Gluonic components of the pion and the transition form factor γ ∗ γ ∗ → π 0
aa r X i v : . [ h e p - ph ] J a n FTUV-09-1208
Gluonic components of the pion and the transitionform factor γ ∗ γ ∗ → π a,b ∗ and V. Vento b † (a) Bogoliubov Laboratory of Theoretical Physics,Joint Institute for Nuclear Research, Dubna, Moscow region, 141980 Russia (b)
Departament de F´ısica Te`orica and Institut de F´ısica Corpuscular,Universitat de Val`encia-CSIC, E-46100 Burjassot (Valencia), Spain
Abstract
We propose an effective lagrangian for the coupling of the neutral pion with gluonswhose strength is determined by a low energy theorem. We calculate the contribution ofthe gluonic components arising from this interaction to the pion transition form factor γ ∗ γ ∗ → π using the instanton liquid model to describe the QCD vacuum. We find thatthis contribution is large and might explain the anomalous behavior of the form factorat large virtuality of one of the photons, a feature which was recently discovered by theBaBar Collaboration.Pacs: 12.38.Lg, 13.40.GpKeywords: pion, gluon component, instanton, form factor ∗ [email protected] † [email protected] he implications, of recent data by the BaBar Collaboration [1] on the transition formfactor γ ∗ γ → π , in our understanding of the structure of the pion are being widely dis-cussed [2, 3, 4, 5, 6]. A possible scenario to explain these data consists in assuming a flatshape for pion distribution amplitude [3, 4], supported by some low energy models likethe Nambu-Jona-Lasinio model [7, 8] and instanton based non-local chiral quark model[9],and its detailed behavior under Quantum Chromodynamics (QCD) evolution, and alarge mass cut-off added to the quark propagator [3], signalling a peculiar behavior of the qq wave function [10]. Never mind the various explanations, what appears evident is thatthe BaBar data[1], if confirmed, are in contradiction with most model predictions basedon the factorization approach to exclusive reactions at large momentum transfer [11] andthis apparent lack of perturbative factorization motivates the present investigation. Inthis Letter we suggest an alternative nonperturbative explanation to the BaBar resultsbased on the existence of additional contributions to the pion form factor never previ-ously considered. These contributions arise from the admixture of gluonic components,associated to nonperturbative properties of the QCD vacuum, which provide a stronginteraction with two photons.Let us propose a low-energy effective π interaction with gluons of the following form L effπgg = − f π G π α s π G aµν e G aµν . (1)Such type of Lagrangian density, describing the interaction of a pseudoscalar meson withgluons, was introduced many years ago by Cornwall and Soni [12] to derive Witten’srelation between the η ′ mass and the topological susceptibility, in a world without lightquarks [13] χ N f =0 t = − f π M η ′ + M η − M K ) , (2)where f π = 92 . M X the mass of the indicated particles,the topological susceptibility is given by χ N f =0 t = i Z d x < | T { Q ( x ) Q (0) }| > G , (3)and Q ( x ) = α s π G aµν ( x ) e G aµν ( x ) (4)is the topological charge density.The effective pion-gluon interaction, Eq.1, is the analogue of the pion-quark effectiveinteraction L effπqq = − f π M q ¯ qiγ ~τ q · ~π, (5)giving the pion coupling to the quarks.To derive the decay constant f π G , which sets the scale of the gluon nonperturbativeinteraction with the neutral pion, we will use a low energy theorem (LET) [14] < | α s π G aµν e G aµν | π > = 12 m d − m u m d + m u f π M π . (6)1e stress that this matrix element is rather big due to the large light quark mass ratio[15] z = m u m d = 0 . − . . (7)By using the effective interaction Eq.1 and the LET Eq.6 we get f π G = − z )(1 − z ) χ N f =0 t f π M π f π . (8)As it was to be expected, the strength of the coupling of the neutral pion to gluons isrelated to the violation of isospin symmetry and proportional to the difference of the d-and u-quark masses 1 f π G ∝ m d − m u . (9)For the value of the mass ratio m u /m d shown in Eq.7 which is the one allowed by theParticle Data Group, one obtains R = f π G f π ≃ . − . . (10) q q I ( ¯ I ) π π I ( ¯ I ) q q p p Figure 1: Gluonic contribution to the pion transition form factor. Symbol I ( ¯ I ) denotesinstanton(antiinstanton).Let us next calculate the contribution arising from the interaction given by Eq.1 tothe transition form factor of two photons to the pion γ ∗ ( q ) γ ∗ ( q ) → π ( p ), where q and q are the photon momenta and q + q = p . We consider the case where all virtualitiesof the incoming and the outcoming particles are in the Euclidean domain Q = − q ≥ Q = − q ≥ P = − p ≥
0. The Lagrangian density Eq.1 describes the pion interactionwith soft gluons. Such gluons should interact with photons through nonperturbative QCDinteractions. We use the instanton liquid model (ILM) for the QCD vacuum to calculatethe interaction of two photons with the gluonic component of the pion. The ILM is oneof the most successful models for the description of nonperturbative QCD effects (seereviews [16, 17]). Within the ILM the single instanton contribution to the pion formfactor coming from the interaction Eq.1 is associated to the diagrams in Fig.1.2he amplitude for the π − γ ∗ γ ∗ interaction via an instanton with center z has thefollowing form T µν ( p, q , q ) = ǫ µναβ X i e i f π G π Z n I ( ρ ) ρ dρ × Z d z Z d z Z d x Z d ye − ipz e iq x e iq y Q I (¯ z )) × h ¯ x h ¯ y ∆ (cid:26) ( h ¯ y ∆ α ¯ y β − h ¯ x ∆ β ¯ x α ) + h ¯ x h ¯ y ¯ x α ¯ y β (cid:27) , (11)where Q I (¯ z )) = 6 ρ π (¯ z + ρ ) (12)is the topological charge density of the instanton in the space-time point z , n I ( ρ ) is theinstanton density, ρ is the instanton size, ∆ = ¯ x − ¯ y , h ¯ x = 1 / (¯ x + ρ ), h ¯ y = 1 / (¯ y + ρ )and the notation ¯ w ≡ w − z for any variable w has been introduced. The sum runs overthe light quark flavors, i.e. i = u, d, s . To get Eq.11 we have used the correlator of twoelectromagnetic currents in the instanton field obtained by Andrei and Gross [18]. Theirresult was corrected by a color factor (see [19]). The first term in the last line of theequation is coming from the quark nonzero modes in the instanton field and the last termarises from the interference between nonzero and zero modes [18].The final result for the gluon contribution to the pion transition form factor inducedby instantons is F ( P , Q , Q ) Ig = 4 < e >f π R Z dρn I ( ρ ) ρ S ( ρ, P , Q , Q ) (13)where S ( ρ, P , Q , Q ) = Φ ( √ z ) Z dt { I ( t, z , z , z ) + (1 − t ) I ( t, z , z , z ) } , (14) I ( t, z , z , z ) = Z ∞ dα α ( α + 1)Φ ( Z ( α, t, z , z , z ))( α + 1 − t ) Z ( α, t, z , z , z ) , (15)and Z ( α, t, z , z , z ) = q ( α + 1)( tαz + tz + (1 − t ) z ) / ( α + 1 − t ) . (16)The functions Φ ( z ) = z K ( z )2 , Φ ( z ) = zK ( z ) (17)behave as Φ , ( z ) → z →
0. In Eqs.13-16 the notations are z = Q ρ , z = Q ρ , z = P ρ and < e > = P i e i .For an estimate we use Shuryak‘s version of the ILM [20], where the density is givenby n I ( ρ ) = n δ ( ρ − ρ c ) (18)and n ≈ / f m − , ρ c ≈ / f m. (19)3ithin this simple model for the instanton distribution the result for the form factor is F ( P , Q , Q ) Ig = 4 < e > f I π f π R S ( ρ c , P , Q , Q ) , (20)where f I = π n ρ c is so-called instanton packing fraction in the QCD vacuum.It should be pointed out that in spite of the smallness of instanton packing fraction f I ≈ .
06, using the single instanton approximation as above is only valid for valuesof the momentum transfers Q , Q ≫ /R I , where R I ≈ ρ c is the distance betweenthe instantons in the ILM. For smaller photon virtualities it is necessary to include thecontributions arising from multiinstanton configurations. With an average size of theinstanton in the QCD vacuum as in Eq.19 for the region Q , Q ≥ /ρ c ≥ µ = 0 . , i.e. z , ≥
1, the validity of a single instanton approximation is assured.The calculation above was done for the case when all external momenta are Euclidean.In order to compare with BaBar data we have to perform an analytic continuation of thepion virtuality to the physical point of the pion on-shell P → − m π − iǫ . An inspectionof the integrals in Eqs.14,15 shows that the dominant contribution to the form factor at m π /Q , ≪ t ≈
0, due to the pole at Z = 0.Assuming the following behavior of the function Φ ( Z ) ≡ ZK ( Z ) ∼ m π we obtain following closed form formulas for the realand imaginary parts of the flavor singlet part of form factor Re ( F ( m π , Q , Q ) Ig ) ≃ < e > f I π f π R ×{ z [ z log ( z ) /z + log ( z )( log ( z /z ) −
1) + Li (( z − z ) /z )]( z − z ) + z [ z log ( z ) − π / − log ( z − z ) / − log ( z )(1 − log ( z ) / z − z ) + z [ Log ( z ) log (( z / ( z − z )) − Li ( z / ( z − z ))]( z − z ) − log ( z /z ) log ( m π ρ c ) z − z } , (21) Im ( F ( m π , Q , Q ) Ig ) ≃ < e > f I πf π R log ( z /z ) z − z . (22)The imaginary part of form factor arises because the pion may decay in this calculationinto a quark-antiquark pair since confinement, which forbids this decay, is not explicitlyimplemented. However, the net contribution of the imaginary part to total transitionform factor in the BaBar kinematics is very small. These formulas are useful to extractthe behavior of the transition form factor with Q . The exact numerical analysis will bedescribed below. For definiteness we consider the case z > z In the limit Q ≫ Q which is valid for BaBar kinematics, the formulas for the realand the imaginary parts simplify, Re ( F ( m π , Q , Q )) Ig ≈ < e > f I π f π R [ log ( Q /m π ) log ( Q /Q ) + π / ρ c Q , (23) Im ( F ( m π , Q , Q )) Ig ≈ < e > f I πf π R log ( Q /Q ) ρ c Q . (24)4t follows from Eqs.23,24 that the flavor singlet gluon induced part of the form factorhas a dependence on the large photon virtuality Q proportional to log ( Q ) /Q , whichis much stronger than that of the flavor nonsinglet part, which in most of the modelsis of the form 1 /Q . The additional feature of this new contribution is its strong chiralenhancement since the massless logs appear governed by the pion mass as log ( Q /m π ).For symmetric kinematics Q = Q = Q the result is Re ( F S ( Q )) Ig ≈ < e > f I π f π R (3 + 2 log ( Q /m π )) ρ c Q , (25) Im ( F S ( Q )) Ig ≈ < e > f I πf π Rρ c Q . (26)Having determined the dependence of the virtuality in the approximation Eq. 21, weproceed to study the exact numerical calculation which includes the effect of the functionalform of ZK ( Z ). Before we do so, we would like to point out that the exact calculationleads to a smaller (by about 50%) result, compared to the approximate calculation. Thisfactor can be absorbed in the uncertainties of the vacuum model associated with thepoor knowledge of the instanton distribution (about 30%), and the additional uncertaintycoming from the value of the pion coupling to gluons Eqs.9 and 10 (about a factor 2) dueindeterminacy in the ratio of u- and d- quark masses, Eq.7.We compare our result with the BaBar data. Before we do so some caveats have tobe expressed since the comparison is not direct. The BaBar experiment, only measuresthe virtuality of one of the photons in the interval Q = 4 −
40 GeV . They only put anupper limit on the virtuality for the second photon, Q < .
18 GeV . Finally, they usea model for the form factor to extract the value at the real photon point Q = 0. Thus,a direct comparison of our results with the BaBar data is not possible. Moreover, ourcalculation only represents the flavor singlet contribution to the form factor, therefore wehave to add a flavor nonsinglet part. We take in the estimate shown in Fig.2 for the flavornonsinglet part the corresponding to a vector meson dominance (VMD) model, i.e. F ( Q , Q ) V MDq = 14 π f π Q /M ρ )(1 + Q /M ρ ) . (27)In order to compare our results with the BaBar data we perform an extrapolation oftheir results from Q = 0 to Q = 0 .
35 GeV . In Fig.2 we compare our calculation withthe extrapolation of the BaBar data described by [1] Q | F BaBarexp ( Q ) | = A (cid:18) Q GeV (cid:19) β , (28)where A ≃ .
182 and β ≃ .
25. This function has been continued to the point µ (0.35GeV in our case) following the VMD model, Q | F BaBar ( Q , µ ) | = Q | F BaBarexp ( Q ) | µ /M ρ . (29)The bands in the figure represent our uncertainties, both in the vacuum model and in thecoupling constant, as mentioned before. 5 Q (GeV )Q | F(Q , m ) | Figure 2: Contributions to the pion transition form factor compared with the extrapo-lation of the BaBar data to Q = µ = 0 .
35 GeV (solid line): gluonic contribution inour model including uncertainties (lower band), conventional VMD contribution (dashedline) line, and their sum (upper band).It should be mentioned that the behavior of the gluonic part of the form factor asfunction of Q is determined by shape of the decay of the non-zero modes in the instantonfield (see Eq.11). At the same time it is well known that the VMD-like behavior of theflavor nonsinglet part of pion form factor can be easily reproduced within a non local chiralquark model based on the quark zero-modes dominance in the instanton vacuum[21]. Dueto the weaker decay of the quark nonzero modes with respect to zero modes one can expecta harder Q dependence of the flavor singlet part of the form factor in comparison withthe flavor nonsinglet part. Such tendency is seen in Fig.2. Indeed, the gluonic part isdescribed well by a fast increasing function of Q , Eq.23. That function shows a similarbehavior, as function of Q , as the BaBar data, Eq.28. Contrary to the gluonic part ofthe form factor, its flavor nonsinglet part has a conventional 1 /Q behavior at large Q ,Fig.2. Taking into account some uncertainties in our estimates related to the poorly knownratio of the u- and d- quark masses, Eq.7, as well as uncertainties in the parameters of theinstanton model, we conclude, that the new contribution related to the gluonic componentof pion might explain the anomalous behavior of the pion transition form factor found bythe BaBar Collaboration. One should be aware that these contributions are beyond theOperator Product Expansion and their QCD evolution is non trivial [22].It is evident that such type of contribution should be present also for the η and η ′ mesons. In this case one should carefully take into account the effects of their largermasses and their strong mixing (see recent discussion in the papers [23, 24]). We alsowould like to point out that the strong Q dependence of the gluonic part of the formfactor opens a new possibility to disentangle particles with dominant gluonic content,i.e. glueballs (see review [25]), in γ ∗ γ ∗ collision. These tasks are the subject of futureinvestigations.In summary, the BaBar data [1] point towards a breaking of perturbative factorization.6his has led us to investigate possible non perturbative mechanism that contribute to thepion transition form factor. We have shown that a nonzero interaction of neutral pionwith gluons arising from isospin violation, i.e. m u = m d , induces a large contributionto this form factor at large virtuality of one of the photons. More sophisticated modelsfor the instanton density and probably multiinstanton contribution might bring the valuecloser to the observed one. Acknowledgments
We are grateful to A.E. Dorokhov and S. Noguera, for very useful discussions. NIKthanks the Departament de F´ısica Te`orica Universitat de Val`encia for the hospitalityand the University of Valencia for a Visiting Professor appointment. This work wassupported in part by HadronPhysics2, a FP7-Integrating Activities and InfrastructureProgram of the European Commission under Grant 227431, by the MICINN (Spain) grantFPA2007-65748-C02-01, by GVPrometeo2009/129, by the RFBR grant 10-02-00368-a andby Belarus-JINR grant.
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