[a]Shohini Bhattacharya

# GPDs in asymmetric frames

Krzysztof Cichy  Martha Constantinou  Jack Dodson  Xiang Gao  Andreas Metz
Swagato Mukherjee  Aurora Scapellato  Fernanda Steffens  Yong Zhao

###### Abstract

It is often taken for granted that Generalized Parton Distributions (GPDs) are
defined in the "symmetric" frame, where the transferred momentum is
symmetrically distributed between the incoming/outgoing hadrons. However, such
frames pose computational challenges for the lattice QCD practitioners. In
these proceedings, we lay the foundation for lattice QCD calculations of GPDs
in "asymmetric" frames, where the transferred momentum is not symmetrically
distributed between the incoming/outgoing hadrons. The novelty of our work
relies on the parameterization of the matrix elements in terms of Lorentz-
invariant amplitudes, which not only helps in establishing relations between
the said frames but also helps in isolating higher-twist contaminations. As an
example, we focus on the unpolarized GPDs for spin-1/2 particles.

##  1 Introduction

Generalized Parton Distributions (GPDs) are the 3D generalizations of the
collinear Parton Distribution Functions (PDFs) [1, 2]. There are several
motivations to study GPDs:

  * •

For $\xi=0$ the Fourier transforms of the GPDs are related to the impact-
parameter distributions which provide information about the three-dimensional
distribution of partons — (one-dimensional) longitudinal momentum
distribution; (two-dimensional) transverse spatial distribution, see for
example Ref. [3].

  * •

Twist-2 GPDs are related to the total angular momentum of partons [1].

  * •

One should look for other ways to access GPDs because of the challenges
involved in their extraction through the processes of Deep Virtual Compton
Scattering (DVCS) [2] and meson production [4]. Challenges are caused by the
sensitivity of differential cross-sections to only $x$-integrals of GPDs, and
not GPDs themselves [1, 2]. Therefore, it is desirable to extract the
$x$-dependence of the GPDs from first principles within Lattice QCD. However,
for a very long time this was not possible because of time-dependence of these
quantities. As a result, all of the lattice calculations were limited to the
calculations of lowest Mellin moments of the GPDs, see Ref. [5]. In 2013,
there was a path-breaking proposal by X. Ji to calculate instead auxiliary
quantities called "quasi-GPDs" [6, 7, 8]. This approach relies on the
extraction of matrix elements for boosted hadrons involving spatially-
separated fields. Ever since this proposal, enormous progress has taken place,
see some reviews [9, 10, 11]. In fact, Ref. [12] provides the first-ever
lattice-QCD results of the unpolarized and helicity GPDs of the nucleon from
the quasi-distribution approach. Lattice QCD calculations have the potential
to not only provide insight into the experimentally-inaccessible features of
GPDs, but also help in extracting the "full" GPDs from the existing
experimental data.

##  2 Formalisms to calculate GPDs in asymmetric frames

###  2.1 Frames: Symmetric and asymmetric

Figure 1: Graphical representation of the two frames employed in this work.
Left plot: Symmetric frame. Right plot: Asymmetric frame.

The most widely used frame of reference to calculate GPDs is the symmetric
frame. For this frame, the momentum transfer is symmetrically distributed
between the incoming ($p_{i}$) and the outgoing hadrons ($p_{f}$) (see left
plot of Fig. 1). However, one can also think of a frame where the momentum
transfer is not equally shared between the incoming and outgoing hadrons, but
is rather exclusively applied to the incoming hadron (see, right plot of Fig.
1). Such a frame is known as an asymmetric frame.

Lattice calculations of GPDs has primarily been confined to symmetric frames.
However, such frames pose serious computational challenges because they
require separate calculation for each values of the momentum transfer
($\Delta$), resulting in increased computational costs. So the question that
we strive to address in this work is: Can we lay a formalism to systematically
perform lattice calculations of GPDs in asymmetric frames (which is expected
to be computationally less expensive)? In this work, we argue that there are
two approaches to solving this question. In the first approach, we will show
that it is possible to relate the two frames via an appropriate Lorentz
transformation. In the second approach, we will propose a Lorentz-covariant
decomposition of the lattice matrix elements in terms of Lorentz-invariant
(frame-independent) amplitudes. These amplitudes will then be used to make
connections between the two frames. As a byproduct, we will show that this
approach helps in identifying higher-twist contaminations which may be present
in quasi-GPDs at finite values of momentum.

###  2.2 Lorentz transformation approach

In this section, we explain the Lorentz transformation (LT) approach. First,
it is straight-forward to realize that a LT along the $z$-direction is not
optimal for lattice calculations because this requires a spatial operator
distance (say $z=(0,0_{\perp},z^{3}\neq 0)$) to pick up a temporal component
(that is $z\xrightarrow[]{\text{LT}}(z^{0}\neq 0,0_{\perp},z^{3})$). However,
a LT applied to any direction transverse to the $z$-axis does not change the
spatial nature of operator distances. This transformation is called as the
"transverse boost". We explain this by considering a transverse boost in the
$x$-direction and for the simplest case of zero skewness. The logic can be
generalized for any general transverse boost and for arbitrary values of
skewness.

We begin by relating the incoming state in the two frames,
$p_{i}^{s}=(E_{i}^{s},-\Delta^{1,s}/2,0,P^{3})$ and
$p_{i}^{a}=(E_{i}^{a},-\Delta^{1,a},0,P^{3})$. LT provides $p^{s}=\Lambda_{\rm
LT}\,p^{a}$,

$\displaystyle\begin{pmatrix}E_{i}^{s}\\\\[2.84544pt]
p_{i}^{1,s}\\\\[2.84544pt] p_{i}^{2,s}\\\\[2.84544pt]
p_{i}^{3,s}\end{pmatrix}$
$\displaystyle=\begin{pmatrix}\gamma&-\gamma\beta&0&0\\\\[2.84544pt]
-\gamma\beta&\gamma&0&0\\\\[2.84544pt] 0&0&1&0\\\\[2.84544pt]
0&0&0&1\end{pmatrix}\times\begin{pmatrix}E_{i}^{a}\\\\[2.84544pt]
-\Delta^{1,a}\\\\[2.84544pt] 0\\\\[2.84544pt] P^{3}\end{pmatrix}\,.$ (1)

This gives,

$\displaystyle E_{i}^{s}$
$\displaystyle=\gamma(E_{i}^{a}+\beta\Delta^{1,a})\,,$ (2)

and,

$\displaystyle p_{i}^{1,s}$ $\displaystyle=-\gamma(\beta
E_{i}^{a}+\Delta^{1,a})\quad\rightarrow\quad\Delta^{1,s}=2\gamma(\beta
E_{i}^{a}+\Delta^{1,a})\,.$ (3)

Similarly, the outgoing state in the two frames,
$p_{f}^{s}=(E_{f}^{s},\Delta^{1,s}/2,0,P^{3})$ and
$p_{f}^{a}=(E_{f}^{a},0,0,P^{3})$ can also be related. (Keep in mind that the
energies of the incoming and outgoing states are different in the asymmetric
frame.) We then find,

$\displaystyle E_{i}^{s}$ $\displaystyle=\gamma E_{f}^{a}\,,$ (4)

and,

$\displaystyle p_{f}^{1,s}$ $\displaystyle=-\gamma\beta
E_{f}^{a}\quad\rightarrow\quad\Delta^{1,s}=-2\gamma\beta E_{f}^{a}\,.$ (5)

From Eqs. (2) and (4), we find,

$\displaystyle\beta$
$\displaystyle=-\bigg{(}\dfrac{E_{i}^{a}-E_{f}^{a}}{\Delta^{1,a}}\bigg{)}\,.$
(6)

From Eqs. (3) and (5), we find,

$\displaystyle\beta$
$\displaystyle=-\dfrac{\Delta^{1,a}}{E_{i}^{a}+E_{f}^{a}}\,.$ (7)

Then, Eqs. (6) and (7) imply,

$\displaystyle\Delta^{1,a}=\sqrt{(E_{i}^{a})^{2}-(E_{f}^{a})^{2}}\,.$ (8)

Hence, $\beta$ can be written as,

$\displaystyle\beta=-\sqrt{\dfrac{E_{i}^{a}-E_{f}^{a}}{E_{i}^{a}+E_{f}^{a}}}<0\,.$
(9)

This implies $\Delta^{0,a}<0$, and

$\displaystyle\gamma=\dfrac{1}{\sqrt{1-\beta^{2}}}=\sqrt{\dfrac{E_{i}^{a}+E_{f}^{a}}{2E_{f}^{a}}}\,.$
(10)

Therefore, by using the expressions for $(\beta,\gamma)$, we can write down
uniquely the symmetric frame variables $(E_{i}^{s},\Delta^{1,s})$ in terms of
the asymmetric frame variables $(E_{i}^{a},E_{f}^{a},\Delta^{1,a})$: The
energy should be,

$\displaystyle E_{i}^{s}=\gamma
E_{f}^{a}=\sqrt{\dfrac{E_{f}^{a}(E_{i}^{a}+E_{f}^{a})}{2}}\,,$ (11)

and the transverse-momentum transfer,

$\displaystyle\Delta^{1,s}$ $\displaystyle=-2\gamma\beta E_{f}^{a}\,,$
$\displaystyle\text{or,}\quad\Delta^{1,s}$
$\displaystyle=2\sqrt{\dfrac{E_{f}^{a}(E_{i}^{a}-E_{f}^{a})}{2}}=2\sqrt{\dfrac{E_{f}^{a}}{2(E_{i}^{a}+E_{f}^{a})}}\,\Delta^{1,a}\,.$
(12)

We repeat that the above method can be generalized for
$\vec{\Delta}_{\perp}=(\Delta^{1},\Delta^{2})$ and for arbitrary values of
skewness.

Now that we have sketched the idea of how to relate the kinematical variables
between the two frames, we proceed to understand how the matrix elements
defining quasi-GPDs transform between the two frames. For this purpose, we
focus on spin-0 particles such as the pion. (The method can be generalized for
spin-1/2 particles.) The (unpolarized) pion GPD is defined as,

$\displaystyle F^{\mu}(z,P,\Delta)$ $\displaystyle=\langle
p_{f}|\bar{q}(-\tfrac{z}{2})\gamma^{\mu}\,{\cal
W}(-\tfrac{z}{2},\tfrac{z}{2})q(\tfrac{z}{2})|p_{i}\rangle\,.$ (13)

Here, ${\cal W}$ is a straight Wilson line required to make the correlator
gauge invariant. Historically, (unpolarized) quasi-GPDs have been defined
through matrix elements of the operator $\gamma^{0}$, see for instance Refs.
[13, 12]. By applying the transverse boost Eq. (1), we find that the matrix
element $\langle..\gamma^{0}..\rangle$ in the symmetric frame can be expressed
in terms of matrix elements of different operators
$\langle..(\gamma^{0}+\gamma^{1})..\rangle$ in the asymmetric frame,

$\displaystyle\langle p_{f}|\bar{q}(-\tfrac{z}{2})\gamma^{0}\,{\cal
W}(-\tfrac{z^{3}}{2},\tfrac{z^{3}}{2})\,q(\tfrac{z}{2})|p_{i}\rangle^{s}$
$\displaystyle=\gamma\langle p_{f}|\bar{q}(-\tfrac{z}{2})\gamma^{0}\,{\cal
W}(-\tfrac{z^{3}}{2},\tfrac{z^{3}}{2})\,q(\tfrac{z}{2})|p_{i}\rangle^{a}$
$\displaystyle-\gamma\beta\langle
p_{f}|\bar{q}(-\tfrac{z}{2})\gamma^{1}\,{\cal
W}(-\tfrac{z^{3}}{2},\tfrac{z^{3}}{2})\,q(\tfrac{z}{2})|p_{i}\rangle^{a}.$
(14)

This equation simply reflects how the $0^{\rm th}$ component of a 4-vector
changes under the Lorentz transformation Eq. (1). Therefore, this implies that
a transverse boost that fixes $(\beta,\gamma)$ (Eqs. (9) and (10)) allows for
an exact calculation of quasi-GPDs in the symmetric frame through matrix
elements of the asymmetric frame. However, Eq. (14) also shows that a quasi-
GPD defined through the operator $\gamma^{0}$ is not Lorentz invariant. In the
limit of a large momentum, we recover,

$\displaystyle\lim_{P^{3}\rightarrow\infty}\langle..\gamma^{0}..\rangle^{s}$
$\displaystyle\,\approx\,\langle..\gamma^{0}..\rangle^{a}+\mathcal{O}\bigg{(}\dfrac{1}{P^{3}}\bigg{)}\langle..\gamma^{1}..\rangle^{a}\,\rightarrow\,\langle..\gamma^{0}..\rangle^{a}\,,$
(15)

which means that the contribution from the matrix element
$\langle..\gamma^{1}..\rangle$ maybe viewed as a power correction at finite
values of momentum $P^{3}$.

###  2.3 Amplitude approach: Spin-$\boldsymbol{1/2}$ particles

In this section, we explain the amplitude approach through the example of
spin-1/2 particles, such as the proton. (We refer to Ref. [14] for details on
spin-0 particles.) As a first step, we build a Lorentz-covariant decomposition
of the vector matrix element in terms of the available vectors
$(P^{\mu},z^{\mu},\Delta^{\mu})$. By considering constraints from parity, we
find that the general structure of the vector matrix element involves eight
linearly-independent Dirac structures multiplied by eight Lorentz-invariant
(frame-independent) amplitudes,

$\displaystyle F^{\mu}(z,P,\Delta)$
$\displaystyle=\bar{u}(p_{f},\lambda^{\prime})\bigg{[}\dfrac{P^{\mu}}{m}A_{1}+mz^{\mu}A_{2}+\dfrac{\Delta^{\mu}}{m}A_{3}+im\sigma^{\mu
z}A_{4}+\dfrac{i\sigma^{\mu\Delta}}{m}A_{5}$
$\displaystyle+\dfrac{P^{\mu}i\sigma^{z\Delta}}{m}A_{6}+mz^{\mu}i\sigma^{z\Delta}A_{7}+\dfrac{\Delta^{\mu}i\sigma^{z\Delta}}{m}A_{8}\bigg{]}u(p_{i},\lambda)\,.$
(16)

Here
$\sigma^{\mu\nu}\equiv\tfrac{i}{2}(\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu})$,
$\sigma^{\mu z}\equiv\sigma^{\mu\rho}z_{\rho}$,
$\sigma^{\mu\Delta}\equiv\sigma^{\mu\rho}\Delta_{\rho}$,
$\sigma^{z\Delta}\equiv\sigma^{\rho\tau}z_{\rho}\Delta_{\tau}$,
$z\equiv(z^{0}=0,z_{\perp}=0_{\perp},z^{3}\neq 0)$. (For a derivation of Eq.
(2.3), we refer to Ref. [15]. See also Ref. [16] where the vector matrix
element has been parameterized in the momentum space for a straight Wilson
line.) For brevity, we use the compact notation $A_{i}\equiv A_{i}(z\cdot
P,z\cdot\Delta,\Delta^{2},z^{2})$, with $A_{i}$’s being the Lorentz-invariant
amplitudes whose arguments are functions of Lorentz scalars111In the
literature, the amplitudes have also been called generalized Ioffe time
distributions (ITDs) [13]..

For spin-$1/2$ particles, the vector matrix element can be parameterized in
terms of two light-cone GPDs $H$ and $E$ [17],

$\displaystyle F^{+}(z,P^{s/a},\Delta^{s/a})$
$\displaystyle=\bar{u}^{s/a}(p_{f}^{s/a},\lambda^{\prime})\bigg{[}\gamma^{+}H(z,P^{s/a},\Delta^{s/a})$
$\displaystyle\hskip
85.35826pt+\frac{i\sigma^{+\mu}\Delta^{s/a}_{\mu}}{2m}E(z,P^{s/a},\Delta^{s/a})\bigg{]}u^{s/a}(p^{s/a}_{i},\lambda)\,.$
(17)

By using $\mu=+$ in Eq. (2.3), followed by a subsequent change of basis, it is
possible to map the $A_{i}$’s onto the $H$ and $E$ GPDs in Eq. (17). The
results are,

$\displaystyle H(z,P^{s/a},\Delta^{s/a})$
$\displaystyle=A_{1}+\dfrac{\Delta^{+,s/a}}{P^{+,s/a}}A_{3}\,,$ (18)
$\displaystyle E(z,P^{s/a},\Delta^{s/a})$
$\displaystyle=-A_{1}-\dfrac{\Delta^{+,s/a}}{P^{+,s/a}}A_{3}+2A_{5}+2P^{+,s/a}z^{-}A_{6}+2\Delta^{+,s/a}z^{-}A_{8}\,.$
(19)

Keep in mind that the arguments of the $A_{i}$’s for light-cone GPDs have no
dependence on $z^{2}$. Also, $z^{\mu}=(0,z^{-},0_{\perp})$ and
$\Delta^{+}/P^{+}=z\cdot\Delta/z\cdot P$, etc. Thus, it is possible to write
the above expressions in a Lorentz invariant way as,

$\displaystyle H(z\cdot P^{s/a},z\cdot\Delta^{s/a},(\Delta^{s/a})^{2})$
$\displaystyle=A_{1}+\dfrac{\Delta^{s/a}\cdot z}{P^{s/a}\cdot z}A_{3}\,,$ (20)
$\displaystyle E(z\cdot P^{s/a},z\cdot\Delta^{s/a},(\Delta^{s/a})^{2})$
$\displaystyle=-A_{1}-\dfrac{\Delta^{s/a}\cdot z}{P^{s/a}\cdot
z}A_{3}+2A_{5}+2P^{s/a}\cdot zA_{6}+2\Delta^{s/a}\cdot zA_{8}\,.$ (21)

This means the light-cone GPDs are frame-independent as long as the Lorentz
scalars $(z\cdot P^{s/a},z\cdot\Delta^{s/a},(\Delta^{s/a})^{2})$ are the same
in the two frames.

Next, we turn to the quasi-GPDs $\mathcal{H}$ and $\mathcal{E}$, which
historically have been defined in terms of matrix elements of $\gamma^{0}$
operator as [18, 19],

$\displaystyle F^{0}(z,P^{s/a},\Delta^{s/a})$ $\displaystyle=\langle
p_{f}^{s/a},\lambda^{\prime}|\bar{q}(-\tfrac{z}{2})\gamma^{0}q(\tfrac{z}{2})|p^{s/a}_{i},\lambda\rangle$
$\displaystyle=\bar{u}^{s/a}(p_{f}^{s/a},\lambda^{\prime})\bigg{[}\gamma^{0}{\cal{H}}^{s/a}_{0}(z,P^{s/a},\Delta^{s/a})$
$\displaystyle\hskip
85.35826pt+\frac{i\sigma^{0\mu}\Delta^{s/a}_{\mu}}{2m}{\cal{E}}^{s/a}_{0}(z,P^{s/a},\Delta^{s/a})\bigg{]}u^{s/a}(p^{s/a}_{i},\lambda)\,.$
(22)

If we use $\mu=0$ in Eq. (2.3), then after performing a change of basis it is
possible to map the $A_{i}$’s onto the quasi-GPDs in Eq. (22). The relations
in the symmetric frame read,

$\displaystyle{\cal{H}}^{s}_{0}(z,P^{s},\Delta^{s})$
$\displaystyle=A_{1}+\dfrac{\Delta^{0,s}}{P^{0,s}}A_{3}-\dfrac{m^{2}\Delta^{0,s}z^{3}}{2P^{0,s}P^{3,s}}A_{4}+\bigg{[}\dfrac{(\Delta^{0,s})^{2}z^{3}}{2P^{3,s}}-\dfrac{\Delta^{0,s}\Delta^{3,s}z^{3}P^{0,s}}{2(P^{3,s})^{2}}-\dfrac{z^{3}(\Delta^{s}_{\perp})^{2}}{2P^{3,s}}\bigg{]}A_{6}$
$\displaystyle+\bigg{[}\dfrac{(\Delta^{0,s})^{3}z^{3}}{2P^{0,s}P^{3,s}}-\dfrac{(\Delta^{0,s})^{2}\Delta^{3,s}z^{3}}{2(P^{3,s})^{2}}-\dfrac{\Delta^{0,s}z^{3}(\Delta^{s}_{\perp})^{2}}{2P^{0,s}P^{3,s}}\bigg{]}A_{8}\,,$
(23) $\displaystyle{\cal{E}}^{s}_{0}(z,P^{s},\Delta^{s})$
$\displaystyle=-A_{1}-\dfrac{\Delta^{0,s}}{P^{0,s}}A_{3}+\dfrac{m^{2}\Delta^{0,s}z^{3}}{2P^{0,s}P^{3,s}}A_{4}+2A_{5}+\bigg{[}-\dfrac{(\Delta^{0,s})^{2}z^{3}}{2P^{3,s}}+\dfrac{P^{0,s}\Delta^{0,s}\Delta^{3,s}z^{3}}{2(P^{3,s})^{2}}+\dfrac{z^{3}(\Delta^{s}_{\perp})^{2}}{2P^{3,s}}$
$\displaystyle-\dfrac{2z^{3}(P^{0,s})^{2}}{P^{3,s}}\bigg{]}A_{6}+\bigg{[}-\dfrac{(\Delta^{0,s})^{3}z^{3}}{2P^{0,s}P^{3,s}}+\dfrac{(\Delta^{0,s})^{2}\Delta^{3,s}z^{3}}{2(P^{3,s})^{2}}+\dfrac{\Delta^{0,s}z^{3}(\Delta^{s}_{\perp})^{2}}{2P^{0,s}P^{3,s}}-\dfrac{2z^{3}P^{0,s}\Delta^{0,s}}{P^{3,s}}\bigg{]}A_{8}\,.$
(24)

On the other hand, the relations in the asymmetric frame read,

$\displaystyle{\cal{H}}^{a}_{0}(z,P^{a},\Delta^{a})=A_{1}+\dfrac{\Delta^{0,a}}{{P}^{0,a}}A_{3}-\bigg{[}\dfrac{m^{2}\Delta^{0,a}z^{3}}{2{P}^{0,a}{P}^{3,a}}-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\dfrac{m^{2}\Delta^{0,a}\Delta^{3,a}z^{3}}{4{P}^{0,a}({P}^{3,a})^{2}}\bigg{]}A_{4}$
$\displaystyle+\bigg{[}\dfrac{(\Delta^{0,a})^{2}z^{3}}{2{P}^{3,a}}-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\dfrac{(\Delta^{0,a})^{2}\Delta^{3,a}z^{3}}{4({P}^{3,a})^{2}}-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\dfrac{{P}^{0,a}\Delta^{0,a}\Delta^{3,a}z^{3}}{2({P}^{3,a})^{2}}-\dfrac{z^{3}(\Delta^{a}_{\perp})^{2}}{2{P}^{3,a}}\bigg{]}A_{6}$
$\displaystyle+\bigg{[}\dfrac{(\Delta^{0,a})^{3}z^{3}}{2{P}^{0,a}{P}^{3,a}}-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\dfrac{(\Delta^{0,a})^{3}\Delta^{3,a}z^{3}}{4{P}^{0,a}({P}^{3,a})^{2}}-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\dfrac{(\Delta^{0,a})^{2}\Delta^{3,a}z^{3}}{2({P}^{3,a})^{2}}-\dfrac{z^{3}(\Delta^{a}_{\perp})^{2}\Delta^{0,a}}{2{P}^{0,a}{P}^{3,a}}\bigg{]}A_{8}\,,$
(25)
$\displaystyle{\cal{E}}^{a}_{0}(z,P^{a},\Delta^{a})=-A_{1}-\dfrac{\Delta^{0,a}}{{P}^{0,a}}A_{3}-\bigg{[}-\dfrac{m^{2}\Delta^{0,a}z^{3}}{2{P}^{0,a}{P}^{3,a}}-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\bigg{(}\dfrac{m^{2}z^{3}}{{P}^{3,a}}-\dfrac{m^{2}\Delta^{0,a}\Delta^{3,a}z^{3}}{4{P}^{0,a}({P}^{3,a})^{2}}\bigg{)}\bigg{]}A_{4}+2A_{5}$
$\displaystyle+\bigg{[}-\dfrac{(\Delta^{0,a})^{2}z^{3}}{2{P}^{3,a}}-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\bigg{(}\dfrac{{P}^{0,a}\Delta^{0,a}z^{3}}{{P}^{3,a}}-\dfrac{(\Delta^{0,a})^{2}\Delta^{3,a}z^{3}}{4({P}^{3,a})^{2}}\bigg{)}-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\bigg{(}\dfrac{2z^{3}({P}^{0,a})^{2}}{{P}^{3,a}}$
$\displaystyle-\dfrac{{P}^{0,a}\Delta^{0,a}\Delta^{3,a}z^{3}}{2({P}^{3,a})^{2}}\bigg{)}+\dfrac{z^{3}(\Delta^{a}_{\perp})^{2}}{2{P}^{3,a}}\bigg{]}A_{6}+\bigg{[}-\dfrac{(\Delta^{0,a})^{3}z^{3}}{2{P}^{0,a}{P}^{3,a}}-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\bigg{(}\dfrac{(\Delta^{0,a})^{2}z^{3}}{{P}^{3,a}}-\dfrac{(\Delta^{0,a})^{3}\Delta^{3,a}z^{3}}{4\overline{P}^{0,a}({P}^{3,a})^{2}}\bigg{)}$
$\displaystyle-\dfrac{1}{(1+\tfrac{\Delta^{3,a}}{2{P}^{3,a}})}\bigg{(}\dfrac{2z^{3}{P}^{0,a}\Delta^{0,a}}{{P}^{3,a}}-\dfrac{(\Delta^{0,a})^{2}\Delta^{3,a}z^{3}}{2({P}^{3,a})^{2}}\bigg{)}+\dfrac{z^{3}(\Delta^{a}_{\perp})^{2}\Delta^{0,a}}{2{P}^{0,a}{P}^{3,a}}\bigg{]}A_{8}\,.$
(26)

However, one can think of other definitions of quasi-GPDs. For this purpose,
we recall the position-space matching relation between, for instance, light-
cone GPD $H$ and quasi-GPD $\mathcal{H}$ [13]:

$\displaystyle{\cal{H}}\big{(}z\cdot P,-2\xi(z\cdot
P),\Delta^{2},z^{2},\mu^{2}\big{)}$
$\displaystyle=\int_{-1}^{1}du\,\bar{C}\,(u,z\cdot
P,\xi,z^{2},\mu^{2})\,H\big{(}u(z\cdot P),-2u\xi(z\cdot
P),\Delta^{2},\mu^{2}\big{)}\,.$ (27)

Here, $\bar{C}$ is the pertubatively-calculable matching coefficient [13] and
$\mu$ is the renormalization scale in the $\overline{\rm MS}$ scheme. At
leading order in $\alpha_{s}$, the above formula indicates that ${\cal{H}}$
collapses to $H$ in the light-cone limit $z^{2}\to 0$,

$\displaystyle\lim_{z^{2}\to 0}{\cal{H}}(z\cdot
P,z\cdot\Delta,\Delta^{2},z^{2})=H(z\cdot P,z\cdot\Delta,\Delta^{2},0)+{\cal
O}(\alpha_{s})\,.$ (28)

Therefore, a natural way to define the quasi-GPDs ${\cal{H}}$ and ${\cal{E}}$
is through a Lorentz-invariant generalization of the light-cone definitions in
Eqs. (20) and (21) to $z^{2}\neq 0$, i.e.,

$\displaystyle{\cal H}(z\cdot
P^{s/a},z\cdot\Delta^{s/a},(\Delta^{s/a})^{2},z^{2})$
$\displaystyle=A_{1}+\dfrac{\Delta^{s/a}\cdot z}{P^{s/a}\cdot z}A_{3}\,,$ (29)
$\displaystyle{\cal E}(z\cdot
P^{s/a},z\cdot\Delta^{s/a},(\Delta^{s/a})^{2},z^{2})$
$\displaystyle=-A_{1}-\dfrac{\Delta^{s/a}\cdot z}{P^{s/a}\cdot
z}A_{3}+2A_{5}+2P^{s/a}\cdot zA_{6}+2\Delta^{s/a}\cdot zA_{8}\,,$ (30)

where now the arguments of the $A_{i}$’s have a non-zero dependence on
$z^{2}$. We expect the definitions in Eqs. (29-(30) to have two advantages:
First, these definitions may converge faster to the light-cone GPDs because of
the similarities in their functional forms with their (respective) light-cone
GPDs. (Such a statement is inspired from Ref. [20], where similar arguments
were made for the quasi-PDFs. See also the next paragraph for explicit
explanations.) Second, these definitions differ from their light-cone GPDs by
frame-independent power corrections; contrast with historic definitions which
are frame-dependent.

We now discuss in detail the various definitions of quasi-GPDs: We notice that
for finite values of the momentum, the historic definitions of quasi-GPDs
(${\cal{H}}^{s/a}_{0}(A_{i};z)\,,{\cal{E}}^{s/a}_{0}(A_{i};z)$) in Eqs.
(2.3)-(2.3) involve additional amplitudes that are not present in the light-
cone GPDs, Eqs. (20)-(21). This is not the case for the Lorentz-invariant
definitions of quasi-GPDs (${\cal{H}}(A_{i};z)\,,{\cal{E}}(A_{i};z)$) in Eqs.
(29)-(30). (Note that this is different from the (unpolarized) quasi-PDF case
where arguments were made in favor of $\gamma^{0}$ (against $\gamma^{3}$)
because of the absence of such additional amplitudes relative to the
(unpolarized) light-cone PDF case [20].) Therefore, the additional amplitudes
in (${\cal{H}}^{s/a}_{0}(A_{i};z)\,,{\cal{E}}^{s/a}_{0}(A_{i};z)$) may be
viewed as contaminations from explicit power corrections, which one would have
to suppress by going to larger and larger values of momentum. Hence, we
believe that (${\cal{H}}(A_{i};z)\,,{\cal{E}}(A_{i};z)$) may converge
relatively faster to their (respective) light-cone GPDs, simply because of the
absence of such additional amplitudes. (Of course,
(${\cal{H}}(A_{i};z)\,,{\cal{E}}(A_{i};z)$) also have power corrections, but
they are implicit within the amplitudes themselves. Our argument above is for
the power corrections that are explicit.) Our reasoning is perhaps too simple
and for sure needs further substantiation. In fact, it may be that the actual
convergence of the various definitions of quasi-GPDs is determined by the
underlying dynamics. Note that the Lorentz non-invariance of the historical
definitions of quasi-GPDs implies that the basis vectors
$(\gamma^{0},i\sigma^{0\Delta^{s/a}})$ do not form a complete set for
spatially-separated bi-local operators for finite values of momentum.
Therefore, we can argue that the Lorentz-invariant definitions are in fact
just a redefinition of quasi-GPDs in terms of a suitable linear combination of
operators (which turns out to be $\gamma_{\perp}$) that make them functions of
Lorentz scalars [14].

In Ref. [14] and [21], we compare numerically the different definitions of
quasi-GPDs for $\xi=0$ to get an idea about the relative size of power
corrections. Finally, we remark on the matching coefficient for the different
definitions of quasi-GPDs: It is known that the GPD matching coefficient for
the operator $\gamma^{0}$ reduces to that for the corresponding PDF when
$\xi=0$, even if $t\neq 0$ [13]. The PDF matching coefficient for $\gamma^{0}$
is for the amplitude $A_{1}$, which is also the only contributing amplitude to
the LI definition of the GPD when $\xi=0$. Therefore, the matching
coefficients for the $\gamma^{0}$ and the LI definitions of the GPDs are
equal. We will elaborate this point more, including the general case of
$\xi\neq 0$, in a forthcoming publication.

##  3 Summary

In these proceedings, we have laid down the theoretical tools to perform
lattice QCD calculations of GPDs in asymmetric frames. We have highlighted two
approaches to performing such calculations:

  * •

Lorentz transformation (LT) approach (Sec. 2.2): We have shown that there
exists a LT called the "transverse boost" (transverse with respect to the
Wilson Line) that allows one to uniquely relate the kinematical variables as
well as the matrix elements in the two frames.

  * •

Amplitude approach (Sec. 2.3): We have proposed a Lorentz-covariant
decomposition of the vector matrix element in terms of Lorentz-
invariant/frame-independent amplitudes. The amplitudes can be used as tools to
relate the two frames. This approach also shows that at finite values of the
boost momentum the historic definitions of quasi-GPDs (defined through
$\gamma^{0}$) have additional amplitudes that are not present in the light-
cone limit. This motivates us to come up with alternative definitions of
quasi-GPDs that may potentially converge faster. One such candidate can be the
case where one chooses the same functional form as the light-cone GPDs
subjected to include $z^{2}\neq 0$. Naively, because of the similarity in the
functional forms (or because of the absence of additional amplitudes), one may
expect such a definition of quasi-GPD to converge faster to the light-cone
GPD. Such a definition is also frame-independent, contrary to the historic
definitions.

Acknowledgements

This material is based upon work supported by the U.S. Department of Energy,
Office of Science, Office of Nuclear Physics through Contract No. DE-
SC0012704, No. DE-AC02-06CH11357 and within the framework of Scientific
Discovery through Advance Computing (SciDAC) award Fundamental Nuclear Physics
at the Exascale and Beyond (S. B. and S. M.). K. C. is supported by the
National Science Centre (Poland) grants SONATA BIS no. 2016/22/E/ST2/00013 and
OPUS no. 2021/43/B/ST2/00497. M. C., J. D. and A. S. acknowledge financial
support by the U.S. Department of Energy, Office of Nuclear Physics, Early
Career Award under Grant No. DE-SC0020405. J. D. also received support by the
U.S. Department of Energy, Office of Science, Office of Nuclear Physics,
within the framework of the TMD Topical Collaboration. The work of A. M. has
been supported by the National Science Foundation under grant number
PHY-2110472, and also by the U.S. Department of Energy, Office of Science,
Office of Nuclear Physics, within the framework of the TMD Topical
Collaboration. F. S. was funded by by the NSFC and the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation) through the funds
provided to the Sino-German Collaborative Research Center TRR110 “Symmetries
and the Emergence of Structure in QCD” (NSFC Grant No. 12070131001, DFG
Project-ID 196253076 - TRR 110). YZ was partially supported by an LDRD
initiative at Argonne National Laboratory under Project No. 2020-0020.
Computations for this work were carried out in part on facilities of the USQCD
Collaboration, which are funded by the Office of Science of the U.S.
Department of Energy. This research used resources of the Oak Ridge Leadership
Computing Facility, which is a DOE Office of Science User Facility supported
under Contract DE-AC05-00OR22725. This research was supported in part by
PLGrid Infrastructure (Prometheus supercomputer at AGH Cyfronet in Cracow).
Computations were also partially performed at the Poznan Supercomputing and
Networking Center (Eagle supercomputer), the Interdisciplinary Centre for
Mathematical and Computational Modelling of the Warsaw University (Okeanos
supercomputer), and at the Academic Computer Centre in Gdańsk (Tryton
supercomputer). The gauge configurations have been generated by the Extended
Twisted Mass Collaboration on the KNL (A2) Partition of Marconi at CINECA,
through the Prace project Pra13_3304 “SIMPHYS". Inversions were performed
using the DD-$\alpha$AMG solver [22] with twisted mass support [23].

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