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Generalized parton distributions of the pion in a covariant Bethe-Salpeter model and light-front models T. Fredericoa , E. Paceb , B. Pasquinic and G. Salm`ed a

Dep. de F´ısica, Instituto Tecnol´ogico de Aeron´autica, 12.228-900 S˜ ao Jos´e dos Campos, S˜ ao Paulo, Brazil b

Dipartimento di Fisica, Universit`a di Roma ”Tor Vergata” and Istituto Nazionale di Fisica Nucleare, Sezione Tor Vergata, Via della Ricerca Scientiﬁca 1, I-00133 Roma, Italy

c

Dipartimento di Fisica Nucleare e Teorica, Universit`a degli Studi di Pavia and Istituto Nazionale di Fisica Nucleare, Sezione di Pavia, Italy

d

Istituto Nazionale di Fisica Nucleare, Sezione di Roma, P.le A. Moro 2, I-00185 Roma, Italy

The generalized parton distributions of the pion are studied within diﬀerent light-front approaches for the quark-hadron and quark-photon vertices, exploring diﬀerent kinematical regions in both the valence and nonvalence sector. Moments of the generalized parton distributions which enter the deﬁnition of generalized form factors are also compared with recent lattice calculations.

1. Introduction Generalized parton distributions (GPDs) represent a key concept for understanding the hadron structure [1–5]. They unify the information encoded in electromagnetic (e.m.) form factors (FFs) and ordinary parton distributions, supplementing them with the possibility to access new aspects of the hadron structure. In particular, the pion GPDs represent an important test ground for model calculations aiming to a detailed description of hadron structure, and this explains the wealth of papers devoted to such a task [6– 16]. Here we review the results of Ref. [16] for the calculation of the pion GPDs in three relativistic models which explore diﬀerent kinematical regions in a complementary way. In particular, the ﬁrst model is a covariant analytic model, based on 4D Ans¨ atze for the pion BetheSalpeter amplitude (BSA), which allows to explore the whole kinematical domain in the valence and non-valence sector. The other two models are constrained to either the valence or nonvalence regions. In the non-valence region we adopt a model based on a microscopical vector0920-5632/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2010.02.041

meson model dressing for the quark-photon vertex and a phenomenological Ansatz for the 3D light-front (LF) projection of the pion and vectormeson BSAs. Finally, in the valence region we discuss a third model constructed within the LF relativistic Hamiltonian dynamics. After a short introduction about the general formalism for the deﬁnition of the pion GPDs, in Sect. 2 we present the main features of the three models, referring to [16] for a more detailed discussion. In the ﬁnal section, we show the model results for the pion GPDs, together with the comparison between our model predictions and recent lattice results for the ﬁrst moments of GPDs entering the deﬁnition of generalized form factors (GFFs). 2. GPDs in covariant and light-Front relativistic models GPDs are deﬁned as the non-forward (p = p ) matrix elements of light-cone bilocal operators separated by a light-like distance, i.e. G Γ = p , π ± | OΓ |p, π ± ,

(1)

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with

OΓ =

z− z− dz − ixP + z− ¯ e ψq (− , 0⊥ )Γψq ( , 0⊥ ). 4π 2 2 (2)

In Eq. (2), P = (p + p )/2 is the average pion momentum and the operator Γ is a matrix in the Dirac space which selects diﬀerent spin polarizations of the quark ﬁelds. In particular, for Γ = γ + one has the unpolarized quark GPD H q , while Γ = iσ +i projects on the transverse polarization of quarks and deﬁnes the chiral-odd GPD ETq . Because of Lorentz invariance the GPDs can only depend on three kinematical variables, i.e. the (average) quark longitudinal momentum fraction x = k + /P + , the invariant momentum square t = Δ2 ≡ (p − p)2 , and the skewness parameter ξ = −Δ+ /(2P + ). In addition, there is an implicit scale dependence in the deﬁnition of GPDs corresponding to the factorization scale μ2 . The variable x allows one to single out two kinematical regions. The ﬁrst region corresponds to the valence contribution and is given by the union of the interval x ∈ [−1, −|ξ|] (for an active antiquark) and x ∈ [|ξ|, 1] (for an active quark). In the Fockspace expansion of the pion state, this region is described by matrix elements with the same number of partons in the initial and ﬁnal states. The second region corresponds to x ∈ [−|ξ|, |ξ|], and is associated with the non-valence contribution involving non-diagonal matrix elements between parton conﬁgurations with Δn = 2. In the forward case, p = p , both Δ and ξ are zero, and H reduces to the usual parton distribution function, while ET vanishes for time-reversal invariance. Moments in the momentum fraction x play an important role in the theory of GPDs. Weighting Eq. (1) with integer powers of x and integrating over x, the quark operator OΓ reduces to a local operator and the corresponding matrix elements can be parametrized in terms of GFFs, i.e. +1 n dx xn−1 H q (x, ξ, t) = (2ξ)2i Aqn,2i (t),

−1

+1 −1

i=0

dx xn−1 ETq (x, ξ, t) =

n i=0

(2ξ)2i BTq n,2i (t). (3)

In Eqs. (3), the lowest moment n = 1 of the unpolarized GPD yields the pion e.m. FF, while the Fourier transform of BTq 1,0 (t) in the impactparameter space determines the dipole-like distortion of the quark density in the transverse plane due to the transverse spin-structure of the quark in the pion. The second Mellin moments of unpolarized GPDs can be related to the GFFs of the energy-momentum tensor of QCD. The starting point of our approach is the Mandelstam formula [17] for the quark correlator in Eq. (2), giving for the u-quark unpolarized GPD d4 k H u (x, ξ, t) = −ı Nc R δ(P + x − k + ) 2(2π)4 ×Λ(k − P, p )Λ(k − P, p) Δ Δ ×T r S(k − P )γ 5 S(k + )γ + S(k − )γ 5 , 2 2 (4) where Nc = 3 is the number of colors, R = 2m2 /fπ2 , with fπ the pion decay constant, and m and S(p) are the mass and the Dirac propagator of the constituent quark (CQ), respectively. In Eq. (4), γ5 Λ(k, p) is the pion vertex function deduced from a simple eﬀective quarkpion Lagrangian [18] . In the following, we will explore diﬀerent approximations to model the momentum-dependent part of the vertex function. In a ﬁrst covariant analytic model, Λ is assumed to be a symmetric function of the two quark momenta with the following form Λ(k − P, p) = C

1 2

(k − Δ/2) − m2R + ı 1

×

2

(P − k) − m2R + ı

.

(5)

A diﬀerent choice, based on the sum instead of the product of the two terms in Eq. (5), was adopted in Ref. [19] for the calculation of the e.m. FF and further discussed in the case of the GPDs in Ref. [16]. However, the product form (5) provides a more realistic transverse-momentum falloﬀ, leading to a more favourable comparison with the experimental data for the e.m. FF at

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high-momentum transfer and also satisfying the support conditions for the parton distribution. Once the CQ mass is ﬁxed and the constant C in Eq. (5) is constrained through the charge normalization, the only free parameter of the model is the regulator mass mR , which is ﬁtted to the experimental value of fπ . The projection into the valence and non-valence contributions to the GPDs is obtained after integration of Eq. (4) over the LF energy k − , fully taking into account the pole structure of both the Dirac propagators and the vertex functions. A second covariant model is introduced by following the approach of Ref. [20] for the calculation of the e.m. FFs both in the spacelike and timelike region. Starting from the same formal expression of Eq. (4) for the GPD, we introduce the following new ingredients: i) instead of the bare quark-photon vertex, γ μ , a dressed quarkphoton vertex Γμ (k, Δ), modeled through a microscopical vector meson (VM) dominance approach, and ii) phenomenological Ans¨atze for the BSAs in the valence and non-valence regions. Another basic diﬀerence with respect to the previous analytic model is that only the simple analytic structure of the Dirac propagators is retained, i.e. the analytic structure is disregarded in the BSAs of both i) the initial and ﬁnal pion and ii) the VM dressing of the quark-photon vertex. This approximation turns out to be a very eﬀective one in the calculation of the e.m. FF only in the Δ⊥ = 0 frame [21]. which will be also adopted for the present calculation of the GPDs. In the valence sector, after integrating over the LF energy, the resulting momentumdependent part of the 3D BSA of the pion and vector mesons are approximated with light-front wave functions (LFWFs) which are eigenstates of the CQ square mass operator of Ref. [22]. In the non-valence region, there is also a pion nonvalence component describing the emission (absorption) of a pion by a quark. Assuming a vanishing pion mass, such a process can be described using a constant interaction, with a coupling constant ﬁxed by the normalization of the pion FF. Furthermore, in the limit mπ = 0, only the pair-production mechanism is contributing to the GPD. In this term, for mπ = 0 one has only in-

stantaneous contributions produced by the standard LF decomposition of the propagator ( i.e. − S(k) = (/ k on + m)/[k + (k − − kon + ı)] + γ + /2k + ). In order to model the instantaneous vertex functions, we put Λist ≈ CΛf ull , where the constant C is thought to roughly describe the eﬀects of the short-range interactions. Indeed, we use the relative weight CV M /Cπ as free parameter. Finally, a third model calculation is based on a light-front Hamiltonian (LFH) approach based on a Poincar`e covariant description of the pion. In particular, the rotational covariance is fulﬁlled through the introduction of the Melosh rotations and the proper deﬁnition of the total intrinsic angular momentum. Such a model allows us to explore only the valence region, and therefore will be discussed just for the ξ = 0 kinematics. In such a frame, the GPDs can be expressed as overlap of LFWFs, given by the product of the momentumdependent part of the wave function in the initial and ﬁnal state with a spin-dependent part as dictated from the proper Melosh transformations. For the momentum-dependent part we adopt a gaussian form [23], with the quark mass and the gaussian width ﬁtted to the pion charge radius and decay constant. 3. Results In Fig. 1 we show the unpolarized GPD H in the (x, t) plane at ﬁxed |ξ| = 1, for the isoscalar u + d and isovector u − d quark combination, comparing the results for the covariant analytic model with a product-form for the BSA with the phenomenological BS model with dressed quarkphoton vertex. The value |ξ| = 1 corresponds to the contribution of the pair-production mechanism in the whole range of x. The general shape for the GPDs in the two models show similar features, with the collinear peak at x ≈ 1 increasing at higher values of |t|. As discussed in Ref. [16], the covariant analytic model exhibits an overall agreement also with the LFH model at ξ = 0, and can be used at any value of x, ξ, t for interpolating between the other two phenomenological models. In particular, at the crossing point of the valence and non-valence region, x = |ξ|, the covariant analytic model predicts a smooth transition, due to

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267

Figure 1. Upper left (right) panel: isoscalar (isovector) unpolarized GPD from the covariant analytic model with the product-form for the BSA (Eq. (5)) at |ξ| = 1 and mπ = 0. Lower panels: the same as in the upper panels for the microscopic model with dressed photon-quark vertex. On the z-axis the ratio with respect to Fmon = 1/(1 + |t|/m2ρ ) is presented. The ﬁgure is adapted from Ref. [16].

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T. Frederico et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 264–269

I=0 Figure 2. Left panel: the ratio AI=0 2,0 (t)/A2,0 (0) as a function of t. Solid line: product-form for the pion BSA, Eq. (5), and mπ = 0. Dotted line: the same as the solid line, but with mπ = 140 MeV. Dashed line: LFH model. with a gaussian pion wave function and the proper Melosh rotations. Shaded area: I=0 results from lattice QCD [24]. Right panel: the same as the left panel, but for AI=0 2,2 (t)/A2,2 (0). The ﬁgure is adapted from Ref. [16].

the continuity of the model. In Fig. 2 we show results for the ratios of I=0 I=0 I=0 the GFFs AI=0 2,0 (t)/A2,0 (0) and A2,2 (t)/A2,2 (0) which are evolution-scale independent [13]. The predictions from the covariant analytic model for two diﬀerent values of the pion mass are shown for both GFFs, while the LFH model can I=0 be only applied for AI=0 2,0 (t)/A2,0 (0) at ξ = 0. The dashed band shows recent lattice results de2 scribed through a monopole form 1/(1 − t/M2,i ), as obtained in Ref. [24]. In particular, we used M2,0 = 1.329 ± 0.058 GeV and M2,2 = 0.89 ± 0.25 GeV, corresponding to an analysis of the lattice data that satisﬁes the low energy theorem, i.e. I=0 AI=0 2,0 (0) = −4A2,2 (0). Our model predictions are overall consistent with the lattice results, except for small values of |t|. A better description of the low |t| region could be obtained by incorporating in our phenomenological models interaction terms responsible for the conﬁnement. On the other side, the large uncertainties in the lattice results for AI=0 2,2 do not allow us to elaborate too much on the comparison between the diﬀerent predictions. Finally, in Fig. 3 we show the density for transversely polarized quark in the impact-parameter space b⊥ . Such a density is deﬁned as [25,26] ρq ( b⊥ ) =

Aq1,0 ( b2⊥ ) si ij bj⊥ ∂ − B q ( b2 ), (6) 2 2mπ ∂b2⊥ T 1,0 ⊥

where the GFFs in the impact-parameter space are obtained by Fourier transform of Eqs. (3). In Eq. (6), the monopole distribution associated to H is distorted by a dipole term proportional to ET in the case of transversely polarized quark. The results in Fig. 3 correspond to the LFH model and exhibit a clear correlation between quark spin and transverse space. The average sideway shift amounts to by⊥ u = BTu 1,0 /(2mπ Au1,0 ) = 0.197 fm. Remarkably, this value is of the same strength as the dipole-like distortion in the density of transversely polarized quarks in an unpolarized nucleon, i.e. by⊥ u = BTu 1,0 /(2mN Au1,0 ) = 0.209 fm, as obtained in a LFH model for the nucleon [27]. These results are also supported from recent lattice calculations [26,28], giving by⊥ u = 0.151(24) fm and by⊥ u = 0.154(6) fm for the pion and nucleon, respectively. Acknowledgments This work was partially supported by the Brazilian agencies CNPq and FAPESP and by Ministero della Ricerca Scientiﬁca e Tecnologica. It is also part of the Research Infrastructure Integrating Activity “Study of Strongly Interacting Matter” (acronym HadronPhysics2, Grant Agreement n. 227431) under the Seventh Framework Programme of the European Community.

T. Frederico et al. / Nuclear Physics B (Proc. Suppl.) 199 (2010) 264–269

Figure 3. Density in the impact-parameter space of transversely polarized u quark in π ± as predicted from the LFH model.

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