Bled Workshops in Physics Vol. 16, No. 1 p. 20
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015
Comparing mesons and WLWL TeV-resonances*
Antonio Dobadoa, Rafael L. Delgadoa, Felipe J. Llanes-Estradaa and Domenec Espriub
a Dept. Fisica Teorica I, Univ. Complutense, 28040 Madrid, Spain b Institut de Ciencies del Cosmos (ICCUB), Marti Franques 1, 08028 Barcelona, Spain
Abstract. Tantalizing LHC hints suggest that resonances of the Electroweak Symmetry Breaking Sector might exist at the TeV scale. We recall a few key meson-meson resonances in the GeV region that could have high-energy analogues which we compare, as well as the corresponding unitarized effective theories describing them. While detailed dynamics may be different, the constraints of unitarity, causality and global-symmetry breaking, incorporated in the Inverse Amplitude Method, allow to carry some intuition over to the largely unmeasured higher energy domain. If the 2 TeV ATLAS excess advances one such new resonance, this could indicate an anomalous q q W coupling.
1 Non-linear EFT for WlWl and hh
The Electroweak Symmetry Breaking Sector of the Standard Model (SM) has a low-energy spectrum composed of the longitudinal W±, ZL and the Higgs-like h bosons. Various dynamical relations suggest that the longitudinal gauge bosons are a triplet under the custodial SU(2)c, and h is a singlet. This is analogous to hadron physics where pions fall in a triplet and the n meson is a singlet. The global symmetry breaking pattern, SU(2) x SU(2) —» SU(2)c is shared between the two fields.
The resulting effective Lagrangian, employing Goldstone bosons w a ~ WL, ZL as per the Equivalence Theorem (valid for energies sufficiently larger than MW, MZ), in the non-linear representation, is [1-3],
L = 2
1 1 h /hx 2 1 +2a—+b -v V v
\ 1
3^3^ j Sij + v"2 ) +
v4	v4	v4
+ 3^H3^h3v^i3v +	(1)
This Lagrangian is adequate to explore the energy region 1-3 TeV > 100 GeV, and contains seven parameters. Their status is given in [1] and basically amounts to
Talk delivered by Felipe J. Llanes-Estrada
Comparing mesons and WLWL TeV-resonances
21
a € (0.88,1.3) (1 in the SM), b G (-1,3) (1 in the SM) and the other, NLO parameters (vanishing in the SM) largely unconstrained. This is a reasonably manageable Lagrangian for LHC exploration of electroweak symmetry breaking in the TeV region, before diving into the space of the fully fledged effective theory [3].
Partial wave scattering amplitudes in perturbation theory A{ (s) = A J O' (s) + ajlo' (s)... for ww and hh, have been reported to NLO in [4]. For example, the LO amplitudes of I = 0,1 and 2, and the ww —» hh channel-coupling one are
A0(s) = ^(1 - a2)s A1(s) = 96^(1 - a2)s 1	/3
A0(s) = -32nV2(1 - a2)s M°(s) = ^(a2 - b)s
and we see how any small separation of the parameters from the SM value a2 = b = 1 leads to energy growth, and eventually to strong interactions. To NLO, the amplitudes closely resemble those of chiral perturbation theory
A JO+NLO' (s) = Ks + (V) + D log ^ + E log -) s2	(2)
with a left cut carried by the Ds2 log s term, a right cut in the Es2 log(-s) term, and the Ks + Bs2 tree-level polynomial. B, D and E can be found in [4] and satisfy perturbative renormalizability (in the chiral sense).
2 Resonances
The perturbative amplitudes in Eq. (2) do not make sense for large s (TeV-region) where they violate unitarity ImAj = |Aj |2, relation satisfied only order by order in perturbation theory, namely ImAjLO' = |ajO'|2.
In hadron physics, the solution is to construct new amplitudes that satisfy unitarity exactly and reproduce the effective theory at low energy (see the lectures [5]) via dispersive analysis. This combination of dispersion relations with effective theory exploits all model-independent information in the two-body experimental data, and is known in both the electroweak symmetry breaking sector and the QCD sector of the Standard Model [7]. A salient example is the NLO Inverse Amplitude Method,
A	K°')2	/3)
J A(LO' A(NLO'	(3)
AIJ - AIJ
a simple formula that can be rigorously generalized to two channels of massless particles by upgrading the various A to matrices. The denominator of Eq. (3) allows for scattering resonances (poles in the 2nd Riemann sheet).
In meson physics, the most salient elastic resonance of the nn system is the isovector p(770) meson, that dominates low-energy dipion production in most experiments; for example, its prominence in COMPASS data [6] is visible in the
22
Felipe J. Llanes-Estrada et al.
^14000
a>
5=12000
®
S10000
c
® 8000 6000 4000 2000
0^05
1 1.5 2 2.5 3 3.5 inv. mass of both rc'rc0 system [GeV/c 2|
10.8-¿0.60.40.2-
0,
0
— ' \ ...
a = 0 a- = +0.005
2
s (TeV2)
4	5
Fig. 1. Left: the physical p in the COMPASS nn spectrum. (Reprinted from [6]. Copyright 2008, AIP Publishing LLC). Right: a possible equivalent WW, WZ state for various a4, a5.
a4 = a5 = 0
a, = +0.005 a. =0
4	5
a4 = -0.005 a5 = 0
a4 = 0 a5 = -0.005
2
3
left plot of figure 1. Independently of particular technicolor models, values of a4 and a5 at the 10-4-10-3 level produce a p-like meson of the electroweak sector in the TeV region. The right panel of figure 1 demonstrates this.
The central attraction of the nuclear potential suggested the introduction of a scalar a meson in the nn spectrum whose existence was long disputed but that is now well established [8]. In addition to detailed dispersive analysis, it gives strength to the low-energy nn spectrum if the p channel is filtered out by cautious quantum number choice, such as J/^ —» wnn that forces the pion subsystem to have positive charge conjugation because the other two mesons both have C = (—1). An analysis of BES data by D. Bugg is shown in the left plot of fig. 2. The right plot shows the equivalent resonance in ww ~ WLWL, that appears for
1=-0.80.60.40.2-
■ IAM
----N/D
------K-improved
0
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
E (TeV)
Fig. 2. Left: nn spectrum with positive charge conjugation clearly showing an enhancement at low invariant mass, related to the fo(500) (or a) meson; (Reprinted from [9] with permission. Copyright 2008, AIP Publishing LLC). Right: IJ = 00 ww scattering in the IAM and other unitarization methods producing an equivalent electroweak resonance.
Comparing mesons and WLWL TeV-resonances
21
a = 1 and/or b = a2 (if the resonance is induced by b alone it is a pure coupled channel one [4], that also has analogues in hadron physics, though less straightforward ones).
The same BES data also reveals another salient meson resonance, the f2 (1270). Partial waves with J = 2 cannot be treated with the NLO IAM, as A0 LO = 0 but a similar structure has been obtained with the N/D or K-matrix unitarization methods, and we show it in figure 3.
0.8-L
0.6-
0.4-
0.2-
N/D ■ K-improved
M i i ¡i i i i i i i i
! ¡ i i i '
! '/ i \i
¡ A.
0
0 0.5 1
1.5 2 2.5 E (TeV)
3 3.5
Fig. 3. Left: generating an IJ = 02 resonance in the electroweak sector is possible with adequate values of a4, a5. Right: positive values of a2 — 1 also generate an isotensor I = 2 resonance, though this is more disputed [2]. In hadron physics the isotensor wave is repulsive, and thus, not resonant.
1
4
3 ATLAS excess in two-jet events
Renewed interest in TeV-scale resonances is due to a possible excess in ATLAS data [10] plotted in figure 4 together with comparable, older CMS data [11] that does not show such an enhancement. The excess is seen in two-jet events tagged as vector boson pairs by invariant mass reconstruction (82 and 91 GeV respectively). The experimental error makes the identification loose, so that the three-channels cross-feed and we should not take seriously the excess to be seen in all three yet. Because WZ is a charged channel, an I = 0 resonance cannot decay there. Likewise ZZ cannot come from an I = 1 resonance because the corresponding Clebsch-Gordan coefficient (1010|10) vanishes. A combination of both I = 0,1 could explain all three signals simultaneously (as would also an isotensor I = 2 resonance).
A relevant relation imported from hadron physics that the IAM naturally incorporates restricts the width of a vector boson. This one-channel KSFR relation [12] links the mass and width of the vector resonance with the low-energy constants v and a in a quite striking manner,
rIAM = Mfe(1 - a2) .	(4)
22
Felipe J. Llanes-Estrada et al.
For M ~ 2 TeV and r ~ 0.2 TeV (see fig. 4), we get a ~ 0.73 which is in tension with the ATLAS-deduced bound aha > 0.88 at 4-5ct level; Eq. (4) predicts that a 2 TeV J = 1 resonance, with current low-energy constants, needs to have r < 50 GeV, a fact confirmed by more detailed calculations [1,2]. However scalar resonances tend to be substantially broader.
1000
o
100
^ 10 rn 1 0.1
	
: • ! •	
■ ■	
" ! i	
1
1.5
Ejj (TeV)
2.5
Fig. 4. Left: replot of the ATLAS data [10] for WZ -> 2 jet, with a slight excess at 2 TeV (also visible in the other isospin combinations WW and ZZ, not shown). The jet analysis is under intense scrutiny [15]. Right: equivalent CMS data [11] with vector-boson originating jets. No excess is visible at 2 TeV (though perhaps some near 1.8-1.9 TeV).
10000
2
3
The cross section for the reaction pp —» W+Z+X for a given WZ Mandelstam s, and with the E2 total energy in the proton-proton cm frame, can be written [13] in standard LO QCD factorization as
dads
dxu
dxa5(s - xuxaEt20tf(xu)f(xa)a(ud^ w+z) .
(5)
The parton-level cross section a is calculated, with the help of the factorization theorem, from the effective Lagrangian in Eq. (1) above. Following [13], we would
Fig. 5. Production of a pair of Goldstone bosons by ud annihilation through a W-meson and anomalous BSM vertex enhancing it.
expect an amplitude (from the left diagram of figure 5) given by M = uytM-ig^v/8)2(i/q2)(k1 — k2)^ in perturbation theory. Further, dispersive analysis reveals the need of a vector form factor in the presence of strong final state rescattering, to guarantee Watson's final state theorem; the phase of the production amplitude must be equal to that of the elastic ww scattering amplitude. If the later is represented by the Inverse Amplitude Method, the form factor in the
Www vertex is FV(s) =
A
(i )
(s)
. (0)r
The resulting cross section [13] was
found to be slightly below the CMS bound, and perhaps insufficient to explain

-1
0
0
+
1
Comparing mesons and WLWL TeV-resonances
21
the possible ATLAS excess. With current precision this statement should not be taken to earnestly, but it is nonetheless not too soon to ask ourselves what would happen in the presence of additional non-SM fermion couplings.
Thus, an original contribution of this note is to add to Eq. (1) a term 1
Lfermion anomalous = 'Lw	(6)
(for a derivation see, e.g. [14]). The parton level cross-section is then dt"
an
1 g4 sin2 0(l + ^S) Vv(s)|2 ,
cm
64n2 s 32
(7)
and if Si = 0 additional production strength appears in the TeV region. The sign of this Si might be determined from the line shape due to interference with the background [16].
4 Conclusion
110
1.5 2 E (TeV)
110
110
110"
110
110-5
310
Fig. 6. Tree-level W production of œœ [13] with final-state resonance; non-zero parameters are a=0.9, b= a2, a4 = 7x 10-4 (at h = 3 TeV). Also shown is a CMS cross-section upper bound (see fig. 4). This can be exceeded with the 5i coupling of Eq. (6).
The 13 TeV LHC run II entails larger cross sections and allows addressing the typical tr, p-like ww resonances, at the edge of the run I sensitivity limit as shown in fig. 6. The large rate at which such a resonance would have to be produced to explain the ATLAS excess (at the 10fbarn level [17]) is a bit puzzling, though it can be incorporated theoretically with the Si parameter. Hopefully this ATLAS excess will soon be refuted or confirmed. In any case, the combination of effective theory and unitarity that the IAM encodes is a powerful tool to describe data up to E = 3 TeV in the electroweak sector if new, strongly interacting phenomena
1 This is only one of the possible additional operators. There is a second one with R fields, and several custodially breaking others. The gauge-invariant version of Eq. (6) actually modifies the fermion-gauge coupling by a factor (1+S1): this cannot be excluded because it would be the quantity that is actually well measured in | decay. The triple gauge
boson vertex would then need not coincide with this coupling. However the latter is much less precisely known and there is room for deviations at the 5-10% level.
22	Felipe J. Llanes-Estrada et al.
appear, with only few independent parameters. The content of new, Beyond the Standard Model theories, can then be matched onto those parameters for quick tests of their phenomenological viability.
Acknowledgements
FLE thanks the organizers of the Bled workshop "Exploring hadron resonances" for hospitality and encouragement. Work supported by Spanish Excellence Network on Hadronic Physics FIS2014-57026-REDT, and grants UCM:910309, MINECO:FPA2014-53375-C2-1-P, FPA2013-46570,2014-SGR-104, MDM-2014-0369; its completion was possible at the Institute for Nuclear Theory of the Univ. of Washington, Seattle, with DOE support.
References
1.	R.L.Delgado, A.Dobado and F.J.Llanes-Estrada, J. Phys. G 41, 025002 (2014); ibid, JHEP 1402,121 (2014); R.L.Delgado, A.Dobado, M.J.Herrero and J.J.Sanz-Cillero, JHEP 1407, 149 (2014).
2.	P. Arnan, D. Espriu and F. Mescia, arXiv:1508.00174 [hep-ph]. D. Espriu and F. Mescia, Phys. Rev. D 90, 015035 (2014). D. Espriu, F. Mescia and B. Yencho, Phys. Rev. D 88, 055002 (2013). D. Espriu and B. Yencho, Phys. Rev. D 87, no. 5, 055017 (2013).
3.	R. Alonso et al, JHEP 1412, 034 (2014); G.Buchalla, O.Cata, A.Celis and C.Krause, arXiv:1504.01707 [hep-ph].
4.	R. L. Delgado, A. Dobado and F. J. Llanes-Estrada, Phys. Rev. Lett. 114, 221803 (2015).
5.	T. N. Truong, EFI-90-26-CHICAGO, EP-CPT-A965-0490, UCSBTH-90-29, C90-01-25.
6.	F. Nerling [COMPASS Collaboration], AIP Conf. Proc. 1257, 286 (2010) [arXiv:1007.2951 [hep-ex]].
7.	A. Dobado, M. J. Herrero and T. N. Truong, Phys. Lett. B 235,129 (1990); A. Dobado and J. R. Pelaez, Nucl. Phys. B 425, 110 (1994) [Nucl. Phys. B 434, 475 (1995)] [hep-ph/9401202].
8.	J. R. Pelaez, Phys. Rept.(in press) arXiv:1510.00653 [hep-ph].
9.	D. V. Bugg, AIP Conf.Proc.1030,3 (2008) [arXiv:0804.3450 hep-ph].
10.	G. Aad etal. [ATLAS Collaboration], arXiv:1506.00962 [hep-ex].
11.	V. Khachatryan et al. [CMS Collaboration], JHEP 1408,173 (2014).
12.	R. L. Delgado, A. Dobado and F. J. Llanes-Estrada, Phys. Rev. D 91, 075017 (2015).
13.	A. Dobado, F. K. Guo and F. J. Llanes-Estrada, Commun. Theor. Phys. (in press), arXiv:1508.03544 [hep-ph].
14.	E. Bagan, D. Espriu and J. Manzano, Phys. Rev. D 60,114035 (1999).
15.	D. Goncalves, F. Krauss, M. Spannowsky, arXiv:1508.04162 [hep-ph].
16.	C. H. Chen and T. Nomura, arXiv:1509.02039 [hep-ph].
17.	ATLAS contribution to the 3rd Annual LHC Physics Conference, St. Petersburg, 31/8 to 5/9 2015, ATLAS-CONF-2015-045.
Bled Workshops in Physics Vol. 16, No. 1 p.l
A
Proceedings of the Mini-Workshop
Exploring Hadron Resonances
Bled, Slovenia, July 5 - 11, 2015

Resonance states and branching ratios from a time-dependent perspective.
Ido Gilary
Shulich Faculty of Chemistry, Technion, Haifa, 3200003, Israel
The spectrum of a given Hermitian quantum mechanical system can be generally separated into a discrete part containing the bound states and a continuous part of scattering states above the threshold. These states are solutions to the time-independent Schrodinger equation (TISE) with the corresponding boundary conditions. The discrete nature of the bound spectrum enables the characterization of the stable part of the studied system along with its physical properties based on its energy levels. The continuum can be used to characterize the system by probing it through scattering experiments. In this context instead of discrete energy levels the discussion is usually shifted to resonances in the scattering profile of the studied system. These appear as sharp features in the energy profile of the interaction.
Wavepacket dynamics in the continuum of meta-stable open quantum systems reveals that in the interaction region the evolution resembles that of bound states. There is, however, one difference where a bound system preserves probability in a meta-stable one when we observe decay in time. When the decay from the interaction region tends to follow an exponential form then the energy content of the localized part of the wavepacket assumes a constant value. This value is complex where the real part represents the energy of a resonance of the system and the imaginary part is related to the width of this resonance. The dynamics outside the interaction region exhibit a spatial exponential increase which drops off at the edge of the escaping wavefront. The velocity of the escaping part of the wavepacket has the momentum corresponding to the average energy inside. Similar dynamics is observed when scattering a wavepacket off a potential at a resonant energy. Initially the arriving wavepacket populates a resonant boundlike state inside the interaction region and consequently the formed meta-stable state decays.
The time-dependent dynamics observed in meta-stable systems demonstrates the properties of both stationary bound and scattering states. This type of dynamics usually occurs due to either the shape of the potential of interaction where the variation can lead to confinement of finite time or due the coupling of a bound state in an closed channel with the continuum of an open channel. The first type of states is often termed shape-type resonance whereas the second type is usually called Feshbach-type resonances. The above discussion suggests that the essence of the dynamics can be captured by solving a time independent equation with appropriate boundary conditions. Such boundary conditions allow only outgo-
2	Ido Gilary
ing flux. Solving the TISE with outgoing boundary conditions leads to solutions with complex momentum. This makes the energies of these states complex just as the portrayed dynamics and it also displays the asymptotic divergence which was observed.
In order to be able to calculate the energies of the resonance states one needs to be able to fix the asymptotic divergence. This can be achieved in various ways which all lead to a non-Hermitian Hamiltonian. Some of the techniques used are: (1) scaling of the coordinate by a complex factor (complex scaling); (2) addition of a complex absorbing potential at the asymptotes far from the interaction region; (3) Feshabch projection formalism which separates between the spaces of localized and scattering states; (4) Siegert pseudo states which satisfy the required boundary condition on a given surface but lead to a quadratic eiegnvalue problem.
The use of Non-Hermitian Hamiltonians readily yields the information regarding the lifetime of a given meta-stable state in addition to its energy. On the other hand, it leads to some complications due the non-Hermiticity. First of all, the resonance states are not orthogonal with respect to the conventional scalar Dirac product. Instead one needs to find and additional set of states which are orthogonal to them. These are the eigenstates of the Hermitian conjugate Hamiltonian which physically are their time-reversed counterparts. The two biorthogonal states form together the resolution of the identity. Another aspect is the loss of the probabilistic interpretation due to the non-Hermiticty. This can be amended by redefining the inner product based on the bi-orthogonal set of states. By doing so the probabilistic interpretation is retained along with the non-unitary evolution resulting from the decay of the system. This allows to describe very complicated dynamics in the continuum based on dynamics of several resonance states alone.
In decaying few-body systems there are often several channels open to decay. In such case the decay rate of the resonance contains contributions due to the flux in each of the open channels. When considering the above mentioned outgoing boundary conditions one finds that the momentum of the outgoing flux in each channel depends on the resonance energy and the threshold energy of the given channel. When following the wavepacket dynamics in such systems one observes that at every channel the wavepacket leaks at the velocity given by the momentum in that channel. Consequently all the information regarding the partial widths to the different channels and their branching ratios can be extracted from the stationary resonance wavefunction. All that is needed in order to evaluate the branching ratios is the complex amplitude of the resonance wavefunction at the asymptotes and the momentum at every given channel which is obtained from the difference between the resonance energy and the channel's threshold energy.
Resonance states and branching ratios from a time-dependent perspective.
3
References
1.	Shachar Klaiman and Ido Gilary, "On Resonance: A First Glance into the Behavior of Unstable States." In: Advances in Quantum Chemistry, 63. "Unstable States in the Continuous Spectra, Part II: Interpretation, Theory and Applications", Edited by Cleanthes A. Nicolaides and Erkki Brandas. San Diego: Academic Press, pp. 131 (2012).
2.	Tamar Goldzak, Ido Gilary and Nimrod Moiseyev, "Evaluation of partial widths and branching ratios from resonance wave functions." Physical Review A. 82,052105 (2010).
102 Povzetki v slovenščini
Analitična zgradba neperturbativnih kvarkovih propagatorjev in mezonskih procesov
Dalibor Kekeza in Dubravko Klabučarb a Rugjer Boškovič Institute, Bijenička č. 54,10000 Zagreb, Croatia
b Physics Department, Faculty of Science, Zagreb University, Bijenička č. 32, Zagreb 10000, Croatia
Raziskujemo analitično zgradbo nekaterih nastavkov za kvarkove propagatorje v neperturbativnem področju kromodinamike. Ce izberemo fizikalno motivirano parametrizacijo masne funkcije M(p2) oblečenih kvarkov, odvisne od gibalne količine in z določeno analitično zgradbo, je skrajno tezavno napovedati in obvladati analitično zgradbo ustreznega neperturbativnega kvarkovega propagatorja. Tudi problem Wickove rotacije, ki povezuje izrazavo v prostoru Minkowskega in Evklida, je skrajno tezaven v neperturbativnem območju. Izpeljemo obliko propagatorja, ki omogoča Wickovo rotacijo in dopusča enakovredne račune v prostoru Minkowskega in Evklida. Kljub preprostosti nudi ta model dober kvalitativen in semikvantitativen opis nekaterih pročesov z psevdoskalarnimi mezoni.
Primerjava med mezoni in resonancami WLWL pri energijah več TeV
Antonio Dobadoa, Rafael L. Delgadoa, Felipe J. Llanes-Estradaa and Domenec Espriub
a Dept. Fisica Teorica I, Univ. Complutense, 28040 Madrid
b Institut de Ciencies del Cosmos (ICCUB), Marti Franques 1, 08028 Barcelona, Spain.
Mikavni signali z Velikega hadronskega trkalnika (LHC) namigujejo, da morda obstajajo v področju zloma elektro-sibke simetrije resonance v območju več TeV. Spomnimo na nekajključnih resonanc mezon-mezon v območju GeV, ki bi utegnile imeti analogne resonance pri visokih energijah in nam sluzijo za primerjavo, hkrati z odgovarjujočo unitarizirano efektivno teorijo. Ceprav je podrobna dinamika lahko različna, pa zahteve po unitarnosti, kavzalnosti in globalnem zlomu simetrije (z uporabo metode inverzne amplitude) dovoljujejo prenos intuicije v večinoma neizmerjeno območje visokih energij. Ce bo povečano stevilo dogodkov na ATLASU okrog 2 TeV podprlo tako novo resonančo, to lahko pomeni anomalno sklopitev qqW.
Resonance v konstituentnem kvarkovem modelu.
R. Kleinhappel and W. Plessas
Theoretical Physics, Institute of Physics, University of Graz, A-8010 Graz, Austria
Na kratko poročamo o danasnjem opisu barionskih resonanc v realističnem modelu s konstituentnimi kvarki, v katerem običajno obravnavamo resonance kot