\section{Introduction}

The characteristics of the top-quark, mainly the total decay width which is one of the fundamental property of top physics
is measured with very good precision from the partial decay width $\Gamma(t \to Wb)$. In addition, its huge mass
$m_t = 173.0\pm 0.4\hspace{0.8mm}GeV$ \cite{Data2018}, as well as its anomalous couplings to bosons in the $t\bar t \gamma$,
$t\bar t Z$, $t\bar t g$, $Wtb$, $\bar t t H$ and $tq\gamma$ vertices, have turned the top-quark into one of the most attractive particles
for new physics searches. Measurements of the properties of the top-quark offer an interesting probe to understanding the electroweak sector \cite{CPYuan}
and physics Beyond the Standard Model (BSM). These and other characteristics have led to developing your own physics program for the top-quark
for present and future $pp$, $e^-p$ and  $e^+e^-$ colliders. Therefore, top-quark physics is one of the most attractive topics at the Large
Hadron Collider (LHC), as well at the High-Luminosity Large Hadron Collider (HL-LHC) and High-Energy Large Hadron Collider (HE-LHC). However,
at the post LHC era a very attractive and interesting option to study the physics of the top-quark, mainly its anomalous couplings is through
the future electron-proton $(e^-p)$ hybrid colliders, such as the Future Circular Collider Hadron Electron (FCC-he) \cite{FCChe,Fernandez,Fernandez1,Fernandez2,Huan,Acar}.
The $e^-p$ colliders will open up new perspective in the field of fundamental physics, especially for particle physics. Several potential
features in favor of this type of electron-proton colliders are the following: 1) Would represent the high resolution collider, with the
cleaner environment for exploring the substructure and dynamics inside matter, with unmatchable precision and sensitivity. 2) The center-of-mass
energies are much higher than that of the future International Linear Collider (ILC) and the Compact Linear Collider (CLIC). 3) With concurrent
$e^-p$ and $pp$ operation, the FCC-he would transform the LHC into an energy frontier accelerator facility. 4) With very precise strong
and electroweak interaction measurements and with suppressed backgrounds from strong interactions, the $e^-p$ results would make the FCC-he
a much more powerful search and measurement laboratory than present laboratories based on $pp$ collisions. 5) The joint $pp + e^-p$ facility
can become a Higgs bosons and top-quark factory for study the physics of both with an unprecedented impact. 6) For it's high-energy,
high-luminosity and while maintaining a very clean experimental environment, the FCC-he has an outstanding opportunity to discover
new physics BSM, such as Higgs sector, top-quark physics, exotic Higgs, dark matter, heavy neutrino, matter-antimatter asymmetry
and possible discoveries in QCD. All these topics are being studied very actively. In conclusion, the physics program of high-energy
and high-luminosity $e^-p$ + $pp$ collisions are vast and contain many unique opportunities. It provides high precision of Higgs boson,
top-quark, QCD and electroweak physics complementary to $e^+e^-$ colliders. Furthermore, $e^-p$ is an attractive, realistic option for
a future energy frontier collider for particle physics. For an exhaustive study on the physics and detector design concepts see Refs. \cite{FCChe,Fernandez,Fernandez1,Fernandez2,Huan,Acar}.

In the first instants of the creation of the universe, the Big Bang should have created equal amounts of matter and antimatter.
However, there is an unexplained dominance of matter over antimatter observed in the universe. CP violation offers one explanation
for the asymmetry in baryonic matter, however, there are not enough current observed sources of CP violation to account for the
total matter-antimatter asymmetry. For this reason, it is necessary to study new sources of CP violation. About this topic, the
Electric Dipole Moments (EDM) are very sensitive to CP violation in the quark and lepton sectors. EDM searches are then in the
ideal situation of that an observation in the next generation of experiments would be a clear indication of new physics BSM.
Under this perspective, the Anomalous Magnetic Dipole Moment (AMDM) and EDM of the top-quark are currently under intense scrutiny
from the point of view: theoretical, phenomenological and experimental. Within the scope of this project, CP violation can be
parameterized by the presence of anomalous couplings in the $t\bar t\gamma$ vertex of top quark production.

In this study, we focus on AMDM $(\hat a_V)$ and the EDM $(\hat a_A)$ of the top-quark. Since the AMDM and EDM of the top-quark is
chirality changing it can be significantly enhanced compared to AMDM and EDM of light fermions by the large top coupling.

For the dominant $e^-p \to e^-\bar b \to \bar t \nu_e \gamma \to \bar t(\to W^- \to (qq', l^- \bar\nu_l)+b) \nu_e\gamma$ production channel
considered here, we find that the proposed FCC-he with $\sqrt{s}=7.07, 10\hspace{0.8mm}TeV$, ${\cal L}=50, 100, 300, 500, 1000\hspace{0.8mm}fb^{-1}$
can probe the dipole moments of the top-quark with good sensitivity. We focus on two different signals: (i) the hadronic channel with
polarized electron beam for $P_{e^-}=-80, 0, 80\hspace{0.8mm}\%$, and (ii) the leptonic channel with polarized electron beam for
$P_{e^-}=-80, 0, 80\hspace{0.8mm}\%$ in the $\bar t \nu_e \gamma$ final state. We show that sensitivity measuring expected on the
electromagnetic anomalous couplings in the $t\bar t\gamma$ vertex can be probed at $95\%$ Confidence Level (C.L.) with center-of-mass
energies $\sqrt{s}=7.07, 10\hspace{0.8mm}TeV$ and integrated luminosity ${\cal L}=1000\hspace{0.8mm}fb^{-1}$ at FCC-he.

\begin{table}[!ht]
\caption{Sensitivities achievable on the electromagnetic dipole moments of the top-quark in the SM and in different processes and colliders.}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline\hline
{\bf Model}  &    {\bf Sensitivity of the SM}                 & {\bf C. L.}  &  {\bf References}\\
\hline
SM                                                            & $a_t= 0.02$, $d_t < 10^{-30} ({\rm ecm})$   & $68 \%$  & \cite{Benreuther},
\cite{Hoogeveen,Pospelov,Soni}  \\
\hline
\hline
\hline
{\bf Model}  &    {\bf Theoretical sensitivity: $\hat a_V$, $\hat a_A$}        & {\bf C. L.}  &  {\bf Reference}\\
\hline
Top-quark pair production at LHC                              &   $ (-0.041, 0.043), (-0.035, 0.038) $ & $68 \%$    & \cite{Juste} \\
\hline
$t\bar t\gamma$ production at LHC                             &   $ (-0.2, 0.2), (-0.1, 0.1) $         & $90 \%$    & \cite{Baur} \\
\hline
Radiative $b\to s\gamma$ transitions at Tevatron and LHC      &   $ (-2, 0.3), (-0.5, 1.5) $           & $90 \%$    & \cite{Bouzas} \\
\hline
Process $pp \to p\gamma^*\gamma^*p\to pt\bar t p $ at LHC     &   $ (-0.6389, 0.0233), (-0.1158, 0.1158) $  & $68 \%$   & \cite{Sh} \\
\hline
Measurements of $\gamma p \to t\bar t$ at LHeC                &   $ (-0.05, 0.05), (-0.20, 0.20)    $       & $90 \%$   & \cite{Bouzas1} \\
\hline
Top-quark pair production $e^+e^- \to t\bar t$  at ILC        &   $ (-0.002, 0.002), (-0.001, 0.001) $       & $68 \%$  & \cite{Aguilar} \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

AMDM and EDM searches of the top-quark are performed in the Standard Model (SM) and on a variety of physics processes.
The sensitivity estimated on the AMDM and EDM of the top-quark in the SM, as well as in different processes and colliders
are reported in Table I. Other direct collider probes of the AMDM and EDM have been studied extensively   \cite{Ibrahim,Atwood,Polouse,Choi,Polouse1,Aguilar0,Amjad,Juste,Asner,Abe,Aarons,Brau,Baer,Grzadkowski:2005ye,murat,Billur}.

Plan of the article is as follows: In Section II, we introduce the top-quark effective electromagnetic interactions. In Section III,
we sensitivity measurement on top-quark anomalous electromagnetic couplings through $e^-p \to e^-\bar b \to \bar t \nu_e \gamma \to
\bar t(\to W^- \to (qq', l^- \bar\nu_l)+b) \nu_e\gamma$ signal. Finally, we present our conclusions in Section IV.

\section{Single top-quark production via the process $e^-p \to e^-\bar b \to \bar t \nu_e \gamma$}

\subsection{Effective Lagrangian of $t\bar t \gamma$ interaction  of the top-quark}

The SM predicts CP violation outside the K, D and B meson systems is small to be observed. However, in some extensions of the SM,
CP violation might be considerably enhanced, especially in the presence of heavy particles as the top quark. In particular,
CP-violating EDM of the top-quark could be enhanced.

Single top-quark production processes are sensitive to the anomalous couplings in the $t\bar t \gamma$ vertex. Furthermore,
since the top-quark lifetime is shorter than the timescale of spin decoherence induced by QCD, its decay products preserve
information of its polarization by the production mechanism. This provides additional powerful tools in the search
for BSM physics in single top-quark studies.

On the other hand, due to the absence so far of any signal of new heavy particles decaying into top-quark, an attractive
approach for describing possible new physics effects in a model-independent way is based on effective Lagrangian. In this
approach, all the heavy degrees of freedom are integrated out leading to obtain the effective interactions between the SM
particles. This is justified because the related observables have not shown any significant deviation from the SM predictions
so far. The Lagrangian describing interaction of the anomalous $tt\gamma$ coupling including the SM contribution and terms BSM
which are related to new physics has the structure:

\begin{equation}
{\cal L}_{eff}={\cal L}^{(4)}_{SM} + \frac{1}{\Lambda^2}\sum_n \Bigl[C_n{\cal O}^{(6)}_n + C^*_n{\cal O}^{\dagger(6)}_n \Bigr].
\end{equation}

\noindent Here, ${\cal L}_{eff}$ is the effective Lagrangian gauge-invariant which contains a series of dimension-six operators
built with the SM fields, ${\cal L}^{(4)}_{SM}$ is the renormalizable SM Lagrangian of dimension-four, $\Lambda$ is the scale at
which new physics expected to be observed, $C_n$ are Wilson coefficients which are dimensionless coefficients and ${\cal O}^{(6)}_n$
represents the dimension-six gauge-invariant operator. The ${\cal O}^{(6)}_n$ operator and the unknown coefficients $C_n$, combined
with $\Lambda^{-2}$, produce the non-standard coupling constants, that is generated anomalous contributions to the photon-top-quark
interaction vertex which is similar in structure to radiative corrections in the SM.

The most general Lagrangian term that one can write for the $t\bar t\gamma$ coupling up to dimension-six gauge invariant
operators \cite{Sh,Kamenik,Baur,Aguilar,Aguilar1} is:

\begin{equation}
{\cal L}_{t\bar t\gamma}=-g_eQ_t\bar t \Gamma^\mu_{ t\bar t  \gamma} t A_\mu,
\end{equation}

\noindent this equation includes the SM coupling and contributions from dimension-six effective operators. $g_e$ is the
electromagnetic coupling constant, $Q_t$ is the top-quark electric charge and the Lorentz-invariant vertex function
$\Gamma^\mu_{t\bar t \gamma}$ is given by:

\begin{equation}
\Gamma^\mu_{t\bar t\gamma}= \gamma^\mu + \frac{i}{2m_t}(\hat a_V + i\hat a_A\gamma_5)\sigma^{\mu\nu}q_\nu.
\end{equation}

\noindent $m_t$ and $q$ are the mass of the top-quark and the momentum transfer to the photon, respectively. The $\hat a_V$
and $\hat a_A$ couplings in Eq. (3) are real and related to the AMDM and the EDM of the top-quark. These couplings
$\hat a_V$ and $\hat a_A$ are directly related to the $a_t$ and $d_t$, via the relations:

\begin{eqnarray}
\hat a_V&=&Q_t a_t,  \\
\hat a_A&=&\frac{2m_t}{e}d_t.
\end{eqnarray}

As shown in Refs. \cite{Buhmuller,Aguilar2,Antonio,Grzadkowski:2005ye}, among operators of dimension-six there exist only two relevant
for the $t\bar t\gamma$ interaction operator:

\begin{eqnarray}
{\cal O}_{uW}^{33}&=&\bar q_{L3}\sigma^{\mu\nu}\tau^a t_{R}{\tilde \phi} W_{\mu\nu}^{a}+{\mbox{h.c.}},\\
{\cal O}_{uB\phi}^{33}&=&\bar q_{L3}\sigma^{\mu\nu}t_{R}{\tilde \phi} B_{\mu\nu}+{\mbox{h.c.}},
\end{eqnarray}

\noindent where the index 3 means the 3rd quark generation, $\bar q_{L3}$ is the quark field, $\sigma^{\mu\nu}$ are the
Pauli matrices, ${\tilde \phi}=i\tau_2\phi^*$, $\phi$ is the SM Higgs doublet, $W_{\mu\nu}^{a}$ and $B_{\mu\nu}$
are the $U(1)_Y$ and $SU(2)_L$ gauge field strength tensors which are defined as:

\begin{eqnarray}
B_{\mu\nu}&=&\partial_\mu B_\nu-\partial_\nu B_\mu,  \\
W^a_{\mu\nu}&=&\partial_\mu W^a_\nu-\partial_\nu W^a_\mu-g\epsilon^{abc}W^b_\mu W^c_\nu,
\end{eqnarray}

\noindent with $a,b,c=1,2,3$. From the parametrization given by Eq. (3), and from the operators of dimension-six given in
Eqs. (6) and (7) after replacing $<{\tilde \phi}> \to \frac{1}{\sqrt{s}}$ give rise to the corresponding CP even ${\hat a_V}$
and CP odd ${\hat a_A}$ couplings:

\begin{eqnarray}
\hat a_V=\frac{2 m_t}{e} \frac{\sqrt{2}\upsilon}{\Lambda^{2}} Re\Bigl[\cos\theta _{W} C_{uB\phi}^{33} + \sin\theta _{W} C_{uW}^{33}\Bigr],\\
\hat a_A=\frac{2 m_t}{e} \frac{\sqrt{2}\upsilon}{\Lambda^{2}} Im\Bigl[\cos\theta _{W} C_{uB\phi}^{33} + \sin\theta _{W} C_{uW}^{33}\Bigr],
\end{eqnarray}

\noindent which are related to the AMDM and EDM of the top-quark. The ${\hat a_V}$ and ${\hat a_A}$ couplings contain $\upsilon=246$ GeV,
the breaking scale of the electroweak symmetry, $\sin\theta _{W} (\cos\theta _{W})$, the sine(cosine) of the weak mixing angle
and $\Lambda$ is the new physics scale.

\subsection{Cross-section of the $e^-p \to e^-\bar b \to \bar t \nu_e \gamma \to \bar t(\to W^- \to (qq',  l^- \bar\nu_l)+b) \nu_e\gamma$ signal}

The FCC-eh project \cite{FCChe,Fernandez,Fernandez1,Fernandez2,Huan,Acar} offer $e^-p$ collisions at TeV-scale center-of-mass energy
and luminosities of order 1000 times larger than that of Hadron-Electron Ring Accelerator (HERA), the first and to date the only
lepton-hadron collider worldwide, providing fascinating probes of QCD and hadron structure as well as a novel configuration for Higgs
boson, top-quark and BSM physics.

The physics highlights are available from combining the capabilities of these facilities they were already mentioned in the introduction.
However, a general aspect is to maximize the BSM physics search potential at high energies by exploiting the unique capabilities of
an $e^-p$ collider.

The most significant top-quark production processes at the $e^-p$ colliders are single top-quark, $t\bar t$, and associated $tW$
production. In Ref. \cite{Bouzas1} is shown the values of the associated cross-sections for these processes. The main source of
production is single-top via the charged current $W$ t-channel \cite{Moretti}, whereas for the signals, $t\bar t$ and $tW$, the
cross-section is minor. In addition, taking into account the advantage of an experimental cleaner environment than the $pp$ colliders,
we can anticipate a potential efficiency of these colliders to study the top-quark physics.

The deep inelastic $e^-p \to e^-\bar b \to \bar t \nu_e \gamma \to \bar t(\to W^- \to (qq',  l^- \bar\nu_l)+b) \nu_e\gamma$
scattering process is measured at FCC-he via the exchange of a $W^{\pm}$ boson in charged current scattering
as shown in Figs. 1 and 2.

For the calculation of the charged current cross-section, we consider the CTEQ6L1 PDFs \cite{CTEQ6L1} and we apply the following detector
acceptance cuts on the pseudorapidity of the photon and the transverse momentum of the photon and the neutrino, respectively, to reduce the
background and to optimize the signal sensitivity:

\begin{eqnarray}
|\eta^{\gamma}|&<& 2.5,   \nonumber \\
p^\gamma_T &>& 20\hspace{0.8mm}GeV,  \\
p^{(\nu)}_T &>& 20\hspace{0.8mm}GeV. \nonumber
\end{eqnarray}

The proposed FCC-he is well-suited for discovering physics BSM and for precisely unraveling the structure of the fundamental physics
with unpolarized and polarized electron beam. In particle physics, polarization refers to the extent to which a particle spin is aligned
along a particular direction. The t-channel single-top-quarks of the process $e^-p \to e^-\bar b \to \bar t \nu_e \gamma$ are produced
with a strong degree of polarization along the direction of the momentum of the light spectator quark, whose direction then defines the
top-quark spin axis. Regarding this subject, the physics in study can be maximized by the use of polarized beams. In this paper shows
the important role of polarized beam and summarizes the benefits obtained from polarizing the electron beam. The polarized $e^-$ beam,
combined with the clean experimental environment provided by the FCC-he, will allow to improve strongly the potential of searches for the
dipole moments, which opens the possibility to resolve shortcomings of the SM. With these arguments, we consider polarized
electron beam in our study. The formula for the total cross-section for an arbitrary degree of longitudinal $e^-$ beams
polarization is given by \cite{XiaoJuan}:

\begin{eqnarray}
\sigma_{e^-_r}= \sigma_{e^-_0}\cdot (1-P_{e^-_r}),  \hspace{7mm}
\sigma_{e^-_l} + \sigma_{e^-_r} &=& 2\sigma_{e^-_0},
\end{eqnarray}

\noindent where $\sigma_{e^-_r}$, $\sigma_{e^-_l}$ and $\sigma_{e^-_0}$ represent the right, left and without electron
beam polarization, respectively and $P_{e^-}$ is the polarization degree of the electron.

We have implemented $t\bar t \gamma$ effective coupling corresponding given by the Lagrangian (2) in CalcHEP \cite{Belyaev}
to compute the tree level amplitudes relevant for the process. The partonic cross-section is convoluted with CTEQ6L1
PDFs \cite{CTEQ6L1}. Finally, we use CalcHEP to compute numerically the cross-section $\sigma(\sqrt{s}, \hat a_V, \hat a_A, P_{e^-})$
as a function of the center-of-mass energy and effective couplings. We displayed the $7.07\hspace{0.8mm}TeV$ and $10\hspace{0.8mm}TeV$
cross-section of the $2 \to 3$ process $e^-p \to e^-\bar b \to \bar t \nu_e \gamma$ with $P_{e^-}=0\%$, $P_{e^-}=-80\%$ and
$P_{e^-}=80\%$ in Eqs. (14)-(25) for the FCC-he:\\

$i)$ Total cross-section for $\sqrt{s}=7.07\hspace{0.8mm} TeV$ and $P_{e^-}=0\%$:

\begin{eqnarray}
\sigma(\hat a_V)&=&\Bigl[(0.0236)\hat a^2_V +(0.0000489)\hat a_V  +0.737 \Bigr] (pb),   \\
\sigma(\hat a_A)&=&\Bigl[(0.0236)\hat a^2_A + 0.737 \Bigr] (pb).
\end{eqnarray}

$ii)$ Total cross-section for $\sqrt{s}=10\hspace{0.8mm} TeV$ and $P_{e^-}=0\%$:

\begin{eqnarray}
\sigma(\hat a_V)&=&\Bigl[(0.0593)\hat a^2_V +(0.000618)\hat a_V  + 1.287 \Bigr] (pb),   \\
\sigma(\hat a_A)&=&\Bigl[(0.0593)\hat a^2_A + 1.287 \Bigr] (pb).
\end{eqnarray}

$iii)$ Total cross-section for $\sqrt{s}=7.07\hspace{0.8mm} TeV$ and $P_{e^-}=80\%$:

\begin{eqnarray}
\sigma(\hat a_V)&=&\Bigl[(0.00472)\hat a^2_V +(0.0000109)\hat a_V  +0.148 \Bigr] (pb),   \\
\sigma(\hat a_A)&=&\Bigl[(0.00472)\hat a^2_A + 0.148 \Bigr] (pb).
\end{eqnarray}

$iv)$ Total cross-section for $\sqrt{s}=10\hspace{0.8mm} TeV$ and $P_{e^-}=80\%$:

\begin{eqnarray}
\sigma(\hat a_V)&=&\Bigl[(0.0119)\hat a^2_V +(0.000127)\hat a_V  +0.257 \Bigr] (pb),   \\
\sigma(\hat a_A)&=&\Bigl[(0.0119)\hat a^2_A + 0.257 \Bigr] (pb).
\end{eqnarray}

$v)$ Total cross-section for $\sqrt{s}=7.07\hspace{0.8mm} TeV$ and $P_{e^-}=-80\%$:

\begin{eqnarray}
\sigma(\hat a_V)&=&\Bigl[(0.0423)\hat a^2_V +(0.000417)\hat a_V  + 1.328 \Bigr] (pb),   \\
\sigma(\hat a_A)&=&\Bigl[(0.0423)\hat a^2_A + 1.328 \Bigr] (pb).
\end{eqnarray}

$vi)$ Total cross-section for $\sqrt{s}=10\hspace{0.8mm} TeV$ and $P_{e^-}=-80\%$:

\begin{eqnarray}
\sigma(\hat a_V)&=&\Bigl[(0.107)\hat a^2_V +(0.00196)\hat a_V  +2.315 \Bigr] (pb),   \\
\sigma(\hat a_A)&=&\Bigl[(0.107)\hat a^2_A + 2.315 \Bigr] (pb).
\end{eqnarray}

Our results given by Eqs. (14)-(25) show the effect of taking $-80\%$ beam polarization for electron, which results in the
enhancement of the SM and non-SM single-top production cross-section as the cross-section scales as $(1 + P_{e^-})$, $P_{e^-}$
being the degree of polarization of the electron.

The variation of the single top-quark production cross-section with the effective $t\bar t \gamma$ couplings, $\hat a_V$
or $\hat a_A$ and taking one anomalous coupling at a time are shown in Figs. 3-6. The curves depict the cross-section for
$e^-p \to e^-\bar b \to \bar t \nu_e \gamma$ from the $80\%$, $-80\%$ polarized and unpolarized $e^-$ beam, respectively,
and we fixed the energy of the $e^-$ beam $E_e = 250 \hspace{0.8mm}GeV$, $500\hspace{0.8mm}GeV$ and the energy of the $p$
beam $E_p = 50\hspace{0.8mm}TeV$. These figures have shown
a stronger dependence of the cross-section on the anomalous coupling $\hat a_V(\hat a_A)$ in the range allowed by these
parameters. Our results indicate that considering the proposed $10\hspace{0.8mm}TeV$ energy, detector acceptance cuts (see Eq. (12))
and $-80\%$ electron polarization, the cross-sections as a function of $\hat a_V$ or $\hat a_A$ are higher.
For instance, the cross-section projected is $\sigma(\hat a_V, -80\%)=(1.66)\sigma(\hat a_V, 0\%)$ for $\sqrt{s}= 7.07\hspace{0.8mm}TeV$,
while $\sigma(\hat a_V, -80\%)=(1.78)\sigma(\hat a_V, 0\%)$ for $\sqrt{s}= 10\hspace{0.8mm}TeV$, that is, there is an improvement
in the cross-section by a factor of 1.66 (1.78) for the polarized case with respect to the case unpolarized. Similar results are
obtained for $\sigma(\hat a_A, 80\%)$.

The cross-sections for the energies $\sqrt{s}= 7.07\hspace{0.8mm}TeV$ and $10\hspace{0.8mm}TeV$ are shown in Figs. 7 and 8.
As can be seen from Figs. 7-8, the surfaces $\sigma(e^-p \to \bar t \nu_e \gamma)$ as functions of $\hat a_V$ and $\hat a_A$
have extreme points: maximum and minimum. The minimum value corresponds to the SM, while the maximum value corresponds to the
anomalous contribution, which is consistent with Eqs. (14)-(25). In both figures, the cross-section depends significantly on
the observables $\hat a_V$ and $\hat a_A$.

With the purpose of comparison and analysis, we compare our results for the anomalous couplings $\hat a_V$ and $\hat a_A$ with those
quoted in the papers \cite{Sh,Bouzas} (see also Table I). The authors of Ref. \cite{Bouzas}, specifically discussed the bounds on
the AMDM and EDM of the top-quark that can be obtained from measurements of the semi-inclusive decays $B \to X_s\gamma$, and of
$t\bar t\gamma$ production at the Tevatron and the LHC. Performing their analysis they find that the AMDM is bounded by
$-2 < \kappa < 0.3$  whereas the EDM is bound by $-0.5 < \tilde\kappa < 1.5$, respectively. For our case, we consider the process
of single anti-top-quark production through charged current with the $e^-p \to e^-\bar b \to \bar t \nu_e \gamma \to \bar t(\to W^- \to
(qq',  l^- \bar\nu_l)+b) \nu_e\gamma$ signal. We based our results on the data at $\sqrt{s}=10\hspace{0.8mm}TeV$,
${\cal L}=1000\hspace{0.8mm}fb^{-1}$, $\delta_{sys}=0\%$, $P_{e^-}=0\%$ and $95\%\hspace{0.8mm}C.L.$, we obtain $\hat a_V=(-0.2308, 0.2204)$,
$\hat a_A=|0.2259|$ and $\hat a_V=(-0.3067, 0.2963)$, $\hat a_A=|0.3019|$ for the hadronic and leptonic modes. Although the conditions
for the study of the dipole moments of the top-quark through the $b\to s\gamma$ transitions at Tevatron and LHC, and
$e^-p \to e^-\bar b \to \bar t \nu_e \gamma \to \bar t(\to W^- \to (qq',  l^- \bar\nu_l)+b) \nu_e\gamma$
are different, our results are competitive with respect to the results reported in Ref. \cite{Bouzas}.
More recently, using the process $pp\to p\gamma^* \gamma^* p \to pt\bar t p$, a detailed study on the top-quark anomalous couplings
$\hat a_v$ and $\hat a_A$ for LHC at $14\hspace{0.8mm}TeV$ with $300\hspace{0.8mm} fb^{-1}$ of data \cite{Sh} is done. The $68\%$ C.L.
bounds that they have obtained are found to be in the intervals of $\hat a_V= (−0.6389, 0.0233)$ and $\hat a_A=(−0.1158, 0.1158)$.
From the comparison of our study via the process $e^-p \to e^-\bar b \to \bar t \nu_e \gamma \to \bar t(\to W^- \to (qq',  l^- \bar\nu_l)+b) \nu_e\gamma$
at the FCC-he, with respect to the process $pp\to p\gamma^* \gamma^* p \to pt\bar t p$ at the LHC, our results indicate a significant improvement
in the measurements for $\hat a_V$ and $\hat a_A$. Additionally, it is noteworthy that with our process the total cross-sections is a factor
${\cal O}(10^3)$ between $pp\to p\gamma^* \gamma^* p \to pt\bar t p$ and $e^-p \to e^-\bar b \to \bar t \nu_e \gamma \to \bar t(\to W^- \to (qq',  l^- \bar\nu_l)+b) \nu_e\gamma$, indicating that our results project 3 orders of magnitude more higher than the reported in Ref. \cite{Sh}.
These predictions indicate that the sensitivity on the anomalous couplings $\hat a_V$ and $\hat a_A$ can be measured better at the FCC-he
by a few orders of magnitude in comparison with the predictions of the LHC.

\section{Model-independent sensitivity estimates on the $\hat a_V$ and $\hat a_A$}

In Tables II-VII is shown the results for the  model-independent sensitivity achievable at $95\%$ C.L. for the non-standard couplings
$\hat a_V$ and $\hat a_A$ obtained from an analysis of the process $e^-p \to e^- \bar b \to \bar t \nu_e \gamma$ at the FCC-he.
At the FCC-he, we assume the center-of-mass energies $\sqrt{s}=7.07, 10\hspace{0.8mm}TeV$ and luminosities ${\cal L}= 50, 100, 300, 500, 1000\hspace{0.8mm}fb^{-1}$ with unpolarized and
polarized electron beam $P_{e^-}=-80\%, 0\%, 80\%$. Additionally, we impose the acceptance cuts for the FCC-he given by Eq. (12) and
take into account the systematic uncertainties $\delta_{sys}=0\%, 3\%, 5\%$.

In order to extract the expected sensitivity at $95\%$ C.L. on the effective operators couplings $\hat a_V$ and $\hat a_A$, we compute
$\sigma_{BSM}(\hat a_V, \hat a_A)$ cross-section of the process $e^-p \to e^- \bar b \to \bar t \nu_e \gamma$ as function of the effective
couplings as discussed in the previous section, and we assume the measured cross-section to coincide with the SM predictions and we construct
the following $\chi^2$ function:

 \begin{equation}
\chi^2(\hat a_V, \hat a_A)=\biggl(\frac{\sigma_{SM}-\sigma_{BSM}(\sqrt{s}, \hat a_V, \hat a_A, P_{e^-})}{\sigma_{SM}\sqrt{(\delta_{st})^2
+(\delta_{sys})^2}}\biggr)^2.
\end{equation}

\noindent $\sigma_{SM}$ is the cross-section of the SM and $\sigma_{BSM}(\sqrt{s}, \hat a_V, \hat a_A, P_{e^-})$ is the total cross-section
containing contributions from the SM and BSM, while $\delta_{st}=\frac{1}{\sqrt{N_{SM}}}$ and $\delta_{sys}$ are the statistical and systematic
uncertainties, respectively. The number of events $N_{SM}$ for the process $e^-p \to e^- \bar b \to \bar t \nu_e \gamma$
is calculated by $N_{SM}={\cal L}_{int}\times \sigma_{SM} \times BR(\bar t \to W^-b)\times BR(W^- \to qq' (l^-\nu_l))\times  \epsilon_{b-tag}$, where ${\cal L}_{int}$ is the integrated FCC-he luminosity and  $b$-jet tagging efficiency is $\epsilon_b=0.8$ \cite{atlas}.
The top-quark decays weakly and almost $100\%$ to a $W$ boson and $b$ quark, specifically $\bar t\to \bar bW^-$, where the $W$
boson decays to either hadronically ($W \to qq'$) or leptonically ($W\to l^-\nu_l$), with a Branching Ratio of: $BR(W \to q q')=0.674$
for hadronic decay, $BR(W \to l\nu_l)(l=e, \mu)=0.213$ for light leptonic decays and $BR(W \to \tau\nu_\tau)=0.113$ \cite{Data2018}.

Next, we present the sensibility measurement for the anomalous couplings $\hat a_V$ and $\hat a_A$ as is shown in Tables II-VII
which are obtained for $\sqrt{s}=7.07, 10\hspace{0.8mm}TeV$, ${\cal L}= 50-1000\hspace{0.8mm}fb^{-1}$ and $P_{e^-}=-80\%, 0\%, 80\%$,
where only one coupling at a time is varied.

From Tables II-VII, the results for the dipole moments $\hat a_V$ and $\hat a_A$, for specific values of $\sqrt{s}= 10\hspace{0.8mm}TeV$,
${\cal L}=1000\hspace{0.8mm}fb^{-1}$, $P_{e^-}= -80\%, 0\%, 80\%$ and $\delta_{sys}=0\%$ are as follows:\\

$i)$ Sensitivity on $\hat a_V$ and $\hat a_A$ for $\sqrt{s}=10\hspace{0.8mm} TeV$, $P_{e^-}=-80\%$ and BR($W^- \to$ hadronic):

\begin{eqnarray}
-0.2041 \leq & \hat a_V & \leq 0.1858, \hspace{3mm}   \mbox{$95\%$ C.L.}, \\
-0.1939 \leq & \hat a_A & \leq 0.1939, \hspace{3mm}   \mbox{$95\%$ C.L.}.
\end{eqnarray}

$ii)$ Sensitivity on $\hat a_V$ and $\hat a_A$ for $\sqrt{s}=10\hspace{0.8mm} TeV$, $P_{e^-}=-80\%$ and BR($W^- \to$ leptonic):

\begin{eqnarray}
-0.2695 \leq & \hat a_V & \leq 0.2512, \hspace{3mm}   \mbox{$95\%$ C.L.}, \\
-0.2592 \leq & \hat a_A & \leq 0.2592, \hspace{3mm}   \mbox{$95\%$ C.L.}.
\end{eqnarray}

$iii)$ Sensitivity on $\hat a_V$ and $\hat a_A$ for $\sqrt{s}=10\hspace{0.8mm} TeV$, $P_{e^-}=0\%$ and BR($W^- \to$ hadronic):

\begin{eqnarray}
-0.2308 \leq & \hat a_V & \leq 0.2204, \hspace{3mm}   \mbox{$95\%$ C.L.}, \\
-0.2259 \leq & \hat a_A & \leq 0.2259, \hspace{3mm}   \mbox{$95\%$ C.L.}.
\end{eqnarray}

$iv)$ Sensitivity on $\hat a_V$ and $\hat a_A$ for $\sqrt{s}=10\hspace{0.8mm} TeV$, $P_{e^-}=0\%$ and BR($W^- \to$ leptonic):

\begin{eqnarray}
-0.3067 \leq & \hat a_V & \leq 0.2963, \hspace{3mm}   \mbox{$95\%$ C.L.}, \\
-0.3019 \leq & \hat a_A & \leq 0.3019, \hspace{3mm}   \mbox{$95\%$ C.L.}.
\end{eqnarray}

$v)$ Sensitivity on $\hat a_V$ and $\hat a_A$ for $\sqrt{s}=10\hspace{0.8mm} TeV$, $P_{e^-}=80\%$ and BR($W^- \to$ hadronic):

\begin{eqnarray}
-0.3428 \leq & \hat a_V & \leq 0.3321, \hspace{3mm}   \mbox{$95\%$ C.L.}, \\
-0.3371 \leq & \hat a_A & \leq 0.3371, \hspace{3mm}   \mbox{$95\%$ C.L.}.
\end{eqnarray}

$vi)$ Sensitivity on $\hat a_V$ and $\hat a_A$ for $\sqrt{s}=10\hspace{0.8mm} TeV$, $P_{e^-}=80\%$ and BR($W^- \to$ leptonic):

\begin{eqnarray}
-0.4563 \leq & \hat a_V & \leq 0.4456, \hspace{3mm}   \mbox{$95\%$ C.L.}, \\
-0.4505 \leq & \hat a_A & \leq 0.4505, \hspace{3mm}   \mbox{$95\%$ C.L.}.
\end{eqnarray}

A direct comparison of Eqs. (27)-(38) for $\hat a_V$ and $\hat a_V$ shown that the sensitivity is increases up to $12\%$ for the case
with $P_{e^-}=- 80\%$ and BR($W^- \to$ hadronic, leptonic) that for the case with $P_{e^-}=0\%$ and BR($W^- \to$ hadronic, leptonic).
While from Eqs. (27)-(30) and (35)-(38) the sensitivity is increases up to $67\%$ for the case with $P_{e^-}=-80\%$ and
BR($W^- \to$ hadronic, leptonic) with respect to the case with $P_{e^-}=80\%$ and BR($W^- \to$ hadronic, leptonic), respectively.

For $P_{e^-}=0\%$ and the hadronic channel of the W boson, Figs. 9-10 show the prospects of the sensitivity on the electromagnetic anomalous
couplings $\hat a_V$ and $\hat a_V$ at the FCC-he. Also, in order to obtain the plots we have assumed that $\sqrt{s}=7.07, 10\hspace{0.8mm}TeV$
and ${\cal L}= 50, 250, 1000\hspace{0.8mm}fb^{-1}$ at the $95\%$ C.L.. Of these contours plots, our forecast for the future sensitivity of the
observables $\hat a_V$ and $\hat a_A$ are based on the process $e^-p \to e^- \bar b \to \bar t \nu_e \gamma$, as well as in the future projections
of the FCC-he for $\sqrt{s}$ and ${\cal L}$. The regions allowed for the top-quark AMDM, EDM and collider sensitivity are colored in
pink, blue, purple, respectively, while the prediction corresponds to the SM can be obtained from Eqs. (14)-(25). The $95\%$ C.L. regions
for each of these couplings separately are $\hat a_V  \hspace{1mm}\in \hspace{1mm}[-0.25, 0.25]$, $\hat a_A \hspace{1mm}\in \hspace{1mm}[-0.29, 0.29]$
for $\sqrt{s}=7.07\hspace{0.8mm}TeV$ and ${\cal L}= 250 \hspace{0.8mm}fb^{-1}$. In addition, $\hat a_V  \hspace{1mm}\in \hspace{1mm}[-0.20, 0.20]$,
$\hat a_A \hspace{1mm}\in \hspace{1mm}[-0.20, 0.20]$ for $\sqrt{s}=10\hspace{0.8mm}TeV$ and  ${\cal L}= 1000 \hspace{0.8mm}fb^{-1}$. These results
are consistent with those shown in Tables IV and V. It is worth mentioning that, the results obtained in Tables IV and V, as well as the corresponding
results obtained through the contours Figs. 9 and 10, in some cases are more sensitive than the reported ones in Table I. In particular, an improvement
is reachable in comparison with the constraints obtained from the radiative $b \to s\gamma$ transitions at Tevatron and LHC \cite{Bouzas} and
$pp\to p\gamma^* \gamma^* p \to pt\bar t p$ \cite{Sh} searches mentioned in Table I and subsection B.

\section{Conclusions}

As mentioned above, due to its sizable $t\bar t\gamma$ coupling the top-quark is one of the most attractive particles and provides
one of the most convincing  alternatives to probe new physics BSM, such as AMDM ($\hat a_V$) and EDM ($\hat a_A$). Furthermore,
the EDM is particularly interesting because it is very sensitive to possible new sources of CP violation in the quark and lepton sectors.

In this paper, we have studied the potential of the FCC-he for sensitivity measuring expected on the electromagnetic anomalous couplings
in the $t\bar t \gamma$ vertex. We consider the single top-quark production mode $e^-p \to e^- \bar b \to \bar t \nu_e \gamma$ with $W$
boson exchange via the $t$-channel, that is through charged current production. This channel has the largest cross-section and is hence
the dominant production mode of single top-quark \cite{Sun}. Our study is based on the projections for the
center-of-mass energies $\sqrt{s}$, the integrated luminosity ${\cal L}$ and the polarization electron beam of the FCC-he.
Additionally, we take into account kinematic cuts and systematic uncertainties $\delta_{sys}=0\%, 3\%, 5\%$. The cut based
optimization at $7.07\hspace{0.8mm}TeV$ and $10\hspace{0.8mm}TeV$, involves a set of selection cuts in various
kinematic variables, carefully chosen with the criterion of not being built with kinematic properties of one or part of the
decay products of the top-quark, that could in principle bias the sensitivity to the anomalous couplings. These kinematic variables
selected for these cuts include $\eta^{\gamma}$, $p^\gamma_T$ and $p^{(\nu)}_T$ as defined in Eq. (12). The final results
of this optimization for the resulting cross-section of the $e^-p \to e^- \bar b \to \bar t \nu_e \gamma$ signal and
prospective sensitivities for the dipole moments $\hat a_V$ and $\hat a_A$ indicated in Figs. 3-10 and Tables II-VII
as well as in Eqs. (14)-(25) and (27)-(38) imply that the process $e^-p  \to \bar t \nu_e\gamma$ at FCC-he is an excellent
option for probing the physics of the top-quark. This makes a future $e^-p$ collider an ideal tool to study the electromagnetic
properties of the top-quark through the $t\bar t \gamma$ vertex.

From Tables III, V and VII, the sensitivity estimated on dipole moments of the top-quark are
$\hat a_V=[-0.2308, 0.2204]$, $|\hat a_A|=0.2259$ at $95\%$ C.L. in the hadronic channel with unpolarized electron beam $P_{e^-}=0\%$.
In the case with polarized electron beam for $P_{e^-}=80\%$ and $P_{e^-}=-80\%$ are $\hat a_V=[-0.3428, 0.3321]$, $|\hat a_A|=0.3371$
and $\hat a_V=[-0.2041, 0.1858]$, $|\hat a_A|=0.1939$ at $95\%$ C.L. The corresponding results for the leptonic channel with
$P_{e^-}=0\%, 80\% -80\%$ are $\hat a_V=[-0.3067, 0.2963]$, $|\hat a_A|=0.3019$, $\hat a_V=[-0.4563, 0.4456]$, $|\hat a_A|=0.4505$
and $\hat a_V=[-0.2695, 0.2512]$, $|\hat a_A|=0.2592$, respectively. The results for $\hat a_V$ and $\hat a_A$ in the leptonic channel
are weaker by a factor of 0.75 than those corresponding to the hadronic channel. From these results, we find that the sensitivity estimated
on dipole moments of the top-quark are of the same order of magnitude as those reported in Table I and Refs. \cite{Ibrahim,Atwood,Polouse,Choi,Polouse1,Aguilar0,Amjad,Juste,Asner,Abe,Aarons,Brau,Baer,Grzadkowski:2005ye,murat,Billur}.
In particular, of the comparison with the constraints obtained from the radiative $b \to s\gamma$ transitions at Tevatron and LHC \cite{Bouzas}
and the process $pp\to p\gamma^* \gamma^* p \to pt\bar t p$ at LHC \cite{Sh}, our results are more sensitive. Given these prospective
sensitivities, we highlight that the FCC-he is the potential top-quark factory that is particularly well suited to sensitivity study
on its dipole moments with cleaner environments.

Summarizing, the FCC-he offers us significant opportunities to study the anomalous couplings of the quark-top. However, more
extensive studies on the theoretical, phenomenological and experimental level they are needed. These new possibilities for
investigating the electromagnetic properties of the top-quark will eventually open new avenues in the understanding of the
quark-top physics, as well as new physics BSM.

\vspace{1.5cm}

\begin{center}
{\bf Acknowledgments}
\end{center}

A. G. R. and M. A. H. R. acknowledge support from SNI and PROFOCIE (M\'exico).

\vspace{1.5cm}

\begin{table}[!ht]
\caption{Sensitivities on the AMDM $\hat a_V$ and the EDM $\hat a_A$ of the top-quark through the process
$e^-p  \to e^-\bar b \to \bar t \nu_e\gamma$ at the FCC-he.}
\begin{center}
 \begin{tabular}{|cc|cc|cc|}
\hline\hline
\multicolumn{6}{|c|}{ $\sqrt{s}=$ 7.07 TeV,  \hspace{5mm}  $P_{e^-} = 0 \%$, \hspace{5mm}  $95\%$ C.L.} \\
\hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{Hadronic} & \multicolumn{2}{c|}{Leptonic} \\
\hline
\cline{1-6}
${\cal L} \, (fb^{-1})$  & \hspace{0.5cm} $ \delta_{sys}$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm} &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm}  &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm} \\
\hline
50  &  $0\%$   & [-0.6599, 0.6578]    &    0.6587  &  [-0.8815, 0.8794] &  0.8802  \\
50  &  $3\%$   & [-1.3758, 1.3737]    &    1.3742  &  [-1.4138, 1.4118] &  1.4123  \\
50  &  $5\%$   & [-1.7606, 1.7585]    &    1.7589  &  [-1.7792, 1.7772] &  1.7776  \\
\hline
100 &  $0\%$   & [-0.5551, 0.5530]    &    0.5539  &  [-0.7414, 0.7393] &  0.7401  \\
100 &  $3\%$   & [-1.3666, 1.3645]    &    1.3651  &  [-1.3864, 1.3843] &  1.3849  \\
100 &  $5\%$   & [-1.7563, 1.7542]    &    1.7546  &  [-1.7657, 1.7636] &  1.7640  \\
\hline
300  &  $0\%$   & [-0.4221, 0.4199]   &    0.4208  &  [-0.5636, 0.5615] &  0.5624  \\
300  &  $3\%$   & [-1.3604, 1.3583]   &    1.3589  &  [-1.3672, 1.3651] &  1.3656  \\
300  &  $5\%$   & [-1.7534, 1.7513]   &    1.7517  &  [-1.7565, 1.7545] &  1.7549  \\
\hline
500 &  $0\%$   & [-0.3715, 0.3695]    &    0.3704  &  [-0.4962, 0.4941] &  0.4949  \\
500 &  $3\%$   & [-1.3591, 1.3571]    &    1.3576  &  [-1.3632, 1.3612] &  1.3617  \\
500 &  $5\%$   & [-1.7528, 1.7507]    &    1.7511  &  [-1.7547, 1.7526] &  1.7530  \\
\hline
1000 &  $0\%$   & [-0.3126, 0.3105]   &    0.3115  &  [-0.4174, 0.4153] &  0.4162  \\
1000 &  $3\%$   & [-1.3582, 1.3561]   &    1.3567  &  [-1.3602, 1.3582] &  1.3587  \\
1000 &  $5\%$   & [-1.7523, 1.7503]   &    1.7507  &  [-1.7533, 1.7512] &  1.7516  \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}[!ht]
\caption{Sensitivities on the AMDM $\hat a_V$ and the EDM $\hat a_A$ of the top-quark
through the process $e^-p  \to e^-\bar b \to \bar t \nu_e\gamma$ at the FCC-he.}
\begin{center}
 \begin{tabular}{|cc|cc|cc|}
\hline\hline
\multicolumn{6}{|c|}{ $\sqrt{s}=$ 10 TeV, \hspace{5mm} $P_{e^-} = 0 \%$, \hspace{5mm}  $95\%$ C.L.} \\
\hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{Hadronic} & \multicolumn{2}{c|}{Leptonic} \\
\hline
\cline{1-6}
${\cal L} \, (fb^{-1})$  & \hspace{0.5cm} $ \delta_{sys}$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm} &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm}  &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm} \\
\hline
50  &  $0\%$   & [-0.4824, 0.4719] & 0.4777 & [-0.6428, 0.6324] & 0.6384 \\
50  &  $3\%$   & [-1.1429, 1.1326] & 1.1391 & [-1.1618, 1.1514] & 1.1579 \\
50  &  $5\%$   & [-1.4667, 1.4563] & 1.4632 & [-1.4757, 1.4653] & 1.4722 \\
\hline
100 &  $0\%$   & [-0.4065, 0.3961] & 0.4017 & [-0.5414, 0.5309] & 0.5368 \\
100 &  $3\%$   & [-1.1386, 1.1281] & 1.1347 & [-1.1482, 1.1378] & 1.1443 \\
100 &  $5\%$   & [-1.4647, 1.4542] & 1.4612 & [-1.4692, 1.4588] & 1.4657 \\
\hline
300 &  $0\%$   & [-0.3101, 0.2997] & 0.3052 & [-0.4126, 0.4022] & 0.4079 \\
300 &  $3\%$   & [-1.1356, 1.1252] & 1.1317 & [-1.1388, 1.1284] & 1.1349 \\
300 &  $5\%$   & [-1.4633, 1.4529] & 1.4598 & [-1.4648, 1.4544] & 1.4613 \\
\hline
500 &  $0\%$   & [-0.2735, 0.2631] & 0.2686 & [-0.3638, 0.3534] & 0.3589 \\
500 &  $3\%$   & [-1.1349, 1.1246] & 1.1311 & [-1.1369, 1.1265] & 1.1330 \\
500 &  $5\%$   & [-1.4629, 1.4526] & 1.4595 & [-1.4639, 1.4535] & 1.4604 \\
\hline
1000 &  $0\%$   & [-0.2308, 0.2204] & 0.2259 & [-0.3067, 0.2963] & 0.3019 \\
1000 &  $3\%$   & [-1.1345, 1.1241] & 1.1306 & [-1.1355, 1.1251] & 1.1316 \\
1000 &  $5\%$   & [-1.4628, 1.4524] & 1.4593 & [-1.4632, 1.4528] & 1.4597 \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{Sensitivities on the AMDM $\hat a_V$ and the EDM $\hat a_A$ of the top-quark
through the process $e^-p  \to e^-\bar b \to \bar t \nu_e\gamma$ at the FCC-he.}
\begin{center}
\begin{tabular}{|cc|cc|cc|}
\hline\hline
\multicolumn{6}{|c|}{$\sqrt{s}=$ 7.07 TeV, \hspace{5mm} $P_e = -80 \%$, \hspace{5mm}  $95\%$ C.L.} \\
\hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{Hadronic} & \multicolumn{2}{c|}{Leptonic} \\
\hline
\cline{1-6}
${\cal L} \, (fb^{-1})$  & \hspace{0.5cm} $ \delta_{sys}$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm} &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm}  &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm} \\
\hline
50  &  $0\%$   & [-0.5756, 0.5574]   &    0.5696  & [-0.7694, 0.7512]  & 0.7612  \\
50  &  $3\%$   & [-1.3809, 1.3627]   &    1.3685  & [-1.4029, 1.3848]  & 1.3904  \\
50  &  $5\%$   & [-1.7727, 1.7545]   &    1.7582  & [-1.7832, 1.7651]  & 1.7687  \\
\hline
100 &  $0\%$   & [-0.4835, 0.4654]   &    0.4789  & [-0.6469, 0.6288]  & 0.6401  \\
100 &  $3\%$   & [-1.3757, 1.3575]   &    1.3633  & [-1.3869, 1.3688]  & 1.3746  \\
100 &  $5\%$   & [-1.7702, 1.7521]   &    1.7558  & [-1.7756, 1.7574]  & 1.7611  \\
\hline
300 &  $0\%$   & [-0.3659, 0.3477]   &    0.3639  & [-0.4910, 0.4729]  & 0.4864  \\
300 &  $3\%$   & [-1.3722, 1.3540]   &    1.3599  & [-1.3760, 1.3579]  & 1.3637  \\
300 &  $5\%$   & [-1.7686, 1.7504]   &    1.7541  & [-1.7704, 1.7522]  & 1.7559  \\
\hline
500 &  $0\%$   & [-0.3209, 0.3027]   &    0.3203  & [-0.4316, 0.4134]  & 0.4280  \\
500 &  $3\%$   & [-1.3715, 1.3533]   &    1.3592  & [-1.3738, 1.3556]  & 1.3615  \\
500 &  $5\%$   & [-1.7683, 1.7501]   &    1.7538  & [-1.7693, 1.7512]  & 1.7549  \\
\hline
1000 &  $0\%$  & [-0.2678, 0.2496]   &    0.2694  & [-0.3618, 0.3436]  & 0.3599  \\
1000 &  $3\%$  & [-1.3709, 1.3528]   &    1.3586  & [-1.3721, 1.3539]  & 1.3598  \\
1000 &  $5\%$  & [-1.7680, 1.7499]   &    1.7536  & [-1.7686, 1.7504]  & 1.7541  \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{Sensitivities on the AMDM $\hat a_V$ and the EDM $\hat a_A$ of the top-quark
through the process $e^-p  \to e^-\bar b \to \bar t \nu_e\gamma$ at the FCC-he.}
\begin{center}
\begin{tabular}{|cc|cc|cc|}
\hline\hline
\multicolumn{6}{|c|}{ $\sqrt{s}=$ 10 TeV, \hspace{5mm} $P_e = -80 \%$, \hspace{5mm}  $95\%$ C.L.} \\
\hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{Hadronic} & \multicolumn{2}{c|}{Leptonic} \\
\hline
\cline{1-6}
${\cal L} \, (fb^{-1})$  & \hspace{0.5cm} $ \delta_{sys}$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm} &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm}  &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm} \\
\hline
50  &  $0\%$   & [-0.4210, 0.4027]   &    0.4103  & [-0.5595, 0.5411]  & 0.5482  \\
50  &  $3\%$   & [-1.1423, 1.1239]   &    1.1287  & [-1.1529, 1.1346]  & 1.1394  \\
50  &  $5\%$   & [-1.4679, 1.4496]   &    1.4531  & [-1.4729, 1.4546]  & 1.4581  \\
\hline
100 &  $0\%$   & [-0.3556, 0.3372]   &    0.3450  & [-0.4719, 0.4536]  & 0.4609  \\
100 &  $3\%$   & [-1.1399, 1.1215]   &    1.1263  & [-1.1453, 1.1269]  & 1.1317  \\
100 &  $5\%$   & [-1.4668, 1.4484]   &    1.4520  & [-1.4693, 1.4509]  & 1.4545  \\
\hline
300 &  $0\%$   & [-0.2724, 0.2541]   &    0.2621  & [-0.3609, 0.3425]  & 0.3503  \\
300 &  $3\%$   & [-1.1382, 1.1198]   &    1.1246  & [-1.1400, 1.1216]  & 1.1264  \\
300 &  $5\%$   & [-1.4660, 1.4477]   &    1.4512  & [-1.4669, 1.4485]  & 1.4520  \\
\hline
500 &  $0\%$   & [-0.2409, 0.2226]   &    0.2307  & [-0.3187, 0.3004]  & 0.3083  \\
500 &  $3\%$   & [-1.1379, 1.1195]   &    1.1243  & [-1.1389, 1.1206]  & 1.1254  \\
500 &  $5\%$   & [-1.4659, 1.4475]   &    1.4510  & [-1.4664, 1.4480]  & 1.4515  \\
\hline
1000 &  $0\%$  & [-0.2041, 0.1858]   &    0.1939  & [-0.2695, 0.2512]  & 0.2592  \\
1000 &  $3\%$  & [-1.1376, 1.1192]   &    1.1240  & [-1.1382, 1.1198]  & 1.1246  \\
1000 &  $5\%$  & [-1.4657, 1.4474]   &    1.4509  & [-1.4660, 1.4476]  & 1.4512  \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{Sensitivities on the AMDM $\hat a_V$ and the EDM $\hat a_A$ of the top-quark
through the process $e^-p  \to e^-\bar b \to \bar t \nu_e\gamma$ at the FCC-he.}
\begin{center}
\begin{tabular}{|cc|cc|cc|}
\hline\hline
\multicolumn{6}{|c|}{ $\sqrt{s}=$ 7.07 TeV, \hspace{5mm} $P_e = 80 \%$, \hspace{5mm}  $95\%$ C.L.} \\
\hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{Hadronic} & \multicolumn{2}{c|}{Leptonic} \\
\hline
\cline{1-6}
${\cal L} \, (fb^{-1})$  & \hspace{0.5cm} $ \delta_{sys}$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm} &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm}  &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm} \\
\hline
50  &  $0\%$   & [-0.9861, 0.9838]   &    0.9842  & [-1.3174, 1.3150]  & 1.3152  \\
50  &  $3\%$   & [-1.4429, 1.4406]   &    1.4406  & [-1.5905, 1.5882]  & 1.5881  \\
50  &  $5\%$   & [-1.7936, 1.7915]   &    1.7913  & [-1.8772, 1.8749]  & 1.8746  \\
\hline
100 &  $0\%$   & [-0.8294, 0.8271]   &    0.8276  & [-1.1079, 1.1056]  & 1.1059  \\
100 &  $3\%$   & [-1.4019, 1.3996]   &    1.3997  & [-1.4874, 1.4851]  & 1.4852  \\
100 &  $5\%$   & [-1.7731, 1.7708]   &    1.7705  & [-1.8177, 1.8153]  & 1.8151  \\
\hline
300 &  $0\%$   & [-0.6305, 0.6282]   &    0.6289  & [-0.8421, 0.8398]  & 0.8404  \\
300 &  $3\%$   & [-1.3725, 1.3702]   &    1.3702  & [-1.4046, 1.4023]  & 1.4024  \\
300 &  $5\%$   & [-1.7588, 1.7565]   &    1.7563  & [-1.7744, 1.7721]  & 1.7719  \\
\hline
500 &  $0\%$   & [-0.5550, 0.5527]   &    0.5535  & [-0.7413, 0.7389]  & 0.7396  \\
500 &  $3\%$   & [-1.3663, 1.3640]   &    1.3641  & [-1.3861, 1.3838]  & 1.3839  \\
500 &  $5\%$   & [-1.7559, 1.7536]   &    1.7534  & [-1.7654, 1.7630]  & 1.7628  \\
\hline
1000 &  $0\%$  & [-0.4669, 0.4646]   &    0.4654  & [-0.6236, 0.6212]  & 0.6219  \\
1000 &  $3\%$  & [-1.3617, 1.3594]   &    1.3595  & [-1.3718, 1.3695]  & 1.3696  \\
1000 &  $5\%$  & [-1.7537, 1.7514]   &    1.7512  & [-1.7585, 1.7562]  & 1.7559  \\
\hline\hline
\end{tabular}
\end{center}
\end{table}

\begin{table}
\caption{Sensitivities on the AMDM $\hat a_V$ and the EDM $\hat a_A$ of the top-quark
through the process $e^-p  \to e^-\bar b \to \bar t \nu_e\gamma$ at the FCC-he.}
\begin{center}
\begin{tabular}{|cc|cc|cc|}
\hline\hline
\multicolumn{6}{|c|}{ $\sqrt{s}=$ 10 TeV, \hspace{5mm} $P_e = 80 \%$, \hspace{5mm} $95\%$ C.L.} \\
\hline
\multicolumn{2}{|c|}{} & \multicolumn{2}{c|}{Hadronic} & \multicolumn{2}{c|}{Leptonic} \\
\hline
\cline{1-6}
${\cal L} \, (fb^{-1})$  & \hspace{0.5cm} $ \delta_{sys}$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm} &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm}  &
\hspace{1.5cm} $\hat a_V$ \hspace{1.5cm}  &
\hspace{0.5cm} $|\hat a_A|$ \hspace{0.5cm} \\
\hline
50  &  $0\%$   & [-0.7189, 0.7083]   &    0.7129  & [-0.9589, 0.9482]  & 0.9526  \\
50  &  $3\%$   & [-1.1770, 1.1663]   &    1.1703  & [-1.2567, 1.2460]  & 1.2499  \\
50  &  $5\%$   & [-1.4835, 1.4728]   &    1.4765  & [-1.5256, 1.5149]  & 1.5185  \\
\hline
100 &  $0\%$   & [-0.6054, 0.5947]   &    0.5995  & [-0.8072, 0.7965]  & 0.8011  \\
100 &  $3\%$   & [-1.1563, 1.1456]   &    1.1497  & [-1.2003, 1.1896]  & 1.1936  \\
100 &  $5\%$   & [-1.4733, 1.4627]   &    1.4663  & [-1.4953, 1.4846]  & 1.4882  \\
\hline
300 &  $0\%$   & [-0.4613, 0.4506]   &    0.4555  & [-0.6146, 0.6039]  & 0.6087  \\
300 &  $3\%$   & [-1.1419, 1.1312]   &    1.1352  & [-1.1576, 1.1469]  & 1.1509  \\
300 &  $5\%$   & [-1.4665, 1.4558]   &    1.4594  & [-1.4739, 1.4633]  & 1.4669  \\
\hline
500 &  $0\%$   & [-0.4067, 0.3959]   &    0.4009  & [-0.5416, 0.5308]  & 0.5357  \\
500 &  $3\%$   & [-1.1389, 1.1282]   &    1.1323  & [-1.1485, 1.1378]  & 1.1419  \\
500 &  $5\%$   & [-1.4651, 1.4544]   &    1.4581  & [-1.4696, 1.4589]  & 1.4626  \\
\hline
1000 &  $0\%$  & [-0.3428, 0.3321]   &    0.3371  & [-0.4563, 0.4456]  & 0.4505  \\
1000 &  $3\%$  & [-1.1367, 1.1259]   &    1.1300  & [-1.1415, 1.1309]  & 1.1348  \\
1000 &  $5\%$  & [-1.4640, 1.4533]   &    1.4570  & [-1.4663, 1.4556]  & 1.4593  \\
\hline\hline
\end{tabular}
\end{center}
\end{table}


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

Quantum Monte Carlo (QMC) methods can accurately calculate the electronic structure of real materials\cite{QMCreview,NigUmr-BOOK-99,Kolorenc11}. The two most commonly used QMC methods for zero temperature calculations are variational Monte Carlo (VMC), which can compute expectation values of operators for optimized trial wave functions, and fixed-node diffusion Monte Carlo (DMC), which improves upon VMC results by using the imaginary-time evolution operator to project the trial wave function onto the ground state subject to the fixed-node boundary condition\cite{Anderson77}. QMC has been used to calculate a variety of properties such as cohesive energies, defect formation energies, and phase transition pressures\cite{Yao96, Gaudoin02, Hood03, Maezono03, Needs03, Alfe04b, Alfe05a, Alfe05b, Drummond06, Batista06, Maezono07, Pozzo08, Kolorenc08, Sola09, Hennig10, Driver10, Maezono10, Parker11, Abbasnejad12, Schwarz12, Hood12, Azadi13, Ertekin13, Shulenburger13, Chen14, Benali14, Azadi14, Foyevtsova14}. The accuracy is limited mostly by the fixed-node approximation\cite{Anderson77, Parker11} and the computational power required to reduce statistical uncertainty (the subject of this paper).

Minimizing the time for a QMC calculation of a property (e.g., energy) to a given statistical accuracy requires minimizing the evaluation cost of the orbitals -- used in the trial wave function $\Psi({\bf R})$ -- at each sampling point ${\bf R}$ of the electron coordinates.  The QMC energy, $E_{\rm QMC}$, is a weighted average of the {\it local energy},
\begin{equation}
  E_L(\Rvec) = \frac{H \Psi(\Rvec)}{\Psi(\Rvec)},
\end{equation}
at $N_{\rm MC}$ stochastically-chosen configurations:
\begin{equation} E_{\rm QMC} = \frac{1}{N_{\rm MC}}
	\sum_{i=1}^{N_{\rm MC}} w_i E_L(\Rvec_i).  \label{qmc_energy}
\end{equation}

The statistical uncertainty in $E_{\rm QMC}$ is proportional to $1/\sqrt{N_{\rm MC}}$.  Thus, repeated evaluation of the wave function $\Psi(\Rvec)$ and the Hamiltonian $H$ acting on the wave function, which requires both the wave function and its first and second derivatives, reduces the statistical uncertainty in the calculated property.  The root-mean-square fluctuation of the local energy in VMC
\begin{equation}
  \sigma_{\rm VMC} =
  \sqrt{\frac{1}{N_{\rm MC}} \sum_{i=1}^{N_{\rm MC}} (E_L(\Rvec_i)-E_{\rm
      VMC})^2} \label{qmc_sigma}
\end{equation}
indicates the quality of $\Psi(\Rvec)$ because the individual local energies equal the average when $\Psi(\Rvec)$ is an exact eigenfunction of $H$.  QMC simulations frequently use the Slater-Jastrow form of the wave function\cite{QMCreview}, $\Psi(\Rvec) = J(\Rvec) D(\Rvec)$, where $J(\Rvec)$ is a Jastrow factor\cite{JastrowFactor} (in this work, a simple electron-electron Jastrow with no free parameters is used to impose the electron-electron cusp condition) and $D(\Rvec)$ is a Slater determinant\cite{SlaterDeterminant} of single-particle orbitals.

The orbitals used in QMC wave functions typically come from density-functional or Hartree-Fock calculations and, in periodic systems, are Bloch functions of the form
\begin{equation}
  \phi_{n \kvec} ({\bf r}} \def\kvec{{\bf k}} \def\xt{\tilde{x}) = u_{n \kvec}({\bf r}} \def\kvec{{\bf k}} \def\xt{\tilde{x}) e^{i\kvec\cdot{\bf r}} \def\kvec{{\bf k}} \def\xt{\tilde{x}},
\end{equation}
where $u_{n \kvec}({\bf r}} \def\kvec{{\bf k}} \def\xt{\tilde{x})$ has the periodicity of the crystal lattice, $n$ is the band index, and $\kvec$ the crystal momentum.  The periodic function, $u_{n \kvec}({\bf r}} \def\kvec{{\bf k}} \def\xt{\tilde{x})$, is represented by a linear combination of basis functions.  Frequently QMC calculations are performed using simulation cells larger than the primitive cell to reduce Coulomb finite-size errors.  However, since $u_{n \kvec}({\bf r}} \def\kvec{{\bf k}} \def\xt{\tilde{x})$ is periodic in the primitive cell, representing it by basis-function expansions in just the primitive cell is sufficient to simulate larger cells.

The computational cost per $N$-electron Monte Carlo move of evaluating the Slater determinant is ${\it O}(N^3)$, when spatially-extended basis functions are used to represent the orbitals, since $N$ orbitals are evaluated for each of the $N$ electrons, and each orbital is a sum over $O(N)$ basis functions.  Spatially-localized basis functions avoid the linear scaling of the number of basis functions with system size since only those basis functions that are non-zero at a given point contribute to the wave function value at that point, resulting in ${\it O}(N^2)$ scaling.

Planewaves, despite their undesirable scaling, are a popular choice for basis functions for the density-functional and Hartree-Fock methods because of their desirable analytic properties.  The advantage of a planewave representation is that planewaves form an orthogonal basis, and, in the infinite sum, a complete single-particle basis.  Thus, adding more planewaves to a truncated basis (as is always used in practice) systematically improves the wave function representation towards the infinite single-particle basis limit.  The energy of the highest frequency planewave included in the sum, the cutoff energy $E_{\rm cut} = \hbar^2 {G_{\rm cut}}^2 / 2\,m_{\rm e}$,
characterizes a given truncated planewave basis by setting the smallest length scale about which the wave
function has information. Thus, a planewave-based orbital $\phi_{\rm PW}$ is a sum over each planewave ${\bf G}$ below the cutoff multiplying a real- or complex-valued coefficient $c_{{\bf G} n {\bf k}}$ unique to that planewave, the band index $n$, and the crystal momentum ${\bf k}$:

\begin{equation}
  \label{eq:pw_orbital}
  u_{n {\bf k}}({\bf r}) = \phi_{\rm PW}(\bf{r}) = \sum_{\bf G} c_{{\bf G} n {\bf k}} \exp(\imath {\bf G}\cdot {\bf r}).
\end{equation}

Williamson {\it et al.}\cite{Williamson01} first applied the pp-form spline interpolation method to approximate planewave-based orbitals by localized basis functions in QMC calculations.  They report an $O(N)$ reduction in the time scaling.  Alf\`e and Gillan\cite{Alfe04} introducing the related method of B-spline approximation in QMC, report significant reduction in the calculation time while maintaining planewave-level accuracy.

This work compares the three methods previously applied to QMC (pp-splines\cite{Williamson01}, interpolating B-splines\cite{Esler_unpub}, and smoothing B-splines\cite{Hernandez97, Alfe04}) with a fourth method (Lagrange polynomials) originally implemented by one of us in QMC but heretofore unpublished.  Section~\ref{sec:methods} introduces, compares and contrasts the four methods.  Section~\ref{sec:accuracy} compares the accuracy of the polynomial methods in reproducing the QMC energies and fluctuations in the local energy relative to the corresponding values from the planewave expansion.  Section~\ref{sec:accuracy} also studies whether it is advantageous to construct separate approximations for the gradient and the Laplacian of the orbitals.  Section~\ref{sec:speed} compares the time cost in QMC calculations of polynomial methods and planewave expansions.  Section~\ref{sec:memory} compares the memory requirements of the polynomial methods and planewaves.  Section~\ref{sec:conclusions} concludes that higher accuracy and lower memory requirement make smoothing B-splines, with a separate approximation for the Laplacian, the best choice.  The appendix describes the details of the approximating polynomials.

\section{Methods} \label{sec:methods}

The four methods of approximating the planewave-represented single-particle orbitals with polynomials that we study in this paper are: interpolating Lagrange polynomials\cite{Lagrange}, interpolating piecewise-polynomial-form splines (pp-splines)\cite{deBoor01} (often simply called interpolating splines), and basis-form splines (B-splines) (both interpolating\cite{B_spline,deBoor01} and smoothing\cite{Hernandez97,Alfe04}).  For the pp-splines, we employ the Princeton \textsc{pspline} package\cite{PrincetonSpline}, and, for interpolating B-splines, the \textsc{einspline} package\cite{Einspline}, interfaced to the \textsc{champ} QMC program\cite{Cha-PROG-XX}.

{\it Common features.}  The methods share several aspects.  They construct the orbital approximation by a trivariate polynomial tensor product.  Each of the methods can employ polynomials of arbitrary order, $n$.  We use cubic polynomials, the customary choice.  The methods transform the cartesian coordinates to reduced coordinates prior to evaluating the polynomial approximation (see Eq.~(\ref{eq:reducedcoordinate}) and the Coordinate paragraph).  They use a grid of real-space points with associated coefficients and have a natural grid spacing defined by the highest-energy planewave in the planewave sum representing the orbital (see Eq.~(\ref{eq:natural_spacing}) and the Grid paragraph).  They share two possibilities for evaluating the required derivatives of a polynomial-represented function: (1) derivatives of the polynomials or (2) separate polynomial approximations of the planewave-represented derivatives.

{\it Distinctive features.}  As Lagrange-interpolated functions have discontinuous derivatives at the grid points, ensuring continuity in the derivative-dependent energies requires a separate interpolation for each of the components of the gradient and for the Laplacian of the orbitals, increasing the memory requirement by a factor of five.  In contrast to Lagrange interpolation, splines of degree $n$ have continuous derivatives up to order $n-1$ at the grid points, and, thus, the gradient and Laplacian of the splined function can approximate the gradient and Laplacian of the planewave sum, though this choice leads to a loss of accuracy. Spline functions have two free parameters in each dimension that are used to set the boundary conditions.  Since planewave-based orbitals are periodic, we choose the boundary conditions to have matching first and second derivatives at the boundaries in each dimension.  The formulation of both B-splines and pp-splines may be either interpolating (exact function values at the grid points)\cite{Einspline} or smoothing\cite{Hernandez97,Alfe04}. Smoothing splines are advantageous when the data is noisy, but this is not the case in our application.  Instead our rationale is the following: the planewave coefficients of each orbital specify that orbital, and the particular form of smoothing spline we use\cite{Hernandez97,Alfe04} is constructed to exactly reproduce the nonzero coefficients (see~\ref{smoothing_spline}).  Since fixing the values at the grid points and specifying the boundary conditions uniquely determines the interpolating spline function, interpolating B-splines and pp-splines yield identical function values\cite{deBoor01}. Due to the reduced number of coefficients stored per point, interpolating B-splines are preferable to pp-splines provided the time required for their evaluation is not larger than for pp-splines.

{\it Grid.}  Each of the interpolation methods permits either uniform or nonuniform grids.  For simplicity, we employ uniform grids, but the number of grid points in each dimension needs not be the same.  The highest-energy planewave in the planewave sum representing a given orbital defines a natural maximal grid spacing, above which short length scale information is lost.  One point per maximum and minimum of the highest-energy planewave $G_{\rm max}$, or two points per wavelength, $\lambda_{\rm min} = 2 \pi / G_{\rm max}$, defines this natural spacing $h_{\rm natural}$:
\begin{equation}
  h_{\rm natural} =
  \frac{\lambda_{\rm min}}{2} = \frac{\pi}{G_{\rm max}}.
  \label{eq:natural_spacing}
\end{equation}

{\it Coordinates.}  To simplify the form of the polynomials for the evaluation of the splines, we formulate the methods such that the point where the function is evaluated lies in the interval $[0,1)$ in each dimension. The spline evaluation for a given point requires
the coefficients at the four neighboring grid points in each dimension and requires
transforming the Cartesian coordinates to reduced coordinates. The
primitive-cell lattice vectors of the crystal ${\bf a}_i$ need not be orthogonal.
The reduced vector, ${\tilde{\bf r}} = (\tilde{r}_1,\tilde{r}_2,\tilde{r}_3) =
(\tilde{x},\tilde{y},\tilde{z})$, corresponding to the Cartesian vector
${\bf r}} \def\kvec{{\bf k}} \def\xt{\tilde{x}$, is
\begin{eqnarray}
	\tilde{r}_i = {N_{{\rm grid},i}}} \def\beq{\begin{eqnarray} \left( {\left( \vector{a}^{-1} \vector{r} \right)}_i -
	\lfloor {\left( \vector{a}^{-1} \vector{r} \right)}_i \rfloor \right) - \\ \nonumber
      \lfloor {N_{{\rm grid},i}}} \def\beq{\begin{eqnarray} \left(
	{\left( \vector{a}^{-1} \vector{r} \right)}_i - \lfloor {\left( \vector{a}^{-1} \vector{r} \right)}_i \rfloor \right) \rfloor,
	\label{eq:reducedcoordinate}
\end{eqnarray}
where $\lfloor \, \rfloor$ is the floor function, which returns the integer part of a number and $\bf a$ is the $3\times 3$ matrix of lattice vectors ${\bf a}_i$.  Multiplying the Cartesian coordinate by ${\bf a}^{-1}$ transforms it to crystal coordinates. Subtracting the integer part of the crystal coordinates forces the coordinate inside the primitive cell, restricting the magnitude of the coordinate values to be between zero and one.  Multiplying the crystal coordinates by the number of grid points $N_{{\rm grid},i}$ along the $i^{th}$ lattice vector transforms the crystal coordinates into units of the grid point interval.  Subtracting the integer part of the interval-unit crystal coordinate (i.e., the index of the grid point smaller than and closest to $r_i$) yields the reduced coordinate ${\tilde{\bf r}}$, each component of which is in the interval $[0,1)$.  To obtain the Cartesian coordinate gradient and Laplacian starting from the reduced coordinate gradient and Hessian requires we use the chain rule, yielding ${\bf \nabla}\phi({\bf r}) = {N_{{\rm grid},i}}} \def\beq{\begin{eqnarray} ({\bf a}^{-1})^{\rm T} \tilde{\bf \nabla}\phi(\tilde{\bf r})$ and $\nabla^{2}\phi({\bf r}) = {N_{{\rm grid},i}}} \def\beq{\begin{eqnarray}^2 \sum_{i,j}(({\bf a}^{-1})({\bf a}^{-1})^{\rm T})_{ij} \partial^{2}\phi(\tilde{\bf r})/\partial \tilde{r}_i \partial \tilde{r}_j$. \ref{sec:explicitforms} gives the explicit forms of the approximating functions.

\section{Results}
\label{sec:results}

Three quantities compare the performance of Lagrange interpolation, pp-spline interpolation, B-spline interpolation and B-spline smoothing, within quantum Monte Carlo calculations of periodic systems: (i) the {\it accuracy} in reproducing the planewave orbital values, (ii) the {\it speedup} from the planewave-based calculation, and (iii) the computer {\it memory} required.

The results presented in this section are obtained for single-particle orbitals at the $\Gamma$ point, that are obtained with the LDA exchange-correlation functional in density functional theory and Troullier-Martins pseudopotentials\cite{TroullierMartins91}. \footnote{Mg -- valence configuration: $2s^22p^63d^{0.1}$, cutoff radii: $= 1.20, 1.50, 1.80 \:{\rm a.u.}$ respectively.  O -- valence configuration: $2s^22p^43d^0$, cutoff radii $= 1.0,1.0,1.0\:{\rm a.u.}$ respectively with Hamann's generalized state method\cite{Hamann89} for the d-channel.  Si -- valence configuration: $3s^23p^23d^0$, cutoff radii $= 2.25,2.25,2.25\:{\rm a.u.}$ respectively.  Al -- valence configuration: $3s^23p^13d^0$, cutoff radii $= 2.28,2.28,2.28\,{\rm a.u.}$ respectively with the generalized state method for the d-channel.}

\subsection{Accuracy} \label{sec:accuracy}

To understand the accuracy of the polynomial approximation to the planewave sum in the context of QMC, we compare the error in the orbital value, gradient of the orbital, Laplacian of the orbital, the total VMC energy, and the root-mean-square fluctuation in the VMC local energy relative to the corresponding quantity computed using the planewave sum.  Increasing the planewave cutoff tests the accuracy of the approximations as the planewave basis becomes more complete.
\footnote{Reducing the grid spacing for fixed planewave cutoff results in all quantities converging to their planewave values for the selected cutoff\cite{supplementary}}.

We find that the choice of $k$-point in our test does not affect the conclusions as similar results were obtained for various $k$-points ($\Gamma$, L, and X high-symmetry points in diamond Si) and different simulation cell sizes (2, 8, 16, and 32-atom cells in diamond Si).  However, significant differences in accuracy of the approximations occur for the three different materials tested, diamond Si, fcc Al and rock-salt MgO\cite{supplementary}. Since the approximation methods show the greatest differences from each other for the case of MgO in the rock-salt structure, we focus here on those results.

\begin{figure}
    \begin{center}
    \includegraphics[width=\columnwidth]{accuracy_cutoff_all_MAE_over_MAV+uncertainty_MgO_rock-salt_2_Gamma.eps}
     \end{center}
     \caption{(Color online) Relative mean absolute error of the
     orbitals, gradient of the orbitals, and Laplacian of the orbitals as a
     function of planewave cutoff in rock-salt MgO (chosen for greatest
     contrast in results) at natural grid spacing for each of the approximation
     methods:  Lagrange polynomials (squares), pp-splines and interpolating
     B-splines (diamonds), and smoothing B-splines (triangles).  Error bars indicate
     one standard deviation from the mean of the data point.  The error in the
     gradient and Laplacian of the orbitals is the error in a direct approximation
     of the gradient and Laplacian, respectively, of the orbital, not the gradient
     or Laplacian of an approximation of the orbital.  The large fluctuation in the
     error of the Laplacian is due to the small, stochastically chosen sample used
     to calculate the mean.  Smoothing B-splines show the smallest error relative to
     the planewave value at all grid spacings although the statistical uncertainty
     in the average over the components of the gradient obscures that result.
     } \label{fig:orbdorbddorb_error}
 \vskip 9mm
 \end{figure}

\begin{figure} \begin{center}
    \includegraphics[width=\columnwidth]{accuracy_cutoff_VMC_E_MgO_rocksalt_2_Gamma.eps}
    \end{center}
    \caption{(Color online) VMC total energy as a function of planewave basis
    cutoff in rock-salt MgO for each of the approximation methods at the
    natural grid spacing for that cutoff.  Error bars on points indicate one
    standard deviation of statistical uncertainty in the
    total energies.  Smoothing B-splines with separate approximations for
    the orbitals and the Laplacian of the orbitals lie within one standard
    deviation of statistical uncertainty of the planewave value for all
    cutoffs tested. (Note: Lagrange polynomials appear on the figure only at
    a cutoff of 250~Ha [$\approx 6800$ eV]).
} \label{fig:energy_with_cutoff}
\vskip 9mm
\end{figure}

\begin{figure}
  \begin{center}
    \includegraphics[width=\columnwidth]{sigma_cutoff_VMC_MgO_rocksalt_2_Gamma.eps}
  \end{center}
  \caption{(Color online) Root-mean-square fluctuation of the VMC local energy,
   $\sigma$, as a function of planewave basis cutoff in rock-salt MgO for each of
   the approximation methods at the natural grid spacing for that cutoff.  Error
   bars on points indicate one standard deviation of statistical uncertainty in
   the values.  Once the planewave value has converged, near 120~Ha ($\approx3270$
   eV), the smoothing B-splines with a separate approximation for the Laplacian is
   within statistical uncertainty of the planewave $\sigma$.  (Note: Lagrange
   polynomials and smoothing B-splines without separate approximation for the
   Laplacian appear on the figure only at a cutoff of 250~Ha[$\approx6800$
   eV]---interpolating B-splines without separate interpolation for the Laplacian
   do not appear at all on this scale)}
  \label{fig:sigma_with_cutoff}
\vskip 9mm
\end{figure}

Figure~\ref{fig:orbdorbddorb_error} shows the relative mean absolute error in the orbital, its gradient, and its Laplacian as a function of the planewave cutoff for the four approximation methods at natural grid spacing in rock-salt MgO.  The relative mean absolute error is the mean absolute error divided by the mean absolute value computed from the planewave sum.  Since pp-splines and interpolating B-splines give identical function values\cite{deBoor01}, they lie on a single curve and set of points.

For each of the approximations, the gradient and Laplacian used by the QMC calculation can be obtained either by taking derivatives of the polynomial-approximated orbital, or, by constructing separate polynomial approximations for the gradient and the Laplacian of the planewave sum.  The central and lower panels of Figure~\ref{fig:orbdorbddorb_error} show the accuracy of separate approximations of the gradient and Laplacian of the planewave sum.  When separately approximating any derivatives of the orbitals, the resulting energy need not be an upper bound to the true energy, but the separate approximations recover the planewave value in the limit of infinite basis set.

Spline interpolation is more accurate than Lagrange interpolation for all planewave cutoffs.  Splines utilize all the tabulated function values (a global approximation) to enforce first and second derivative continuity across grid points, whereas Lagrange interpolation uses just the closest 64 points (a local approximation) and has derivative discontinuities at the grid points.  This leads to larger fluctuations in the error of Lagrange interpolation compared to splines.

Figures~\ref{fig:energy_with_cutoff} and~\ref{fig:sigma_with_cutoff} show the quantities of importance to QMC calculations, the total VMC energy $E_\mathrm{VMC}$ and the standard deviation of the local energy $\sigma_\mathrm{VMC}$, respectively, as a function of planewave cutoff in rock-salt MgO for the four approximations.  The deviations of $E_\mathrm{VMC}$ and $\sigma_\mathrm{VMC}$ from the planewave values reflect the errors in the orbitals, their gradients and Laplacian.  Smoothing B-splines are more accurate than interpolating splines, which in turn are more accurate than Lagrange interpolation.  Furthermore, separately approximating the Laplacian in the spline approximations significantly improves the accuracy of $E_\mathrm{VMC}$ and $\sigma_\mathrm{VMC}$.  The standard deviation of the local energy $\sigma_\mathrm{VMC}$ is more sensitive than the total energy $E_\mathrm{VMC}$ to the errors in the approximations because errors in the local energies partially cancel when averaging the local energy to obtain $E_\mathrm{VMC}$.
In all systems tested, convergence of $E_\mathrm{VMC}$ to within 1~mHa is observed for planewave cutoff energies 9-25~Ha smaller than for the convergence of $\sigma_\mathrm{VMC}$ to the same level.

\subsection{Speedup}
\label{sec:speed}
Figure~\ref{fig:speed} illustrates that the three methods of polynomial approximation speed up the planewave calculation by the same factor which scales as $O(N)$.  Tests on three different computer platforms (3.0 GHz Intel Pentium 4, 2.4 GHz Intel Xeon, 900 MHz Intel Itanium 2) show that the time scaling (in seconds) with the number of atoms $N$ for the approximating polynomials is of the order of $10^{-4}\,N^2 + 10^{-6}\,N^3$ compared to the scaling for planewaves of $10^{-3}\,N^3$.  The difference in computational time between the Lagrange polynomials and the B-splines is less than 10\% and varies between the different computers.

While pp-splines store eight coefficients at each grid point, and Lagrange interpolation and B-splines store just one, all methods require accessing the same number of coefficients (64 for cubic polynomials in 3D) from memory for each function evaluation.  However, Lagrange and B-splines access one coefficient from each of the 64 nearest-neighbor points whereas pp-splines access eight coefficients from each of the eight nearest-neighbor points.  This reduction in accessed neighbor points could make pp-splines faster since the data access is more local.  However, the calculations show that, for the implementation of pp-splines used here\cite{PrincetonSpline}, no speedup occurs in practice.  Additionally, further optimization of the smoothing B-splines routines has reduced the time scaling prefactor by an order of magnitude.

\begin{figure}
  \begin{center}
	\includegraphics[width=\columnwidth]{speed_tMC_Natoms_Si_diamond.eps}
  \end{center}

  \caption{(Color online) VMC time per Monte Carlo step versus number of atoms.
  All three approximations speed up the planewave calculation by a factor of
  $O(N)$ with nearly the same prefactor, recommending all approximation
  methods equally {\em on the basis of speed}.
  Further optimization of the smoothing B-splines routines has reduced the time
  scaling prefactor by an order of magnitude from the data shown here.
  The similarity in form of the approximation methods
  (see Eqs.~(\ref{eq:lagrangeinterpolation3}),~(\ref{eq:pp-splineinterpolation3}),~(\ref{eq:b-splineapproximation}))
  and the fact that all methods need to retrieve a similar number of
  coefficients from memory
  account for the similarity in evaluation speed
  despite differences in approximation properties.
  }
  \label{fig:speed}
\end{figure}

\subsection{Memory}
\label{sec:memory}
At the natural grid spacing, the polynomial approximations store a total number of values equal to or greater than the number of planewaves.  \footnote{If the planewave expansion includes only planewaves that lie within a parallelepiped defined by three reciprocal lattice vectors, then the number of grid points at natural spacing (see Eq.~(\ref{eq:natural_spacing})) equals the number of planewave coefficients.  However, it is customary to include planewaves in a sphere up to some energy cutoff in the planewave sum, in which case the number of grid points is larger than the number of planewave coefficients.
For example, in a cubic lattice, the ratio of the number of grip points at natural
grid spacing to the number of planewaves is the ratio of the volume of a cube
to the volume of the inscribed sphere, namely $6/\pi$.
} Trivariate, cubic pp-splines store eight values per grid point for each function, namely the function values, the three second derivatives along each direction, three mixed fourth-order derivatives, and one mixed sixth-order derivative (see \citep{PrincetonSpline}] or Eq.~(\ref{eq:pp-splinecoefficients3}) for details).  Lagrange interpolation and B-splines store only one value per grid point for each function.  In the case of Lagrange interpolation, the stored values are the function values, whereas, for B-splines, the stored values are the derived B-spline coefficients (see \ref{sec:explicitforms}).

All the approximations can obtain the gradient and the Laplacian by either taking appropriate derivatives of the splined functions or by generating separate approximations for the gradient and the Laplacian.  Separate approximations for the Laplacian increase the memory requirement by a factor of two, and, separate approximations for the Laplacian and the gradient increase the memory requirement by a factor of five.  Since the gradient and Laplacian of the Lagrange interpolation are not continuous, we always use separate approximations for the Laplacian and the gradient when using Lagrange interpolation.  For splines, the increased accuracy achieved by using separate approximations warrants using separate approximations for the Laplacian but not for the gradient.

\section{Conclusions}
\label{sec:conclusions}
The four polynomial approximation methods -- interpolating Lagrange polynomials, interpolating pp-splines, interpolating B-splines, and smoothing B-splines -- speed up planewave-based quantum Monte Carlo (QMC) calculations by $O(N)$, where $N$ is the number of atoms in the system.  At natural grid spacing, smoothing B-splines are more accurate than interpolating splines, which are in turn more accurate than Lagrange interpolation for all planewave cutoff values tested.  Separately approximating the Laplacian of the orbitals results in the total energy and root-mean-square fluctuation of the local energy to be closest to the values obtained using the planewave sum.  High accuracy and low memory requirement make smoothing B-splines, with the Laplacian splined separately from the orbitals, the best choice for approximating planewave-based orbitals in QMC calculations.

\section{Acknowledgments}
This work was supported by the Department of Energy Basic Energy Sciences, Division of Materials Sciences (DE-FG02-99ER45795 and DE-FG05-08OR23339), and the National Science Foundation (CHE-1112097 and DMR-1056587).  Computational resources were provided by the Ohio Supercomputing Center, the National Center for Supercomputing Applications, and the National Energy Research Scientific Computing Center (supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC02-05CH11231).  We thank Neil Drummond, Ken Esler, Jeongnim Kim, Mike Towler, and Andrew Williamson for helpful discussions, Ken Esler for recommending and helping with implementation of his Einspline library, and Jos{\'e} Lu{\'i}s Martins for the use of his pseudopotential generation and density functional programs.

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