Calculations of form factors in different forms of relativistic kinematics
are presented. They involve the instant, front and point forms. In the two
first cases, different kinematical conditions are considered while in the
latter case, both a Dirac-inspired approach and a hyperplane-based one are
incorporated in our study. Numerical results are presented for the pion form
factors with emphasis on both the low and high Q**2 range. A new argument is
presented, explaining why some approaches do considerably much better than
other ones whenonly a single-particle current is considered.Comment: To appear in the proceedings of MENU04 (Beijing, Aug. 29- Sept. 4,
  2004

B. DESPLANQUES

Desplanques B.

Keister B. D.

English

arXiv

Crossref

NUCLEON AND PION FORM FACTORS IN DIFFERENT FORMS OF RELATIVISTIC KINEMATICS

Calculations of form factors in different forms of relativistic kinematics are presented. They involve the instant, front and point forms. In the two first cases, different kinematical conditions are considered while in the latter case, both a Dirac-inspired approach and a hyperplane-based one are incorporated in our study. Numerical results are presented for the pion form factors with emphasis on both the low and high Q2 range. A new argument is presented, explaining why some approaches do considerably much better than other ones when only a single-particle current is considered

Desplanques, B.

Hal - Université Grenoble Alpes

Nucleon and pion form factors in different forms of relativistic kinematics

HAL-IN2P3

Nucleon and pion form factors in different forms ofrelativistic kinematicsB. DesplanquesTo cite this version:B. Desplanques. Nucleon and pion form factors in different forms of relativistic kinematics.Aug 2004, Beijing, 20, pp.1601-1606, 2005, <10.1142/S0217751X05023050>. <hal-00003237>HAL Id: hal-00003237https://hal.archives-ouvertes.fr/hal-00003237Submitted on 8 Nov 2004HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.ccsd-00003237, version 1 - 8 Nov 2004November 8, 2004 15:5 WSPC/INSTRUCTION FILE paper.hyper27699International Journal of Modern Physics Ac© World Scientific Publishing CompanyNUCLEON AND PION FORM FACTORS IN DIFFERENT FORMSOF RELATIVISTIC KINEMATICSB. DESPLANQUESLaboratoire de Physique Subatomique et de Cosmologie (UMR IN2P3/CNRS-UJF-INPG),F-38026 Grenoble Cedex, FranceCalculations of form factors in different forms of relativistic kinematics are presented.They involve the instant, front and point forms. In the two first cases, different kinemati-cal conditions are considered while in the latter case, both a Dirac-inspired approach anda hyperplane-based one are incorporated in our study. Numerical results are presentedfor the pion form factors with emphasis on both the low and high Q2 range. A newargument is presented, explaining why some approaches do considerably much betterthan other ones when only a single-particle current is considered.Keywords: 03.65.Pm; 21.45.+v; 13.40.Gp1. IntroductionThe knowledge of hadron form factors, especially for the nucleon and the pion ones,represents an important source of information about the structure of the systemsunder consideration. By varying the momentum transfer, large as well as smalldistances can be explored, allowing one to learn about hadronic physics in theperturbative and non-perturbative regimes of QCD and its modelization. Involvinglarge momentum transfers, the above study supposes that a reliable implementationof relativity is made. This is mandatory if some information about the hadronicstructure is to be looked for from experiments.There are many ways to implement relativity in the description of properties ofa few-body system. The most ambitious one is based on field theory but, at present,its use for the nucleon form factor is hardly conceivable. A quite different approachinvolves relativistic quantum mechanics (RQM), which contrary to the previousone, assumes a fixed number of degrees of freedom. Less fundamental, this one ishowever more adapted when a modelization of hadrons from constituent particlesis used, as most often done. Following Dirac,1 many approaches along these lineshave been proposed depending on the symmetry properties of the hypersurface onwhich physics is described. This reflects in the construction of the Poincare´ groupgenerators, which drop accordingly into dynamical and kinematical operators.When calculating properties such as form factors, all approaches should convergeto a unique answer but, of course, some may be more convenient in that the bulkcontribution is produced by a one-body current. In other ones, large contributions1November 8, 2004 15:5 WSPC/INSTRUCTION FILE paper.hyper276992 B. Desplanquesfrom two- or many body-currents may be required, possibly obscuring conclusionsabout the physics. This requires that independent studies be performed to establishthe respective advantage of various approaches by comparing their predictions.Those studies that will be presented here have largely been motivated by thesuccessful description of the nucleon form factors in the “point-form” approach2while a standard front-form one3 is failing in the same conditions. Adding to thispuzzling situation, it is noticed that accounting for the well known physics underly-ing the vector-meson-dominance phenomenology, ignored in the former case, wouldreduce the discrepancy in the latter one. For the present purpose however, we willconsider a system simpler than the nucleon, namely the pion. Apart from the factthat there is an evident logics in considering systems with increasing complexity, thesmallness of the pion mass in comparison with the sum of the constituent massesturns out to considerably enhance the differences between various approaches. Thiscan contribute to sharpen the conclusions.The plan of the paper is as follows. In the second section, we precise the ingre-dients entering the calculation of form factors in different kinematics of relativiticquantum mechanics (RQM): instant, front and point forms. For each approach dif-ferent cases, described in the text, are considered. Results for both the charge andscalar pion form factors are presented and discussed in the third section. Some at-tention is given to their asymptotic behavior. The fourth section is devoted to theconclusion where the role of the space-time translation invariance is evoked. Due tolimited space, we skip many details and refer to published works for them.4,5,62. Different forms of relativistic quantum mechanics: a few pointsE  , Pi i e  , pp E  , Pf fe  , pf fie  , piqFig. 1. Contribution to the form factor in the single-particle approximationIn order to calculate form factors of a given system, two ingredients are needed:the relation between the momenta of its constituents and the total momentum,which characterizes each RQM approach (see kinematics in Fig. 1), and a solutionof a mass operator, which can be chosen as independent of the approach. They aresuccessively discussed in the following.For the two-body system of interest here, the relation between the momenta ofits constituents and the total momentum takes a unique form. This one reads:~p1 + ~p2 − ~P =~ξξ0(e1 + e2 − EP ), (1)November 8, 2004 15:5 WSPC/INSTRUCTION FILE paper.hyper27699Forms of Relativistic Kinematics 3where the 4-vector, ξµ, is representative of the symmetry properties (if any) evi-denced by the hypersurface which physics is described on. Accordingly, the Poincare´group generators, Pµ (P 0, ~P ) and Mµν ( ~K, ~J) drop into dynamical or kinematicalones.7 This character together with the 4-vector ξµ are precised below:- instant form: t = τ, ξ0 = 1, ~ξ = 0 ;dynamical: P 0, ~K, kinematical: ~P , ~J ,- front form: t− ~n · ~x = τ, ξ0 = 1, ~ξ = ~n ,where ~n is a unit vector with a fixed direction, generally chosen opposite to thez-axis orientation (ξ2 = 0);dynamical: P 0−P z, J⊥, kinematical: P0+P z, P⊥, Jz, Kz, K⊥−zˆ×J⊥,- Dirac’s inspired point form: t2 − ~x2 = τ, ξ0 = u0 = 1, ~ξ = ~u ,5where ~u is a unit vector that points to any direction, consistently with theabsence of a particular 3-direction on a hyperboloid (ξ2 = 0);dynamical: P 0, ~P , kinematical: ~K, ~J .An “instant-form” approach “which displays the symmetry properties inherentlypresent in the point-form” one has been proposed.8 The Poincare´ group generators,Pµ and Mµν , have respectively a dynamical and a kinematical character, as for theDirac’s point form. However, as noticed by Sokolov,9 it implies physics describedon an hyperplane perpendicular to the velocity of the system under consideration(hypersurface v ·x = τ). It therefore differs from the Dirac’s one. This “point form”,which has been referred to in many recent applications,2,10,11,12,13 evidences spe-cific features. Contrary to the other approaches mentioned above, the 4-vector, ξµ,depends on the properties of the system (ξµ ∝ Pµ). This approach is also on adifferent footing with other respects.5For the mass operator, we refer to an equation used in our previous works4,5,6with appropriate changes due to the 1/2-spin of the constituents.13 For our purpose,which is mainly to compare different approaches between themselves rather than toexperiment, we include in the interaction a confining potential with string tension,σs.t. = 1GeV/fm and a gluon exchange one with strength αs = 0.35. This lastcontribution is of relevance to test the ability of RQM approaches in reproducing theexpected QCD asymptotic behavior of form factors. As been noticed,14 this behavioris closely related to the most singular part of the interaction at short distances. Thequark and pion masses are taken as mq = 0.3GeV and Mpi = 0.14GeV.Expressions of the single-particle contribution to form factors in the spinlesscase have been given elsewhere (see for instance Ref. 6). They can be expressed asan integral over the spectator-particle momentum. Interestingly, they take a uniqueform in most cases, which allows one to discard major biases in the comparisonof different approaches. Their derivation supposes to express the momenta of theconstituents, ~pi,f and ~p in Fig. 1 in terms of the total momentum, ~P , the internalvariable appearing in the mass operator, ~k, and the 4-vector, ξµ. The relation of ~p ’sto the ~k variable assumes a Lorentz-type transformation while fulfilling Eq. (1). Itis nothing but the Bakamjian-Thomas one in a particular case.15November 8, 2004 15:5 WSPC/INSTRUCTION FILE paper.hyper276994 B. Desplanques0 0.05 0.1 0.15 0.2Q2 [(GeV/c)2]00.20.40.60.81F 1(Q2 )F.F. (perp.)I.F. (Breit frame)D.P.F.I.F.+F.F. (parallel)"P.F."m=0.3 GeV, M=0.14 GeV0 2 4 6 8 10Q2 [ (GeV/c)2]10-610-410-2100Q2  F 1(Q2 ) [(GeV/c)2 ]F.F. (perp.)I.F. (Breit frame)D.P.F.I.F.+F.F. (parallel)"P.F."m=0.3 GeV,  M=0.14 GeVFig. 2. Pion charge form factor at low and high Q2 together with experimental data3. Results for the charge and scalar pion form factorsThe pion has two form factors: the charge one, F1(Q2), for which measurements areavailable16,17,18 and a scalar one, F0(Q2), which, in absence of an appropriate probe,is unknown but can be nevertheless useful for a comparison of different approaches.The low and high Q2 behaviors of F1(Q2), in relation with the charge radius or theasymptotic behavior, are of special interest. Moreover, as the instant- and front-form form factors are not Lorentz invariant, they can be considered for variouskinematical configurations. Besides the standard ones (respectively Breit frame andq+ = 0), we consider both of them for a parallel kinematics and |~Pi + ~Pf | → ∞,where they coincide. We also consider results in two point-form approaches, whichcontrary to the other forms, are Lorentz invariant.Results for F1(Q2) are presented in Fig. 2. They clearly fall into two sets: thestandard instant- and front-form form factors that are relatively close to experimentand the other ones that are far apart. Looking in detail at these last ones, it isfound that they roughly depend on the momentum transfer Q through the quantity(2e¯k/Mpi)Q, hence a charge radius scaling like the inverse of the pion mass, whichexplains the rapid fall off of the corresponding form factors at low Q2 (a rapid fall offis also found in truncated field-theory calculations19,20,21). At higher Q2, it soundsthat the Q−2 asymptotic behavior is reached. Actually, examination of the chargeform factor at much higher Q2 indicates that the behavior is Q−4 (see Fig. 3). Theconsideration of the scalar form factor, F0(Q2), is especially useful here. As Fig. 3shows, this form factor has the QCD-expected Q−2 asymptotic behavior, indicatingthat there is nothing wrong with the solution of the mass operator we used. Inorder to get the right power-law behavior for the charge form factor, we consideredthe contribution of two-body currents (pair-type term). This one, determined sothat to reproduce the full Born amplitude,22 has been calculated in the instantform. As Fig. 3 shows, it provides the right Q−2 asymptotic behavior. Moreover,the coefficient has the expected expression (up to a numerical factor).November 8, 2004 15:5 WSPC/INSTRUCTION FILE paper.hyper27699Forms of Relativistic Kinematics 50 20 40 60 80 100Q2 [(GeV/c)2]10-710-510-310-1101Q2 F 0(Q2 )  [(GeV/c)2]F.F. (perp.)I.F. (Breit frame)D.P.F.I.F.+F.F. (parallel)"P.F."m=0.3 GeV, M=0.14 GeV0 20 40 60 80 100Q2 [(GeV/c)2]00.20.40.60.8Q2 F 1(Q2 ) [(GeV/c)2 ]F.F. (perp.)I.F. (Breit frame)I.F. (B.f.) + "pair"m=0.3 GeV, M=0.14 GeVFig. 3. Pion scalar and charge form factor in the asymptotic regime4. Conclusion and prospectThe examination of the pion charge form factor calculated from a single-particlecurrent in different RQM approaches shows unambiguously that results fall into“good” and “bad” ones. The conclusion is not to be affected by refining the physicaldescription as the discrepancy in the last case reaches huge factors. It fully confirmsthe conclusions achieved in the spinless case whose physical description is simpler.6The QCD asymptotic behavior is obtained from two-body currents.Lorentz invariance is often advertised as a validity criterion of some approach.This view is not however supported by present point-form results, which explicitlyevidence the above invariance property. Moreover, the violation of Lorentz invari-ance, as measured from the rather small discrepancy between the standard instant-and front-form results, does not seem to be necessarily large. Another criterion hastherefore to be found. In a field-theory approach, the 4-momentum is conservedat the vertex representing the interaction of constituents with the external probe.This cannot be generally fulfilled in RQM approaches at the operator level (unlessmany-body currents are considered). One can however require that the property beverified at the level of the matrix element. Considering this weaker argument, it al-lows one to account for the observed discrimination of results into “good” and “bad”ones. As the 4-momentum conservation stems from Poincare´ space-time translationinvariance, fulfilling this property could be the relevant criterion. The above in-variance may also be the important symmetry whose violation is suggested by thepeculiar behavior of some form factors in the limit of a zero-mass system.6,22Poincare´ space-time translation invariance implies relations such as:23[Pµ , Jν(x)]= −i ∂µ Jν(x), (2)which can be used for a quantitative check. Considering a single-particle current,it is found that the equality is satisfied at the matrix-element level for the stan-dard instant- and front-form results. It is violated in all the other cases. Skippingdetails, one finds that the l.h.s. and r.h.s respectively involve the quantities Q andNovember 8, 2004 15:5 WSPC/INSTRUCTION FILE paper.hyper276996 B. Desplanques(2e¯k/Mpi) Q. The extra factor at the r.h.s., (2e¯k/Mpi), which is the same as the oneexplaining the discrepancy between the “good” and “bad” form factors in Fig. 2,provides a measure of the violation of Poincare´ space-time translation invariance.To get rid of it, interaction currents should be considered. Schematically, their effectcould combine with the kinetic energy term, 2 e¯k, at the numerator of the abovefactor so that the overall factor be 1 (using 2 e¯k + V =Mpi).22Coming back to the motivation of the present work, it is noticed that the successof the point-form description of the nucleon form factors is mostly due to a factorsimilar to the above one. As a more complete calculation is expected to removethis factor, a situation similar to the standard front-form calculation should berecovered. We believe it is doubly fortunate. The description of the nucleon formfactor could now incorporate the well known vector-meson dominance phenomeno-logy. The difficulty to reconcile the point-form descriptions of the nucleon and pionform factors, respectively good2 and bad,13 vanishes.While Lorentz invariance has often been advocated in calculating form factors,Poincare´ space-time translation invariance could be a more relevant property.AcknowledgementsWe are very grateful to A. Amghar, T. Melde, S. Noguera and L. Theußl for dis-cussions or comments at various steps of the present work.References1. P.A.M. Dirac, Rev. Mod. Phys. 21, 392 (1949).2. R.F. Wagenbrunn, et al., Phys. Lett. B511, 33 (2001).3. F. Cardarelli, E. Pace, G. Salme`, S. Simula, Phys. Lett. B357, 267 (1995).4. A. Amghar, B. Desplanques, L. Theußl, Nucl. Phys. A714, 213 (2003).5. B. Desplanques, nucl-th/0405059, Nucl. Phys. (accepted).6. B. Desplanques, preprint, nucl-th/0407074.7. B.D. Keister and W. Polyzou, Adv. Nucl. Phys. 20, 225 (1991)8. B. Bakamjian, Phys. Rev. 121, 1849 (1961).9. S.N. Sokolov, Theor. Math. Phys. 62, 140 (1985).10. T.W. Allen, W.H. Klink, Phys. Rev. C58, 3670 (1998).11. T.W. Allen, W.H. Klink, W.N. Polyzou, Phys. Rev. C63, 034002 (2001).12. B. Desplanques, L. Theußl, Eur. Phys. J. A13, 461 (2002).13. A. Amghar, B. Desplanques, L. Theußl, Phys. Lett. B574, 201 (2003).14. C. Alabiso, G. Schierholz, Phys. Rev. D10, 960 (1974).15. B. Bakamjian, L.H. Thomas, Phys. 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