Pion-Photon Transition Distribution Amplitudes
aa r X i v : . [ h e p - ph ] M a y Pion-Photon Transition DistributionAmplitudes
A. Courtoy ∗ , † and S. Noguera ∗ , ∗∗ ∗ Departamento de F´ısica Te´orica and Instituto de F´ısica Corpuscular,Universidad de Valencia-CSIC,E-46100 Burjassot (Valencia), Spain. † E-mail: [email protected] ∗∗ E-mail: [email protected]
Abstract.
The newly introduced Transition Distribution Amplitudes (TDAs) are discussedfor the π - γ transitions. Relations between π - γ and γ - π TDAs for different cases are given.Numerical values for the π - γ TDAs in different models are compared. GPD’s features areextended to TDAs and the role of PCAC highlighted. We give hints for the evaluation ofcross sections for meson pair production in our approach.
Keywords:
TDA, GPD, parton distributions, pion
PACS:
1. INTRODUCTION
One of the main open questions in particle physics is the understanding of thestructure of hadrons in terms of quarks. An important tool for such a purpose isprovided by hard processes. The large virtuality Q involved in these processesallows the factorization of their amplitudes into hard and soft contributions. Thehard contribution to the scattering amplitude is known from perturbative QCDbut the interesting quantity unveiling the structure of hadrons is the soft or non-perturbative contribution. In Deep Inelastic Scattering (DIS) this nonperturbativequantity is expressed in terms of the parton distribution functions (PDF). PDFscan be expressed as the Fourier transform of a bilocal current between the sameinitial and final hadronic state. Generalized parton distributions (GPD) [1, 2, 3]extend this concept to off-diagonal matrix elements of the same currents. GPD gov-ern exclusive processes with the same hadron in the initial and final state in thesoft part of the process but with momentum transfer different from zero. Deeplyvirtual Compton scattering (DVCS) is a tipical example of processes governed byGPDs. Recently a further “generalization” to transition distribution amplitudes(TDA) in which the initial and the final state in the soft part of the amplitude aredifferent has been introduced. They have been defined for processes like hadronannihilation into two photons and backward VCS in the kinematical regime wherethe virtual photon is highly off-shell but with small momentum transfer t [4] H ¯ H → γ ∗ γ → e + e − γ and γ ∗ H → Hγ , (1) ( P ) M h γ ∗ ( q ′ ) γ ( P ′ ) e + e − T DA M ( q ) φ M FIGURE 1.
Factorization of the amplitude for the process (1). with H a hadron, or exclusive meson pair production in γ ∗ γ scattering in the samekinematical regime [5] γ ∗ L γ → M ± π ∓ , (2) M being either ρ L or π . Advocating [4] that the factorization theorems for exclusiveprocesses can be extended to the processes under consideration, i.e. (1) and (2),the amplitude can be factorized as shown in Fig. 1 with the TDAs, describingthe π → γ transition, being the Fourier transform of the matrix element of bilocalcurrents at a light-like distance.The nonperturbative nature of the distribution functions imposes the use ofeffective theories, models or phenomenological parametrizations. In Ref. [6] thecalculation of the π - γ TDAs in a covariant Bethe-Salpeter approach has beendefined. All the invariances of the problem are hence preserved, e.g. gauge andtranslational invariance. For the numerical evaluation, the Nambu - Jona-Lasinio(NJL) model for the description of the pion has been used. The Pauli-Villarsregularization scheme has been chosen because it is Lorentz invariant.Estimates for the π - γ TDAs have been given in Ref. [7] and a calculation hasbeen performed in the Spectral Quark Model (SQM) [8]. Both studies parametrizethe TDAs by means of double distributions [1]. A detailed comparison of the SQMand the NJL models in the determination of the pion GPD can be found in Ref. [9].Finally in Ref. [10] TDAs have been calculated in a non-local chiral quark model,confirming the results of the previous calculations.In the following Section we introduce the vector and axial TDAs emphasizingtheir connection to the pion transition form factors, appearing in the radiativepion decay [11]. The presence of an additional contribution due to PCAC isexplicitly shown. Numerical results and a comparison between the different modelsare presented in Section 3.
2. DEFINITION OF THE PION-PHOTON TDAS
General arguments, such as Lorentz invariance, lead to some important propertiesof GPDs. Taking their first Mellin moment, one can relate GPDs to the corre-
IGURE 2.
Pion pole contribution between the axial current (the cross) and the incomingpion-photon vertex. sponding form factors through the sum rules. Also their higher Mellin momentsare polynomials in the skewness variable ξ by Lorentz invariance.Similarly, as a consequence of Lorentz invariance, TDAs are constrained bysum rules and polynomial expansions. The first Mellin moments of π - γ transitiondistribution functions are related to the vector and axial-vector transition formfactors, F V and F A , through the sum rules. The definition of these form factors isgiven from the vector and axial-vector currents [11] h γ ( p ′ ) | ¯ q (0) γ µ τ − q (0) | π ( p ) i = − i e ε ν ǫ µνρσ p ′ ρ p σ F V ( t ) m π , (3) h γ ( p ′ ) | ¯ q (0) γ µ γ τ − q (0) | π ( p ) i = e ε ν (cid:16) p ′ µ p ν − g µν p ′ .p (cid:17) F A ( t ) m π + e ε ν ( p ′ − p ) µ p ν √ f π m π − t − √ f π g µν ! , (4)with f π = 92 . ε = 1 and τ − = ( τ − i τ ) /
2. All the structure of thedecaying pion is included in the form factors F V and F A . The vector current onlycontains a Lorentz structure associated with the F V form factor. The axial formfactor F A also gives the structure of the pion but contains additional terms requiredby electromagnetic gauge invariance. The second term on the right-hand side ofEq. (4), which corresponds to the axial current for a point-like pion, contains a pionpole coming from the pion inner bremsstrahlung: the incoming pion and outgoingphoton couple with the axial current through a virtual pion (Fig. 2) as required bythe Partial Conservation of the Axial Current (PCAC). The third term in Eq. (4)is a pion-photon-axial current contact term, proportional to f π g µν .Going to the TDAs we introduce the light-cone coordinates v ± = ( v ± v ) / √ ~v ⊥ = ( v , v ) for any 4-vector v µ . We define P =( p + p ′ ) / p ′ − p, therefore P = m π / − t/ t = ∆ . The skewness variable describes the loss of momentum in the light frontdirection of the incident pion, i.e. ξ = ( p − p ′ ) + / P + . Its value ranges between t/ (2 m π − t ) < ξ <
1. Negative values of the skewness variable can be allowed.Regarding the real photon polarization, ε ν , we have the transverse condition ε.p ′ = 0and an additional gauge fixing condition. We assume that this condition is suchthat ε + /P + is kinematically higher twist. The standard gauge fixing conditions, ε = 0 or ε + = 0, both satisfy the previous requirement. To leading twist, the TDAsre therefore defined Z dz − π e ixP + z − h γ ( p ′ ) | ¯ q (cid:18) − z (cid:19) γ + τ − q (cid:18) z (cid:19) (cid:12)(cid:12)(cid:12) π + ( p ) E(cid:12)(cid:12)(cid:12)(cid:12) z + = z ⊥ =0 = i e ε ν ǫ + νρσ P ρ ∆ σ V π + → γ ( x, ξ, t ) √ f π , Z dz − π e ixP + z − h γ ( p ′ ) | ¯ q (cid:18) − z (cid:19) γ + γ τ − q (cid:18) z (cid:19) (cid:12)(cid:12)(cid:12) π + ( p ) E(cid:12)(cid:12)(cid:12)(cid:12) z + = z ⊥ =0 = e (cid:16) ~ε ⊥ · ~ ∆ ⊥ (cid:17) A π + → γ ( x, ξ, t ) √ f π + e ( ε · ∆) 2 √ f π m π − t ǫ ( ξ ) φ x + ξ ξ ! , (5)with ǫ ( ξ ) equal to 1 for ξ > , and equal to − ξ <
0. Here V ( x, ξ, t ) and A ( x, ξ, t )are respectively the vector and axial TDAs. Hence the axial matrix element containsthe axial TDA and the pion pole contribution that has been isolated in a modelindependent way [7, 6, 12]. The latter term is parametrized by a point-like pionpropagator multiplied by the distribution amplitude (DA) of an on-shell pion, φ ( x ). Notice that the pion DA obeys the normalization condition R dx φ ( x ) = 1;the connection through the sum rules of Eq. (5) with Eqs. (3) and (4) is thereforeobvious.The contribution of a pion pole is not a new feature of large-distance distri-butions. TDAs like GPDs are low-energy quantities in QCD though their degreesof freedom are quarks and gluons. One thus expects chiral symmetry to manifestitself, what implies a matching between the degrees of freedom of parton distribu-tions and the low-energy degrees of freedom such as pions. Actually, in the region x ∈ [ −| ξ | , | ξ | ], the emission of a q ¯ q pair from the initial state can be assimilated toa meson distribution amplitude.Here we have defined the TDAs in the particular case of a transition from a π + to a photon, parametrizing the processes given by Eq. (1). Symmetries relate thelatter distributions to TDAs involved in other processes. For instance, we couldwish to study the γ - π − TDAs entering the factorized amplitude of the process (2).Unifying our notations with the notations of Ref. [5], we define the γ - π ± TDAs Z dz − π e ixP + z − D π ± ( p ) (cid:12)(cid:12)(cid:12) ¯ q (cid:18) − z (cid:19) γ + τ ± q (cid:18) z (cid:19) | γ ( p ′ ε ) i (cid:12)(cid:12)(cid:12)(cid:12) z + = z ⊥ =0 = i e ε ν ǫ + νρσ P ρ ( p − p ′ ) σ V γ → π ± ( x, − ξ, t ) √ f π , Z dz − π e ixP + z − D π ± ( p ) (cid:12)(cid:12)(cid:12) ¯ q (cid:18) − z (cid:19) γ + γ τ ± q (cid:18) z (cid:19) | γ ( p ′ ε ) i (cid:12)(cid:12)(cid:12)(cid:12) z + = z ⊥ =0 = − e (cid:18) ~ε ⊥ · ( ~p ⊥ − ~p ′⊥ ) (cid:19) A γ → π ± ( x, − ξ, t ) √ f π dπ + γ γduπ + FIGURE 3.
Diagrams contributing to pion-photon TDA, shown here for an active u -quark.The diagram on the right corresponds to a pion “rescattering” . ± e ( ε · ( p − p ′ )) 2 √ f π m π − t ǫ ( − ξ ) φ x + ξ ξ ! , (6)where we have preserved the definition ξ = ( p − p ′ ) + / ( p + p ′ ) + given before theEq. (5).Time reversal relates the π + - γ TDAs to γ - π + TDAs in the following way D π + → γ ( x, ξ, t ) = D γ → π + ( x, − ξ, t ) , (7)where D = V, A . And
CP T relates the presently calculated TDAs to their analogfor a transition from a photon to a π − V π + → γ ( x, ξ, t ) = V γ → π − ( − x, − ξ, t ) ,A π + → γ ( x, ξ, t ) = − A γ → π − ( − x, − ξ, t ) . (8)
3. DISCUSSION
Basic properties of GPDs and TDAs, like sum rules and polynomiality are relatedto Lorentz and gauge symmetries. Therefore, we need a method of calculation whichpreserves these properties. The main problem here is the description of hadrons,as bound states of quarks, preserving these symmetries. One solution is to usea field theroretical formalism, with a covariant Bethe-Salpeter approach for thedescription of the hadrons. In this formalism, GPDs and TDAs are integrals overthe Bethe-Salpeter amplitudes. This method has been developed in Refs. [6, 13]for local lagrangians and in Refs. [14, 15] for non-local lagrangians. In the caseof pions, the simplest realistic model which realizes these ideas is the NJL modelwithin the Pauli-Villars regularization scheme. We will therefore use this model forthe discussion of the calculation.We consider that the process is dominated by the handbag diagram. Each TDAhas two related contributions [6], depending on which quark ( u or d ) of the pion isscattered off by the deep virtual photon. The leading contributions to the handbagdiagram are depicted in Fig. 3 for an active u -quark, with the diagram on theight corresponding to a coupling of the bilocal current to a quark-antiquark pair.The vector TDA receives contribution only from the first type of diagram, i.e.the diagram on the left of Fig. 3. In the case of the axial TDA, a contribution inthe −| ξ | < x < | ξ | region arises from the second diagram of Fig. 3. This secondcontribution comes from the re-scattering of a q ¯ q pair in the pion channel. Itcontains the pion pole which, according to Eq. (5), must be subtracted in order toobtain the axial TDA.We can express both, V ( x, ξ, t ) and A ( x, ξ, t ), TDAs as the sum of the active u -quark and the active ¯ d -quark distributions. The first contribution will be pro-portional to the d ’s charge, Q d , and the second contribution to the u ’s charge, Q u .Therefore, we can write D π + ( x, ξ, t ) = Q d d π + u → d ( x, ξ, t ) + Q u d π + ¯ d → ¯ u ( x, ξ, t ) , (9)with D = V, A and d = v, a . Isospin relates these two contributions. For the vec-tor TDA, we obtain v π + ¯ d → ¯ u ( x, ξ, t ) = v π + u → d ( − x, ξ, t ). For the axial TDA we have a π + ¯ d → ¯ u ( x, ξ, t ) = − a π + u → d ( − x, ξ, t ) where the minus sign is originated in the change inhelicity produced by the γ operator. The d π + u → d ( x, ξ, t ) contribution is non-vanishingin the region −| ξ | < x < d π + ¯ d → ¯ u ( x, ξ, t ) in the region − < x < | ξ | . Given therelation (9), the support of the whole TDA, V π + ( x, ξ, t ) or A π + ( x, ξ, t ), is therefore x ∈ [ − , fordifferent values of the momentum transfer t . The mass of the pion, being either m π = 0 MeV in the chiral limit or m π = 140 MeV, does not significantly influencethe result. A ξ -symmetry is observed for V ( x, ξ, t ) in the chiral limit: the vectorTDA is an even function of the skewness variable so that we show the results forpositive ξ only. At the contrary, the shape of the axial TDA depends on the sign ofthe skewness variable. The two distinct behaviours are shown in Fig. 4. For positive ξ , the contribution coming from the second diagram of Fig. 3 is dominant andproduces the maxima around x = 0. For negative values of the skewness variable,the contribution of both diagrams have opposite signs, as shown in Fig. 5. GivenEq. (9), we can say that isospin relates the value of the vector and axial TDAs inthe | ξ | < x < − < x < −| ξ | regions, V ( x, ξ, t ) = − V ( − x, ξ, t ) & A ( x, ξ, t ) = 12 A ( − x, ξ, t ) , | ξ | < x < . (10)The factor 1 / u and d quarks.We observe in Fig. 4 that our TDAs satisfy these relations. It must be realized thatthe relation (10) cannot be changed by evolution. For numerical predictions, the standard values of the parameters given in Ref. [16] are used.
Vector TDA (m π =140 MeV and t=-0.5 GeV ) ξ =0.25 ξ =0.5 ξ =0.75 -0.010-0.005 0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040-1.0 -0.5 0.0 0.5 1.0x Axial TDA (m π =0 MeV and t=-0.5 GeV ) ξ =-0.7425 ξ =-0.2475 ξ =0.2475 ξ =0.7425 FIGURE 4.
Vector and axial TDA in the NJL model. The amplitudes are lower for higher( − t ) values, as it can be inferred from the decreasing of the form factors with ( − t ), connected tothe TDAs through the sum rules. The obtained TDAs should obey the sum rules, as already mentioned. For theform factors given in the NJL model (see Appendix of Ref. [6]), the sum rules Z − dx D π + ( x, ξ, t ) = √ f π m π F D ( t ) , (11)with D = V, A , are recovered. In particular we obtain the value F π + V (0) = 0 . t = 0, which is in agreement with the experimentalvalue F V (0) = 0 . ± .
008 given in [17]. We also obtain F π + A (0) = 0 . t = 0, which is about twice the value F π + A (0) = 0 . ± . ξ Z − dx x n − D ( x, ξ, t ) = n − X i =0 C Dn,i ( t ) ξ i , (12)where C Dn,i ( t ) are the generalized form factors. The relation (12) has been numer-ically verified. In the chiral limit, we have numerically found that the odd powersin ξ go to zero for the polynomial expansion of the vector TDA.Other studies of pion-photon TDAs have already been done [7, 8, 10]. In Refs. [7,8], double distributions have been used. Therefore polynomiality is implemented bydefinition and cannot be considered a result. The aim of the author of Ref. [7] is toprovide some estimates of the vector and axial TDAs on the basis of the positivitybounds. In this way we must compare only the order of magnitude of the obtainedamplitudes, which is similar to ours.The vector and axial TDAs calculated in the SQM [8], NJL model [6] and non-local chiral quark model [10] are compared in Fig. 6. The authors of the firstreference use the so-called asymmetric notation. The comparison is here awkward Axial TDA m =140 MeV and t = -0.5 GeV A ( x ,, t ) x =0.25 =-0.5 FIGURE 5.
Contributions to the axial TDA for both positive ( ξ = 0 . , solid line) and negative( ξ = − . , dashed line) values of the skewness variable. In each case, and in the x ∈ [ −| ξ | , | ξ | ]region, the contribution coming from the first diagram of Fig. 3 is represented by the dashed-dotted lines and the non-resonant part of the second diagram of Fig. 3 is represented by thedotted lines. since the authors define, after their Eq. (3), ζ = ( p γ − p π ) + /p + π while, after theirEq. (4), the definition − ζ = ( p γ − p π ) .n is given. We nevertheless decide to usethe standard relation between their asymmetric notations and our symmetricones [3]. Their functions V SQM and A SQM corresponds to our d π + → γ given in Eq. (9).The normalization condition is different from the one used here and the resultsquoted by these authors must be corrected, for the vector TDA, by a factor48 π √ f π F V (0) /m π ∼
10 before comparison. From Fig. 6 we conclude that there isa qualitative and quantitative agreement, for the vector TDA, between the resultsof Ref. [8] and those obtained in the NJL model. Regarding the axial TDA weobserve, in addition to the normalization factor 48 π √ f π F A (0) /m π ∼
10, a changein the global sign due to different definitions. In the first version of [8] the axialTDA for positive values of ξ is given . It coincides with the results in the NJLmodel, as observed in Fig. 6. Surprisingly, the result presented in Ref. [8] coincideswith our result for negative ξ (perhaps due to the change in the definition of ξ mentioned above).In Ref. [10] the TDAs are calculated in three different models. The first one isa local model which pole structure has some similarity with the one of the NJLmodel. The two other models, i.e. semi-local and full non-local, follow the resultsof the local one. The most prominent difference between the results obtained in There is a typographic error in Eq. (23) of the first version of this reference, where a factor M V / he non-local and the NJL models is the appearence of important odd powers in ξ in the polynomial expansion of the vector TDA. We know from Ref. [14] that, fornon-local models, there are additional contributions to those calculated in Ref. [10].In the case of PDFs, disregarding these contributions can produce small isospinviolations [15]. It can therefore be considered that, on that point, the results ofRef. [10] must be confirmed. For numerical comparison, the results obtained in[10] must be corrected by a factor 2 π √ f π F V (0) /m π ∼ .
45 due to the use ofa different normalization condition. For the axial TDA, there is a change in thedefinition of the skewness variable between the caption of Figs. 3, 5 and Fig. 9 ofRef. [10]. We observe, see Fig. 6, that our results in the NJL model coincide withthe results of the latter reference if the convention of the caption of their Fig. 9 ischosen.It can eventually be concluded that there is no disagreement between thedifferent studies concerning the π - γ TDAs besides the ambiguity in the definition ofthe skewness variable. Calculations of Refs. [8, 10] are performed in the chiral limit,where ξ runs from − ξ . On the other hand, the NJL calculation [6] is given for physicalpion mass. In this case, the kinematics of the process imposes t/ (2 m π − t ) < ξ < ξ = t/ (2 m π − t ), preventing us from going through unphysicalvalues of ξ . Moreover, the sum rule Eq. (11) for both the vector and axial TDAsare here satisfied for physical values of ξ , and broken in the unphysical regions ξ < t/ (2 m π − t ) and ξ >
1. We therefore conclude that the choice of sign in Ref. [6]is consistent and gives a guideline for comparing with other models.The π - γ vector and axial Transition Distribution Amplitudes have been defined.The results in different models have been discussed. In particular, the numericalresults of Ref. [8, 6, 10] have been compared. The use of a fully covariant and gaugeinvariant approach guaranties that all the fundamental properties of the TDAs willbe recovered, in particular, the right support, i.e. x ∈ [ − , t -independent double distributions, in a firstapproach, and, in a second, the t -dependent results of Ref. [7]. In the line ofsight of this reference, similar estimates for the meson pair production in hard γ ∗ γ scattering using the results for the TDAs cited above could be given [18]. Inparticular, it would be worth investigating the model independence of the structureindependent terms [12] and then estimating the pion pole contribution to the crosssection [18]. ACKNOWLEDGMENTS
This work has been supported by the Sixth Framework Program of the EuropeanCommision under the Contract No. 506078 (I3 Hadron Physics); by the MEC(Spain) under the Contract FPA 2007-65748-C02-01 and the grant AP2005-5331 V - > ( x , , - . ) x =0.5 Non Local =-0.5 non Local = 0.5 NJL and SQM -1.0 -0.5 0.0 0.5 1.0-0.020.000.020.04 A - > ( x , , - . ) x =0.5 Non Local =0.5 NJL =0.5 SQM =-0.5 Non Local =-0.5 NJL =-0.5 SQM FIGURE 6.
Comparison of the π + → γ TDAs for t = − . of Refs. [8, 10] for m π = 0MeV and Ref. [6] for m π = 140 MeV. On the left, we have: the vector TDA for ξ = ± . χ QM calculation for ξ = 0 . ξ = − . ξ as discussed in the text. and by EU FEDER. Feynman diagrams drawn using JaxoDraw [19]. REFERENCES
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