DESY-05-081 hep-ph/0505269

Generalized parton distributions for the pion

[0.3em] in chiral perturbation theory

M. Diehl, A. Manashov and A. Schäfer

Deutsches Elektronen-Synchroton DESY, D-22603 Hamburg, Germany

Institut für Theoretische Physik, Universität Regensburg, D-93040 Regensburg, Germany

Department of Theoretical Physics, Sankt-Petersburg State University, St.-Petersburg, Russia

Abstract

[10pt] Generalized parton distributions provide a unified parameterization of hadron structure and allow one to combine information from many different observables. Lattice QCD calculations already provide important input to determine these distributions and hold the promise to become even more important in the future. To this end, a reliable extrapolation of lattice calculations to the physical quark and pion masses is needed. We present an analysis for the moments of generalized parton distributions of the pion in one-loop order of chiral perturbation theory.

## 1 Introduction

Generalized parton distributions (GPDs) provide general parameterizations for non-forward hadronic matrix elements, see e.g. the original work [1, 2, 3] and the recent reviews [4, 5, 6]. They contain the usual parton distribution functions and elastic form factors, and they are closely related with light-cone wave functions and distribution amplitudes. GPDs can be analyzed using standard operator product expansion techniques [1, 7], and factorization theorems have been proven for many different processes. In addition, evolution equations have been derived to next-to-leading order accuracy, and generally speaking the GPD formalism has reached a similar level of stringency as the QCD description of inclusive deep inelastic scattering. Finally, it has become clear that GPDs contain information which cannot be deduced directly from any experiment, most notably the orbital angular momentum carried by partons  and the transverse structure of hadrons in the impact parameter plane . On the experimental side, data which is especially suited to constrain the form of GPDs has been obtained at DESY  and Jefferson Lab .

An important problem hampering the practical realization of the physics potential of GPDs is that, typically, measurements are only sensitive to convolutions of GPDs, such that the assumed functional form enters crucially into the extraction of the distributions themselves. The situation is further complicated by potentially large higher-twist contributions in some processes. Therefore, the possibility to obtain independent information from lattice QCD, which allows one to directly calculate -moments of GPDs, is especially interesting. Pioneering lattice calculations for nucleon GPDs have already been performed some time ago . These calculations are being improved and their scope is being systematically widened , also in the direction of analyzing other hadrons, in particular the pion. Calculations for the pion are being performed by the QCDSF collaboration, using dynamical improved Wilson fermions, but have to be made at unphysically large quark masses to save on computing time. The reliable extrapolation to the chiral limit, i.e. to physical quark masses, requires precise knowledge on the functional form of the quark mass dependence of moments of GPDs. Close to the chiral limit, this knowledge is provided by chiral perturbation theory (ChPT), and in recent years the combination of ChPT and lattice QCD has in fact proven to be very powerful in obtaining precise lattice predictions. Obviously the ChPT analysis should be performed in parallel to the lattice calculations, and this is the objective of our paper.

In this Letter we present an analysis for the moments of pion GPDs in one-loop order of ChPT. The paper is organized as follows. In Section 2 we recall the definition of the pion GPDs and the expression of their moments in terms of the matrix elements of local twist-two operators. In Sections 3 and 4 we construct the relevant twist-two operators in the first two orders of ChPT, and in Section 5 we calculate the one-loop contributions to their matrix elements in the pion. We summarize our findings in Section 6.

## 2 Definitions and properties of pion GPDs

The pion GPDs can be introduced as the pion matrix elements of nonlocal operators. As is done in lattice QCD calculations, we assume isospin symmetry to be exact, neglecting the difference between - and -quark masses. We shall study the isoscalar and isovector quark GPDs, which are respectively defined as

 δabHI=0(x,ξ,t) = 12∫dλ2πeixλ(Pu)⟨πb(p′)|¯q(−12λu)uμγμq(12λu)|πa(p)⟩, (1) iϵabcHI=1(x,ξ,t) = 12∫dλ2πeixλ(Pu)⟨πb(p′)|¯q(−12λu)uμγμτcq(12λu)|πa(p)⟩, (2)

where denotes the doublet of - and -quark fields and the Pauli matrices in isospin space.  is a lightlike auxiliary vector, and we use the standard notations for the kinematical variables

 P=p+p′2,Δ=p′−p,t=Δ2,ξ=−Δu2Pu. (3)

In terms of the distributions for individual quark flavors defined in  we have and . As usual, Wilson lines between the quark fields are to be inserted in (1) and (2) if one is not working in the light-cone gauge .

Time reversal and charge conjugation symmetry result in the following properties for the pion GPDs:

 HI(x,ξ,t) = HI(x,−ξ,t), (4) HI(x,ξ,t) = (−1)1−IHI(−x,ξ,t). (5)

The distributions can be analytically continued to positive and are real-valued for , as discussed in [13, 14].

The -moments of the GPDs are related to the matrix elements of the local isoscalar and isovector operators

 Osμ1μ2…μn=S12¯qγμ1iD\raisebox3.44pt$↔$μ2…iD\raisebox3.44pt$↔$μnq,Oaμ1μ2…μn=S12¯qγμ1iD\raisebox3.44pt$↔$μ2…iD\raisebox3.44pt$↔$μnτaq, (6)

where and all fields are to be taken at the same space-time coordinate. denotes symmetrization and subtraction of trace terms for all uncontracted Lorentz indices. Defining form factors

 ⟨πb(p′)|Osμ1μ2…μn|πa(p)⟩ = δabn∑k=0,2,…AI=0n,k(t)SΔμ1…ΔμkPμk+1…Pμn, (7) ⟨πb(p′)|Ocμ1μ2…μn|πa(p)⟩ = iϵabcn∑k=0,2,…AI=1n,k(t)SΔμ1…ΔμkPμk+1…Pμn, (8)

where the operators are to be taken at zero space-time position and where time reversal invariance limits to even values, one finds

 ∫1−1dxxn−1HI(x,ξ,t)=n∑k=0,2,…(2ξ)kAIn,k(t). (9)

Note that because of the symmetry relation (5) the moments with odd in (9) vanish for and those with even vanish for . The form factors are real-valued in the region . Information on the lowest moments

 ∫1−1dxHI=1(x,ξ,t) = AI=11,0(t), (10) ∫1−1dxxHI=0(x,ξ,t) = AI=02,0(t)+4ξ2AI=02,2(t) (11)

will soon be accessible from the lattice for pion masses down to values  MeV  . Note that the first moment of is related to the pion electromagnetic form factor as and the second moment of with the form factors in the parameterization 

 ⟨πb(p′)|TQμν|πa(p)⟩=δab[2PμPνθ2(t)+12(gμνΔ2−ΔμΔν)θ1(t)], (12)

where is the quark part of the energy-momentum tensor. Comparison with (7) gives and .

An analogous discussion can be given for the gluon GPD, which is defined by

 δabHgπ(x,ξ,t)=1Pu∫dλ2πeixλ(Pu)⟨πb(p′)|uαGαμ(−12λu)uβGμβ(12λu)|πa(p)⟩ (13)

in the convention of . It has the same symmetry properties as , and its moments are analogs of the moments , with which they mix under renormalization scale evolution. For ease of notation we will in the following concentrate on the quark sector, bearing in mind that our results for the isoscalar form factors readily generalize to their gluonic counterparts.

## 3 Tensor operators in ChPT

We use the standard power-counting of ChPT, where denotes a generic pion energy or momentum. The Goldstone boson fields are collected in the matrix-valued field , and the leading-order Lagrangian, which is of order , reads 

 L(2)ππ = 14F2tr(DμUDμU†+χ†U+χU†) (14)

for two light flavors, with the covariant derivative and the external field tensor defined as

 DμU = ∂μU−irμU+iUlμ, (15) χ = 2B0(s+ip). (16)

Here and denote external scalar and pseudoscalar fields, which count as quantities of order . Furthermore, and are external right- and left-handed vector fields with intrinsic chiral power . The two leading-order parameters appearing in (14) are the pion decay constant (normalized to  MeV) and the two-flavor chiral condensate , both evaluated in the chiral limit . Throughout this work we use the non-linear representation for the pion fields ,

 U=exp{iτaπa/F}. (17)

Left and right chiral rotations of the nonlinear pion field

 U→VRUV†L (18)

induce the following transformation of the external fields:

 rμ → VRrμV†R+iVR∂μV†R,χ→VRχV†L, (19) lμ → VLlμV†L+iVL∂μV†L.

One method to calculate the matrix elements of the operators and from (6) in ChPT would be to introduce corresponding tensor currents as external sources in the Lagrangian, in generalization of the vector currents and in (14). For tensors of higher rank, this becomes however increasingly awkward. Instead, we will directly match the operators in (6) onto operators with pion degrees of freedom. The expression of the isovector vector current is for instance well-known to be 

 Oaμ=12¯qγμτaq ≃ Vaμ=−i4F2tr{τa(U†∂μU+U∂μU†)} (20)

to leading order in the chiral expansion. Despite the progress in deriving chiral dynamics directly from QCD (see e.g. ) this matching problem is far from being solved in general. The so-called low-energy constants appearing in ChPT can in principle be determined by fits to either experimental or lattice data, but only a limited number of them are known in practice. What we can however do is to construct for a given quark or gluon operator in QCD all operators in ChPT with the same symmetry. Their matching coefficients typically are linear combinations of low-energy constants and can be fitted directly to lattice data at different unphysical pion masses, provided they are sufficiently close to the chiral limit. In Section 5 we will determine the functional dependence on of the form factors required for such a fit at one-loop level in ChPT.

For the discussion of chiral symmetry it is useful to consider the left- and right-handed tensor operators

 (OLij)μ1…μn = S¯qjγμ11−γ52iD\raisebox3.44pt$↔$μ2…iD\raisebox3.44pt$↔$μnqi, (21) (ORij)μ1…μn = S¯qjγμ11+γ52iD\raisebox3.44pt$↔$μ2…iD\raisebox3.44pt$↔$μnqi, (22)

which take values in two-dimensional isospin space. The operators in (6) are then given by

 Osμ1…μn=12tr{OLμ1…μn+ORμ1…μn},Oaμ1…μn=12tr{τa(OLμ1…μn+ORμ1…μn)}. (23)

Note that for ease of notation we shall use the same symbol for the quark operators in QCD and for the operators representing them in ChPT. The left- and right-handed operators behave like

 OLμ1…μn→VLOLμ1…μnV†L,ORμ1…μn→VRORμ1…μnV†R (24)

under chiral rotations and are related as by the parity transformation . To construct a linearly independent set of operators for the expansion of , let us inspect the building blocks at hand. For the purpose of calculating matrix elements between pion states we can set the external fields , and to zero, and keep the scalar field as a nonzero constant field implementing the explicit breaking of chiral symmetry by the quark masses. Since we assume exact isospin symmetry, this simply gives , where is the bare pion mass in ChPT. For constructing the operator this leaves us with the objects , and , with the additional condition . It is convenient to introduce the following quantities

 Lμ=U†∂μU,χL±=χ†U±U†χ, (25)

which transform covariantly under left rotations and are invariant under right rotations. The product of some number of and with any number of additional derivatives acting on them transforms covariantly under left rotations. Since for , one can eliminate all derivatives acting on . Similarly, taking into account the identities

 ∂μLν−∂νLμ=[Lμ,Lν],∂μ∂μLν=∂ν∂μLμ+[∂μLν,Lμ]+[Lν,∂μLμ] (26)

and the equation of motion for the field ,

 ∂μLμ=−14(2χL−−{1}\kern-2.5pt% \hbox{l}trχL−), (27)

one can trade the tensors for the symmetric and traceless ones,

 Lν1…νn=S∂ν1…∂νn−1Lνn. (28)

A linearly independent set of operators transforming like under chiral rotations can hence be constructed from products of the fields and the tensors . In the ChPT expansion of they can be multiplied by chirally invariant operators, which can in turn be constructed from the traces of products of and .111Equivalently, one may take the traces of products of and introduced below. In each term of the expansion, some Lorentz indices of the tensors are equal to and all others are contracted. Since and are of order , each contraction of the indices, as well as the insertion of a field increases the chiral counting by a factor .

At leading order of ChPT the operator can hence be represented as the sum of products of tensors and their traces, with no contracted Lorentz indices. It is convenient to contract free indices with the light-line vector , introducing the notation

 OLn(u)=OLμ1…μnuμ1…uμn,Lk(u)=Lμ1…μkuμ1…uμk. (29)

The operator is then expanded on terms of the form

 Lk1(u)…Lkj1(u)tr{Lkj1+1(u)…Lkj2(u)}tr{Lkj2+1(u)…Lkj3(u)}… (30)

and explicitly reads

 OLn(u)=cnLn(u)+n−1∑k=1cn,kLk(u)Ln−k(u)+…, (31)

where we used that and have omitted terms containing three or more tensors. Such terms do in fact not contribute to the two-pion matrix elements in leading order and next-to-leading order of ChPT. To see this, notice that the expansion of into pion fields starts as

 Lμ=iτa(∂μπa/F−ϵabcπb∂μπc/F2+23πb(πa∂μπb−πb∂μπa)/F3+O(π3)∂μπ). (32)

Terms with three or more tensors hence do not contribute to two-pion matrix elements at Born level. At one loop, they contribute only through the tadpole diagram of Fig. 1a, where three or four pion fields from the inserted operator then carry a derivative contracted with the light-like vector . The numerator in the corresponding Feynman integral thus has the form , where and is the loop momentum. For odd the integral vanishes because the integrand is antisymmetric in , whereas for even the integral is proportional to by Lorentz invariance, and hence zero as well.

Repeating our discussion for the right-handed operators , with

 Rμ=U∂μU†,χR±=χU†±Uχ† (33)

and definitions analogous to (28) and (29), the tensor operator reads at order in ChPT

 On(u)=cn(Ln(u)+Rn(u))+n−1∑k=1cn,k(Lk(u)Ln−k(u)+Rk(u)Rn−k(u))+…, (34)

where the dots denote terms which do not contribute to two-pion matrix elements to accuracy. The coefficients of the and terms must be equal because of parity invariance. Using the hermiticity and time reversal properties of following from (21), (22) and the corresponding properties of and , one finds

 (cn)∗=−cn, (cn)∗=(−1)ncn, (35) (cn,k)∗=cn,n−k, (cn,k)∗=(−1)ncn,k. (36)

Neglecting terms which do not contribute to two-pion matrix elements one derives for the isoscalar and isovector operators at order

 Osn(u) = n/2∑k=1an,k(Lak(u)Lan−k(u)+Rak(u)Ran−k(u)),              (n even) (37) Oan(u) = ibn,0(Lan(u)+Ran(u)) (38) + εabc(n−1)/2∑k=1bn,k(Lbk(u)Lcn−k(u)+Rbk(u)Rcn−k(u)),     (n odd)

where is defined by and in analogy. The expansion coefficients and are linear combinations of the coefficients in (34),

 an,k = cn,k+cn,n−kfor k≥1, (39) bn,k = i(cn,k−cn,n−k)for k≥1,bn,0=−icnk. (40)

Because of the symmetry relations (35), (36) one has for odd and for even , and the nonzero coefficients and are real. To obtain simple expressions for the form factors in (9), we rearrange derivatives and obtain for the isosinglet operator

 Osn(u)=F2n−2∑k=0,2,…~an,k(iu∂)k(La(u)(2iu∂\raisebox3.44pt$↔$)n−k−2La(u)+Ra(u)(2iu∂\raisebox3.44pt$↔$)n−k−2Ra(u)) (41)

with

 A(0)n,k=2n−k[~an,k−2−~an,k],(n and k even, k≤n) (42)

where we set . To ease the notation we have omitted the isospin label, implying here and in the following if the first index of a form factor is even, and if it is odd. The inverse of (42) reads

 ~an,k=−k∑m=02m−nA(0)n,m=n∑m=k+22m−nA(0)n,m.(n, m and k even, k≤n−2) (43)

From (42) we also see that at order the sum of the isoscalar form factors with given vanishes,

 n∑k=0,2,…2kA(0)n,k=0,(n even) (44)

which according to (9) gives the soft pion theorem first derived in . Recalling our discussion at the end of Section 2, we emphasize at this point that the ChPT equivalent of the twist-two gluon operators has the same form as (41), of course with different matching coefficients. Analogs of (42) to (44) therefore hold for the appropriate moments of the gluon GPD .

Rewriting (38) in a form similar to (41) gives for the isovector operator

 Oan(u) = ~bn,n−1(iu∂)n−1Va(u) (45) + iF2ϵabcn−3∑k=0,2~bn,k(iu∂)k(Lb(u)(2iu∂\raisebox3.44pt$↔$)n−k−2Lc(u)+Rb(u)(2iu∂\raisebox3.44pt$↔$)n−k−2Rc(u)),

where the are linear combinations of the in (38). Here

 Va(u)=uμVaμ=−i2F2(La(u)+Ra(u)) (46)

is the isovector vector current already given in (20), which also implies . For the isovector form factors one derives at

 A(0)n,k=2n−k[~bn,k−~bn,k−2],~bn,k=k∑m=0,2,…2m−nA(0)n,m,(n odd, k even, k≤n−1) (47)

where we set . Note that the isovector analog of (44),

 n−1∑k=0,2,…2kA(0)n,k=2n~bn,n−1,(n odd) (48)

does not have a vanishing right-hand side. The coefficient is however related to another observable, namely the one-pion matrix element of the isovector axial twist-two operator

 Oa5,n(u)=12tr{τa(ORn(u)−OLn(u))}=12¯quμγμγ5(iuD\raisebox3.44pt$↔$)n−1τaq, (49)

which to leading order of ChPT has the form

 Oa5,n(u)=~bn,n−1(iu∂)n−1Aa(u)+…, (50)

where is the isovector axial current and the dots denote terms which do not contribute to the one-pion matrix element to accuracy for reasons analogous to those we discussed after (32). One readily finds at order . The pion distribution amplitude is defined as

 12∫dλ2πeixλ(pu)/2⟨πb(p)|¯q(−12λu)uμγμγ5τaq(12λu)|0⟩=−iδabFπϕπ(x), (51)

where is the momentum fraction of the quark in the pion and . Here is the physical value of the pion decay constant, given in one-loop ChPT as 

 Fπ=F(1−m216π2F2logm2μ2+m2F2lr4(μ)+O(p4)), (52)

where denotes the renormalization scale and is a renormalized low-energy constant. Taking -moments of the distribution amplitude, one obtains the matrix elements of , so that to order one has

 Bndef=12n∫1−1dxxn−1ϕπ(x)=~bn,n−1.(n odd) (53)

Combining this with (48) and going from moments to -space, we obtain the soft pion theorem from . Let us now consider the order corrections to the one-pion matrix elements. A nonanalytic dependence on the pion mass is generated by one-loop graphs with the insertion of the leading-order operator (50), which according to (32) contains terms with three pion fields. Since for different the operators in (50) only differ by their coefficients and the number of overall derivatives , the one-loop corrections of the one-pion matrix elements for all are universal and hence the same as the corrections to the isovector axial current, given by (52). We can therefore write

 ⟨πb(p)|Oa5,n(u)|0⟩=−iδab(pu)nFπ~bn,n−1+O(m2) (54)

to order accuracy, where the terms are generated by tree-level insertions of higher-order terms in the ChPT expansion of , which can be constructed in analogy to those in Section 4. As a consequence, the relation (53) does not receive nonanalytic corrections in the pion mass at next-to-leading order in ChPT, and one has up to terms of order . Together with the corrections of the form factors , this will allow us to discuss the one-loop corrections to the soft pion theorem in Section 5. We note that to the order given, the bare pion mass in (52) and (54) can be replaced with its renormalized value , given by 

 m2π=m2(1+m232π2F2logm2μ2+2m2F2lr3(μ)+O(p4)). (55)

Let us finally show that our representations (41) and (45) of the operators and in ChPT are equivalent to those given in  without derivation. There, the matching was done for the nonlocal operators appearing in the definition of GPDs. For the left-handed current we find in our notation

 ¯qj(−12λu)uμγμ1−γ52qi(12λu) (56) ≃

where and , and where the dots denote terms not contributing to two-pion matrix elements at order . Due to time reversal invariance the function is even in . Taylor expansion gives the local operators

 ¯qjuμγμ1−γ52(iuD\raisebox3.44pt$↔$)n−1qi ≃

where all fields are taken at space-time position zero and the dots have the same meaning as before. By repeated use of the identity we can rewrite

 U†(u∂\raisebox3.44pt$↔$)nU=⎧⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎨⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎩n−2∑m=0,2,…(12u∂)m[L(u∂\raisebox3.44pt$↔$)n−m−2L]+O(L3)(n even)n−3∑m=0,2,…(12u∂)m[L(u∂\raisebox3.44pt$↔$)n−m−2L]+(12u∂)n−1L+O(L3)(n odd) (58)

and then project out the isoscalar part for even and the isovector part for odd. Repeating the same arguments for the right-handed operators, we recognize precisely the operators that appear in (41) and (45).

## 4 Tensor operators at order O(p2)

The number of independent terms in the ChPT expansion of operators rapidly grows with the order. Henceforth we shall consider only the terms which contribute to two-pion matrix elements at tree-level. Taking into account the discussion in the previous section, one finds that the corrections of order to the operators can be cast into the form

 ΔOsn(u) = 12aχntr{χL−Ln(u)+χR−Rn(u)}+12n/2∑k=1aχn,ktr{Lk(u)Ln−k(u)+Rk(u)Rn−k(u)}trχL+ (59) + 12n/2∑k=0Δan,ktr{Lρ,k(u)Lρn−k(u)+Rρ,k(u)Rρn−k(u)}

for and

 ΔOan(u) = ibχn,0(Lan(u)+Ran(u))trχL++εabc(n−1)/2∑k=1bχn,k(Lbk(u
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