DESY-06-123 hep-ph/0608113

Chiral perturbation theory for
nucleon generalized parton distributions

M. Diehl$\mskip 1.5mu{}^{1}$, A. Manashov$\mskip 1.5mu{}^{2,3}$ and A.
Schäfer$\mskip 1.5mu{}^{2}$

1 Theory Group, Deutsches Elektronen-Synchroton DESY, 22603 Hamburg, Germany
2 Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg,
Germany
3 Department of Theoretical Physics, Sankt-Petersburg State University,
St.-Petersburg, Russia

Abstract
We analyze the moments of the isosinglet generalized parton distributions $H$,
$E$, $\tilde{H}$, $\tilde{E}$ of the nucleon in one-loop order of heavy-baryon
chiral perturbation theory. We discuss in detail the construction of the
operators in the effective theory that are required to obtain all corrections
to a given order in the chiral power counting. The results will serve to
improve the extrapolation of lattice results to the chiral limit.

##  1 Introduction

In recent years one has learned that many aspects of hadron structure can be
described in the unifying framework of generalized parton distributions
(GPDs). This framework allows one to combine information which comes from very
different sources in an efficient and model-independent manner. The field was
pioneered in [1, 2, 3] and has evolved to considerable complexity, reviewed
for instance in [4, 5, 6, 7]. As GPDs can be analyzed using standard operator
product expansion techniques [1, 8], their moments can be and have been
calculated in lattice QCD [9]. Lattice calculations of well-measured
quantities can be used to check the accuracy of the method, which may then be
employed to evaluate quantities that are much harder to determine
experimentally. This complementarity is especially valuable in the context of
GPDs, because experimental measurements as e.g. in [10] may not be sufficient
to determine these functions of three kinematic variables in a model-
independent way. Moreover, several moments of GPDs admit a physically
intuitive interpretation in terms of the spatial and spin structure of
hadrons, see e.g. [2, 11, 12, 13].

A notorious problem of lattice QCD is the need for various extrapolations from
the actual simulations with finite lattice spacing, finite volume and
unphysically heavy quarks to the continuum, infinite volume and physical quark
masses. Simple phenomenological fits are often still sufficient in view of the
general size of uncertainties, but with increasing numerical precision more
reliable methods have to be applied. Chiral perturbation theory (ChPT)
provides such a method [14]. Describing the exact low-energy limit of QCD it
predicts the functional form for the dependence of observables on the finite
volume and the pion mass [15] and also the finite lattice spacing [16]. At a
given order in the expansion parameter, ChPT defines a number of low-energy
constants which determine each of these limits. Some of these constants are
typically known from independent sources, and the remaining ones have to be
determined from fits to the lattice data. The task of ChPT is thus to provide
the corresponding functional expressions for a sufficient number of
observables. In this paper we contribute to this endeavor by analyzing the
moments of the isoscalar nucleon GPDs $H$, $E$, $\tilde{H}$ and $\tilde{E}$ in
one-loop order.

The analysis of pion GPDs in ChPT has been performed in several papers [17,
18, 19]. In the case of the nucleon GPDs, the chiral corrections have been
calculated for the lowest moments [20, 17, 21] in the framework of heavy-
baryon ChPT, which performs an expansion in the inverse nucleon mass $1/M$.
Due to the kinematic limit taken in this scheme, the sum and difference of the
incoming and outgoing nucleon momenta $p^{\mu}$ and $p^{\prime\mu}$ are of
different order in $1/M$. As a consequence, the $n$th moment of a nucleon GPD
contains terms up to $n$th order in the $1/M$ expansion. Given the rapidly
growing number of low-energy constants in higher orders of ChPT, it has been
assumed that the chiral corrections can only be calculated for the terms of
lowest order in $1/M$, i.e. for the form factors accompanied by the smallest
number of vectors $(p^{\prime}-p)^{\mu}$. This would be a serious setback for
the program sketched above. The aim of the present paper is to show that the
situation is much better. In particular, we find that the corrections of order
$O(m_{\pi})$ and $O(m_{\pi}^{2})$ to _all_ form factors parameterizing the
moments of chiral-even isoscalar nucleon GPDs come from one-loop diagrams in
ChPT and the corresponding higher-order tree-level insertions.

This paper is organized as follows. In Section 2 we recall the relation
between moments of nucleon GPDs and matrix elements of twist-two operators and
rewrite it in a form suitable for the $1/M$ expansion. In Section 3 we discuss
the construction of twist-two operators in heavy-baryon ChPT and give a
general power-counting scheme for their contribution to a given nucleon matrix
element. In Sections 4 and 5 we identify the operators that contribute to
moments of GPDs at lowest order in the chiral expansion and give the results
of the corresponding loop calculations. We summarize our findings in Section
6.

##  2 Generalized parton distributions in the nucleon

The nucleon GPDs can be introduced as matrix elements of nonlocal operators.
In this paper we limit ourselves to the chiral-even isoscalar quark GPDs,
which are defined by

$\displaystyle\int\frac{d\lambda}{4\pi}\,e^{ix\lambda(aP)}\langle
p^{\prime}|\,\bar{q}(-{\textstyle\frac{1}{2}}\lambda
a)\,\not{a}\,q({\textstyle\frac{1}{2}}\lambda a)\,|p\rangle$
$\displaystyle=\frac{1}{2aP}\,\bar{u}(p^{\prime})\left[\not{a}\,H(x,\xi,t)+\frac{i\sigma^{\mu\nu}a_{\mu}\Delta_{\nu}}{2M}\,E(x,\xi,t)\,\right]u(p)\,,$
$\displaystyle\int\frac{d\lambda}{4\pi}\,e^{ix\lambda(aP)}\langle
p^{\prime}|\,\bar{q}(-{\textstyle\frac{1}{2}}\lambda
a)\,\not{a}\gamma_{5}\,q({\textstyle\frac{1}{2}}\lambda a)\,|p\rangle$
$\displaystyle=\frac{1}{2aP}\,\bar{u}(p^{\prime})\left[\not{a}\gamma_{5}\,\widetilde{H}(x,\xi,t)+\frac{a\Delta}{2M}\gamma_{5}\,\widetilde{E}(x,\xi,t)\right]u(p)\,,\hfill$
(1)

where a sum over $u$ and $d$ quark fields on the l.h.s. is understood, so that
$H=H^{u}+H^{d}$ etc. Here $a$ is a light-like auxiliary vector, $M$ is the
nucleon mass, and we use the standard notations for the kinematical variables

$\displaystyle P=\frac{1}{2}(p+p^{\prime}),$
$\displaystyle\qquad\Delta=p^{\prime}-p,$ $\displaystyle\qquad t=\Delta^{2},$
$\displaystyle\qquad\xi=-\frac{\Delta a}{2Pa}\,.$ (2)

As usual, Wilson lines between the quark fields are to be inserted in (2) if
one is not working in the light-cone gauge $a^{\mu}A_{\mu}=0$. The $x$-moments
of the nucleon GPDs are related to the matrix elements of the local twist-two
operators

$\displaystyle\mathcal{O}_{\mu_{1}\mu_{2}\ldots\mu_{n}}=\mathbf{S}\,\bar{q}\gamma_{\mu_{1}}i{D^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu_{2}}\ldots
i{D^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu_{n}}\,q\,,$
$\displaystyle\widetilde{\mathcal{O}}_{\mu_{1}\mu_{2}\ldots\mu_{n}}=\mathbf{S}\,\bar{q}\gamma_{\mu_{1}}\gamma_{5}\,i{D^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu_{2}}\ldots
i{D^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu_{n}}\,q\,,$
(3)

where
${D^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}^{\mu}=\frac{1}{2}({D^{\raisebox{2.41112pt}{$\scriptstyle\rightarrow$}}}{}^{\mu}-{D^{\raisebox{2.41112pt}{$\scriptstyle\leftarrow$}}}{}^{\mu})$
and $\mathbf{S}$ denotes the symmetrization of all uncontracted Lorentz
indices and the subtraction of traces, e.g.
$\mathbf{S}\,t_{\mu\nu}={\textstyle\frac{1}{2}}(t_{\mu\nu}+t_{\nu\mu})-\frac{1}{4}\mskip
1.5mug_{\mu\nu}\mskip 1.5mut^{\lambda}{}_{\lambda}$ for a tensor of rank two.
It is convenient to contract all open Lorentz indices with the auxiliary
vector $a$,

$\displaystyle{\mathcal{O}}_{\mu_{1}\ldots\mu_{n}}\leavevmode\nobreak\
\to\leavevmode\nobreak\ {\mathcal{O}}_{n}(a)=a^{\mu_{1}}\ldots
a^{\mu_{n}}\,{\mathcal{O}}_{\mu_{1}\ldots\mu_{n}}\,,$ (4)

and in analogy for $\widetilde{\mathcal{O}}$. The matrix elements of the
operators (3) can be parameterized as [4, 6]

$\displaystyle\langle{p^{\prime}|\mathcal{O}_{n}(a)|p}\rangle$
$\displaystyle=\sum_{\begin{subarray}{c}k=0\\\
{\mathrm{even}}\end{subarray}}^{n-1}(aP)^{n-k-1}\,(a\Delta)^{k}\,\,\bar{u}(p^{\prime})\left[\not{a}\,A_{n,k}(t)+\frac{i\sigma^{\mu\nu}a_{\mu}\Delta_{\nu}}{2M}\,B_{n,k}(t)\right]u(p)$
$\displaystyle\quad+\bmod(n+1,2)\,(a\Delta)^{n}\,\frac{1}{M}\,\bar{u}(p^{\prime})u(p)\,C_{n}(t)\,,$
$\displaystyle\langle{p^{\prime}|\widetilde{\mathcal{O}}_{n}(a)|p}\rangle$
$\displaystyle=\sum_{\begin{subarray}{c}k=0\\\
{\mathrm{even}}\end{subarray}}^{n-1}(aP)^{n-k-1}\,(a\Delta)^{k}\,\,\bar{u}(p^{\prime})\left[\not{a}\gamma_{5}\,\widetilde{A}_{n,k}(t)+\frac{a\Delta}{2M}\gamma_{5}\,\widetilde{B}_{n,k}(t)\right]u(p)\,.$
(5)

The moments of the above GPDs are polynomials in $\xi^{2}$,

$\displaystyle\int_{-1}^{1}dx\,x^{n-1}\,H(x,\xi,t)$
$\displaystyle=\sum_{\begin{subarray}{c}k=0\\\
{\mathrm{even}}\end{subarray}}^{n-1}(2\xi)^{k}\,A_{n,k}(t)+\mathrm{mod}(n+1,2)\,(2\xi)^{n}C_{n}(t)\,,$
$\displaystyle\int_{-1}^{1}dx\,x^{n-1}\,E(x,\xi,t)$
$\displaystyle=\sum_{\begin{subarray}{c}k=0\\\
{\mathrm{even}}\end{subarray}}^{n-1}(2\xi)^{k}\,B_{n,k}(t)-\mathrm{mod}(n+1,2)\,(2\xi)^{n}C_{n}(t)\,,$
$\displaystyle\int_{-1}^{1}dx\,x^{n-1}\,\widetilde{H}(x,\xi,t)$
$\displaystyle=\sum_{\begin{subarray}{c}k=0\\\
{\mathrm{even}}\end{subarray}}^{n-1}(2\xi)^{k}\,\widetilde{A}_{n,k}(t)\,,$
$\displaystyle\int_{-1}^{1}dx\,x^{n-1}\,\widetilde{E}(x,\xi,t)$
$\displaystyle=\sum_{\begin{subarray}{c}k=0\\\
{\mathrm{even}}\end{subarray}}^{n-1}(2\xi)^{k}\,\widetilde{B}_{n,k}(t)\,.$ (6)

The restriction to even $k$ in (2) and (2) is a consequence of time reversal
invariance.

To calculate the chiral corrections to the nucleons form factors we shall use
the formalism of heavy-baryon chiral perturbation theory, which treats the
nucleon as an infinitely heavy particle and performs a corresponding non-
relativistic expansion [22]. The evaluation of nucleon form factors in heavy-
baryon ChPT is simplified if one works in the Breit frame [23]. It is defined
by the condition $\vec{P}=0$, so that the incoming and outgoing nucleons have
opposite spatial momenta ${\vec{p}\,}^{\prime}=-\vec{p}=\vec{\Delta}/2$ and
the same energy $p_{0}^{\prime}=p_{0}^{\phantom{{}^{\prime}}}=M\gamma$, where

$\gamma=\sqrt{1-\Delta^{2}/4M^{2}}\,.$ (7)

In the heavy-baryon formalism the baryon has a additional quantum number, the
velocity $v$, which in the Breit frame is $v=(1,0,0,0)$. The incoming and
outgoing nucleon momenta are thus given by $p=M\gamma v-\Delta/2$ and
$p^{\prime}=M\gamma v+\Delta/2$.

The Dirac algebra simplifies considerably in the heavy-baryon formulation. All
Dirac bilinears can be expressed in terms of the velocity $v_{\mu}$ and the
spin operator

$S_{\mu}=\frac{i}{2}\gamma_{5}\sigma_{\mu\nu}\,v^{\nu}.$ (8)

Using that $(v\Delta)=(vS)=0$, one finds in particular

$\displaystyle\overline{u}(p^{\prime})u(p)$
$\displaystyle=\gamma\,\bar{u}_{v}(p^{\prime})\,u_{v}(p)\,,\phantom{\frac{1}{M}}$
$\displaystyle\overline{u}(p^{\prime})\gamma_{\mu}u(p)$
$\displaystyle=v_{\mu}\,\bar{u}_{v}(p^{\prime})\,u_{v}(p)+\frac{1}{M}\,\bar{u}_{v}(p^{\prime})\,[S_{\mu},(S\Delta)]\,u_{v}(p)\,,$
$\displaystyle\frac{i}{2M}\,\overline{u}(p^{\prime})\,\sigma_{\mu\nu}\Delta^{\nu}u(p)$
$\displaystyle=v_{\mu}\frac{\Delta^{2}}{4M^{2}}\,\bar{u}_{v}(p^{\prime})\,u_{v}(p)+\frac{1}{M}\,\bar{u}_{v}(p^{\prime})\,[S_{\mu},(S\Delta)]\,u_{v}(p)\,,$
$\displaystyle\overline{u}(p^{\prime})\gamma_{\mu}\gamma_{5}u(p)$
$\displaystyle=2\gamma\,\bar{u}_{v}(p^{\prime})S_{\mu}u_{v}(p)+\frac{\Delta_{\mu}}{2M^{2}(1+\gamma)}\,\bar{u}_{v}(p^{\prime})\,(S\Delta)\,u_{v}(p)\,,$
$\displaystyle\overline{u}(p^{\prime})\gamma_{5}u(p)$
$\displaystyle=\frac{1}{M}\,\bar{u}_{v}(p^{\prime})\,(S\Delta)\,u_{v}(p)\,,$
(9)

where the spinors

$\displaystyle u_{v}(p)$
$\displaystyle=\mathcal{N}^{-1}\,\frac{1+\not{v}}{2}\,u(p),$ $\displaystyle
u_{v}(p^{\prime})$
$\displaystyle=\mathcal{N}^{-1}\,\frac{1+\not{v}}{2}\,u(p^{\prime})$ (10)

with

$\mathcal{N}=\sqrt{\frac{M+vp}{2M}}=\sqrt{\frac{M+vp^{\prime}}{2M}}=\sqrt{\frac{1+\gamma}{2}\rule{0.0pt}{13.50008pt}}$
(11)

are normalized as
$\bar{u}_{v}(p,s^{\prime})\,u_{v}(p,s)=2M\delta_{s^{\prime}s}$. With (2) one
obtains the following representation for the matrix elements (2) in the Breit
frame:

$\displaystyle\langle{p^{\prime}|\mathcal{O}_{n}(a)|p}\rangle$
$\displaystyle=\sum_{k=0}^{n}(M\gamma)^{n-k-1}\,(av)^{n-k}\,(a\Delta)^{k-1}\,$
$\displaystyle\quad{}\times\bar{u}_{v}(p^{\prime})\,\Big{[}(a\Delta)\,E_{n,k}(t)+\gamma\,[(aS),(S\Delta)]\,M_{n,k-1}(t)\Big{]}\,u_{v}(p)\,,$
$\displaystyle\langle{p^{\prime}|\widetilde{\mathcal{O}}_{n}(a)|p}\rangle$
$\displaystyle=\sum_{k=1}^{n}(M\gamma)^{n-k}\,(av)^{n-k}\,(a\Delta)^{k-1}\,$
$\displaystyle\quad{}\times\bar{u}_{v}(p^{\prime})\left[2\gamma\mskip
1.5mu(aS)\,{\widetilde{E}}_{n,k-1}(t)+\frac{(a\Delta)(S\Delta)}{2M^{2}}\,{\widetilde{M}}_{n,k-1}(t)\right]u_{v}(p)\,,\hskip
40.00006pt$ (12)

with

$\displaystyle E_{n,k}(t)$
$\displaystyle=A_{n,k}(t)+\frac{\Delta^{2}}{4M^{2}}B_{n,k}(t)\hskip
20.00003pt\text{for\leavevmode\nobreak\ }k<n\,,$ $\displaystyle E_{n,n}(t)$
$\displaystyle=\gamma^{2}C_{n}(t)\,,$ $\displaystyle M_{n,k}(t)$
$\displaystyle=A_{n,k}(t)+B_{n,k}(t)\,,$ $\displaystyle\widetilde{E}_{n,k}(t)$
$\displaystyle=\widetilde{A}_{n,k}(t)\,,$
$\displaystyle{\widetilde{M}}_{n,k}(t)$
$\displaystyle=\frac{1}{1+\gamma}\,\widetilde{A}_{n,k}(t)+\widetilde{B}_{n,k}\,.$
(13)

The definition of the $E_{n}$ and ${\widetilde{E}}_{n}$ is conventional but
might be confusing as $E_{n}$ is not the $n$th moment of $E(x,\xi,t)$ etc. We
nevertheless use this notation, in order to make it easier to compare our
results with those in the literature. Notice that according to (2) the terms
with $E_{n,k}$ in (2) are only nonzero if $k$ is even, whereas those with
$M_{n,k-1}$, $\smash{\widetilde{E}}_{n,k-1}$ and
$\smash{\widetilde{M}}_{n,k-1}$ are only nonzero if $k$ is odd. We will
evaluate these form factors in heavy-baryon ChPT. It is straightforward to
transform back to the original form factors using

$\displaystyle A_{n,k}(t)$
$\displaystyle=\frac{1}{\gamma^{2}}\left[E_{n,k}(t)-\frac{\Delta^{2}}{4M^{2}}M_{n,k}(t)\right]\,,$
$\displaystyle B_{n,k}(t)$
$\displaystyle=\frac{1}{\gamma^{2}}\,\Big{[}M_{n,k}(t)-E_{n,k}(t)\Big{]}\,,$
$\displaystyle\widetilde{B}_{n,k}(t)$
$\displaystyle=\widetilde{M}_{n,k}(t)-\frac{1}{1+\gamma}\,\widetilde{E}_{n,k}(t)\,.$
(14)

##  3 Twist-two matrix elements in heavy-baryon ChPT

Heavy-baryon ChPT combines the techniques of chiral perturbation theory and of
heavy-quark effective field theory [22] (for a detailed review see Ref. [24]).
The effective Lagrangian describes the pion-nucleon interactions in the limit
when $m_{\pi},q\ll M$, where $q$ is a generic momentum. In this situation the
velocity $v$ of the nucleon is preserved in the process. One introduces the
nucleon field with velocity $v$ as [22]

$N(x)=e^{-iM_{0}vx}\bigl{(}N_{v}(x)+n_{v}(x)\bigr{)}\,,$ (15)

where $M_{0}$ is the nucleon mass in the chiral limit. The fields $N_{v}(x)$,
$n_{v}(x)$ respectively contain the large and small components of the nucleon
field and satisfy $\not{v}N_{v}=N_{v}$, $\not{v}n_{v}=-n_{v}$. Their Fourier
transform depends on the residual nucleon momentum, i.e. the original nucleon
momentum minus $M_{0}v$. Integrating out the field $n_{v}(x)$, one obtains an
effective Lagrangian for the pion-nucleon system which involves the nucleon
field $N_{v}(x)$ and pion field $\pi(x)$,

${\mathcal{L}}_{\mathrm{eff}}={\mathcal{L}}_{\pi}+{\mathcal{L}}_{\pi N}\,,$
(16)

where

$\displaystyle{\mathcal{L}}_{\pi}$
$\displaystyle={\mathcal{L}}_{\pi}^{(2)}+{\mathcal{L}}_{\pi}^{(4)}+\ldots\,,$
$\displaystyle{\mathcal{L}}_{\pi N}$ $\displaystyle={\mathcal{L}}_{\pi
N}^{(1)}+{\mathcal{L}}_{\pi N}^{(2)}+\ldots$ (17)

are expanded in powers of $q$. The explicit expressions for the lowest-order
terms read [24]

$\displaystyle{\mathcal{L}}^{(2)}_{\pi}$
$\displaystyle=\frac{F^{2}}{4}\mathop{\rm
Tr}\nolimits\left(\partial_{\mu}U\partial^{\mu}U^{\dagger}+(\chi^{\dagger}U+U^{\dagger}\chi)\right)\,,$
$\displaystyle{\mathcal{L}}^{(1)}_{\pi N}$
$\displaystyle=\overline{N}_{\\!v}\,\Big{\\{}i\mskip
1.5mu(v\nabla)+g_{A}(Su)\Big{\\}}\,N_{v}\,,$
$\displaystyle{\mathcal{L}}^{(2)}_{\pi N}$
$\displaystyle=\overline{N}_{\\!v}\left\\{\frac{(v\nabla)^{2}-\nabla^{2}}{2M_{0}}-\frac{ig_{A}}{2M_{0}}\,\bigl{\\{}(\nabla
S),(vu)\bigr{\\}}+c_{1}\mathop{\rm Tr}\nolimits\Big{(}u^{\dagger}\chi
u^{\dagger}+u\chi^{\dagger}u\Big{)}\right.$ $\displaystyle\hskip
22.0pt\left.{}+\Big{(}c_{2}-\frac{g_{A}^{2}}{8M_{0}}\Big{)}(vu)^{2}+c_{3}\,u_{\mu}u^{\mu}+\Big{(}c_{4}+\frac{1}{4M_{0}}\Big{)}[S^{\mu},S^{\nu}]\,u_{\mu}u_{\nu}\right\\}N_{v}$
(18)

with $U=u^{2}=\exp\\{i\pi^{a}\tau^{a}/F\\}$, the covariant derivative
$\nabla_{\mu}=\partial_{\mu}+\Gamma_{\mu}$, and

$\displaystyle\Gamma_{\mu}$
$\displaystyle=\frac{1}{2}\left(u^{\dagger}\partial_{\mu}u+u\partial_{\mu}u^{\dagger}\right)=\frac{i}{4F^{2}}\,\epsilon^{abc}\,\pi^{a}\,\partial_{\mu}\pi^{b}\tau^{c}+O(\pi^{4})\,,$
$\displaystyle u_{\mu}$
$\displaystyle={i}\left(u^{\dagger}\partial_{\mu}u-u\partial_{\mu}u^{\dagger}\right)\hskip
4.09024pt=-\frac{1}{F}\,\partial_{\mu}\pi^{a}\tau^{a}+O(\pi^{3})\,.$ (19)

The trace $\mathop{\rm Tr}\nolimits$ and the Pauli matrices $\tau^{a}$ refer
to isospin space. As is done in current lattice QCD calculations, we assume
isospin symmetry to be exact here, neglecting the difference between $u$\- and
$d$-quark masses. The leading-order parameters appearing in (3) are the pion
decay constant $F$ (normalized to $F\approx 92\mbox{\leavevmode\nobreak\
MeV}$) and the nucleon axial-vector coupling $g_{A}$, both taken in the chiral
limit. The field $\chi$ implements the explicit breaking of chiral symmetry by
the quark masses, and in the isospin limit can be replaced by $\chi\to
m^{2}\,\hbox{{1}\kern-2.5pt\hbox{l}}$, where $m$ is the bare pion and 1l the
unit matrix in isospin space. Estimates of the low-energy constants $c_{i}$ in
the second-order Lagrangian ${\mathcal{L}}^{(2)}_{\pi N}$, which are of order
$1/M$, can be found in [25]. We note that ${\mathcal{L}}^{(2)}_{\pi N}$
induces corrections to the nucleon propagator, which we treat as insertions on
a nucleon line. They read $-i\,\bigl{(}(vl)^{2}-l^{2}\bigr{)}\big{/}(2M_{0})$
and $4i\mskip 1.5muc_{1}m^{2}$, where $l$ is the residual nucleon momentum,
and are to be multiplied with a nucleon propagator $i/(vl+i\mskip 1.5mu0)$
from ${\mathcal{L}}^{(1)}_{\pi N}$ on either side. The pion-nucleon vertices
following from ${\mathcal{L}}^{(2)}_{\pi N}$ can be found in Appendix A of
[24].

In the following subsection we discuss how to construct the operators in the
effective theory that match the twist-two quark operators (3). Nucleon matrix
elements in the Breit frame are then obtained as [26]

$\langle{p^{\prime}|\mathcal{O}|p}\rangle=\mathcal{N}^{2}Z_{N}\;\overline{u}_{v}(p^{\prime})\,G_{\mathcal{O}}(r^{\prime},r)\,u_{v}(p)\,,$
(20)

with the spinors $u_{v}$ and normalization $\mathcal{N}$ given in (10) and
(11). Here $G_{\mathcal{O}}(r^{\prime},r)$ is the truncated Green function for
external heavy-baryon fields $\overline{N}_{\\!v}$, $N_{v}$ and the operator
$\mathcal{O}$ in the effective theory. The residual momenta of the incoming
and outgoing nucleon are given by

$\displaystyle r$ $\displaystyle=p-M_{0}\mskip 1.5muv=wv-\Delta/2\,,$
$\displaystyle r^{\prime}$ $\displaystyle=p^{\prime}-M_{0}\mskip
1.5muv=wv+\Delta/2$ (21)

with

$w=M(\gamma-1)+\delta M=-\frac{\Delta^{2}}{8M}-4c_{1}m^{2}+O(q^{3})\,,$ (22)

where $\delta M=M-M_{0}$ is the nucleon mass shift. Finally, $Z_{N}$ is the
heavy-baryon field renormalization constant,

$Z_{N}=1-\frac{3m^{2}g_{A}^{2}}{2\mskip 1.5mu(4\pi
F)^{2}}-\frac{9m^{2}g_{A}^{2}}{4\mskip 1.5mu(4\pi
F)^{2}}\log\frac{m^{2}}{\mu^{2}}-8m^{2}\,d_{28}^{\mskip
1.5mur}(\mu)+O(q^{3})\,,$ (23)

where $d_{28}^{\mskip 1.5mur}(\mu)$ is a low-energy constant in the Lagrangian
$\mathcal{L}_{\pi N}^{(3)}$. As explained in [27] the corresponding operator
is required for renormalization but does not appear in physical matrix
elements. The value of $d_{28}^{\mskip 1.5mur}(\mu)$ can therefore be chosen
freely (with different choices resulting in different values for other low-
energy constants), and in [26] it was chosen such that it compensates the
$\log(m^{2}/\mu^{2})$ term in (23) at the _physical_ value of $m$. Since we
are interested in the pion mass dependence of matrix elements, we must
explicitly keep the logarithmic term in $Z_{N}$. For further discussion we
refer to Section 3.2.

###  3.1 Construction of effective operators

We now discuss how to construct the isoscalar local twist-two operators in the
effective theory that match the quark-gluon operators $\mathcal{O}(a)$ defined
in (3) and (4). The relevant operators in the effective theory can be divided
into two groups: operators ${\mathcal{O}}_{\pi}$ which contain only pion
fields (and couple to the nucleon via pion loops) and operators
${\mathcal{O}}_{\pi N}$ which are bilinear in the nucleon field. The matching
of operators thus takes the form

$\displaystyle{\mathcal{O}}(a)$
$\displaystyle\,\cong\,{\mathcal{O}}_{\pi}(a)+{\mathcal{O}}_{\pi N}(a)\,,$
$\displaystyle\widetilde{\mathcal{O}}(a)$
$\displaystyle\,\cong\,\widetilde{\mathcal{O}}_{\pi N}(a)\,,$ (24)

where we have taken into account that there is no isoscalar pion operator of
negative parity (i.e. no $\widetilde{\mathcal{O}}_{\pi}(a)$). The pion
isoscalar operators ${\mathcal{O}}_{\pi}(a)$ have been analyzed in several
papers [28, 17, 18, 19] and we shall simply use their results.

Let us now list the building blocks for constructing the operators
${\mathcal{O}}_{\pi N}(a)$ and $\widetilde{\mathcal{O}}_{\pi N}(a)$, which we
collectively denote by $Q(a)$, omitting the subscript $\pi N$ for ease of
writing. They should be bilinear in the nucleon field and should be tensors
that have $n$ indices contracted with the auxiliary vector $a$ according to
(4). To build tensors we have the following objects with Lorentz indices at
our disposal: the velocity vector $v_{\mu}$, the spin vector $S_{\mu}$, the
derivative $\partial_{\mu}$, and the antisymmetric tensor
$\epsilon_{\mu\nu\lambda\rho}$. We recall that any Dirac matrix structure can
be reduced to an expression containing the spin operator $S_{\mu}$, and that
the metric tensor $g_{\mu\nu}$ can be omitted in the construction because the
twist-two operators are traceless. Using the identities

$\displaystyle\\{S_{\mu},S_{\nu}\\}$
$\displaystyle=\frac{1}{2}(v_{\mu}v_{\nu}-g_{\mu\nu})\,,$
$\displaystyle[S_{\mu},S_{\nu}]$
$\displaystyle=i\epsilon_{\mu\nu\lambda\rho}\,v^{\lambda}S^{\rho}$ (25)

we can impose that $S_{\mu}$ should appear at most linearly, or quadratically
as the commutator $[S_{\mu},S_{\nu}]$. Concerning the derivative
$\partial_{\mu}$, we find it useful to have it acting either on single
nonlinear pion fields $u$, $u^{\dagger}$ in the combinations $\Gamma_{\mu}$ or
$u_{\mu}$ given in (3), or as a total derivative on the product of all fields,
or in the antisymmetric form
${\partial^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu}={\textstyle\frac{1}{2}}({\partial^{\raisebox{2.41112pt}{$\scriptstyle\rightarrow$}}}{}_{\mu}-{\partial^{\raisebox{2.41112pt}{$\scriptstyle\leftarrow$}}}{}_{\mu})$
on the product of all fields to its right or to its left. This will make it
easy to keep track of factors $\Delta_{\mu}$ in the corresponding matrix
elements, which play a particular role as we shall see. To give operators with
the correct chiral transformation behavior, the derivative
${\partial^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}$ must
appear in the covariant combination
${\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu}={\partial^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu}+\Gamma_{\mu}$.
The fields and derivatives used in our construction are then any number of
$u_{\mu}$,
${\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu}$ and
$\chi_{\pm}=u^{\dagger}\chi u^{\dagger}\pm u\chi^{\dagger}u$ between the
nucleon fields $\overline{N}_{\\!v}$ and $N_{v}$, and any number of total
derivatives $\partial_{\mu}$ acting on the operator as a whole. In the sense
of (3) we henceforth refer to $\partial_{\mu}$,
${\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu}$ and
$u_{\mu}$ as “derivatives”. They have chiral dimension 1, whereas $\chi_{\pm}$
has chiral dimension 2 and will not appear at the order of the chiral
expansion we limit ourselves to.

We can decompose the pion-nucleon operators $Q_{n}(a)$ as

$\displaystyle Q_{n}(a)=\sum_{k=0}^{n}M^{n-k-1}\,(av)^{n-k}Q_{n,k}(a)\,,$ (26)

where $Q_{n,k}(a)=a_{\mu_{1}}\ldots a_{\mu_{k}}\,Q_{n}^{\mu_{1}\ldots\mu_{k}}$
does not contain any factors $(av)$. The $k$ external vectors $a$ in
$Q_{n,k}(a)$ can be contracted only with derivatives $\partial_{\mu}$,
${\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu}$,
$u_{\mu}$ and the spin vector $S_{\mu}$, or with the antisymmetric tensor.
There can be at most one factor $(aS)$ as discussed after (25), so that
$Q_{n,k}(a)$ has to contain at least $k-1$ derivatives. We can hence
write111Instead of $M$ one could also use $M_{0}$ or $F$ in (26) and (27),
since all are of the same order in chiral power counting. We find powers of
$M$ most convenient, because they also appear in the form factor
decompositions (2).

$Q_{n,k}=MQ_{n,k,-1}+Q_{n,k,0}+\frac{1}{M}\,Q_{n,k,1}+\ldots\,,$ (27)

where the operator $Q_{n,k,i}$ has chiral dimension $k+i$. Note that due to
parity the number of factors $S_{\mu}$, $u_{\mu}$ and
$\epsilon_{\mu\nu\lambda\rho}$ must be even for $\mathcal{O}$ and odd for
$\widetilde{\mathcal{O}}$. We remark that the contraction of $a$ with the
$\epsilon$-tensor involves at least two derivatives, given that we chose to
replace its simultaneous contraction with $v_{\lambda}$ and $S_{\rho}$ by
$[S_{\mu},S_{\nu}]$ using (25). As a consequence, the antisymmetric tensor
does not appear in the operators with lowest chiral dimension for a given $k$.

###  3.2 Tree-level insertions

At tree level, the matrix elements of the effective operators between two
nucleon states are easy to calculate. Since $u_{\mu}$ and $\Gamma_{\mu}$
involve at least one or two pion fields according to (3), derivatives in the
effective operators are to be replaced as $\partial_{\mu}\to i\Delta_{\mu}$,
$u_{\mu}\to 0$, and
${\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu}\ \to-
iwv_{\mu}$ with $w$ given in (22). Notice that, while generically the
derivative
${\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu}$
counts as $O(q)$ in the chiral expansion, the kinematics of the external
nucleon momenta forces $wv_{\mu}$ to be of order $O(q^{2})$. As a result, the
leading-order contributions of the operator $Q_{n,k}$ to the form factors in
(2) come from the terms with maximum number of factors $\Delta_{\mu}$ and no
factor $wv_{\mu}$. With (26) one readily obtains

$\displaystyle\langle{p^{\prime}|\mathcal{O}_{n,k}(a)|p}\rangle$
$\displaystyle\,\stackrel{{\scriptstyle\mathrm{LO}}}{{=}}\,(a\Delta)^{k-1}\,\bar{u}_{v}(p^{\prime})\left[(a\Delta)\,E_{n,k}^{(0)}+[(aS),(S\Delta)]\,M_{n,k-1}^{(0)}\right]u_{v}(p)\,,$
$\displaystyle\langle{p^{\prime}|\widetilde{\mathcal{O}}_{n,k}(a)|p}\rangle$
$\displaystyle\,\stackrel{{\scriptstyle\mathrm{LO}}}{{=}}\,(a\Delta)^{k-1}\,\bar{u}_{v}(p^{\prime})\left[\mskip
1.5mu2M(aS)\,{\widetilde{E}}_{n,k-1}^{(0)}+\frac{(a\Delta)(S\Delta)}{2M}\,{\widetilde{M}}_{n,k-1}^{(0)}\right]u_{v}(p)\,,$
(28)

where the superscript on each form factor indicates the term of order
$O(q^{0})$ in its chiral expansion. At this order, the form factors $E_{n,k}$
and $M_{n,k-1}$ of the vector GPD are related to the matrix element of the
operator $\mathcal{O}_{n,k,0}$, since the nucleon matrix element of the
operator $\mathcal{O}_{n,k,-1}$ is zero at tree level. As explained above,
this operator contains a factor $(aS)$, which due to parity must be
accompanied by the axial field $u_{\mu}$ and hence does not contribute to
tree-level matrix elements without external pions. For the axial vector GPDs
one finds that the form factor ${\widetilde{E}}_{n,k-1}$
(${\widetilde{M}}_{n,k-1}$) receives its leading contribution from the
operator $\widetilde{\mathcal{O}}_{n,k,-1}$
($\widetilde{\mathcal{O}}_{n,k,1}$), given the required number of factors
$\Delta_{\mu}$ in (3.2).

Beyond leading order, tree-level insertions contribute to the form factors
starting at order $O(q^{2})$. Contributions proportional to $\Delta^{2}$ are
due to operators with $\partial^{2}$ or to a factor $w$ from operators with
${\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}$, or to the
kinematic factors $\gamma$ in (2) and $\mathcal{N}$ in (20). Contributions
proportional to $m^{2}$ are due to operators with $\chi_{+}$ or with
${\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}$ and from
the wave function renormalization constant $Z_{N}$ in (20). In the results of
the following sections we explicitly include the terms proportional to
$g_{A}^{2}$ in the expansion (23) of $Z_{N}$, whereas contributions from
$d_{28}^{\mskip 1.5mur}$ are lumped into the coefficients describing the
$m^{2}$ corrections due to tree-level insertions.

###  3.3 Loop contributions

Let us now consider a loop diagram with the insertion of the operator
$Q_{n}(a)$. One easily finds that the term $M^{n-k-1}\,(av)^{n-k}\,Q_{n,k}(a)$
in the sum (26) can contribute to the form factors in (2) which are
accompanied by at least $n-k$ powers of $(av)$, i.e. to $E_{n,m}$,
$M_{n,m-1}$, $\widetilde{E}_{n,m-1}$ and $\widetilde{M}_{n,m-1}$ with $m\leq
k$. Chiral counting determines which terms can contribute to a given order.
Namely, the contribution of the operator $Q_{n,k,i}$ in a loop diagram has
chiral dimension

$D_{k,i}=4L+(k+i)+\sum_{j=1}^{N_{\pi}}\dim V_{\pi}(j)+\sum_{j=1}^{N_{\pi
N}}\dim V_{\pi N}(j)-2I_{\pi}-I_{N}\,,$ (29)

where $L$ is the number of loops and $(k+i)$ is the chiral dimension of the
operator insertion. $V_{\pi}(j)$ and $V_{\pi N}(j)$ respectively denote the
$j$th vertex from $\mathcal{L}_{\pi}$ and $\mathcal{L}_{\pi N}$ in the graph.
$N_{\pi}$ and $N_{\pi N}$ are the corresponding total numbers of vertices, and
$I_{\pi}$ and $I_{N}$ are the numbers of pion and nucleon propagators.222Note
that a nucleon propagator correction from a higher-order Lagrangian counts as
one (nucleon-nucleon) vertex with two nucleon propagators on either side, see
the discussion after (3). Using the relation $L=I_{\pi}+I_{N}-N_{\pi}-N_{\pi
N}$ (see e.g. [24]) and the fact that for our specific diagrams $I_{N}=N_{\pi
N}$, we can rewrite this expression as a sum of positive terms, which makes it
easy to identify the different contributions at a given order:

$D_{k,i}=2L-1+k+(i+1)+\sum_{j=1}^{N_{\pi}}\,\bigl{(}\dim{V_{\pi}}(j)-2\bigr{)}+\sum_{j=1}^{N_{\pi
N}}\,\bigl{(}\dim V_{\pi N}(j)-1\bigr{)}\,.$ (30)

For each vertex we can insert either the lowest or any higher order, i.e.
$\dim V_{\pi}(j)=2,4,...$ and $\dim V_{\pi N}(j)=1,2,...$. Note that a loop
diagram with chiral dimension $D_{k,i}$ generates contributions to a nucleon
matrix element of order $O(q^{\mskip 1.5mud})$ with $d\geq D_{k,i}$. This is
on one hand because of the explicit factors $\mathcal{N}$ and $Z_{N}$ in (20),
and on the other hand because the sum $r^{\mu}+r^{\prime\mu}=2wv^{\mu}$ is of
order $O(q^{2})$ and thus one order higher than the generic power associated
with residual nucleon momenta.

The form factors enter a matrix element multiplied by factors $(a\Delta)$ or
$(S\Delta)$ as given in (2). Taking these into account, one finds that the
chiral correction from $Q_{n,k,i}$ to $E_{n,m}$ and $M_{n,m-1}$ has at least
order $D_{k,i}-m$, while for the form factors $\widetilde{E}_{n,m-1}$ and
$\widetilde{M}_{n,m-1}$ it has at least order $D_{k,i}-m+1$ and $D_{k,i}-m-1$,
respectively. This is a main result of our paper and allows one to determine
which operators need to be considered to calculate the corrections to a form
factor to a given order in the chiral expansion. Because $D_{k,i}$ contains a
term $k$ and because of the constraint $k-m\geq 0$, the number of loops and
the order of the chiral Lagrangian required to calculate the lowest-order
corrections for a given form factor do _not_ grow with $m$. Instead, a growing
number of factors $\Delta_{\mu}$ accompanying a form factor in the nucleon
matrix element requires a growing number of derivatives in the operator
$Q_{n,k}$.

As an application of our general result we find that the form factors
$E_{n,m}$ and $M_{n,m-1}$ can receive corrections starting at order

  * •

$O(q)$ from one-loop diagrams with insertion of the operator $Q_{n,m,-1}$ and
leading-order (LO) pion-nucleon vertices,

  * •

$O(q^{2})$ from the one-loop diagrams with insertion of the operators
$Q_{n,m,0}$ and $Q_{n,m+1,-1}$ and LO pion-nucleon vertices, and from one-loop
diagrams with insertion of the operator $Q_{n,m,-1}$ and one next-to-leading
order (NLO) pion-nucleon vertex or nucleon propagator correction.

In turn, the form factor $\widetilde{E}_{n,m-1}$ receives corrections starting
at order $O(q^{2})$ from one-loop diagrams with leading-order vertices and
insertion of the operator $Q_{n,m,-1}$. For $\widetilde{M}_{n,m-1}$ the
discussion of corrections up to order $O(q^{2})$ is more involved and will be
given in Section 4.1.

To conclude the discussion of power counting, we consider the contribution to
the form factors $E_{n,m}$ and $M_{n,m-1}$ of loop graphs with the insertion
of the pion operators $\mathcal{O}_{\pi}(a)$, see (24). Repeating the above
argument and taking into account that now $I_{N}=N_{\pi N}-1$, one finds that
such diagrams have chiral dimension

$D_{\pi}=2L-1+\dim{\mathcal{O}_{\pi}}+\sum_{j=1}^{N_{\pi}}\,\bigl{(}\dim{V_{\pi}(j)}-2\bigr{)}+\sum_{j=1}^{N_{\pi
N}}\,\bigl{(}\dim V_{\pi N}(j)-1\bigr{)}\,.$ (31)

Given that the leading operator $\mathcal{O}_{\pi}^{n}(a)$ contributing to
$\mathcal{O}_{n}(a)$ has the chiral dimension $n$, one finds that it can
contribute to the form factors $E_{n,m}$ and $M_{n,m-1}$ starting at order
$O(q^{n-m+1})$. Note that because of charge conjugation invariance the
isoscalar pion operators $\mathcal{O}_{\pi}^{n}(a)$ have even $n$ and that due
to time reversal invariance the form factors $E_{n,k}$ and $M_{n,k}$ vanish
for odd $k$. Together with our power-counting formula one thus finds that
$E_{n,n}$ gets contributions from $\mathcal{O}_{\pi}^{n}(a)$ starting at order
$O(q)$ and $M_{n,n-2}$ starting at order $O(q^{2})$. All other corrections
from operators $\mathcal{O}_{\pi}(a)$ to form factors $E_{n,k}$ and $M_{n,k}$
start at $O(q^{3})$.

Let us now take a closer look at the one-loop graphs with pion-nucleon
operator insertions, which are shown in Fig. 1. With our construction of
operators explained in Section 3.1 we can readily analyze the origin of
factors $\Delta_{\mu}$, whose number determines to which form factor a graph
will contribute. Using $(v\Delta)=(vS)=0$ and the form (3) of the LO and NLO
pion-nucleon Lagrangian, we find that the numerators of the loop integrals are
composed as specified in Table 1. The denominators of the pion and nucleon
propagators respectively are $(l^{2}-m^{2}+i\mskip 1.5mu0)$ and $(lv+w+i\mskip
1.5mu0)$, so that the loop integration turns tensors $l_{\mu_{1}}\ldots
l_{\mu_{j}}$ into tensors constructed from $v_{\mu}$ and $g_{\mu\nu}$. A
factor $\Delta_{\mu}$ that can be contracted with $a^{\mu}$ or $S^{\mu}$ (i.e.
is not contracted to $\Delta^{2}$) can hence only originate from total
derivatives $\partial_{\mu}$ in the operator insertion and from an NLO pion-
nucleon vertex or nucleon propagator correction. We will see that this reduces
considerably the number of operators contributing to the leading chiral
corrections of nucleon GPDs.

Table 1:  Four-vectors and their products appearing in the numerators of the loop graphs of Fig. 1. $NN$ vertices (arising from nucleon propagator corrections) are not explicitly shown in the graphs. derivatives in operator insertion | $\partial_{\mu}$ | $\Delta_{\mu}$
---|---|---
| ${\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{}_{\mu}$ |  $l_{\mu}$ and $wv_{\mu}$
| $u_{\mu}$ | $l_{\mu}$
vertices |  $\pi NN$ at LO | $Sl$
|  $\pi NN$ at NLO | $(vl)(Sl)\pm(vl)(S\Delta)$
|  $NN$ at NLO | $(vl)^{2}-l^{2}\pm l\Delta-\Delta^{2}/4$

Figure 1:  One-loop graphs with the insertion of a pion-nucleon operator
$\mathcal{O}_{\pi N}(a)$ or $\widetilde{\mathcal{O}}_{\pi N}(a)$, denoted by a
black blob. Not shown is the analog of graph c with residual momentum
$l+wv+\Delta/2$ of the intermediate nucleon line.

##  4 Chiral corrections up to order $O(q^{2})$

###  4.1 Axial-vector operators

Using the formalism developed in the previous section, we now evaluate the
form factors up to relative order $O(q^{2})$. Let us start by giving the
operators in $\widetilde{O}_{n,k,i}$ that have the maximum number of total
derivatives $\partial_{\mu}$ contracted with $a^{\mu}$ or $S^{\mu}$. It will
turn out that these are required to produce the factors of $(a\Delta)$ and
$(S\Delta)$ in the form factor decomposition (2). With the constraints of
parity invariance, we find

$\displaystyle\widetilde{\mathcal{O}}_{n,k,-1}(a)$
$\displaystyle=\tilde{b}_{n,k}\,(ia\partial)^{k-1}\,\overline{N}_{\\!v}(aS)N_{v}+\ldots\,,$
$\displaystyle\widetilde{\mathcal{O}}_{n,k,1}(a)$
$\displaystyle=\tilde{c}_{n,k}\,(ia\partial)^{k}(i\partial_{\mu})\,\overline{N}_{\\!v}\,S^{\mu}N_{v}+\ldots\,,$
(32)

where the $\ldots$ stand for operators with fewer total derivatives. One has
${\widetilde{E}}_{n,k-1}^{(0)}=\tilde{b}_{n,k}/2$ and
${\widetilde{M}}_{n,k-1}^{(0)}=2\tilde{c}_{n,k}$ for the tree-level
contributions at order $O(q^{0})$. From the time-reversal constraints on the
form factors it follows that the low-energy constants $\tilde{b}_{n,k}$ and
$\tilde{c}_{n,k}$ are zero for even $k$.

As derived in Section 3.3, the leading chiral corrections to
$\widetilde{E}_{n,k-1}$ come from one-loop graphs with LO pion-nucleon
vertices and the operator $\widetilde{O}_{n,k,-1}$. Since this operator does
not contain pion fields, one needs to calculate only graph a in Fig. 1. One
finds

$\widetilde{E}_{n,k}(t)=\widetilde{E}_{n,k}^{(0)}\left\\{1-\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\left[\log\frac{m^{2}}{\mu^{2}}+1\right]\right\\}+\widetilde{E}_{n,k}^{(2,m)}m^{2}+\widetilde{E}_{n,k}^{(2,t)}t+O(q^{3})\,,$
(33)

where the terms going with $m^{2}$ and $t$ originate from tree-level
insertions as discussed at the end of Section 3.2. Here and in the following
we use the subtraction scheme of [14] for the loop graphs, subtracting
$1/\epsilon+\log(4\pi)+\psi(2)$ for each $1/\epsilon$ pole in $4-2\epsilon$
dimensions. The renormalization scale is denoted by $\mu$, and the $\mu$
dependence of the logarithm in (33) cancels against the $\mu$ dependence of
$\widetilde{E}_{n,k}^{(2,m)}$, which we have not displayed for simplicity.
Note that the bare parameters $m$, $F$, $g_{A}$ can be replaced with their
counterparts at the physical point within the precision of our result. Since
the nonanalytic corrections in (33) are independent of the moment indices $n$
and $k$, they apply to the entire nucleon GPD $\widetilde{H}(x,\xi,t)$,

$\displaystyle\widetilde{H}(x,\xi,t)=\widetilde{H}^{(0)}(x,\xi)\left\\{1-\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\left[\log\frac{m^{2}}{\mu^{2}}+1\right]\right\\}+m^{2}\mskip
1.5mu\widetilde{H}^{(2,m)}(x,\xi)+t\mskip
1.5mu\widetilde{H}^{(2,t)}(x,\xi)+O(q^{3})\,.$ (34)

Let us now consider the chiral corrections for $\widetilde{M}_{n,k-1}$. It
follows from (2) that the relevant diagrams have to produce a factor
$(a\Delta)^{k}(S\Delta)$. By power counting, the form factor
$\widetilde{M}_{n,k-1}$ could receive corrections of order $O(q^{0})$ from
diagrams with LO vertices and the operator insertion
$\widetilde{\mathcal{O}}_{n,k,-1}$. Similarly, corrections of order $O(q)$
could come from the diagrams with LO vertices and insertion of
$\widetilde{\mathcal{O}}_{n,k+1,-1}$ or $\widetilde{\mathcal{O}}_{n,k,0}$, and
from diagrams with insertion of $\widetilde{\mathcal{O}}_{n,k,-1}$ and one NLO
pion-nucleon vertex or nucleon propagator correction. One finds no operator in
$\widetilde{\mathcal{O}}_{n,k,0}$ that has $k$ or more partial derivatives
contracted with $a^{\mu}$ or $S^{\mu}$, and the same holds of course for
$\widetilde{\mathcal{O}}_{n,k,-1}$. According to our discussion in Section 3.3
the graphs just discussed can thus produce at most $k$ vectors $\Delta_{\mu}$
(not counting those appearing in $\Delta^{2}$) and hence do not contribute to
$\widetilde{M}_{n,k-1}$. At order $O(q^{2})$ there is a number of
possibilities:

  1. 1.

graphs with LO vertices and insertion of $\widetilde{\mathcal{O}}_{n,k+2,-1}$,
$\widetilde{\mathcal{O}}_{n,k+1,0}$ or $\widetilde{\mathcal{O}}_{n,k,1}$. The
insertion of $\widetilde{\mathcal{O}}_{n,k+1,0}$ does not produce a sufficient
number of factors $\Delta_{\mu}$, whereas insertion of
$\widetilde{\mathcal{O}}_{n,k+2,-1}$ gives a factor $(a\Delta)^{k+1}(aS)$,
which contributes to the form factor $\widetilde{E}_{n,k+1}$ but not to
$\widetilde{M}_{n,k-1}$. A correction to $\widetilde{M}_{n,k-1}$ is obtained
from insertion of the operator $\widetilde{\mathcal{O}}_{n,k,1}$ given in
(4.1), which already provides the tree-level term of this form factor. Only
the loop graph in Fig. 1a is nonzero for this insertion, and the result is
analogous to the one for the contribution of
$\widetilde{\mathcal{O}}_{n,k,-1}$ to $\widetilde{E}_{n,k-1}$.

  2. 2.

graphs with one NLO vertex or propagator correction and insertion of
$\widetilde{\mathcal{O}}_{n,k+1,-1}$ or $\widetilde{\mathcal{O}}_{n,k,0}$.
Insertion of $\widetilde{\mathcal{O}}_{n,k,0}$ does again not provide enough
factors of $\Delta_{\mu}$, whereas graphs with
$\widetilde{\mathcal{O}}_{n,k+1,-1}$ give zero due to time reversal
invariance. This can be seen by direct calculation, or by noting that
$\widetilde{M}_{n,k-1}$ is only nonzero for odd $k$, whereas the coefficient
$\tilde{b}_{n,k+1}$ is only nonzero for even $k$, as remarked below (4.1).

  3. 3.

graphs with insertion of $\widetilde{\mathcal{O}}_{n,k,-1}$ and $(i)$ two
loops with LO vertices, or $(ii)$ one loop with two NLO pion-nucleon vertices
or nucleon propagator corrections, or $(iii)$ one loop with one NNLO pion-
nucleon vertex or nucleon propagator correction, or $(iv)$ one loop with a
pion propagator correction from $\mathcal{L}_{\pi}^{(4)}$. The operator
insertion provides $k-1$ factors of $\Delta_{\mu}$, so that two more factors
must be provided by the vertices or propagator corrections (without being
contracted to $\Delta^{2}$). This is not possible in case $(i)$, because the
LO pion-nucleon vertices only involve pion momenta and the pion momenta in a
two-loop graph can be parameterized such that they are independent of $\Delta$
(as in the one-loop graphs of Fig. 1). Likewise, a pion propagator correction
in case $(iv)$ does not depend on $\Delta$ and can therefore not contribute.
In cases $(ii)$ and $(iii)$ one obtains nonzero contributions from the graph
in Fig. 1a. The NNLO vertices and propagator corrections follow from the
Lagrangian $\mathcal{L}_{\pi N}^{(3)}$ given in [27]. We find that the only
term providing the two required factors of $\Delta_{\mu}$ is the $\pi NN$
vertex generated by

$-\frac{g_{A}}{4M_{0}^{2}}\,\overline{N}_{\\!v}\,\Big{\\{}({\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftarrow$}}}{}S)(u{\nabla^{\raisebox{2.41112pt}{$\scriptstyle\rightarrow$}}}{})+({\nabla^{\raisebox{2.41112pt}{$\scriptstyle\leftarrow$}}}{}u)(S{\nabla^{\raisebox{2.41112pt}{$\scriptstyle\rightarrow$}}}{})\Big{\\}}\,N_{v}\,.$
(35)

Note that this vertex does not introduce a new low-energy constant, similarly
to the term proportional to $g_{A}$ in $\mathcal{L}_{\pi N}^{(2)}$, which
generates the $\pi NN$ coupling at NLO. These terms arise from the $1/M_{0}$
expansion of the leading-order relativistic pion-nucleon Lagrangian
$\overline{N}\,(i\not{\nabla}-M_{0}+{\textstyle\frac{1}{2}}g_{A}\not{u}\gamma_{5})\,N$,
see e.g. [24].

Putting everything together, we obtain

$\displaystyle\widetilde{M}_{n,k}(t)$
$\displaystyle=\widetilde{M}_{n,k}^{(0)}\,\left\\{1-\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\left[\log\frac{m^{2}}{\mu^{2}}+1\right]\right\\}-\widetilde{E}_{n,k}^{(0)}\;\frac{m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}$
$\displaystyle\quad{}+\widetilde{M}_{n,k}^{(2,m)}m^{2}+\widetilde{M}_{n,k}^{(2,t)}t+O(q^{3})\,,$
(36)

where the terms going with $m^{2}$ and $t$ are due to tree-level insertions.
With (33), (2) and (2) one can write for the isoscalar quark GPD
$\widetilde{E}(x,\xi,t)$

$\displaystyle\widetilde{E}(x,\xi,t)$
$\displaystyle=\widetilde{E}^{(0)}(x,\xi)\,\left\\{1-\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\left[\log\frac{m^{2}}{\mu^{2}}+1\right]\right\\}-\widetilde{H}^{(0)}(x,\xi)\,\frac{m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}$ $\displaystyle\quad{}+m^{2}\mskip
1.5mu\widetilde{E}^{(2,m)}(x,\xi)+t\mskip
1.5mu\widetilde{E}^{(2,t)}(x,\xi)+O(q^{3})\,.$ (37)

###  4.2 Vector operators

The analysis of the vector operators proceeds along similar lines. The
operator $\mathcal{O}_{n,k,0}$ reads

$\mathcal{O}_{n,k,0}(a)=b_{n,k}\,(ia\partial)^{k}\,\overline{N}_{\\!v}\,N_{v}+c_{n,k}\,(ia\partial)^{k-1}\,(i\partial_{\mu})\,\overline{N}_{\\!v}\,[(Sa),S^{\mu}]N_{v}+\ldots\,,$
(38)

where the $\ldots$ denote operators with fewer total derivatives. One finds
$E_{n,k}^{(0)}=b_{n,k}$ and $M_{n,k-1}^{(0)}=-c_{n,k}$ for the leading-order
tree-level insertions, which implies $b_{n,k}=c_{n,k+1}=0$ for odd $k$.

According to (2) the graphs that give chiral corrections to $E_{n,k}$ or
$M_{n,k-1}$ must produce $k$ factors of $\Delta_{\mu}$ contracted with
$a^{\mu}$ or $S^{\mu}$. With the constraints from parity invariance one finds
that the operator $\mathcal{O}_{n,k,-1}$ does not contain terms which have
$k-1$ or more total derivatives contracted with $a^{\mu}$ or $S^{\mu}$. With
the results of Section 3.3 this implies that $E_{n,k}$ and $M_{n,k-1}$ do not
receive corrections from pion-nucleon operator insertions at order $O(q)$, and
that corresponding corrections at order $O(q^{2})$ can only come from the
diagram in Fig. 1a with LO vertices. One finds that the one-loop contribution
to the form factor $E_{n,k}$ is canceled by the terms proportional to
$g_{A}^{2}$ in the wave function renormalization constant (23). For the form
factor $M_{n,k}$ one obtains a correction

$M_{n,k}^{(0)}\left\\{1-\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}\right\\}\,.$ (39)

We note that for $n=1$, $k=0$ this implies a chiral logarithm for the
isoscalar magnetic form factor $G_{M,s}(t)$,

$G_{M,s}(t)=\mu_{s}^{(0)}\left\\{1-\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}\right\\}+G_{M,s}^{(2,m)}m^{2}+G_{M,s}^{(2,t)}t+O(q^{3})\,,$
(40)

where $\mu_{s}^{(0)}$ is the isoscalar magnetic moment of the nucleon in the
chiral limit and where we have added analytic terms due to tree-level
insertions. The form (40) is consistent with the result of the relativistic
calculation in [29].

Figure 2:  One-loop graphs with the insertion of the pion operator
${\mathcal{O}}_{\pi}^{n}(a)$, denoted by a black blob.

One finally has to evaluate corrections due to the diagrams in Fig. 2 with
insertion of the pion operator ${\mathcal{O}}_{\pi}^{n}(a)$, where $n$ is
even. We use the representation of this operator given in [19],333Note that
the normalization of the twist-two operators (3) used here differs from that
in [19] by a factor of 2. The coefficients $\tilde{a}_{n,k}$ have the same
normalization here and in [19].

${\cal O}_{\pi}^{n}(a)=F^{2}\,\sum_{\begin{subarray}{c}k=0\\\
\mathrm{even}\end{subarray}}^{n-2}\tilde{a}_{n,k}\,(ia\partial)^{k}\mathop{\rm
Tr}\nolimits\left[(aL)\,(2ia{\partial^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{})^{n-k-2}(aL)+(aR)\,(2ia{\partial^{\raisebox{2.41112pt}{$\scriptstyle\leftrightarrow$}}}{})^{n-k-2}(aR)\right]$
(41)

with $L_{\mu}=U^{\dagger}\mskip 1.5mu\partial_{\mu}U$ and
$R_{\mu}=U\partial_{\mu}U^{\dagger}$. As discussed in Section 3.3, the
corrections to $M_{n,k}$ start at order $O(q^{2})$ for $k=n-2$ and at higher
order otherwise. They are due to diagrams with LO vertices, so that only the
graph in Fig. 2a contributes. This is because the leading-order $\pi\pi NN$
vertex corresponds to an isovector transition of the nucleon, as follows from
(3) and the expansion of $\Gamma$ in (3). Combining the result with the
correction in (39) and adding analytic terms from tree-level insertions, we
obtain

$M_{n,k}(t)=M_{n,k}^{(0)}\left\\{1-\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}\right\\}+\delta_{k,n-2}^{\phantom{0}}\,M_{n}^{(2,\pi)}(t)+M_{n,k}^{(2,m)}m^{2}+M_{n,k}^{(2,t)}t+O(q^{3})\,,$
(42)

where $k$ is even and

$\displaystyle M_{n}^{(2,\pi)}(t)$ $\displaystyle=\frac{3g_{A}^{2}}{(4\pi
F)^{2}}\,\sum_{\begin{subarray}{c}j=0\\\
\mathrm{even}\end{subarray}}^{n-2}{\tilde{a}}_{n,n-j-2}\int_{-1}^{1}d\eta\left[\frac{\partial^{2}}{\partial\eta^{2}}\,\eta^{j}(1-\eta^{2})\right]m^{2}(\eta)\,\log\frac{m^{2}(\eta)}{\mu^{2}}$
$\displaystyle=\frac{3g_{A}^{2}}{(4\pi
F)^{2}}\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}j(j-1)\,A_{n,n-j}^{\pi\mskip
1.5mu(0)}\int_{-1}^{1}d\eta\,\eta^{j-2}\,m^{2}(\eta)\,\log\frac{m^{2}(\eta)}{\mu^{2}}$
(43)

with

$m^{2}(\eta)=m^{2}-\frac{t}{4}\mskip 1.5mu(1-\eta^{2})\,.$ (44)

Here $A_{n,k}^{\pi\mskip 1.5mu(0)}$ is the chiral limit of the form factors
$A_{n,k}^{\pi}(t)$ describing the moments of the pion isoscalar GPD,

$\displaystyle\int_{-1}^{1}dx\,x^{n-1}H_{\pi}^{I=0}(x,\xi,t)=\sum_{\begin{subarray}{c}k=0\\\
\mathrm{even}\end{subarray}}^{n}(2\xi)^{k}A_{n,k}^{\pi}(t).$ (45)

The relation to the low-energy constants $\tilde{a}_{n,k}$ reads [19]

$\displaystyle A_{n,k}^{\pi\mskip
1.5mu(0)}=2^{n-k}\,\Big{[}\tilde{a}_{n,k-2}-\tilde{a}_{n,k}\Big{]}\,,$ (46)

which implies

$\displaystyle\tilde{a}_{n,n-k}$
$\displaystyle={}-\sum_{\begin{subarray}{c}j=k\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}A_{n,n-j}^{\pi\mskip 1.5mu(0)}\hskip
20.00003pt\text{for\leavevmode\nobreak\ }k>0\,,$
$\displaystyle\sum_{\begin{subarray}{c}j=0\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}A_{n,n-j}^{\pi\mskip 1.5mu(0)}$
$\displaystyle=0\,.$ (47)

The corrections to $E_{n,k}$ start at order $O(q)$ for $k=n$ and at order
$O(q^{3})$ or higher otherwise. At one-loop order we obtain $O(q)$ corrections
to $E_{n,n}$ from graphs involving only LO vertices. Corrections of order
$O(q^{2})$ involve either graphs with one NLO vertex or propagator correction,
or graphs with LO vertices and the subleading part $wv$ of the residual
nucleon momenta, see the discussion after (30). Our final result including
analytic terms from tree-level insertions is

$\displaystyle
E_{n,k}(t)=E_{n,k}^{(0)}+\delta_{n,k}^{\phantom{0}}\Bigl{[}E_{n}^{(1,\pi)}(t)+E_{n}^{(2,\pi)}(t)\Bigr{]}+E_{n,k}^{(2,m)}m^{2}+E_{n,k}^{(2,t)}t+O(q^{3})\,,$
(48)

where the order $O(q)$ correction reads

$\displaystyle E_{n}^{(1,\pi)}(t)$ $\displaystyle={}-M(2m^{2}-t)\,\frac{3\pi
g_{A}^{2}}{8\mskip 1.5mu(4\pi F)^{2}}\,\sum_{\begin{subarray}{c}j=0\\\
\mathrm{even}\end{subarray}}^{n-2}{\tilde{a}}_{n,n-j-2}\,\int_{-1}^{1}d\eta\,\frac{\eta^{j}(1-\eta^{2})}{m(\eta)}$
$\displaystyle=M(2m^{2}-t)\,\frac{3\pi g_{A}^{2}}{8\mskip 1.5mu(4\pi
F)^{2}}\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}\,A_{n,n-j}^{\pi\mskip
1.5mu(0)}\int_{-1}^{1}d\eta\,\frac{1-\eta^{j}}{m(\eta)}\,,$ (49)

and the order $O(q^{2})$ term is

$\displaystyle E_{n}^{(2,\pi)}(t)=$ $\displaystyle\leavevmode\nobreak\
\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}\;\sum_{\begin{subarray}{c}j=0\\\
\mathrm{even}\end{subarray}}^{n-2}\tilde{a}_{n,n-j-2}$
$\displaystyle+\frac{6}{(4\pi F)^{2}}\,\sum_{\begin{subarray}{c}j=0\\\
\mathrm{even}\end{subarray}}^{n-2}\tilde{a}_{n,n-j-2}\int_{-1}^{1}d\eta\,\eta^{j}(1-\eta^{2})\,\Biggl{\\{}\frac{g_{A}^{2}}{32}\left[2t\left(\log\frac{m^{2}(\eta)}{\mu^{2}}+1\right)-\frac{(t-2m^{2})^{2}}{m^{2}(\eta)}\,\right]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ +M\left[c_{1}\mskip
1.5mum^{2}\left(\log\frac{m^{2}(\eta)}{\mu^{2}}+1\right)-\frac{3}{4}c_{2}\mskip
1.5mum^{2}(\eta)\,\log\frac{m^{2}(\eta)}{\mu^{2}}-c_{3}\mskip
1.5mum^{2}(\eta)\left(\log\,\frac{m^{2}(\eta)}{\mu^{2}}+\frac{1}{2}\right)\right]\Biggr{\\}}$
$\displaystyle=$ $\displaystyle{}-\frac{3m^{2}g_{A}^{2}}{2\mskip 1.5mu(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}j\,A_{n,n-j}^{\pi\mskip 1.5mu(0)}$
$\displaystyle-\frac{6}{(4\pi F)^{2}}\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}\,A_{n,n-j}^{\pi\mskip
1.5mu(0)}\int_{-1}^{1}d\eta\,(1-\eta^{j})\,\Biggl{\\{}\frac{g_{A}^{2}}{32}\left[2t\left(\log\,\frac{m^{2}(\eta)}{\mu^{2}}+1\right)-\frac{(t-2m^{2})^{2}}{m^{2}(\eta)}\,\right]$
$\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ +M\left[c_{1}\mskip
1.5mum^{2}\left(\log\frac{m^{2}(\eta)}{\mu^{2}}+1\right)-\frac{3}{4}c_{2}\mskip
1.5mum^{2}(\eta)\,\log\frac{m^{2}(\eta)}{\mu^{2}}-c_{3}\mskip
1.5mum^{2}(\eta)\left(\log\frac{m^{2}(\eta)}{\mu^{2}}+\frac{1}{2}\right)\right]\Biggr{\\}}$

The integrals over $\eta$ in (4.2), (4.2) and (4.2) are elementary, but we
have not found a simple closed form of the result for general $n$. In the next
section we give explicit results for the values and derivatives at $t=0$ of
the form factors.

Our result for the form factor $\widetilde{M}_{n,k}(t)$ disagrees with [30],
where it was taken for granted that the only operators which contribute at
order $O(q^{2})$ are those which already appear at tree-level in the same form
factor. As our analysis shows, this holds indeed in many cases but not in all.
For all other form factors our results agree with [30] where
comparable.444Note that [30] gives the correction to $E_{n,n}$ at order $O(q)$
but not at order $O(q^{2})$. For $n=2$ our results for the vector operators
also agree with those of Belitsky and Ji [21].555When comparing results, one
must take into account that [21] uses $\overline{\mathrm{MS}}$
renormalization, where for each pole in $4-2\epsilon$ dimensions one subtracts
$1/\epsilon+\log(4\pi)+\psi(1)$, whereas we use the scheme of [14] and
subtract $1/\epsilon+\log(4\pi)+\psi(2)$.

##  5 Results for moments of GPDs

We now transform the results of the previous section to the basis of the form
factors $A_{n,k}$, $B_{n,k}$, $C_{n}$ and $\widetilde{A}_{n,k}$,
$\widetilde{B}_{n,k}$ corresponding to moments of GPDs in the conventional
parameterization. We give the values and derivatives of these form factors at
$t=0$, which allows us to obtain closed expressions. Furthermore, these
quantities are of most immediate interest in studies of GPDs on the lattice.

With our results (33), (4.1), (42), (48) and the conversion formulae (2), (2)
one obtains for the form factors at $t=0$

$\displaystyle\widetilde{A}_{n,k}(0)$
$\displaystyle=\widetilde{A}_{n,k}^{(0)}\,\left\\{1-\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\left[\log\frac{m^{2}}{\mu^{2}}+1\right]\right\\}+\widetilde{A}_{n,k}^{(2,m)}m^{2}+O(m^{3})\,,$
$\displaystyle\widetilde{B}_{n,k}(0)$
$\displaystyle=\widetilde{B}_{n,k}^{(0)}\,\left\\{1-\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\left[\log\frac{m^{2}}{\mu^{2}}+1\right]\right\\}-\widetilde{A}_{n,k}^{(0)}\;\frac{m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}+\widetilde{B}_{n,k}^{(2,m)}m^{2}+O(m^{3})\,,$
$\displaystyle A_{n,k}(0)$
$\displaystyle=A_{n,k}^{(0)}+A_{n,k}^{(2,m)}m^{2}+O(m^{3})\,,$ $\displaystyle
B_{n,k}(0)$
$\displaystyle=B_{n,k}^{(0)}-\bigl{(}A_{n,k}^{(0)}+B_{n,k}^{(0)}\bigr{)}\,\frac{3m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}+\delta_{k,n-2}^{\phantom{0}}\,M_{n}^{(2,\pi)}(0)+B_{n,k}^{(2,m)}m^{2}+O(m^{3})\,,$
$\displaystyle C_{n}(0)$
$\displaystyle=C_{n}^{(0)}+E_{n}^{(1,\pi)}(0)+E_{n}^{(2,\pi)}(0)+C_{n}^{(2,m)}m^{2}+O(m^{3})$
(51)

with coefficients related to those in Section 4 by
$\widetilde{A}_{n,k}^{(0)}=\widetilde{E}_{n,k}^{(0)}$,
$\widetilde{B}_{n,k}^{(0)}=\widetilde{M}_{n,k}^{(0)}-{\textstyle\frac{1}{2}}\widetilde{E}_{n,k}^{(0)}$,
$A_{n,k}^{(0)}=E_{n,k}^{(0)}$, $B_{n,k}^{(0)}=M_{n,k}^{(0)}-E_{n,k}^{(0)}$,
$C_{n\phantom{l}}^{(0)}=E_{n,n\phantom{l}}^{(0)}$ and by analogous relations
for the coefficients with superscript $(2,m)$. Setting $m$, $g_{A}$, $F$ to
their physical values and choosing $\mu=M$, one finds that the corrections
from loop graphs with nucleon operator insertions in (5) are moderately large,
with $3m^{2}g_{A}^{2}\,(4\pi F)^{-2}\,[\mskip 1.5mu\log(m^{2}/\mu^{2})+1\mskip
1.5mu]\approx-0.20$ and $m^{2}g_{A}^{2}\,(4\pi
F)^{-2}\log(m^{2}/\mu^{2})\approx-0.09$. In the case of $B_{n,k}$ this loop
correction can be substantial if $|B_{n,k}|\ll|A_{n,k}|$, which is empirically
found for the electromagnetic form factors (i.e. the case $n=1$) and also in
lattice evaluations [9] for the moments with $n=2$. The contributions to
$B_{n,n-2}(0)$ and $C_{n}(0)$ from loop graphs with pion operator insertions
are

$\displaystyle M_{n}^{(2,\pi)}(0)$ $\displaystyle=\frac{6m^{2}g_{A}^{2}}{(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}j\,A_{n,n-j}^{\pi\mskip 1.5mu(0)}\,,$
$\displaystyle E_{n}^{(1,\pi)}(0)$ $\displaystyle=\frac{3\pi\mskip
1.5mumMg_{A}^{2}}{2\mskip 1.5mu(4\pi F)^{2}}\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}\,\frac{j}{j+1}\,A_{n,n-j}^{\pi\mskip
1.5mu(0)}\,,$ $\displaystyle E_{n}^{(2,\pi)}(0)$
$\displaystyle={}-\frac{3m^{2}g_{A}^{2}}{2\mskip 1.5mu(4\pi
F)^{2}}\,\log\frac{m^{2}}{\mu^{2}}\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}j\,A_{n,n-j}^{\pi\mskip 1.5mu(0)}$
$\displaystyle{}+\frac{12m^{2}}{(4\pi
F)^{2}}\,\left\\{\frac{g_{A}^{2}}{8}-M\left[c_{1}\left(\log\frac{m^{2}}{\mu^{2}}+1\right)-\frac{3}{4}c_{2}\mskip
1.5mu\log\frac{m^{2}}{\mu^{2}}-c_{3}\left(\log\frac{m^{2}}{\mu^{2}}+\frac{1}{2}\right)\right]\right\\}\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}\frac{j}{j+1}\,A_{n,n-j}^{\pi\mskip
1.5mu(0)}\,.$

Setting $M$, $m$, $g_{A}$, $F$ to their physical values, choosing $\mu=M$, and
taking the estimates $c_{1}\approx-0.9\mbox{\leavevmode\nobreak\ GeV}^{-1}$,
$c_{2}\approx 3.3\mbox{\leavevmode\nobreak\ GeV}^{-1}$,
$c_{3}\approx-4.7\mbox{\leavevmode\nobreak\ GeV}^{-1}$ from [25] we find
$M_{2}^{(2,\pi)}(0)\approx-0.27\,A_{2,0}^{\pi\mskip 1.5mu(0)}$ and
$E_{2}^{(1,\pi)}(0)+E_{2}^{(2,\pi)}(0)\approx(0.12+0.17)\,A_{2,0}^{\pi\mskip
1.5mu(0)}$. At the physical point, the order $O(m)$ correction is hence not
very large. The full size of the order $O(m^{2})$ corrections depends of
course on the analytic terms in (5), whose values are not known.

The derivatives of the form factors at $t=0$ obtain nonanalytic contributions
only from the pion operator insertions. Writing
$\partial_{t}A(0)=\bigl{[}\frac{\partial}{\partial t}A(t)\bigr{]}_{\,t=0}$
etc. we have

$\displaystyle\partial_{t}\widetilde{A}_{n,k}(0)$
$\displaystyle=\widetilde{E}_{n,k}^{(2,t)}+O(m)\,,$
$\displaystyle\partial_{t}\widetilde{B}_{n,k}(0)$
$\displaystyle=\widetilde{M}_{n,k}^{(2,t)}-\widetilde{E}_{n,k}^{(2,t)}\big{/}2-\widetilde{E}_{n,k}^{(0)}\mskip
1.5mu\big{/}(32M^{2})+O(m)\,,$ $\displaystyle\partial_{t}A_{n,k}(0)$
$\displaystyle=E_{n,k}^{(2,t)}-\bigl{(}M_{n,k}^{(0)}-E_{n,k}^{(0)}\bigr{)}\big{/}(4M^{2})+O(m)\,,$
$\displaystyle\partial_{t}B_{n,k}(0)$
$\displaystyle=\delta_{k,n-2}\;\partial_{t}M_{n}^{(2,\pi)}(0)+M_{n,k}^{(2,t)}-E_{n,k}^{(2,t)}+\bigl{(}M_{n,k}^{(0)}-E_{n,k}^{(0)}\bigr{)}\big{/}(4M^{2})+O(m)\,,$
$\displaystyle\partial_{t}C_{n}(0)$
$\displaystyle=\partial_{t}E_{n}^{(1,\pi)}(0)+\partial_{t}E_{n}^{(2,\pi)}(0)+E_{n,n}^{(2,t)}+E_{n,n}^{(0)}\big{/}(4M^{2})+O(m)$
(53)

with

$\displaystyle\partial_{t}M_{n}^{(2,\pi)}(0)$
$\displaystyle={}-\frac{3g_{A}^{2}}{(4\pi
F)^{2}}\left[\log\frac{m^{2}}{\mu^{2}}+1\right]\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}\frac{j}{j+1}\,A_{n,n-j}^{\pi\mskip
1.5mu(0)}\,,$ $\displaystyle\partial_{t}E_{n}^{(1,\pi)}(0)$
$\displaystyle={}-\frac{M}{m}\,\frac{\pi g_{A}^{2}}{8\mskip 1.5mu(4\pi
F)^{2}}\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}\,\frac{j\,(5j+14)}{(j+1)(j+3)}\,A_{n,n-j}^{\pi\mskip
1.5mu(0)}\,,$ $\displaystyle\partial_{t}E_{n}^{(2,\pi)}(0)$
$\displaystyle={}-\frac{3g_{A}^{2}}{4\mskip 1.5mu(4\pi
F)^{2}}\,\left[\log\frac{m^{2}}{\mu^{2}}+3\right]\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}\frac{j}{j+1}\,A_{n,n-j}^{\pi\mskip
1.5mu(0)}\,$ $\displaystyle{}+\frac{2}{(4\pi
F)^{2}}\,\left\\{\frac{g_{A}^{2}}{8}+M\left[c_{1}-\frac{3}{4}c_{2}\left(\log\frac{m^{2}}{\mu^{2}}+1\right)-c_{3}\left(\log\frac{m^{2}}{\mu^{2}}+\frac{3}{2}\right)\right]\right\\}\,\sum_{\begin{subarray}{c}j=2\\\
\mathrm{even}\end{subarray}}^{n}2^{-j}\frac{j\,(j+4)}{(j+1)(j+3)}\,A_{n,n-j}^{\pi\mskip
1.5mu(0)}\,.$

Note that in the chiral limit the derivative
$\partial_{t}B_{n,n-2}(0)\sim\partial_{t}M_{n}^{(2,\pi)}(0)$ diverges as
$\log(m^{2}/\mu^{2})$ and
$\partial_{t}C_{n}(0)\sim\partial_{t}E_{n}^{(1,\pi)}(0)$ as $1/m$. With the
parameters specified above, one finds $\partial_{t}M_{2}^{(2,\pi)}(0)\approx
1.7\mbox{\leavevmode\nobreak\ GeV}^{-2}\,A_{2,0}^{\pi\mskip 1.5mu(0)}$ and
$\partial_{t}E_{2}^{(1,\pi)}(0)+\partial_{t}E_{2}^{(2,\pi)}(0)\approx-(2.5+1.2)\mbox{\leavevmode\nobreak\
GeV}^{-2}\,A_{2,0}^{\pi\mskip 1.5mu(0)}$. Numerically, the term
$\partial_{t}E_{n}^{(1,\pi)}(0)$ is thus important but not extremely large at
the physical point.

##  6 Summary

Using heavy-baryon chiral perturbation theory, we have calculated the chiral
corrections up to order $O(q^{2})$ for the form factors which parameterize
moments of nucleon GPDs. We have restricted ourselves to vector and axial-
vector quark distributions in the isosinglet combination. Our results
generalize trivially to the corresponding gluon GPDs, which have the same
quantum numbers and therefore the same corresponding operators in the
effective theory (except for the values of the matching constants). Our method
is also applicable to operators of different tensor or flavor structure.

The moments of GPDs contain terms of different order in $1/M$, ranging from
$M^{n-1}$ to $M^{-1}$. We have shown that, due to the way in which factors
$v_{\mu}$ and $\Delta_{\mu}$ arise in the calculation, the number of loops and
the order in the expansion of the effective Lagrangian required to calculate a
form factor to a given order $O(q^{\mskip 1.5mud})$ does not grow with the
number of factors $\Delta_{\mu}$ that accompany the form factor in the nucleon
matrix element. A general power-counting formula is given after (30). In the
case of the form factors $\widetilde{M}_{n,k}(t)$, calculation of the order
$O(q^{2})$ correction requires the pion-nucleon Lagrangian up to third order.

We have found that the form factors $\widetilde{E}_{n,k}(t)$ and
$\widetilde{M}_{n,k}(t)$ receive corrections of order $O(q^{2})$ which, apart
from analytic terms, are independent of the moment indices and independent of
$t$. The same holds for the one-loop corrections to $M_{n,k}(t)$ with $k<n-2$,
whereas the corresponding corrections for $E_{n,k}$ with $k<n$ are zero. The
form factors $M_{n,n-2}$ receive additional corrections at order $O(q^{2})$
from one-loop graphs with the insertion of pion operators, and $E_{n,n}$
receives corresponding contributions starting at order $O(q)$.

For the form factors parameterizing moments of isoscalar GPDs, we find that
$B_{n,k}$, $\widetilde{A}_{n,k}$ and $\widetilde{B}_{n,k}$ at $t=0$ receive
nonanalytic corrections of the form $m^{2}\log(m^{2}/\mu^{2})$ from loops with
nucleon operator insertions. No such corrections are found for $A_{n,k}$ and
$C_{n}$. The form factors $B_{n,n-2}$ at $t=0$ receive in addition
$m^{2}\log(m^{2}/\mu^{2})$ corrections from loop graphs with pion operator
insertions, and the corresponding nonanalytic contributions to $C_{n}$ give a
term proportional to $m$. To leading chiral order, loop graphs with pion
operator insertions are the only source of nonanalytic $m^{2}$ dependence for
the derivatives of the form factors at $t=0$. The derivative of $M_{n,n-2}$
diverges like $\log(m^{2}/\mu^{2})$ in the chiral limit, and the derivative of
$C_{n}$ like $1/m$.

## Acknowledgments

We are grateful to U.-G. Meißner for clarifying discussions and valuable
remarks on the manuscript. This work is supported by the Helmholtz
Association, contract number VH-NG-004. A.S. acknowledges the hospitality of
INT, Seattle, USA (program INT-06-1: “Exploration of Hadron Structure and
Spectroscopy using Lattice QCD”), where part of this work was done.

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