[a]K. Utku Can

# The Compton amplitude and nucleon structure functions

for the QCDSF/UKQCD Collaboration

###### Abstract

Structure functions are the essential objects for understanding the deep
inelastic scattering processes, providing valuable insight into the partonic
structure of hadrons. A direct calculation of the Compton amplitude provides a
complementary way to accessing the structure functions, circumventing the
operator mixing and renormalisation issues of the standard operator product
expansion approach. We describe the connection between the Compton amplitude
and the structure functions, and describe a Feynman-Hellmann approach to
calculate the amplitude directly. As an application, we extract the moments of
transverse and longitudinal proton structure functions and study the power
corrections.

##  1 Introduction

Calculating the structure functions from first principals poses several
challenges for lattice QCD practitioners. Traditionally, lattice calculations
make use of the operator product (OPE) expansion. However, it is known that in
the OPE approach, contributions of leading-twist operators are inseparably
connected with the contributions from operators of higher twist, due to
operator mixing and renormalisation [1]. In recent years, the focus has been
on light-cone parton distribution functions (PDFs) accessible from the quasi-
PDF approach introduced by Ji [2], which enables a direct investigation of the
$x$-dependence of parton distributions. A detailed account of the quasi-PDF
and related approaches is given in recent reviews [3, 4] and presented in
plenary talks at the lattice conferences [5, 6], highlighting the immense
efforts and the progress of the lattice community. The operator mixing issue,
however, remains a concern [1, 7, 8]. The majority of the investigations are
limited to the leading-twist contributions, with fewer works on twist-3
contributions [9, 10, 11].

In this contribution we describe an alternative and complementary approach
that is being pursued by the QCDSF/UKQCD Collaboration, which is to directly
calculate the forward Compton amplitude on the lattice in the space-like
region. While the Compton amplitude is a 4-point correlation function, via an
application of the Feynman-Hellmann approach we reduce this problem to a more
straightforward analysis of 2-point functions. By working with the physical
amplitude, the operator mixing and renormalization issues, and the restriction
to light-cone operators are circumvented. Given the Compton amplitude is known
sufficiently accurately, we can expect to estimate the power corrections in
structure functions, i.e. quantify the target mass corrections and estimate
the contributions from higher-twist operators, which could be useful for
global PDF analyses. In principle, the $x$-dependence of the structure
functions can be recovered [12], although in practice this requires tackling
an inverse-problem [13]. Although our focus is the Compton amplitude in
forward kinematics, this approach is applicable to off-forward kinematics
enabling an investigation of the generalised parton distributions [14, 15,
16].

We give an overview of the relation between the Compton amplitude and the
moments of structure functions in Section 2, followed by a summary of the
application of the Feynman-Hellmann theorem in Section 3. The technical
details follow in Sections 4 and 5. We present some selected results on the
moments of nucleon structure functions and discuss the power corrections in
Section 6.

##  2 The Compton tensor and the moments of structure functions

The starting point is the forward Compton amplitude described by the time
ordered product of electromagnetic currents sandwiched between nucleon states,

$\displaystyle T_{\mu\nu}(p,q)=$ $\displaystyle\int d^{4}z\,e^{iq\cdot
z}\rho_{ss^{\prime}}{\left\langle
p,s^{\prime}\left|\mathcal{T}\left\\{\mathcal{J}_{\mu}(z)\mathcal{J}_{\nu}(0)\right\\}\right|p,s\right\rangle},$
(1)

where $p$ is the momentum and $s$ is the spin of the nucleon, $q$ is the
momentum of the virtual photon, and $\rho$ is the polarisation density matrix.
For parity-conserving processes that involve conserved currents the Compton
tensor is parametrised in terms of four Lorentz-invariant scalar functions,
$\mathcal{F}_{1}$, $\mathcal{F}_{2}$, $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$
as follows

$\displaystyle T_{\mu\nu}(p,q)$
$\displaystyle=T_{\\{\mu\nu\\}}(p,q)+T_{[\mu\nu]}(p,q)$ (2) $\displaystyle
T_{\\{\mu\nu\\}}(p,q)$
$\displaystyle=\left(-g_{\mu\nu}+\frac{q_{\mu}q_{\nu}}{q^{2}}\right)\mathcal{F}_{1}(\omega,Q^{2})+\left(p_{\mu}-\frac{p\cdot
q}{q^{2}}q_{\mu}\right)\left(p_{\nu}-\frac{p\cdot
q}{q^{2}}q_{\nu}\right)\frac{\mathcal{F}_{2}(\omega,Q^{2})}{p\cdot q}$ (3)
$\displaystyle T_{[\mu\nu]}(p,q)$
$\displaystyle=i\varepsilon^{\mu\nu\alpha\beta}\frac{q_{\alpha}}{p\cdot
q}\left[\mathcal{G}_{1}(\omega,Q^{2})s_{\beta}+\mathcal{G}_{2}(\omega,Q^{2})\left(s_{\beta}-\frac{s\cdot
q}{p\cdot q}p_{\beta}\right)\right],$ (4)

where we have separated the symmetric and antisymmetric parts, and
$Q^{2}=-q^{2}$ and $\omega=2(p\cdot q)/Q^{2}$. Here, $\varepsilon^{0123}=1$,
and $s_{\mu}$ is the spin vector of the polarised target satisfying
$s^{2}=-M^{2}$ ($M$ being the mass of the nucleon) and $s\cdot p=0$.

The Compton structure functions are related to the corresponding ordinary
structure functions via the optical theorem, which states

$\displaystyle\operatorname{Im}\mathcal{F}_{1,2}(\omega,Q^{2})=2\pi
F_{1,2}(x,Q^{2}),$ (5)
$\displaystyle\operatorname{Im}\mathcal{G}_{1,2}(\omega,Q^{2})=2\pi
g_{1,2}(x,Q^{2}).$ (6)

Making use of analyticity, crossing symmetry and the optical theorem, we can
write dispersion relations for $\mathcal{F}$ and $\mathcal{G}$ and connect
them to the inelastic structure functions,

$\displaystyle\overline{\mathcal{F}}_{1}(\omega,Q^{2})=2\omega^{2}\int_{0}^{1}dx\frac{2x\,F_{1}(x,Q^{2})}{1-x^{2}\omega^{2}-i\epsilon},\quad\mathcal{F}_{2}(\omega,Q^{2})=4\omega\int_{0}^{1}dx\,\frac{F_{2}(x,Q^{2})}{1-x^{2}\omega^{2}-i\epsilon},$
(7)
$\displaystyle\mathcal{G}_{1}(\omega,Q^{2})=4\omega\int_{0}^{1}dx\frac{g_{1}(x,Q^{2})}{1-x^{2}\omega^{2}-i\epsilon},\quad\mathcal{G}_{2}(\omega,Q^{2})=4\omega\int_{0}^{1}dx\,\frac{g_{2}(x,Q^{2})}{1-x^{2}\omega^{2}-i\epsilon},$
(8)

where we will use
$\overline{\mathcal{F}}_{i}(\omega,Q^{2})=\mathcal{F}_{i}(\omega,Q^{2})-\mathcal{F}_{i}(0,Q^{2})$
throughout to denote a once subtracted function. Additionally, a once-
subtracted dispersion relation for the longitudinal structure function
$F_{L}(x)$ is written as,

$\overline{\mathcal{F}}_{L}(\omega,Q^{2})\equiv\mathcal{F}_{L}(\omega,Q^{2})+\mathcal{F}_{1}(0,Q^{2})=\frac{8M_{N}^{2}}{Q^{2}}\int_{0}^{1}dxF_{2}(x,Q^{2})+2\omega^{2}\int_{0}^{1}dx\frac{F_{L}(x,Q^{2})}{1-x^{2}\omega^{2}-i\epsilon},$
(9)

where,

$\displaystyle F_{L}(x,Q^{2})$
$\displaystyle=\left(1+\frac{4M_{N}^{2}}{Q^{2}}x^{2}\right)F_{2}(x,Q^{2})-2xF_{1}(x,Q^{2}),$
(10)

with $M_{N}$ the mass of the nucleon. Note that a subtraction is necessary,
given the high-energy behaviour of $F_{1}$. Although we are only concerned
with subtracting it away, understanding the subtraction function is an
interesting subject in itself. Related discussions on the subtraction function
can be found in [17, 18, 19, 20]. As $Q^{2}\to\infty$, Equation 10 reduces to
the familiar Callan-Gross relation, $F_{L}(x)\to F_{2}(x)-2xF_{1}(x)$, which
vanishes in the quark-parton model.

Expanding the integrands in Equations 7, 8 and 9 at fixed $Q^{2}$ as a
geometric series, the Compton structure functions can be expressed as infinite
sums over the Mellin moments of the inelastic structure functions,

$\displaystyle\overline{\mathcal{F}}_{1}(\omega,Q^{2})$
$\displaystyle=\sum_{n=1}^{\infty}2\omega^{2n}M^{(1)}_{2n}(Q^{2}),$
$\displaystyle\text{with}\;M^{(1)}_{2n}(Q^{2})=2\int_{0}^{1}dx\,x^{2n-1}F_{1}(x,Q^{2}),$
(11) $\displaystyle\mathcal{F}_{2}(\omega,Q^{2})$
$\displaystyle=\sum_{n=1}^{\infty}4\omega^{2n-1}M^{(2)}_{2n}(Q^{2}),$
$\displaystyle\text{with}\;M^{(2)}_{2n}(Q^{2})=\int_{0}^{1}dx\,x^{2n-2}F_{2}(x,Q^{2}),$
(12) $\displaystyle\overline{\mathcal{F}}_{L}(\omega,Q^{2})$
$\displaystyle=\sum_{n=1}^{\infty}2\omega^{2n}M^{(L)}_{2n}(Q^{2}),$
$\displaystyle\text{with}\;M^{(L)}_{2n}(Q^{2})=\int_{0}^{1}dx\,x^{2n-2}F_{L}(x,Q^{2}),$
(13) $\displaystyle\mathcal{G}_{1,2}(\omega,Q^{2})$
$\displaystyle=\sum_{n=1}^{\infty}4\omega^{2n-1}\tilde{M}^{(1,2)}_{2n}(Q^{2}),$
$\displaystyle\text{with}\;\tilde{M}^{(1,2)}_{2n}(Q^{2})=\int_{0}^{1}dx\,x^{2n-2}g_{1,2}(x,Q^{2}).$
(14)

We note that the physical moments $M_{2n}$ that appear in Equations 11, 12, 13
and 14 are dominated by their leading-twist contributions, i.e. the moments of
PDFs, at asymptotically large $Q^{2}$.

Finally, the unpolarised Compton structure functions $\mathcal{F}_{1}$ and
$\mathcal{F}_{2}$ are accessed from the symmetric part of the Compton tensor
via,

$\displaystyle\mathcal{F}_{1}(\omega,Q^{2})$
$\displaystyle=T_{\\{33\\}}(p,q),$
$\displaystyle\text{for}\,\mu=\nu=3\,\text{and}\,p_{3}=q_{3}=0,$ (15)
$\displaystyle\frac{\mathcal{F}_{2}(\omega,Q^{2})}{\omega}$
$\displaystyle=\frac{Q^{2}}{2E_{N}^{2}}\left[T_{\\{00\\}}(p,q)+T_{\\{33\\}}(p,q)\right],$
$\displaystyle\text{for}\,\mu=\nu=0\,\text{and}\,p_{3}=q_{3}=q_{0}=0,$ (16)

and the longitudinal Compton structure function $\mathcal{F}_{L}$ is
constructed as,

$\mathcal{F}_{L}(\omega,Q^{2})=-\mathcal{F}_{1}(\omega,Q^{2})+\frac{\omega}{2}\mathcal{F}_{2}(\omega,Q^{2})+\frac{2M_{N}^{2}}{Q^{2}}\frac{\mathcal{F}_{2}(\omega,Q^{2})}{\omega}.$
(17)

We discuss how to access $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ in Section 5.

##  3 The Feynman-Hellmann technique

An analysis of the Compton amplitude requires the evaluation of lattice
4-point correlation functions. However, this is not an easy task given the
rapid deterioration of the signal for large time separations and the
contamination due to excited states. The application of the Feynman-Hellmann
theorem reduces the problem to a simpler analysis of 2-point correlation
functions using the established techniques of spectroscopy. Our implementation
of the second order Feynman-Hellmann method is presented in detail in [21].
Here, we outline its main aspects.

We modify the fermion action with the following perturbing term,

$S(\lambda)=S+\lambda\int
d^{3}z\operatorname{cos}({\mathbf{q}}\cdot{\mathbf{z}})\,\mathcal{J}_{\mu}(z),$
(18)

where $\lambda$ is the strength of the coupling between the quarks and the
external field, $\mathcal{J}_{\mu}(z)=Z_{V}\bar{q}(z)\gamma_{\mu}q(z)$ is the
renormalised electromagnetic current coupling to the quarks along the $\mu$
direction, ${\mathbf{q}}$ is the external momentum inserted by the current and
$Z_{V}$ is the renormalization constant for the local electromagnetic current.
The perturbation is introduced on the valence quarks only, hence only quark-
line connected contributions are taken into account in this work. For the
perturbation of valence and sea quarks see [22].

The main strategy to derive the relation between the energy shift and the
matrix element is to work out the second-order derivatives of the two-point
correlation function with respect to the external field from two complementary
perspectives. A two-point correlation function projected to definite momentum
in the presence of an external field is defined as,

$G_{\lambda}^{(2)}({\mathbf{p}};t;{\boldsymbol{\Gamma}})\equiv\int
d^{3}xe^{-i{\mathbf{p}}\cdot{\mathbf{x}}}{\boldsymbol{\Gamma}}\langle\Omega_{\lambda}|\chi({\mathbf{x}},t)\bar{\chi}(0)|\Omega_{\lambda}\rangle,$
(19)

where ${\mathbf{\Gamma}}$ is the spin-parity projection matrix and
$|\Omega_{\lambda}\rangle$ is the vacuum in the presence of the external
field. The asymptotic behaviour of the correlator at large Euclidean time
takes the familiar form,

$G^{(2)}_{\lambda}({\mathbf{p}};t;{\boldsymbol{\Gamma}})\simeq
A_{\lambda}({\mathbf{p}})e^{-E_{N_{\lambda}}({\mathbf{p}})\,t},$ (20)

where $E_{N_{\lambda}}({\mathbf{p}})$ is the energy of the ground state
nucleon in the external field and $A_{\lambda}({\mathbf{p}})$ the
corresponding overlap factor. Differentiating the perturbed nucleon correlator
(Equation 20) with respect to energy, one finds a distinct temporal signature
for the second-order energy shift,

$\displaystyle\frac{\partial^{2}G^{(2)}_{\lambda}({\mathbf{p}};t)}{\partial\lambda^{2}}\bigg{|}_{\lambda=0}=$
$\displaystyle\left(\frac{\partial^{2}A_{\lambda}({\mathbf{p}})}{\partial\lambda^{2}}-tA({\mathbf{p}})\frac{\partial^{2}E_{N_{\lambda}}({\mathbf{p}})}{\partial\lambda^{2}}\right)e^{-E_{N}({\mathbf{p}})t},$
(21)

where we have assumed that first-order perturbations of the energy vanish, as
ensured by avoiding Breit-frame kinematics. The derivatives of
$A_{\lambda}({\mathbf{p}})$ and $E_{N_{\lambda}}({\mathbf{p}})$ are assumed to
be evaluated at $\lambda=0$. The first term corresponds to the shift in the
overlap factor and the second order energy shift is identified in the
$t$-enhanced (or time-enhanced) term.

By differentiating Equation 19 twice with respect to $\lambda$ in the path
integral formalism and evaluated at $\lambda\to 0$, we find

$\displaystyle\left.\frac{\partial^{2}G^{(2)}_{\lambda}({\mathbf{p}};y)}{\partial\lambda^{2}}\right|_{\lambda=0}=$
$\displaystyle\int
d^{3}x\,e^{-i{\mathbf{p}}\cdot{\mathbf{x}}}{\mathbf{\Gamma}}\left[{\left\langle\chi({\mathbf{x}},t)\overline{\chi}(0)\left(\frac{\partial
S(\lambda)}{\partial\lambda}\right)^{2}\right\rangle}+\cdots\right],$ (22)

where ellipsis denote the terms that do not lead to a time-enhanced term.
Inserting the explicit form of the external electromagnetic current (Equation
18) and following the algebra, we arrive at,

$\displaystyle\left.\frac{\partial^{2}G^{(2)}_{\lambda}({\mathbf{p}};t)}{\partial\lambda^{2}}\right|_{\lambda=0}=tA({\mathbf{p}})\frac{e^{-E_{N}({\mathbf{p}})t}}{2E_{N}({\mathbf{p}})}\left\langle
N({\mathbf{p}})\left|\int d^{4}z\left(e^{iq\cdot z}+e^{-iq\cdot
z}\right)\mathcal{J}_{\mu}(z)\mathcal{J}_{\mu}(0)\right|N({\mathbf{p}})\right\rangle+\ldots,$
(23)

where the subleading terms are suppressed by the ellipsis.

Finally, matching the time-enhanced terms of this form with Equation 21, one
arrives at the desired relation between the energy shift and the matrix
element describing the Compton amplitude,

$\left.\frac{\partial^{2}E_{N_{\lambda}}({\mathbf{p}})}{\partial\lambda^{2}}\right|_{\lambda=0}=-\frac{T_{\mu\mu}(p,q)+T_{\mu\mu}(p,-q)}{2E_{N}({\mathbf{p}})},$
(24)

where $T$ is the Compton amplitude defined in Equation 1. Equation 24 is the
principal relation that we use to access the flavour-diagonal, i.e. $uu$ and
$dd$, pieces of the symmetric part (Equation 3) of the Compton amplitude.

The above derivation can be generalised to mixed currents by including
additional perturbing terms in Equation 18 to study the flavour-mixed, i.e.
$ud$, piece, and the antisymmetric part (Equation 4) of the Compton tensor. On
that account, we make the modifications,

$\displaystyle S(\lambda)$ $\displaystyle=S+\lambda_{1}\int
d^{3}z\operatorname{cos}({\mathbf{q}}\cdot{\mathbf{z}})\,\mathcal{J}_{\mu}(z)+\lambda_{2}\int
d^{3}y\operatorname{cos}({\mathbf{q}}\cdot{\mathbf{y}})\,\mathcal{J}_{\mu}(y),$
(25) $\displaystyle S(\lambda)$ $\displaystyle=S+\lambda_{1}\int
d^{3}z\operatorname{cos}({\mathbf{q}}\cdot{\mathbf{z}})\,\mathcal{J}_{\mu}(z)+\lambda_{2}\int
d^{3}y\operatorname{sin}({\mathbf{q}}\cdot{\mathbf{y}})\,\mathcal{J}_{\nu}(y),$
(26)

to access the flavour-mixed and flavour-diagonal antisymmetric pieces of the
amplitude, respectively. Consequently, expressions analogous to Equation 24
are,

$\displaystyle\left.\frac{\partial^{2}E_{N_{\lambda}}({\mathbf{p}})}{\partial\lambda_{1}\partial\lambda_{2}}\right|_{{\boldsymbol{\lambda}}=0}=-\frac{T_{\mu\mu}(p,q)+T_{\mu\mu}(p,-q)}{2E_{N}({\mathbf{p}})},$
(27)
$\displaystyle\left.\frac{\partial^{2}E_{N_{\lambda}}({\mathbf{p}})}{\partial\lambda_{1}\partial\lambda_{2}}\right|_{{\boldsymbol{\lambda}}=0}=\frac{T_{\mu\nu}(p,q)-T_{\mu\nu}(p,-q)}{2E_{N}({\mathbf{p}})},$
(28)

corresponding to the modifications in Equations 25 and 26, respectively, where
the crossing relations $T_{\mu\mu}(p,q)=T_{\mu\mu}(p,-q)$ and
$T_{\mu\nu}(p,q)=-T_{\mu\nu}(p,-q)$ ensure that we have non-vanishing Feynman-
Hellmann relations.

##  4 Extracting the energy shifts

We can expand the perturbed energy in the limit $\lambda\to 0$,

$\displaystyle E_{N_{\lambda}}({\mathbf{p}})$
$\displaystyle=E_{N}({\mathbf{p}})+\lambda\left.\frac{\partial
E_{N_{\lambda}}({\mathbf{p}})}{\partial\lambda}\right|_{\lambda=0}+\frac{\lambda^{2}}{2!}\left.\frac{\partial^{2}E_{N_{\lambda}}({\mathbf{p}})}{\partial\lambda^{2}}\right|_{\lambda=0}+\mathcal{O}(\lambda^{3})$
(29) $\displaystyle=E_{N}({\mathbf{p}})+\Delta
E^{e}_{N_{\lambda}}({\mathbf{p}})+\Delta E^{o}_{N_{\lambda}}({\mathbf{p}}),$
(30)

considering a single current insertion (e.g. Equation 18), where we have
collected the terms even ($e$) and odd ($o$) in $\lambda$ to all orders in
writing the second line. A similar Taylor expansion of the perturbed energy
can be written for the double current insertion (e.g. Equation 25),

$E_{N_{\lambda}}({\mathbf{p}})=E_{N}({\mathbf{p}})+\Delta
E^{eo}_{N_{\lambda}}({\mathbf{p}})+\Delta
E^{oe}_{N_{\lambda}}({\mathbf{p}})+\Delta
E^{ee}_{N_{\lambda}}({\mathbf{p}})+\Delta E^{oo}_{N_{\lambda}}({\mathbf{p}}),$
(31)

where the term of interest is,

$\Delta
E^{oo}_{N_{\lambda}}({\mathbf{p}})=\lambda_{1}\lambda_{2}\left.\frac{\partial^{2}E_{N_{\lambda}}({\mathbf{p}})}{\partial\lambda_{1}\partial\lambda_{2}}\right|_{\lambda=0}+\mathcal{O}(\lambda_{1}\lambda_{2}^{3})+\mathcal{O}(\lambda_{1}^{3}\lambda_{2}),$
(32)

with respect to Equations 27 and 28.

We construct the ratios,

$\displaystyle\mathcal{R}^{e}_{\lambda}({\mathbf{p}};t;\Gamma_{4})$
$\displaystyle\equiv\frac{G^{(2)}_{+\lambda}({\mathbf{p}};t;\Gamma_{4})G^{(2)}_{-\lambda}({\mathbf{p}};t;\Gamma_{4})}{\left(G^{(2)}({\mathbf{p}};t;\Gamma_{4})\right)^{2}}\xrightarrow{t\gg
0}A_{\lambda}({\mathbf{p}})e^{-2\Delta E^{e}_{N_{\lambda}}({\mathbf{p}})\,t},$
(33)
$\displaystyle\mathcal{R}^{oo}_{\lambda}({\mathbf{p}};t;{\boldsymbol{\Gamma}})$
$\displaystyle\equiv\frac{G^{(2)}_{(+\lambda,+\lambda)}({\mathbf{p}};t;{\boldsymbol{\Gamma}})G^{(2)}_{(-\lambda,-\lambda)}({\mathbf{p}};t;{\boldsymbol{\Gamma}})}{G^{(2)}_{(+\lambda,-\lambda)}({\mathbf{p}};t;{\boldsymbol{\Gamma}})G^{(2)}_{(-\lambda,+\lambda)}({\mathbf{p}};t;{\boldsymbol{\Gamma}})}\xrightarrow{t\gg
0}A_{\lambda}({\mathbf{p}})e^{-4\Delta
E^{oo}_{N_{\lambda}}({\mathbf{p}})\,t},$ (34)

in order to extract the second-order energy shifts, where
$A_{\lambda}({\mathbf{p}})$ is the overlap factor, which is irrelevant for the
rest of the discussion. These ratios isolate the energy shifts only at even
orders of $\lambda$. Here, $G^{(2)}_{\lambda}$ and
$G^{(2)}_{(\lambda_{1},\lambda_{2})}$ are the perturbed two-point functions
with $|\lambda_{1}|=|\lambda_{2}|=|\lambda|$, and $G^{(2)}$ is the unperturbed
one. The spin-parity projection matrices are defined as
$\Gamma_{4}\equiv(1+\gamma_{4})/2$ for an unpolarised positive-parity nucleon,
and
$\Gamma_{+}^{\hat{e}}\equiv\Gamma_{4}(1+{\boldsymbol{\hat{e}}}\cdot{\boldsymbol{\gamma}}\gamma_{5})/2$
for a spin-up positive-parity nucleon polarised along the $\hat{e}$ direction.
Equation 33 is used for the flavour-diagonal pieces of the symmetric part of
the Compton amplitude, i.e. $\mathcal{F}^{(uu,dd)}_{(1,2)}$, while Equation 34
with ${\boldsymbol{\Gamma}}=\Gamma_{4}$ is used for the flavour-mixed piece,
i.e. $\mathcal{F}^{ud}_{(1,2)}$.

##  5 Separating the $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$

Accessing the polarised Compton structure functions $\mathcal{G}_{1}$ and
$\mathcal{G}_{2}$ is more involved. By choosing the currents along the
$\mu=1$, and $\nu=2$ directions, and adopting the kinematics $q_{1}=p_{1}=0$,
we cancel the symmetric part (Equation 3) and isolate the spin dependent part
of the tensor,

$T_{[12]}(p,q)=i\varepsilon^{1230}\frac{q_{3}}{p\cdot
q}\left[\mathcal{G}_{1}(\omega,Q^{2})s_{0}+\mathcal{G}_{2}(\omega,Q^{2})\left(s_{0}-\frac{s\cdot
q}{p\cdot q}p_{0}\right)\right],$ (35)

where we are left with the choices $\alpha=3$ and $\beta=0$ to have a non-
vanishing amplitude since $q_{0}=0$ by construct in the Feynman-Hellmann
approach. We rearrange Equation 35 by using the four-spin vector in a boosted
frame,

$s^{\mu}({\mathbf{p}})\equiv\left(\frac{{\mathbf{{\boldsymbol{\hat{e}}}}}\cdot{\mathbf{p}}}{M_{N}},{\mathbf{{\boldsymbol{\hat{e}}}}}+\frac{{\mathbf{{\boldsymbol{\hat{e}}}}}\cdot{\mathbf{p}}}{M_{N}(E_{N}({\mathbf{p}})+M_{N})}{\mathbf{p}}\right),$
(36)

where ${\mathbf{{\boldsymbol{\hat{e}}}}}=\sigma M_{N}\hat{n}$, with $\hat{n}$
the quantisation axis and $\sigma=+1$ for spin up, along with the
substitutions $\varepsilon^{0123}=1$, $p_{0}=E_{N}({\mathbf{p}})$, $a\cdot
q\equiv a^{\mu}q_{\mu}=-{\mathbf{a}}\cdot{\mathbf{q}}$, and
$\omega=2({\mathbf{p}}\cdot{\mathbf{q}})/Q^{2}$, into a more compact form,

$T^{\hat{e}}_{[12]}(p,q)=C_{1}({\mathbf{{\boldsymbol{\hat{e}}}}},{\mathbf{p}},{\mathbf{q}})\frac{\mathcal{G}_{1}(\omega,Q^{2})}{\omega}+C_{2}({\mathbf{{\boldsymbol{\hat{e}}}}},{\mathbf{p}},{\mathbf{q}})\frac{\mathcal{G}_{2}(\omega,Q^{2})}{\omega},$
(37)

with the coefficients,

$\displaystyle
C_{1}({\mathbf{{\boldsymbol{\hat{e}}}}},{\mathbf{p}},{\mathbf{q}})$
$\displaystyle=i\frac{2q_{3}}{Q^{2}}\frac{{\mathbf{{\boldsymbol{\hat{e}}}}}.{\mathbf{p}}}{M_{N}},$
(38) $\displaystyle
C_{2}({\mathbf{{\boldsymbol{\hat{e}}}}},{\mathbf{p}},{\mathbf{q}})$
$\displaystyle=i\frac{2q_{3}}{Q^{2}}\left[\frac{{\mathbf{{\boldsymbol{\hat{e}}}}}.{\mathbf{p}}}{M_{N}}-E_{N}({\mathbf{p}})\left(\frac{{\mathbf{{\boldsymbol{\hat{e}}}}}.{\mathbf{p}}}{M_{N}(E_{N}({\mathbf{p}})+M_{N})}+\frac{{\mathbf{{\boldsymbol{\hat{e}}}}}\cdot{\mathbf{q}}}{{\mathbf{p}}\cdot{\mathbf{q}}}\right)\right].$
(39)

Note that we have introduced a superscript $\hat{e}$ in Equation 37 to keep
track of the spin polarisations. Once we extract the energy shifts, $\Delta
E_{N_{\lambda}}^{oo}({\mathbf{p}})$, using Equation 34 with
${\boldsymbol{\Gamma}}=\Gamma_{+}^{\hat{e}}$ and $\hat{e}$ along the $\hat{y}$
and $\hat{z}$ directions, and determine the Compton amplitude using Equation
28, we separate $\mathcal{G}_{1}$ and $\mathcal{G}_{2}$ by solving the system
of linear equations,

${\mathbf{T}}={\bf C}\,{\mathbf{g}},$ (40)

where ${\mathbf{T}}=[T^{\hat{y}}_{[12]},T^{\hat{z}}_{[12]}]^{\intercal}$,
${\mathbf{g}}=[\mathcal{G}_{1},\mathcal{G}_{2}]^{\intercal}/\omega$, and the
coefficient matrix

$\bf{C}=\begin{pmatrix}C_{1}(\hat{y},{\mathbf{p}},{\mathbf{q}})&C_{2}(\hat{y},{\mathbf{p}},{\mathbf{q}})\\\
C_{1}(\hat{z},{\mathbf{p}},{\mathbf{q}})&C_{2}(\hat{z},{\mathbf{p}},{\mathbf{q}})\end{pmatrix}.$
(41)

##  6 Selected results and discussion

We first present our results for the unpolarised Compton amplitude. Our
simulations are carried out on QCDSF/UKQCD-generated $2+1$-flavour gauge
configurations. Two ensembles are used with volumes $V=[32^{3}\times
64,48^{3}\times 96]$, and couplings $\beta=[5.50,5.65]$ corresponding to
lattice spacings $a=[0.074,0.068]\,{\rm fm}$, respectively. Quark masses are
tuned to the $SU(3)$ symmetric point where the masses of all three quark
flavours are set to approximately the physical flavour-singlet mass,
$\overline{m}=(2m_{s}+m_{l})/3$ [23, 24], yielding
$m_{\pi}\approx[470,420]\,{\rm MeV}$. Up to $\mathcal{O}(10^{4})$ and
$\mathcal{O}(10^{3})$ measurements are performed by employing multiple sources
on the $32^{3}\times 64$ and $48^{3}\times 96$ ensembles, respectively.

We obtain amplitudes for several values of current momentum, $Q^{2}$, in the
range $1.5\lesssim Q^{2}\lesssim 7$ GeV2. Multiple $\omega$ values are
accessed at each simulated value of ${\mathbf{q}}$ by varying the nucleon
momentum ${\mathbf{p}}$, which allows for a mapping of the $\omega$ dependence
of the Compton structure functions. For each $\omega$, we extract the energy
shifts from the ratios defined in Equations 33 and 34 for two $|\lambda|$
values. We determine the fit windows by a covariance-matrix based $\chi^{2}$
analysis where we choose the ranges that have $\chi^{2}_{dof}\sim 1.0$.
Effective mass plots for the ratios are shown in Figure 1 for some selected
cases.

Figure 1: Effective mass plots of the ratios (Equation 33) for the amplitudes
$T_{33}$ (left) and $T_{00}+T_{33}$ (right). Fit windows, along with the
extracted energy shifts with their $1\sigma$ uncertainty, are shown by the
shaded bands. We are showing the results obtained on the $48^{3}\times 96$
ensemble for the $uu$ piece, for
$({\mathbf{p}},{\mathbf{q}})=((0,1,0),(5,3,0))\,\left(\frac{2\pi}{L}\right)$
corresponding to $\omega=0.18$ at $Q^{2}\sim 4.9\,{\rm GeV}^{2}$. Figures
taken from [25].

We perform polynomial fits of the form, $\Delta
E_{N_{\lambda}}({\mathbf{p}})=\lambda^{2}\left.\frac{\partial^{2}E_{N_{\lambda}}({\mathbf{p}})}{\partial\lambda^{2}}\right|_{\lambda={\mathbf{0}}}+\mathcal{O}(\lambda^{4})$,
to determine the second order energy shift. Equation 32 also reduces to this
form since we set $|\lambda_{1}|=|\lambda_{2}|=|\lambda|$ in our simulations.
The unperturbed energy, $E_{N}$, and odd-order lambda terms
($\mathcal{O}(\lambda)$, $\mathcal{O}(\lambda^{3})$, $\dots$) are removed by
construction in the ratios. Given the smallness of our $\lambda$ values,
higher order $\mathcal{O}(\lambda^{4})$ terms are heavily suppressed, hence
the fit form reduces to a simple one parameter polynomial. We show
representative cases for the $\lambda$ fits in Figure 2. We confirm the
suppression of the $\mathcal{O}(\lambda^{4})$ term, and the absence of
$\lambda$-odd terms, by including $\mathcal{O}(\lambda)$,
$\mathcal{O}(\lambda^{3})$, and $\mathcal{O}(\lambda^{4})$ terms separately in
the fit. We find that any residual contamination has a negligible effect
compared to the statistical error on the extracted amplitudes.

Figure 2: $\lambda$ dependence of the energy shifts for the $uu$ and $dd$
pieces of the $T_{33}$ and $T_{00}+T_{33}$ amplitudes. Results are from the
$48^{3}\times 96$ ensemble with the same kinematics given in Figure 1. Figure
taken from [25].

The above analysis is performed to map out the $\omega$ dependence of the
Compton structure functions given in Equations 15 and 16 for each $Q^{2}$
value that we study. $\mathcal{F}_{L}(\omega,Q^{2})$ is constructed according
to Equation 17. We show the $\omega$ dependence of the Compton structure
functions, along with their fit curves, in Figure 3 for a representative case
of $Q^{2}\sim 4.9\,{\rm GeV}^{2}$ calculated on the $48^{3}\times 96$
ensemble.

Figure 3: $\omega$ dependence of the Compton structure functions
$\overline{\mathcal{F}}_{1}$, $\mathcal{F}_{2}$, and
$\overline{\mathcal{F}}_{L}$ at $Q^{2}\sim 4.9\,{\rm GeV}^{2}$ for the $uu$
(top), $dd$ (middle) and $ud$ (bottom) contributions. Coloured shaded bands
show the fits with their 68% credible region of the highest posterior density.
Points are displaced for clarity. Figure taken from [25].

The first few Mellin moments of $F_{1}$, $F_{2}$ and $F_{L}$ are determined by
performing a simultaneous fit to $\overline{\mathcal{F}}_{1}$ and
$\mathcal{F}_{2}$ in a Bayesian framework at each $Q^{2}$ value. We use
Equation 11 for $\overline{\mathcal{F}}_{1}$ and express $\mathcal{F}_{2}$ in
terms of the independently positive definite moments of $F_{1}$ and $F_{L}$,

$\displaystyle\frac{\mathcal{F}_{2}(\omega)}{\omega}=\frac{\tau}{\left(1+\tau\,\omega^{2}\right)}\sum_{n=0}^{\infty}4\omega^{2n}\left[M_{2n}^{(1)}+M_{2n}^{(L)}\right],$
(42)

where $\tau=Q^{2}/4M_{N}^{2}$, $M^{(1)}_{0}(Q^{2})=0$, and
$M^{(L)}_{0}(Q^{2})=\frac{4M_{N}^{2}}{Q^{2}}M^{(2)}_{2}(Q^{2})$. The intercept
at $\omega=0$ is proportional to the lowest moment of $F_{2}$, i.e.
$M_{2}^{(2)}(Q^{2})$. We truncate the series at $n=4$ (inclusive). No
dependence on higher-order terms is seen. We sample the moments from uniform
distributions with bounds $M_{2}(Q^{2})\in[0,1]$ and
$M_{2n}(Q^{2})\in[0,M_{2n-2}(Q^{2})]$, for $n>1$, to enforce the monotonic
decreasing nature of the moments, $M_{2}(Q^{2})\geq
M_{4}(Q^{2})\geq\cdot\cdot\cdot\geq M_{2n}(Q^{2})\geq\cdot\cdot\cdot\geq 0$,
for $uu$ and $dd$ contributions separately. Note that the positivity bound
does not hold for the $ud$ contributions but they are constrained by
$\left|M_{2n}^{ud}(Q^{2})\right|^{2}\leq
4M_{2n}^{uu}(Q^{2})M_{2n}^{dd}(Q^{2})$, since the total inclusive cross
section (hence each moment) is positive for any value of the quark charges and
at all kinematics. The sequences of individual $uu$, $dd$, and $ud$ moments
are selected according to the standard probability distribution,
$\operatorname{exp}(-\chi^{2}/2)$, where,
$\chi^{2}=\sum_{\mathcal{F}}\sum_{i}\left[\mathcal{F}^{\text{model}}_{i}-\mathcal{F}^{\text{obs}}(\omega_{i})\right]^{2}/\sigma^{2}$,
is the $\chi^{2}$ function with $\sigma^{2}$ the diagonal elements of the full
covariance matrix. Here, $\mathcal{F}$ stands for $\overline{\mathcal{F}}_{1}$
and $\mathcal{F}_{2}$, and the indices $i$, $j$ run through all the $\omega$
values and flavour-diagonal and mixed-flavour pieces. We account for the
correlations between the data points by a bootstrap analysis. Fits depicting
the extraction of the moments are also shown in Figure 3 by shaded bands for a
representative case.

We show the lowest moments of $F_{2}$ for proton in Figure 4 as a function of
$Q^{2}$. Note that the moments of the proton are constructed via
$M_{2,p}^{(2,L)}=\frac{4}{9}M_{2,uu}^{(2,L)}+\frac{1}{9}M_{2,dd}^{(2,L)}-\frac{2}{9}M_{2,ud}^{(2,L)}$.
Also shown are the experimental determinations of the Cornwall-Norton moments
of $F_{2}$ [26]. We see a remarkable agreement, although we should note that
our systematics are not fully accounted for yet.

Figure 4: $Q^{2}$ dependence of the lowest moments of proton $F_{2}$. Filled
stars are the experimental Cornwall-Norton moments of $F_{2}$ [26]. Figure
taken from [25].

The Compton amplitude encompasses all power corrections, therefore it is
possible to estimate the leading power correction (i.e. twist-4) by studying
the $Q^{2}$ behaviour of the moments in a twist expansion,

$M_{2,h}^{(2)}(Q^{2})=M_{2,h}^{(2)}+C_{2,h}^{(2)}/Q^{2}+\mathcal{O}(1/Q^{4}),$
(43)

where $h\in\\{uu,dd,ud,p\\}$. Utilising only the $M_{2}^{(2)}(Q^{2})$ moments
obtained on the $48^{3}\times 96$ ensemble, we study the power corrections
down to $Q^{2}\approx 1.5\;{\rm GeV}^{2}$. Our fit for proton is shown in
Figure 4. The extracted values for $M_{2,h}^{(2)}$ and $C_{2,h}^{(2)}$ are
collected in Table 1. Although we focus on the proton due to availability of
experimental data, it is possible to estimate the moments for neutron or the
isovector $u-d$ combination since we have the contributions of different
flavour components. Power corrections are a combination of target mass
corrections, pure higher-twist terms, and the elastic contributions. These
effects can be disentangled further, for instance by determining the elastic
contributions from form factors [27, 28], and employing Nachtmann moments [29]
to account for the target mass corrections, along with including the
logarithmic evolution of moments in Equation 43. We leave such an
investigation to a future study.

Table 1:  Asymptotic values of the moments and the coefficients of the leading power correction terms. We quote the power corrections at the scale of the nucleon mass $Q^{2}=M_{N}^{2}$. $h$ | $M_{2,h}^{(2)}$ | $C_{2,h}^{(2)}/M_{N}^{2}$
---|---|---
$uu$ | 0.268(13) | 0.206(24)
$dd$ | 0.146(7) | 0.024(14)
$ud$ | 0.000(0) | 0.007(3)
$p$ | 0.135(6) | 0.091(11)

In Figure 5, we show the lowest (Cornwall-Norton) moments of $F_{L}$ in
comparison to the experimentally determined Nachtmann moments [30]. With our
current precision, we are able to set an upper bound for the moments that is
compatible with the experimental moments.

The leading twist part of $F_{L}$ is related to $F_{2}$ in leading-order in
$\alpha_{s}$. In terms of moments this relation reads [31],

$M_{2,p}^{(L),{\rm
twist-}2}(Q^{2})=\frac{4}{9\pi}\alpha_{s}(Q^{2})M_{2,p}^{(2),{\rm
twist-}2}(Q^{2}),$ (44)

where we have replaced the leading-twist moment on RHS with
$M_{2,p}^{(2)}(Q^{2})$ from the current work as an approximation. We use the
value of $\alpha_{s}(Q^{2})$ determined in the $\overline{MS}$ scheme at
$\mu=Q^{2}$ at the four-loop order. The $Q^{2}$ behaviour is in good agreement
with experimental points as shown in Figure 5. Improving the precision in
future studies, would help to resolve the difference between the direct
determination and twist-2 part of the lowest few moments of $F_{L}$ and reveal
the higher-twist effects.

Figure 5: $Q^{2}$ dependence of the lowest moments of proton $F_{L}$ (Direct).
Open squares are the experimental Nachtmann moments of $F_{L}$ [30]. We also
show the moments (twist-2) determined via the relation, Equation 44. Twist-2
points are displaced for clarity. Figure taken from [25].  Figure 6:  The
polarised Compton structure function $\mathcal{G}_{1}(\omega,Q^{2})/\omega$ as
a function of $\omega$. We show the preliminary result obtained on the
$48^{3}\times 96$ ensemble at $Q^{2}\sim 4.9\,{\rm GeV}^{2}$.

Finally, in Figure 6 we show the the $uu$ and $dd$ pieces of the polarised
$\mathcal{G}_{1}$ Compton structure function, along with the isovector $uu-dd$
combination, as a function of $\omega$. We extract the first few Mellin
moments using Equation 14 by following the same Bayesian analysis performed
for the unpolarised case. Shaded bands in Figure 6 depict the resulting fits.

The lowest moment of $g_{1}$ is a particularly interesting phenomenological
quantity. It is directly related to the matrix elements of the axial current
at leading-twist. For instance, the leading-twist contribution of the lowest
isovector moment $\tilde{M}_{2,uu-dd}^{(1)}$ provides a complementary approach
to determining the nucleon isovector axial charge $g_{A}$ via the Bjorken sum
rule [32, 33], or alternatively, the QCD effective charge [34, 35]. However,
our current results on polarised structure functions are very preliminary.
They are obtained on a single ensemble at a single photon virtuality,
$Q^{2}\sim 4.9\,{\rm GeV}^{2}$ with limited statistics in an exploratory
simulation. Although these are encouraging results, more progress is needed.

##  7 Summary and outlook

The Compton amplitude approach has reached a certain maturity where it is
possible to directly investigate the structure functions including the effects
beyond leading twist. We have overviewed the relations between the forward
Compton amplitude and the structure functions, and described a novel extension
of the Feynman-Hellmann techniques that simplifies the calculation of the
amplitude. We showed the versatility of this approach by calculating the
moments of transverse and longitudinal proton structure functions along with
their $Q^{2}$ dependence. This allows us to study the power corrections for
the first time in a lattice calculation. Currently our calculations involve
configurations with two different lattice spacings and volumes, all at the
$SU(3)$ symmetric point. Calculations on additional ensembles that cover a
range of lattice spacings and pion masses are required to fully account for
systematic effects.

We are working towards extending our formalism to include the spin-dependent
structure functions. Our preliminary results are encouraging. Additionally,
accessing the parity violating structure function $F_{3}$, by considering weak
currents is an exciting future direction.

## Acknowledgments

I thank the organisers of “The 39th International Symposium on Lattice Field
Theory” for the invitation. I would also like to thank my colleagues from the
QCDSF/UKQCD collaboration, this work would not have been possible without
their contributions. The numerical configuration generation (using the BQCD
lattice QCD program [36])) and data analysis (using the Chroma software
library [37]) was carried out on the DiRAC Blue Gene Q and Extreme Scaling
(EPCC, Edinburgh, UK) and Data Intensive (Cambridge, UK) services, the GCS
supercomputers JUQUEEN and JUWELS (NIC, Jülich, Germany) and resources
provided by HLRN (The North-German Supercomputer Alliance), the NCI National
Facility in Canberra, Australia (supported by the Australian Commonwealth
Government) and the Phoenix HPC service (University of Adelaide). KUC is
supported by the Australian Research Council grants DP190100297 and
DP220103098. For the purpose of open access, the authors have applied a
Creative Commons Attribution (CC BY) licence to any Author Accepted Manuscript
version arising from this submission.

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