Threshold Pion Electroproduction at Large Momentum Transfers
We consider pion electroproduction close to threshold for in the region GeV on a nucleon target. The momentum transfer dependence of the S-wave multipoles at threshold, and , is calculated in the chiral limit using light-cone sum rules. Predictions for the cross sections in the threshold region are given taking into account P-wave contributions that, as we argue, are model independent to a large extent. The results are compared with the SLAC E136 data on the structure function in the threshold region.
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Threshold pion photo- and electroproduction , is a very old subject that has been receiving continuous attention from both experimental and theoretical side for many years. From the theory point of view, the interest is because in the approximation of the vanishing pion mass chiral symmetry supplemented by current algebra allow one to make exact predictions for the threshold cross sections, known as low-energy theorems (LET) KR (); Nambu:1997wa (); Nambu:1997wb (). As a prominent example, the LET establishes a connection between charged pion electroproduction and the axial form factor of the nucleon. In the real world the pion has a mass, , and the study of finite pion mass corrections to LET was a topical field in high energy physics in the late sixties and early seventies before the celebrated discovery of Bjorken scaling in deep–inelastic scattering and the advent of QCD, see, in particular, the work by Vainshtein and Zakharov Vainshtein:1972ih () and a monograph by Amaldi, Fubini and Furlan AFF () that addresses many of these developments.
Twenty years later, a renewed interest to threshold pion production was trigged by the extensive data that became available on Mazzucato:1986dz (); Beck:1990da () and, most importantly, , at the photon virtuality GeV Welch:1992ex (). At the same time, the advent of chiral perturbation theory (CHPT) has allowed for the systematic expansion of low–energy physical observables in powers of the pion mass and momentum. In particular classic LET were reconsidered and rederived in this new framework, putting them on a rigorous footing, see Bernard:1995dp () for an excellent review. The new insight brought by CHPT calculations is that certain loop diagrams produce non-analytic contributions to scattering amplitudes that are lost in the naive expansion in the pion mass, e.g. in Vainshtein:1972ih (); Scherer:1991cy (). By the same reason, the expansion at small photon virtualities has to be done with care as the limits and do not commute, in general Bernard:1992ys (). The LET predictions including CHPT corrections seem to be in good agreement with experimental data on pion photoproduction Drechsel:1992pn (). Experimental results on the S-wave electroproduction cross section for GeV are consistent with CHPT calculations as well, Bernard:1992rf (); Bernard:1995dp (), and cannot be explained without taking into account chiral loops.
The rapid development of experimental techniques is making possible to study threshold pion production in high-energy experiments and in particular electroproduction with photon virtuality in a few GeV range. Such experiments would be a major step forward and require very fine energy resolution in order to come close to the production threshold to suppress the P-wave contribution of the multipole. Various polarisation measurements can be especially helpful in this respect. We believe that such studies are feasible on the existing and planned accelerator facilities, especially at JLAB, and the task of this paper is to provide one with the necessary theoretical guidance.
In the traditional derivation of LET using PCAC and current algebra is not assumed to be small but the expansion in powers of the pion mass involves two parameters: and Vainshtein:1972ih (); Scherer:1991cy (). The appearance of the second parameter in this particular combination reflects the fact that, for finite pion masses and large momentum transfers, the emitted pion cannot be ’soft’ with respect to the initial and final state nucleons simultaneously. For the threshold kinematics, this affects in particular the contribution of pion emission from the initial state PPS01 () and in fact is nothing but the nucleon virtuality after the pion emission, divided by . It follows that the LET are formally valid (modulo CHPT loop corrections Bernard:1995dp ()) for the momentum transfers as large as where CHPT is no more applicable, at least in its standard form. To the best of our knowledge, there has been no dedicated analysis of the threshold production in the GeV region, however.
For the LET break down: the initial state pion radiation occurs at time scales of order rather than necessitating to add contributions of hadronic intermediate states other than the nucleon. Finally, for very large momentum transfers, the situation may again become tractable as one can try to separate contributions of ’hard’ scales as coefficient functions in front of ’soft’ contributions involving small momenta and use current algebra (or CHPT) for the latter but not for the amplitude as a whole.
This approach was pioneered in the present context in Ref. PPS01 () where it was suggested that for asymptotically large the standard pQCD collinear factorisation technique Efremov:1979qk (); Lepage:1980fj () becomes applicable and the helicity-conserving multipoles can be calculated (at least for ) in terms of chirally rotated nucleon distribution amplitudes. In practice one expects that the onset of the pQCD regime is postponed to very large momentum transfers because the factorisable contribution involves a small factor and has to win over nonperturbative “soft” contributions that are suppressed by an extra power of but do not involve small coefficients.
The purpose of this paper is to suggest a realistic QCD-motivated model for the dependence of both transverse and longitudinal S-wave multipoles at threshold in the region GeV that can be accessed experimentally at present or in near future. In Ref. Braun:2001tj () we have developed a technique to calculate baryon form factors for moderately large using light-cone sum rules (LCSR) Balitsky:1989ry (); Chernyak:1990ag (). This approach is attractive because in LCSR “soft” contributions to the form factors are calculated in terms of the same nucleon distribution amplitudes (DAs) that enter the pQCD calculation and there is no double counting. Thus, the LCSR provide one with the most direct relation of the hadron form factors and distribution amplitudes that is available at present, with no other nonperturbative parameters.
The same technique can be applied to pion electroproduction. In Ref. Braun:2006td () the relevant generalised form factors were estimated in the LCSR approach for the range of momentum transfers GeV. For this work, we have reanalysed the sum rules derived in Braun:2006td () taking into account the semi-disconnected pion-nucleon contributions in the intermediate state. We demonstrate that, with this addition, the applicability of the sum rules can be extended to the lower region and the LET are indeed reproduced at GeV to the required accuracy . The results presented in this work essentially interpolate between the large- limit considered in Braun:2006td () and the standard LET predictions at low momentum transfers.
The presentation is organised as follows. Section 2 is introductory and contains the necessary kinematics and notations. In Section 3 we define two generalised form factors that contribute to pion electroproduction at the kinematic threshold, explain the relation to S-wave multipoles and suggest a model for their dependence based on LCSR. The details of the LCSR calculation are presented in the Appendix. In Section 4 we suggest a simple model for the electroproduction close to threshold, complementing the S-wave form factor-like contributions by P-wave terms corresponding to pion emission in the final state that can be expressed in terms of the nucleon electromagnetic form factors. In this framework, detailed predictions are worked out for the differential cross sections from the proton target and also for the structure functions measured in the deep-inelastic scattering experiments. The comparison with SLAC E136 results Bosted:1993cc () is presented. The final Section 5 is reserved for a summary and conclusions.
Ii Kinematics and Notations
For definiteness we consider pion electroproduction from a proton target
Basic kinematic variables are
The identification of the momenta is clear from Eq. (II); is the nucleon and the pion mass, respectively. In what follows we neglect the electron mass and the difference of proton and neutron masses.
The differential cross section for electron scattering in laboratory frame is equal to
where the sum (average) over the polarisations is implied, , and being the polar and azimuthal angles of the pion in the final nucleon-pion c.m. frame, respectively, the electromagnetic current is defined as
and is the kinematic factor related to the c.m.s. momentum of the subprocess in the final state:
Alternatively, instead of the polar angle dependence, one could use the Mandelstam -variable of the subprocess
where is the c.m.s. momentum in the initial state:
Traditionally one writes the electron scattering cross section in (2.3) in terms of the scattering cross section for the virtual photon
is the virtual photon flux and
In turn, it is convenient to separate an overall kinematic factor in the virtual photon cross section
For unpolarised target can be written as a sum of contributions
We will also use the notation
for the corresponding partial cross sections. The invariant functions etc. depend on the invariants of the subprocess only; in the last term in (2.13) is the beam helicity.
Iii Generalised form factors
Pion electroproduction at threshold from a proton target can be described in terms of two generalised form factors Braun:2006td () in full analogy with the electroproduction of a spin-1/2 nucleon resonance:
The form factors and are real functions of the momentum transfer and can be related to the S-wave transverse and longitudinal multipoles:
Here is the photon energy in the c.m. frame (at threshold). For physical pion mass both form factors are finite at . However, develops a singularity at in the chiral limit . The differential cross section at threshold is given by
The LET KR (); Nambu:1997wa (); Nambu:1997wb () can be formulated for the form factors directly; the corresponding expressions can be read e.g. from Ref. Scherer:1991cy (). Neglecting all pion mass corrections one obtains
where and are the Sachs electromagnetic form factors of the proton and neutron, respectively, and the axial form factor induced by the charged current; is the axial coupling. In this expression the terms in and correspond to the pion emission from the initial state whereas the contribution of (Kroll-Ruderman term KR ()) is due to the chiral rotation of the electromagnetic current. The correspondence between and becomes especially simple to this accuracy:
In the photoproduction limit one obtains and so that many more are produced at threshold compared to , in agreement with experiment.
As already mentioned, although LET were applied historically to small momentum transfers GeV their traditional derivation using PCAC and current algebra does not seem to be affected as long as the emitted pion remains ’soft’ with respect to the initial state nucleon. Qualitatively, one expects from (3.3) that the production cross section increases rapidly with whereas the cross section, on the contrary, decreases since contributions of and have opposite sign. We are not aware of any dedicated analysis of the threshold pion production data in the GeV region, however. Such a study can be done, e.g., in the framework of global partial wave analysis (PWA) of and scattering (cf. Drechsel:1998hk (); Arndt:2002xv (); Arndt:2006ym (); Drechsel:2007if ()) and to our opinion is long overdue.
For the LET break down: the initial state pion radiation occurs at time scales of order rather than necessitating to add contributions of all hadronic intermediate states other than the nucleon. In perturbative QCD one expects that both form factors scale as at asymptotically large momentum transfers. In particular is calculable in terms of pion-nucleon distribution amplitudes using collinear factorisation PPS01 (). In Ref. Braun:2006td () we have suggested to calculate the form factors and using the LCSR. The motivation and the theoretical foundations of this approach are explained in Braun:2006td () and do not need to be repeated here. The starting point is the correlation function
where is a suitable operator with nucleon quantum numbers, see a schematic representation in Fig. 1.
When both the momentum transfer and the momentum flowing in the vertex are large and negative, the main contribution to the integral comes from the light-cone region and the correlation function can be expanded in powers of the deviation from the light cone. The coefficients in this expansion are calculable in QCD perturbation theory and the remaining matrix elements can be identified with pion-nucleon distribution amplitudes (DAs). Using chiral symmetry and current algebra these matrix elements can be reduced to the usual nucleon DAs. On the other hand, one can represent the answer in form of the dispersion integral in and define the nucleon contribution by the cutoff in the invariant mass of the three-quark system, the so-called interval of duality (or continuum threshold). This cutoff does not allow large momenta to flow through the -vertex so that the particular contribution shown in Fig. 1 is suppressed if becomes too large. Hence the large photon momentum has to find another way avoiding the nucleon vertex, which can be achieved by exchanging gluons with large transverse momentum between the quarks. In this way the standard pQCD factorisation arises: leading pQCD contributions correspond to three-loop corrections in the LCSR approach. For not so large , however, the triangle diagram in Fig. 1 actually dominates by the simple reason that each hard gluon exchange involves a small factor which is a standard perturbation theory penalty for each extra loop.
The LCSR for pion electroproduction involve a subtlety related to the contribution of semi-disconnected pion-nucleon contributions in the dispersion relation. In Ref. Braun:2006td () such contributions were neglected, the price being that the predictions could only be made for large momentum transfers of order GeV. For the purpose of this paper we have reanalysed the sum rules derived in Braun:2006td () taking into account the semi-disconnected pion-nucleon contributions explicitly, see Appendix A. We demonstrate that, with this modification, the sum rules can be extended to the lower region so that the LET expressions in (3.3) are indeed reproduced at GeV to the required accuracy .
Note that the LCSR calculation is done in the chiral limit, we do not address finite pion mass corrections in this study. Beyond this, accurate quantitative predictions are difficult for several reasons, e.g. because the nucleon distribution amplitudes are poorly known. In order to minimize the dependence of various parameters in this work we only use the LCSR to predict certain form factor ratios and then normalise to the electromagnetic nucleon form factors as measured in experiment, see Appendix A for the details.
The sum rules in Braun:2006td () have been derived for the proton target but can easily be generalised for the neutron as well, which only involves small modifications. We have done the corresponding analysis and calculated the generalised form factors for the threshold pion electroproduction both from the proton, , and the neutron, , . The results are shown in Fig. 10 and Fig. 11, respectively.
The resulting LCSR-based prediction for the S-wave multipoles for the proton target is shown by the solid curves in Fig. 2. The four partial waves at threshold that are related to the generalised form factors through the Eq. (3.4) are plotted as a function of , normalised to the dipole formula
This model is used in the numerical analysis presented below. It is rather crude but can be improved in future by calculation of radiative corrections to the sum rules and if lattice calculations of the parameters of nucleon DAs become available. To give a rough idea about possible uncertainties, the “pure” LCSR predictions (all form factors and other input taken from the sum rules) are shown by dashed curves for comparison.
Iv Moving away from threshold
We have argued that the S-wave contributions to the threshold pion electroproduction are expected to deviate at large momentum transfers from the corresponding predictions of LET and suggested a QCD model that should be applicable in the intermediate region. In contrast, we expect that the P-wave contributions for all are dominated in the limit by the pion emission from the final state nucleon (see also PPS01 ()). Adding this contribution, we obtain a simple expression for the amplitude of pion production close to threshold, :
Hereafter and are the Dirac and Pauli electromagnetic form factors of the proton, and are the isospin coefficients.
The separation of the generalised form factor contribution and the final state emission in (IV) can be justified in the chiral limit but involves ambiguities in contributions . We have chosen not to include the term in the numerator of the proton propagator in the second line in (IV) so that this contribution strictly vanishes at the threshold. In addition, we found it convenient to include the term in the Lorentz structure that accompanies the form factor in order to make the amplitude formally gauge invariant. To avoid misunderstanding, note that our expression is not suitable for making a transition to the photoproduction limit in which case, e.g. pion radiation from the initial state has to be taken in the same approximation to maintain gauge invariance.
The amplitude in Eq. (IV) does not take into account final state interactions (FSI) which can, however, be included in the standard approach based on unitarity (Watson theorem), writing (cf. e.g. Drechsel:1998hk ())
where is the pion-nucleon elastic scattering amplitude (for a given isospin channel) with the S-wave phase shift and inelasticity parameter . We leave this task for future, but write all expressions for the differential cross sections and the structure functions for generic complex and so that the FSI can eventually be incorporated. Of course, FSI in P-wave also have to be added.
Using Eq. (IV) one can calculate the differential virtual photon cross section (2.12), (2.13). The complete expressions for the invariant functions are rather cumbersome but are simplified significantly in the chiral limit and assuming . We obtain
The measurements of the differential cross sections at large in the threshold region would be very interesting as the angular dependence discriminates between contributions of different origin. In our approximation (exactly) which is because we do not take into account the D-wave. Consequently, to our accuracy the contribution to the cross section is absent so that its measurement provides one with a quantitative estimate of the importance of the D-wave terms in the considered range. Also note that the single spin asymmetry contribution involves imaginary parts of the generalised form factors that arise because of the FSI (and are calculable, at least in principle). The numerical results shown below are obtained using exact expressions for ; the difference is less than 20% in most cases. Strictly speaking, this difference is beyond our accuracy although one might argue that kinematic factors in the calculation of the cross section should be treated exactly.
In fact the curve appears to be practically linear and there is no azimuthal angle dependence. This feature is rather accidental and due to an almost complete cancellation of the contributions to from and for the chosen value of . It is very sensitive to the particular choice of model parameters and does not hold in the general case.
The integrated cross section (in units of GeV) as a function of for GeV (lower curve) and GeV (upper curve) is shown in Fig. 4. The predicted scaling behaviour
is consistent with the SLAC measurements of the deep-inelastic structure functions Bosted:1993cc () in the threshold region that we are going to discuss next.
To avoid misunderstanding we stress that the estimates of the cross sections presented here are not state-of-the-art and are only meant to provide one with the order-of-magnitude estimates of the threshold cross sections that are to our opinion most interesting. These estimates can be improved in many ways, for example taking into account the energy dependence of the generalised form factors generated by the FSI and adding a model for the D-wave contributions. The model can also be tuned to reproduce the existing lower and/or larger experimental data. A more systematic approach could be to study the threshold production in the framework of global PWA of and scattering using QCD-motivated S- and P-wave multipoles and the D- and higher partial waves estimated from the analysis of the resonance region (cf. Drechsel:1998hk (); Arndt:2002xv (); Arndt:2006ym (); Drechsel:2007if ()) where there is high statistics.
V Structure Functions
The deep-inelastic structure functions and are directly related to the total cross section of the virtual photon–proton interaction. For the longitudinal photon polarisation one obtains
and for the transverse
Here we introduced the Bjorken variable
It is customary to write the total cross section in terms of the structure function and , the ratio of the longitudinal to transverse cross sections:
In the threshold region , , the structure functions can be calculated starting from the amplitude in Eq. (IV). In particular for we obtain
Similar to the differential cross sections, expressions for the structure functions are simplified considerably in the chiral limit and assuming : we have to retain the kinematic factor but can neglect the pion mass corrections and the difference whenever possible. The results are
where, for completeness, we included the polarised structure functions and . Note that in this limit the contributions and can be identified with the pure S-wave and P-wave, respectively. Numerically, the difference between the complete expressions like the one in (5.4) and the ones in the chiral limit in (5.5) is less than 20% and, strictly speaking, beyond our accuracy.