University of Ljublana Faculty of Mathematics and Physics Simon Sirca The axial form factor of the nucleon from coincident pion electro-production at low Q2 octoral dissertatio Supervisor: prof. dr. Milan Potoka Cosupervisor: prof. dr. Thomas Walche Ljubana, 199 To my mother, and other teachers. I would like to thank my supervisor Prof. Dr. Milan Potokar for giving me the opportunity to face the challenges of experimental nuclear physics as a member of the Al Collaboration at the Nuclear Physics Institute of the Mainz University and for his continuous interest in my work. My experimental co-supervisor Prof. Dr. Thomas Walcher deserves my gratitude for his confidence in assigning to me the 'hot topic' of the axial form factor, and for many food-forthought discussion and goal-oriented incentives. lam also indebted to Prof. Dr. Reiner Neuhausen wh constantly assisted in making my visits to Mainz possible, and to Dr habil. Giinther Rosner for help during the experiment and data analysis. I thank my theoretical co-supervisor Prof. Dr. Bojan Golli for keeping a watchful ey over my work. Ever since my final year as an undergraduate, his instinct and experience in physics were often stepping-stones that protected me from wading the stream of erroneous ideas, pitfalls and dead-alleys of calculations. lam also grateful to Prof. Dr. Manuel Fiolhais who enabled me to visit the University ofCoimbr twice, and to Dr Pedro Alberto and Dr. Jose Amoreira, who succeeded in making these stays even more fruitful and pleasant The experimental part of my work could not have been started, let alon accomplished, without the help of all members oftheAl Collaboration. I hav repeatedly resorted to this source of knowledge, which seems to embody the Feynman's vision of the "expanding frontier of ignorance". But my particular thanks go to Dipl Phys. Arnd Liesenfeld, Dr. Michael Distler, Dr. Richard Florizone Dr. Harald Merkel, Dr. Ulrich Miiller, Dipl. Phys. Axel Wagner Dipl. Phys. Ralph Bohm, and mag. Klemen Bohinc for helping me, everyone in his own way. Arnd Liesenfeld is an outstanding exception I wish to highlight — his immense patience, his rich experience, and his readiness to help show even the Feynman's frontier in a friendlier perspective. At the end, I owe gratitude to my family and ask their forgiveness for me sacrificing much of the spare time to this work instead of to them. Abtract Nearthreshold electro-production of charged pions on protons at low Q is a contemporary precise tool to study the axial-vector form factor of the nucleon, which is intimately related t calculations of chiral perturbation theory and to chiral quark models of nucleon structure. This thesis discusses the acquisition and the analysis of the p(e,e'7t+)n data obtained in a coincidence experiment performed with the Al Collaboration at the Nuclear physics institut of the Mainz University, at W = 1125MeV and at Q2 = 0.117, 0.195 and 0.273 (GeV/c)2. The transverse and the longitudinal crosssections are determined from the measured crosssections using the Rosenbluth method, and the axial form factor with the corresponding axial mass pa rameter are extracted from the transverse part. In addition, the axial form factor of the nucleon is calculated in the framework of the quark-level linear sigma model and the chromodielectri model, where centre-of-mass and recoil effects are eliminated. Keywords: electro-production of pions on the nucleon, coincidence experimen with electrons, structure functions, axial form factor, chiral perturbation theory, linear sigma model, chromodielectric model CMS corrections, recoil corrections. PACS (9: 13.60.Le, 14.20.Dh, 12.39.-x, 12.39.Fe . v ontent . Introductio 2. General formalism 2.1 The expression for the reaction crosssectio ..................... 2. Motivation for the separation of the structure functions and extraction of Ga • • • 11 . Experimental setup 1 3.1 The MaMi-B accelerator................................. 16 3. The hydrogen cryotarget and the beam wobbler................... 17 3. Magnetic spectrometers and detector systems..................... 2 3. Electronics......................................... 2 4. Data analysi 2 4.1 Coincidence tim..................................... 2 4. Missing mas ....................................... 31 4. Further background reduction.............................. 3 4. Detector efficiencies.................................... 34 4. Correction factors..................................... 37 4. Luminosity......................................... 4 4.7 Spectrometer acceptances................................ 4 4. The differential crosssection............................... 4 5. Extractin Ga(c|2) from experiment 4 5.1 Early theorie....................................... 4 5. The DT model....................................... 4 5. Chiral perturbation theor................................ 51 6. Calculation of the nucleon axial form facto 5 6.1 The axial-vector coupling constan........................... 5 6. Definition of GA(q ................................... 5 6. Ga(c) in the MIT bag and in models with a scalar confining potentia...... 5 6. GA(q2)inthLSandtheCD............................ 5 7. Summary and outlook Appndi Extracting M from (anti)neutrino scattering experiment Multipole expansio Isospin decompositio Corrections of radiation losses in p 7 The DHKT mode The Lagrangians of the L and the C The model wave-function of the L and the C 8 Evaluation o GA(q2) in the L and the CM technical detail 8 l The quark contribution to the term with jo(qt").................... 8 The quark contribution to the term with J2(qt").................... 8 The meson contribution to the term with jo(qr).................... 91 The meson contribution to the term with J(qr).................... 9 The norm overlap..................................... 94 Reference 1 Introducton Der wahre Weg geht iiber ein Seil das nicht in der Hohe gespannt ist, sondern knapp iiber dem Boden Es scheint mehr bestimmt stolpern zu machen, als begangen z werden Franz Kafk For the past four decades, nuclear and particle physics research focuses on coincidenc experiments, in which the target nuclei are probed by either hadron or electron beams. In these measurements, the scattered projectile is detected in coincidence with the ejected particle or the residual nucleus. Since the hadronic crosssections are usually large, hadronic probes were successfully used in the early coincidence measurements, even though the accelerator duty factors were rather low and the underlying interaction was poorly known. In a typica experiment of the time, e. g. A(p, pp)A—1, the scattered protons were detected in coincidenc with the protons ejected from the nucleus A [1] But the physical picture of such processes was obscured by the ignorance of both strong interaction matrix elements, the studied one (in the output channel) as well as the one tha represents the excitation of the system under study (in the entrance channel). This meant tha the extracted physical knowledge largely depended on the ability of certain models to describe the strong interaction. Coincidence measurements were therefore mostly limited to processes of minimal complexity, for example, to quasi-elastic scattering, where particular kinematica conditions can be chosen in which the hadronic projectile probes almost free nucleons within the nucleus; at the same time, the interaction between the ejected nucleon and the residual nu cleus is negligible in the first approximation. It should be stressed, however, that the accuracy of the experiments does grow over years and that the sophistication of the models gradually improves. For example, recent theoretical analyses of the p(p, p'7r+)n process [2] based on the one-pion exchange approximation parameterised in terms of the DWBA t-matrix for the firs step of pp —> nA++(pA+) — np7t+, agrees remarkably well with the accurate coincidence dat of [3] 1. Inroduio Nevertheless, experiments with electron beams that evolved later on introduced a majo improvement in the field. Electrons are point-like particles without internal structure or excited states. In addition, the electro-magnetic interaction of the electrons with the probed particles i well understood, and has a relatively weak coupling constant. Contrary to hadronic projectile probing only the nuclear surface, electrons therefore penetrate deeply into the target nucleu and interact without disturbing other nuclear constituents. The interaction of an electron with a light nucleus can be visualised as an exchange of a single virtual photon, and first orde perturbation approach is usually sufficient to describe the electro-magnetic parts of the reactio amplitudes in the interpretation of experimental results. Electron scattering experiments can be classified into two groups, depending on whether any hadrons are detected in coincidence with the scattered electron or not. In inclusive mea surements, the final nuclear state is not unique. Since the scattered electron is the only detected particle, several nuclear states are effectively summed over in the crosssection. In exclusive experiments, the scattered electron is detected in coincidence with one or more ejected or re coiled hadrons, and only a specific final state is considered. Since the electro-magnetic inter action crosssections are small, coincidence experiments with electrons were impossible in the early days of pulsed, low dutyfactor accelerators, even though these provided high peak cur rents. The detectors and spectrometers used to identify and analyse the reaction products were not capable of handling these short, intense particle bursts. In addition, they had small angula and momentum acceptances. Exclusive measurements of one nucleon knock-out reactions A(e, e'p)B [4] or pion electro production A(e, e7r)B [5] were among the first modern electron coincidence experiments. In the (e, e'p) measurements, in which the momentum of the ejected proton is determined simultaneously with the momentum of the electron, the extracted reaction amplitudes directly reflec the Fourier transforms of the corresponding part of the nuclear wavefunction. If the incomin electron also excites one of the target nucleons into an excited state, the measured final hadron can be linked to the decay of the corresponding resonant state and to its propagation through the nuclear medium. Moreover, the coincidence cross-section allows access to a narrower se of matrix elements (with their magnitudes and relative phases) which are unattainable in th inclusive crosssection, and thereby convey a richer information on the nuclear structure. Similarly, coincidence reactions like N(e, e7t)N offer an insight into the structure of the nu cleon. The pioneering coincidence experiments of pion electro-production on the nucleon were carried out about three decades ago at DESY in Hamburg, in Saclay, Frascati, Bonn, Manchester and other European laboratories (see [6] and references quoted in section 2.2 for a review). Un fortunately, the experimental data were burdened with large statistical and systematical uncer tainties for the reasons enumerated above. Today, new electron accelerators with high dutyfactors or continuous electron beams (MIT-Bates, NIKHEF, MaMi, TJNAF) and highly efficient detector systems enable us to significantl improve the accuracy of the experiments. Among the observables that can be measured with these modern setups, or re-measured with much lower uncertainties, the electroweak form fac tors of the nucleon are among the most relevant. Simultaneously, quantum chromodynamics (QCD) evolved as the fundamental theory of the strong interactions [7], giving birth or inspiring several effective theories and models of hadron structure. In particular, the chiral pertur bation theory (xPT) recently emerged as an effective field theory of the standard model below the chiral symmetry breaking scale [8]. Modern detector setups allow us to access previously unattainable kinematical regions of the nucleon-pion processes, in which the perturbative QCD approach is inappropriate, so these new measurements may serve not only as constraints o the existing models of the nucleon structure, but also as a testing ground of xPT predictions. Particularly the axial form factor of the nucleon recently received intense renewed attention because of sizeable inconsistencies in the world supply of data available to date. There ar basically two methods to determine this form factor. One set of experimental data comes from measurements of quasi-elastic (anti)neutrino scattering on protons [9,10,11], deuterons [12,13 14, 15, 16] and other nuclei (Al, Fe) [17, 18] or composite targets like freon [19, 20, 21, 22] and propane [22, 23]. The procedure 1 followed in the extraction of the axial form factor is to fit th q2-dependence of the (anti)neutrino-nucleon crosssection do^ dq (q)±B(q w in which the axial form factor GA(q2) is contained in the bilinear forms A(q2), B(q2) and C(q2) of the nucleon form factors Fi, Y2 and Ga itself. The latter is assumed to be the only unknown quantity, parameterised empirically in terms of an 'axial mass' Ma (or, equivalently, in term of an 'axial radius' ta = \/T2/Ma) a (q A(g2) ____________ gA(0) (1-q (1.1 where Qa[0) = 1.2670±0.0035 [25] is the axial coupling constant. Figure 1.1 shows the available supply of values for Ma obtained from these studies. References [17,19,20,23] reported severe uncertainties in either knowledge of the incident neutrino flux or reliability of the theoretica input needed to subtract the background from genuine elastic events (both of which gradually improved in subsequent experiments). Their reported values fall well outside the range o values known today and exhibit very large statistical and systematical errors. Following th data selection criteria of the PDG ([24], p. 9) they were excluded from this compilation. Argonne ( Argonne ( CERN ( Argonne ( CERN ( BNL ( BNL ( Argonne ( Eernniab ( BNL ( BNL ( Avi 1969) 1973) 1977) 1977) 1979) 1980) 1981) 1982) 1983) 1986) ^^ 1987) ^~ erage r 18] 14] 1] [ [2 [9 15] 13] 16] 10] 1] 1 1.2 M, [GeV] Figure 11: The axial mas Ma as extracted from quasi-elastic neutrino and anti-neutrino scaterin experiments (the ferens for the individual data points ar given on the right) The weighted average with statistical and systematical errors (if thy have been specified eparately) added in qudrature i Ma = (1017 ± 0019) GeV, or (1017 ± 0023) GeV using the scaleerror averaging recommnded b [24]. A brief descriptio given in Appendix A 1. Another body of data comes from charged pion electro-production on protons [26, 27, 28 29, 30, 31, 32, 33, 34, 35] not far above the pion production threshold. As opposed to neutrin scattering studies which can be covered by the basic Cabibbo theory, extracting the axial form factors from electron scattering measurements requires a more profound theoretical picture o the process, involving specific models of the nucleon structure; the situation is reviewed in chapter 5. The results of various approaches are shown in figure 1.2. Note again that reference [3637] were omitted from the fit for lack of reasonable compatibility with other results. Frascati (1970) 8] GE" = 0 Frascati (1972) _^ DESY (1973) -------»------ [30] Daresbury (1975) SP -------,------ [31] DR FPV ----------o---------- BNR —0— Daresbury (1976) SP ^_ DR r^ BNR ----------3---------- DESY (1976) [33] Kharkov (1978) ----0---- 6] Osson (1978) [27 Sacay (1993) [35] Average T 0.9 1 1.1 1.2 MA [GeV] Figure 1.2: The axial mass Ma as extracted from charged pion electro-production experiments (the rerences for th individual data points ar given on the right). The weighted average with statistical and systematical errors added in quadrature (if they have been specified separately) is Ma = (1068 ± 0015) GeV, or (1068 ± 0017) GeV using th scalederror averaging recommended b [24. Although the results of the individual axial form factor determinations deviate substan tially from one another, there seems to be a significant difference of AMa = (0.051 ± 0.024) GeV between the neutrino and electron scattering weighted averages. But it seems obvious from fig ures 1.1 and 1.2 that at least in the older experiments the systematical errors were grossly under estimated and that the ±0.024 GeV deviation of AMa is too small. The scaled-error weighted averaging of the PDG gives a larger deviation of ±0.028 GeV. Similarly, making an 'iterative' weighted average of the uncertainties by using the deviations of individual values of Ma from the calculated weighted mean, we obtain Ma = (1.017 ± 0.029) GeV from neutrino scatterin experiments. From electron scattering experiments, we get Ma = (1068 ± 0.023) GeV, and s the difference is AMA = (0.051 ± 0.037) GeV. This 'axial mass discrepancy' and its inconclusive uncertainty were in the focus of our in vestigation of the reaction p(e, e'7t+)n, and one of our principal aims was to perform a mea surement accurate enough to show whether the discrepancy is genuine and, according to th result, to either make a claim in favour or against the 0[q2) prediction of xPT for AMa. Th meaning and possible physical background of this discrepancy are discussed in section 2.2 (and later in section 5.3), and the pion electro-production formalism is reviewed at the beginning o chapter 2. The experimental part of this thesis deals with the p(e, e'7t+)n coincidence experiment per formed by the Al Collaboration at the Institute for Nuclear Physics, University of Mainz. In this experiment, our intention was the extraction of the q2-dependence of the nucleon axial form factor by means of an effectiv Lagrangian model [38] based on the formalis of [40 41] 1. Inroduio and [42]. The measurement was performed at the invariant mass of W 1125 MeV and at vi tual photon four-momentum transfers q2 of —3, —5 and —7 fm~ , in order to be able to perfor a q2-fit of the model calculation to the data. The experimental equipment used in the experiment (the electron accelerator, the cryogeni liquid hydrogen target, the magnetic spectrometers and the detector systems) is described in chapter 3. Chapter 4. is entirely devoted to the data analysis. Several cuts and correction factor were applied to the raw experimental data to generate the missing mass spectra containing true coincident events within their peaks. These spectra are normalised to the simulated detector ac ceptances and the calculated luminosities to yield the reaction crosssections in each measure setting. For each q2, the transversal and longitudinal parts are separated from the measure crosssections by the Rosenbluth technique. The axial form factor and the pion charge form factor can then be extracted from the q2-dependence of these crosssections using specific theo retical models, and some of the past attempts using data from inclusive electron scattering an pion electro-production on proton targets are reviewed in chapter 5. Emphasis is finally give to the effectiv Lagrangian approach used in our own analysis. A great deal of motivation to study the nucleon axial form factor originates in the 'axia mass discrepancy' described above, but the knowledge of GA(q2) can also assist us in find ing out its proper theoretical understanding, and in particular of the axial coupling constan gA = 9a(0)- Chapter 6. therefore starts with a historically annotated formal introduction t the axial coupling constant and reviews some general theoretical concepts and problems. In the second part of the chapter, we calculate Ga(q2) in the framework of the linear a-mode (characterised by an extraordinary strong pion cloud surrounding the quark core), and the chi ral chromodielectric model (which possesses a peculiar QCDinspired, dynamically generated binding field for quarks), and investigate to what extent the elimination of spurious centre-of mass motion and recoil effects improves the agreement of the model calculations with data. Chapter 7. summarises the results of the thesis. 2 eneral formal We can describe all reactions of the A(e, e'X)B kind in which the scattered electron is detected in coincidence with the ejected particle X (e. g. one nucleon knockout reactions or pion electro production off nucleons), with similar formal tools. This chapter describes the formalism o electron coincidence reactions, based on the assumptions that electrons may be described b plane waves and that a single virtual photon is exchanged between the scattering electron and the hadronic system. This planewave Born approximation, which is believed to be appropriate for electrons scattering on light nuclei, enables us to separate the physically interesting conten of the electro-magnetic interaction in the hadronic vertex from the well known interaction i the electron vertex. First I display the kinematics of the p(e, e7t+)n reaction. Then I show that a general expres sion for the differential reaction cross section can be obtained by contracting the leptonic an hadronic tensors, which are bilinear forms of electron and hadron electro-magnetic transition currents. The separation of the transverse and longitudinal parts of the measured differentia reaction crosssection was one of the main goals of this thesis, and a commonly used procedur to perform the separation is presented next. At the end of the chapter I try to explain why both experimental and theoretical studies of pion electro-production off nucleons are currently highly physically motivated. All derivations within this chapter are largely in the spirit of [43, 44]. The conventions o jorken and Drell [45] and Drechsel and Tiator [38] are adopted throughout the thesis. 2. The expression for the reaction cross-sectio If the electro-magnetic process of pion electro-production off nucleon e(pe) (pi) - [v n{pn) (pf) is treated in the plane-wave Born approximation, it can be visualised as shown in figure 2.1 where momentum four-vectors of all the particles involved are listed. The incoming electron in the state | e) goes to th final state | e') whereas the target nucleon | i) absorbs the virtua photon and becomes | f). When the virtual photon energy transfer exceeds the pion production threshold, a pion | n} can be emitted from the nucleon. The virtual photon then transfers energ and momentum q = (o>, q) = pe — P — V Vn ~ Pi to the nucleon w = Ee - E = E En - Ei I = P - P = V ~ P 2. General formali The differential reaion roseion in the laboratory rame for such a reaion can be wten in the standard form [45] 1 "F= m d3pd3p d3p 4 (4] where |veil = | = |pe|/Ee is the relative velocity between the target nucleon and the incoming electron, l-Mgl is the (complex) square of the invariant matrix element for the process un der consideration, the 6'4^ function expresses overall energy and momentum conservation an where phase spaces of all outgoing particles were taken into account. When neither a polarised incoming electron beam is prepared nor the polarisation of the outgoing particles is measured the reaction cross-section has to be averaged over the initial and summed over the final electron and nucleon spins, which is denoted by th symbol Figur 21: Th eaction e e'7 in the planwave Born approximation. In an exclusive reaction like p(e, e7t)n only the scattered electron and the outgoing pion are measured in the final state. The recoiled neutron is not detected. Integrating the expression for da first over pf and then over \pn\ we eliminate the angular and momentum dependence o the crosssection with respect to the recoiled particle. This integration introduces recoil facto fe into dcr, so tha da M Pe O (27r)5M wher culJ-Elqlcose M and 0^ is the angle between q and pn. The coordinate system used to discuss p(e,e7r)n reaction kinematics is shown in figure 2.2. Momenta of the incoming and of the outgoin electron define the scattering plane. The reaction plane, tilted with respect to the scattering plane at an angle of qb^, is in turn spanned by the momenta of the virtual photon an th outgoing pion. To a reasonably good approximation of the pion electro-production process, the electrons interact with the hadronic system by exchanging a single virtual photon. The Lorentzinvariant matrix element Ma can therefore generally be written as a product of the electron electromagnetic current, the photon propagator, and the hadronic electro-magnetic transition current In the conventions used we hav = [-u^^u^pe.e)] ^f- [er(q)] (2.1 2. Scattering plan rP Figur 22: Th coodinat sysem use t escribe e e'7n eaction where Ue are standard Dirac spinors for electrons with four-momenta pe and spin se. Here P(q)fi is the four-vector of the electro-magnetic transition current for the hadronic system receiving four-momentum q from the virtual photon (see (B.l)). The square of the invariant matrix element, averaged over initial spins and summed over final spins, is equal to the con tractio }M J^ WP;Pe,e)H^(q) of the leptonic and hadronic tensor (P;P) = ^iv btP ) ]* (P )Y(p ) ] The leptonic tensor is exactly calculable i QED. In the case when polarisation of the fina electrons is not measured, it is equal to [45] lP Pe> eJ — Pe Ve y In this case the differential reaction crosssection in the laboratory system is equal t do M pe whe O^ 67M (qf (2. and where we defined vo = (Eg + Ee) — | q |. Individual vKs are called electronic factors and the corresponding TlgS are the hadronic structure functions. The labels k denote the longitudina and transverse components of the polarisation of the virtual photon, which in turn correspond to the components of the hadronic transition current with respect to the direction of q. In th 2. General formali extreme relativistic limit q2 : rewrite (2.2) a da 4EeE sin jde an vo = 4EeE cos2 jQ, and it is instructive t O 1 sil M T 67 wher e is the electron scattering angle and the ter in brackets is the usual Mott crosssection. The essential advantage of the factorised notation is that all dependence on the kinematica settings of a chosen measurement is carried by the electronic factors whereas the dynamics and the physical content of the process under consideration are stored in the hadronic structur functions. These are bilinear forms of components of the hadronic current four-vector, which can be directly extracted from an experiment by choosing appropriate kinematical condition (energy of the incoming and of the outgoing electrons, electron scattering angle). In the cas when we are not concerned about spins of the final particles and an unpolarised electron beam is used, only four electronic factors multiplying four corresponding hadronic structure func tions enter the formalism. In the laboratory system we then have the electronic factor vT — q q ta -j vl = q2 lql q2 VLT = w q2 0 q2 VTT = |q? ucture functions ar u(q) ir(q) K |p(q) ^ -2Rp*(q)(J(q) .R (q)(q)]. We have written the components of the hadronic transition current four-vecto = (p, J) in th (P> ]±> Jz) basis and used current conservatio (q) ^p(q) Iql In the treatment of pion electro-production we usually do not define hadronic states with respect to the laboratory frame, but with respect to the centre-of-mass system (CMS) of th final hadrons; in our case, this is the system of the ejected pion and the recoiled neutron (from now on, we shall label quantities in the CMS with a *). The transition to the CMS involves Lorentz boost along q and it can be shown [44] that this amounts to replacing the energies and momenta in the laboratory frame with their CMS values and to multiplying the electronic facto vl by (W/Mp)2 and vlt by W/Mp. The form of the Lorentz scalars is not affected, although th energy and momentum transfers o> and q in the laboratory frame are replaced by their CM counterparts (the polar angle 9^ is also changed to 0*, while the azimuthal angle cjj^ = (L>*) In pion electro-production off nucleons we can safely set Mn = Mp M so that the factor /47tW2 and e2 - 4ncx can be absorbed in the structure functions. It is also customary t 2. General formali express all kinemacal fators wth degrees of ransvese and longtudinal polasaion of th virtual photo [6] 1 |q *2 and to define th virtual photon flux 2? l It can be understood as the number of virtual photons exchanged between the hadronic system and the electron beam, emitted into the infinitesimal energy interval dEe and the solid angl interval dOe. Here QY = (W2 - M2)/2M = WQ*/M is the energy which a real photon should have had in order to excite a pion-nucleon final state with an invariant mass of W (the quantity QY is also known as the equivalent photon energy). The differential reaction crosssection ca then be rewritten in a factorised form in the CMS a da da da (2. The separation of the electronic factors and structure functions differs slightly from (2.2), bu its physical content is the same. The response of the hadronic system in the observed proces is contained in the differential reaction crosssection da dO L W2Z L{l)ReW ^-W in which the structure functions still encapsulate the components of physicall interestin hadronic currents = 4ttW Bilinear combinations Wp of hadronic currents carry an implicit angular dependence on the azimuthal angle cjj = <$>n and on the polar (scattering) angle 9. For our purposes, it is con venient to write out the (^-dependence explicitly and to suppress th 9*-dependence for th moment Let us renam i{wvr "(WXX- W*zz = Rl ReW = RLT cos^n Wyy) = RTT cos 2^ an = (Q ) da/ {T, LT, TT}, so that finally da dO dO (1 ^f cos^ cos 2n. (2. 2. General formali Motiat te saton o te stct ct d extcton o It is shown in appendix B, where the structure functions are expanded in terms of the electro production multipole amplitudes and of the pion scattering angl 0*, that in the first order o the multipole expansion dcu/dQ.^ ~ sin 0* and dajj/dQ.^ ~ sin 0*. This means that a mea surement in parallel kinematics, in which the centre of the hadronic spectrometer's acceptanc is aligned with q (sin 0 = 0) enables us to disentangle the linear combinatio ^ (2-5 from the measured cross-section, whereas the two interference cross sections vanish. At chosen energy and momentum transfers cu and q|, the transversal and longitudinal cross sections ca be separated by measurements at different values of L by means of a straightline fit The L/T separation of the electro-production crosssections closely resembles the 'classi cal' Rosenbluth separation in inclusive electron scattering experiments N(e, e')N. In order to increase the accuracy of the separation, the eL has to span a lever arm as large as possible. But usually the range of its values is kinematically constrained by the mutual placement of the electron and the hadron spectrometer and by the maximal energy of the electron beam. Large values of eL correspond to smaller electron scattering angles, which are usually difficult to achieve experimentally due to the proximity of the exit beam pipe. On the other hand, smal values of L can be reached at large scattering angles, where the virtual photon flux Vv strongly decreases . Especially in the early days of low dutyfactor accelerators and detectors of mod erate performance, results from electron coincidence experiments therefore suffered from larg statistical and systematical uncertainties. In spite of that, numerous measurements of charged pion electro-production were perfor med already in the 1960s and 1970s at the accelerator facilities in Hamburg, Daresbury, Frascati and Saclay: total or inclusive crosssection for p(e, e')p in the region of the A resonance [46,47 48]; total crosssection for p(e, e')p in the vicinity of the pion production threshold [26, 28, 29 37] and in the A resonance region A [49]; the differential reaction cross-section for p(e,e'p)7t° [50, 51, 52, 53, 54, 55, 56, 57] and p(e, e'7t+)n [51, 58] near the A resonance; and the differentia reaction crosssection for p(e, e'+)n close to threshol [530, 59, 60] All these experiments, exclusively employing proton targets, were mostly aimed at mea surements of angular distributions of the recoiled protons. Although electron accelerators with high peak beam intensities were used, coincidence measurements were strongly impaired b sudden particle bursts in the detectors due to the accelerators' pulsed mode of operation. In addition, experimental requirements in pion electro-production are not the same as in (e, e'p) reactions: for example, to discern among tv^ and e±, one needs highly efficient Cerenkov de tectors in both spectrometer arms. Poor statistics practically did not allow the separation o electro-production cross-sections. For these reasons, pion electro-production experiments were completely halted after 1978. Figure 2.3, containing all available data on transverse and longi tudinal cross-sections for p(e, '+)n to that time, shows a notable exception of [5] (and, condi tionally, of [60]) 2The magnetic spectrometers ued to detect and anayse the particles emerging from a coincidence reactio possess only finite momentum and angular acceptances. Although it is of no significance to the 'point-wise' L/T separation described here, one has to be aware that the extracted cross-sections are therefore implicitly averaged over certain kinematical variables, e. g. over dispersive and non-dispersive angles. If one wants to avoid this by narrowing the acceptances one either needs to increase the total measurement time or cope with larger statitical error 2. General formali d *2 dO - - tt--------1----------r d dO .2 0. 0. 0 -q2 [GeV/c2] -q2 [GeV2/c2] Figure 23: Longitudinal and transverse cross-sections da^/dD.^ and da-f/dD.^, extracted from wo pee'7T+)n coincidence experiments performed before 1978: o [5, • [60 (for the lattr, the dffL wa deermined by assuming a fixed calculated value of doj)- Both measurements wre don in parall kinematics (9* = 0 with W = 1175 MeV The crossections ar in units of usr With modern, high duty-factor electron accelerators and with new, large-acceptance and highresolution magnetic spectrometers, coincidence electro-production experiments were re born. In particular, coincidence pion electro-production experiments on the proton newly be came interesting and feasible. For example, the interest in the L/T separation in charged pio electro-production was revived by the fact that in the charged channel, e + p —> e' + n + 7r+ the transverse crosssection close to threshold is dominated by the electric dipole amplitud (see appendix B and appendix C for its definition), which can be directly relate V2(^ to the axial couping contan A of the nucleon. A(0) and to the axial form fator Ga( ) = 9a(a(0) Although the axial-vector coupling constant appears to be a textbook topic, its experimenta value seems to have been stabilised only recently. In fact, as pointed out by [64], the value o the axial coupling 'constant' increased for as much as 7% since the early measurements in 1959, until the modern measurements in which the total relative experimental errors do no even exceed 0.5 % [65, 66, 67, 68, 69]. The issue at stake is nontrivial since pinning down the precise value of Qa{0) also enables us to give a better estimate of the validity of the Goldberger Treiman relation [61] A(0)=tN(0) (2.6 where g^NNtO) is the pion-nucleon coupling constant and in = 92.4 MeV is the pion decay constant ([25], p. 353.) The Goldberger-Treiman relation is mandated by the chiral symmetry of QCD and is generally believed to hold to a level better than 10%. Since the g^NN in (2.6 should be evaluated at zero momentum transfer (which is unphysical), it is more appropriat to examine the Goldberger-Treiman discrepanc 1 MgA(0) 1 gTtN(Q) tN (2.7 Unfortunately, the chiral perturbation theory does not predict An. At the 0(p) of the gen eral 7rN Lagrangian, (2.6) remains exact; but even at the C(p3), pion loops do not modify the C(p2) expression and the value of An is essentially an input to the theory [62]. However the Dashen-Weinstein theore [63] relates An to the discrepancies defined analogously for th 2. General formali ev an A — ev proceses. It can be expresed as a 'sum rule' n = / f k m md f „ m m 9ak 9ikn ----------- —F~----------- tNN tN whe fKgAKN fKgiK and where gAKN(in-K) = —13.5, giKN(Tn-K) = 4.3, g^(0) — —0.72 and g^(0) = 0.34 as quoted in [64]. Further consideration depends on the value of g^NN^n)- With the 'traditional' valu of 13.4 [70], we get An = 0.041 o m/(m ma — 48, whereas with the new value of 13.05 [71], we obtain An = 0.015 or ms/(m.u + d) 17. The latter result strongly favours th conventional xPT picture predicting 2ms/(mu + m^) = 25 to the generalised version predicting ras/( + m^) < 25. This is where the value of gA(0) comes into play: the smaller value o QnNN^n) and the current average experimental value of Qa(0) = 1.2670 ± 0.0035 [25], hint a a validity of (2.6) to a level o 1 %, which is much smaller than the usually quoted 10 %. Similar historical developments could be observed in the understanding of the axial-vector form factor of the nucleon (see chapter 6. for a review). In the past few decades, its q2 dependence emerged from a lively interplay between new experiments and theoretical model which were initially formulated for photo-production of massless particles. The statements about low energy photo-production of massless charged pions can be traced back to the Kroll Ruderman result [72]. Considering only conservation of electro-magnetic current, they obtained the threshold electric dipole amplitude in the limit of vanishing pion mass (physically, ran = uM. with u 0.15) and neglecting the terms linear in the photon momentum | q | (| q | u) In this limit, the charged pion photo-production amplitude is fixed by the pion charg kr egnNN n-> g^N 87(1 jt) 87 ( and correspondingly vanishes in the case of neutral pion photo-production. The theorem was subsequently extended to virtual photons by Nambu, Lurie and Shrauner [73, 74], who ob tained the general result for the isospin (—) threshold electric dipole amplitud « = 0,) = 1 q2 nl ^{G*(" ^2?^'} and its momentum deviation 6. In first order of the magnet optics the dipole magnets are the only dispersive elements. Ideally, this means that particles wit equal initial coordinates but different momenta will be displaced at the exit of the magnet. On the other hand, trajectories of particles with equal momenta, but different initial angles, wil be focused to the same point. In reality, in terms of transfer matrix formalism, the first orde matrix elements () (magnification) an 6) (dispersion) are large, whereas (0) is small The quadrupole magnets are the first magneto-optical elements following the entrance collimators of spectrometers A and C. They are non-dispersive to the first order, and couple dis persive coordinates x and 0 and non-dispersive coordinates y and (j) through (x|x), (x|0), (0|x) (0|0), (u|u), [y\$>), [4>\y] and (4)|4)) matrix elements, whereas there are no matrix elements in volving 6 to first order. The quadrupoles are positioned in such a way that particle trajectories are defocused in the dispersive plane and focused in the non-dispersive directions. Thereby transversal angular acceptance of these spectrometers is significantly increased. If kinematica settings allow us to use them as hadron spectrometers, their high non-dispersive angular accep tance can play a key role in studying angular distributions of the produced hadrons. However there is a certain trade-off involved in gaining large angular and momentum acceptances: i leads to large angular divergences (as much as ±12 with respect to the reference trajectory) o particle rays at the exit of the second dipole. Consequently, the tracking and the time-of-fligh detectors positioned close to the focal planes of the spectrometers have to be relatively large. The nonzero inclination of either the entrance or the exit pole faces with respect to the ref erence trajectory contributes to dispersive matrix elements involving x and 0. Quadrupole defocusing in the dispersive plane is therefore compensated by the edges of the dipole mag nets, inclined with respect to the normal axis of incidence. The inclination angles are such tha defocusing in the dispersive plane is minimised, whereas focusing in the non-dispersive direc tion remains preserved. Additionally, the edges of the dipole magnets are slightly curved, and this curvature introduces additional sextupole strengths. The sextupole magnet between the quadrupole and the first dipole with a curved entrance boundary is used to diminish spherical aberrations in the non-dispersive plane due to second-order (2) and (2) matrix elements. Spectrometers A and C operate in the pointto-point imaging mode in the dispersive plane ((x|0) = 0 is necessary for high momentum resolution) and in the parallelto-point mode in the non-dispersive planes (() = 0 enables high angular resolution). Relatively small spa 3. Expeimental setu tial (vertex) resolution of spectrometers A and C is compensated for by spectrometer B whic operates in the pointto-point imaging mode in both planes, having a very good angular and momentum resolutions, but smaller angular and momentum acceptances (see table 3.1). Th magnetic system of spectrometer B is only one dipole ('clamshell') magnet with inclined and curved edges. Only very slight entrance and exit inclinations and curvatures were necessary t eliminate second- and higher-order aberrations (see [89, 90, 91] for details) The magnetic field in the interior of the spectrometers is measured with builtin Hall an NMR probes. Each dipole magnet is equipped with one Hall probe and four NMR probes, each covering a certain range of field values from 0.09 to 2.1 T. Since the NMR probes in spectrometers A and C are attached to the inner walls (approximately 60 mm from the spectrometers' magnetic mid-planes), the measured values differ from the values encountered by particles in the vicinity of magnetic mid-planes. Although the corrections to the magnetic field B do no exceed «2-104 T for reference fields below « 1.2 T or « 10-3 T for fields above « 1.2 T, they were taken into account by gauging th NMR probes in terms of polynomials in B [92] Figure 6 mgnetic system a the dector pge o spetrmeter A (te magnetic ystem of spectromr C is basically the sam as in spectrometer A scaled by a factor of 13/17, whereas spectromee B has only one dipole magnet) All thre spectrometers can be rotated around a common point (in the lower riht corner of the figur bove whic th scatering chamber containing the target ell i installe. The magnetic field of spectrometer B is too inhomogeneous for the NMR probes to operate (to reach the resonance, the inhomogeneities of the magnetic field density 6B/B should no exceed the level of 2.5 • 105). To overcome this problem, a miniature printedcircuit board quadrupole 'magnet' was placed around the NMR probe with a field gradient opposite to th dipole field gradient, effectively diminishing the local inhomogeneity to less than 1 • 10 [92]. The uncertainty of the field measurements with the Hall probes is much higher than th uncertainty of the NMR field measurements, which is in fact smaller than the energy sprea AE/E« 2-104 of the incoming electron bea [93] 3. Expeimental setup When charged particles traverse the spectrometer's magnetic system, they enter the detector packages. In the focal plane of the spectrometers and parallel to it, there are two pairs of ver tical drift chambers, followed by two layers of scintillation counters and a Cerenkov detecto at the top of the package. Figure 3.6 shows the position of the detector package installed in spectrometer A and figure 3.7 shows the same detector package in some more detail (all figure of the detector systems were made by A. Liesenfeld of the Al Collaboration) Figure 37: The detector sytem spectrometer A (the detector sytem o spectrometer B is qualit tively identical). Particl trajectores are dtermined by two pairs of vertical drift hambers; particles ar identiie and distinguished by wo segmented scintillation counters an the Ceenkov dtector Vertical drift chamber When the particles pass through the magnetic system, they traverse the focal plane. In all three spectrometers, the focal plane is tilted by an angle of about 55 with respect to the horizonta plane, and is traversed by the particles at incidence angles between 33 and 54. The averag useful area of the focal plane is approximately 185 cm x 38 cm in spectrometer A and 190 cm x 11 cm in spectrometer B. Drift chambers installed in the focal plane enable us to reconstruc the particle impact points and directions of their flight. Vertical drift chambers are most suited to this purpose: due to the characteristic configuration of the electric field, electrons fro th electron avalanches drift vertically, i e. perpendicular to the chamber plane. Vertical drift chambers play a key role in determining the reaction point in the target cell Four chamber layers, in which sequences of signal wires are rotated relative to one another [91,94, 95], enable us do determine two points of the trajectory and therefore the particle fligh direction. Knowing the imaging properties of the magnetic systems it is then possible to accu rately reconstruct the coordinates of the particle and its momentum on target. The resolution design values from table 3.1 can be achieved with spatial resolution of the vertical drift cham bers of < 200 um in the dispersive direction and < 400 u in the non-dispersive direction. 3. Expeimental setu Scintillation count The scintillation counters and the Cerenkov detectors constitute the triggering system of th magnetic spectrometers. The scintillators are used to identify the particles and to distinguish physically interesting events from the unwanted background (accidental coincidences from concurrent processes, detector noise, cosmic rays and other radiation background). At th same time, scintillation counters are used in all timing tasks: they are used in determination of particles' time of flight from the target to the focal plane and to trigger, time or gate the signals from other components of the detector package and of the data acquisition system. Moreover, high quality of triggering signals from the scintillation counters is crucial for a goo coincidence time resolution. Figure 3.8: Scintillation counters of spectromete A. e dE layr i sed t distinuis protns f minimum ionising particles The ToF layer i use to measur th ti of fliht The scintillation counter package consists of two layers, as shown in figure 3.8. The firs layer (in the particle's flight direction) is the 3 mm thick dE layer, which is used to distinguish protons from minimum ionising electrons, positrons and pions. In this layer, protons generally deposit much more energy than the minimum ionising particles and the two families of parti cles can later be separated by appropriate cuts in the ADC spectrum. The dE layer is followed by the 10 mm thick ToF layer made of a much faster scintillation material as it is used to mea sure the time of flight. Both planes are segmented into paddles: light pulses from each paddle are read out by two photo-multipliers attached to the both sides of the paddle. As an excep tion, the narrower paddles of spectrometer B are read out only on one side. Segmentation o scintillator layers into paddles improves the timing resolution within a large detection volum since path lengths of light in the paddles is shortened. In our experiment, the scintillation counters together with the coincidence electronics [84] are used to separate positrons and pions from protons. The scintillation counters are not suit able to further distinguish positrons from pions since in our energy range of a fe 100 Me both are minimum ionising. For this purpose, we have to use Cerenkov detector Cerenkov detector The Cerenkov detectors are used to distinguish between electrons or positrons and pions. Since in our energy range, all these are minimum ionising particles, scintillation counters are not ca pable of their discrimination. In magnetic spectrometers of the Al Collaboration, the Cerenkov detectors are placed behind the scintillator arrays. The active radiator volume is filled wit Freon with an index of refraction of about n = 10012 (figure 3.9) 3. Expeimental setup _________________________________________________________ Figure 39: Èrenkov detector in spectrometer A. The Èerenkov radiation generated in the radiator ga (the trapezoidal activ volume in the lower part of the figure) is gathered by mirrors, mounte at the top of the assmbly and dected into photo-multiplers at it ees The momentum threshold for Èerenkov radiation is about 10MeV/c for electrons or posi trons and 2.4GeV/c for pions. In our energy range, only electrons or positrons can trigger Èerenkov signal, whereas the pions can not. Particles with |3n ~ 1 induce Èerenkov radiation directed predominantly into an angle of 0 = arccos 1/|3n ~ 0, i. e. practically along the particle trajectory. Èerenkov light radiated from the active volume is collected by mirrors at the top o the detector assembly, and focused onto photocathodes of photo-multipliers at its edges. To control the operation of the scintillation counters and the Èerenkov detectors, a specia laser monitoring system was developed [96]. During the measurement, pulses of laser light can be guided by optical fibres and fed to the scintillator paddles. Incident light pulses simulat the passage of particles through the scintillator layers. These 'pseudo-events' can then be use to monitor individual parts of the detector system and of the coincidence electronics. 3. Electronics After traversing the drift chambers, particles coming from the target first hit the dE scintillator plane and then the ToF plane, and finally reach the Èerenkov detectors. The physical processe in all these detectors lead to signals which constitute a part of the raw detector data. It is th duty of the electronic circuitry to read out, convert, convey, process and store these data. The trigger system complementing it is responsible for the decision on when (i. e. for which events these actions should take place. On all three spectrometers, the local logic circuitry is done i an almost identical manner (see figure 3.10 and [84]) Signals of the individual scintillator and Èerenkov photo-multipliers are first split into two branches. The signals from the first branch are led to the leading-edge discriminators, wherea the signals from the other branch are led via delay lines to the ADCs. These delayed signal arrive at the ADC inputs exactly at the time when the signals of the first branch will hav worked their way through the rest of the circuitry and eventually generate an interrupt to trigger the digitalisation process in the ADCs and the readout sequence. On each spectrometer the signals from the discriminators are given to the leftright coincidence units and then to th spectrometer LU. 3. Expeimental setu z> C r a R L Ih-i- • * lh* » • • • Figure 3.10: Trigger electronics of spectrometer A (similar on spectrometers B and C). A particle is shown traversing the detectors: • - VDC signal wires of all four VDC wire planes, dE, ToF - scintillator planes, Cer - Cerenkov detector, Top - top scintillator. The detector signals are handled by: D - leading-edge discriminators, & - logical units (OR, AND, or programmable gates), PLU - Programmable Logical nit, uB - u-Busy module ADCs and TDCs. Various delay lines are not shown (see text for details) rimnt et The basis of the PLU opation is the so-called 'lookup mechanism. At the time when the strobe signal is provided, the PLU inputs are read out and interpreted as a memory addres containing the appropriate output pattern on the output. During this operation, the PLU is insensitive to its input until the unit is cleared by its sync-output. This unambiguous input output mapping is achieved by preprogramming. In he standard operation mode, the spec trometer PLU is programmed as de_AND_tof _OR_ch_OR_top, meaning that coincidences be tween at least one paddle in the dE and at least one paddle in he ToF layer are required In thi mode, the strobe for the PLU must essentially be defined by ToF and it is critical hat he To and he strobe signals coincide in time jjj j {|| Figure 3.11: Coincidence electronics. The Coinc PLU is fed by single-arm signals A Si, B Si and C Si of spectrometers A, B and C (in addition, single-arm signals are prescaled (P) on the Coinc PLU input), and by signals A uB, B uB and C uB of (iBusy modules from individual spectrometers. The event module synchronised by A Clk measures dead time by the Dead subunit, triggers A, B and C interrupts, stop the spectrometer TDCs, supplies event info in case of a vali event an starts an stop the coincidence TDC (see text for details) After the spectrometer PLUs, and in the most general case of triple coincidences, we have three signals leaving the individual spectrometer PLUs: the not-scaled singles or shortly sin gles A, singles B, and singles C. These are led to the coincidence PLU, in addition to the scaled singles which first enter the prescalers where they can be optionally scaled down. The coin cidence PLU (figure 3.11) also needs a strobe signal defining exactly when the input will be mapped to output. The strobe for the coincidence PLU comes from ORed signals of individual spectrometer PLUs. The timing is adjusted in such a way that Spec B wins if there was a signa from Spec B; as a consequence the timing after the coincidence PLU is determined by Spec and the signals for Spec A and Spec C need to be retimed 3 3Retiming (as an example we consider here only retiming for Spec A) effectively means that the nocaled single from Spec A have to be ANDed with the interrupt signal for Spec A. It is crucial for Spec A and Spec C signals to be retimed. In particular the stop signal for the VDC TDCs started by the dicriminated wire signals must originat 2 rimnt et The coincidnce PU works he same way as the individual spectrometer PLUs: it uses the lookup mehanis to determine the output from the given state at its input. The out put depends on ho he coincidence PLU is programmed. The standard programming is triple_w_double_sc_w_single_sc, meaning that in general, triple coincidences, (option ally down-scaled) double coincidences and (optionally down-scaled) single events will be pro cessed and passed on. As soon as the coincidence PLU generates an interrupt (i. e. recognises a valid event), the event counter of the event module is incremented. At the same time, the interrupt signal and the information on the event type are distributed from the master spectrometer to slave event modules on other spectrometers involved. These interrupts actually trigger the readou process. Next, the uBusy flip-flops on the spectrometers are set; this halts further data taking while the electronics is busy. On each spectrometer involved, te interrupt signal also generate a gate for he detector ADCs and starts the ADC timer. The photo-multiplier signals from the detectors that were hit by the traversing particles, have exactly reached the ADC inputs by now after being appropriately delayed (see above). Within the time gate opened by the interrupt hese signals are integrated and the digitised values are fed into the data stream. Similarly the TDCs of the VDCs hat were started by the discriminated signals from the signal wires, ar stopped by the retimed spectrometer interrupt. In addition, there are TDCs that measure th time of flight from the dE to the ToF plane and from ToF to the Cerenkov detector, and these ar also started and stopped analogously. Only when all these tasks are properly done and dat acquired, the spectrometer p.Busy flip-flop is reset, thereby unlocking the coincidence PLU, an furher data taking is allowed. exactly from the photo-multiplier signal belonging to the correspnding scint Spec A and Spec C interrupts are not directly appliable; they mu be retimed 4 Data analysis In this chpter I review the analysis of the data for p(e,e'7t+)n, measured at the invarian mass of W = 1125MeV, at virtual photon four-momentum transfers of q2 = — 5fm~ an —7fm~ . The data analysis effectively amounts to deducing physically relevant informatio pertaining to the interaction point (like particles' momenta or emission angles) from the raw data (e. g. wire chamber drift times, scintillator detector hit patterns and signal levels, and Cerenkov detector signals). From the particles' momenta, further physically relevant spectr (e. g. the missing mass or the neutron recoil energy) can be generated and used for particl identification and background reduction. 4. Coincidence time The timing of a double-arm coincidence experiment employs the discriminated photo-multipli er signals of the spectrometers' scintillator detectors. The primary means used to isolate tru coincident events from the data is the coincidence time spectrum, which should essentially contain the differences between the times of flight of the associated particles through the first and he second spectrometer This time difference is digitised by he coincidence TDC module which is started by one of the spectrometers and stopped by the oher. In our experiment spectrometer A was always used to start and spectrometer B to stop TDC and the tempora sequence of he timing signals can be viewed on as in figure 4.1 I I I ^___________±______________J I I Figure 4.1: The temporal sequence of the timing signals from spectrometers A and B. The signal from spectrometer A opens the coincidence gate of width T; all signals from spectrometer B arriving within this interval (say, after t trigger coincidenc signal. The value of tc is then digitised by the TDC module 2 3 _______________________________________________________________ Data alsi Ideally, the difference between the time of flight of an electron throuh one spectromete and the time of flight of a pion through the other spectrometer, both coming from the sam reaction p(e,e'7)n, should be constant for all events. However, due to particle momentum dependence of the time of flight, there are certain deviations from a constant difference. Nevertheless, a genuine coincident pion should trigger a signal with a fixed time relation to he signa triggered by the corresponding electron, whereas an uncorrelated pion and electron should no have any particular signature. In practice, the histogram of all events in tc (the coincidence time spectrum) therefore exhibits a peak at a certain average value of tc corresponding to true coin cidences and a continuous background of accidental coincidences spanning the whole width of the coincidence gate. The observed width of the background of accidental coincidence is equal to the width of the coincidence gate T. For purposes of our experiment, it was set t T = 80 ns in order to accommodate a broad range of pions' times of fliht w -L 1000 Z3 o 800 600 400 200 ~ 0 200 400 600 800 1000 1200 1400 Raw tc [TDC channe] Figure 4.2: The raw coincidence time TDC spectrum for kinematics 437. One channel corresponds t 100 ps. The covered range is equal to the width of the coincidence gate, i. e. about 8 n an the FWH of the peak is 6. ns The raw coincidence time spectrum in figure 42 is an accumulation of raw coincidenc signals and has a large width. The observed width is mainly due to different velocities an different path lengths (and therefore varying times of flight) of particles from the target to the ToF scintillator layers. In the case of spectrometer A, for example, the path length differences can be as high as ±1.5 m with respect to he central trajectory. The raw time peak is addition ally smeared because of the processes in the scintillators (time required by light travelling from the particle impact point towards the photo-cathodes of the photo-multipliers, intrinsic resolu tion of the photo-multipliers) and in the detector electronics (modules, cables and delays). The broadening of the coincidence peak can be reduced by software corrections. Since the optica properties of the spectrometers are well known [91] and since the VDCs provide enough information to reconstruct the particle trajectories, the path length differences relative to the centra trajectories and the corresponding 'correction' to the coincidence time can be calculated. In ad dition he particle's point of impact in he scintillator paddle can be determined and correcte for The correction of he coincidence time due to different path lengths has an astonishin influence on the width of the coincidence peak. The effect can be demonstrated by comparin two dimensional spectra of the dispersive coordinate x in the spectrometer focal plane vs. th coincidence time without and with he time of flight correction (figure 4.3 for (228 ±17) MeV/c pions in spectrometer B). Pions with the highest momenta rea the upper edge of the foca plane (x —> 1000 mm) as mu as 14 ns later han the pions wi he lowest momenta reachin Data alysi the lower edge (x —> — 1000 mm). By correcting for this differece, we make the pions 'sem' a if they all passed the focal plane at almost equal times (note hat such a correction als has t be done in he electron ar of he setup) j= 1000 1000 800 1000 1000 40 20 0 20 40 t0 (uncorr) [ns] o o 600 - 400 200 aa 2000 - 1500 - 1000 500 40 20 0 20 40 tc (uncorr) [ns] 40 20 0 20 40 L [ns] Figure 4.3: Correlation between the coincidence time and the dispersive coordinate x in the focal plane of spectrometer in kinematics 437: al - without and a2 - with the path length difference correction. The FWHM of the corrected coincidence peak is 1.6 ns (b2) compared to the uncorrected width of 6. n (bl) Note that the individual paddle tim offsets were already accounte for It is imperative to correct the coincidence time spectrum for all described effects and thereb optimise the coincidence time resolution. Namely as stressed in (3.1), it is necessary to keep th ratio between the true and accidental coincidences as high as possible, especially in coincidenc experiments with small reaction cross-sections. Otherwise, the coincidence peak can be masked by te accidental background and he examined reaction can not be reliably isolated. As it wil be shown in section 4.2, a cut in the coincidence time spectrum enables us to make a har separation between he true coincidences and background events 4. Missing mass Another means of identifying true coincident events and distinguishing them from accidenta coincidences is the missing energy or he missing mass spectrum. In nuclear reactions of the A(e, e'p) A — 1 type, the missing energy is defined as the energy which can not be directly measured in the coincidence experiment and is therefore "missing" in the overall energy balance The energy and momentum conservation Ee + Ea = Eg+Ep + EA-i andp+p = Pe+Pp+P-1 for su a reaction express ha TA-T Data alsi The kinetic energy Tp = (pp + M2)1/—M of the knocked-out proton is known, since we directl measure its momentum p , whereas the kinetic energy of he residual nucleus wih momentu Pa-i = q — p can be indirectly "measured" by Ta-i = ((q-P + mL1 1/ M This description is very illustrative in the case of one proton nock-out reactions where Em is also equal to the difference between the binding energies of the residual and the targe nucleus, and equal to the sum of the separation energy of the proton in the target nucleus and the excitation energy of the residual nucleus. But in an A(e, e'p)A — 1 process, the proton i already in he nucleus just befor he knock-out, wherea in p(e,+)n, he pion may hav first to be formed, and the descron is ot as so we ed E corresponding neutron quantitie nd p as Em Ee - E + Mp - E 1 - V = V In our data analysis, we therefore used an event-by-event reconstruction of he fourvector (Em,Pm) = (cu + Mp — Eyt, q—), and the true coincidences were observed in the accumulate missing mass distributio Mm = (E-|1/-M (41 which was offs by around zero. for practica reasns and was cnseuently expected to be centere The missing mass spectrum for al coincident events of kinematics 437 witht any cuts o corrections is shown in figure 44a. The signal-to-noise ratio is poor, and a cut in the coincidenc time is applied to separate the true coincident events from the accidental ones. The resultin spectrum is shown in figure 4.4b (dark-shaded) The lighthaded spectrum corresponds t accidental coincidences an has to be subtracted. -I. I 1-4 * -* ^ iii 'P'H'M**-. 50 25 0 25 50 Mm [MeV] 50 25 0 25 50 Mm [MeV] Figure 4.4: Th missing mass spectrum for kinematics 437: a - without any cuts, b - true coincidences within the —2. ns < tc < 2. ns cut (dark shading, corresponding to figure 43b2) and accidental coin- cidences within he (—30ns < tc < factor of 2 x (3 - 10)/(2 x 2.5) 10n || (10ns < tc < 30ns) cut, correspondingly scaled down b (ligh shading in the same figure) Data alysi _______________________________________________________________ 4. Further background reductio In the measurement of p(e,e'7t+)n, three additional sorts of positively charged particles can enter the pion arm and be misidentified as pions: the positrons originating from the production of ee+ pairs in the target, the muons stemming fro harged pion decay in flight, and protons recoiling fro he p(e, e')p elastic scattering. Positro ackgroun The positron background was eliminated by cuts in the Cerenkov detector energy spectrum. The positrons give a clear Cerenkov ADC energy signal of a very typical shape. This is illustrated by the bump in figure 4.5b for kinematics 648 wih spectrometer B as the pion spectrom eter. In this kinematics, particles with the ADC energies above 1337 (in units of ADC hannels were identified as positrons and the corresponding events were rejected. C O 1500 1000 500 10 0 10 ~ 1400 1600 1800 tc [nsl ECer [ADC channe] Figure 4.5: The anatomy of a coincidence time spectrum for kinematics 648: a, upper histogram - n cuts a lower histogram - only events with E/ > 337, multiplied by 100, b - summed signal from th Cerenkov detector of spectrometer (the pion spectrometer) Particles that can be falsely identified as charged pions can in principle also partly be seen in the coincidence time spectrum. The upper histogram in figure 4.5a was generated withou cuts, while the lower histogram (scaled by 100) contains only the events with E^er > 1337 i the pion spectrometer (i. e. positrons). The lower histogram is faintly peaked at —6.8 ns, i agreement with the calculated tx0F(e+) — tToF^"1") — —7ns for the average flight path throug the spectrometer of 12m and 227MeV/c momentum. But observe that this small background is almost completely masked by the statistical fluctuations of the accidental coincidences, s that even if the positron signal were less 'flat' it can only be removed by he Cerenkov cut an not by he cut in timing. Muo contamiatio The muon contamination, as opposed to the rather insignificant positron background, can b seen in the coincidence time spectrum, but it is impossible to isolate it with the time resolution we have. The left side-peak of the upper histogram in figure 4.5a corresponds to muons wit momentum close to 227MeV/c, for this momentum is estimated to give txoF(M-+) — tx0F(7t+) —2.5 ns But he muon peak and he dominating pion peak overlap and hey can not be clearly 7t' (xl00) J00 zoo 100 Wt**1^^ 3 Data alsi separated. The muon contamiation was therefore etained within he coincidence time cut but was later determined by a computer simulatio an subtracted from oher events in procedure described in section 45 Proto ackgroun The proton background can be eliminated by a safe cut in the ADC energy spectrum of the d scintillator layer in the pion spectrometer. The minimum ionising particles (pions, muons and positrons) deposit little energy in the scintillator material and contribute to the lower part o the energy distribution, whereas the protons deposit more energy and contribute to the upper part. The proton background was therefore removed by an appropriate cut in each kinematica setting. The sharp spikes of the energy spectrum originate fro he ADC overflows in uppermost channels of he measured range m 5000 c zj o o +000 3000 2000 1000 "0 200 400 600 800 1000 1200 EdE[ADC channe] Figure 4.6: The ADC energy spectrum of the dE scintillator in spectrometer B, summed over all paddles for kinematics 648. Protons deposit more energy in the scintillator material than the minimum ionising particles and accumulate in the upper part of th spectrum. In this example, th cut was applied at EdE = 730 4. Detecto fficiencies VDC effciencie There are two kinds of efficiencies that can be defined for a vertical rift chambr package each consisting of four wire planes (see page 23 for description of the hardware). The singl wire efficiency histogram for a VDC is generated from the 'number of wire' and the 'tagge wire' histograms. For each event, the two extreme wires (e. g. wire #100 and wire #105 of th total of 6 neighbouring wires in that event) that fired in each VDC layer are taken as reference wires All the wires, i. e. the extreme two and the wires between them are marked as 'tagged' and checked for hits. For the wires that fired, an entry is added at the appropriate place in he 'number of wire' histogram. The single wire efficiency histogram is obtained by dividing the accumulated 'number of wire' histogram by the accumulated 'tagged wire' histogram. Fig ure 4.7 is an example of a single wire efficiency histogram for the x.1 layer of spectrometer A (for kinematics 437) with an average at 95.46 % (typically average single wire efficiencies bette han 85 % could be achieved for he x layers and better han 95 % for he s layers) Data alysi ">, 100 o c CD 'o t 80 60 40 20 100 200 300 400 500 # of wire Figure 4.7: The single wire efficiency histogra for the xl layer of spectrometer (kinematics 437) Th hole in channels #32, #46 #223 and #389 correspon to wires without any response If the VDC layers acted as independent detectors, a single layer efficiency of 95 % for al four layers would mean an overall VDC efficiency of only (95 %)4 ~ 81 %. However, the VDC overall efficiency is defined as the ratio of all hose events for which the particle trajectory could be successfully reconstructed, to all events that passed the VDC layers. To safely reconstruc he trajectory of a particle, at least three wires in the x-layers (xl and /or x2) and at least thre wires in the s-layers (si and/or s2) should fire. Trajectories can therefore not be reconstructed (or can be only inaccurately estimated) for events in which this 'joint' multiplicity lies below 6 and only suh events diminih the overall efficiencies of he VDCs. It is worthwhile to note that even if average single-wire efficiencies for one of the paired (xl,x2) or (sl,s2) layers drop to as much as 70%, the overall efficiency of the drift chamber does not deteriorate significantly since the complementary layer usually supplies the relevan wire information. On the other hand, events with small multiplicities in a single layer or event in whih a whole layer failed to respond can have a very poor trajectory reconstruction. For all settings but 219, 500 and 834 (where an overall VDC efficiency of 100 % was as sumed), the overall efficiencies were determined from the measured single-wire efficiencies by a computer simulation. The chamber layers of given efficiencies (e. g. that in figure 4.7) were 'scanned' by particles impinging at different dispersive coordinates x and dispersive angles 0 whic were both varied in accordance with the measured distributions. The overall efficienc was then determined by dividing the number of events that met the '3 + 3' criterion, with th total number of simulated events. As it has been expected hese efficiencies were always ver close to 100 % and are listed in table 41 Table 4.1: Overall efficiencies of the vertical drift chambers of spectrometers A and B. For settings 219 500 and 834 [84], the overall efficiencies were estimated to be 100 . For all other settings, th efficiencies were determined by a computer simulation (see text for details) Kiemti 22 37 742 25 457 648 LVD. LVD. 98.82 99.86 99.63 99.79 99.89 99.87 99.70 99.93 99.99 99.92 99.94 99.94 Data alsi cintillat etct cici The efficiencies of the scintillator detectors were investigated in [84], where he three-detecto method had been used to determine the efficiency of the dE layer (with the VDCs and th ToF layer as reference detectors) and of th ToF layer (with the dE layer and the Cerenkov detector acting as reference detectors) of each spectrometer. The sensitive area of the scintillator layers was 'scanned' by elastically scattered electrons, and only events lying within the targe and nominal spectrometer acceptances were used for normalisation. Table 4.2 summarises results. Table 4.2: Efficiencies of the scintillator detectors of spectrometer A and B. In the present analysis, th overall efficiency was Lint = 99.53 x 99. = 98.1 Oal cint. 99.75 ±0.01 99.78 ±0.06 99.53 ±0.07 int. 99.70 ±0.01 99.48 ±0.50 99.18 ±0.51 The small overall efficiency detriorations of 0.47% for sctrometer A and 0.82% for spec trometer B originate entirely in the imperfect junctions of he paddle pairs (see figure 3.8) and can not be eliminated without overlapping paddles in each plane. However, overlapping wa avoided since it would lower he reliability of he particle identification using energy loss. Cerekov detctor effciencie Only electrons trigger a signal in the Cerenkov detectors (cf subsection 33) so that the Ceren kov efficiency for the detection of electrons is equal to N^et/Ns/ where N^et is the number o electrons actually seen by the detector and Ns is the number of electrons that traversed it and had energies large enough to cause the Cerenkov effect. The veto efficiency, on the other hand equals 1 — N^/Ngn, where Nsn is the number of particles traversing the detector which should not cause the Cerenkov effect, and N^e is the number of particles that were nevertheless seen by the detector. The efficiencies of the Cerenkov detectors were tuned to have at least eigh photoelectrons reaching he first dynode of one of the hoto-multipliers per event, and wer studied in [86] applying the three-detector method with the ToF scintillators at the lower sid and the Top scintillators on the upper side of Cerenkov counters The results are shown i table 43 Coicidence effciency The coincidence efficiency of a double coincidence experiment can be defined as the ratio of the H(e,e'p) coincidence cross-section to the H(e,e') elastic scattering (inclusive) cross-section. If both reactions are measured at equal kinematical conditions, the cross-sections should be equal In oher words, the coincidence efficiency measures the ability of the event-builder to combine synchronise, and process two completely independent spectrometer data streams. This was studied in detail in a previous work [84], and we used the value of eCoinc = 0.996 quoted there The systematical uncertainty of the overall detector (VDCs, scintillator counters an Cerenkov detectors) and coincidence efficiency was estimated to be 1 for all settings [84] Data alysi Table 4.3 Efficiencies of the Èerenkov detectors of spectrometer A and B for the identification of electrons and for veto operation (pions). For details of the analysis, see [86]. In our measurement, the Èerenkov detectors were not used as an activ part of the trigger-system, but only in the offline analysis Particle Electr Veto (pins er_ 99.98 100.00 . 99.97 100.00 4.5 Correction factors Dad t corrction The overall dead time of a coincidence setup originates in individual dead times of the de tectors, readout and trigger electronics, and data acquisition software. In our experiment, the major contribution to the overall dead time comes from the software, by far exceeding the in trinsic detector dead times. The dead time correction factor for a coincidence measuremen wi spectrometers A and B can be expressed as K^d = 1(1 — Ldead) her tded *dA + *d tdAB . TlA"tA + B^ , „ ^ ded-------- ----------7--------------- + -----------------7--------------------- (42 T Trun Tun The tdA/ tdB and tdAB in the first term of 4.2 denote the data acquisition dead times of th individual spectrometers and of the coincidence setup, respectively. The second contribution corresponds to he trigger electronics (mostly due to the coincidence PLU module) dead times of ttA = ttB = ttc = 90 ns per event (settings 437, 648, 457 and 259; in all other settings, 500 ns for each of the single events nA and tlb or coincident events txab- Overall dead times tdead and total runtimes trun for all measured runs are listed in table 44 The systematical uncertainty o he dead time correction is 0.5 [84] Table 4.4: Overall dead times t^ea^, total run-times tmn (aH in [s]), Ldead and Kded f°r a^ measure settings Note that the entry for setting 834 [84] corresponds only to the measurement parallel to q. Kinematic 219 500 834 229 437 742 259 457 648 -ded 1991 1051 610 1992 623 735 2837 1536 2234 1 86494 48628 20215 99950 44926 19403 159744 109716 48534 ded 0.023 0.022 0.030 0.020 0.014 0.038 0.018 0.014 0.046 ded 1.024 1.022 1.031 1.020 1.014 1.039 1.018 1.014 1.048 Pion cay rrcti By far the largest correction fctor to he measured umber of events origiates in the chaged pion decay n+ —> u"1-^ in fliht from he target to the detectors The decay follows the simpl 3 Data alsi ti l -S/It (3 where l^ (E/TrL7T)T7T37:c = (pn/m^cJT^c is the pion decay length, with the pion lifetim (in the frame where the pion is at rest) of t^ = 26.03 ns ([24], p. 321). This amounts to l^ ~ 11 m for a typical pion momentum of p^ = 200 MeV/c, meaning that during their flight through e. g. spectrometer B with the reference trajectory length of Lref = 12.03m, two hirds of pion decay (the situation is not much more favourable in spectrometer A with Lref = 10.75m). The scintillator planes are then hit by the remaining pions and by the fraction of he muons ha were not stopped in the collimator slits or internally in the spectrometer walls. The problem is that muons with momenta very close to the pion momenta can not be distin guihed from the pions by either time of flight differences or differences in the ADC spectrum of the scintillators. The muons represent a certain contamination of the pions, which can only be determined by a computer simulation. We used the RAYTRACE computer code [97] fo tracing particle rays throu optical systems that was later upgraded for pion decay [98] o o 250 200 150 7 100 50 U jiy|yj|y|| k ^ 100 100 x [cm] Figure 4.8: Muon contamination in the focal plane of spectrometer B in kinematics 457: a - in th dispersive coordinate x, b - in th non-dispersive coordinate y. The light shaded histograms correspon to the pions an the dark shade one to the muon from th pion deay Figure 4.8 shows the simulated distributions of pions and muons from the pion decays, in the dispersive focal-plane coordinate x and in the non-dispersive coordinate y of spectrom eter B with the central momentum of 227MeV/c (the distribution in the dispersive angle 9 roughly resembles that in x, whereas the distribution in the non-dispersive angle 4> looks sim ilar to that in y). In a previous work [84], the pion decay correction was determined in two steps of the simulation. First, initial pion rays were randomly and uniformly distributed over the spectrometer acceptance and transported to the focal plane, allowing pions to decay un derway Cuts were hen applied in the four focal plane distributions (x, 9, y and c|j) that corre sponded to pions; the muon contamination R^ in he focal plane was then identified with the remaining muons surviving all four cuts, i. e. R^ = N|X/(N7T + N^). This contamination wa then subtracted from the number of detected coincident events. The result was hen multiplie by the correction factor Kdecay, obtained from (4.3) with s = Lrf In this work, a more accurate approach is attempted, largely mandated by the use of the ex tended liquid hydrogen target and motivated by improved means of particle tracing and back-tracing. Again we first randomly generated a sample of pion rays originating at he target, bu wi linear and angular distributions corresponding to he actual extension of he target an Data alysi 3 to the beam wobbling amplitudes acually used in the experiment. All the particles eaching the focal plane, the pions as well as the muons, were then back-traced to the target. A sample result of a back-tracing procedure is shown in figure 4.9 for kinematics 457, showing origina pion distributions, back-traced pion distributions, and back-traced muon distributions in fou target coordinates. The cuts can then be applied at the target. The ratio Kdeca of the number of he pions hat survived the cut, to the original number of pions, reduced for the muon contamination R^ within the cuts, was finally interpreted as the total pion decay correction factor Table 4.5 summarises the correction factors for all measured kinematical settings. The system atical uncertainty of (1 — RpjKdecay was estimated to be 1.6 % for settings 834 and 500,1.8 fo setting 219, 0.6 % for setting 229 and 0.5 % for all other settings. o o 400 350 = 300 /jpvpwif|!Pi"TT 250 p I 4 200 -_ 150 A 100 A 50 0 10 0 10 600 500 400 300 200 100 - j^^^^l^^^^^^ 50 ^oMiMmmm c5T[%] 50 0„ [mr] ifi -L 8000 o 7000 O r^ j 6000 '- h 5000 4 -, -= 4000 4 -_ 3000 4 1 ~- 2000 4 4 1000 4 4 4 , , , I , , , , I , 1 , , , , F 5 .5 5 y0 [cm] Figure 4.9: Pion and muon distributions after back-tracing the particles from the focal plane o spectrometer B back to the target (kinematics 457, extended target with a wobbled beam), in: a - kinetic energy deviation 6T = (T — Tref)/Tref, b - dispersive angle on target, c - vertex on target, and d - non-dispersive angle on target. Unshaded histograms correspond to pion rays entering the simulation, ligh shaded histogram to bak-traced pions and dark shaded histograms to back-traced muons ati l cti The electrons and the pions of the p(e,e'7t+)n reaction interact with the target protons and with the electron clouds of the target atoms, and thereby lose a fraction of their energy b radiating additional real or virtual photons. Since these radiation losses can not be directl measured, the reconstructed energy transfer, momentum transfer, missing mass and related distributions become distorted: they ehibit radiative tails, populated by events in which som of he particles involved lost a part of heir energy Three mehanisms of energy losses ar 4 Data alsi involved in the p(e,e'7t+)n reaction: internal Bremsstrhl external Bremsstrhl a target ionisation (see Appendix D for details). Table 45: Simulated muon contamination R^ and the pion decay correction factors Kdecay f°r a^ mea_ sured settings. Note that in settings 219, 500 and 834, the tota correction factor (1 — R^K^™, was determined from the simulated pion and muon distributions in the focal plane, whereas for all other settings, it wa extracted from the same distributions back-trace to the target Kinematic 219 500 834 229 437 742 259 457 648 de 0.072 3.116 0.148 2.953 0.148 2.891 0.010 2.910 0.017 2.909 0.111 2.512 0.017 2.668 0.018 2.660 0.018 2.674 The energy loss correction can be done by counting the number of events 1M (AMm) withi a certain interval AMm above the peak of the backgroundfree spectrum of Mm — Mn an multiplying it by the correction factor Krad(AMm) = KSchw(AMm) Kbr(AMm) KLand(AM corresponding to the 'cut-off energy AMm. The 1M (AMm) Krad (AMm) is then interpreted as th 'true' number of coincident events, i. e. events that would have been observed, had there been no energy losses. This approach was followed by [84] for settings 834, 500 and 219. In the en ergy ranges of our experiment, the electrons lose their energy almost exclusively by Schwinger radiation. Bremsstrahlung of pions is further suppressed by a factor of trig/m^ and may be ne glected. The pions only lose energy through ionisation losses, but for our kinematical setting and typical cut-off energies of ~ 10 MeV, the correction factors are close to 1. Energy losses due to Landau straggling can be neglected for target thicknesses smaller than 0.05 of the targe material's radiation length. The overall correction factor Krad was 1.227 for setting 834, 1.182 for 500, and 1.156 for 219 [84] and its systematical uncertainty was estimated to be 2 %. But the problem is hat the energy losses of he particles do not map trivially to the corre sponding changes in the missing mass, since Mm (4.1) is a non-linear function of Ee, Ee and E^ For example it is obvious that an incoming electron's energy loss of 10 MeV which is also 'i herited' by the exchanged virtual photon, will not necessarily correspond to a shift of 10 MeV in the distribution of Mm. We therefore incorporated the radiative correction into the acceptanc simulation (see subsection 4.7 for details). The advantage of his approach is the possibility to properly handle the particles' energy losses in the target and in the variety of other material of entrance and exit foils. In addition, beam wobbling amplitudes were equal to those actually used in the experiment, so that the detector acceptances were consistently folded into the ra diative tails. The systematical uncertainty of the radiative loss correction was estimated to b 3.9 % (setting 259), 2.8 % 457, 742), 2.6 % (437), 1.2 (229) and 1.0 (648). 4.6 Luminosit The integrated luminosity is defined as the product of the number of target nuclei per uni surface N and the number of electrons Ne impinging on he target in a time span T Ne = ^i rT dt. eo e0 Determining the integrated luminosity therefore effectively amounts to measuring the integra charge accumulated on he target The bea current is measured with the probes describe Data alysi in section 3.2, and the elpsed time information is kpt in the runtime and realtime scaler controlled by the |u.Busy module; during data taking, these scalers are read out and their value directly entered into the data stream. To calculate the number of target nuclei per unit surfac from a given target density p and average target length x, we us xN A where Na is the Avogadro number and A is the mass number. Because the beam wobble was used at all times during our experiment, illuminating different portions of the target fo different deflection amplitudes (see figure 3.3), the target length has to be averaged over thes amplitudes Table 46 shows he values determined for he two target cell types Table 4.6: Average target cell length x, target density p and the number of target nuclei per unit surfac for two type of LH2 target cells used in the experiment (t A = 2.000; tt A =008 Kinematic (219,500,834)t (229,742) tt 437,259,457,648)tt x [cm] p[g/cm3] Nt[1022/cm2] 1.868±0.5 0.1374±0.5 7.731 ±0.7 1.868±0.5 0.0708±0.5 7.730±0.7 4.886±0.3 0.0708±0.5 20.67±0.6 In the coincidence experiment we performed, the data streams of individual spectrometer were combined into a single event stream in the final stage of the data acquisition system (se figure 4.10 for an illustration of he time sequence of spectrometer data streams as seen by acquisition software) 1 2 3 4 5 6 10 11 12 14 13 . 3456 3457 3458 3460 • • • • 9 13 16 19 ..... Run start Run stop Figure 4.10: Spectrometer data streams in an experiment utilising three spectrometers Bullets represen single events and numbers represent their sequential numbers. For example, events #15 and #3458 are AB coincidences, and event #19 is an ABC coincidence. Only the time period during which spectrometer A's and spectrometer B's data streams overlap can be used fo the evaluation of luminosity in an AB coincidence experiment The time intervals called 'Pre A' and 'Post C in the igure, during which the actual data taking has not started or finished yet, are irrelevant. But the differences between the starts and stop of the individual spectrometer data streams ('No A' and 'No B' in the figure), are important t the luminosity calculation for an AB-coincidence experiment. Since there can obviously be no coincident events during these (relatively short) acquisition starting and stopping procedures' dead times, only the time when spectrometer streams 'overlap' may be used for the luminos ity calculation. Correspondingly only those events fro he data stream may be used in analysis that fit into he time window after he 'No A' and before the 'No B'. The average elec tron beam currents, the total accumulated charges and integrated luminosities for all measure settings, and heir systematical uncertainties are listed in table 4.7 Table 4.7: Average electron beam currents Te, total accumulated charges Qtt and integrated luminosities L for all measured settings (the entry for setting 834 [84] corresponds only to the measurement parallel to q). Note that for settings 437, 259, 457 and 648, L ^ NtQtot/eo since the small corrections due to the data acquisition dead times mentioned above have already been taken into account. The systematical uncertainties of Q an originate in the uncertaintie of the Forster probes an in the fluctuation of th target density Kinematic [uA] [As] L[104/c 219 33.6 3.106±0.5 14.985±0.9 500 33. 1.628±0. 7.855±0.9 834 14. 0.2899±1.1 1.399±1. 229 33.9 3.824±0.3 18.451 ±0.6 437 19. 0.8716±0.03 11.368±0. 742 21. 0.4783±0. 2.308±0.7 259 23.1 3.662±0.02 47.768± 0.5 457 16. 1.843±0.03 24.043±0. 648 25.1 1.215±0.01 15.844±0. 4.7 pectroeter acceptances When two spectrometers are used in a coincidence experiment, the ranges of the kinemati cal variables they cover (the so-called nominal spectrometer acceptances as given for instance in table 3.1) are in general much smaller than the ranges of the same variables that occur at the target. Specific geometries and experimental conditions (different spectrometer angles collimators, scattering cell types, beam wobbling, or imperfect detector efficiencies) and the energy-momentum constraints of the reaction being studied further narrow the kinematica 'slit' in which coincidence events can be observed. Unfortunately, this 'slit' is almost impossible to calculate analytically since the mapping of the boxlike nominal acceptances to the actua distribution of the events as seen by the spectrometers is very complex. The only possibility to determine the actual acceptance is to use a computer simulation. Single-arm events are gener ated at the target in kinematical ranges exceeding the nominal spectrometer acceptances, and individual particle rays are checked for momentum and angles at the spectrometers. For a suf ficiently large number of tries, e ratio between the accepted coincidence events 1M and the events generated at the target lMtg approahes he ratio of he actual acceptance to the nomina acceptance In he case of p(e, e;7r+)n dE ^ AE (4 The acceptance integra reflects only the geometry of the setup and energy-momentum c servation, but does not have any further physical content, and has to be divided out from the observed spectra In he analysis of our experiment he particles' radiation losses he calcula Data alysi tion f the virtual photon fux rv and the frame transformation dD.n implied in he st from (2.2) to (2.3) were also included in the acceptance simulation. First, a valid reaction point is randomly generated according to the beam wobbling amplitudes equal to those used in the experiment. When the reaction point is known he length of the path (and the corresponding energy loss) of the incoming electron through the target cel can be calculated. Second, the scattered electron is generated. One randomly selects its dispersive and non-dispersive angle from ranges AOg = A sin d'e Aty'e that slihtly exceed the nomina angular acceptance of the electron spectrometer, and a value of q2 from a sufficiently broad in terval, so hat the momentum of the scattered electron (which can be calculated from the angles and from q2) lies within AL'e. One then checks whether the outgoing electron passes the colli mator: if it does, the lengh of its path throuh the target cell is calculated and a correspondin energy loss is forced. If the momentum of the electron still fits into the momentum bite of spectrometer, one has a valid electron; in all other circumstances the event is rejected. The calculation now proceeds in the centre-ofmass frame. If the scattered electron wa accepted, he virtual photon flux Vv is calculated from the energy-loss-subtracted Ee, Eg an q and the event is weighted by TvAEe AOe times Afl* = AO* = A sin 0* Ac|)* and normalised to the total number of tries (this very last step can be done at any point). Since the four-vector of the photon and the target proton are known the magnitudes of the CMS three-momenta in p* = p* + p* = 0 can be calculated from the yv + p —> n + 7t+ kinematics. Finally, the spherica angles of he neutron are randomly selected, the neutron is boosted from he centre-of-mas frame to the laboratory frame, and the pion four vector is calculated from pn. If the outgoing pion then passes the collimator, the leng of its path through the target cell is calculated and corresponding energy loss is forced. If the momentum of the pion fits into the momentum bit of the spectrometer, one has a valid coincidence event, which receives an additional weight due to vertex corrections (see appendix D for details). In all other circumstances, the whole event i rejected. To be able to compare the simulated particle distributions to the ones obtained fro the analysis program, the energy losses of the event are corrected for in he last step of simulation. The results of the simulation are summarised in table 4.8 Table 48: The acceptance integral (44), expressed in the centre-of-mass frame of the final hadrons, in units of [10~9sr2] for the measurements at q —0.195 and —0.273 GeV /c. The energy losses of th particles involved were included in the simulation: the (5) value includes all events in the simulated missing mass spectrum, whereas the (10) value implies a Mm 10 MeV cut actually used in the analysis For the calculation of the acceptance integral at q = —0.117GeV/c where a different procedure ha been adopte to handle the particles' energy losses, refer to [84] Kinematic 229 437 742 259 457 648 TvdEednedn*(50) vdEednedn*(io) 0.7308 0.7245 1.6339 1.5319 5.9777 5.5835 0.7826 0.7426 1.4803 1.3638 3.0083 2.7192 4. The differential cross-sectio To determine the reaction cross-section (2.2) for p(e, e7t+)n, we count all 'true' events (i. e. back ground free pion-electron coincident events) and normalise their number to the luminosity and to the acceptance covered by the spectrometer setup. The 'true' number of events Nrue is iso lated fro he measured number of events Nexp by eliminating he background N^ (throu 4 Data alsi subtraction of accidental coincidences and performing cuts to dispose of the remaining b ground) and by applying he correction factors described in sections 4.3 to 4.5 (N N) (1 de de The luminosity L = NtQtot/eo is determined from the properties of the target and from th measurement of the total accumulated charge, as described in section 4.6. The sole quantity that can exclusively be obtained by a computer simulation, is the actual physical detection volume available to the particles in the final state. The cross-section, averaged over he nomina acceptance na = AEg AL AO, is then dE ^ Since the reaction cross-section for p(e, er+)n also assumes a factorised for (23) he hysi cally relevant differential cross-section da/dfl* can be calculated from TVdE ( As indicated in (4.5) ad discussed in section 4.7, he virtual photon flux factor ad th Jacobian determinant ]n were directly included in the acceptance calculation, in which eac accepted event is appropriately weihted. The measured cross-sections are listed in table 49 Result an ysi Table 4.9: Measured centre-of-mass cross-sections for the p(ee'7T+)n reaction at W = 1125MeV and four-momentum transfers q2 of -0117 (settings 29, 500, 834), -95 (229, 437, 742), an 273 GeV (259,457, 648) See also table 21 Kiemti dadO* [Mb/sr] Stat erro [fib/sr] yst erro [ub/sr] 219 500 834 5.96 8.40 11.14 0.14 (2.3% 0.11 (1.3% 0.08 (0.7% 0.19(3.2% 0.31 (3.7% 0.41 (3.7% 229 437 742 4.69 5.61 7.73 0.10(2.2% 0.10(1.8% 0.12(1.6% 0.08 (1.8% 0.16(2.9% 0.24(3.1% 259 457 648 3.55 4.16 5.13 0.05 (1.5% 0.06 (1.4% 0.06(1.1% 0.15(4.1% 0.13(3.1% 0.09 (1.7% Data alysi Since all meaurements were peformed in parallel kinematics, the interference cros-section vanish, and the transverse and the longitudinal cross-sections could be separated by applying the Rosenbluh method to data points at constant q2, but various es (see section 2.2). The slope and the u-axis intercepts of the straight-line fits of cross-sections from (2.5) were identified wi the longitudinal and the transverse cross-sections, respectively. Figures 4.11 and 4.12 show the results of the fits and the results for the separated transverse and longitudinal cross-section in dependence of q, including the predictions of the models of Drechsel and Tiator (DT) ([38] section 52) and of Drehsel, Hanstein Kamalov and Tiator (DHKT) ([39] appendix E) ^ c X b x [Ge dox/dO* i-qiv*) da/dO* [ubsr] [ubsr] -0.117 4.160 ±0.165 8.94 ± 0.254 -0.195 3.208 ±0.149 5.933 ± 0.307 -0.273 2.452 ± 0.094 4.038 ± 0.201 0.2 0.4 0.6 0.8 Figure 411: Least-squares straightline fits to the measured cross sections at constant values of q2 as functions of the virtual photon polarisation e. The slopes of the fits are proportional to the longitudinal and the y-axis intercepts of the fits to the transverse cross-sections (see (2.5)). The smaller error bar correspon to statistical, the larger one to the sum of statistical an systematical uncertainties a. DT. dcrT(0) fixed, M, fitted DT. daT(0), MA fitted DHKT. 0.2 0.3 0.4 q2 [GeV2/c2] 0.3 0.4 q2 [GeV2/c2] Figure 4.12: Separated a - transverse and b - longitudinal cross-sections at W 1125MeV, together with the q-dependence predicted by the DT and DHKT models. Full curves: DT model fit to the data points with dcr^O fixed by photo-production data (see also figure 5.3) and M.^ as the fit parameter; dashed curves: DT model fit with do) as the second fit parameter; dotted curves: DHKT model Data alsi In the nalysis of the q -dependence of the rosectis, rting to an effective La grangian model was inevitable, since the values of an W in the experiment were stil too high for the current development stage of the xPT. The DT model we used in the anal ysis is a gauge-invariant model for charged pion electro-production in the region below A-resonance. In this model, he procedure of gauge-invariance restoration influences only the longitudinal cross-section, whereas the transverse part containing the axial form factor remain unaffected. The q2-dependence of the form factor (or the corresponding value of Ma) can henc be extracted from the fit of he calculated transverse cross-section to he experimental data wi an ehanced sensitivity. Two approaches were attempted. In the first approach only the axial mass was varied, whereas the cut-off energy of the pio (monopole) form factor was set to An = 0.682 GeV. The value of daj at q 0 was fixe to 7.4(ib/sr, indicate by the photo-production dispersion analysis of [101]. In the second approach, we used the same value of A^, but treated do"x(0) as an independent fit variable yielding the value of dor(0) = (7.05 ± 0.54) |u.b/sr. In both methods, the resulting best-fi parameters for the transverse part were then used in the fit of he longitudinal part. Using the first and preferred method, we find Ma = (1.073 ± 0.016) GeV, corresponding to (r2 )V2 = (0.637 ± 0.010) fm. From the longitudinal part, we obtain An = (0.673 ± 0.018) GeV corresponding t r1/ = (0.718 ±0.019) fm. Using the second method, we get Ma = (1.105 ± 0.059) GeV or (r2 )V = (0.618 ± 0.033) fm, and A„ = (0.685 ± 0.019) GeV or (r)1^ = (0.706 ± 0.019) fm. The results indicate that our extracted value of MA = (1.073 ± 0.016) GeV is (0.056 ± 0.028) GeV larger than the axial mass Ma = (1.017±0.023) GeV known from neutrino scatterin experiments. Our value essentially overlaps with the scaled-error weighted average M (1.068 ± 0.017) GeV of all older pion electro-production experiments. If we append it to th database, the weighted average increases to Ma = (1.070 ± 0.012) GeV (see the correspondin figure 71) and he 'axial mass discrepancy' becomes AM = (0.053 ± 0.026) GeV. xtractin (q r experiment The theoretical its in figure 4.12 are meeting points of experimental data and theoretical models, and exploit the q2-dependence of the cross-section to probe the matrix element of he mode axial current and equivalently the axial form factor In the past few decades, the knowledge on the q2-behaviour of the axial form factor emer ged from an interplay between experimental improvements and gradual sophistication of the oretical models. But apart from the uncertainties related to the estimates of centre-of-mas motion and recoil effects (which will be referred to in the next chapter), these models were facing another correction problem. Early attempts to describe low energy photo-production o massless charged pions can be traced back to the Kroll-Ruderman formula (2.8). Their resul was subsequently extended to virtual photons by Nambu, Lurie and Shrauner [73, 74], who calculated he isospin-odd electric dipole amplitude at hrehold (29). Expanded to the orde ofq O = °V) = fL {i T JSP h1} + oi")} <« where is the isovector anomalous magnetic moment of the nucleon (see also section 5.3). (In turn he longitudinal s-wave multipole L^+ additionally appearing in electro-production con tains the isoscalar anomalous magnetic moment and the pion form factor F^q2).) This resul can be used to extract the axial radius ta from experimental data on threshold pion electro production. Namely, since in the immediate vicinity of the threshold only s-wave multipole contribute to the cross-section (23), it can be expressed in a simplified form dO _ r it /cn according to (B.10) and (B.ll) 4. For the p(e,)n reaction, E^+ is the dominant multipol containing (rA). The experimental value for he hreshold cross-section was therefore obtaine by extrapolation from a range of energies W down to threshold W = M + m^, and (r2 could be isolated from the q2-slope of he cross-section using (5.1) and (5.2). The problem is tha (5.1) was also formulated for unphysical pions with zero four-momentum (and therefore zero mass), and to be able to confront the experiment, it had to be extrapolated into the hysica region with m^ / 0 and p / 0 There were several attempts to tackle his problem. 4When a specfic physical channel is cd, approprte in comon of he mulipol appear (52) (see appendi C for the definitions) 4 5 Extractin Ga(c|2) fro experimen 5. Ealy theories Fur Ian, Paver and Verzegnassi (FPV) were among the first authors who tried to extend (5.1) t hysical pions. Their method was based on current algebra and on the approach of [117], i hich the generic physical reaction amplitude F(v = Ppp^u = p2 = m) was derived from he soft-pion amplitude F(0,0) by means of a dispersion relation F(v,)=F(0,0) + ^ where the integration path y lies in he (v, u)-plane. Such dispersion relations are a means to use analytical continuations of the physical amplitudes into the complex plane in order to b able to connect them to other observables (e. g. to connect ImF(u') to the total cross-section using the optical theorem [103]). It was hoped that the integration could be performed in su a way that the dispersion corrections would be small. It was shown [75, 76, 77, 6] hat corrections to the transverse multipole (51 Eft^, q) = EL'NLS(0, q) + fiELW q hi vanish fo ra^ —» 0, show up at a level of 10 to 30 % at the most. The prospects were much worse for the corrections to the longitudinal multipoles Eo+, and at that time, this trig gered further experimental efforts to accurately separate he electro-production cross-section at hrehold and extract G(c) from he transverse part. ^ 1 ex < 0.8 0.6 0.4 0.2 "0 0.2 0.4 0.6 0.8 1 q2 [GeV2/c2] Figure 5.1: Available experimental data for Gj^iq), extracted from pion electro-production experiment in the vicinity of the pion production threshold. Note that in experiments wher more theoretical model (FPV, BNR, DR) were used to extract G a all results are shown in the figure. The curves show prediction of some of the quark models: MIT bag model or the Cloudy bag model (thin full curve), model with a confining potential of the form ~ r3 without CMS corrections (upper dashed line) and with CMS corrections (lower dashed line) [115], Skyrme model (upper dotted line), and Skyrme model with vector mesons (lower dotted line) [118, 119] The upper and the lower thick full curves correspond to dipol parameterisations of Ga(c|2) with M 110 an OOGeV, respectively. Experimental data are from [26 2829 30 31 3233 35 37] Sound objections to the FPV approach were risen by Benfatto, Nicolo, and Rossi (BNR), who claimed hat he mass extrapolation of the amplitudes could not be trusted and that the dis fu 5 Extractin Ga(c|2) from experimen 4 persive orrections do not necessarily remain small [80,81] To diminish the problem, they sep arated the total electro-production amplitude into a term that was expected to be well-behaved under mass extrapolation, and into a 'Born' term. For the first term the FPV result was essen tially kept, whereas the second term was fixed by gauge invariance. The drawback of the BNR theory, as pointed out by [117], was that in addition to the basic current algebra commutators, specific phenomenological Lagrangian had to be chosen to describe he Born terms, where pion charge form factor F7(q2) had to be introduced. Another approach was adopted by Dombey and Read (DR) [78, 79] who noted that he dis persion relations used in the description of pion electro-production were incompatible with t requirements of current algebra. Although it seems natural to expect hat the threshold Eq+ am plitude for charged pion electro-production can be identified with the s or u-channel nucleon pole terms, and hence depends on the proton charge form factor, current algebra asserts ha the axial form factor is the most important. The mismatch could be traced back to the fact tha dispersion calculations used the pseudo-scalar 7tNN coupling, whereas current algebra used the pseudo-vector coupling meeting the requirement of PCAC and directly leading to a contac (or seagull) term for yvN —> Nr7t upon minimal substitution. This term could be identifie with the part of the current algebra amplitude containing Ga(c|2). The extrapolation from the soft-pion limit to the physical region occurs hrou his correspondence and Ga(c2) remain he relevant form factor 5. The D ode The D model of Drechsel and Tiator [38, 85] we used in the extraction of Ga(c|2) from ou experimental data is based on the effective Lagrangian models of [40, 41]. It is an earlier an simplified version of the unitary isobar model discussed in appendix E. The electro-production amplitude is treated in two parts: he part whih describes he cou pling of the virtual photon to the nucleon or to the pion, and the part which describes the 7tN vertex. The nNN vertex is more involved, since it embodies the strong interaction part of the electro-production process At low energies it can be well described by the pseudo-vector (PV couplin __ Vftji which also reproduces PCAC and is consistent with the low-energy theorems and chiral per turbation theory [8] to the leading order. In the one-photon approximation, the total electro production amplitude is a coherent sum of the non-resonant Born terms and he resonant term with nucleon and meson resonances in the intermediate states (figure 52) here for factor are inserted at he photonhadron vertices. The problem is that the gauge invariance of the electro-magnetic current qM^JM. = 0, inher ent to the PV coupling, can only be maintained if the pion and the axial-vector form factors are set equal to the Dirac isovector form factor, Ga(c|2) = E^q2) = EJ(q2) = 1/(1 — q2/A^)2 Insertion of form factors with different q2-behaviours spoils the gauge invariance (i. e. the cur rent conservation), and it is crucial to cure that Gauge invariance can be restored by includin additional gauge terms in th hadronic curren J q where the subtracted term is purely longitudinal since according to figure 2.2, q = (tu,0,0,|q|) The subtraction effectively modifies only he longitudinal part of the cross-section and corre 5 5 Extractin Ga(c|2) fro experimen + _ l 71 Figure 5.2: Decomposition of the total charged pion electro-production amplitude into non-resonant Born terms: a - s-channel term, b - u-channel term, c - tchannel (pion pole) term, d - contact (seagull) term, and resonant terms: e - s-channel A-exchange term, f - vector meson exchange term. The full circles in a, b, c and d indicate the insertions of the appropriate form factors, and ellipses stand for inclusions of higher resonances. The contact term containing the axial fo factor G(c) follows directly from the pseudo-vecto coupling upon minimal substitution. spondingly, influences the weigt of the pinole term nd he extracted value arame terising he monopole pion form factor (q) = V(l-q). This procedure does not affect the transverse part of the cross-section. This allows us to appl form factors with different q-dependencies: we used the cut-off parameter of A^ = 0.843 GeV for the dipole isovector Dirac form factors, whereas the cut-off parameters of he dipole GA(q2) and of the monopole E^q) form factors were fitted to the transverse and to the longitudina cross-sections, respectively, as described on page 46. The A-resonance exchange term is by itsel gaugeinvariant, q^^ 0. In his sense, the parameterisation of he for factor of the yNA vertex is irrelevant to the determination of GA(q)- The value of dor at q2 = 0 was fixed by extrapolating our transverse cross-section (i. e th Eo(n7t+) amplitude) to the photo-production angular distribution dff/dO* at Qn = 0°. Du to the lack of reliable near-threshold data for charged pion photo-production, the multipol analyses maintained by the SAID group [99] (figure 5.3) are rather imprecise in this energ region and could not be used to this purpose. We used a value of 7.4 ub/sr, strongly favoured by the photo-production dispersion analysis of [101]. The corresponding value of Eq is also well supported by the studies of he GD sum rule [102] and by th Kroll-Ruderman theorem. Due to the cancellations between the terms including only I > 1 partial waves and interference terms with te s-wave amplitude Eo+, the model transverse cross-section is predom inantly sensitive to the Eo+(n7t+) amplitude and therefore to GA(q2) or, if the dipole parameterisation (1.1) is used, to Ma- In the case of the longitudinal cross-section, he sensitivity on the corresponding open quantity, the pion form factor, is more intricate. The s-waves con tribute only 10 to the longitudinal part, and the pion form factor appears only at the order o 5 Extractin Ga(c|2) from experimen 1 " ¦ ¦ 1 1 1 1 mil 1 i I T " ?, 1 1 1 1 1 i i i Figure 5.3: Determination of the transverse pion electro-production cross-section at th origin from pion photo-production data by extrapolating the multipole analyses of angular distributions to 0°. Only those entries of the SAID database [99] corresponding to 1100 MeV < W < 1150 MeV were used. Th full curve is the result of the SAID multipole analysis, and the dotted curve is the prediction of the DHKT model described in appendix E. Note, however, that the SAID result was not used in our analysis for lack of reliable data and systematical uncertainties of the SAID multipole analysis in the threshold region (e. g. questionable treatment of multipoles in the vicinity of the charged and neutral pion thresholds). The value of d(x) represents the solution of the Dirac equation for massless quark moving freely in an infinitely deep spherically symmetric potential well (als called a bag) of radius R. Wit Mx (r c,spin-isospi the solutions for component are f(r) = AHEjo(Er) and g(r) = — TVEji (Er) where M is th normalisation constant fixed by the conservation of probability, J0(f2 + g2)rdr = 1. I the lowest quark radial state, E = Eo with o)q = EoR — 2.0428. The axial current A^ (x) = i[>(x)yhY52Ti[>(x)©(R — r) differs from zero only in the interior of the bag and we ge ( ;(r) 0 ( about 14% below the experimental value. Models with confining potentials (or, equivalently with the mass term M(r)) of the form M(r) = Crn and the integration range correspondingly extended to oo, give values remarkably close to the experimental value: n = gives gA = 1 and n = 3 gives gA = 1.21 [106] One of the inherent difficulties in the calculation of gA in the framework of quark models is the spurious centre-of-mass motion. But with the exception of the non-relativistic version o the harmonic oscillator quark model, the centre-of-mass motion can not be explicitly separated from the relative motion of the quarks, and corrections are mostly only approximate. Centre of-mass corrections to the order of C(p2) can be incorporated into the MIT bag model or into any relativistic quark model invoking Dirac-like wave-functions containing upper and lowe components f and g by using the wave-packet formalism suggested i 107. For the matri element of the n —» p axial-vector transition current one obtain MlT.Ofp2) 9a (p2) ---------------------- mn (r ; instead of (6.3), where typical (p2) is of the order of 0.1 GeV2. The problem in this procedur is that once a certain momentum P is projected out of the three-quark wavefunction alulati t cl xi frm fct _____________________________________ [108], the resulting wave-function does become translationally invariant in the sense that th centre-of-mass moves as an exp(iPr) plane wave, but the Lorentz invariance, and therefore the conservation of the electro-magnetic current, are spoiled. The reason is that the lower spino components are not treated properly by such projection. The technique is therefore limited to conditions in which the nucleon as a whole moves non-relativistically, i. e. to cases where P|2 — 0 and n^Y^ — 0 at the bag boundary where t is an outward normal fourvector Mx)^y5^Mx) hW5ib M(r) (MIT bag model), (scalar confining potential) alulati th cl xi f fct 4>(r) ~ (1 H-rivr) exp(—m^rj/r2. This ha a important consequenc for th pioni contributio to the axial coupling constant defined by ^ (|V[o(r)]T| analogously to (6.1) Usi t identity [ o(r the surprising resul ] = (o(r) [or/R) and ji (a>or/R), respectively). The reason for this behaviour is the linear dependence of the axial current (6.4) on the pion field n[x) Higher-order corrections in powers of n are expected to be small: for instance, if the Weinber representation of the pion fiel 115] 7 t/(1 + 7i2/)~'i/2 is used and the correspondin pion part of the axial curren w(, M is expanded in terms of n, the contribution of the 7I3 ter to gL turns out to be negligible. The situation is essentially different in models in which the ind^7ij[x) term in the axial current is replaced by u[x)d^n^[x) — 7tj(x)9H0"(x), where a(x) is allowed to vary in space. In these chiral soliton models the nucleon is described in terms of a quark core coupled to the meson cloud in which the a and the n fields appear symmetrically as chiral partners. In the chiral soliton models, the basic ingredients are represented by interacting dynamical fields constrained b the equations of motion, contrary to the potential models in which parameters are fixed in advance. One of the prices we have to pay for this liberty is the value of gA which generally overestimates the experimental value typically b 0 o 61.4 the the CM The constraints of chiral symmetry in the world of hadrons composed of light quarks have a firm theoretical and phenomenological background. If the masses of the u and d quark were zero, the left-handed and right-handed components of the quark fields in QCD would decouple and maintain separate 'left' and 'right' invariances, and QCD would possess a chiral SU(2)l x SU(2)R (or, equivalently, SU(2)y x SU(2)a) symmetry 6. Even with the mu = m^ / 0, the chiral SU(2) remains a fairly accurate symmetry of QCD, which nevertheless does no emerge in the energy spectrum of the physical hadrons: there is no parity doubling among the lowestlying hadron states. What we do observe is that the masses in the multiplets (e. g. o [n+, 7t°, n~) or (p, n) are nearly equal. We can therefore conclude that the chiral SU() is broken to the vectorial isospin SU(2)y (in other words, the SU(2)a remains hidden). The LSM and the CDM are characteristic representatives of phenomenological quark soli ton models of the nucleon implementing the concept of chiral symmetry and containing th Wi i alulati t cl xi f fct mechanism of its breakdown at the quark level (see appendix F for the basics of the models). In these models, the nucleon is described in terms of a core consisting of three u/d valence quark coupled to a-meson and pion fields The axial current operator has the for w(x) = ^My^iTMx) ff(x)3(x) -7(x)3a(x) (6 where iK*) arc the quark spinors and the |TT) ('hh' stands for 'hedge hog') in which the bare quark core |B) is coupled to the cloud of a-mesons I) and pions |TT) (see (G.5)), do not correspond to physical states, since spin and isospin are not good quantu numbers: the meson part of the baryon wave-function is a superposition of the meson vacuum and components with one, two, or more mesons, and the quark part is also a superposition o a three-quark state with the quantum numbers of the nucleon, and the three-quark state with the quantum numbers of the A. In other words, the model wave-function emerges as a soliton which is neither an eigenstate of angular momentum nor isospin, and therefore breaks the ro tational (spin and isospin) invariance of the Lagrangian. Although the expectation value of th angular momentum operato between hedgehog states vanishes we hav M>hhl : J : liphh) + 0 . The physical model states of the nucleon and th A can be derived from the model states | iKh by means of the Peierls-Yoccoz projection [11 IJTMMt) = pLMt|iW where P^MM is the projection operator yielding a state of definite spin and isospin. Because o the grand-spin symmetry of the hedgehog state (see appendix G), only one of the projection onto spin J or isospin T is sufficient, since a projection onto J automatically projects also ont T = J, and vice versa. The projector which projects a state with angular momentum J an isospin T = J from the hedgehog is given b ^mt ("DMt ^ 3H V*^Mj ) R() (66 where CI = (a, |3,y) are the Euler angles, V'M K) are the Wigner functions and R(fl) is th rotation operator. We consider only states with M = — Mj and use the shorthand notation Pj^ M = PjM, where Pj!^ _M = PlJM m- The projected baryon states obtained in this projection can now be used in the calculation of physical observables The expectation value correspond ing to an arbitrary operato O is (JTMMj I I JTMMj) = <^hh}M M^hh) • Thus to calculate gA in the LSM or the CDM one has to evaluate the expectation value of (6 between nucleon states i e lll HH alulati th cl xi f fct In both models, the resulting expressions for t th pi ntributi t t xi coupling constant have the same algebraic for !? (m) a 5 3 4N [ 1/ drr (u ±v) A d( drr - (7 (68 where Nn is the calculated number of pions before the projection and T-\/2 and T3/2 are th overlaps of the unprojected and the projected nucleon states (see appendix G). The factors i front of the integrals originate in the spin/isospin structure of the model wave-function and are fixed. The radial functions u(r), v(r), (r) (see (G.l), (G.3) and (G.4) for thei definitions) depend on the choice of the quark-meson coupling constant g, and are determine variationally. In the LSM, a typical result with g = 5.0 is gA = Q^+Q^ = °-961 +0-823 = 1.783 In the CDM, where the pion field is relatively weak since the quarks are basically bound by th X field alone, we obtain qa = L a 1 0 1 with a typical couplin constant o / = OGeV. 6 efnitio o (q2) Formally, the axial-vector form factor is defined as the coefficient of the axial-vector term in th general Lorentz decomposition 7 of the matrix elemen (Nf(p)|(0)|(puf(pf (q)T (q (q • Y5y(p of the space par of the axial-vector curren A1, where j is the isospin index, q = pi — pf i th four-momentum transfer, and the spinors u; and Uf satisfy the Dirac equation. Here Ga(c) = 9A(q2)/9A(0) is the axial-vector form factor, Gp(q2) is the induced pseudo-scalar, and Gx(q2) is the pseudo-tensor form factor (see also appendix A). Requiring that the axial current is Hermitian and that its matrix element is invariant with respect to time reversal, we get Gj = 0 This equation is the starting point for all model calculations of the axial form factor: its RHS is fixed by the Lorentz and Dirac properties of the nucleon spinors whereas the model wave functions and the model axial current operator enter on its LHS 6 (c2) i t MIT a a i odels with scalar confinin potentia For a nucleon consisting of a core of three point-like u and d quarks confined in a scalar potential, expression (63) can easily be generalised to q /0. Without centre-of-mass corrections we obtain , . , ,\l, , ^i h(qr) )o(qr) [(r)-g(r)j (q (r) where f(r) and g(r) are the upper and lower components of the spinor that solves the Dira equation for massless quarks in the confining potential M(r) In the MIT bag model where th 7It can be shown [12] that 12 derent independent Lorent axial-vectors cn be constrcted from the pertaining four-vectors q = (pj—Pf) and (pj+Pf), and from the matrices y|x, 75 and c^"" = (i/2) [y^y^ ]. Using the properties of the y-matrices and the Dirac equation, the number of independent axialvectors reduces to 3: yy qy and ffw which are alo used here (see alo (Al)) alulati t cl xi f fct single quark wave-function is completely determined by its eigenenergy o>o (and also in th CBM, where the same holds and where the pion contribution to Ga(q2) vanishes just as it di in gA) the result is even simple (q (a l M i (q qr where x = a>or/R. Attempts to eliminate the centre-of-mass motion and to accoun for recoi effects in Ga(c|2) face the same problems as for the gA- When corrections are applied [115], th general trend is to bring the calculated q-dependence into a much better agreement with th experiment (see figur 1) 6 GA (q2) in the LS and the CD The most convenient reference frame to calculate the nucleon form factors in the LSM and th CDM is the hadron Breit frame in which the momenta of the target and the recoil nucleon ar anti-parallel and equal in magnitude Pi = (E.^) Pf = (E,l) E = ( 1/, and therefore q = (0, q), i. e. the energy transfer is zero and q general expression for the axial current matrix element reduces t . I this frame, th (l)|(0)|(l)= TrGA(q (q (q (9 where <7l = q(aq),