Anomalous quark-gluon chromomagnetic interaction and high energy ρ -meson electroproduction
aa r X i v : . [ h e p - ph ] F e b Anomalous quark-gluon chromomagnetic interactionand high energy ρ -meson electroproduction Nikolai Korchagin a, , Nikolai Kochelev a, , Nikolai Nikolaev b,c, (a) Bogoliubov Laboratory of Theoretical Physics, Joint Institute for Nuclear Research,Dubna, Moscow region, 141980, Russia (b)
L.D. Landau Institute for Theoretical Physics,st. Kosygina 2, 119334 Moscow, Russia (c)
Forschungszentrum, Institute f¨ur Kernphysik, Postfach 1913, D-52425 J¨ulich,Germany
Abstract
It is shown that existence of a large anomalous chromomagnetic moment of quarkinduced by non-perturbative structure of QCD leads to the additional contribution toexclusive ρ -meson electroproduction off proton target. The significant contribution com-ing from new type of quark-gluon interaction to the ρ -meson production cross sectionfor both transversal and longitudinal polarization of virtual photon is found. Such non-perturbative contribution together with conventional perturbative two-gluon exchangeallows us to describe the experimental data at low Q for transversal polarization. How-ever, in the longitudinal polarization case there is still some discrepancy with the data.The possible source of this deviation is discussed.Pacs: 24.85.+p, 12.38.-t, 12.38.Mh, 12.39.MkKeywords: quarks, gluons, non-perturbative QCD, vector meson, electroproduction [email protected] [email protected] [email protected] Introduction
One of salient features of perturbative high-energy QCD is conservation of the s -channelhelicity of quarks. On the other hand, the QCD vacuum possesses a nontrivial topologicalstructure - instantons are an extensively studied example (for the reviews [1, 2]). Suchtopological fluctuations generates the celebrated multiquark t’Hooft interaction which isresponsible, for example, for the solution of U A (1) problem in QCD [3]. Additionally,instantons were shown to generate a non-trivial spin-flip, i.e., s-channel helicity non-conserving, quark-gluon interaction [4]. This interaction can be described in terms of ananomalous chromomagnetic moment of quarks (ACMQ) complementary to the perturba-tive Dirac coupling. Novel contributions from this non-perturbative interaction to the softPomeron, gluon distribution in the nucleon and sizable spin effects in strong interactionshave already been discussed in the literature [4, 5, 6, 2]. A magnitude of the ACMQ canbe related to the instanton density in the QCD vacuum.The exclusive vector meson electroproduction is a unique testing ground of the s -channel helicity properties of the quark-gluon coupling. Specifically, s -channel helicitynon-conserving transitions from photons to vector mesons are possible even within theperturbative QCD for a fundamental reason that a helicity of mesons can be different froma sum of the quark and antiquark helicities (see [7] and references therein). The ACMQwould introduce an extra contribution to both the s -channel helicity conserving and non-conserving vector meson production amplitudes. Although it comes from manifestly softregion, a direct evaluation of such a contribution is needed.Helicity properties of electroproduced vector mesons have been extensively studied inthree experiments at HERA DESY, i.e. by the H1, ZEUS and HERMES collaborations.Although gross features of these data are well consistent with pQCD-based theoreticalpredictions (see review [7] and recent development in [8],[9] ), there remain several openissues. An outstanding problem is a large relative phase of the leading helicity amplitudesfor production of longitudinal and transverse ρ ’s and φ ’s, which has been observed by theHERMES and H1 Collaborations [10, 11, 12]. It is definitely larger than the predictionbased on the handbag mechanism [13] and also in the conventional pQCD pomeron-basedcolor dipole approach [14] (see discussion in [7]). Besides that, pQCD-driven approachesseem to fail with the experimentally observed Q dependence of the ratio σ L /σ T in full ex-perimentally studied range of Q : theoretical calculations substantially overestimate thisratio at large Q . Finally, it is important to evaluate an impact of new non-perturbativemechanism on the transition from real to virtual electroproduction.In the present paper we study exclusive ρ - meson electroproduction off protons withallowance for the novel s -channel helicity non-conservation mechanism induced by theanomalous chromomagnetic moment of quarks. We focus on the simplest observables, σ L and σ T , a calculation of the full set of helicity amplitudes will be reported elsewhere.1 Anomalous quark-qluon chromomagnetic interac-tion
In the most general case, the interaction vertex of massive quark with gluon can be writtenin the following form: V µ ( k , k , κ ) t a = − g s t a [ F ( k , k , κ ) γ µ + σ µν κ ν m F ( k , k , κ )] , (1)where the first term is a conventional perturbative QCD quark-gluon vertex and the secondterm comes from non-perturbative sector of QCD. In Eq.1 the form factors F , describea nonlocality of the non-perturbative interaction, k , are the momenta of incoming andoutgoing quarks, respectively, and κ = k − k , m is the quark mass, and σ µν = ( γ µ γ ν − γ ν γ µ ) /
2. In what follows, we focuse on effects of the novel color chromomagnetic vertexand keep F ( k , k , κ ) = 1. The anomalous quark chromomagnetic moment (AQCM)equals µ a = F (0 , , . In the earlier paper [4] it was shown that instantons, a strong vacuum fluctuationsof gluon fields with nontrivial topology, generate an ACMQ which is proportional to theinstanton density µ a = − π Z dρn ( ρ ) ρ α s ( ρ ) . In terms of the average size of instantons ρ c and the dynamical quark mass m in non-perturbative QCD vacuum one finds [5] µ a = − π ( mρ c ) α s ( ρ c ) , (2)which exhibits a strong sensitivity of the ACMQ to a dynamical mass of quarks. To thisend we emphasize an implicit assumptions that quarks are light, ie., the above estimatesof ACQM are valid for u, d, s , while for heavy quarks ACQM vanishes. For the averageinstanton size ρ − c = 0 . m = 170 M eV in the mean field(MF) approximation to m = 345 MeV within Diakonov-Petrov (DP) model. The QCDstrong coupling constant at the instanton scale can be evaluated as α s ( ρ c ) ≈ . , and the resulting AQCM for light quarks is numerically quite large: µ MFa ≈ − . , µ DPa ≈ − . . Recently, an AQCM of similar magnitude has been obtained within the Dyson-Schwingerequation approach to non-perturbative QCD (see discussion and references in[15]).The form factor F ( k , k , κ ) suppresses the AQCM vertex at short distances whenthe respective virtualites are large. Within the instanton model its explicit is related toFourier-transformed quark zero-mode and instanton fields and reads F ( k , k , κ ) = µ a Φ q ( | k | ρ/ q ( | k | ρ/ F g ( | κ | ρ ) , q ( z ) = − z ddz ( I ( z ) K ( z ) − I ( z ) K ( z )) ,F g ( z ) = 4 z − K ( z )are the where I ν ( z ), K ν ( z ), are the modified Bessel functions and ρ is the instanton size.Recent discussion has shown [5] that the ACQM contribution complements the pQCDevaluations of the total quark-quark cross section in a way which improves constituentquark model description of high energy nucleon-nucleon total cross section. Furthermore,such model provide the soft contribution to the gluon distribution in the nucleon whichis consistent with initial conditions to a DGLAP evolution of phenomenological PDFs.A purpose of the present communication is to explore the effects of ACQM in elastic(diffractive) electroproduction of ρ mesons. Driven by VBKL considerations, high energy diffractive production of vector mesons isusually desctribed by an exchange of a color-singlet two-gluon tower in the t -channel. Thecorresponding diagrams are presented in Fig.1. By using the Sudakov expansion for the p p − ∆ q q + ∆ κ − ∆ / κ + ∆ / kq − k a ) b ) c ) d ) Figure 1: The diagrams which contribute to high energy exclusive vector meson electro-production off proton by two gluon exchange. Here the blob stands for the generalizedquark-gluon vertex Eq.1. 3omenta of proton and virtual photon p µ = p ′ µ + m p s q ′ µ , q µ = q ′ µ − xp ′ µ , q ′ = p ′ = 0 , (3)where Q = − q , x = Q s ≪ s = 2( p ′ q ′ ), the quark momentum k in the quark loop,gluon momentum κ , and momentum transfer to proton ∆ (see Fig.1) can be presented inthe following form k µ = yp ′ µ + zq ′ µ + ~k µ ,κ µ = αp ′ µ + βq ′ µ + ~κ µ , (4)∆ µ = δp ′ µ + σq ′ µ + ~ ∆ µ , where any vector ~l is transversal part of four-vector l µ which satisfies the relation ~l · q ′ = ~l · p ′ = 0.Hereafter we follow the k -factorization analyis devloped in [16, 18, 17]. The polariza-tion vectors for virtual photon e and vector meson V are e T µ = ~e µ ,e Lµ = 1 Q ( q ′ µ + xp ′ µ ) ,V T µ = ~V µ + 2( ~ ∆ ~V ) s p ′ µ , (5) V Lµ = 1 M q ′ µ + ~ ∆ − M s p ′ µ + ~ ∆ µ ! where M is the mass of ¯ qq pair on mass-shell: M = ~k + m z (1 − z ) (6)The imaginary part of the amplitude takes the form A ( x, Q , ~ ∆) = − is c V √ πα em π Z dzz (1 − z ) Z d ~kψ ( z, ~k ) Z d ~κ~κ α S F ( x, ~κ, ~ ∆) ×× (cid:20) − zz I ( a ) ~k a + m + z (1 − z ) Q + I ( b ) ~k b + m + z (1 − z ) Q + (7)+ I ( c ) ~k c + m + z (1 − z ) Q + z − z I ( d ) ~k d + m + z (1 − z ) Q (cid:21) . Here α em is the fine-structure constant, c V = 1 / √ ρ meson wave function, F ( x, ~κ, ~ ∆) is the differential gluon density and ψ ( z, ~k ) islight-cone wave function of the ρ -meson, for which we us a simple parameterization ψ = c exp (cid:18) − a p (cid:19) = c exp (cid:18) − a (cid:18) ~k + 14 (2 z − M (cid:19)(cid:19) (8)4here p is 3-dimensional relative momentum of quarks in pair. Two constants a and c were fixed by the normalization of wave function and decay width Γ( ρ → e + e − ), we find a = 3 .
927 GeV − , c = 17 . α s ( q ) = 4 π q + m g ) / Λ QCD ) , (9)where Λ QCD = 0 .
28 GeV and the value m g = 0 .
88 GeV imposes the infrared freezing at α s (1 /ρ c ) ≈ π/ q is the maximum virtuality of momentum which inserted tovertex, i.e. q = Max( k , k , κ ). I ( i ) is the trace over quark line in i ) diagram from Fig.1 divided by 2 s . It is involveda three parts: I ( i ) = I ( i ) pert + I ( i ) cm + I ( i ) mix . (10)Below we are using the following notation for the transverse momentum of quark ~k a = ~k − (1 − z ) ~ ∆ ~k b = ~k − (1 − z ) ~ ∆ + ~κ + 12 ~ ∆ ~k c = ~k − (1 − z ) ~ ∆ − ~κ + 12 ~ ∆ (11) ~k d = ~k + z ~ ∆Such shift of momentum is needed for keeping transverse momentum of quarks insertedto meson vertex to be equal ~k for all diagrams in order to take out ψ ( z, ~k ) as commonmultiplier. Besides, [ ~a~b ] = a x b y − a y b x .The formula for T T transition in case when both of quark-gluon vertices are pertur-bative is I ( c ) pert ( T → T ) = h ( ~e~V ∗ )( m + ~k~k c ) + ( ~V ∗ ~k )( ~e~k c )(1 − z ) − ( ~e~k )( ~V ∗ ~k c ) i . (12)For pure perturbative vertices there is the relation between different contributions I ( b ) pert = I ( c ) pert = − − zz I ( a ) pert = − z − z I ( d ) pert subject to a proper substitution of the loop quark momenta the relevant diagrams ofFig.1.When both quark-gluon vertices come from nonperturbative ACQM, we find I ( a ) cm ( T → T ) = − z − z h ( ~e~V ∗ )( m + ~k~k a ) + ( ~V ∗ ~k )( ~e~k a )(1 − z ) − ( ~e~k )( ~V ∗ ~k a ) i ~κ ×× F ( k I avg , , κ ) F ( k II avg , , κ ) . (13)Here we notice that one of quarks is always on mass-shell, and k I,II avg stand for thevirtuality of the off-mass shell quark. To obtain formula for I ( d ) cm one should perform thesubstitutions z/ (1 − z ) → (1 − z ) /z and ~k a → ~k d in above expression. The AQCMcontribution from graph in Fig.1c is I ( c ) cm ( T → T ) = h(cid:0) (1 − z ) ( ~V ∗ ~k )( ~k c ~e ) − ( ~k~k c )( ~e~V ∗ ) + ( ~k~e )( ~V ∗ ~k c ) (cid:1) ~κ ++ m (cid:0) ~e~κ )( ~V ∗ ~κ ) − ( ~e~V ∗ ) ~κ (cid:1)i F ( k I avg , , κ ) F ( k II avg , , κ ) . (14)5he replacement ~k c → ~k b leads to the formula for I ( b ) cm .The interference of the pQCD and ACQM vertices gives I ( c ) mix ( T → T ) = m h ( ~V ∗ ~k )( ~κ~e )(1 − z ) − [ ~e~κ ][ ~V ∗ ~k ] − ( ~e~k c )( ~κ~V )(1 − z ) − [ ~e~k c ][ ~κ~V ∗ ] i ×× ( F ( k II avg , , κ ) − F ( k I avg , , κ ))(15)The I ( b ) mix is obtained from the above by substitution k c → k b . The remaining amplitudesare I ( a ) mix ( T → T ) = zm − z h ( ~V ∗ ~k )( ~κ~e )(1 − z ) − [ ~e~κ ][ ~V ∗ ~k ] − ( ~e~k a )( ~κ~V )(1 − z ) − [ ~e~k a ][ ~κ~V ∗ ] i ×× (cid:0) F ( k II avg , , κ ) − F ( k I avg , , κ ) (cid:1) (16) I ( d ) mix ( T → T ) = (1 − z ) mz h ( ~V ∗ ~k )( ~κ~e )(1 − z ) + [ ~e~κ ][ ~V ∗ ~k ] − ( ~e~k d )( ~κ~V ∗ )(1 − z ) + [ ~e~k d ][ ~κ~V ∗ ] i ×× (cid:0) F ( k II avg , , κ ) − F ( k I avg , , κ ) (cid:1) . (17)For the longitudinal polarization we obtain I ( c ) pert ( L → L ) = − QM z (1 − z ) I ( c ) cm ( L → L ) = − QM z (1 − z ) ~κ F ( k I avg , , κ ) F ( k II avg , , κ ) . (18)And I ( b ) = I ( c ) = − − zz I ( a ) = − z − z I ( d ) with corresponding ~k i . In the longitudinal casethe contribution from the interference of pQCD and AQCM vertices vanishes I ( a ) mix ( L → L ) = I ( b ) mix = I ( c ) mix = I ( d ) mix = 0 . (19)A common feature of nonperturbative approaches is a fast running dynamical massof constiituent quarks which drops to a small current quark mass at large virtualities.In principle, the running of the quark masses affect the Q dependence of vector mesonproduction observables. However, such an fill-fledged involved calculation is beyond thescope of the present communication. Here we only recall that according to the colortransparency considerations the vector meson production amplitudes are dominated bythe components of the vector meson wave function taken at transverse size, i.e., thescanning radius r S ∼ / p Q + m V [19, 20, 7]. Consequently, the virtuality of quarks is ∝ ( Q + m V ), and arguably one can model an approach to the pQCD regime at largevirtuality of the photon making use of a simple approximation m ( Q ) = m (0)1 + Q /m V , (20)where m (0) = 345 MeV and m V = 770 MeV6 Discussion of the results for σ L and σ T The final result for cross sections is presented in Fig.2 in comparison with data obtainedby H1 and ZEUS Collaborations. Helicity flip transitions are found to give a very smallcontribution to the total production cross section, see also discussion in [7]: σ T = σ T → T + σ T → L ≈ σ T → T (21) σ L = σ L → L + σ L → T ≈ σ L → L . H1 ZEUS L [ nb ] Q [GeV ] -2 -1 H1 ZEUS T [ nb ] Q [GeV ] Figure 2: Cross sections of ρ meson exclusive electroproduction: left panel is for longitu-dinal virtual photon polarization and right panel for transversal photon polarization. Thesolid line is the calculation with AQCM, dashed line is the result of pQCD contributionwith running quark mass, Eq.20, and dotted line is pQCD calculation with fixed quarkmass m q = 220 MeV [17, 18]. Experimental points are taken from [21] for H1 and from[22] for ZEUS Collaborations.The longitudinal cross-section σ L is free of the interference of the pQCD and AQCMvertices. Allowance for AQCM effects slightly enhances σ L in the region of non-perturbativesmall Q , but still the cross section is overestimated in low Q region. There is one caveat,though: we evaluated the pQCD contribution using the unintegrated gluon density whichhas been tuned to the experimental data on the proton structure function. To be moreaccurate, one must reanalyze the structure function data with allowance for the AQCM ef-fect, arguably that would lower somewhat the resulting pQCD contribution to σ L bringingthe theoretical curve closer to the experimental data points.The case of the transverse cross section σ T is more subtle. In this case the effect ofthe pQCD-AQCM interference is quite substantial. As a matter of fact, the resulting de-structive interference numerically takes over the pure AQCM contribution and lowers σ T compared to the pure pQCD contribution. It is well understood that σ T is more suscep-tible to the non-perturbative effects in comparison with σ L . Indeed, the non-perturvativecorrections to σ T die out substantially slower than corrections to σ L .7igure 3: Additional contribution to exclusive electroproduction of the ρ meson comingfrom quark-antiquark exchange between chromomagnetic vertices.To this end we notice that we only treated a leading in 1 /N c restricted class of the in-stanton induced non-perturbative QCD interactions which are reducible to the anomalouschromomagnetic quark-glion vetrex. The full-fledged t’Hooft’s-like multipartonic interac-tion [3] gives rise to a more complicated diagrams, examples are shown in Fig.3. Theissue of such contributions will be considered elsewhere [23]. We also have checked thepossible effect of introduction of the form factor into pQCD vertex which cuts low transfermomentum region where one-gluon exchange picture looks questionable. It leads to thedecreasing of the both longitudinal and transverse cross sections at low Q . In this casewe observe significant improvement of agreement with experiment of our calculation for σ L , but agreement of calculated σ T with data becomes worse.As we emphasized, the ACQM vertex manifestly violates the quark s -channel helicityconservation. Arguably, the effects of ACQM will be stronger in the helicity flip am-plitudes of vector meson electroproduction. The numerical results for full spin desnitymatrix of diffractive ρ -mesons will be presented elsewhere. The authors are very grateful to I. O. Cherednikov, A.E. Dorokhov, E.A. Kuraev andL.N. Lipatov for useful discussions. This work was supported in part by RFBR grant10-02-00368-a, by Belarus-JINR grant, and by Heisenberg-Landau program.
References [1] T. Sch¨afer and E.V. Shuryak, Rev. Mod. Phys. (1998) 1323.[2] D. Diakonov, Prog. Par. Nucl. Phys. (2003) 173.[3] G.t Hooft, Phys. Rev. D32 (1976) 3432.[4] N. I. Kochelev, Phys. Lett.
B426 (1998) 149.[5] N. Kochelev, Phys. Part. Nucl. Lett. (2010) 326. [arXiv:0907.3555 [hep-ph]].[6] N. I. Kochelev, hep-ph/9707418. 87] I. P. Ivanov, N. N. Nikolaev and A. A. Savin, Phys. Part. Nucl. (2006) 1.[8] I. V. Anikin, A. Besse, D. Y. Ivanov, B. Pire, L. Szymanowski, S. Wallon, Phys. Rev. D84 (2011) 054004. [arXiv:1105.1761 [hep-ph]].[9] I. V. Anikin, D. Y. Ivanov, B. Pire, L. Szymanowski, S. Wallon, Nucl. Phys.
B828 (2010) 1. [arXiv:0909.4090 [hep-ph]].[10] A. Airapetian et al. [HERMES Collaboration], Eur. Phys. J.
C62 (2009) 659.[arXiv:0901.0701 [hep-ex]].[11] W. Augustyniak, A. Borissov and S. I. Manayenkov,arXiv:0808.0669 [hep-ex].[12] C. Adloff et al. [H1 Collaboration], Eur. Phys. J.
C13 (2000) 371.[13] S. V. Goloskokov and P. Kroll, Eur. Phys. J.
C53 (2008) 367.[14] J. Nemchik, N. N. Nikolaev, E. Predazzi and B. G. Zakharov, Z. Phys.
C75 (1997)71.[15] L. Chang, C. D. Roberts, P. C. Tandy, [arXiv:1107.4003 [nucl-th]].[16] E. V. Kuraev, N. N. Nikolaev, B. G. Zakharov, JETP Lett. (1998) 696-703.[hep-ph/9809539].[17] I. P. Ivanov, N. N. Nikolaev, Phys. Atom. Nucl. (2001) 753; Phys. Rev. D65 (2002) 054004.[18] I. P. Ivanov, [hep-ph/0303053].[19] N. N. Nikolaev, Comments Nucl. Part. Phys. (1992) 41.[20] B. G. Kopeliovich et al. Phys. Lett.
B309 (1993) 179.[21] F. D. Aaron et al. [ H1 Collaboration ], JHEP1005