# Magnetic polarisability of the nucleon using a Laplacian mode projection

Ryan Bignell<EMAIL_ADDRESS>Waseem Kamleh  Derek Leinweber
Special Research Centre for the Subatomic Structure of Matter (CSSM),
Department of Physics, University of Adelaide, Adelaide, South Australia 5005,
Australia

###### Abstract

Conventional hadron interpolating fields, which utilise gauge-covariant
Gaussian smearing, are ineffective in isolating ground state nucleons in a
uniform background magnetic field. There is evidence that residual Landau mode
physics remains at the quark level, even when QCD interactions are present. In
this work, quark-level projection operators are constructed from the SU(3)
$\times$ U(1) eigenmodes of the two-dimensional lattice Laplacian operator
associated with Landau modes. These quark-level modes are formed from a
periodic finite lattice where both the background field and strong
interactions are present. Using these eigenmodes, quark-propagator projection
operators provide the enhanced hadronic energy-eigenstate isolation necessary
for calculation of nucleon energy shifts in a magnetic field. The magnetic
polarisability of both the proton and neutron is calculated using this method
on the $32^{3}\times 64$ dynamical QCD lattices provided by the PACS-CS
Collaboration. A chiral effective-field theory analysis is used to connect the
lattice QCD results to the physical regime, obtaining magnetic
polarisabilities of $\beta^{p}=2.79(22)\\!\left({}^{+13}_{-18}\right)\times
10^{-4}$ fm3 and $\beta^{n}=2.06(26)\\!\left({}^{+15}_{-20}\right)\times
10^{-4}$ fm3, where the numbers in parentheses describe statistical and
systematic uncertainties.

###### pacs:

13.40.-f, 12.38.Gc, 12.39.Fe

††preprint: ADP-20-6/T1116

##  I Introduction

The magnetic polarisability describes the response of a system of charged
particles to an external magnetic field. The study of nucleon polarisabilities
is an area of key experimental and theoretical interest Shanahan  _et al._
(2017); Lensky  _et al._ (2018); Sokhoyan  _et al._ (2017); Mornacchi (2018,
2019); Pasquini  _et al._ (2019), and the magnetic polarisabilities are key
quantities in this area. Measurement of magnetic polarisabilities is difficult
MacGibbon  _et al._ (1995); Blanpied  _et al._ (2001) and improvement in
experimental measurements is evident in recent years Tanabashi  _et al._
(2018); McGovern  _et al._ (2013); Pasquini  _et al._ (2019). Lattice QCD can
play an important role in making predictions in this area.

The uniform background field method has been used successfully to calculate
magnetic moments Martinelli  _et al._ (1982); Bernard  _et al._ (1982a); Lee
_et al._ (2005); Parreno  _et al._ (2017) of hadrons and the magnetic
polarisability of neutral particles such as the neutral pion Alexandru  _et
al._ (2016); Bignell  _et al._ (2019) and neutron Bignell  _et al._ (2018);
Chang  _et al._ (2015). Herein, we present new lattice QCD techniques that
enable an investigation of the proton polarisability in an accurate manner.

An uniform external magnetic field is induced through the introduction of an
exponential $U(1)$ phase factor on the gauge links across the lattice. This
external field changes the energy of the nucleon according to the energy-field
relation Martinelli  _et al._ (1982); Bernard  _et al._ (1982b); Burkardt  _et
al._ (1996); Tiburzi and Vayl (2013); Primer  _et al._ (2014); Chang  _et al._
(2015); Bignell  _et al._ (2018)

$E(B)=m+\vec{\mu}\cdot\vec{B}+\left(2\,n+1\right)\frac{\left|qe\,B\right|}{2\,m}-\frac{4\,\pi}{2}\,\beta\,B^{2}+\ldots$
(1)

where the nucleon has mass $m$ and magnetic moment and magnetic polarisability
$\vec{\mu}$ and $\beta$ respectively. The Landau energy term Itzykson and
Zuber (1980) is proportional to $\left|qe\,B\right|$. There is in principle an
infinite tower of Landau levels,
$\left(2\,n+1\right)\,\left|qe\,B\right|/2\,m$ for $n=0,1,2,\dots$ which poses
an additional complication for charged hadrons such as the proton.

While the extraction of the magnetic polarisability seems simple $-$ simply
fit the linear and quadratic coefficients of the energies of Eq.
$\left(\ref{eqn:EofB}\right)$ as a function of field strength $-$ this
approach is problematic as the magnetic polarisability appears at second-order
in the energy of the nucleon Burkardt  _et al._ (1996); Tiburzi and Vayl
(2013); Primer  _et al._ (2014); Chang  _et al._ (2015); Bignell  _et al._
(2018). The contribution of the magnetic polarisability to the nucleon energy
is necessarily small compared to the overall energy of the particle if the
energy expansion of Eq. $\left(\ref{eqn:EofB}\right)$ is to have small
$\mathcal{O}\left(B^{3}\right)$ contributions.

Three-dimensional gauge-covariant Gaussian smearing Gusken (1990) on the quark
fields at the source has been shown to efficiently isolate the nucleon ground
state in pure QCD calculations Durr  _et al._ (2008); Mahbub  _et al._ (2012).
This is not the case when a uniform background field is present; the magnetic
field breaks three-dimensional spatial symmetry and introduces electromagnetic
perturbations into the dynamics of the charged quarks, thus altering the
physics present. When QCD interactions are absent, each quark will have a
Landau energy proportional to its charge. In the presence of QCD these quarks
hadronise, such that in the confining phase the Landau energy will correspond
to that of the composite hadron.

It is clear that in the confining phase the effects of the QCD and magnetic
interactions compete with each other. Previous studies have demonstrated that
Landau physics remains relevant even when QCD interactions are present
Bruckmann  _et al._ (2017); Bignell  _et al._ (2018). This leads us to the
idea of using quark operators on the lattice that capture both of these
forces. In particular, we have the freedom of choosing asymmetric source and
sink operators in order to construct correlation functions that provide better
overlap with the energy eigenstates of the nucleon in a background magnetic
field.

The two-dimensional $U(1)$ Laplacian is associated with the Landau modes of a
charged particle in a magnetic field. In our previous study of the neutron
Bignell  _et al._ (2018) we considered a quark sink projection based on these
two-dimensional $U(1)$ eigenmodes on gauge-fixed QCD fields. In the present
study we explore the use of a projector derived from the eigenmodes of the
two-dimensional lattice SU(3) $\times$ U(1) Laplacian operator. The use of a
fully gauge-covariant eigenmode-projected quark sink which encapsulates both
QCD and Landau level physics eliminates the need for gauge fixing. We find the
use of the SU(3) $\times$ U(1) modes as a quark projection operator to be
effective in isolating the ground state of the proton in an external magnetic
field, enabling an accurate determination of the proton magnetic
polarisability.

The presentation of this research is as follows. Section II briefly describes
our implementation of a uniform magnetic field. Section III describes the
process by which the magnetic polarisability can be extracted from nucleon two
point correlation functions while Section IV describes the smeared source and
SU(3) $\times$ U(1) projected sink used to isolate the nucleon ground states
at non-zero magnetic field strengths. The results at several quark masses are
presented in Section V, and these are used to inform the chiral extrapolations
to the physical regime of Section VI. Section VII summarises conclusions.

##  II Background field method

The following background field method is used to introduce a constant magnetic
field along a single axis. This technique is derived first in the continuum
where a minimal electromagnetic coupling is added to form the covariant
derivative

$\displaystyle D_{\mu}=\partial_{\mu}+i\,qe\,A_{\mu},$ (2)

where $qe$ is the charge of the fermion field and $A_{\mu}$ is the
electromagnetic four potential. On the lattice, the equivalent modification is
to multiply the QCD gauge links by an exponential phase factor

$\displaystyle U_{\mu}(x)\rightarrow U_{\mu}(x)\,e^{(i\,a\,qe\,A_{\mu}(x))}.$
(3)

As $\vec{B}=\vec{\nabla}\times\vec{A}$, a uniform magnetic field along the
$\hat{z}$ axis is obtained via

$\displaystyle B_{z}$ $\displaystyle=\partial_{x}\,A_{y}-\partial_{y}\,A_{x}.$
(4)

To give a constant magnetic field of magnitude $B$ in the $+\hat{z}$ direction
on the lattice we exploit both $A_{x}$ and $A_{y}$. Throughout the lattice we
set $A_{x}=-B\,y$. To maintain the constant magnetic field across the
$\hat{y}$ edge of the lattice where periodic boundary conditions are in
effect, we set $A_{y}=+B\,N_{y}\,x$ along the $\hat{y}$ boundary $y=N_{y}$.
This then induces a quantisation condition for the uniform magnetic field
strength Primer  _et al._ (2014)

$\displaystyle qe\,B\,a^{2}=\frac{2\,\pi\,k}{N_{x}\,N_{y}}.$ (5)

Here $a$ is the lattice spacing, $N_{x}$ and $N_{y}$ are the spatial
dimensions of the lattice, and $k$ an integer specifying the field strength in
terms of the minimum field strength.

In this work the field quanta $k$ is in terms of the charge of the down quark,
i.e., $k=k_{d}$ and $q=-1/3$

$\displaystyle q_{d}e\,B\,a^{2}=\frac{2\,\pi\,k_{d}}{N_{x}\,N_{y}}.$ (6)

Hence a field with $k_{d}=1$ will be in the $-\hat{z}$ direction. The magnetic
field experienced by a baryon is defined to be $k_{B}=-3\,k_{d}$.

##  III Magnetic Polarisability

The naive process of fitting to Eq. $\left(\ref{eqn:EofB}\right)$ as a
function of field strengths is not a viable method with which to extract the
magnetic polarisability. Instead a ratio of correlation functions is
constructed to isolate the energy shift in a manner enabling correlated QCD
fluctuations to be reduced. To form this ratio, we define the spin-field
antialigned two-point correlation function

$\displaystyle G_{\upharpoonleft\\!\downharpoonright}(B)=G(+s,-B)+G(-s,+B).$
(7)

and the spin-field aligned correlator by

$\displaystyle G_{\upharpoonleft\\!\upharpoonright}(B)=G(+s,+B)+G(-s,-B).$ (8)

Here spin-up/down is represented by $\left(+s/-\\!s\right)$ respectively and
the magnetic field orientation along the spin quantisation direction,
$\hat{z}$, by $\left(\pm B\right)$. These spin-field antialigned and aligned
correlators form an improved unbiased estimator for the required correlation
functions as they are averages over the required spin and field combinations.

The ratio required to isolate the magnetic polarisability of Eq.
$\left(\ref{eqn:EofB}\right)$ draws on Eqs. $\left(\ref{eqn:Gupdown2}\right)$
and $\left(\ref{eqn:Gupup2}\right)$ along with the spin-averaged zero-field
correlator $G(0,t)$

$\displaystyle
R(B,t)=\frac{G_{\upharpoonleft\\!\upharpoonright}(B,t)\,G_{\upharpoonleft\\!\downharpoonright}(B,t)}{G(0,t)^{2}}.$
(9)

The zero-field correlator subtracts the mass term from the total energy of the
anti-aligned and aligned contributions while the contribution from the
magnetic moment term of Eq. $\left(\ref{eqn:EofB}\right)$ is removed by the
product of the spin-field antialigned and aligned correlators. This product
yields an exponent of the sum of the aligned and antialigned energy shifts

$\displaystyle\frac{\delta
E_{{\upharpoonleft\\!\upharpoonright}}\left(B\right)+\delta
E_{{\upharpoonleft\\!\downharpoonright}}\left(B\right)}{2}=\frac{\left|qe\,B\right|}{2\,M}-\frac{4\,\pi}{2}\,\beta\,\left|B\right|^{2}+\mathcal{O}\left(B^{4}\right).$
(10)

Thus the desired energy shift is

$\displaystyle\delta E\left(B,t\right)$
$\displaystyle=\frac{1}{2}\,\frac{1}{\delta\,t}\,\log\left(\frac{R\left(B,t\right)}{R\left(B,t+\delta\,t\right)}\right)$
(11)
$\displaystyle\rightarrow\frac{\left|qe\,B\right|}{2\,M}-\frac{4\,\pi}{2}\,\beta\,\left|B\right|^{2}+\mathcal{O}\left(B^{4}\right),$
(12)

for large Euclidean time. For the neutrally charged neutron,
$\left|qe\,B\right|=0$ and hence the Landau level term vanishes, providing
direct access to the polarisability. For the proton this term makes an
important contribution and we investigate its magnitude in a variety of fits.

###  III.1 Fitting

For a charged hadron such as the proton, the energy shift for the
polarisability is as specified by Eq. $\left(\ref{eqn:esubm}\right)$ which has
a term linear in $B$ and a term quadratic in $B$. As such, an appropriate fit
as a function of field strength has both a linear and a quadratic dependence

$\displaystyle\delta E\left(k_{d}\right)=c_{1}\,k_{d}+c_{2}\,k_{d}^{2}$ (13)

where we fit as a function of $k_{d}$, the integer magnetic flux quanta in Eq.
$\left(\ref{eqn:qc}\right)$. $c_{1}$ and $c_{2}$ are fit parameters with the
units of $\delta E\left(k_{d},t\right)$. This is in contrast to the neutron
where only a quadratic term is required Bignell  _et al._ (2018).

As $c_{1}$ is a free parameter, fitting in this manner allows the charge of
the proton to be non-unitary. Thus, we also consider the linear constrained
energy shift

$\displaystyle\delta
E\left(k_{d}\right)-\frac{\left|e\,B\right|}{2\,M}=c_{2}\,k_{d}^{2}=-\frac{4\,\pi}{2}\,\beta\,\left|B\right|^{2}+\mathcal{O}\left(B^{4}\right).$
(14)

Here the known linear term has been explicitly subtracted with $q=1$ such that
only a term quadratic in $B$ is fitted. This constrains the charge of the
proton to $q_{p}=1$.

The quantisation condition of Eq. $\left(\ref{eqn:qc}\right)$ provides

$\displaystyle\beta=-2\,c_{2}\,\alpha\,q_{d}^{2}\,a^{4}\,\left(\frac{N_{x}\,N_{y}}{2\,\pi}\right)^{2},$
(15)

where $\alpha=1/137\dots$ is the fine structure constant.

The energy shifts which are fit in Eq. $\left(\ref{eqn:EmmCon}\right)$ must be
well determined at all field strengths. We apply the single state ansatz,
requiring that a constant plateau fit can be found as a function of Euclidean
lattice time $t$. Correlations between adjacent Euclidean time slices are
considered through the use of the full covariance matrix $\chi^{2}_{dof}$
which is estimated via the jackknife method Efron (1979). The resulting fit as
a function of field strength must then also fit the energy shifts with an
acceptable $\chi^{2}$ per degree of freedom.

Fit windows were kept consistent across the three non-zero field strengths
considered where possible. Where this proved difficult, the fit windows
between field strengths were allowed to vary in a monotonic manner, with the
lowest field strength having the longest fit region. Through this process we
ensure that the lowest lying state is isolated for each energy shift.

##  IV Quark Operators

The use of asymmetric source and sink operators enable the construction of
nucleon correlation functions which have greater overlap with the lowest
energy eigenstates of the nucleon in a background magnetic field. This is
important as the energy shift required for the magnetic polarisability is
small compared to the total energy. The signal becomes disguised by noise at
late Euclidean time.

The quark source is constructed using three-dimensional gauge-invariant
Gaussian smearing Gusken (1990) as is common practice in lattice QCD Durr  _et
al._ (2008); Mahbub  _et al._ (2012) while the quark sink uses the SU(3)
$\times$ U(1) eigenmode quark-projection method discussed below.

###  IV.1 SU(3) $\times$ U(1) eigenmode projection

For a charged particle in a uniform magnetic field the lattice Landau levels
are eigenmodes of the (2D) $U(1)$ Laplacian. Here we wish to construct a fully
gauge-covariant quark sink projection operator that encompasses QCD as well as
the electromagnetic potential. To do so, we calculate the low-lying eigenmodes
$\ket{\psi_{i}}$ of the two-dimensional lattice Laplacian

$\displaystyle\Delta_{\vec{x},\vec{x}^{\prime}}=4\,\delta_{\vec{x},\vec{x}^{\prime}}-\\!\sum_{\mu=1,2}U_{\mu}(\vec{x})\delta_{\vec{x}+\hat{\mu},\vec{x}^{\prime}}+U^{\dagger}_{\mu}(\vec{x}-\hat{\mu})\delta_{\vec{x}-\hat{\mu},\vec{x}^{\prime}},$
(16)

where $U_{\mu}\left(\vec{x}\right)$ are the full SU(3) $\times$ U(1) gauge
links as applied in the full lattice QCD calculation. We can then define a
projection operator by truncating the completeness relation

$\displaystyle 1=\sum_{i=1}\,\ket{\psi_{i}}\bra{\psi_{i}}.$ (17)

In the pure $U(1)$ case, the lowest Landau level on the lattice has a
degeneracy equal to the magnetic flux quanta $|k|$ given in Eq. 5, providing a
natural place to truncate the above sum. The introduction of the QCD
interactions into the Laplacian causes the $U(1)$ modes associated with the
different Landau levels to mix, such that it is no longer possible to clearly
identify the modes associated with the lowest Landau level at small field
strengths. Instead, we simply choose a fixed number $n>|k|$ modes to project.
This truncation has a similar effect to performing (2D) smearing, by filtering
out the high frequency modes. Indeed, we find that for small values of $n$ the
projected hadron correlator becomes noisy, just as it does when performing
large amounts of sink smearing.

Figure 1: The proton energy shift $\delta E\left(k_{d},t\right)$ of Eq.
$\left(\ref{eqn:logR}\right)$ for three field strengths using $64$ (orange
diamond) and $96$ (blue square) eigenmodes in the projection operator of Eq.
$\left(\ref{eqn:coordP_n}\right)$. The $m_{\pi}=0.702$ GeV ensemble is shown.

The eigenmode truncation is chosen to be sufficiently large so as to avoid
introducing large amounts of noise into the two-point correlation function,
but also small enough to place a focus on the low-energy physics relevant to
isolating the magnetic polarisability. Truncation at $32$, $64$ and $96$ modes
are investigated in a manner similar to that at the source. While $32$ modes
was not effective, it can be observed in Figure 1 that both $64$ and $96$
eigenmodes produce consistent behaviour in the proton energy shift.

Due to the two-dimensional nature of the Laplacian, the low-lying eigenspace
is calculated independently for each $(z,t)$-slice on the lattice.
Consequently, we can interpret the four-dimensional coordinate space
representation of an eigenmode

$\displaystyle\braket{\vec{x},t\,}{\,\psi_{i,\vec{B}}}=\psi_{i,\vec{B}}(x,y\,|\,z,t),$
(18)

as selecting the two-dimensional coordinate space representation
$\psi_{i,\vec{B}}(x,y)$ from the eigenspace belonging to the corresponding
$(z,t)$-slice of the lattice. The four-dimensional coordinate space
representation of the projection operator follows,

$\displaystyle
P_{n}\left(\vec{x},t;\vec{x}^{\prime},t^{\prime}\right)=\sum_{i=1}^{n}\,\braket{\vec{x},t\,}{\,\psi_{i,\vec{B}}}\braket{\psi_{i,\vec{B}}\,}{\,\vec{x}^{\prime},t^{\prime}}\,\delta_{zz^{\prime}}\,\delta_{tt^{\prime}}.$
(19)

The Kronecker delta functions in the definition above ensure that the outer
product is only taken between eigenmodes from the same subspace (i.e. the
projector acts trivially on the $(z,t)$ coordinates).

This projection operator is then applied at the sink to the quark propagator
in a coordinate-space representation as

$\displaystyle
S_{n}\left(\vec{x},t;\vec{0},0\right)=\sum_{\vec{x}^{\prime}}\,P_{n}\left(\vec{x},t;\vec{x}^{\prime},t\right)\,S\left(\vec{x}^{\prime},t;\vec{0},0\right).$
(20)

We select $n=96$ modes for our analysis.

Using the SU(3) $\times$ U(1) eigenmode quark-projection operator and a tuned
smeared source produces nucleon correlation functions at non-trivial field
strengths where the proton is in the QCD ground state and the $n=0$ lowest
lying Landau level approximation is justified. The energy shifts required by
Eqs. $\left(\ref{eqn:logR}\right)$ and $\left(\ref{eqn:esubm}\right)$ display
good plateau behaviour, as exhibited in Figures 1 and 6.

###  IV.2 Source Smearing

Whilst we attempt to encapsulate the quark-level physics of the
electromagnetic interaction at the sink, we use a smeared source to provide a
representation of the QCD interactions with the intent of isolating the QCD
ground state. A broad range of smearing levels are examined at zero field
strength, $B=0$ in order to do this.

The effective mass at $B=0$ was investigated for each ensemble and the
smearing which produces the earliest onset of plateau behaviour is chosen.
This $B=0$ effective mass is shown in Figure 2 for $m_{\pi}=0.702$ GeV where
the optimal smearing of $150$ sweeps is chosen. On the lightest ensemble
considered, at $m_{\pi}=0.296$ GeV as shown in Figure 3, the choice is not as
obvious. In this case the full set of correlation functions at each finite
field-strength is run for each smearing and these results examined.

Figure 2: Proton zero-field effective masses from smeared source to SU(3)
$\times$ U(1) eigenmode projected sink correlators using various levels of
covariant Gaussian smearing at the source on the $m_{\pi}=0.702$ GeV ensemble.
The source is at $t=16$. Figure 3: Proton zero-field effective masses from
smeared source to SU(3) $\times$ U(1) eigenmode projected sink correlators
using various levels of covariant Gaussian smearing at the source on the
$m_{\pi}=0.296$ GeV ensemble. The source is at $t=16$. Figure 4: Proton
$k_{d}=2$ anti-aligned effective energies from smeared source to SU(3)
$\times$ U(1) eigenmode projected sink correlators using various levels of
covariant Gaussian smearing at the source on the $m_{\pi}=0.296$ GeV ensemble.
The source is at $t=16$.

This reveals a particularly interesting problem with large amounts of smearing
which can be seen in the anti-aligned energy shown in Figure 4. The anti-
aligned energy is examined in preparation its use in the energy shift ratio of
Eq. $\left(\ref{eqn:R-2}\right)$ and has spin and magnetic field anti-aligned
as in Eq. $\left(\ref{eqn:Gupdown2}\right)$.

When the source is excessively smeared, the larger field strengths couple
preferentially to higher Landau levels rather than the lowest. This is evident
in how the $350$ sweeps effective energy differs from the other smearings in
both value and slope. This difference is close to the difference between
Landau levels for the proton, i.e. for

$\displaystyle E_{n}^{2}\left(B\right)\sim
m^{2}+\left|e\,B\right|\,\left(2\,n+1\right),$ (21)

the difference can be determined by considering the relativistic energy
difference
$E^{2}_{1}\left(B\right)-E^{2}_{0}\left(B\right)=2\,\left|e\,B\right|.$ The
difference of squares can be factored as

$\displaystyle\left(E_{1}\left(B\right)-E_{0}\left(B\right)\right)\,\left(E_{1}\left(B\right)+E_{0}\left(B\right)\right)$
$\displaystyle=2\,\left|e\,B\right|.$ (22)

Defining
$\Delta\,E_{10}\left(B\right)=E_{1}\left(B\right)-E_{0}\left(B\right)$ as the
energy difference visible in Figure 4, we obtain a quadratic form

$\displaystyle\Delta\,E_{10}\left(B\right)\,\left(\Delta\,E_{10}\left(B\right)+2\,E_{0}\left(B\right)\right)-2\,\left|e\,B\right|=0.$
(23)

Recalling that the field strength experienced by the proton is related to that
of the down quark by $k_{B}=-3\,k_{d}$, the appropriate values for Figure 4
are

$\displaystyle E_{0}\left(k_{d}=2\right)$ $\displaystyle\sim 0.9\text{ GeV}$
$\displaystyle\left|e\,B\left(k_{d}=2\right)\right|$ $\displaystyle\sim
0.522\text{ GeV}^{2},$

and the energy difference between these two Landau levels is
$\Delta\,E_{10}\left(k_{d}=2\right)\sim 0.46$ GeV. This is consistent with the
difference between smearings visible in Figure 4.

The smeared source examination is followed at each of the quark masses where
for masses $m_{\pi}=702$, $570$, $411$, $296$ MeV, optimal smearings of
$N_{sm}=150$, $175$, $300$, $250$ respectively are obtained.

An advantage of using the $U(1)\times SU(3)$ Laplacian projector is that it is
well defined at zero magnetic field strength, where the $U(1)$ field is equal
to unity. This means that the fluctuations at finite $B$ and $B=0$ are
strongly correlated, such that they cancel out when taking the ratio of the
correlators in Eq. $\left(\ref{eqn:R-2}\right)$, providing an improved signal
in comparison to the $U(1)$ projection. This improvement does come at a
computational cost, as the $U(1)\times S\\!U(3)$ Laplacian eigenmodes must be
calculated on every configuration. Using the SU(3) $\times$ U(1) eigenmode
quark-projection operator and a tuned smeared source produces nucleon
correlation functions at non-trivial field strengths where the proton is in
the QCD ground state and the $n=0$ lowest lying Landau level approximation is
justified. This is demonstrated by the energy shifts required by Eqs.
$\left(\ref{eqn:logR}\right)$ and $\left(\ref{eqn:esubm}\right)$, which
display good plateau behaviour as exhibited in Figures 1 and 6.

###  IV.3 Hadronic Landau Projection

Figure 5: Lowest-lying U(1) eigenmode probability densities of the lattice
Laplacian operator in a constant background magnetic field oriented in the
$\hat{z}$ direction are plotted as a function of the $x,y$ coordinates. As
$k_{B}=-3\,k_{d}$, the degenerate eigenmodes for the sixth quantised field
strength relevant to the proton for $k_{d}=2$ are displayed in a linear
combination that maximises the overlap of the first mode with the source. The
origin is at the centre of the $x-y$ plane.

The SU(3) $\times$ U(1) eigenmode projection technique defined above is
relevant to Landau effects at the quark-level. As the proton is a charged
hadron it will also experience Landau level behaviour in a magnetic field. The
Landau-level physics at the hadronic level is easier to capture due to the
colour-singlet nature of the proton. We can simply use the eigenmodes of the
$U(1)$ Laplacian, with a well defined degeneracy for the lowest Landau level
Kamleh  _et al._ (2018); Bignell  _et al._ (2018). Below we describe in detail
the prescription for the hadronic Landau level projection.

To study hadronic two-point correlation functions in the zero-field case one
calculates the momentum-projected correlator

$\displaystyle
G\left(\vec{p},t\right)=\sum_{\vec{x}}\,\text{e}^{-i\,\vec{p}\cdot\vec{x}}\,\Braket{\Omega}{T\left\\{\chi\left(\vec{x},t\right)\,\bar{\chi}\left(\vec{0},t\right)\right\\}}{\Omega},$
(24)

where $\chi$ and $\bar{\chi}$ are appropriate interpolating fields.

This standard approach of a three-dimensional Fourier projection is not
appropriate for the proton when the uniform background magnetic field is
present. The presence of the background field causes the energy eigenstates of
the charged proton to no longer be eigenstates of the $p_{x},p_{y}$ momentum
components. Hence, we instead project the $x,y$ dependence of the two-point
correlator onto the lowest Landau level, $\psi_{\vec{B}}\left(x,y\right)$,
explicitly, and also select a specific value for the $z$ component of
momentum,

$\displaystyle G\left(p_{z},\vec{B},t\right)=\sum_{r}\,$
$\displaystyle\psi_{\vec{B}}\left(x,y\right)\,\text{e}^{-i\,p_{z}\,z}$
$\displaystyle\times\Braket{\Omega}{T\left\\{\chi\left(r,t\right)\,\bar{\chi}\left(\vec{0},t\right)\right\\}}{\Omega}.$
(25)

In the continuum limit, the lowest Landau mode has a Gaussian form,
$\psi_{\vec{B}}\sim\text{e}^{-\left|qe\,B\right|\,\left(x^{2}+y^{2}\right)/4}$.
However, in a finite volume the periodicity of the lattice causes the wave
function’s form to be altered Tiburzi and Vayl (2013); Bignell  _et al._
(2017). As such, we instead calculate the lattice Landau eigenmodes using the
two-dimensional U(1) gauge-covariant lattice Laplacian in an analogous way to
Eq. $\left(\ref{eqn:2DLap}\right)$ Kamleh  _et al._ (2018); Bignell  _et al._
(2018). Here $U_{\mu}$ contains only the $U(1)$ phases appropriate to the
background magnetic field quantised on the lattice.

The correlator projection is then onto the space spanned by the degenerate
modes $\psi_{i,\vec{B}}$ associated with the lowest lattice Landau level
available to the proton

$\displaystyle G\left(p_{z},\vec{B},t\right)=\sum_{r}\,\sum_{i=1}^{n}\,$
$\displaystyle\psi_{i,\vec{B}}\left(x,y\right)\,\text{e}^{-i\,p_{z}\,z}$
$\displaystyle\times\Braket{\Omega}{T\left\\{\chi\left(r,t\right)\,\bar{\chi}\left(\vec{0},t\right)\right\\}}{\Omega}.$
(26)

The degeneracy of the lowest-lying Landau mode is given by the magnetic-field
quanta $\left|k_{d}\right|$.

In evaluating Eq. $\left(\ref{eqn:19Draft}\right)$, we also consider the case
of fixing $n=1$, such that only the first eigenmode having the best overlap
with the source is considered. Assuming the source
$\rho(x,y)=\delta_{x,0}\delta_{y,0}$ is located at the origin, the overlap
with the first mode $i=1$ is optimised through a rotation of the $U(1)$
eigenmode basis that maximises the value of
$|\braket{\rho\,}{\,\psi_{i=1,\vec{B}}}|^{2}$. An optional phase can be
applied so that $\psi_{i=1,\vec{B}}(0,0)$ is purely real at the source point.

In most cases the results are almost indistinguishable and we proceed with
$n=\left|3\,k_{d}\right|$. The only exception is for the ensemble with
$m_{\pi}=0.411$ GeV. Here the $i=1$ mode alone provides superior results. As
discussed in Section II, for the proton $k_{B}=-3\,k_{d}$ and therefore the
degeneracy is $n=\left|3\,k_{d}\right|$. Figure 5 illustrates the six
degenerate modes associated with $k_{d}=2$.

More generally, this hadronic eigenmode-projected correlator offers superior
isolation of the ground state for the proton Tiburzi and Vayl (2013) and is
crucial for the identification of constant plateaus in the energy shift of Eq.
$\left(\ref{eqn:logR}\right)$.

###  IV.4 Simulation Details

Table 1: Lattice simulation parameters with corresponding statistics used. $\kappa_{ud}$ | a (fm) |  $m_{\pi}$ (GeV) | $L^{3}\times T$ | $m_{\pi}\,L$ | N${}_{\text{src}}$
---|---|---|---|---|---
0.13700 | 0.1023 | 0.702 | $32^{3}\times 64$ | 11.6 | 5
0.13727 | 0.1009 | 0.570 | $32^{3}\times 64$ | 9.4 | 4
0.13754 | 0.0961 | 0.411 | $32^{3}\times 64$ | 6.4 | 4
0.13770 | 0.0951 | 0.296 | $32^{3}\times 64$ | 4.5 | 7

The $2+1$ flavour dynamical gauge configurations provided by the PACS-CS Aoki
_et al._ (2009) collaboration through the ILDG Beckett  _et al._ (2011) are
used in this work. These configurations span a variety of masses, allowing a
chiral extrapolation to be performed. A non-perturbatively improved clover
fermion action and Iwasaki gauge action provide a physical lattice spacing of
$a=0.0907(13)$ fm. Four different ensembles are considered, corresponding to
four different pion masses. The lattice spacing for each ensemble was set
using the Sommer scale with $r_{0}=0.49$ fm. The details of each of these
ensembles, including the pion mass and statistics used can be found in Table
1. Fixed boundary conditions in the time direction are used and the source
placed at $N_{t}/4=16$. Source locations are then systematically varied to
produce large distances between adjacent sources Bignell  _et al._ (2018).

We calculate correlation functions at four distinct magnetic-field strengths -
including zero. Propagators at ten nonzero field-strengths are calculated to
achieve this at $e\,B=\pm 0.087$, $\pm 0.174$, $\pm 0.261$, $\pm 0.348$, $\pm
0.522$ GeV2; corresponding to $k_{d}=\pm 1,\,\pm 2,\,\pm 3,\,\pm 4,\,\pm 6$ in
Eq. $\left(\ref{eqn:qc}\right)$.

The additive mass renormalisation due to the Wilson term Brandt  _et al._
(2016); Bali  _et al._ (2018) is removed through use of the Background-Field-
Corrected-Clover action Bignell  _et al._ (2019). As the background field is
known analytically; a tree-level contribution of $F^{B}_{\mu\nu}$ from the
background field can be included in the clover term of the fermion action,
avoiding the non-perturbative improvement coefficient, $C_{SW}$. This removes
the non-physical magnetic-field induced additive mass renormalisation due to
the Wilson term of the fermion action.

The configurations used are electro-quenched; the magnetic field exists only
for the valence quarks of the hadron. While it is possible to include the
background field on each configuration Fiebig  _et al._ (1989) this requires a
separate Monte Carlo simulation for each field strength and so is
prohibitively expensive. Performing separate calculations would also remove
the correlated QCD fluctuations between finite-field and zero-field
correlation functions, reducing the efficacy of Eq.
$\left(\ref{eqn:R-2}\right)$.

##  V Results

The formalism for extracting the magnetic polarisability of the nucleons using
lattice QCD and the background field method has now been established. Hadronic
Landau level projected correlation functions are computed at several non-zero
field strengths using a specialised SU(3) $\times$ U(1) quark sink. Thus
hadronic as well as quark level Landau energy level effects are considered in
order to isolate the ground state energy of the nucleon in an external
magnetic field. A tuned smeared source provides a good representation of the
QCD ground state.

Figure 6: The magnetic polarisability effective energy shift, $\delta
E\left(B,t\right)$ of Eq. $\left(\ref{eqn:logR}\right)$ for the
$m_{\pi}=0.570$ GeV proton as a function of Euclidean time (in lattice units),
using a smeared source and the SU(3) $\times$ U(1) quark-eigenmode projection
technique. Results for field strengths $k_{d}=1,2,3$ are shown. The selected
fits for this ensemble and the $\chi^{2}_{dof}$ are also illustrated. Figure
7: The magnetic polarisability effective energy shift, $\delta
E\left(B,t\right)$ of Eq. $\left(\ref{eqn:logR}\right)$ for the
$m_{\pi}=0.296$ GeV proton as a function of Euclidean time (in lattice units),
using a smeared source and the SU(3) $\times$ U(1) quark-eigenmode projection
technique. Results for field strengths $k_{d}=1$ and $2$ are shown. The
selected fits for this ensemble and the $\chi^{2}_{dof}$ are also illustrated.

The effectiveness of this approach is visible in Figures 6 and 7 where the
energy shift required to extract the magnetic polarisability of the proton is
plotted for the $m_{\pi}=0.570$ GeV and $m_{\pi}=0.296$ GeV ensembles
respectively. The effective energy shifts display good plateau behaviour
across all three non-zero field strengths. This is a common feature across the
heavier three quark masses considered. Good plateaus can be found at all three
field strengths.

At the lightest quark mass considered, the third field strength does not
present good plateau behaviour. As such only the first two field strengths are
considered.

The fit function of Eq. $\left(\ref{eqn:EmmCon}\right)$ is applied to the non-
zero energy shifts produced by Eq. $\left(\ref{eqn:logR}\right)$. The constant
Euclidean time fits to Eq. $\left(\ref{eqn:logR}\right)$ are selected using a
strict $\chi^{2}_{dof}$ criteria, where we require $\chi^{2}_{dof}\leq 1.2$.
This ensures single-state dominance in the energy shift.

In determining the optimal Euclidean-time fit windows to use, each possible
fit window across all three field strengths is considered. Where all the
Euclidean-time plateau and field-strength dependent fits are acceptable,
magnetic polarisability values are calculated from the quadratic coefficient
$c_{2}$ of Eq. $\left(\ref{eqn:EmmCon}\right)$.

Where this fitting process yields no or only a small number of acceptable fit
windows, we also allow the Euclidean-time fit window at each field strength to
vary. These fits must still have a common fit end point but the fit start
point is allowed to increase in a monotonic manner with increasing field
strength. The smallest field strength must have the longest fit window, the
second field strength the second longest and similarly for further field
strengths. This fitting process expands the fit window parameter space
available and is particularly helpful at lighter quark masses.

Figure 8: Constrained quadratic fits of the energy shift to the field quanta
at each quark mass for the proton.

At each pion mass considered, the linearly constrained quadratic fits of Eq.
$\left(\ref{eqn:EmmCon}\right)$ are determined. The final fits are presented
in Figure 8. In every case, the full covariance matrix based $\chi^{2}_{dof}$
indicates an acceptable fit with the charge of the proton constrained to one.

We have also assumed higher-order terms of the expansion of Eq.
$\left(\ref{eqn:EofB}\right)$ are negligible. In order to check the validity
of this assumption, a linearly constrained quadratic $+$ quartic fit
incorporating a $c_{4}\,k_{d}^{4}$ term is performed. We find this quartic
term provides no additional information, and similar magnetic polarisabilities
are observed. When the unconstrained linear $+$ quadratic fit is considered;
the linear coefficient $c_{1}$ produces a charge value $q$ in agreement with
one.

Table 2: Magnetic polarisability values for the neutron and proton at each quark mass considered. The number of sources considered for each quark mass varies as described in Table 1. The numbers in parentheses describe statistical uncertainties. $m_{\pi}$ (GeV) | $\beta^{n}\left(\text{fm}^{3}\times 10^{-4}\right)$ | ${}^{n}\chi^{2}_{dof}$ | $\beta^{p}\left(\text{fm}^{3}\times 10^{-4}\right)$ | ${}^{p}\chi^{2}_{dof}$
---|---|---|---|---
0.702 | 1.91(12) | 0.85 | 1.90(19) | 0.96
0.570 | 1.66(10) | 0.88 | 1.87(18) | 1.10
0.411 | 1.53(29) | 0.74 | 1.98(21) | 0.67
0.296 | 1.27(37) | 0.91 | 1.93(22) | 0.33

This is the first time that Euclidean-time plateau fits have been successfully
constructed for the proton’s magnetic polarisability energy shift. Thus the
SU(3) $\times$ U(1) quark level eigenmode projection technique is effective in
isolating the energy shifts required to access the magnetic polarisability of
the proton to be well determined.

Figure 9: The magnetic polarisability effective energy shift, $\delta
E\left(B,t\right)$ of Eq. $\left(\ref{eqn:logR}\right)$ for the
$m_{\pi}=0.570$ GeV neutron as a function of Euclidean time (in lattice
units), using a smeared source and the SU(3) $\times$ U(1) quark-eigenmode
projection technique. Results for field strengths $k_{d}=1,2,3$ are shown. The
selected fits for this ensemble and the $\chi^{2}_{dof}$ are also illustrated.
Figure 10: The magnetic polarisability effective energy shift, $\delta
E\left(B,t\right)$ of Eq. $\left(\ref{eqn:logR}\right)$ for the
$m_{\pi}=0.296$ GeV neutron as a function of Euclidean time (in lattice
units), using a smeared source and the SU(3) $\times$ U(1) quark-eigenmode
projection technique. Results for field strengths $k_{d}=1$ and $2$ are shown.
The selected fits for this ensemble and the $\chi^{2}_{dof}$ are also
illustrated.

Figure 11: Quadratic fits of the energy shift to the field quanta at each
quark mass for the neutron.

A similar method is followed for the neutron. This time a standard Fourier
transform at the hadron sink projects to zero momentum, and as the neutron is
overall charge less the subtraction process of Eq.
$\left(\ref{eqn:EmmCon}\right)$ is not required. Figures 9 and 10 display
representative Euclidean-time fits to the effective energy shifts of the
neutron, analogous to Figures 6 and 7 for the proton.

The final quadratic fits for the neutron are displayed in Figure 11. From the
quadratic term of the fit in Eq. $\left(\ref{eqn:EmmCon}\right)$, the magnetic
polarisability can be found using Eq. $\left(\ref{eqn:betaConv}\right)$.
Polarisability results for both the proton and neutron are summarised in Table
2.

For the neutron field-strength dependent fit on the $m_{\pi}=0.702$ GeV
ensemble, only the first two non-zero field strengths are used. Fitting to the
third requires the additional quartic term and produces a magnetic
polarisability value that agrees with a single quadratic fit to the first two
field strengths. The neutron polarisability energy shift at the largest field
strength considered suffers from the same signal-to-noise problem as that of
the proton and is hence not used to extract the magnetic polarisability.

The neutron magnetic polarisabilities obtained herein are in good agreement
with those obtained in Ref. Bignell  _et al._ (2018) on the same ensembles.
There, a $U(1)$-based Landau-mode projection technique was applied to Landau-
gauge fixed quark propagators. We find the SU(3) $\times$ U(1) eigenmode quark
projection technique to be similarly successful in isolating the neutron
ground state in a background magnetic field. Now, for the first time, a
unified method for extracting both proton and neutron magnetic
polarisabilities has been presented. It is anticipated that the mesons and
hyperons will also be tractable using this approach.

##  VI Chiral Extrapolation

Figure 12: The leading-order meson loop contribution to the magnetic
polarisability of the nucleon. Figure 13: The next to leading-order meson loop
contribution to the magnetic polarisability of the nucleon, allowing
transitions to nearby strongly coupled baryons.

To connect lattice results to the physical regime, chiral effective-field
theory $\left(\chi EFT\right)$ provides a powerful tool. This analysis is a
generalisation of Ref. Hall  _et al._ (2014) with modifications arising from
the consideration of both the proton and neutron.

The chiral expansion considered is

$\displaystyle\beta^{B}\left(m_{\pi}^{2}\right)=$
$\displaystyle\sum_{M}\,\beta^{M\,B}\left(m_{\pi}^{2},\Lambda\right)+a_{0}\left(\Lambda\right)+a_{2}\left(\Lambda\right)\,m_{\pi}^{2}$
$\displaystyle+$
$\displaystyle\sum_{M,\,B^{\prime}}\,\beta^{M\,B^{\prime}}\left(m_{\pi}^{2},\Lambda\right)+\mathcal{O}\left(m_{\pi}^{3}\right),$
(27)

where $a_{0}\left(\Lambda\right)$ and $a_{2}\left(\Lambda\right)$ are residual
series coefficients Young  _et al._ (2002) constrained by our infinite volume
corrected lattice QCD results and $\Lambda$ is a renormalisation scale. The
leading-order loop contributions $\beta^{M\,B}$ and $\beta^{M,\,B^{\prime}}$
are shown in Figures 12 and 13. Figure 13 allows for transitions of the baryon
$B$ to nearby strongly coupled baryons, $B^{\prime}$, with mass splitting
$\Delta$, via a meson, $M$, loop whereas Figure 12 does not encounter any
baryon mass-splitting effects.

In the heavy-baryon approximation Jenkins and Manohar (1991) appropriate for a
low energy expansion, these have integral forms Hall  _et al._ (2014)

$\displaystyle\beta^{M\,B}\left(m_{\pi}^{2},\Lambda\right)=\frac{e^{2}}{4\,\pi}\frac{1}{288\,\pi^{3}\,f_{\pi}^{2}}\chi_{M\,B}\int\text{d}^{3}k\,\frac{\vec{k}^{2}\,u^{2}\\!\left(k,\Lambda\right)}{\left(\vec{k}^{2}+m_{M}^{2}\right)^{3}},$
(28)

and for $\Delta=m_{B^{\prime}}-m_{B}\neq 0$

$\beta^{M,\,B^{\prime}}\left(m_{\pi}^{2},\Lambda\right)=\frac{e^{2}}{4\,\pi}\frac{1}{288\,\pi^{3}\,f_{\pi}^{2}}\chi_{M\,B^{\prime}}\int\text{d}^{3}k\,u^{2}\\!\left(k,\Lambda\right)\\\
\times\frac{\omega_{\vec{k}}^{2}\,\Delta\,\left(3\,\omega_{\vec{k}}+\Delta\right)+\vec{k}^{2}\,\left(8\,\omega_{\vec{k}}^{2}+9\,\omega_{\vec{k}}\,\Delta+3\,\Delta^{2}\right)}{8\,\omega_{\vec{k}}^{5}\,\left(\omega_{\vec{k}}+\Delta\right)^{3}},$
(29)

respectively. Here $\omega_{\vec{k}}=\sqrt{\vec{k}^{2}+m_{M}^{2}}$ is the
energy carried by the meson $M$ which has three-momentum $\vec{k}$,
$f_{\pi}=92.4$ MeV is the pion decay constant and $u\left(k,\Lambda\right)$ is
a dipole regulator

$\displaystyle
u\left(k,\Lambda\right)=\frac{1}{\left(1+\vec{k}^{2}/\Lambda^{2}\right)^{2}},$
(30)

which ensures that only soft momenta flow through the effective-field theory
degrees of freedom.

The renormalised low-energy coefficients of the chiral expansion are formed
from the residual series coefficients $a_{0}\left(\Lambda\right)$,
$a_{2}\left(\Lambda\right)$ and the analytic contributions of the loop
integrals Young  _et al._ which also depend on $\Lambda$. The full details of
the renormalisation procedure are provided in the Appendix of Ref. Young  _et
al._ . The standard coefficients for full QCD, $\chi_{M\,B}$ and
$\chi_{M\,B^{\prime}}$ reflect photon couplings to the intermediate meson.

The loop integral of Eq. $\left(\ref{eqn:bpN}\right)$ for $\beta^{M\,B}$
contains the leading nonanalytic contribution proportional to $1/m_{M}$. For
finite $B^{\prime}-B$ mass splitting, $\Delta=m_{B^{\prime}}-m_{B}$, the loop
integral of Eq. $\left(\ref{eqn:bpD}\right)$ accounts for transitions to
nearby strongly coupled baryons $B^{\prime}$ and contributes a nonanalytic
logarithmic contribution proportional to
$\left(-1/\Delta\right)\,\log{\left(m_{M}/\Lambda\right)}$, to the chiral
expansion.

Here we consider mesons $M=\pi$ and $\eta$ for nucleon transitions with $B=n$
or $p$ for the integral of Eq. $\left(\ref{eqn:bpN}\right)$. While the total
charge of these mesons is zero, it is important to consider their
contributions in assessing the contribution of sea-quark-loops. The
$\eta^{\prime}$-meson is much heavier and thus its contribution is suppressed
and safely neglected.

We consider transitions to the baryons $B^{\prime}=\Sigma,\,\Lambda,\,\Delta$
and $\Sigma^{*}$ with mesons $M=\pi,\,\eta$ and $K$. These transitions are
accounted for by Eq. $\left(\ref{eqn:bpN}\right)$ with the appropriate mass
splittings and couplings

Our lattice QCD results are electroquenched - they do not include
contributions of photon couplings to disconnected sea-quark loops of the
vacuum. Disconnected sea-quark loops form part of the full meson dressing of
$\chi EFT$ and thus it is necessary to model the corrections associated with
their absence in the lattice QCD calculations. Hence the standard coefficients
for full QCD $\chi_{M\,B}$ and $\chi_{M\,B^{\prime}}$ are altered to account
for partial quenching effects Detmold  _et al._ (2006) as explained in Ref.
Hall  _et al._ (2014) for the neutron. The proton is briefly discussed below
while the neutron follows the analysis in Ref. Hall  _et al._ (2014); Bignell
_et al._ (2018).

###  VI.1 Partially quenched $\chi$EFT

In order to model the corrections to account for partial quenching effects,
the contribution of each quark-flow diagram is separated into ’valence-
valence’, ’valence-sea and ’sea-sea’ contributions. Each of these describes
the coupling of the two photons to the valence or sea quarks available in the
intermediate state mesons. All possible quark-flow diagrams for the
$p\rightarrow p\,\pi^{0}$ channel are constructed in Figure 14 without
attaching external photons to the meson. As there is baryon no mass splitting,
this is an example of Figure 12.

As Figures 14a and 14b have both sea and valence quark lines of the
intermediate meson, photon lines may be attached to the valence or sea-quark
lines of the intermediate meson. Hence they may contribute to all three
sectors. This is in contrast to Figure 14c which contains only valence quarks
and hence contributes only to the valence-valence sector. The contributions to
each sector is proportional to the quark charges, i.e. for Figure 14b the
leading non-analytic term of the chiral expansion has coefficents

$\displaystyle\chi_{v-v}$ $\displaystyle\propto q_{u}^{2},$ (31)
$\displaystyle\chi_{v-s}$ $\displaystyle\propto 2\,q_{u}\,q_{\overline{u}}$
(32) $\displaystyle\chi_{s-s}$ $\displaystyle\propto q_{\overline{u}}^{2},$
(33)

where Eq. $\left(\ref{eqn:factwo}\right)$ reflects the two orderings of the
photon couplings available.

(a) The down quark loop diagram where the two photons can couple to valence-
valence, sea-sea or valence-sea quarks. (b) The up quark loop diagram where
the two photons can couple to valence-valence, sea-sea or valence-sea quarks.

(c) The quark-flow diagram where the two photons can couple only to valence
quarks.

Figure 14: Decomposition of the process $p\rightarrow p\,\pi_{0}$ into its
possible one-loop quark-flow diagrams. The configuration of the two photon
couplings to the valence and/or sea quarks determines the coefficients of
partially quenched chiral perturbation theory.

While the sum of the valence-valence, valence-sea and sea-sea contributions is
zero for this process due to the neutrality of the $\pi^{0}$ meson; the
valence-sea and sea-sea terms are not present in the lattice QCD simulation
and hence must be accounted for.

The sea-sea disconnected sea-quark-loop flow for Diagram 14a can be isolated
by temporarily replacing the down-quark loop with a strange quark Leinweber
(2004). This provides a coupling strength

$\displaystyle\chi_{s-s}^{diag(b)}\propto\,q^{2}_{\overline{d}}\,\chi^{2}_{K^{0}\Sigma^{+}}=q^{2}_{\overline{d}}\,\,2\,\left(D-F\right)^{2}.$
(34)

Repeating the above procedure for the up-quark loop of Diagram 14b one finds

$\displaystyle\chi_{s-s}^{diag(c)}\propto$
$\displaystyle\,q_{\overline{u}}^{2}\,\left(\chi^{2}_{K^{+}\Lambda}+\chi^{2}_{\Sigma^{0}K^{+}}\right)$
$\displaystyle=$ $\displaystyle
q_{\overline{u}}^{2}\,\left(\frac{1}{3}\,\left(3\,F+D\right)^{2}+\left(D-F\right)^{2}\right).$
(35)

The components of the $p\rightarrow p\,\pi^{0}$ channel which have a
disconnected sea-quark loop have been identified and hence the sea-sea
contributions have been calculated. The same process may be performed for the
valence-sea contributions. As the total contribution is known from standard
$\chi$PT, the remaining valence-valence contribution which includes the
connected quark-flow diagram of Diagram 14(a) is also known. All such channels
for the integral processes described by Eq.
$\left(\ref{eqn:chiEFT:fitfunc}\right)$ are investigated using the
diagrammatic procedure described above for $p\rightarrow p\,\pi^{0}$.

Table 3: Chiral coefficients for the leading-order loop integral contributions
for the proton.

Process Total Valence-sea Sea-sea $p\rightarrow N\,\pi$ $p\rightarrow
p\,\pi_{0}$ $0$
$\frac{3}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)+2\,q_{d}\,q_{\overline{d}}\,\chi^{2}_{K^{0}\,\Sigma^{+}}\right)$
$\frac{3}{6}\,\left(q_{\overline{u}}^{2}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)+q_{\overline{d}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{+}}\right)$
$p\rightarrow p\,\eta$ $0$
$\frac{1}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)+2\,q_{d}\,q_{\overline{d}}\,\chi^{2}_{K^{0}\,\Sigma^{+}}\right)$
$\frac{1}{6}\,\left(q_{\overline{u}}^{2}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)+q_{\overline{d}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{+}}\right)$
$p\rightarrow p\,\eta^{\prime}$ $0$
$\frac{2}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)+2\,q_{d}\,q_{\overline{d}}\,\chi^{2}_{K^{0}\,\Sigma^{+}}\right)$
$\frac{2}{6}\,\left(q_{\overline{u}}^{2}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)+q_{\overline{d}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{+}}\right)$
$p\rightarrow n\,{\pi^{+}}$ $\chi^{2}_{{\pi^{+}}\,n}$
$2\,q_{u}\,q_{\overline{d}}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)$
$q_{\overline{d}}^{2}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)$
$p\rightarrow p^{+}\,\pi^{-}$ $0$
$2\,q_{d}\,q_{\overline{u}}\,\chi^{2}_{K^{0}\,\Sigma^{+}}$
$q_{\overline{u}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{+}}$ $p\rightarrow\Sigma\,K$
$p\rightarrow\left(\Sigma^{0},\Lambda\right)\,{K^{+}}$
$\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}$
$2\,q_{u}\,q_{\overline{s}}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)$
$q_{\overline{s}}^{2}\,\left(\chi^{2}_{{K^{+}}\,\Sigma^{0}}+\chi^{2}_{{K^{+}}\,\Lambda}\right)$
$p\rightarrow\Sigma^{+}\,K^{0}$ $0$
$2\,q_{d}\,q_{\overline{s}}\,\chi^{2}_{K^{0}\,\Sigma^{+}}$
$q_{\overline{s}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{+}}$ $p\rightarrow\Delta\,\pi$
$p\rightarrow\Delta^{0}\,{\pi^{+}}$ $\chi^{2}_{{\pi^{+}}\,\Delta^{0}}$
$2\,q_{u}\,q_{\overline{d}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}$
$q_{\overline{d}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}$
$p\rightarrow\Delta^{++}\,\pi^{-}$ $\chi^{2}_{\pi^{-}\,\Delta^{++}}$
$2\,q_{d}\,q_{\overline{u}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}$
$q_{\overline{u}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}$
$p\rightarrow\Delta^{+}\,\pi_{0}$ $0$
$\frac{3}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}+2\,q_{d}\,q_{\overline{d}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}\right)$
$\frac{3}{6}\,\left(q_{\overline{u}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}+q_{\overline{d}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}\right)$
$p\rightarrow\Delta^{+}\,\eta$ $0$
$\frac{1}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}+2\,q_{d}\,q_{\overline{d}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}\right)$
$\frac{1}{6}\,\left(q_{\overline{u}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}+q_{\overline{d}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}\right)$
$p\rightarrow\Delta^{+}\,\eta^{\prime}$ $0$
$\frac{2}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}+2\,q_{d}\,q_{\overline{d}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}\right)$
$\frac{2}{6}\,\left(q_{\overline{u}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}+q_{\overline{d}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}\right)$
$p\rightarrow\Sigma^{*}\,K$ $p\rightarrow\Sigma^{*\,0}\,{K^{+}}$
$\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}$
$2\,q_{u}\,q_{\overline{s}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}$
$q_{\overline{s}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}$
$p\rightarrow\Sigma^{*\,+}\,K^{0}$ $0$
$2\,q_{d}\,q_{\overline{s}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}$
$q_{\overline{s}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,+}}$

Table 4: Chiral coefficients for the leading-order loop integral contributions
for the neutron.

Process Total Valence-sea Sea-sea $n\rightarrow N\,\pi$ $n\rightarrow
n\,\pi_{0}$ $0$
$\frac{3}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}+2\,q_{d}\,q_{\overline{d}}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)\right)$
$\frac{3}{6}\,\left(q_{\overline{u}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}+q_{\overline{d}}^{2}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)\right)$
$n\rightarrow n\,\eta$ $0$
$\frac{1}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}+2\,q_{d}\,q_{\overline{d}}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)\right)$
$\frac{1}{6}\,\left(q_{\overline{u}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}+q_{\overline{d}}^{2}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)\right)$
$n\rightarrow n\,\eta^{\prime}$ $0$
$\frac{2}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}+2\,q_{d}\,q_{\overline{d}}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)\right)$
$\frac{2}{6}\,\left(q_{\overline{u}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}+q_{\overline{d}}^{2}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)\right)$
$n\rightarrow p\,\pi^{-}$ $\chi^{2}_{\pi^{-}\,p}$
$2\,q_{d}\,q_{\overline{u}}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)$
$q_{\overline{u}}^{2}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)$
$n\rightarrow n^{-}\,\pi^{+}$ $0$
$2\,q_{u}\,q_{\overline{d}}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}$
$q_{\overline{d}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}$ $n\rightarrow\Sigma\,K$
$n\rightarrow\left(\Sigma^{0},\,\Lambda\right)\,K^{0}$ $0$
$2\,q_{d}\,q_{\overline{s}}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)$
$q_{\overline{s}}^{2}\,\left(\chi^{2}_{K^{0}\,\Sigma^{0}}+\chi^{2}_{K^{0}\,\Lambda}\right)$
$n\rightarrow\Sigma^{-}\,K^{-}$ $\chi^{2}_{{K^{+}}\,\Sigma^{-}}$
$2\,q_{u}\,q_{\overline{s}}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}$
$q_{\overline{s}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{-}}$
$n\rightarrow\Delta\,\pi$ $n\rightarrow\Delta^{0}\,\pi_{0}$ $0$
$\frac{3}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}+2\,q_{d}\,q_{\overline{d}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}\right)$
$\frac{3}{6}\,\left(q_{\overline{u}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}+q_{\overline{d}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}\right)$
$n\rightarrow\Delta^{0}\,\eta$ $0$
$\frac{1}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}+2\,q_{d}\,q_{\overline{d}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}\right)$
$\frac{1}{6}\,\left(q_{\overline{u}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}+q_{\overline{d}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}\right)$
$n\rightarrow\Delta^{0}\,\eta^{\prime}$ $0$
$\frac{2}{6}\,\left(2\,q_{u}\,q_{\overline{u}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}+2\,q_{d}\,q_{\overline{d}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}\right)$
$\frac{2}{6}\,\left(q_{\overline{u}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}+q_{\overline{d}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}\right)$
$n\rightarrow\Delta^{+}\,\pi^{-}$ $\chi^{2}_{\pi^{-}\,\Delta^{+}}$
$2\,q_{d}\,q_{\overline{u}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}$
$q_{\overline{u}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}$
$n\rightarrow\Delta^{-}\,{\pi^{+}}$ $\chi^{2}_{{\pi^{+}}\,\Delta^{-}}$
$2\,q_{u}\,q_{\overline{d}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}$
$q_{\overline{d}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}$
$n\rightarrow\Sigma^{*}\,K$ $n\rightarrow\Sigma^{*\,0}\,K^{0}$ $0$
$2\,q_{d}\,q_{\overline{s}}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}$
$q_{\overline{s}}^{2}\,\chi^{2}_{K^{0}\,\Sigma^{*\,0}}$
$n\rightarrow\Sigma^{*\,-}\,{K^{+}}$ $\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}$
$2\,q_{u}\,q_{\overline{s}}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}$
$q_{\overline{s}}^{2}\,\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}$

Table 5: $SU(3)$ flavour coupling coefficients for the chiral effective field
theory analysis. The header row indicates the intermediate baryon species in
the meson-baryon loop dressing. Through conservation of quark flavour, one can
identify the baryon which is being dressed.

$\Sigma^{*}$ $\Delta$ $\Sigma$ $\Lambda$ $N$
$\chi^{2}_{{K^{+}}\,\Sigma^{*\,-}}$ $\frac{4}{9}\,\mathcal{C}^{2}$
$\chi^{2}_{\pi^{-}\,\Delta^{+}}$ $\frac{4}{9}\,\mathcal{C}^{2}$
$\chi^{2}_{{K^{+}}\,\Sigma^{-}}$ $2\,\left(D-F\right)^{2}$
$\chi^{2}_{K^{0}\,\Lambda}$ $\frac{1}{3}\,\left(D+3\,F\right)^{2}$
$\chi^{2}_{{\pi^{+}}\,n}$ $2\,\left(D+F\right)^{2}$
$\chi^{2}_{K^{0}\,\Sigma^{*\,0}}$ $\frac{2}{9}\,\mathcal{C}^{2}$
$\chi^{2}_{{\pi^{+}}\,\Delta^{-}}$ $\frac{4}{3}\,\mathcal{C}^{2}$
$\chi^{2}_{K^{0}\,\Sigma^{0}}$ $\phantom{2\,}\left(D-F\right)^{2}$
$\chi^{2}_{{K^{+}}\,\Lambda}$ $\frac{1}{3}\,\left(D+3\,F\right)^{2}$
$\chi^{2}_{\pi^{-}\,p}$ $2\,\left(D+F\right)^{2}$
$\chi^{2}_{K^{0}\,\Sigma^{*\,+}}$ $\frac{4}{9}\,\mathcal{C}^{2}$
$\chi^{2}_{{\pi^{+}}\,\Delta^{0}}$ $\frac{4}{9}\,\mathcal{C}^{2}$
$\chi^{2}_{{K^{+}}\,\Sigma^{0}}$ $\phantom{2\,}\left(D-F\right)^{2}$
$\chi^{2}_{{K^{+}}\,\Sigma^{*\,0}}$ $\frac{2}{9}\,\mathcal{C}^{2}$
$\chi^{2}_{\pi^{-}\,\Delta^{++}}$ $\frac{4}{3}\,\mathcal{C}^{2}$
$\chi^{2}_{K^{0}\,\Sigma^{+}}$ $2\,\left(D-F\right)^{2}$

Having performed this procedure for each relevant channel, the coefficients
used when fitting the lattice QCD results reflect the absence of the
disconnected sea-quark-loop contributions and can be determined by subtracting
the valence-sea and sea-sea contributions from the total contribution in
Tables 3 and 4 for the proton and neutron respectively. The numerical value of
the coefficients can be found in Table 5 where the standard values of
$g_{A}=1.267$ and $\mathcal{C}=1.52$ with $g_{A}=D+F$ and the SU(6) symmetry
relation $F=\frac{2}{3}\,D$ are used.

The regulator mass, $\Lambda=0.80$ GeV Wang  _et al._ (2009a); Young  _et al._
(2002); Leinweber  _et al._ (2005, 2006); Wang  _et al._ (2009b) is chosen in
anticipation of accounting for the missing disconnected sea-quark-loop
contributions in the lattice QCD calculations. This regulator mass enables
corrections to the pion cloud contributions associated with missing
disconnected sea-quark-loop contributions as it defines a pion cloud
contribution to masses Young  _et al._ (2002), magnetic moments Leinweber  _et
al._ (2005) and charge radii Wang  _et al._ (2009a). The nucleon core
contribution is insensitive to sea-quark-loop contributions at this regulator
mass Wang  _et al._ (2014).

To consider the effect of the finite-volume of the lattice, we replace the
continuum integrals of the chiral expansion with sums over the momenta
available on the periodic lattice. It is important to note that the lattice
volume is slightly different on each of the four lattice ensembles used due to
our use of the Sommer scale. To produce inite-volume corrected (FVC) results,
$\beta^{FVC}_{v-v}$, we take the difference between these sums and the
continuum integrals

$\displaystyle\beta^{FVC}_{v-v}\left(m_{\pi}^{2}\right)=\beta^{lat.}_{v-v}\left(m_{\pi}^{2}\right)$
$\displaystyle-\left(\sum_{M}\,\beta_{SUM}^{M\,B}\left(m_{\pi}^{2},\Lambda^{FV}\right)+\sum_{M,\,B^{\prime}}\beta_{SUM}^{M\,B^{\prime}}\left(m_{\pi}^{2},\Lambda^{FV}\right)\right)$
$\displaystyle+\left(\sum_{M}\beta^{M\,B}\left(m_{\pi}^{2},\Lambda^{FV}\right)+\sum_{M,\,B^{\prime}}\,\beta^{M\,B^{\prime}}\left(m_{\pi}^{2},\Lambda^{FV}\right)\right),$
(36)

where we note that we are correcting for finite-volume only and hence the
coefficients used in evaluating these sums and integrals reflect only valence-
valence contributions. The finite-volume corrections should be independent of
the value of the regulator parameter, $\Lambda^{FV}$, as long as
$\Lambda^{FV}$ is sufficiently large. Here we choose $\Lambda^{FV}=2.0$ GeV
Hall  _et al._ (2010).

The strength of the $\chi PT$ analysis is that the leading and next-to-leading
non-analytic terms of the chiral expansion are model-independent predictions
of chiral perturbation theory. The leading source of uncertainty comes from
the higher order terms in the expansion. We provide an estimation of the
uncertainty in these terms through variation of the regulator parameter
$\Lambda$ over a wide range.

###  VI.2 Analysis

The extrapolation to the physical regime requires that the residual series
coefficients $a_{0}\left(\Lambda\right)$ and $a_{2}\left(\Lambda\right)$ are
constrained by fitting to the finite-volume corrected lattice results

$\displaystyle
a_{0}\left(\Lambda\right)+a_{2}\left(\Lambda\right)\,m_{\pi}^{2}$
$\displaystyle=\beta_{v-v}^{FVC}\left(m_{\pi}^{2}\right)-\sum_{M}\,\beta_{v-v}^{M\,B}\left(m_{\pi}^{2},\,\Lambda\right)$
$\displaystyle-\sum_{M,\,B^{\prime}}\,\beta_{v-v}^{M\,B^{\prime}}\left(m_{\pi}^{2},\,\Lambda\right),$
(37)

where the regulator parameter $\Lambda$ takes the value $\Lambda=0.80$ GeV as
discussed above. After the residual series coefficients have been determined,
the chiral expansion of Eq. $\left(\ref{eqn:chiEFT:fitfunc}\right)$ can be
used to calculate the magnetic polarisability for any value for $m_{\pi}^{2}$.
Valence-sea and sea-sea loop integral contributions are accounted for by using
the chiral coefficients for the “total” process. A physical extrapolation is
produced by setting $m_{\pi}=m_{\pi}^{phys}=0.140$ GeV.

The resulting chiral extrapolation for the proton predicts a magnetic
polarisability of $\beta^{p}=2.79(22)\times 10^{-4}$ fm3 where the numbers in
parentheses represents the statistical uncertainty. By considering the
variation of the regulator parameter over the broad range $0.6\text{
GeV}\leq\Lambda\leq 1.0$ GeV a systematic uncertainty associated with the
higher order terms of the chiral expansion can be reported. Thus the
prediction for the magnetic polarisability of the proton at the physical point
is

$\displaystyle\beta^{p}=2.79(22)\left({}^{+13}_{-18}\right)\times
10^{-4}\text{ fm}^{3}.$

Figure 15: The magnetic polarisability of the proton, $\beta^{p}$, from the
chiral effective field analysis herein $\left(\chi EFT\text{
Prediction}\right)$. Lattice results of this work, finite-volume corrected (FV
Corr.) with total QCD coefficients are compared with experimental measurements
via an infinite-volume (Inf. Vol) chiral extrapolation with total QCD
coefficents. The error bar at the physical point reflects systematic and
statistical uncertainties added in quadrature. Experimental results from the
PDG Tanabashi  _et al._ (2018), McGovern _et al._ McGovern  _et al._ (2013),
Beane _et al._ Beane  _et al._ (2003), Blanpied _et al._ Blanpied  _et al._
(2001), Olmos de León _et al._ de León  _et al._ (2001), MacGibbon _et al._
MacGibbon  _et al._ (1995) and Pasquini _et al._ Pasquini  _et al._ (2019) are
offset for clarity.

Figure 15 highlights a comparision of the chiral extrapolation prediction
produced herein with a selection of recent experimental measurements.
Excellent agreement is seen between the experimental measurements and the
result obtained herein. This highlights the utility of the quark projection
technique and partially quenched chiral effective field theory used herein. It
validates our understanding of QCD behind their development and use.

Figure 16: Finite volume (FV) extrapolations of $\beta^{p}$ with total full
QCD coefficients appropriate for fully dynamical background field lattice QCD
simulations. The infinite-volume case relevant to experiment is also
illustrated. Both finite-volume valence-valence lattice QCD results (Lattice
Points) and their finite-volume (FV Corr.) total QCD corrected values are
illustrated. The Lattice Points are offset for clarity.

The chiral expansion of Eq. $\left(\ref{eqn:chiEFT:fitfunc}\right)$ may also
be used to guide future lattice QCD calculations at a range of lattice volumes
by using the discretised sum forms of the continuum integrals with either
valence-valence or total integral coefficients. Figure 16 shows finite volume
extrapolations of $\beta^{p}$ with total full QCD coefficients for a range of
lattice volumes, $3.0\text{ fm}\leq L_{s}\leq 7.0$ fm and pion masses with
$m_{\pi}\,L_{s}\geq 2.4$. Here the $7.0$ fm result still differs from the
infinite volume result by $\sim 4\%$.

Figure 17: The magnetic polarisability of the neutron, $\beta^{n}$ from our
chiral effective field analysis $\left(\chi EFT\text{ Prediction}\right)$.
Lattice results of this work, finite-volume corrected (FV Corr.) with total
QCD coefficients are compared with experimental measurements via an infinite-
volume (Inf. Vol) chiral extrapolation with total QCD coefficents. The error
bar at the physical point reflects systematic and statistical uncertainties
added in quadrature. Experimental results from Kossert _et al._ Kossert  _et
al._ (2002, 2003), the PDG Tanabashi  _et al._ (2018), Myers _et al._ Myers
_et al._ (2014) and Griesshammer _et al._ Griesshammer  _et al._ (2012) are
offset for clarity.

The same process is used to predict the value for the magnetic polarisability
of the neutron at the physical point to be

$\displaystyle\beta^{n}=2.06(26)\left({}^{+15}_{-20}\right)\times 10^{4}\text{
fm}^{3},$

where the numbers in parentheses represent statistical and systematic errors
respectively. This value is in very good agreement with the value obtained in
our earlier work of Ref. Bignell  _et al._ (2018) where the $U(1)$ Landau
eigenmode projection technique is used with a chiral extrapolation to obtain
$\beta^{n}=2.05(25)(19)\times 10^{4}$ fm3. This agreement indicates the
success of both the $SU(3)\times U(1)$ and $U(1)$ eigenmode quark projection
techniques. Figure 17 presents a comparison of $\beta^{n}$ to recent
experimental measurements where good agreement is also observed.

Figure 18: Finite volume (FV) extrapolations of $\beta^{n}$ with total full
QCD coefficients appropriate for fully dynamical background field lattice QCD
simulations. The infinite-volume case relevant to experiment is also
illustrated. Both finite-volume valence-valence lattice QCD results (Lattice
Points) and their finite-volume (FV Corr.) total QCD corrected values are
illustrated. The Lattice Points are offset for clarity.

In an identical manner to the proton, Figure 18 presents finite-volume
extrapolations with full QCD coefficients as a guide to future lattice QCD
calculations. For both the proton and neutron, the leading and next-to-leading
non-analytic terms of the chiral expansion give rise to a significant
enhancement of the magnetic polarisabilities. In the case of the neutron, this
chiral contribution reverses the trend observed on the lattice. Nevertheless,
the coefficients of these non-analytic terms are model independent and well
known. Figures 16 and 18 highlight the volume dependence of these
contributions and indicate a significant challenge to directly observe these
effects in future lattice QCD calculations. Here we note that the infinite
volume value for the neutron is greater than that of the $L_{s}=7.0$ fm
lattice by $\sim 6\%$.

As the lattice simulations are performed with only a single lattice spacing,
it is not possible to directly quantify an uncertainty associated with taking
the continuum limit. The non-perturbatively improved clover fermion action
used herein has been shown Edwards  _et al._ (1998); Zanotti  _et al._ (2005)
to display excellent scaling behaviour for hadron masses such that the
$\mathcal{O}\left(a^{2}\right)$ corrections are expected to be small relative
to the uncertainties already presented. Indeed Ref. Edwards  _et al._ (1998)
estimates a $<0.5\%$ error at the lattice spacing used in this study.

Our conservative estimates for the systematic error due to the continuum limit
extrapolation have a negligible effect on the final error when added in
quadrature.

###  VI.3 Magnetic polarisability difference $\beta^{p}-\beta^{n}$

The difference between the magnetic polarisability of the proton and the
neutron can provide a test of Reggeon dominance Gasser  _et al._ (2015, 2020).
When Reggeon dominance is assumed, the difference of magnetic polarisabilities
can be predicted using chiral perturbation theory techniques and Baldin sum
rules Baldin (1960); Gasser  _et al._ (2015); Griesshammer  _et al._ (2012).

Figure 19: The difference of the proton and neutron magnetic polarisabilities
$\beta^{p}-\beta^{n}$. Lattice results of this work, finite-volume corrected
(FV Corr.) with full QCD coefficients are compared with experimental
measurements via an infinite-volume (Inf. Vol) chiral extrapolation. The error
bar at the physical point (This Work) reflects systematic and statistical
uncertainties added in quadrature. Experimental results from the PDG Tanabashi
_et al._ (2018) and a Reggeon dominance prediction from Gasser _et al._ Gasser
_et al._ (2015) are offset for clarity.

We calculate the correlated difference between the magnetic polarisability of
the proton and the neutron at each quark mass and then extrapolate to the
physical regime using the formalism already discussed. In taking the
difference, $u$-$d$ symmetry in the leading loop-integral coefficients of the
chiral expansion in full QCD cause the contributions to cancel, leaving a
simple linear extrapolation of our infinite-volume and full QCD corrected
lattice results. The resulting prediction at the physical point is

$\displaystyle\beta^{p}-\beta^{n}=0.80\left(28\right)\left(4\right)\times
10^{4}\text{ fm}^{3},$

where both statistical and systematic errors are indicated respectively. The
central value differs from the difference between the extrapolated values
discussed above due to the removal of rounding errors at each stage of the
calculation. Figure 19 shows an extrapolation of all four lattice mass results
to the physical regime with comparison to the PDG value Tanabashi  _et al._
(2018) and a result derived using Reggeon dominance Gasser  _et al._ (2015).

##  VII Conclusions

The magnetic polarisabilities of the proton and neutron have been calculated
using asymmetric operators at the source and sink. Gauge invariant Gaussian
smearing at the source encodes the dominant QCD dynamics while the SU(3)
$\times$ U(1) eigenmode quark projection technique is used at the sink to
encapsulate the low-lying quark-level Landau physics resulting from the
presence of the uniform magnetic field.

At the hadronic level, it is crucial to use a Landau wave function projection
onto the proton two point correlation function as the proton is charged and
hence experiences Landau level physics in a uniform magnetic field. The
combination of these techniques has enabled constant plateau fits to be found
in the magnetic polarisability energy shift of the proton for the first time.

Furthermore, using the QCD gauge-covariant SU(3) $\times$ U(1) projection we
are simultaneously able to produce magnetic polarisability energy shifts
corresponding to both the neutron and proton ground states in a uniform
background field. This represents a significant advance over the previous
gauge-fixed $U(1)$ quark-level projection used to study the neutron
polarisability.

Connection with experimental results in the physical regime is achieved
through the use of heavy-baryon chiral effective field theory and lattice QCD
simulations at several pion masses. The resulting theoretical prediction for
the magnetic polarisability of the proton is
$\beta^{p}=2.79(22)\\!\left({}^{+13}_{-18}\right)\times 10^{-4}$ fm3 and
$\beta^{n}=2.06(26)\\!\left({}^{+15}_{-20}\right)\times 10^{-4}$ fm3 for the
neutron. These predictions are built upon _ab initio_ lattice QCD simulations
using effective-field theory techniques to account for disconnected sea-quark-
loop contributions, the finite volume of the periodic lattice and an
extrapolation to the light quark masses of nature. These theoretical
predictions are in good agreement with current experimental measurements and
pose an interesting challenge for increased experimental precision.

While we are necessarily in the confining phase of QCD, due to the small
$\vec{B}$ field strengths required for the perturbative energy expansion; from
the success of the SU(3) $\times$ U(1) eigenmode projected quark sink
technique it is clear that the external magnetic field has a significant
effect on the distribution of the quarks within the nucleon.

Our lattice results are electroquenched, they do not directly incorporate the
sea-quark-loop contributions from the magnetic field. Future work would
require a separate Monte Carlo ensemble for each value of $B$ considered and
as such is prohibitively expensive due to a loss of QCD correlations. Another
avenue that could be considered is to investigate the relativistic corrections
to the energy-field expansion of Eq. $\left(\ref{eqn:EofB}\right)$. Here,
improvements in lattice precsion will be required in order to succesfully fit
the energies $\left(E+M\right)$ and construct the relativistic energy shift.

It will be particularly interesting to extend this work to the case of
hyperons. There the increased mass of the strange quark will illustrate
differences between $\Sigma^{+}$ and $p$ or $\Xi^{0}$ and $n$ polarisabilities
and give first insights into the environment sensitivity of quark-sector
contributions to baryon magnetic polarisabilities.

###### Acknowledgements.

It is a pleasure to thank Heinrich Leutwyler for his comments highlighting how
the proton and neutron magnetic polarisability difference provides a test of
Reggeon dominance. We thank the PACS-CS Collaboration for making their $2+1$
flavour configurations available and the ongoing support of the International
Lattice Data Grid (ILDG). This work was supported with supercomputing
resources provided by the Phoenix HPC service at the University of Adelaide.
This research was undertaken with the assistance of resources from the
National Computational Infrastructure (NCI). NCI resources were provided
through the National Computational Merit Allocation Scheme, supported by the
Australian Government through Grants No. LE190100021, LE160100051 and the
University of Adelaide Partner Share. R.B. was supported by an Australian
Government Research Training Program Scholarship. This research is supported
by the Australian Research Council through Grants No. DP140103067,
DP150103164, DP190102215 (D.B.L) and DP190100297 (W.K).

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# The impact of frame quantization on the dynamic range of a one-bit image
sensor

Lucas J. Koerner
Department of Electrical and Computer Engineering
University of St. Thomas
St. Paul, MN 55105
<EMAIL_ADDRESS>

###### Abstract

For a one-bit image sensor, the number of frames captured determines the
quantization step size of the binary rate measurement. The discrete values of
the binary rate set a maximum measurable intensity. In this note, we consider
how this frame quantization impacts the high-end of the dynamic range of a
one-bit image sensor.

_Keywords_ SPAD, Image Sensor, Dynamic Range

##  1 Background

A single frame from a pixel of a one-bit image sensor produces a binary
measurement of either 0 or 1. With zero read noise, the probability that a
single frame detects a 1 (one or more photons) is given by Bernoulli
statistics as $1-e^{-H}$ and the probability of 0 is subsequently $e^{-H}$.
$H$ is termed the exposure and is the the average photon count within the
exposure window. The exposure relates to the photon flux ($\lambda$) and the
exposure time ($T$) as $H=\lambda T$. From the average $\overline{Y}$ over a
sequence of $N$ Bernoulli samples, or frames, ${Y_{1},...,Y_{N}}$ the exposure
is estimated as $\hat{H}=-\ln(1-\overline{Y})$ [1].

##  2 Maximum measurable exposure

Since a sequence of all $1$s estimates an infinite exposure, the largest
finite exposure estimate is found when one of $N$ frames measures a $0$ for a
binary rate of $\overline{Y}=\hat{p}=(N-1)/N$. This maximum exposure estimate
is then

$\hat{H}_{+qntz}=-\ln\left(1-\frac{N-1}{N}\right)=\ln(N).$ (1)

The exposure referred signal to noise ratio (SNR) as a function of the binary
rate $p$, exposure level $H=\lambda T$, and number of frames $N$ is given as
[2, 3]:

$\textit{SNR}(p,H,N)=-\ln(1-p)\sqrt{N}\sqrt{\frac{1-p}{p}}=H\sqrt{N}\sqrt{\frac{e^{-H}}{1-e^{-H}}}.$
(2)

The quantization of the binary rate leads to an SNR at the high-end of the
exposure found by substituting the result of Eq. 1 into Eq. 2 of

$\textit{SNR}_{qtz+}=\ln(N)\sqrt{\frac{N}{N-1}}\approx\ln(N)$ (3)

where the right-most approximation is valid for $N\gg 1$.

One definition of Bernoulli imager dynamic range calculates the ratio of the
two exposures ($H_{-}$ and $H_{+}$) that have an SNR of one [4]. To find these
exposure values, Eq. 2 is rearranged and the equation

$e^{H}-NH^{2}=1$ (4)

is solved numerically. However, the high-end exposure value $(H_{+})$ found is
not measurable as it is beyond the limit set be frame quantization given in
Eq. 1.

Consider the scenario of $N=1000$ frames captured. At an SNR of $1$
$($0\text{\,}\mathrm{d}\mathrm{B}$)$, the high-end exposure value found from
Eq. 4 is $H_{+}=11.85$. However, this exposure level is highly likely to
saturate. With $1000$ frames at an exposure of $H_{+}=11.85$ the likelihood of
measuring all ones is $0.993$ as calculated from $(1-e^{-H})^{N}$. In other
words, the high-end exposure solution ($H_{+}$) with $\textit{SNR}=1$ cannot
be practically measured due to frame quantization and saturation. To measure
this exposure, when frame quantization is considered, Eq. 1 asserts that
$140,085$ frames are required. When a frame constraint of $1000$ is
maintained, Eq. 1 predicts a maximum measurable exposure of
$\overline{H}_{+qntz}=\ln(N)=6.91$ which has a $0.368$ chance of saturation.

The lower end of the exposure range for $\mathit{SNR}=1$ approaches
$H_{-}=\frac{1}{N}$ for $N\gg 1$ and $H\ll 1$. This result is found by
expanding $e^{H}\approx H+1$ and substituting into Eq. 4. This low-end
exposure is measurable as $H_{-}=\sqrt{\frac{1}{N}}=\hat{p}=1/N$ for $N\gg 1$.

Fig. 1 shows the exposure referred SNR versus exposure for captures of $100$
and $1000$ frames. For a given (continuous) exposure level the representation
of Fig. 1 rounds to the closest discrete binary rate ($p$) before evaluating
the SNR using Eq. 2. The curve stops at the maximum measurable exposure near
$\ln(N)$ with an SNR significantly above $0\text{\,}\mathrm{d}\mathrm{B}$.

Figure 1: The SNR of a binary imager using discrete steps of the binary rate
estimate to emphasize the quantization due to the finite number of frames
($N$).

##  3 Dynamic range

Table 1 shows two definitions of dynamic range. The first value is
$\textit{DR}=H_{+}/H_{-}$ while the value that considers frame quantization is
$\textit{DR}_{qtz}=H_{+qtz}/H_{-qtz}$. Looking across the rows of Table 1
shows that the dynamic range measure that considers frame quantization is
consistently lower than the measure that calculates the ratio (or the
logarithmic difference) of the two exposure levels with $\mathit{SNR}=1$. At
low numbers of frames, the deviations between the two dynamic range measures
are most significant, with around $6\text{\,}\mathrm{d}\mathrm{B}$ of
difference when $100$ frames are measured. The last column shows the exposure
level for a 50% change of saturation which is evaluated as

$H_{50\%-sat.}=-\ln\left(1-0.5^{1/N}\right).$ (5)

An exposure limit set by a 50% chance of saturation follows closely with
$H_{+qtz}$.

N | $H_{-}$ | $H_{+}$ | $H_{-qtz}$ | $H_{+qtz}$ |  $DR$ [dB] |  $DR_{qtz}$ [dB] | $H_{50\%-sat.}$
---|---|---|---|---|---|---|---
10 | $1.05\text{\times}{10}^{-01}$ | 5.83 | $1.05\text{\times}{10}^{-01}$ | 2.30 | 34.86 | 26.79 | 2.70
100 | $1.01\text{\times}{10}^{-02}$ | 9.00 | $1.01\text{\times}{10}^{-02}$ | 4.61 | 59.04 | 53.22 | 4.98
1000 | $1.00\text{\times}{10}^{-03}$ | 11.85 | $1.00\text{\times}{10}^{-03}$ | 6.91 | 81.47 | 76.78 | 7.28
10000 | $1.00\text{\times}{10}^{-04}$ | 14.57 | $1.00\text{\times}{10}^{-04}$ | 9.21 | 103.27 | 99.29 | 9.58
100000 | $1.00\text{\times}{10}^{-05}$ | 17.20 | $1.00\text{\times}{10}^{-05}$ | 11.51 | 124.71 | 121.22 | 11.88
Table 1: A comparison of two dynamic range metrics for single photon imagers.
Each row assesses the minimum and maximum exposure and resulting dynamic range
for a particular number of frames, $N$. The last column, $H_{50\%-sat.}$, is
the exposure value with a $50\%$ chance of being saturated (returning a one
for every frame).

##  4 Discussion

The compressive non-linear mapping from binary rate to intensity of one-bit
image sensors expands dynamic range as compared to a linear response. This
compressive transfer response suggests that one-bit sensors do not saturate.
However, the intensity estimate when a one-bit image sensor measures all 1s is
not defined which indicates a saturation intensity. The maximum measurable
intensity level (“just before” saturation) depends on the number of frames
since the binary rate measurement is quantized in steps of $1/N$. In this note
we defined the maximum measurable exposure as that estimated when the binary
rate estimate is $\hat{p}=\frac{N-1}{N}$. The occurrence of saturation after a
sequence of frames is probabilistic, so a precise exposure level that is
guaranteed to saturate does not exist.

Ideally, a binary imager without read noise is operated with a sufficiently
rapid frame rate so that no pixels saturate. However, due to power consumption
and bandwidth constraints, a practical limit on the number of frames captured
may exist. The finite number of frames discretizes the measurement of the
binary rate and limits the maximum measurable exposure before saturation. An
open challenge in the application of binary imagers is accounting for
saturated pixels and defining capture schemes that maximize performance when
the number of frames is constrained.

## References

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