\section{Introduction}
\label{sec:Intro}

While superconducting transition temperatures are non-universal properties, and hence not generally amenable to a simple theoretical analysis, understanding
what physics determines $T_c$
is of self-evident importance.
One approach to this challenge is to focus on a key physical process that contributes to the development of superconducting order, and to formulate an upper bound -- either rigorous or heuristic -- on $T_c$~\cite{McMillan1968,McMillan1968,cohen1972comments,ginzburg1991,emery1995importance,Hazra18,Esterlis2018,esterlis2018bound}.

In this paper, we examine three proposed bounds on $T_c$ that are expressed as a fraction of a measurable physical quantity of a given system: an appropriately defined Fermi temperature, a characteristic phonon frequency, or the zero-temperature superfluid phase stiffness. While these putative bounds are physically motivated, and provide valuable intuition in many cases of practical importance, we show by explicit counter-examples that they can be violated by an arbitrary amount.
In addition to the fundamental importance of these results, we hope they suggest routes to further optimize $T_c$.

\section{Summary of results}
\label{sec:summary}
We briefly summarize our key results here:
\begin{enumerate}
    \item
The notion of an upper bound on $T_c$ in terms of an appropriately defined Fermi energy comes from the fact that, in many situations, as $E_F\rightarrow 0$, the electrons have no kinetic energy. Thus, in this limit, the superfluid stiffness must seemingly go to zero. What sets  $T_c$ in the limit of small $E_F$ is pertinent to moir\'e superconductors~\cite{Cao2018,AY19,Efetov19,SNP20,PJH21,PK21}, where the bands can be tuned to be narrow.
To make this question
precise, we must define
$E_F$ in a strongly interacting system. We propose two such definitions of $E_F$, in terms of (i) the difference in the chemical potential between a system with a given density of electrons and a system with a vanishing density, or (ii) the energy dispersion of a single electron added to the empty system.

We show that there is no general bound on $T_c/E_F$ by either definition, by studying two explicit models. In the first model, a flat band is separated by an energy gap from a broad band with pair-hopping interaction between the two. The second model consists of a pair of
perfectly flat bands with an on-site electron-electron attraction.
We show explicitly that the first model violates any putative $T_c/E_F$ bound when using the first definition of $E_F$ above, while the second model violates the bound using either definition of $E_F$. Both models
have been defined on a two-dimensional lattice for convenience\footnote{ In the context of two-dimensional systems, we identify $T_c$ as the Berezinskii-Kosterlitz-Thouless (BKT) transition temperature.},
but generalized versions of the same models in any $D>1$ can be easily seen to exhibit qualitatively similar behaviors.

\item In two-dimensional systems where $T_c$ is limited by phase fluctuations, an intuitive bound on $T_c$ is given in terms of the zero-temperature phase stiffness, $\rho_s(0)$. This comes from the relation~\cite{NK77} $T_c = \pi\rho_s(T^-_c)/2$, and the (often physically reasonable) assumption that $\rho_s(T)$ is a decreasing function of $T$, and hence $\rho_s(0)\ge\rho_s(T)$.

We construct an explicit counter-example in a two-band model of bosons (or, equivalently, tightly-bound Cooper pairs), where $\rho_s(0)$ can be made arbitrarily smaller than $T_c$. In this model, $\rho_s(0)$ can even vanish while $T_c>0$, implying that there is a reentrant transition into the non-superconducting state below $T_c$.

\item In electron-phonon superconductors, a heuristic bound on $T_c/\omega_0$ ($\omega_0$ being the characteristic phonon frequency) was proposed~\cite{Esterlis2018,esterlis2018bound,chubukov2020eliashberg}. The reasoning behind this bound is that, as the dimensionless electron-phonon coupling constant $\lambda$ increases past an $O(1)$ value, the system tends to become unstable, either towards the formation of localized bipolarons or towards a charge density wave state. At the same time (and relatedly), the phonon frequency is renormalized downward as $\lambda$ increases, suppressing $T_c$.

Here, we construct an explicit $d-$dimensional model where these strong-coupling instabilities are avoided, and $T_c$ increases without bound upon increasing $\lambda$. The model includes $N$ electronic bands interacting with $N^2$ phonon modes.
The model is solvable asymptotically in the large-$N$ limit; then, the famous Allen-Dynes result~\cite{Dynes1975} $T_c \propto \omega_0 \sqrt{\lambda}$ is valid for large $\lambda$,

so long as $\lambda \ll N$,
and hence $T_c/\omega_0$ is unbounded as $N\to \infty$.
 Note that at the heuristic level, it is difficult to identify physical circumstances where more than a few phonons are comparably strongly coupled to the relevant electrons. Nevertheless, our analysis suggests that generically, the larger the number of phonon modes coupled to the electrons, the larger the $\lambda$ at which the suppression of $T_c$ onsets.
\end{enumerate}

\section{Flat band superconductivity:\\ bound on $T_c/T_F$?}
\label{sec:flat}

In most conventional superconductors, $T_c$ is determined by the energy scale associated with electron pairing. On the other hand, across numerous unconventional superconductors, $T_c$ is more strongly sensitive to the ``phase ordering scale"~\cite{emery1995importance}. In this context, an important recent advance is the result by Hazra, Verma, and Randeria (HVR)~\cite{Hazra18} of a rigorous upper bound on $\rho_s(T)$, the temperature-dependent superfluid phase stiffness, in terms of the integral of the optical conductivity over frequency (the optical sum rule). However, since this integral includes all the bands, this upper bound is often of the order of several electron-volts in electronic systems of interest.

At the heuristic level, this bound has been interpreted~\cite{park2021tunable} as implying a bound on $T_c/T_F$, where $T_F=E_F/k_B$ is the Fermi energy in units of temperature.\footnote{For a Galilean invariant system with a parabolic band, HVR express their bound in terms of the Fermi energy.}

At the outset, it is important to define sharply what we mean by $E_F$. A particular protocol that is often adopted in experiments to estimate $E_F$ is to use an effective mass, $m^*$, obtained from quantum oscillations along with an estimate of the Fermi momentum ($k_F$) from a measurement of the the carrier density, and then defining $E_F = k_F^2/(2m^*)$.
This procedure is only possible when there is a nearby Fermi liquid-like state that displays quantum oscillations.

Below, we propose two different definitions of $E_F$, that we can use in settings that do not rely on any underlying assumptions (e.g., that there is a nearby Fermi liquid) and are also amenable to an experimental interpretation. We consider the case in which we add a given density, $n$, of electrons to an insulating reference state. We can define $E_F$ as follows:
\begin{itemize}
\item Starting from our reference state, we set the temperature to zero and consider the change in the chemical potential, $\mu(n,T=0)$, as we fill in $n$ electrons,
\begin{equation}
E_F^{(1)} \equiv \mu(n,T=0) - \mu_0,
\label{eq:EF1}
\end{equation}
as an effective definition of $E_F$. Here $\mu_0 = \lim_{n\to 0} \mu(n,T=0)$. Note that the above definition of $E_F$ includes all many-body corrections, which can furthermore be dependent on the density itself, and does not make any reference to any non-interacting limit.

\item Alternatively, we can define the Fermi energy through  the ``non-interacting" density of states, $\rho(\epsilon)$, for adding a single electron to the insulating system. In this case, the Fermi energy ($E_{F}^{(2)}$) is defined implicitly from the expression
\begin{equation}
n= \int_{\epsilon_{\mathrm{min}}}^{\epsilon_{\mathrm{min}}+E_{F}^{(2)}} d\epsilon\ \rho(\epsilon),
\label{eq:EF2}
\end{equation}
where $\epsilon_{\mathrm{min}}$ is the energy of the ground state with one electron added on top to the insulating reference state. $E_F^{(2)}$ is accessible directly in e.g. STM measurements \cite{Xie2019}.
\end{itemize}

\begin{figure}
	\begin{center}
          \includegraphics[width=0.99\columnwidth]{figures/Efs.pdf}
	\end{center}
	\caption{(a)
	Illustration of the two operational definitions of $E_F$. (a,b) $E_F^{(1)}$ is defined in Eq. \eqref{eq:EF1} as the difference between the chemical potential at
	density $n$ (panel a) and the limiting value of the chemical potential $\mu_0$
	at which $n$ approaches the value, $n_0$, it takes
	in a reference insulating state (panel b).
(c) $E^{(2)}_F$ is defined by Eq. \eqref{eq:EF2} in terms of the density of states $\rho(\varepsilon)$ of a single electron excitation
added to the reference insulating state.}
	\label{fig:Efs}
\end{figure}

Below, we provide model Hamiltonians of interacting electrons in flat bands where the superconducting $T_c$ exceeds the Fermi energy by one or both of the above definitions.
Thus, these models exemplify flat band superconductivity, where $T_c$ is determined entirely by the interaction scale~\cite{Shaginyan1990,Volovik2011,Volovik2013}. Analogous phenomena may also occur in semi-metals \cite{Kopnin2008,Kopnin2010}.

\subsection{Flat band superconductivity induced by a nearby dispersive band}
\label{pair}

We consider a model consisting of a nearly-flat band and a dispersive band.{\footnote{A closely related model \cite{LL21} has recently been studied in the context of superconductivity in twisted bilayer graphene. }}
The single-particle part of the Hamiltonian is given by
\begin{equation}
H_0 = \sum_{\vec{k},{\ell},\sigma} \epsilon_{\ell}(\vec{k}) c_{\vec{k},\ell,\sigma}^\dagger c_{\vec{k},\ell,\sigma}^{\vphantom{dagger}},
\label{eq:H0}
\end{equation}
where $c_{\vec{k},{\ell},\sigma}^\dagger$ creates an electron with quasi-momentum $\vec{k}$ in band ${\ell}=1,2$ and spin polarization $\sigma$. We consider the lower band (${\ell}=1$) to be a flat band with bandwidth, $W_1$, that we will ultimately take to be parametrically small (i.e. $W_1\rightarrow0$). The upper band (${\ell}=2$) is separated from the flat band by an energy gap, ${\Delta}_\textnormal{gap}$, and has a large bandwidth, $W_2\gg W_1$. The bands are topologically trivial and the Wannier functions are tightly localized on the lattice sites.

We now introduce an on-site interaction which scatters a pair of electrons between the flat band and the dispersive band:
\begin{subequations}
\begin{eqnarray}
H_{\textnormal{int}} &=& V \sum_{\bm{R}}\left[ \phi_1^\dagger (\bm{R})\phi_2(\bm{R}) + \textnormal{h.c.}\right], \label{pairint} \\
\phi_{\ell}(\vec{r}) &=& c_{\bm{R},\ell,\downarrow} c_{\bm{R},\ell,\uparrow},
\label{eq:Hint}
\end{eqnarray}
\end{subequations}
where $\bm{R}$ labels a lattice site.
Let us focus on the case where the flat band is half-filled, such that the number of particles per unit cell is $n=1$.

We consider the case where $V\ll \Delta_{\rm{gap}}\ll W_2$. Within mean-field theory, the superconducting transition temperature is given by (see Appendix \ref{app:mft}),
\begin{equation}
T_{\textnormal{MF}} \approx  \left[\frac {\nu_2 V^2}{8}\right]\ \ln \left[\frac {W_2}{\Delta_{\rm{gap}}}\right],
\label{Tmf}
\end{equation}
where we have assumed a constant density of states per unit cell, $\nu_2 = 1/W_2$, in the dispersive band. The zero-temperature phase stiffness is given by (Appendix~\ref{app:mft})
\begin{equation}
\rho_{s}(T=0) \approx \frac{V^2}{8\pi \Delta_{\rm{gap}}}.
\end{equation}
Hence $\rho_s(0)\gg T_{\rm{MF}}$, which implies that phase fluctuations are unimportant in determining $T_c$~\cite{emery1995importance},  i.e. $T_c\approx T_{\rm{MF}}$.

We now examine the Fermi energies $E^{(1,2)}_F$ defined in Eqs.~(\ref{eq:EF1}) and (\ref{eq:EF2}) and compare them to $T_c$. Adding a single particle to the empty system, we find that $E_F^{(2)}\sim W_1 \ll T_c$, and hence $T_c/E_F^{(2)} $ can be made arbitrarily large.  $E_F^{(1)}$ is computed in Appendix~\ref{app:mft} by calculating the chemical potentials at $n=1$ and $n\rightarrow 0$. The result is $E_F^{(1)} = 2 T_{\rm{MF}} \sim T_c$. Hence, in this model, $T_c/E^{(1)}_F = O(1)$. An example of a different model where $T_c/E^{(1)}_F$ is unbounded is presented in the next section.

\subsection{Flat band superconductivity induced by spatial extent of Wannier functions}
\label{chiral}

We now introduce a different model for superconductivity in a narrow band.
The model is defined on a two-dimensional square lattice with two electronic orbitals per unit cell. The Hamiltonian is given by
\begin{subequations}
\label{eq:chiral}
\begin{eqnarray}
    H &=& H_0 + H_U,\label{eq:modHam}\\
    H_0 & = & t \sum_{\substack{\vec{k}\\\gamma=\{{\ell},\sigma\}}} c^{\dagger}_{\vec{k}} \left(\tau_x\sin{\alpha_{\vec{k}}}  + \sigma_z \tau_y\cos{\alpha_{\vec{k}}}  -\mu  \right) c^{\phantom{\dagger}}_{\vec{k}}, \nonumber \\
    H_U &=& - \frac{U}{2} \sum_{\vec{r},{\ell}} \delta n^2_{\vec{r},{\ell}} + V\sum_{\langle \vec{r},\vec{r}' \rangle,{\ell}}  \delta n_{\vec{r},{\ell}} \delta n_{\vec{r}',{\ell}}.
\end{eqnarray}
\end{subequations}
Here, $c^\dagger_{\bm{k}} = (c^\dagger_{\bm{k},1,\uparrow},c^\dagger_{\bm{k},1,\downarrow},c^\dagger_{\bm{k},2,\uparrow},c^\dagger_{\bm{k},2,\downarrow})$, where the operator $c^{\dagger}_{\vec{k},\ell,\sigma}$ creates an electron with momentum $\vec{k}$ in orbital ${\ell}=1,2$ with spin $\sigma=\uparrow,\downarrow$. The Pauli matrices $\tau_{
x,y,z}$ and $\sigma_{
x,y,z}$ act on the orbital and spin degrees of freedom, respectively, and $\delta n_{\vec{r},{\ell}} \equiv \sum_{\sigma} c^{\dagger}_{\vec{r},{\ell},\sigma} c^{\vphantom{\dagger}}_{\vec{r},{\ell},\sigma} - 1$ is the number of particles at site $\vec{r}$ and orbital ${\ell}$, relative to quarter filling ($n=1$). $\langle \vec{r},\vec{r}' \rangle$ denotes nearest-neighbor sites. The single-particle Hamiltonian $H_0$ leads to perfectly flat bands with energies ${\varepsilon}=\pm t$. The function $\alpha_{\vec{k}} \equiv \zeta (\cos{k_x}+\cos{k_y})$ controls the spatial extent of the Wannier functions in each band,  tuned by the dimensionless parameter $\zeta$.\footnote{More specifically, the Wannier functions decay exponentially over a distance proportional to $\zeta$.} Note that there is no obstruction towards constructing exponentially localized Wannier functions for the above model since the bands are topologically trivial.\footnote{This can be seen from the fact that the Berry curvature of the bands is identically zero, since $H_0$ contains only $\tau_{x,y}$ but not $\tau_z$.} The strength of on-site attractive Hubbard interaction is denoted $U>0$, whereas $V>0$ is a nearest-neighbor repulsion.

We are interested in the limit where $T \ll U,\,V\ll \Delta_{\textnormal{gap}}~(=2t)$ and $\zeta\ll 1$.\footnote{An extensive numerical study of the model~\eqref{eq:chiral} beyond this parameter regime will appear in an upcoming publication.} In this limit, we project $H_{U}$ to the lower eigenband. The projected Hamiltonian is expanded in powers of $\zeta$. The average density is set to $n=1$ particles per unit cell.

For $\zeta=0$, the problem effectively reduces to a set of decoupled sites with a strong attractive interaction; the resulting ground state manifold is highly degenerate with local ``Cooper pairs'' but a vanishing phase stiffness.
The linear corrections in $\zeta$ vanish due to a chiral symmetry and the orbital $l$ independent interaction strength $U$.
At second order in $\zeta$, the projected interaction, $\widetilde{H_U}$, contains nearest-neighbor pair-hopping and density-density interactions:
\begin{equation}
\widetilde{H_U} = - \sum_{{\langle}\vec{r},\vec{r}'{\rangle}} \left[J_\perp(\hat{\eta}^x_{\bm{r}} \hat{\eta}^x_{\bm{r}'} + \hat{\eta}^y_{\vec{r}'} \hat{\eta}^y_{\vec{r}}) + J_z \hat\eta^z_{\vec{r}} \hat\eta^z_{\vec{r}'}\right].
\label{xxz}
\end{equation}
Here, we have also introduced the pseudospin operators $\hat\eta^{a}_{\bm{r}}=(\Psi^\dagger_{\vec{r}} \eta^{a} \Psi^{\vphantom{\dagger}}_{\vec{r}})/2$, where $\Psi^\dagger_{\vec{r}}=(\tilde{c}^{\dagger}_{\bm{r},\uparrow}, \tilde{c}^{\phantom{\dagger}}_{\bm{r},\downarrow})$,
$\tilde{c}^\dagger_{\vec{r},\sigma}$ creates an electron with spin $\sigma$ in a Wannier-orbital localized around site $\vec{r}$ in the lower band of $H_0$ (with $\tau_y=-\sigma$ for $\zeta=0$),
and $\eta^a$ are Pauli matrices. The $J_\perp$ and $J_z$ terms correspond to hopping and a nearest-neighbor interaction of the Cooper pairs, respectively, and their strengths are
$J_\perp = \zeta^2(2U+V)/8$ and $J_z = \zeta^2 U/4 - V(2+9 \zeta^2/8)$.

For $V=0$, the system has an emergent $SU(2)$ symmetry that relates the pairing and charge order parameters~\cite{Tovmasyan2018}. This symmetry is weakly broken by terms of order $(U/t)^2$, not included in Eq.~\eqref{xxz}. For $0<V\ll U$, the problem is equivalent to a two-dimensional pseudospin ferromagnet with a weak easy-plane anisotropy. Parameterizing the anisotropy by $\Delta J = J_\perp - J_z$, we can estimate the critical temperature of the BKT transition as~\cite{Nelson1977}
\begin{equation}
T_c \sim \pi J_\perp/\log(\pi J_\perp/\Delta J).
\label{eq:xxz_Tc}
\end{equation}
Note that in the limit $\Delta J\rightarrow 0$ we get $T_c\rightarrow 0$, as required by the Mermin-Wagner-Hohenberg theorem.

We now turn to estimating $E_F^{(1,2)}$. Due to the particle-hole symmetry of the effective Hamiltonian in Eq.~\eqref{xxz}, the chemical potential at $n=1$ (i.e., $\langle \hat{\eta}_{\vec{r}}^z\rangle = 0$) is $\mu(n=1)=0$. In the limit $n\rightarrow 0$, the system consists of dilute Cooper pairs. In this limit, the interactions between the Cooper pairs can be neglected, and the chemical potential is equal to half the energy per Cooper pair: $\mu(n\rightarrow 0) = -(J_\perp - J_z) = -\Delta J$.\footnote{Importantly, for $J_\perp>J_z$, the system does not phase separate at any density.} Therefore, $E_F^{(1)} = \mu(n=1) - \mu(n\rightarrow 0) = \Delta J$. Comparing this to Eq.~\eqref{eq:xxz_Tc}, we find that for $\Delta J \ll J_\perp$, $T_c \gg E_F^{(1)}$. Clearly, since the lower band is completely dispersionless, $E_F^{(2)}=0$. We conclude that $T_c$ can be made arbitrarily larger than the Fermi energy by either of the two definitions of Eqs.~(\ref{eq:EF1}) and (\ref{eq:EF2}).

It is worth noting that, in the parameter regime we are considering, $\rho_s(0)\sim J_\perp \sim \zeta^2 U$. Hence, the delocalization of the Cooper pairs is entirely due to the interactions and the spatial overlap between the Wannier function of the active band, as in  Refs.~\cite{Torma15,Bernevig19,Rossi19,Torma19,JHEBDC,MR21}.{\footnote{The finite value of $\rho_s(0)$ and the associated lower bound as derived in Refs.~\cite{Torma15,Bernevig19,Rossi19,Torma19} is based on the application of BCS mean-field theory.}} Here, however, for $U\gg V$, we get $\Delta J\ll J_\perp$ and hence $\rho_s(0)\gg T_c$ [see Eq.~\eqref{eq:xxz_Tc}].

\begin{figure}
	\begin{center}
          \includegraphics[width=0.99\columnwidth]{Tc_rhos.pdf}
	\end{center}
	\caption{Schematic illustration of the dispersion and momentum distribution function of the bosons in the model with unbounded $T_c/\rho_s(0)$ [Eq.~\eqref{eq:Hb}]. (a) At $T\gtrsim T_c$, the flat band is approximately half filled, and the remaining $n_b-1/2$ bosonic particles per site occupy the dispersive band. (b) At $T=0$, the flat band is completely filled with one boson per site, and hence the superfluid density is proportional to $n_b-1$.}
	\label{fig:Tc_rhos}
\end{figure}

\section{Bound on $T_c/\rho_s(0)$?}
\label{sec:rhos}

In this section, we turn to the question of whether the zero temperature phase-stiffness, $\rho_s(0)$, sets an upper bound on $T_c$ in two dimensions. $\rho_s(0)$ can be extracted directly from a measurement of the London penetration depth ($\lambda_L^2(0)\propto 1/\rho_s(0)$), or from the imaginary part of the low-frequency optical conductivity.

As is well known, in two spatial dimensions, the phase stiffness right below $T_c$ is related to $T_c$
by the inequality\footnote{At a continuous BKT transition, $\rho_s(T\rightarrow T^{-}_c)=2T_c/\pi$. However, if the transition is first order, $\rho_s$ right below $T_c$ can be larger than the universal BKT value. See, e.g., Ref. \cite{Domany1984}.} $\rho_s(T\rightarrow T^{-}_c)\geq 2T_c/\pi$.
However, $T_c$ is not directly related to $\rho_s(0)$.
On physical grounds, it often makes sense to identify $\rho_s(0)$ with a ``phase ordering scale'' that sets an upper limit on $T_c$~\cite{emery1995importance}. This is justified by the fact that $\rho_s(T)$ is usually a monotonically decreasing function of temperature, i.e. $\rho_s(0)\geq \rho_s(T_c)$, and therefore $T_c$ can be bounded from above by $\rho_s(0)$. In conventional superconductors, $\rho_s(0)\gg T_c$, and $T_c$ is almost entirely determined by the pairing scale. In contrast, in underdoped cuprates, $\rho_s(0)$ is close to $T_c$, as illustrated by the famous Uemura plot~\cite{Uemura1989}. This suggests that in these systems, phase fluctuations play an important role in limiting $T_c$~\cite{emery1995importance}.

While this line of reasoning is likely correct in most circumstances, we will show here that  --- as a matter of principle ---  there
  is {\it no} bound on $T_c/\rho_s(0)$. We outline a concrete model where $\rho_s(0)$ can be made arbitrarily smaller than $\rho_s(T_c)$.

Let us begin with a two-dimensional lattice model of two species of (complex) bosons, $b_1,~b_2$,
\begin{subequations}
\begin{eqnarray}
\label{eq:Hb}
H^b &=& H^b_0 + H^b_{\textnormal{int}},\\
H^b_0 &=& \sum_{\alpha,\vec{k}} {\varepsilon}_\alpha(\vec{k})~ b_{\alpha,\vec{k}}^\dagger b^{{^{\vphantom{\dagger}}}}_{\alpha,\vec{k}},
\end{eqnarray}
\end{subequations}
where ${\varepsilon}_2(\vec{k})$ is assumed to have a large bandwidth, $W_2$, and ${\varepsilon}_1 = {\varepsilon}_2(0)-{\varepsilon}_0$ forms a completely flat band at an energy ${\varepsilon}_0$ below the bottom of the ${\varepsilon}_2$ band, i.e. the $b_1$ bosons are completely localized on individual sites. The dispersion of the two species of bosons is illustrated in Fig.~\ref{fig:Tc_rhos}. For the purpose of our discussion here, we can approximate ${\varepsilon}_2(\vec{k})\approx k^2/2m_b$ near the bottom of the broad band. $H^b_{\textnormal{int}}$ includes an on-site (Hubbard) interaction of strength $U_{1,2}$ for the $b_{1,2}$ bosons. We take $U_2 \ll W_2$ whereas $U_1\rightarrow \infty$, implying that the number of $b_1$ bosons on each localized site can only be $0$ or $1$. The total average number of bosons per unit cell is chosen to be $n_b>1$.

 At temperatures near $T_c$, the chemical potential is slightly above the bottom of the broad band. Then, assuming that ${\varepsilon}_0 \ll T$, the average occupation of the localized sites is close to $1/2$ (since the $b_1$ bosons are essentially hard-core bosons at effectively `infinite' temperature), so there are approximately $n_b-1/2$ bosons per unit cell left to occupy the broad band. The critical temperature is $T_c \sim \frac{n_b-1/2}{2 m_b}$, up to logarithmic corrections in $\frac{n_b-1/2}{2 m_b}/U_2$ \cite{popov1972theory,kagan1987influence,Fisher1988}.\footnote{Since we are in two spatial dimensions, in the absence of interactions, $T_c=0$.} The momentum distribution of particles at $T\gtrsim T_c$ is shown schematically in Fig.~\ref{fig:Tc_rhos}a.

On the other hand, at $T=0$, all the localized sites are filled with one boson. The density of bosons in the broad band is thus $n_b-1$, and the superfluid stiffness is $\rho_s\approx \frac{n_b-1}{2 m_b}$. The boson distribution function is illustrated in Fig.~\ref{fig:Tc_rhos}b. Clearly, the ratio $T_c/\rho_s(0)$ can be made arbitrary large by letting $n_b\rightarrow 1^+$. If $1/2<n_b<1$, the ground state is not a superfluid, and there is a reentrant transition into a superconducting state with increasing $T$.

Note that
in our simple model (Eq.~\eqref{eq:Hb}), the numbers of the two boson species are separately conserved. However, we do not expect
the key results to be changed qualitatively
by the addition of a weak hybridization between the two species, that breaks this separate conservation of the two boson numbers.
In particular, a small hybridization
generically produces a perturbative correction
to $T_c$ and $\rho_s(0)$.

Indeed, a mild version of this sort of breakdown of the heuristic bound on $T_c/\rho_s(0)$ has been documented experimentally in Zn-doped cuprates \cite{uchida}. Here, the pristine material comes close to saturating the heuristic bound;  light Zn doping suppresses $T_c$ but apparently suppresses $\rho_s(0)$ more rapidly, leading to a ratio that slightly exceeds the value proposed in Ref. \cite{emery1995importance}. This was explained --- likely correctly --- by the authors of Ref. \cite{uchida} as being due to Zn-induced inhomogeneity of the superfluid response. This explanation is spiritually close to the  model discussed above: Each Zn impurity destroys the superconductor in its vicinity, possibly due to
pinning of local spin-density-wave order \cite{hirota}. In some sense, this can be thought of as a state with localized d-wave pairs near the impurities. This effect depletes the condensate at low temperature, causing a decrease in the superfluid density. However, near $T_c$, this effect weakens, as the spin-density wave order partially melts.
From this perspective, it would be interesting to explore whether this violation can be made parametrically large with increasing Zn concentration - approaching the point at which $T_c\to 0$.

\section{Electron-phonon superconductivity: \\bound on $T_c/\omega_0$?}
\label{sec:el_ph}

Recently, it has been proposed that $T_c$ in an electron-phonon superconductor can never exceed a certain fraction of the characteristic phonon frequency, $\omega_0$~\cite{Esterlis2018,esterlis2018bound}. This putative bound implies that Migdal-Eliashberg (ME) theory~\cite{migdal1958interaction,eliashberg1960interactions,marsiglio2008electron} must break down when the dimensionless electron-phonon coupling $\lambda$ is of order unity~\cite{Esterlis2018,chubukov2020eliashberg}, since according to ME theory, $T_c$ grows without limit with increasing $\lambda$~\cite{Dynes1975}.
In general, the failure of ME theory at $\lambda = O(1)$ is a result of strong-coupling effects: (i) The lattice tends to become unstable for large $\lambda$, resulting in a charge density wave (CDW) transition; (ii) electrons become tightly bound into bipolarons, whose kinetic energy is strongly quenched in the strong coupling limit; and (iii) as $\lambda$ increases, the phonon frequency renormalizes downward by an appreciable amount, $\Delta\omega$, suppressing $T_c$~\cite{Cohen06,note_softening}.

These strong-coupling effects certainly play an important role in limiting $T_c$ in real systems, where it is typically found never to exceed about $0.1\,\omega_0$ across numerous conventional superconductors~\cite{esterlis2018bound}. Determinant Monte Carlo simulations of the paradigmatic Holstein model reveal that ME theory indeed fails for $\lambda=O(1)$, and the maximal $T_c$ is significantly below $\omega_0$~\cite{Esterlis2018}. As we shall now show, however, this is not a rigorous bound on $T_c$.

To demonstrate this, we consider a particular large$-N$ variant of the electron-phonon problem~\cite{Werman2016,werman2017non} with $N-$component electrons and $N\times N-$component (`matrix') phonons, defined on a $d-$dimensional hypercubic lattice. The Hamiltonian is given by
\begin{subequations}
\begin{eqnarray}
H &=& H_e + H_{\textnormal{ph}} + H_{\textnormal{e-ph}}, \label{elph}\\
H_e &=& \sum_{a=1}^{N}\sum_{\vec{r},\vec{r}',\sigma}(-t_{\vec{r}\r'}-\mu\delta_{\vec{r}\r'})c_{\vec{r},\sigma,a}^{\dagger}c^{\vphantom{\dagger}}_{\vec{r}',\sigma,a},\\
H_{\textnormal{ph}} &=& \sum_{a,b=1}^{N}\sum_{\vec{r}}\left(\frac{P_{\vec{r},ab}^{2}}{2M}+\frac{K}{2}X_{\vec{r},ab}^{2}\right),\\
H_{\textnormal{e-ph}} &=& \frac{\alpha}{\sqrt{N}}\sum_{a,b=1}^{N}\sum_{\vec{r},\sigma}X^{\vphantom{\dagger}}_{\vec{r},ab}c_{\vec{r},\sigma,a}^{\dagger}c^{\vphantom{\dagger}}_{\vec{r},\sigma,b}.
\label{eq:alpha}
\end{eqnarray}
\end{subequations}
Here, $c^\dagger_{\vec{r},\sigma,a}$ creates an electron at position $\vec{r}$ with spin $\sigma$
in ``orbital'' $a$. The hopping parameters $t_{\vec{r}\r'}$ and chemical potential $\mu$ are assumed to be identical for all orbitals. We have introduced a real, symmetric matrix of phonon displacements, $\hat{X}_{\vec{r}}$ and their canonically conjugate momenta, $\hat{P}_{\vec{r}}$, with frequency $\omega_{0}=\sqrt{K/M}$, assumed to be much smaller than the Fermi energy. The phonons are taken to be dispersionless for simplicity. The purely on-site electron-phonon coupling constant is denoted $\alpha$, with a $N-$dependent normalization factor that ensures that the model has a finite energy density in the $N\rightarrow\infty$ limit. The dimensionless electron-phonon coupling constant is defined as $\lambda = \alpha^2 \nu(0)/K$, where $\nu(0)$ is the electronic density of states at the Fermi level per orbital.

\begin{figure}
	\begin{center}
          \includegraphics[width=0.99\columnwidth]{ElPhDiags.pdf}
	\end{center}
	\caption{Self-energy for the (a) phonon ($X_{ab}$), and (b) electron ($c_a$) fields, respectively. (c) Bethe-Salpeter equation for the pairing vertex, $\Phi$. Double dashed and solid lines represent the phonon and electron fields. Solid circles (triangles) denote the electron-phonon (pairing) vertex.}
	\label{fig:diags}
\end{figure}

We are interested in the large$-N$ limit of the model defined in Eq.~\eqref{elph}.
Since the number of phonon degrees of freedom is much larger than the number of electron orbitals, the phonon dynamics are essentially unaffected by the coupling to the electrons, even when the electrons are strongly perturbed.
This implies that the strong-coupling effects mentioned above are suppressed, even for
$\lambda \gg 1$.  In particular, as we show in Appendix~\ref{app:stable},
there is no lattice instability or polaron formation as long as $\lambda \ll N$, and the softening of the phonon frequency is only of the order of $\Delta\omega\sim \lambda \omega_0/N$.

To zeroth order in $1/N$, the equations for the electron self-energy and the pairing vertex are exactly those given by Eliashberg theory, whereas the phonon self-energy is of order $1/N$ (see Fig.~\ref{fig:diags}). Thus, the self-consistent equations for the pairing vertex are identical to those of ME theory neglecting the renormalization of the phonons, and hence the result is the same. In particular, for $1 \ll \lambda \ll N$, $T_c \approx 0.1827 \,\omega_0 \sqrt{\lambda}$~\cite{Dynes1975}.
Implicit in the fact that Migdal-Eliashberg theory is exact at $N\to \infty$ is an assurance that there is no suppression of $T_c$ by phase fluctuations.\footnote{In the $d=2$ version of our model, the BKT temperature differs from the mean-field transition temperature only by a $1/N$ correction.}   More specifically, this follows from the observation that the superfluid stiffness is ${\cal O}(N)$.
Thus, $T_c$ is unbounded.

The key ingredient in our model that allows us to take $\lambda>1$ without suffering from lattice instabilities is that the different phonon modes couple to electron bilinears that do not commute with each other [see Eq.~\eqref{eq:alpha}]. This limits the energy gain from distorting a given set of phonon modes when forming a CDW or a polaron bound state, since the resulting perturbations to the electronic Hamiltonian cannot be diagonalized simultaneously. In contrast, the contributions of the individual phonon modes add algebraically in the total dimensionless coupling $\lambda$ that enters the equation for the pairing vertex (the same dimensionless coupling also determines the resistivity in the normal state of this model~\cite{Werman2016}).

It is worth noting that while these considerations may be of some
use in searching for systems with ever higher $T_c$, as a practical matter it may be difficult to significantly violate the proposed heuristic bound.  To achieve  $T_c \approx \omega_0$ requires
 the extremely large value of $\lambda\approx 25$.
 At the same time, to avoid polaron formation requires $N \gg \lambda$, which means the number of distinctly coupled phonon modes would have to be $N^2 \gg \lambda^2 \sim 625$!

\section{Outlook}
\label{outlook}

The notion of a fundamental upper bound on $T_c$ for models of interacting electrons is an attractive concept. In this paper, we have demonstrated that while there are numerous physical settings where such bounds can be formulated at a heuristic level, there exists no fundamental, universal upper bound on $T_c$ in terms of the characteristic energy scales of interest to us, which include an appropriately defined $T_F$, $\rho_s(0)$ and $\omega_0$. We have constructed explicit counter-examples which violate these heuristic bounds by an arbitrary amount.

On the experimental front, it would be  fruitful to look for candidate materials where the heuristic bounds are violated by a large amount.
The fact that these bounds are usually satisfied is to be expected, since although the bounds are not rigorous, the physical reasoning behind them is quite robust.
As our theoretical discussion illustrates, whenever such bounds are violated, there is an interesting underlying physical reason behind the violation; moreover, the mechanisms behind the violation of the heuristic bounds may suggest ways to optimize $T_c$. Our work provides two such examples. Flat band systems with a large spatial extent of the Wannier functions are a promising platform for increasing $T_c$. In electron-phonon systems, the instabilities that limit $T_c$ at large electron-phonon interaction strength can be partially mitigated if the coupling is shared between several active phonon modes that couple to non-commuting electronic operators.

\acknowledgements
We thank Pablo Jarillo-Herrero, Mohit Randeria, and J.M. Tranquada for stimulating discussions. SAK was supported, in part, by the National Science Foundation (NSF) under Grant No. DMR2000987. DC was supported by the faculty startup grants at Cornell University. EB and JH were supported by the European Research Council (ERC) under grant HQMAT (Grant Agreement No. 817799), the Israel-US Binational Science Foundation (BSF), and by a Research grant from Irving and Cherna Moskowitz.


\section{Introduction}
\setcounter{footnote}{0}

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
\cite{GPDs,Ji,R97} and has evolved to considerable complexity,
reviewed for instance in \cite{Ji98,GPV,Diehl03,Belitsky:2005qn}.  As
GPDs can be analyzed using standard operator product expansion
techniques \cite{GPDs,blumlein}, their moments can be and have been
calculated in lattice QCD \cite{QCDSF}.  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 \cite{exp} 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.\ \cite{Ji,Burkardt:2002hr,Polyakov:2002yz,Belitsky:2003nz}.

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 \cite{ChPT}.  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 \cite{chvol} and
also the finite lattice spacing \cite{baer}.  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~\cite{CJi,KP,DMS}.  In the case of the nucleon GPDs, the chiral
corrections have been calculated for the lowest
moments~\cite{BFHM,CJi,BJi} 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 $p'^\mu$ 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
$(p'-p)^\mu$.  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 \emph{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~\ref{sec:definitions}
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~\ref{CHPT} 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~\ref{sec:results} and
\ref{sec:gpd-res} 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~\ref{sec:sum}.

\section{Generalized parton distributions in the nucleon}
\label{sec:definitions}

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
\begin{align}
  \label{quark-gpd}
\int \frac{d \lambda}{4\pi}\, e^{ix \lambda(aP)}
\langle p'|\, \bar{q}(- {\textstyle\frac{1}{2}} \lambda a)\, \slashed{a} \,
  q( {\textstyle\frac{1}{2}} \lambda a) \,|p \rangle
&= \frac{1}{2a P}\, \bar{u}(p') \left[
  \slashed{a}\, H(x,\xi,t) +
  \frac{i \sigma^{\mu\nu} a_{\mu} \Delta_\nu}{2M} \, E(x,\xi,t)\,
\right]  u(p) \,,
\nonumber \\[4mm]
\int \frac{d \lambda}{4\pi}\, e^{ix \lambda(aP)}
  \langle p'|\, \bar{q}(- {\textstyle\frac{1}{2}} \lambda a)\, \slashed{a}\gamma_5 \,
  q( {\textstyle\frac{1}{2}} \lambda a) \,|p \rangle
&= \frac{1}{2a P}\, \bar{u}(p') \left[
  \slashed{a}\gamma_5\, \widetilde{H}(x,\xi,t) +
  \frac{a \Delta}{2M}\gamma_5\, \widetilde{E}(x,\xi,t) \right]  u(p) \,,
\hfill
\end{align}
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
\begin{align}
P=\frac{1}{2} (p+p'),& &\qquad\Delta=p'-p,&
&\qquad t=\Delta^2,& &\qquad \xi=-\frac{\Delta  a}{2P  a} \,.
\end{align}
As usual, Wilson lines between the quark fields are to be inserted in
\eqref{quark-gpd} 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
\begin{align}
\label{operators}
\mathcal{O}_{\mu_1 \mu_2 \ldots \mu_n}=
  \mathbf{S}\, \bar{q} \gamma_{\mu_1}
       i \lrD_{\mu_2} \ldots i \lrD_{\mu_n}\, q \, , & &
\widetilde{\mathcal{O}}_{\mu_1 \mu_2 \ldots \mu_n}=
  \mathbf{S}\, \bar{q} \gamma_{\mu_1}\gamma_5\,
       i \lrD_{\mu_2} \ldots i \lrD_{\mu_n}\, q \,,
\end{align}
where $\lrD^\mu=\frac12(\rD^\mu-\lD^\mu)$ and $\mathbf S$ denotes the
symmetrization of all uncontracted Lorentz indices and the subtraction
of traces, e.g.\ $\mathbf{S}\, t_{\mu\nu} =  {\textstyle\frac{1}{2}}  (t_{\mu\nu} +
t_{\nu\mu}) - \frac{1}{4}\mskip 1.5mu g_{\mu\nu}\mskip 1.5mu
t^{\lambda}{}_{\lambda}$ for a tensor of rank two.  It is convenient to
contract all open Lorentz indices with the auxiliary vector $a$,
\begin{align}
\label{defOa}
{\mathcal O}_{\mu_1\ldots\mu_n} ~\to~ {\mathcal O}_n(a)=
a^{\mu_1}\ldots a^{\mu_n}\, {\mathcal O}_{\mu_1\ldots\mu_n}\,,
\end{align}
and in analogy for $\widetilde{\mathcal{O}}$.  The matrix elements of
the operators \eqref{operators} can be parameterized
as~\cite{Ji98,Diehl03}
\begin{align}
  \label{nucl-mat}
\vev{p'|\mathcal{O}_n(a)|p} &=
\sum_{\substack{k=0\\{\mathrm{even}}}}^{n-1}
  (a P)^{n-k-1}\, (a  \Delta)^k\,\,
\bar u(p') \left[ \slashed{a}\, A_{n,k}(t)
           +\frac{i\sigma^{\mu\nu} a_{\mu}\Delta_\nu}{2M}\, B_{n,k}(t)
\right] u(p)
\nonumber \\
&\quad + \bmod(n+1,2)\, (a\Delta)^n\,
         \frac{1}{M}\,\bar u(p') u(p)\, C_n(t)\,,
\nonumber\\[2mm]
\vev{p'|\widetilde{\mathcal{O}}_n(a)|p} &=
\sum_{\substack{k=0\\{\mathrm{even}}}}^{n-1}
  (a P)^{n-k-1}\, (a  \Delta)^k\,\,
\bar u(p') \left[ \slashed{a}\gamma_5\, \widetilde{A}_{n,k}(t)
           +\frac{a \Delta}{2M}\gamma_5\, \widetilde{B}_{n,k}(t)
\right] u(p) \,.
\end{align}
The moments of the above GPDs are polynomials in $\xi^2$,
\begin{align}
  \label{gpd-mom}
\int_{-1}^1dx\, x^{n-1}\, H(x,\xi,t) &=
\sum_{\substack{k=0\\{\mathrm{even}}}}^{n-1}
(2\xi)^k\, A_{n,k}(t) + \mathrm{mod}(n+1,2)\,(2\xi)^{n} C_{n}(t)\,,
\nonumber \\
\int_{-1}^1dx\, x^{n-1}\, E(x,\xi,t) &=
\sum_{\substack{k=0\\{\mathrm{even}}}}^{n-1}
(2\xi)^k\, B_{n,k}(t) - \mathrm{mod}(n+1,2)\,(2\xi)^{n} C_{n}(t)\,,
\displaybreak \nonumber \\
\int_{-1}^1dx\, x^{n-1}\, \widetilde{H}(x,\xi,t) &=
\sum_{\substack{k=0\\{\mathrm{even}}}}^{n-1}
(2\xi)^k\, \widetilde{A}_{n,k}(t)\,,
\nonumber \\
\int_{-1}^1dx\, x^{n-1}\, \widetilde{E}(x,\xi,t) &=
\sum_{\substack{k=0\\{\mathrm{even}}}}^{n-1}
(2\xi)^k\, \widetilde{B}_{n,k}(t)\,.
\end{align}
The restriction to even $k$ in \eqref{nucl-mat} and \eqref{gpd-mom} 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~\cite{JM}. The evaluation
of nucleon form factors in heavy-baryon ChPT is simplified if one
works in the Breit frame~\cite{B92}.  It is defined by the condition
$\vec{P}=0$, so that the incoming and outgoing nucleons have opposite
spatial momenta ${\vec{p}\,}'=-\vec{p}=\vec{\Delta}/2$ and the same
energy $p_0' =p_0^{\phantom{'}} =M\gamma$, where
\begin{equation}
  \gamma = \sqrt{1-\Delta^2/4M^2} \,.
\end{equation}
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 $p'=M\gamma v+ \Delta/2$.

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
\begin{equation}
S_\mu=\frac{i}{2}\gamma_5\sigma_{\mu\nu}\,v^\nu .
\end{equation}
Using that $(v \Delta)= (v S) =0$, one finds in particular
\begin{align}
  \label{D-b}
\widebar u(p') u(p) &= \gamma\,\bar u_v(p')\, u_v(p) \,,
\phantom{\frac{1}{M}}
\nonumber \\[2mm]
\widebar u(p')\gamma_\mu u(p) &=
v_\mu\, \bar u_v(p')\, u_v(p)+\frac{1}{M}\, \bar u_v(p')\,
  [S_\mu,(S\Delta)]\, u_v(p)\,,
\nonumber \\[2mm]
\frac{i}{2M}\, \widebar u(p')\,\sigma_{\mu\nu}\Delta^\nu u(p)
  &= v_\mu \frac{\Delta^2}{4M^2}\, \bar u_v(p')\,u_v(p)
  +\frac{1}{M}\, \bar u_v(p')\,[S_\mu,(S\Delta)]\, u_v(p) \,,
\nonumber \\[2.5mm]
\widebar u(p')\gamma_\mu\gamma_5 u(p)
  &= 2\gamma\, \bar u_v(p')S_\mu u_v(p)+
\frac{\Delta_\mu}{2M^2(1+\gamma)}\, \bar u_v(p')\,(S\Delta)\, u_v(p)\,,
\nonumber \\[2mm]
\widebar u(p')\gamma_5 u(p) &= \frac{1}{M}\,
  \bar u_v(p')\,(S\Delta)\, u_v(p)\,,
\end{align}
where the spinors
\begin{align}
  \label{spinors}
u_v(p)  &= \mathcal{N}^{-1}\, \frac{1+\slashed{v}}{2}\,u(p) , &
u_v(p') &=\mathcal{N}^{-1}\, \frac{1+\slashed{v}}{2}\,u(p')
\end{align}
with
\begin{equation}
  \label{spinor-norm}
\mathcal{N} = \sqrt{\frac{M+vp}{2M}} = \sqrt{\frac{M+v p'}{2M}}
            = \sqrt{\frac{1+\gamma}{2} \rule{0pt}{1.35em}}
\end{equation}
are normalized as $\bar u_v(p,s')\, u_v(p,s) = 2M \delta_{s's}$.
With~\eqref{D-b} one obtains the following representation for the
matrix elements \eqref{nucl-mat} in the Breit frame:
\begin{align}
  \label{OT}
\vev{p'|\mathcal{O}_n(a)|p} &=
\sum_{k=0}^{n} (M\gamma)^{n-k-1}\, (a v)^{n-k}\, (a \Delta)^{k-1}\,
\nonumber \\
& \quad {}\times \bar u_v(p')\, \Big[ (a \Delta)\, E_{n,k}(t)
   +\gamma\,[(a S),(S\Delta)]\, M_{n,k-1}(t) \Big] \,u_v(p) \,,
\nonumber \\[2mm]
\vev{p'|\widetilde{\mathcal{O}}_n(a)|p} &=
\sum_{k=1}^{n} (M\gamma)^{n-k}\, (a v)^{n-k}\, (a \Delta)^{k-1}\,
\nonumber \\
& \quad {}\times \bar u_v(p') \left[
   2\gamma\mskip 1.5mu (a S)\,{\widetilde E}_{n,k-1}(t)
   +\frac{(a \Delta) (S\Delta)}{2M^2}\,
    {\widetilde M}_{n,k-1}(t) \right] u_v(p) \,, \hspace{4em}
\end{align}
with
\begin{align}
  \label{ff-trafo}
E_{n,k}(t) &= A_{n,k}(t)+\frac{\Delta^2}{4M^2}B_{n,k}(t)
   \hspace{2em}\text{for~} k<n\,,
& E_{n,n}(t) &= \gamma^2 C_n(t)\,,
\nonumber \\[2mm]
M_{n,k}(t)&=A_{n,k}(t)+B_{n,k}(t)\,,
\nonumber \\[2.5mm]
\widetilde{E}_{n,k}(t)&=\widetilde{A}_{n,k}(t)\,,
\nonumber \\[1mm]
{\widetilde M}_{n,k}(t)&= \frac{1}{1+\gamma}\,\widetilde{A}_{n,k}(t)
+\widetilde{B}_{n,k}\,.
\end{align}
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 \eqref{nucl-mat} the terms with $E_{n,k}$ in \eqref{OT}
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
\begin{align}
  \label{ff-inverse}
A_{n,k}(t) &= \frac{1}{\gamma^2} \left[ E_{n,k}(t)
  - \frac{\Delta^2}{4M^2} M_{n,k}(t) \right] \,,
& B_{n,k}(t) &= \frac{1}{\gamma^2}\, \Big[ M_{n,k}(t)
  - E_{n,k}(t) \Big] \,,
\nonumber \\[1mm]
\widetilde{B}_{n,k}(t) &= \widetilde{M}_{n,k}(t)
  - \frac{1}{1+\gamma}\,\widetilde{E}_{n,k}(t) \,.
\end{align}

\section{Twist-two matrix elements in heavy-baryon ChPT}
\label{CHPT}

Heavy-baryon ChPT combines the techniques of chiral perturbation
theory and of heavy-quark effective field theory~\cite{JM} (for a
detailed review see Ref.~\cite{BKM}). 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~\cite{JM}
\begin{equation}
  N(x)=e^{-iM_0 v x}\bigl(N_v(x)+n_v(x)\bigr)\,,
\end{equation}
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 $\slashed{v}N_v=N_v$,
$\slashed{v}n_v=-n_v$.  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 $\pi(x)$,
\begin{equation}
{\mathcal L}_{\mathrm{eff}}={\mathcal L}_{\pi}+{\mathcal L}_{\pi N}\,,
\end{equation}
where
\begin{align}
{\mathcal L}_{\pi}&={\mathcal L}_{\pi}^{(2)}+{\mathcal L}_{\pi}^{(4)}
  +\ldots\,, &
{\mathcal L}_{\pi N}&={\mathcal L}_{\pi N}^{(1)}
  +{\mathcal L}_{\pi N}^{(2)}+\ldots
\end{align}
are expanded in powers of $q$.  The explicit expressions for the
lowest-order terms read \cite{BKM}
\begin{align}
  \label{L2}
{\mathcal L}^{(2)}_{\pi} &= \frac{F^2}{4}
\mathop{\rm Tr}\nolimits\left( \partial_\mu U\partial^\mu U^\dagger
+(\chi^\dagger U+ U^\dagger \chi ) \right)\,,
\nonumber \\[0.4em]
{\mathcal L}^{(1)}_{\pi N} &=
\widebar{N}_{\!v}\, \Big\{ i\mskip 1.5mu (v \nabla)+g_A (Su) \Big\} \,N_v\,,
\nonumber \\[0.1em]
{\mathcal L}^{(2)}_{\pi N} &= \widebar{N}_{\!v} \left\{
  \frac{(v\nabla)^2 -\nabla^2}{2M_0}
  - \frac{ig_A}{2M_0}\, \bigl\{ (\nabla S), (v u)\bigr\}
  + c_1 \mathop{\rm Tr}\nolimits\Big( u^\dagger \chi u^\dagger + u \chi^\dagger u \Big) \right.
\nonumber \\
& \hspace{2.2em} \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
\end{align}
with $U=u^2=\exp\{i\pi^a\tau^a/F\}$, the covariant derivative
$\nabla_\mu=\partial_\mu+\Gamma_\mu$, and
\begin{align}
  \label{VA}
\Gamma_\mu&= \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) \,,
\nonumber \\[0.2em]
u_\mu&= {i}\left(u^\dagger \partial_\mu u-u\partial_\mu u^\dagger\right)
 \hspace{0.95ex}= -\frac{1}{F}\,\partial_\mu \pi^a\tau^a +O(\pi^3) \,.
\end{align}
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~\eqref{L2} are the pion decay constant $F$ (normalized to $F\approx
92 \mbox{~MeV}$) 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 $\chi\to m^2\, \hbox{{1}\kern-.25em\hbox{l}}$, where $m$ is the bare pion and
$\hbox{{1}\kern-.25em\hbox{l}}$ 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
\cite{Meissner:2005ba}.  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 $-i\, \bigl( (vl)^2 - l^2
\bigr) \big/(2 M_0)$ and $4i\mskip 1.5mu c_1 m^2$, where $l$ is the residual
nucleon momentum, and are to be multiplied with a nucleon propagator
$i/(vl + i\mskip 1.5mu 0)$ 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 \cite{BKM}.

In the following subsection we discuss how to construct the operators
in the effective theory that match the twist-two quark operators
\eqref{operators}.  Nucleon matrix elements in the Breit frame are
then obtained as \cite{Steininger:1998ya}
\begin{equation}
  \label{matching}
\vev{p'|\mathcal{O}|p} = \mathcal{N}^2 Z_N\;
  \widebar{u}_v(p')\, G_{\mathcal{O}}(r',r)\, u_v(p) \,,
\end{equation}
with the spinors $u_v$ and normalization $\mathcal{N}$ given in
\eqref{spinors} and \eqref{spinor-norm}.  Here $G_{\mathcal{O}}(r',r)$
is the truncated Green function for external heavy-baryon fields
$\widebar{N}_{\!v}$, $N_v$ and the operator $\mathcal{O}$ in the effective
theory.  The residual momenta of the incoming and outgoing nucleon are
given by
\begin{align}
r  &= p  - M_0\mskip 1.5mu v = wv - \Delta/2 \,, &
r' &= p' - M_0\mskip 1.5mu v = wv + \Delta/2
\end{align}
with
\begin{equation}
  \label{w-def}
w = M (\gamma -1) + \delta M
  = - \frac{\Delta^2}{8M} - 4c_1 m^2 + O(q^3) \,,
\end{equation}
where $\delta M = M-M_0$ is the nucleon mass shift.  Finally, $Z_N$ is
the heavy-baryon field renormalization constant,
\begin{equation}
  \label{ZN}
Z_N = 1 - \frac{3 m^2 g_A^2}{2\mskip 1.5mu (4\pi F)^2}
  - \frac{9 m^2 g_A^2}{4\mskip 1.5mu (4\pi F)^2}
    \log\frac{m^2}{\mu^2} - 8m^2\, d_{28}^{\mskip 1.5mu r}(\mu) + O(q^3) \,,
\end{equation}
where $d_{28}^{\mskip 1.5mu r}(\mu)$ is a low-energy constant in the
Lagrangian $\mathcal{L}_{\pi N}^{(3)}$.  As explained in
\cite{Fettes:1998ud} the corresponding operator is required for
renormalization but does not appear in physical matrix elements.  The
value of $d_{28}^{\mskip 1.5mu r}(\mu)$ can therefore be chosen freely (with
different choices resulting in different values for other low-energy
constants), and in \cite{Steininger:1998ya} it was chosen such that it
compensates the $\log (m^2/\mu^2)$ term in \eqref{ZN} at the
\emph{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~\ref{sec:tree}.

\subsection{Construction of effective operators}
\label{sec: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 \eqref{operators} and \eqref{defOa}. 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
\begin{align}\label{OOa}
{\mathcal O}(a) &\,\cong\,
  {\mathcal O}_\pi(a)+{\mathcal O}_{\pi N}(a)\,, &
\widetilde{\mathcal O}(a) &\,\cong\,
  \widetilde{\mathcal O}_{\pi N}(a) \,,
\end{align}
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~\cite{ASv,CJi,KP,DMS} 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~\eqref{defOa}.  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
\begin{align}
\label{SS}
\{S_\mu,S_\nu\} &= \frac{1}{2} ( v_\mu v_\nu-g_{\mu\nu} ) \,, &
[ S_\mu,S_\nu ] &= i\epsilon_{\mu\nu\lambda\rho}\, v^\lambda S^\rho
\end{align}
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 \eqref{VA}, or as a total derivative
on the product of all fields, or in the antisymmetric form
$\lrpartial_\mu =  {\textstyle\frac{1}{2}}  (\rpartial_\mu - \lpartial_\mu)$ 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 $\lrpartial$ must appear in the covariant combination
$\lrnab_\mu = \lrpartial_\mu + \Gamma_\mu$.  The fields and
derivatives used in our construction are then any number of $u_\mu$,
$\lrnab_\mu$ and $\chi_{\pm} = u^\dagger \chi u^\dagger \pm u
\chi^\dagger u$ between the nucleon fields $\widebar{N}_{\!v}$ and $N_v$, and any
number of total derivatives $\partial_\mu$ acting on the operator as a
whole.  In the sense of \eqref{VA} we henceforth refer to
$\partial_\mu$, $\lrnab_\mu$ 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
\begin{align}
\label{ON}
Q_n(a)= \sum_{k=0}^n M^{n-k-1}\, (a v)^{n-k} Q_{n,k}(a)\,,
\end{align}
where $Q_{n,k}(a) = a_{\mu_1}\ldots a_{\mu_k}\,
Q_{n}^{\mu_1\ldots\mu_k}$ does not contain any factors $(a v)$.  The
$k$ external vectors $a$ in $Q_{n,k}(a)$ can be contracted only with
derivatives $\partial_\mu$, $\lrnab_\mu$, $u_\mu$ and the spin vector
$S_\mu$, or with the antisymmetric tensor.  There can be at most one
factor $(a S)$ as discussed after \eqref{SS}, so that $Q_{n,k}(a)$ has
to contain at least $k-1$ derivatives.  We can hence write\footnote{%
  Instead of $M$ one could also use $M_0$ or $F$ in \protect\eqref{ON}
  and \protect\eqref{ONK}, 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 \protect\eqref{OT}.}
\begin{equation}
  \label{ONK}
Q_{n,k} =M Q_{n,k,-1} +Q_{n,k,0} +\frac{1}{M}\, Q_{n,k,1} +\ldots\,,
\end{equation}
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 \eqref{SS}.  As a consequence,
the antisymmetric tensor does not appear in the operators with lowest
chiral dimension for a given $k$.

\subsection{Tree-level insertions}
\label{sec:tree}

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
\eqref{VA}, derivatives in the effective operators are to be replaced
as $\partial_\mu \to i\Delta_\mu$, $u_\mu\to 0$, and $\lrnab_\mu\ \to
-i wv_\mu$ with $w$ given in \eqref{w-def}.  Notice that, while
generically the derivative $\lrnab_\mu$ counts as $O(q)$ in the chiral
expansion, the kinematics of the external nucleon momenta forces $w
v_\mu$ 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
\eqref{OT} come from the terms with maximum number of factors
$\Delta_\mu$ and no factor $w v_\mu$.  With \eqref{ON} one readily
obtains
\begin{align}
\label{tree}
\vev{p'|\mathcal{O}_{n,k}(a)|p}
  & \,\stackrel{\mathrm{LO}}{=}\,
(a \Delta)^{k-1}\, \bar u_v(p') \left[ (a\Delta)\, E_{n,k}^{(0)}
+ [(a S),(S\Delta)]\, M_{n,k-1}^{(0)} \right] u_v(p)\,,
\nonumber \\[3mm]
\vev{p'|\widetilde{\mathcal{O}}_{n,k}(a)|p}
  & \,\stackrel{\mathrm{LO}}{=}\,
(a\Delta)^{k-1}\, \bar u_v(p')
\left[\mskip 1.5mu 2M (a S)\, {\widetilde E}_{n,k-1}^{(0)}
+ \frac{(a\Delta) (S\Delta)}{2M}\,
{\widetilde M}_{n,k-1}^{(0)} \right] u_v(p)\,,
\end{align}
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 $(a S)$,
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 \eqref{tree}.

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 $\lrnab$, or to the kinematic factors $\gamma$ in
\eqref{OT} and $\mathcal{N}$ in \eqref{matching}.  Contributions
proportional to $m^2$ are due to operators with $\chi_{+}$ or with
$\lrnab$ and from the wave function renormalization constant $Z_N$ in
\eqref{matching}.  In the results of the following sections we
explicitly include the terms proportional to $g_A^2$ in the expansion
\eqref{ZN} of $Z_N$, whereas contributions from $d_{28}^{\mskip 1.5mu r}$ are
lumped into the coefficients describing the $m^2$ corrections due to
tree-level insertions.

\subsection{Loop contributions}
\label{sec:loop}

Let us now consider a loop diagram with the insertion of the operator
$Q_n(a)$.  One easily finds that the term $M^{n-k-1}\,(a v)^{n-k}\,
Q_{n,k}(a)$ in the sum \eqref{ON} can contribute to the form factors
in \eqref{OT} 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 $m\le k$.  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
\begin{equation}
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) -2 I_{\pi} -I_N\,,
\end{equation}
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.\footnote{%
  Note 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
  \protect\eqref{VA}.}
Using the relation $L=I_{\pi}+I_N-N_{\pi}-N_{\pi N}$ (see e.g.\
\cite{BKM}) 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:
\begin{equation}
  \label{pow-count}
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)\,.
\end{equation}
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 $O(q^{\mskip 1.5mu d})$
with $d\ge D_{k,i}$.  This is on one hand because of the explicit
factors $\mathcal{N}$ and $Z_N$ in \eqref{matching}, and on the other
hand because the sum $r^\mu+r'^\mu = 2w v^\mu$ 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 \eqref{OT}.  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 $k-m \ge 0$, the
number of loops and the order of the chiral Lagrangian required to
calculate the lowest-order corrections for a given form factor do
\emph{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
\begin{itemize}
\item $O(q)$ from one-loop diagrams with insertion of the operator
  $Q_{n,m,-1}$ and leading-order (LO) pion-nucleon vertices,
\item $O(q^2)$ 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.
\end{itemize}
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~\ref{sec:axial-results}.

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~\eqref{OOa}.  Repeating the above argument and taking into account
that now $I_N = N_{\pi N} -1$, one finds that such diagrams have
chiral dimension
\begin{equation}
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)\,.
\end{equation}
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.~\ref{fig-1}.  With our
construction of operators explained in Section~\ref{sec:operators} we
can readily analyze the origin of factors $\Delta_\mu$, whose number
determines to which form factor a graph will contribute.  Using $(v
\Delta)= (v S) =0$ and the form \eqref{L2} of the LO and NLO
pion-nucleon Lagrangian, we find that the numerators of the loop
integrals are composed as specified in Table~\ref{tab:vectors}.  The
denominators of the pion and nucleon propagators respectively are
$(l^2 - m^2 +i\mskip 1.5mu 0)$ and $(lv +w +i\mskip 1.5mu 0)$, 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.

\begin{table}
\caption{\label{tab:vectors} Four-vectors and their products appearing
  in the numerators of the loop graphs of Fig.~\protect\ref{fig-1}.
  $NN$ vertices (arising from nucleon propagator corrections) are not
  explicitly shown in the graphs.}
\begin{center}
  \renewcommand{\arraystretch}{1.2}
  \begin{tabular}{lll} \hline\hline
derivatives in operator insertion & $\partial_\mu$ & $\Delta_\mu$ \\
 & $\lrnab_\mu$ & $l_\mu$ ~and~ $w v_\mu$ \\
 & $u_\mu$ & $l_\mu$ \\ \hline
vertices & $\pi NN$ at LO & $Sl$ \\
 & $\pi NN$ at NLO & $(vl) (Sl) \pm (vl) (S\Delta)$ \\
 & $NN$ at NLO & $(vl)^2 -l^2 \pm l\Delta -\Delta^2/4$ \\
\hline\hline
  \end{tabular}
\end{center}
\end{table}

\begin{figure}[t]
\psfrag{l}[c][c][0.8]{$l$}
\psfrag{a}[c][c][0.8]{$wv-\frac{\Delta}2$}
\psfrag{c}[c][c][0.8]{$wv+\frac{\Delta}2$}
\psfrag{e}[c][c][0.8]{$l+wv-\frac{\Delta}2$}
\psfrag{o}[c][c][0.8]{$l+wv+\frac{\Delta}2$}
\psfrag{aa}[b][b][1.0]{a}
\psfrag{bb}[b][b][1.0]{b}
\psfrag{cc}[b][b][1.0]{c}
\centerline{\includegraphics[width=16cm]{loop-graphs-1.eps}}
\caption{\label{fig-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.}
\end{figure}

\section{Chiral corrections up to order $O(q^2)$}
\label{sec:results}

\subsection{Axial-vector operators}
\label{sec:axial-results}

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 \eqref{OT}.  With the constraints of parity invariance,
we find
\begin{align}
\label{Oa-1}
\widetilde\mathcal{O}_{n,k,-1}(a) &= \tilde{b}_{n,k}\,
  (ia\partial)^{k-1}\, \widebar{N}_{\!v} (a S)N_v + \ldots \,,
\nonumber \\[0.3em]
\widetilde\mathcal{O}_{n,k,1}(a) &= \tilde{c}_{n,k}\,
  (ia\partial)^k (i\partial_\mu)\, \widebar{N}_{\!v}\, S^\mu N_v + \ldots \,,
\end{align}
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~\ref{sec:loop}, 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.~\ref{fig-1}.  One finds
\begin{equation}
  \label{res-Et}
\widetilde E_{n,k}(t)=\widetilde
E_{n,k}^{(0)}\left\{1-\frac{3m^2g_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)\,,
\end{equation}
where the terms going with $m^2$ and $t$ originate from tree-level
insertions as discussed at the end of Section~\ref{sec:tree}.  Here
and in the following we use the subtraction scheme of \cite{ChPT} 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 \eqref{res-Et} 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
\eqref{res-Et} are independent of the moment indices $n$ and $k$, they
apply to the entire nucleon GPD $\widetilde H(x,\xi,t)$,
\begin{align}\label{H1loop}
\widetilde H(x,\xi,t)=
\widetilde H^{(0)}(x,\xi)\left\{1-\frac{3m^2g_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) \,.
\end{align}

Let us now consider the chiral corrections for $\widetilde M_{n,k-1}$.
It follows from \eqref{OT} 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~\ref{sec:loop} 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:
\begin{enumerate}
\item 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 \eqref{Oa-1}, which already
  provides the tree-level term of this form factor.  Only the loop
  graph in Fig.~\ref{fig-1}a is nonzero for this insertion, and the
  result is analogous to the one for the contribution of
  $\widetilde\mathcal{O}_{n,k,-1}$ to $\widetilde{E}_{n,k-1}$.
\item 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 \eqref{Oa-1}.
\item 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.~\ref{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.~\ref{fig-1}a.  The NNLO vertices and propagator
  corrections follow from the Lagrangian $\mathcal{L}_{\pi N}^{(3)}$
  given in \cite{Fettes:1998ud}.  We find that the only term providing
  the two required factors of $\Delta_\mu$ is the $\pi NN$ vertex
  generated by
  \begin{equation}
    -\frac{g_A}{4M_0^2}\, \widebar{N}_{\!v}\, \Big\{
    (\lnab S) (u \rnab) + (\lnab u) (S \rnab) \Big\}\, N_v \,.
  \end{equation}
  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 $\widebar{N}\,
  (i\slashed{\nabla} - M_0 + {\textstyle\frac{1}{2}}  g_A \slashed{u} \gamma_5)\, N$, see
  e.g.~\cite{BKM}.
\end{enumerate}
Putting everything together, we obtain
\begin{align}
  \label{res-Mt}
\widetilde M_{n,k}(t) &= \widetilde M_{n,k}^{(0)}\,
\left\{1-\frac{3m^2g_A^2}{(4\pi F)^2}
\left[\log\frac{m^2}{\mu^2}+1\right] \right\}
-\widetilde E_{n,k}^{(0)}\; \frac{m^2g_A^2}{(4\pi F)^2}\,
\log\frac{m^2}{\mu^2}
\nonumber \\[0.4em]
& \quad {}+\widetilde{M}_{n,k}^{(2,m)} m^2
      +\widetilde{M}_{n,k}^{(2,t)} t +O(q^3) \,,
\end{align}
where the terms going with $m^2$ and $t$ are due to tree-level
insertions.  With \eqref{res-Et}, \eqref{gpd-mom} and
\eqref{ff-inverse} one can write for the isoscalar quark GPD
$\widetilde E(x,\xi,t)$
\begin{align}
\widetilde E(x,\xi,t) &= \widetilde E^{(0)}(x,\xi)\,
\left\{1-\frac{3m^2g_A^2}{(4\pi F)^2}
\left[\log\frac{m^2}{\mu^2}+1\right] \right\}
-\widetilde H^{(0)}(x,\xi)\, \frac{m^2g_A^2}{(4\pi F)^2}\,
\log\frac{m^2}{\mu^2}
\nonumber \\[0.4em]
& \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) \,.
\end{align}

\subsection{Vector operators}

The analysis of the vector operators proceeds along similar lines. The
operator $\mathcal{O}_{n,k,0}$ reads
\begin{equation}\label{Otnk}
\mathcal{O}_{n,k,0}(a)= b_{n,k}\, (ia\partial)^k\, \widebar{N}_{\!v}\, N_v
+c_{n,k}\, (ia\partial)^{k-1}\, (i\partial_\mu)\,
\widebar{N}_{\!v}\, [(S a),S^\mu] N_v + \ldots \,,
\end{equation}
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 \eqref{OT} 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~\ref{sec:loop}
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.~\ref{fig-1}a 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 \eqref{ZN}.  For the form factor $M_{n,k}$ one obtains a
correction
\begin{equation}\label{res-M}
M_{n,k}^{(0)}
\left\{1-\frac{3m^2g_A^2}{(4\pi F)^2}\,\log\frac{m^2}{\mu^2}\right\}
\,.
\end{equation}
We note that for $n=1$, $k=0$ this implies a chiral logarithm for the
isoscalar magnetic form factor $G_{M,s}(t)$,
\begin{equation}
  \label{res-GM}
G_{M,s}(t) = \mu_s^{(0)}
  \left\{1-\frac{3m^2g_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) \,,
\end{equation}
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 \eqref{res-GM} is consistent with
the result of the relativistic calculation in \cite{Kubis:2000zd}.

\begin{figure}[t]
\psfrag{a}[c][c][0.9]{$wv-\frac{\Delta}{2}$}
\psfrag{c}[c][c][0.9]{$wv+\frac{\Delta}{2}$}
\psfrag{e}[c][c][0.9]{$l+wv$}
\psfrag{b}[c][c][0.9]{$l+\frac{\Delta}{2}$}
\psfrag{d}[c][c][0.9]{$l-\frac{\Delta}{2}$}
\psfrag{aa}[b][b][1.0]{a}
\psfrag{bb}[b][b][1.0]{b}
\centerline{\includegraphics[width=12cm]{loop-graphs-pi.eps}}
\caption{\label{fig-2} One-loop graphs with the insertion of the pion
operator ${\mathcal O}_{\pi}^n(a)$, denoted by a black blob.}
\end{figure}

One finally has to evaluate corrections due to the diagrams in
Fig.~\ref{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~\cite{DMS},\footnote{%
Note that the normalization of the twist-two operators
\protect\eqref{operators} used here differs from that in
\protect\cite{DMS} by a factor of 2.  The coefficients
$\tilde{a}_{n,k}$ have the same normalization here and in
\protect\cite{DMS}.}
\begin{equation}\label{ss-n}
{\cal O}_\pi^{n}(a)= F^2\,
\sum_{\substack{k=0\\ \mathrm{even}}}^{n-2}
\tilde a_{n,k}\, (i a\partial)^k \mathop{\rm Tr}\nolimits \left[
(aL)\,(2i a \lrpartial)^{n-k-2} (aL)+
(aR)\,(2i a \lrpartial)^{n-k-2} (aR)
\right]
\end{equation}
with $L_\mu= U^\dagger\mskip 1.5mu \partial_\mu U$ and $R_\mu = U
\partial_\mu U^\dagger$.
As discussed in Section~\ref{sec:loop}, 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.~\ref{fig-2}a contributes.  This is because the leading-order
$\pi\pi NN$ vertex corresponds to an isovector transition of the
nucleon, as follows from \eqref{L2} and the expansion of $\Gamma$ in
\eqref{VA}.  Combining the result with the correction in \eqref{res-M}
and adding analytic terms from tree-level insertions, we obtain
\begin{equation}
  \label{res-full-M}
M_{n,k}(t) = M_{n,k}^{(0)}
\left\{1-\frac{3m^2g_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) \,,
\end{equation}
where $k$ is even and
\begin{align}
  \label{result-M2}
M_{n}^{(2,\pi)}(t) &=
\frac{3g_A^2}{(4\pi F)^2}\,
\sum_{\substack{j=0\\ \mathrm{even}}}^{n-2} {\tilde a}_{n,n-j-2}
\int_{-1}^1d\eta
\left[ \frac{\partial^2}{\partial\eta^2}\, \eta^j(1-\eta^2) \right]
m^2(\eta)\, \log\frac{m^2(\eta)}{\mu^2}
\nonumber\\[2mm]
&= \frac{3g_A^2}{(4\pi F)^2}\,
\sum_{\substack{j=2\\ \mathrm{even}}}^{n} 2^{-j} j(j-1)\,
A_{n,n-j}^{\pi\mskip 1.5mu (0)}
\int_{-1}^1d\eta\,\eta^{j-2}\,m^2(\eta)\,\log\frac{m^2(\eta)}{\mu^2}
\end{align}
with
\begin{equation}
m^2(\eta)=m^2- \frac{t}{4}\mskip 1.5mu (1-\eta^2) \,.
\end{equation}
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,
\begin{align}
\int_{-1}^1 dx\, x^{n-1} H_\pi^{I=0}(x,\xi,t)
=\sum_{\substack{k=0\\ \mathrm{even}}}^n (2\xi)^k A_{n,k}^\pi(t) .
\end{align}
The relation to the low-energy constants $\tilde a_{n,k}$
reads~\cite{DMS}
\begin{align}
A_{n,k}^{\pi\mskip 1.5mu (0)}=2^{n-k}\,
  \Big[\tilde a_{n,k-2}-\tilde a_{n,k}\Big]\,,
\end{align}
which implies
\begin{align}
\tilde{a}_{n,n-k} &= {}- \sum_{\substack{j=k\\ \mathrm{even}}}^n
  2^{-j} A_{n,n-j}^{\pi\mskip 1.5mu (0)} \hspace{2em}\text{for~} k>0 \,, &
\sum_{\substack{j=0\\ \mathrm{even}}}^n 2^{-j}
  A_{n,n-j}^{\pi\mskip 1.5mu (0)} &= 0 \,.
\end{align}
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 \eqref{pow-count}.  Our final result including
analytic terms from tree-level insertions is
\begin{align}\label{res-E}
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)\,,
\end{align}
where the order $O(q)$ correction reads
\begin{align}
  \label{result-E1}
E_{n}^{(1,\pi)}(t) &= {}-M (2m^2-t)\,
\frac{3\pi g_A^2}{8\mskip 1.5mu (4\pi F)^2}\,
\sum_{\substack{j=0\\ \mathrm{even}}}^{n-2} {\tilde a}_{n,n-j-2}\,
\int_{-1}^1d\eta\, \frac{\eta^j (1-\eta^2)}{m(\eta)}
\nonumber\\[2mm]
&= M (2m^2-t)\,\frac{3\pi g_A^2}{8\mskip 1.5mu (4\pi F)^2}\,
\sum_{\substack{j=2\\ \mathrm{even}}}^{n}
2^{-j}\,A_{n,n-j}^{\pi\mskip 1.5mu (0)}\int_{-1}^1d\eta\,
\frac{1-\eta^j}{m(\eta)}\,,
\end{align}
and the order $O(q^2)$ term is
\begin{align}
  \label{result-E2}
E_{n}^{(2,\pi)}(t) =& ~\frac{3m^2 g_A^2}{(4\pi F)^2}\,
\log\frac{m^2}{\mu^2}\;
\sum_{\substack{j=0\\ \mathrm{even}}}^{n-2} \tilde a_{n,n-j-2}
\nonumber\\[2mm]
&
+\frac{6}{(4\pi F)^2}\,
\sum_{\substack{j=0\\ \mathrm{even}}}^{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]
\nonumber\\[2mm]
& ~~+ M \left[c_1\mskip 1.5mu m^2 \left(\log\frac{m^2(\eta)}{\mu^2}+1\right)
-\frac{3}{4} c_2\mskip 1.5mu m^2(\eta)\, \log\frac{m^2(\eta)}{\mu^2}
-c_3\mskip 1.5mu m^2(\eta)
  \left(\log\,\frac{m^2(\eta)}{\mu^2}+\frac{1}{2}\right)
\right] \Biggr\}
\nonumber\\[2mm]
=& {}- \frac{3m^2 g_A^2}{2\mskip 1.5mu (4\pi F)^2}\,
\log\frac{m^2}{\mu^2}\,
\sum_{\substack{j=2\\ \mathrm{even}}}^{n}2^{-j} j\,
A_{n,n-j}^{\pi\mskip 1.5mu (0)}
\nonumber\\[2mm]
& -\frac{6}{(4\pi F)^2}\, \sum_{\substack{j=2\\ \mathrm{even}}}^{n}
2^{-j}\,A_{n,n-j}^{\pi\mskip 1.5mu (0)} \int_{-1}^1d\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]
\nonumber\\[2mm]
& ~~+ M \left[c_1\mskip 1.5mu m^2 \left(\log\frac{m^2(\eta)}{\mu^2}+1\right)
-\frac{3}{4}c_2\mskip 1.5mu m^2(\eta)\, \log\frac{m^2(\eta)}{\mu^2}
-c_3\mskip 1.5mu m^2(\eta)
  \left(\log\frac{m^2(\eta)}{\mu^2}+\frac{1}{2}\right)
\right] \Biggr\}
\nonumber \\
\end{align}
The integrals over $\eta$ in \eqref{result-M2}, \eqref{result-E1} and
\eqref{result-E2} 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~\cite{ando}, 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 \cite{ando} where
comparable.\footnote{%
Note that \protect\cite{ando} 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~\cite{BJi}.\footnote{%
When comparing results, one must take into account that
\protect\cite{BJi} 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
\protect\cite{ChPT} and subtract $1/\epsilon+\log(4\pi)+\psi(2)$.}

\section{Results for moments of GPDs}
\label{sec:gpd-res}

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 \eqref{res-Et}, \eqref{res-Mt}, \eqref{res-full-M},
\eqref{res-E} and the conversion formulae \eqref{ff-trafo},
\eqref{ff-inverse} one obtains for the form factors at $t=0$
\begin{align}
  \label{res-gpd-0}
\widetilde{A}_{n,k}(0) &= \widetilde{A}_{n,k}^{(0)}\,
  \left\{1-\frac{3m^2g_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) \,,
\nonumber \\[0.1em]
\widetilde{B}_{n,k}(0) &= \widetilde{B}_{n,k}^{(0)}\,
  \left\{1-\frac{3m^2g_A^2}{(4\pi F)^2}
  \left[\log\frac{m^2}{\mu^2}+1\right] \right\}
  - \widetilde{A}_{n,k}^{(0)}\; \frac{m^2g_A^2}{(4\pi F)^2}\,
    \log\frac{m^2}{\mu^2}
  + \widetilde{B}_{n,k}^{(2,m)} m^2 + O(m^3) \,,
\nonumber \\[0.3em]
A_{n,k}(0) &= A_{n,k}^{(0)} + A_{n,k}^{(2,m)} m^2 + O(m^3) \,,
\nonumber \\[0.3em]
B_{n,k}(0) &= B_{n,k}^{(0)} - \bigl( A_{n,k}^{(0)}
   +B_{n,k}^{(0)}\bigr)\, \frac{3m^2g_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) \,,
\nonumber \\[0.4em]
C_{n}(0) &= C_{n}^{(0)} + E_{n}^{(1,\pi)}(0) + E_{n}^{(2,\pi)}(0)
  + C_{n}^{(2,m)} m^2 + O(m^3)
\end{align}
with coefficients related to those in Section~\ref{sec:results} 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 \eqref{res-gpd-0} are moderately large, with $3 m^2
g_A^2\, (4\pi F)^{-2}\, [\mskip 1.5mu \log(m^2/\mu^2)+1 \mskip 1.5mu] \approx -0.20$
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 \cite{QCDSF} 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
\begin{align}
M_{n}^{(2,\pi)}(0) &= \frac{6m^2 g_A^2}{(4\pi F)^2}\,
  \log\frac{m^2}{\mu^2}\,
  \sum_{\substack{j=2\\ \mathrm{even}}}^{n}
  2^{-j} j\, A_{n,n-j}^{\pi\mskip 1.5mu (0)} \,,
\nonumber \\[0.1em]
E_{n}^{(1,\pi)}(0) &= \frac{3\pi\mskip 1.5mu m M g_A^2}{2\mskip 1.5mu (4\pi F)^2}\,
  \sum_{\substack{j=2\\ \mathrm{even}}}^{n}
  2^{-j}\, \frac{j}{j+1}\, A_{n,n-j}^{\pi\mskip 1.5mu (0)} \,,
\nonumber \\[0.1em]
E_{n}^{(2,\pi)}(0) &= {}- \frac{3m^2 g_A^2}{2\mskip 1.5mu (4\pi F)^2}\,
  \log\frac{m^2}{\mu^2}\,
  \sum_{\substack{j=2\\ \mathrm{even}}}^{n}
  2^{-j} j\, A_{n,n-j}^{\pi\mskip 1.5mu (0)}
\nonumber \\[0.1em]
& \hspace{-3.3em}{}+ \frac{12 m^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_{\substack{j=2\\ \mathrm{even}}}^{n}
    2^{-j} \frac{j}{j+1}\, A_{n,n-j}^{\pi\mskip 1.5mu (0)} \,.
\nonumber \\
\end{align}
Setting $M$, $m$, $g_A$, $F$ to their physical values, choosing
$\mu=M$, and taking the estimates $c_1\approx -0.9\mbox{~GeV}^{-1}$,
$c_2\approx 3.3\mbox{~GeV}^{-1}$, $c_3\approx -4.7\mbox{~GeV}^{-1}$ from
\cite{Meissner:2005ba} 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 \eqref{res-gpd-0}, 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
\begin{align}
\partial_t \widetilde{A}_{n,k}(0) &= \widetilde{E}_{n,k}^{(2,t)}
   + O(m) \,,
\nonumber \\[0.15em]
\partial_t \widetilde{B}_{n,k}(0) &= \widetilde{M}_{n,k}^{(2,t)}
  - \widetilde{E}_{n,k}^{(2,t)} \big/2
  - \widetilde{E}_{n,k}^{(0)} \mskip 1.5mu\big/(32 M^2)
  + O(m) \,,
\nonumber \\[0.15em]
\partial_t A_{n,k}(0) &= E_{n,k}^{(2,t)}
  - \bigl( M_{n,k}^{(0)}-E_{n,k}^{(0)} \bigr) \big/(4 M^2) + O(m) \,,
\nonumber \\[0.15em]
\partial_t B_{n,k}(0) &=
  \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/(4 M^2) + O(m) \,,
\nonumber \\[0.3em]
\partial_t C_{n}(0) &=
  \partial_t E_{n}^{(1,\pi)}(0) + \partial_t E_{n}^{(2,\pi)}(0)
  + E_{n,n}^{(2,t)} + E_{n,n}^{(0)} \big/(4 M^2)
  + O(m)
\end{align}
with
\begin{align}
\partial_t M_{n}^{(2,\pi)}(0) &= {}-\frac{3g_A^2}{(4\pi F)^2}
  \left[\log\frac{m^2}{\mu^2}+1\right]\,
  \sum_{\substack{j=2\\ \mathrm{even}}}^{n}
  2^{-j} \frac{j}{j+1}\, A_{n,n-j}^{\pi\mskip 1.5mu (0)} \,,
\nonumber \\[0.1em]
\partial_t E_{n}^{(1,\pi)}(0) &= {}- \frac{M}{m}\,
  \frac{\pi g_A^2}{8\mskip 1.5mu (4\pi F)^2}\,
  \sum_{\substack{j=2\\ \mathrm{even}}}^{n}
  2^{-j}\, \frac{j\, (5j+14)}{(j+1) (j+3)}\,
  A_{n,n-j}^{\pi\mskip 1.5mu (0)} \,,
\nonumber \\[0.1em]
\partial_t E_{n}^{(2,\pi)}(0) &= {}- \frac{3g_A^2}{4\mskip 1.5mu (4\pi F)^2}\,
  \left[\log\frac{m^2}{\mu^2} +3\right]\,
  \sum_{\substack{j=2\\ \mathrm{even}}}^{n}
  2^{-j} \frac{j}{j+1}\, A_{n,n-j}^{\pi\mskip 1.5mu (0)}\,
\nonumber \\[0.1em]
 & \hspace{-4.2em}{}+ \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_{\substack{j=2\\ \mathrm{even}}}^{n}
  2^{-j} \frac{j\, (j+4)}{(j+1) (j+3)}\, A_{n,n-j}^{\pi\mskip 1.5mu (0)} \,.
\nonumber \\
\end{align}
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 $\partial_t
M_{2}^{(2,\pi)}(0) \approx 1.7\mbox{~GeV}^{-2}\, A_{2,0}^{\pi\mskip 1.5mu (0)}$ and
$\partial_t E_{2}^{(1,\pi)}(0)+\partial_t E_{2}^{(2,\pi)}(0) \approx
-(2.5 + 1.2)\mbox{~GeV}^{-2}\, A_{2,0}^{\pi\mskip 1.5mu (0)}$.  Numerically, the
term $\partial_t E_{n}^{(1,\pi)}(0)$ is thus important but not
extremely large at the physical point.

\section{Summary}
\label{sec:sum}

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
$O(q^{\mskip 1.5mu d})$ 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 \eqref{pow-count}.  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 $1/m$.

\section*{Acknowledgments}

We are grateful to U.-G.\ Mei{\ss}ner 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.