#  40Ca transverse response function from coupled-cluster theory

J. E. Sobczyk  Institut für Kernphysik and PRISMA+ Cluster of Excellence,
Johannes Gutenberg-Universität, 55128 Mainz, Germany  B. Acharya  Physics
Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, USA  Institut
für Kernphysik and PRISMA+ Cluster of Excellence, Johannes Gutenberg-
Universität, 55128 Mainz, Germany  S. Bacca  Institut für Kernphysik and
PRISMA+ Cluster of Excellence, Johannes Gutenberg-Universität, 55128 Mainz,
Germany  Helmholtz-Institut Mainz, Johannes Gutenberg-Universität Mainz,
D-55099 Mainz, Germany  G. Hagen  Physics Division, Oak Ridge National
Laboratory, Oak Ridge, TN 37831, USA  Department of Physics and Astronomy,
University of Tennessee, Knoxville, TN 37996, USA

###### Abstract

We present calculations of the 40Ca transverse response function obtained from
coupled-cluster theory used in conjunction with the Lorentz integral transform
method. We employ nuclear forces derived at next-to-next-to leading order in
chiral effective field theory with and without $\Delta$ degrees of freedom. We
first benchmark this approach on the 4He nucleus and compare both the
transverse sum rule and the response function to earlier calculations based on
different methods. As expected from the power counting of the chiral expansion
of electromagnetic currents and from previous studies, our results retaining
only one-body term underestimate the experimental data for 4He by about
$20\%$. However, when the method is applied to 40Ca at the same order of the
expansion, response functions do not lack strength and agree well with the
world electron scattering data. We discuss various sources of theoretical
uncertainties and comment on the comparison of our results with the available
experiments.

##  I Introduction

A longstanding quest in nuclear physics is to accurately describe a wide range
of nuclear phenomena by treating nuclei as systems of protons and neutrons
interacting among themselves and with external fields. In a so-called _“ab
initio”_ approach, the nuclear many-body Schrödinger equation is solved,
either exactly or using controlled approximations, by employing nuclear forces
that are constrained by the symmetries of quantum chromodynamics, nucleon-
nucleon (NN) scattering experiments, and selected nuclear data Hergert (2020);
Ekström  _et al._ (2023). Modern _ab initio_ computations have increasingly
adopted chiral effective field theory ($\chi$EFT) to derive nuclear forces,
providing a rigorous framework to organize NN, three-nucleon (3N) and many-
nucleon forces, as well as currents that couple nucleons to external fields,
into a low-momentum expansion Epelbaum  _et al._ (2009). The close relation
between the nuclear Hamiltonians and electroweak currents makes $\chi$EFT
studies of electroweak observables very interesting Gardestig and Phillips
(2006); Bacca and Pastore (2014). Extending _ab initio_ studies to electroweak
cross sections is important to test the theory, to estimate theoretical
uncertainties, and to possibly improve on the precision.

Traditionally applied to light and closed-shell medium-mass nuclei, _ab
initio_ methods have recently found success in describing heavy Hu  _et al._
(2022), deformed Caprio  _et al._ (2015); Launey  _et al._ (2016); Novario
_et al._ (2020); Miyagi  _et al._ (2020); Frosini  _et al._ (2022); Hagen  _et
al._ (2022); Ekström  _et al._ (2023), and exotic Bonaiti  _et al._ (2022);
Kaufmann  _et al._ (2020) nuclei. Coupled-cluster theory Coester (1958);
Coester and Kümmel (1960); Kümmel  _et al._ (1978); Mihaila and Heisenberg
(2000); Dean and Hjorth-Jensen (2004); Włoch  _et al._ (2005); Hagen  _et al._
(2008, 2010); Binder  _et al._ (2014); Hagen  _et al._ (2014) played an
important role in these achievements among other computational methods that
exhibit a polynomial scaling with the number of nucleons (see Ref. Hergert
(2020) for a recent review). Extensions of _ab initio_ studies to processes
that involve nuclear excitations in the continuum is an important frontier
Johnson  _et al._ (2020). Recently, the Lorentz integral transform method
Efros  _et al._ (1994) was reformulated within coupled-cluster theory to
generate an approach called LIT-CC which is well suited to study nuclear
responses induced by electroweak probes. Within this approach, the sum over
continuous final states is reformulated in terms of an integral transform and
later inverted to access the response function. The LIT-CC method was
extensively benchmarked on electric dipole excitations of medium-mass nuclei
Bacca  _et al._ (2013a, 2014); Miorelli  _et al._ (2016), but it can be
extended to a variety of electroweak observables.

In addition to improving our understanding of the nuclear dynamics, one of the
main motivations to investigate electroweak response functions is the neutrino
experiments, especially the long-baseline programs, which require reliable
predictions of neutrino-nucleus cross-sections to perform the neutrino energy
reconstruction and, from there, infer the oscillations parameters. The quasi-
elastic peak is the dominating mechanism in the T2K and the future HyperK Abe
_et al._ (2015) experiments, and is also an important reaction channel for
DUNE Acciarri  _et al._ (2015). Presently, the only _ab initio_ predictions of
neutrino cross sections in this region are available from the Green’s Function
Monte Carlo (GFMC) method for light nuclei up to 12C Lovato  _et al._ (2014,
2015, 2018, 2020). Under certain approximations, other methods, such as the
short-time approximation Pastore  _et al._ (2020) or the spectral function
Rocco and Barbieri (2018); Barbieri  _et al._ (2019); Sobczyk  _et al._
(2020), have been proposed to connect the cross sections with _ab initio_
calculations. Recently there has also been progress in computing the 16O
spectral function calculated within coupled-cluster theory Sobczyk and Bacca
(2023).

In this endeavor, electron-scattering experiments play an important role in
guiding theoretical studies Khachatryan  _et al._ (2021). Due to an analogous
structure of electron- and neutrino-nucleus cross-sections, the precise
electron data provide a stringent test for our modeling of the nuclear
dynamics. The Rosenbluth technique allows a separation of the electron-
scattering cross section into the longitudinal and the transverse response
functions, which are driven by the electric charge and the electromagnetic
current operators, respectively. Using the LIT-CC approach, we successfully
described the longitudinal response function at the momentum transfers of a
few hundred MeV in the region medium-mass nuclei Sobczyk  _et al._ (2020,
2021). The final state interactions were consistently taken into account,
leading to a very good agreement with available experimental data on 4He and
40Ca.

In this work, we further extend the LIT-CC method to describe the transverse
response function, which dominates the cross section at backward scattering
angles. We provide the first _ab initio_ results for mass number $A=40$ in the
impulse approximation. This is an important step towards building a complete
and systematic theory for lepton-nucleus scattering.

The article is structured as follows. In Sec. II, we introduce the definition
of the transverse response function and describe the framework in which we
perform calculations. We present our results in Sec. III, starting from the
transverse sum rule for 4He and 40Ca and benchmark with existing calculations.
Next, we present the response functions for these two nuclei up to $400$
MeV/c, commenting on the tensions existing in available experimental data on
40Ca. Finally, we conclude in Sec. IV.

##  II Formalism

The inclusive electron-nucleus scattering cross-section in the Born
approximation can be expressed in terms of longitudinal and transverse
response functions. The focus of this work is the transverse response function
$R_{T}$, which depends on the energy and momentum transfer $(\omega,q)$, and
is defined as

$R_{T}(\omega,q)\\!=\\!\int\\!\\!\\!\\!\\!\\!\\!\sum_{f}|\\!\left\langle\Psi_{f}|\mathbf{J}^{T}(q)|\Psi_{0}\right\rangle\\!|^{2}\,\delta\\!\\!\left(\\!E_{f}+\frac{q^{2}}{2M}-E_{0}-\omega\\!\right)\,.$
(1)

Here, $\mathbf{J}^{T}$ corresponds to the transverse part of the
electromagnetic current operator with respect to the direction of the vector
momentum transfer ${\mathbf{q}}$. In Eq. (1), the nuclear ground state
wavefunction is denoted by $\Psi_{0}$ (with corresponding energy $E_{0}$) and
we sum over all possible final states $\Psi_{f}$ (with energy $E_{f}$) which
obey the energy conservation expressed by the delta function, where $M$ is the
mass of the nucleus and $q=|{\mathbf{q}}|$.

In the impulse approximation, the current operator can be decomposed into the
isoscalar and isovector parts $\mathbf{J}=\mathbf{J}_{s}+\mathbf{J}_{v}$, as

$\displaystyle\mathbf{J}_{s}=\sum_{j=1}^{A}\left(G_{E}^{S}(Q^{2})\frac{\mathbf{{p}}_{j}}{m}-i\,G_{M}^{S}(Q^{2})\frac{\mathbf{q}\times\bm{\sigma}_{j}}{2m}\right)e^{i\mathbf{q}\cdot\mathbf{r}_{j}}\frac{1}{2}\,,$
$\displaystyle\mathbf{J}_{v}=\sum_{j=1}^{A}\left(G_{E}^{V}(Q^{2})\frac{\mathbf{{p}}_{j}}{m}-i\,G_{M}^{V}(Q^{2})\frac{\mathbf{q}\times\bm{\sigma}_{j}}{2m}\right)e^{i\mathbf{q}\cdot\mathbf{r}_{j}}\frac{\tau^{z}{{}_{j}}}{2}\,.$
(2)

Here, one can clearly identify the convection and spin terms, which depend on
the nucleon momenta ${\mathbf{p}}_{j}$ and spins $\bm{\sigma}_{j}$,
respectively. The nucleon mass is indicated with $m$ and its third isospin
component with $\tau^{z}_{j}$, while $G_{E,M}^{S}\equiv
G_{E,M}^{p}+G_{E,M}^{n}$ , $G_{E,M}^{V}\equiv G_{E,M}^{p}-G_{E,M}^{n}$ are the
electric and magnetic form factors related to the proton and neutron form
factors. They are functions of $Q^{2}=q^{2}-\omega^{2}$. However, in our
calculation the dependence on the energy transfer $\omega$ is not known _a
priori_ , so we assume $\omega\approx\sqrt{q^{2}+m^{2}}-m$ which corresponds
to the kinematical region of the quasi-elastic peak. This approximation has
been previously applied in GFMC calculations Lovato  _et al._ (2013, 2020).
While several parameterizations exist for the form factor, in our calculations
we use that from Ref. Kelly (2004).

The transverse response function is known to receive a large contribution from
two-nucleon currents which are required by the continuity equation Bacca and
Pastore (2014). This fact has been confirmed by Green function Monte Carlo
calculations for light nuclei up to 12C using phenomenological potentials and
currents Lovato  _et al._ (2020). It is also expected from the power counting
when performing the chiral expansion of electromagnetic currents, where the
two-body currents appear at the same order as the leading one-body currents in
the counting advocated by Refs. Kolling  _et al._ (2009); Krebs (2020) and are
suppressed by only one chiral order in the power counting adopted by Refs.
Pastore  _et al._ (2008); Acharya and Bacca (2022). In this work we only
include the one-body current contributions as a first step. The study of the
role of two-body currents is left for future work.

###  II.1 Lorentz integral transform

The sum over continuum spectrum of final states in Eq. (1) poses a serious
computational challenge. To overcome it, we follow the Lorentz integral
transform method Efros  _et al._ (2007). First, we calculate the convolution
of the nuclear response function with the Lorentzian kernel

$K(\omega,\sigma)=\frac{\sigma_{I}}{\pi}\frac{1}{(\omega-\sigma_{R})^{2}+\sigma_{I}^{2}}\,$
(3)

as

${\cal L}_{T}(\sigma,q)=\int d\omega R_{T}(\omega,q)K(\omega,\sigma)\,,$ (4)

where $\sigma=\sigma_{R}+i\sigma_{I}$ is a complex parameter. Using Eqs. (1)
and (4), ${\cal L}_{T}(\sigma,q)$ can be formally expressed as the solution of
a “Schrödinger-like” equation with a source term

$(H-E_{0}-\sigma)|\widetilde{\Psi}_{\sigma}(q)\rangle=\mathbf{J}^{T}(q)|{\Psi_{0}}\rangle\,.$
(5)

The latter can be solved with a good many-body bound-state technique, so that
the integral transform of the response function becomes

${\cal
L}_{T}(\sigma,q)=\langle\widetilde{\Psi}_{\sigma}(q)|\widetilde{\Psi}_{\sigma}(q)\rangle$
(6)

and can be numerically evaluated. The response $R_{T}(\omega,q)$ is then
retrieved from ${\cal L}_{T}(\sigma,q)$ through an inversion procedure. In
general, the inversion is an ill-posed problem which requires some additional
assumptions about the shape of the response function Efros  _et al._ (2007).
We invert the transform assuming that the final response function is smooth.
This assumption is fully justified in the region of the quasi-elastic peak.

###  II.2 Coupled-cluster method

In the coupled-cluster method many-body correlations are systematically
included to the wave function through an exponential ansatz $e^{T}$ acting on
a reference state $\Phi_{0}$. The correlation operator $T$ is a linear
expansion in $n$-particle $n$-hole excitation operators with respect to the
reference state,

$T=T_{1}+T_{2}+\ldots+T_{A}$ (7)

The Schrödinger equation, rewritten with the coupled-cluster ansatz, becomes
then

$H_{N}e^{T}|\Phi_{0}\rangle=E_{0}e^{T}|\Phi_{0}\rangle\ \ \ \rightarrow\ \ \\\
\overline{H}_{N}|\Phi_{0}\rangle=E_{0}|\Phi_{0}\rangle\,,$ (8)

where $\overline{H}_{N}\equiv e^{-T}H_{N}e^{T}$ is the similarity transformed
normal ordered Hamiltonian, and $\Phi_{0}$ is the Hartree-Fock solution to a
Hamiltonian containing nucleon-nucleon and three-nucleon forces. The
excitation operator $T$ is usually truncated at some low excitation rank. In
this work, we use $T\approx T_{1}+T_{2}$, which is known as coupled-cluster
with single and double excitations (CCSD) Shavitt and Bartlett (2009). This
approximation is known to account for $\sim 90\%$ of the correlation energy in
the ground state for systems with a well defined Fermi surface Bartlett and
Musiał (2007).

###  II.3 The LIT-CC method

The Lorentz integral transform method has been recently formulated within
coupled-cluster theory Bacca  _et al._ (2013b). In this language, Eq. (6)
becomes

$\displaystyle(\overline{H}_{N}-E_{0}-\sigma)|\widetilde{\Psi}_{\sigma}^{R}(q)\rangle=\overline{\Theta}_{J}|{\Phi_{0}}\rangle\,,$
$\displaystyle\langle\widetilde{\Psi}_{\sigma}^{L}(q)|(\overline{H}_{N}-E_{0}-\sigma)=\langle\Phi_{0}|(1+\Lambda)\,\overline{\Theta}_{J},$
(9)

where $\overline{\Theta}_{J}\equiv e^{-T}\Theta_{J}e^{T}$ is a similarity
transformed operator. Given that coupled-cluster theory is non-Hermitian, we
solve both the left and right equations separately for
$\widetilde{\Psi}_{\sigma}^{L}(q)$ and $\widetilde{\Psi}_{\sigma}^{R}(q)$ to
get

${\cal
L}_{T}(\sigma,q)=\langle\widetilde{\Psi}_{\sigma}^{L}(q)|\widetilde{\Psi}^{R}_{\sigma}(q)\rangle\,.$
(10)

The operator $\Lambda=\Lambda_{1}+\Lambda_{2}+...$ is a de-excitation operator
which has an analogous structure as $T$ operator and accounts for the fact
that the left and right eigenstates are not the same. The equations in (9) are
both equations-of-motion with a source term. They can be solved, after having
solved for the ground-state, using similar techniques to the standard ones to
get the spectrum of excited states within coupled-cluster theory Stanton and
Bartlett (1993); Bacca  _et al._ (2013b).

The operator $\Theta_{J}$ is a multipole, i.e., a spherical rank-$J$ tensor,
of the transverse electromagnetic current, namely
$\mathbf{J}^{T}(q)\equiv\sum_{J=0}^{J_{max}}\Theta_{J}$. Expressions for the
multipoles of the current operators can be found in Ref. Bacca  _et al._
(2007a). After calculating the LITs of Eq. (10) for each rank-$J$ multipole,
we sum them up, and finally perform an inversion of the total integral
transform.

##  III Results

We now present results focusing on the nuclei 4He, which we will use as a
benchmark, and 40Ca. All the results presented in this section are obtained
within a model space of 15 major oscillator shells which proved to be
sufficient for systems and observables we compute Sobczyk  _et al._ (2021). An
additional cut is imposed on the three-body potential $e_{3{\rm
max}}=2n_{1}+l_{1}+2n_{2}+l_{2}+2n_{3}+l_{3}\leq 16$. We check the convergence
of the model space by varying the underlying harmonic oscillator frequency. We
use two different chiral Hamiltonians at the next-to-next-to-leading order:
NNLOsat Ekström  _et al._ (2015) and $\Delta$NNLOGO(450) Jiang  _et al._
(2020). In the latter one, the $\Delta$ degree of freedom is considered
explicitly in the construction of nuclear potential. Both interactions have
the same regulator cut-off of $450$ MeV/c. They predict correct binding
energies and radii for light and medium-mass nuclei, in particular the ones we
consider.

###  III.1 Transverse sum rule

The transverse sum rule is defined in a standard way as

$\displaystyle\text{TSR}(q)=\frac{2m^{2}}{Z\mu_{p}^{2}+N\mu_{n}^{2}}\frac{1}{q^{2}}\Big{(}$
$\displaystyle\langle\Psi_{0}|\mathbf{J}^{T\dagger}\mathbf{J}^{T}|\Psi_{0}\rangle$
$\displaystyle-|\langle\Psi_{0}|\mathbf{J}^{T}|\Psi_{0}\rangle|^{2}\Big{)}\,.$
(11)

with the nucleon magnetic moments $\mu_{n,p}$. The elastic contribution
(second line) is zero for the closed-shell nuclei which we consider. The
normalization is chosen in such a way that $\text{TSR}(q\to\infty)\approx 1$.
We use this limit as a numerical check of the multipole decomposition of the
current. Let us first benchmark our transverse sum rule result for 4He. We
employed the NNLOsat potential with the underlying harmonic oscillator
frequency $\hbar\Omega=16$ MeV, in accordance with our previous benchmarks of
the longitudinal response and Coulomb sum rule Sobczyk  _et al._ (2020).

As shown in Fig. 1, we obtain a very good agreement above $q=200$ MeV/c when
comparing with other simulations which use either AV14+UVII Schiavilla (1989)
or AV18+UIX potentials Carlson  _et al._ (2002), for the variational Monte
Carlo (VMC) and the correlated hyperspherical harmonics (CHH) methods,
respectively. The low-$q$ behavior diverges (because of the $1/q^{2}$ factor
in the definition of the sum rule, see Eq. (III.1)) and differs quite
substantially from the calculation of Ref. Schiavilla (1989) that employed a
phenomenological potential. In the limit of $q\to 0$, the spin part of the
current becomes negligible, while the convection-current contribution becomes
proportional to the kinetic energy (see Eq. (II)). The latter strongly depends
on the employed Hamiltonian, which explains the difference between the
calculations. We performed an internal check of our method, comparing our
result with an average kinetic energy $\langle T\rangle=\int
d^{3}pn(p)p^{2}/2m$ where $n(p)$ is the momentum distribution. We obtained
agreement within a few percent.

Figure 1: 4He results for transverse sum rule for N2LOsat and $\hbar\Omega=16$
MeV compared with results of Refs. Schiavilla (1989); Carlson  _et al._
(2002).

In our previous works we investigated the role played by spurious center-of-
mass states excited by the charge operator and devised a procedure to remove
them. We do not repeat the procedure here, but rather refer the reader to Ref.
Sobczyk  _et al._ (2020) for details. Instead, we present here the result of
an analogous analysis of center of mass effects performed on the transverse
sum rule. We checked the various contributions of the transverse current
operator and found out that, as expected, only the isoscalar electric part
suffers from substantial spurious contributions. The difference between a
calculation with spurious states and one where those are projected out is
shown in Fig. 2. Moreover, this part of the current is an order of magnitude
smaller compared to the isovector one. We conclude that the spurious-states
contamination plays here a negligible role.

Figure 2: The effect of spurious states removal in 4He for the isoscalar
electric part of the current.

Finally, in Fig. 3 we show the TSR result for 40Ca obtained with the NNLOsat
potential using the harmonic oscillator frequency $\hbar\Omega=22$ MeV. Its
behaviour is qualitatively similar to 4He. We also present separately the
electric contribution (shadowed area) which dominates over the magnetic one at
lower $q$. The number of multipoles needed to converge and their relative
strength depends on the size of the nucleus and the momentum transfer. In Fig.
4, we show this dependence for four values of the momentum transfer,
$q=100-400$ MeV/c in case of 40Ca. This information is useful to estimate the
numerical cost for calculating the response functions. The number of
multipoles needed to get a convergent result grows from four for $q=100$ MeV/c
up to twelve for $q=400$ MeV/c (for 4He these are three and six,
respectively).

Figure 3: 40Ca results for the transverse sum rule. We employ the NNLOsat, the
model space of 15 major oscillator shells and $\hbar\Omega=22$ MeV. The
shadowed region shows the contribution of the electric multipoles only
(without the magnetic multipoles). Figure 4: The strength of various
multipoles of the decomposed response as a function of momentum transfer $q$
for 40Ca.

###  III.2 Transverse Response functions

We now turn our attention to the transverse response function, which we
calculate from the inversions of the LITs, using an ansatz for the response
function. We expand it in terms of the following basis functions

$\displaystyle
R_{T}(\omega,q)=\sum_{i=1}^{N}c_{i}f_{i}(\omega;n_{0},\beta_{i})\,,$ (12)
$\displaystyle
f_{i}(\omega;n_{0},\beta_{i})=(\omega-\omega_{th})^{n_{0}}e^{-\frac{(\omega-\omega_{th})}{\beta_{i}}}\,.$

This form of basis functions is motivated by the threshold behavior of the
response function for the deuteron which is known analytically to be
$\omega^{\ell+1/2}$ ($\ell$ is the orbital angular momentum quantum number).
The threshold energy $\omega_{th}$ is nucleus-dependent and in our case
corresponds to the proton separation energy. For various values of
$\sigma_{I}$, $n_{0}$ and $N$, we vary the continuous parameter $\beta_{i}$
over a wide range. For each combination, we search for the optimal $c_{i}$
coefficients minimizing the least square fit.

For all the results in this section we performed a robust analysis, varying
$n_{0}=\\{0.5,1.5\\}$, and the number of basis functions $N=\\{6,7,8\\}$. We
repeated the simulation for two values of $\sigma_{I}$, namely 5 and 10 MeV.
We estimate the inversion uncertainty as the spread between all of the found
optimal solutions.

Figure 5:  4He transverse response function for $q=300$ and $400$ MeV/c using
the NNLOsat interaction compared with GFMC predictions using AV18+IL7
potential Rocco  _et al._ (2018) and hyperspherical harmonics with the MTI-III
interaction Bacca  _et al._ (2007b). Experimental data are taken from Refs.
Dytman  _et al._ (1988) (circles) and Zghiche  _et al._ (1994) (crosses).

In Fig. 5, we present our benchmark results for 4He at the kinematics of
$q=300$ and $400$ MeV/c, comparing them with experimental data and other
theoretical calculations obtained with one-body currents only. The uncertainty
band of our CCSD results comes from the inversion procedure, not from a
variation of the harmonic oscillation frequency, which is kept fixed at
$\hbar\omega$=16 MeV, given that varying it yields negligible differences. The
band is a few percent broad and becomes of the order of $10\%$ only in the
lower part of the spectrum for $q=300$ MeV/c. For both considered momentum
transfers we observe around $20\%$ missing strength with respect to the
experimental data. This fact was already predicted using other nuclear
Hamiltonians and many-body methods and was attributed to the lack of two-body
currents Lovato  _et al._ (2015).

We observe, especially for $q=300$ MeV/c (also visible for $q=400$ MeV/c), an
irregular shape of the quasi-elastic peak. The visible two-peak structure is
qualitatively similar to the one obtained from hyperspherical harmonics
calculations with the MTI-III potential. For the GFMC result we observe a much
smoother quasi-elastic peak with a different low-energy behaviour. We
interpret these differences as a result of using $(i)$ different integral
transforms: the Lorentz (in our case) or Laplace (for the GFMC), $(ii)$
different nuclear forces, as well as $(iii)$ different inversion procedures.
We notice that a similar discrepancy has been already reported for the
longitudinal response function in Ref. Rocco  _et al._ (2018) where the GFMC
results were compared with the hypersherical harmonics using the same
interaction but different integral transforms. To get more insight about the
shape of the response function, in Fig. 6 we split the contributions to the
transverse response function at $q=300$ MeV/c into the electric/magnetic and
isoscalar/isovector parts. This reveals that the the low-energy shoulder
originates from the magnetic part of current and that the isovector
contribution is an order of magnitude larger than the isoscalar.

Figure 6: Response functions on 4He at $q=300$ MeV/c. The isovector (upper
panel) and isoscalar (lower panel) contributions are shown separately for the
electric and magnetic part of the transverse current.

We proceed to the calculation of the transverse response function for 40Ca. In
this case, we use two different chiral Hamiltonians, employed previously for
the longitudinal response function Sobczyk  _et al._ (2021), namely the
NNLOsat Ekström  _et al._ (2015) and $\Delta$NNLOGO(450) Jiang  _et al._
(2020) chiral potentials at next-to-next-to-leading order in a $\chi$EFT with
and without $\Delta$ degrees of freedom, respectively. For the inversion, we
use the theoretical values of the proton separation energy, namely
$\omega_{th}=6.9$ MeV for NNLOsat and $\omega_{th}=8$ MeV for
$\Delta$NNLOGO(450), imposing the response function to be zero below this
threshold. In our studies of the longitudinal response function we found that
the dependence on the $\hbar\Omega$ is negligible when varying it between 16
and 22 MeV. Here, we performed an analogous analysis using $\hbar\Omega=16$
and $22$ MeV and we arrived at the same conclusion for the transverse response
function.

Let us briefly comment on the available experimental data for 40Ca. Two
experiments performed the Rosenbluth separation in the past Williamson  _et
al._ (1997); Meziani  _et al._ (1984) and their results are in tension. In
particular, the authors of Ref. Meziani  _et al._ (1984) found a $25-40\%$
lower Coulomb sum rule. Later, a re-analysis of the world data was performed
by Jourdan Jourdan (1996), who presented the results for three values of
momentum transfer $q=300,\,380,$ and $570$ MeV/c. However, there is not a
single value of momentum transfer, for which the data of all of these three
analyses is available. Therefore, we decided to show our results at several
kinematics against various data-sets.

First we look at $q=300$ and $380$ MeV/c, for which the world data is
available. For $300$ MeV/c, the re-analysis by Jourdan overlaps to great
extent with the original data points from Ref. Williamson  _et al._ (1997),
apart from a few points above 120 MeV. A very good agreement between our
calculation and the data for both kinematics is observed, although somehow
unexpectedly, since for light nuclei the one-body current gives only $70-80\%$
of the strength with respect to the data.

Within our framework we are able to assess several sources of uncertainty,
both coming from the chiral expansion of the employed Hamiltonian, as well as
from the truncations in the many-body method. To estimate systematic
uncertainties from $\chi$EFT, we check the response at $q=300$ MeV/c, for
which we used potentials at two orders: $\Delta$NLOGO(450) and
$\Delta$NNLOGO(450) of the chiral Hamiltonian (see Fig. 8). The difference
between them is at the order of a few percent. This is analogous to our
findings for the longitudinal response function Acharya  _et al._ (2022). We
leave the further studies of the convergence of the many-body method for the
future, since it will require much more expensive calculations including
triple correlations. However, we can say that first pilot runs including
triples excitations do not show any sizable difference in the quasi-elastic
peak. It remains to be seen what the effect of two-body currents will be. Work
to include them is presently ongoing Ach .

Finally, to illustrate the tension between the experiments, in Fig. 9 we show
data for $q=400$ MeV/c Williamson  _et al._ (1997) and $q=410$ MeV/c Meziani
_et al._ (1984) and compare them with our calculations done at $q=400$ MeV/c,
both for the longitudinal and transverse response functions. Experimental
data-sets differ even by $30\%$ in the quasielastic peak, although a $10$ MeV
difference between momentum transfers is not expected to give such a big
effect (rather of the order of a few percent). Our results agree considerably
better with Ref. Williamson  _et al._ (1997). However, the larger prediction
of $R_{T}$ by Ref. Meziani  _et al._ (1984), would be more consistent with the
assumption of large two-body currents contribution.

Figure 7: 40Ca transverse response function for $q=300$ and $380$ MeV/c with
the NNLOsat and $\Delta$NNLOGO(450) potentials. Experimental data are taken
from Williamson  _et al._ (1997) (circles) and Jourdan (1996) (crosses).
Figure 8: Dependence of the transverse response function on the chiral order
of the $\Delta$-full interaction. Experimental data aretaken from Williamson
_et al._ (1997) (circles) and Jourdan (1996) (crosses).

Figure 9: Transverse and longitudinal response functions for 40Ca calculated
for $q=400$, MeV/c for NNLOsat and $\Delta$NNLOGO(450) potentials.
Experimental data for $q=400$ MeV/c taken from Williamson  _et al._ (1997)
(circles) and for $q=410$ MeV/c from Meziani  _et al._ (1984) (squares).

##  IV Conclusions

We presented the first _ab initio_ calculation of the transverse response in
the region of medium-mass nuclei computed within the LIT-CC method. This work
is another step forward in the development of our theoretical program towards
calculating electroweak responses for the future neutrino oscillation
experiments.

We benchmarked our method on 4He where we found a reasonable agreement with
previous calculations and experimental data. Our results underestimate the
data, which is expected since we included only one-body current. Nevertheless,
the transverse sum rule as a function of $q$ has a very similar behavior as
other calculations performed with other methods and phenomenological
potentials.

Next, we performed an analysis of the transverse response function on 40Ca.
Our results stay in a very good agreement with the world data by Jourdan. Our
finding motivates further investigations: both the inclusion of the two-body
currents, as well as the study of truncations in the many-body expansion are
to be performed in the future. We also want to point out that the tension
between the ‘two experimental Rosenbluth data on 40Ca have not yet been fully
understood and clarified. In view of the long-baseline neutrino programs and
of the role electron scattering plays in constraining theoretical predictions,
it is important to shed light on this matter. A new experimental program in
this direction is being initiated in Mainz Dor .

###### Acknowledgements.

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No. 101026014. This work was supported by the Deutsche
Forschungsgemeinschaft (DFG) through the Cluster of Excellence “Precision
Physics, Fundamental Interactions, and Structure of Matter” (PRISMA+ EXC
2118/1) funded by the DFG within the German Excellence Strategy (Project ID
39083149). It is also supported by the U.S. Department of Energy, Office of
Science, Office of Nuclear Physics under Award Number DE-SC0018223 (SciDAC-4
NUCLEI) and the SciDAC-5 NUCLEI collaboration, and by the Office of High
Energy Physics, U.S. Department of Energy, under Contract No. DE-
AC02-07CH11359 through the Neutrino Theory Network Fellowship awarded to BA.
Computer time was provided by the supercomputer MogonII at Johannes Gutenberg-
Universität Mainz, and by the Innovative and Novel Computational Impact on
Theory and Experiment (INCITE) programme. This research used resources of the
Oak Ridge Leadership Computing Facility located at Oak Ridge National
Laboratory, which is supported by the Office of Science of the Department of
Energy under contract No. DE-AC05-00OR22725

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