvezi s tem se pojavi tudi problem hierarhije med skalo elektrosibkih interakcij 100 GeV) in Planckovo skalo Mp, kjer je tezko razumeti, zakaj bi bila masa Higgsovega bozona na sibki skali. To je tudi glavni teoretični argument za veljavnost SM zgolj do skale ~ 1 TeV, ali kvecjemu nekaj TeV. Obstaja mnogo modelov, ki na skali okrog 1 TeV dopolnijo SM in so v literaturi poimenovani nova fizika. V disertaciji obravnavamo efekte supersimetricnih razsiritev z ohranjeno ali krseno parnostjo Rp, modelov z dodatnim umeritvenim bozonom Z', dodatnim singletnim kvarkom, ali singletnim leptokvarkom z nabojem -1/3. 8.2 Fizika tezkih kvarkovskih okusov Eksperimenti v fiziki kvarkovskih okusov testirajo sibke interakcije, kjer lahko pride do sprememb med sestimi okusi kvarkov 1. Za to so posebno pripravni sibki razpadi mezonov, vezanih stanj kvarka in antikvarka. Tezka mezona sta mezona D in B, ki ustrezata tezkima kvarkoma c in b, vezanima z enim od lahkih antikvarkov (u, d, ali s). Torej je kvarkovska sestava tezkih mezonov B = (bq) in D = (cq), kjer q oznacuje lahki antikvark. Masa tezkih mezonov (< 5 GeV) je dva velikostna reda pod sibko skalo, kar omogoca tvorbo parov tezkih mezonov v velikih kolicinah pri relativno nizkih energijah v t.i. tovarnah tezkih mezonov (eksperimenta Belle in BaBar). Ti eksperimenti slonijo na veliki statistiki, ki omogoca dobro natancnost pri 1Barvna in elektromagnetna interakcija ohranjata okus kvarkov. meritvi parametrov fizike okusov — 6 mas kvarkov in matrike Cabibbo-Kobayashi-Maskawa (CKM), ki vsebuje 3 kvarkovske mesalne kote in fazni parameter Pri teoreticni obravnavi razpadov je potrebno upostevati neperturbativne ucinke barvnih interakcij, ki se v sistemu tezkih mezonov lahko poenostavijo zaradi hierarhije med masama tezkega in lahkega kvarka2. Zaradi masivnosti so na voljo mnogi sibki razpadni kanali v vezana stanja lazjih kvarkov ter leptonov. 8.2.1 Kvarkovski okusi v SM Interakcijske clene v Lagrangevi gostoti doloca lokalna umeritvena invarianca pod umeritveno grupo G = SU(3)c x SU(2)W x U(1)Y. Reprezentacije fermionske snovi v SM so leptonske (E, in kvarkovske (Q, uR, dR): E(1, 2)_1/2 = , 1)_1, Q(3, 2)1/6 = uL uL dL uL dL (8.1) ur(3, 1)2/3 = (uR uR uR) , dR(3,1)_1/3 = (dR dR dR) kjer stevilke v oklepajih oznacujejo, v katero reprezentacijo grupe G spada posamezno fermionsko polje. Tako polje E(1,2)_1/2 oznacuje singlet grupe SU(3)c, dublet pod sibko grupo SU(2)W, ter hipernaboj Y = -1/2. Interakcije med fer-mioni SM prenasajo umeritveni bozoni s spinom 1. Eksplicitni masni cleni unicijo umeritveno invarianco SM, zato simetrijo zlomimo s Higgsovim mehanizmom, ki priskrbi mase sibkim umeritvenim bozonom W±, Z ter preko Yukawinih sklopitev tudi mase fermionom. Po rotaciji kvarkovskih polj v masno bazo, nam v interak-cijskih clenih kvarkov z nabitimi sibkimi bozoni ostane unitarna mes alna matrika CKM V s stirimi prostimi parametri: Lkin 9 - —W+ Ui7^PLVijdj + H.c.. (8.2) Izmerjena skoraj s nja diagonalnost matrike CKM nudi razvoj v vrsto (Wolfenstei-nova parametrizacija) po elementu VUc = A / V= 1 2 A A AA3(p - in)^ 1 2 yAA3(1 - p - in) -AA2 AA2 1 + O(A4), (8.3) / kjer je A = 0.226 ± 0.001. Vozlisce nabitega sibkega bozona s sibkim tokom kvarkov poleg spremembe okusa povzroci tudi prehod med spodnjim (naboj -1/3) in zgornjim kvarkom (naboj 2/3). Okus spreminjajoci nevtralni tokovi (FCNC) se v SM pojavijo sele na ravni kvantnih popravkov z virtualnim sibkim bozonom in so dodatno zastrti zaradi unitarnosti matrike CKM preko mehanizma Glashow-Iliopoulos-Maiani (GIM). 2Tezki mezon je analog vodikovemu atomu, kjer lahko maso jedra v prvem približku vzamemo za neskončno. Posledicno so vsi procesi, ki vkljucujejo spremembo okusa, a ne naboja, v SM zelo redki. V modelih nove fizike ne pricakujemo mehanizma analognega GIMu, zato je meritev procesov FCNC lahko okno v efekte nove fizike. 8.2.2 Razvoj produkta operatorjev v sibkih razpadih Okus spreminjajoci kvarkovski procesi vkljucujejo izmenjavo sibkih bozonov mase ~ 100 GeV (mW = 80.4 GeV, mZ = 90.2 GeV), mnogo večje od znacilne energijske skale pri razpadu tezkih mezonov GeV). Izracuni z uporabo elektrosibke teorije postanejo neprakticni, saj je v nizkoenergijskih procesih vzbujanje sibkih prosto-stnih stopenj zastrto. Zelo pripravno je uporabiti efektivno teorijo, kjer obdrzimo le za dani proces relevantne prostostne stopnje, medtem ko propagacijo masivnih prostostnih stopenj (kratkoseznih efektov) skrcimo v tockovno interakcijo. V primeru interakcije 4 fermionov prek izmenjave sibkega bozona imamo v efektivni Lagrangevi gostoti produkt dveh tokov povezanih s propagatorjem sibkega bozona (x,y) (glej enacbo (2.18)) j d4xd4y J-(x)A^ (x,y)J+(y), (8.4) ki ga formalno lahko razvijemo v potencno vrsto okrog tocke mw = Zgornja nelokalna Lagrangeva gostota se tako zapise kot razvoj produkta operatorjev d4xd4y - y) J+(y) + O(1/mW)] , (8.5) kjer je razvidno, daje dominanten prispevek tockovna interakcija 4 fermionov, medtem ko so nelokalni operatorji visjih dimenzij zastrti z visjimi potencami 1/mW. V vodilnem redu razvoja v 1/mW dobimo v Lagrangevi gostoti stiri-fermionske operatorje masne dimenzije 6 Ldim-6 =--TtT^j Ci(^)Qi(u), Qi = (V^ri^i2)(^/i3ri^i4)• (8.6) 2i Brezdimenzijske Wilsonove koeficiente dolocimo iz ujemalnih pogojev (ang. matching conditions), ki zahtevajo enakost s tiritockovnih enodel c no ireducibilnih Gre-enovih funkcij v efektivni in celotni teoriji. V ujemanje obeh teorij lahko vkljucimo tudi virtualne popravke barvnih interakcij. Ti v obravnavo uvedejo renormalizacij-sko skalo ki razmejuje kratkosezne od dolgoseznih efektov. 8.3 Redek proces b ^ dds V SM kvarkovski proces b ^ dds poteka preko izmenjave dveh bozonov W, ki tvorita skatlasti diagram (Slika 8.1). Ta proces je zaradi dvojne vsote po notranjih gornjih kvarkih dvojno-GIM zastrt, kar ga naredi eksperimentalno nevidnega. V skladu s temi pricakovanji obstaja le zgornja meja na razvejitveno razmerje trodelcnega razpada, ki ga na kvarkovskem nivoju sprozi b ^ dds B(B- ^ +) < 9.5 x 10-7 pri 90% intervalu zaupanja . (8.7) dL Slika 8.1: Izmenjava dveh bozonov W, ki sproži proces b ^ dds v SM. V nasprotju z napovedmi SM lahko nekateri modeli nove fizike drastiCno poveCajo pogostost procesa b ^ dds. Analizirali bomo minimalni supersimetriCni model z ohranjeno (MSSM) in krseno (RpMSSM) parnostjo Rp ter model z dodatnim nevtralnim umeritvenim bozonom Z'. 8.3.1 Inkluzivni razpad Za dani proces in studirane modele je efektivna Hamiltonova gostota linearna kombinacija 5 Heff. = ^ \CnOn + CnOn , (8-8) n=1 kjer so Wilsonovi koeficienti Cn z masno dimenzijo —2 mnozijo efektivne operatorje (3.3). V SM je prispevek le k koeficientu C3, ki vsebuje kvarkovska tokova Lorentzove strukture (V — A)M. Perturbativni ucinki barvne interakcije so za operator O3 zanemarljivi. Za inkluzivno razpadno sirino b ^ dds dobimo le SM I2 m5 rSMMl. = 1C3J/* , CfM - 5.3 x 10-13 GeV-2, (8.9) in razvejitveno razmerje je mnogo premajhno za zaznavo v prihodnjih eksperimentih (B(b ^ dds)SM = (8 ± 2) x 10-14). V modelu MSSM je tako kot v SM nenicelen le koeficient C3, ki se v tem primeru inducira preko skatlastih diagramov s skvarki in gluini, katerih prispevki so bili izracunani za AS = 2 procese. Dobimo |CMSSM| < 1.6 x 10-12 GeV-2 ter posledicno za zgornjo mejo razvejitvenega razmerja 7 x 10-13. V MSSM torej ni moznosti, da bi ta proces opazili. Ce v modelu MSSM sprostimo zahtevo po ohranitvi parnosti Rp, ki je za vsak delec definirana kot R = (—(—1)3B+L = { + ; sumy ■ kjer je S spin, L leptonsko stevilo, in B barionsko stevilo delca, dovolimo clene v La-grangevi gostoti, ki krsijo leptonsko ali barionsko stevilo. V tem modelu (RpMSSM) sklopitve Ajfc krsijo leptonsko stevilo in lahko posredujejo okus spreminjajoce nev- tralne tokove med kvarki preko izmenjave snevtrina (Slika 8.2) cR p = - ^ v* 2 ' m2. n=1 CRP = — ^ ^ra21^ra13 n=1 m 2 (8.11) (8.12) Ucinke perturbativnih popravkov barvne interakcije upostevamo z uporabo re- dn bR Slika 8.2: Prehod b ^ dds preko izmenjave snevtrina v modelu RpMSSM. normalizacijske grupe, ki v tem primeru na nivoju razpadne Širine prispeva faktor /qcd, ki je za mase snevtrinov do 1 TeV pribliz no 2.2. -pRP 1 incl. m5/QCDCM 2048n3 |CR |2 + |CR (8.13) Zgornja kombinacija parametrov Ajfc iz drugih procesov se ni znana. V modelu z dodatnim nevtralnim umeritvenim bozonom Z' so FCNC prav tako inducirani na drevesnem redu in njihove prispevke upostevamo v koeficientih Cf = -4^2Gf yBd2L B^, Cf' = -4^2GF y^^ Bf3L, Cf' = -4^2GFyBd2L BdL, ^f' = -4^2GFyBd2i Bjdf. (8.14a) (8.14b) Me s anje operatorjev zaradi barvnih interakcij generira tudi koncen C2, in za mf ' = 500 GeV so koeficienti na skali mase kvarka b sledeci: Cf' (mb) = 0.90C f' Cf' (m6) = 0.47C f' Cf (mb) = 0.81Cf (8.15) (8.16) Na enak nacin se me s ajo koeficienti C,2,3. Meje na zgornje Wilsonove koeficiente bomo izpeljali iz zgornje meje razpadne sirine B- ^ K+n-n-. b d L L 8.3.2 Ekskluzivni razpadi mezona B Za izracun hadronske amplitude moramo upo stevati ucinke neperturbativnih barvnih interakcij pri rac unanju razpadnih amplitud. Efektivno Lagrangevo gostoto je potrebno izvrednotiti med zacetnim in koncnim stanjem hadronov. Najprej uporabimo priblizek saturacije z vakuumom kot edinega vmesnega stanja. Amplituda se tako faktorizira v produkt dveh hadronskih matricnih elementov. V primeru za razpad v tri psevdoskalarne mezone preko operatorja O3 dobimo (P2(P2)P1(P1) I d7(Us I 0) (P(p) I d7^b I B-(pb)) = (8.17) = (t — (s)FfB (s) + (mpl — mp2 )(m£ — mp) F™ (s)F™ (s) — F*** (s)FoPB (s) kjer smo uvedli Mandelstamove kinematske spremenljivke s = (pB — p)2, t = (pB — p1)2 in u = (pB — p2)2. Funkcije F so oblikovni faktorji, ki vsebujejo ucinke neperturbativnih barvnih interakcij. Definirani so v poglavju 2.3.1. Njihove funkcijske odvisnosti od kinematicnega parametra vzamemo iz literature. Za oblikovne faktorje B ^ n(p) uporabimo vrednosti izracunane v relativisticnem konstituen-tnem kvarkovskem modelu [74]. Za prehoda Ds ^ D in K ^ n obstaja napoved narejena v efektivni teoriji tezkih kvarkov in kiralne perturbacijske teorije [75, 76]. Izracunali smo razvejitvena razmerja nekaterih dvo- in trodelcnih razpadnih kanalov mezona B- (Tabela 8.1). Kot je bilo razvidno ze iz inkluzivnih razvejitvenih Razpadni kanal SM MSSM / pMSSM Z' B B B B B B n n K+ n-D-D+ n-K 0 p-K 0 n-K *0 p-K *0 1 x 10-15 6 x 10-21 3 x 10-16 3 x 10-16 5 x 10-16 6 x 10-16 1 x 10-14 6 x 10-20 3 x 10-15 3 x 10-15 5 x 10 6 x 10 -15 -15 omejitev par. omejitev par. 9 x 10-9 4 x 10-9 5 x 10-8 2 x 10-7 5 x 10-8 4 x 10-7 7 5 x 10 6 x 10 -7 Tabela 8.1: Razvejitvena razmerja za AS = —1 hadronske razpade mezona B-, izracunana v SM, MSSM, RpMSSM in modelu z bozonom Z'. Iz eksperimentalne zgornje meje za razpad B(B- ^ n-n-K+) < 9.5 x 10-7 smo dolocili zgornje meje parametrov modela RpMSSM (cetrti stolpec) in Z' (peti stolpec). razmerij, je model MSSM nemogoce opaziti v teh razpadih, saj poveca razvejitvena razmerja glede na SM le za red velikosti. Veliki prispevki so mozni v modelih RpMSSM in Z', kjer smo iz zgornje meje B(B- ^ n-n-K+) < 9.5 x 10-7 dolocili meje na parametre. Za RpMSSM ta meja ustreza V f100 Ge^2 (A' A'* + A' A'* ) m_ VAn31 An12 + An21An13j n=1 x ' < 6.6 x 10 5 medtem ko v modelu Z' izpeljemo R^L R dR , r,dR R dL b12 b13 + b12 b13 dL dL dR dR b12 b13 + b12 b13 < 3.2 x 10-4, < 5.2 x 10-4. (8.18) (8.19a) (8.19b) Neenacbe (8.18) in (8.19) ustrezajo napovedim v zadnjih dveh stolpcih Tabele 8.1. V Z' je najvec moznosti za zaznavo dvodelcnih razpadov v K0*, medtem ko so faktorizabilni prispevki k tem razpadom v RpMSSM nicelni, kanali s K0 pa so za velikostni razred manjsi. s 8.4 Nevtralni tokovi čarobnega kvarka in razpad D ^ Pl+l- Carobni mezoni so edini hadronski sistem za opazovanje nevtralnih kvarkovskih tokov med gornjimi kvarki. Pomemben proces z nevtralnim tokom je c ^ uy, kjer je lahko y realen ali tvori par leptonov t+t_ in z ene proces c ^ ut+t_. V luci nedavno izmerjenih parametrov mes anja nevtralnih mezonov D0 x = Amo =(0.98 ± 0.25) x 10_2, (8.20a) r d Ar y = = (0.83 ± 0.16) x 10_2, (8.20b) 2r d bomo raziskali njihov vpliv na razpade c ^ uy in razpade mezona D v psevdoskalar in leptonski par (D ^ Pt+t_, t = e,p). Razpade bomo analizirali v SM, MSSM, RpMSSM, modelu s singletnim gornjim kvarkom, ter modelu z leptokvarkom. Gornji meji za kanal s pionom in elektronom ali mionom sta znani B(D+ ^ n+e+e_) < 7.4 x 10_6 [108], (8.21a) B(D+ ^ < 3.9 x 10_6 [109]. (8.21b) 8.4.1 Razpad c ^ «y v MSSM V SM ima inkluzivni c ^ uy razvejitveno razmerje r(c ^ U7)/rDo = 2.5 x 10_8 [100]. V minimalnem supersimetricnem SM (MSSM) ta proces posredujejo diagrami z virtualnimi gluini, kjer se okus zamenja v prvem redu masnih vstavkov (ang. mass insertions). Prispevata le masna vstavka (čf2)LR in (5U2)RL [101, 102], ki prav tako generirata mesanje mezonov D0-D0. Iz zahteve, da vakuumsko stanje MSSM ni niti elektri c no niti barvno nabito, dobimo pogoje (5U2)LR,RL < VŠmc/m, [126], kjer q oznacuje maso skvarkov. Primerjava mej iz mesanja in stabilnosti vakuuma za degenerirane mase skvarkov in gluinov je prikazana v Tabeli 8.2. Ce privzamemo mg = mg max |(5 u;)lr,rl| max |(5 U2 )lr,rl| omejitev iz —mD omejitev iz stabilnosti vakuuma 350 GeV 0.007 0.006 500 GeV 0.01 0.004 1000 GeV 0.02 0.002 Tabela 8.2: Zgornje meje na masne vstavke | (5f2)LR,RL| dobljene iz —mD in omejitev na stabilnost vakuuma [126]. mase gluinov in skvarkov mg = mg = 350 GeV in uporabimo mejo iz Tabele 8.2 dobimo v MSSM r(c ^ UY)/rDo < 8 x 10_7, (8.22) kar je en red velikosti nad napovedjo SM, vendar so hadronski razpadi D ^ Vy povsem zasiceni z dolgosez nimi prispevki barvne interakcije. 8.4.2 Kratkosežni prispevki k c ^ uf+£ SM V SM nam proces c ^ generira efektivna Lagrangeva gostota na skali mc kvarka [103] c: eff SM = -^gF A^ + A^ Ci(^c)Q? - Ci(^c)Qi V2 i=1,2 i=1,2 i=3,...,10 (8.23) z operatorji definiranimi na strani 55. Dominantni prispevek je generiran z vstavitvijo operatorja Q2 in dodatnim virtualnim gluonom. KonCna amplituda je sorazmerna z operatorjem Q7 s koeficientom [100, 103] Cff = As(0.007 + 0.020i)(1 ± 0.2). (8.24) Dodaten singleten kvark z nabojem 2/3 V celem razredu modelov [94, 110] nastopajo sibko-singletni kvarki z nabojem 2/3, ki lahko inducirajo okus spreminjajoCe nevtralne tokove na drevesnem redu. Gornji kvarki se mes ajo z mes alno matriko Qj - r3 1 _ 1 - J^/3 = QiLYM= 2UiLYM^ijUjL - 2diLY^diL. (8.25) Prispevek je k dvema operatorjema, ki sta v SM zanemarljiva: 4n VaV^Cg = 4-Quc(4 sin2 - 1), (8.26a) a 4n Vu6Vc*b^Ci0 = — Que . (8.26b) Isti parameter mesalne matrike nastopa v mesanju nevtralnih mezonov in iz izmerjenih parametrov dobimo Quc < 2.8 x 10-4 [116]. MSSM in RpMSSM V MSSM bi naivno pricakovali, da bodo prispevki ojacani pri majhnih q2 zaradi propagatorja fotona, vendar nam za koncna stanja s psevdoskalarjem nastopa se dodaten faktor q2. Zato se tu osredotocimo le na model RpMSSM, kjer zaradi sklopitev Aijk dolnji skvarki d sklapljajo gornje kvarke in leptone. V vodilnem redu razvoja v 1/m| nam ti generirajo efektivna operatorja Q9,10 z Wilsonovima koeficientoma VcbVu6= -VctVu6ČC,10 = 2a2W ( mm^ ) ^2fc^. (8.27) k=1 \ dkR/ Skalarni leptokvark v reprezentaciji (3,1, -1/3) Sklaplja par lepton-kvark, kjer sta oba ali sibka singleta (desnorocna) ali sibka dubleta (levoroc na). Za vse prispevke je potrebno raz s iriti nabor operatorjev tudi na tenzorske tokove, vendar so ti zaradi mocnih mej iz meritev D0 ^ e+e-,p+p-zelo zastrti. Dominantni prispevki so le k operatorjema Q9,10 in njunima kiralno obrnjenima partnerjema Q9,10 VUVWC. = -™o = mW TO (8.28a) 2a — a sin2 6>W m2 = V«* V^C™ = -^-jW -f YflYR**. (8.28b) 2a — a 8.4.3 DolgoseZni prispevki v D ^ P£+£- V amplitudi razpada D ^ lahko dobimo pole v spremenljivki q2 zaradi vezanih stanj, ki se sklapljajo z operatorjema Q1 in Q2. Tak s ni resonan c ni efekti so posledica vezanih stanj barvne interakcije in jih lahko modeliramo z Breit-Wignerjevo obliko. Prispevek vektorskih resonanc, ki razpadajo v par lepton-antilepton zapi semo ald [d ^ KV ^ Kr^+] = q2 _ m^a+ imvTv u(p-)pv(p+), (8.29) kjer sta p± gibalni kolicini leptonov, my, Ty masa in razpadna sirina resonance, in ay prost brezdimenzijski parameter. Prispevajo nevtralne vektorske resonance V = p, 0, za katere lahko parametre ay doloc imo iz eksperimentalno znanih s irin rD^Py in ry. Ker so resonance vse relativno ozke ry ^ my, velja priblizek B [D ^ PV ^ ~ B [D ^ P V ] x B [V ^ . (8.30) 8.4.4 Primerjava resonančnih in kratkoseZnih spektrov Izkaze se, da je model z dodatnim singletnim kvarkom preostro omejen iz meritev parametrov mesanja nevtralnih D mezonov in nanj spekter in razpadna sirina nista obcutljiva. Po drugi strani model RpMSSM in model z leptokvarkom lahko rezulti-rata v opazljivih prispevkih tako v kanalu D+ ^ n+e+e-, kot tudi D+ ^ p-. Sklopitve A', ki nastopajo v razpadu z elektroni v koncnem stanju so omejene iz testov univerzalnosti nabitih tokov in meritev breznevtrinskega dvojnega P razpada. Po drugi strani pa je za kanal z mioni eksperimentalna zgornja meja (8.21b) mocnej s a od mej na relevantne parametre A' v literaturi in iz nje izpeljemo 3 v' v * A22fc A21fc E22k 21k o (m,../100 GeV)* < 10 X 10-3- <-> Za sklopitve leptokvarkov dobimo Y 21 Y11* YL(R) YL(R) < 1.6 10-3 V22 ^12* YL(R) YL(R) (-a/100 GeV)2 ' (-a/100 GeV)2 < 1.0 x 10-3. (8.32) 1.0 1.5 2.0 q2 (GeV2) q2 (GeV2) 1.0 1.5 2.0 q2 (GeV2) ?2 (GeV2) Slika 8.3: Distribucija razpadne sirine po invariantni masi leptonov q2 za razpad D+ ^ n+l+l- (levo) in Ds ^ K+l+l- (desno). V zgornji vrsti je i = e, v spodnji i = Rdece crte predstavljajo resonancni Breit-Wigner prispevek, temno modre crte pa maksimalni prispevek v modelu RpMSSM. Svetle modre crte predstavljajo kratkosezni prispevek SM. 0.0 0.5 2.0 2.5 Za analizo razpadov čarobno-čudnega mezona Ds, v primeru ko je končno stanje K+p+p-, uporabimo mejo (8.31). Za končno stanje z elektroni uporabimo enake sklopitve, kot smo jih za razpade D ^ Pe+e-. Vsa razvejitvena razmerja so zbrana v Tabelah 8.3 in 8.4, značilno ojačanje spektrov izven resonančnega območja pa je vidno na grafih 8.3. razpad eksperiment res. res.+RpMSSM/LQ D+ ^ n+e+e < 7.4 x 10-6 1.7 x 10-6 < 4.2 x 10-6 D+ ^ n+p+p- < 3.9 x 10-6 1.7 x 10-6 omejitev sklopitev A' Tabela 8.3: Primerjava eksperimentalnih mej in predikčij za razvejitvena razmerja razpadov mezona D+. Zadnja dva stolpča vsebujeta napovedi za resonanč ni spekter in čeloten spekter v modelu RpMSSM ali modelu leptokvarkov. razpad eksperiment res. res. + RpMSSM/LQ D+ ^ K+e+e- < 1.6 x 10-3 6.0 x 10-7 < 1.9 x 10-6 D+ ^ K+p+p- < 2.6 x 10-5 6.0 x 10-7 < 1.8 x 10-6 Tabela 8.4: Primerjava eksperimentalnih mej in napovedi za razvejitvena razmerja razpadov mezona D+. Zadnja dva stolpča vsebujeta napovedi za resonanč ni spekter in čeloten spekter v modelu RpMSSM ali modelu leptokvarkov. 8.5 Analiza Dalitzovega diagrama za razpad B ^ KnY Razpad b ^ sy z realnim fotonom v SM poteka preko pingvinskega diagrama, kjer k kromomagnetnem dipolnem operatorju e Q7 = 8^2 [m^v(1 + Y5)b + msš^v(1 - Y5)b] . (8.33) dominantno prispevata kvark t in s ibki bozonom W v zanki. Najbolje raziskan raz-padni kanal za meritev velikosti pripadajočega Wilsonovega koefičienta C7 je razpad B ^ K*y. V tem razdelku bomo analizirali razpad B ^ K^y v kinematičnem območju z energetskim fotonom in enim od lahkih mezonov. Razvejitveni razmerji naslednjih dveh razpadov sta ze izmerjeni B(B0 ^ K0nY) = (7.1-|;0 ± 0.4) x 10-6, (8.34a) B(B+ ^ K+nY) = (7.7 ± 1.0 ± 0.4) x 10-6. (8.34b) 8.5.1 Pristop z efektivnimi teorijami kvantne kromodinamike Za emisijo nizkoenergijskega mezona iz tezkega mezona B uporabimo kiralno per-turbačijsko teorijo tez kih mezonov, kjer amplitude razvijemo v potenč no vrsto po dveh majhnih parametrih: AQCD/mb in p/Ax (p je gibalna količina lahkega mezona, Ax skala zlomitve kiralne simetrije ~ 1 GeV). Vodilni člen v Lagrangevi gostoti te efektivne teorije je Leff = g (Ha(vKbYMY5H6(v)) . (8.35) Natan c na definicija polj te z kih in lahkih mezonov je dana v razdelku 2.3.3. Za preostali nadaljnji razpad tez kega mezona B na energetska 7 in lahki mezon upo stevamo, da sta oba delca v koncnem stanju zaradi njune majhne mase skoraj na svetlobnem stoz cu. V limiti, ko je njuna energija zelo velika, lahko oblikovne funkcije za prehod B * — K preko Q7 izrazimo z oblikovnimi funkcijami vektorskega toka (K (k) I ^ (1 + 75 )hv I B * (v, n )> = (0) [2Me^pa vM fcp n (8.36) + iM2nv _ in • q(Mv + k)v Nas pristop deluje za nizkoenergijski n ali K, kjer sta (Slika 8.4) Feynmanova diagrama za oba primera razli c na. e) K(k) K(k) Slika 8.4: Diagram na levi je vodilni prispevek v kinematicnem obmocju z nizko-energijskim mezonom n, na desni pa vodilni prispevek za nizkoenergijski mezon K. 8.5.2 Fotonski spektri Mo z nost resonancnih prispevkov nudijo vmesna stanja na masni lupini B — K*(1430)y ali B — K|(1780)y, vendar lahko iz znanih razvejitvenih razmerij teh resonanc [28] ugotovimo, daje resonancno ozadje majhno. Prav tako v eksperimentalnih spektrih teh razpadov [146, 148] ni evidentnih resonancnih prispevkov. Napovedi fotonskih spektrov razpadov so dobljene v efektivnem pristopu, opisanem v prejsnjem razdelku. Za opis me s anja n in n' uporabimo pristop [154], kjer je mesalni kot 6 med SU(3) oktetom n8 in singletom ni enak -15.4°. Spektri z n v koncnem stanju so vidni na Sliki 8.5. Ko pointegriramo spektra po obeh obmo cjih (Ek < 1.2 GeV ali < 1.2 GeV) za razpad B0 — K0nY zagotovimo pribli z no 10% razpadne s irine (8.34a). Z narascajoco natan c nostjo eksperimentov bo v prihodnosti moz no studirati velikost C7 tudi v omenjenih kinematicnih obmocjih razpadov B — KnY. 8.6 Ozadje mehkih fotonov v razpadih B — D£v Meritve ekskluzivnih razpadov B — D^v so pomembne za ekstrakcijo elementa VCb matrike CKM, se posebej zavoljo trenutnega razhajanja rezultatov med inkluzivno in ekskluzivno metodo dolo c anja Vcb. Vrednost Vcb postaja dominanten izvor napake dE/dE, [GeV"1] dB/dEr [GeV"1] 1. x 10 " 8. x 10" 6. x 10" 4. x 10" 2. x 10" Ey [GeV] E [GeV] Slika 8.5: Spektra razpada B0 K0nY. Na levi fotonski spekter v obmocju En < 0.8 GeV (kratko crtkana crta), En < 1.0 GeV (neprekinjena crta) in En < 1.2 GeV (dolgo crtkana crta). Na desni iste oznake za nizkoenergijski K, Ek < 0.8,1.0,1.2 GeV. pri teoreticni napovedi parametra eK |6K|« bbkn|Vcb|2A2. (8.37) kjer je A zelo natancno znan kosinus Cabibbovega kota. Napaka parametra vrece Bk iz simulacij kvantne kromodinamike na mrezi je trenutno ~ 5% in postaja primerljiva z napako VCb. Inkluzivna in ekskluzivna metoda imata do velike mere neodvisne sistematicne efekte. Za meritve Vcb v ekskluzivnih razpadih B ^ Dlv je v prihajajocih tovarnah okusa pricakovana relativna natancnost ~ 1%, kjer lahko pridejo do izraza majhni sistematicni efekti. Ker je tipicen eksperiment obcutljiv le na fotone do neke najmanjse energije, se lahko za foton y pod to mejo zgodi, da eksperiment napacno prepozna razpad B ^ DIvy kot B ^ Dlv. 8.6.1 Enodelčna vmesna stanja Amplitude za radiacijski razpad se zapise kot A(B(p) ^ D(p')i(pi)iž(fc)Y(e, q)) = (8.38) eGF VC6 Fv (t) e*M U(pž ) [ — 2^ Ym (/ + / + mi) + V^ — A^j yv (1 — Y5)v(k), kjer je Fv(t) matricni element vektorskega toka za prehod B ^ D in ustreza izsevu fotona iz leptona, medtem ko je tenzor — hadronski korelator elektromagnetnega in sibkega toka. Divergentni prispevki (odvisni le od nabojev hadronov) se v limiti EY ^ 0 pokrajsajo z virtualnimi elektromagnetnimi popravki na nivoju razpadne sirine, medted ko so strukturno odvisni prispevki v limiti EY ^ 0 koncni. V hadronskem korelatorju nam vmesna enodelcna stanja generirajo pole v invari-antnih masah. V nasem primeru bo zaradi majhne razlike mas med stanji D0* in D0 dominantno prispeval ravno D0*, ki lahko nastane na masni lupini, tako da nam generira resonancno obliko amplitude i (D(p') | J | D*(p' + q)) (D*(p' + q) | Vv | B(p)). (p' + q)2 — mD* + im-D* Td* ' ' 8.6.2 Spekter mehkega fotona Prehod D*0 ^ D0y v realni foton nam določa neperturbativni parameter gDo*DoY, ki je bil izracunan na mreži, kot so bile tudi oblikovne funkcije (aksialno-)vektorskega prehoda B ^ D* (za nabor oblikovnih funkcij glej razdelek 2.3.1). Spekter se v celoti nahaja pod 350 MeV, z energijo večine fotonov v intervalu od 100 do 200 MeV. Ce bi bil torej eksperiment slep za fotone pod 350 MeV, bi v vzorec dogodkov 0.14 : o.i2; o.io; o.o8; o.o6; o.o4; o.o2; 0.00 : 0 Slika 8.6: Delez napacno prepoznanih radiacijskih dogodkov kot funkcija najmanjse se zaznavne energije fotona. B ^ D£v zajel se vse dogodke B ^ D*£v ^ Dy£v. Na Sliki 8.6 je razvidno, da resolucija eksperimenta 80 MeV ustreza priblizno 1% relativni napaki na Vcb samo zaradi mehkih fotonov. 8.7 Zaključek Fizika okusov, eksperimentalna in teoreticna, je igrala pomembno vlogo pri preverbi Yukawinega sektorja standardnega modela. Mehanizem mesanja kvarkovskih okusov Cabibbo-Kobayashi-Maskawa je potrjen na nivoju 10% natancnosti in jasno je, da so efekti nove fizike velikostnega reda tipicno nekaj odstotkov ali manj. V disertaciji smo analizirali razpade tezkih mezonov, za katere je pricakovana razpadna sirina v okviru standardnega modela majhna, in so tako bolj obcutljivi za prispevke nove fizike. V tem kontekstu smo izpostavili modele nove fizike in razpadne kanale, ki se ponujajo moznost za eksperimentalna iskanja. V drugem delu disertacije smo predlagali metodi za preverjanje napovedi samega standardnega modela, ki lahko kaj kmalu pokazejo na nekonsistenco meritev z napovedmi standardnega modela in tako posredno kazejo na novo fiziko. Fizika okusov bo igrala pomembno vlogo tudi v interpretaciji podatkov iz Velikega hadronskega tr-kalnika (LHC), saj bo potrebno za odkritje novih delcev na LHC njihov vpliv najti in preveriti tudi v virtualnih efektih, kjer do izraza pridejo drugi nabori parametrov nove fizike. V tem oziru pricakujemo komplementarnost prihodnjih tovarn okusa in velikega hadronskega trkalnika v iskanju fizike onkraj standardnega modela. IZJAVA O AVTORSTVU Spodaj podpisani Nejc Kos nik izjavljam, dela in raziskav. Ljubljana, 16. april 2010 da je pricujoc a disertacija plod lastnega Nejc Kosnik