Also at ]RCPTM, Joint Laboratory of Optics of Palacký University and Institute
of Physics of Czech Academy of Sciences, 17. listopadu 12, 771 46 Olomouc,
Czech Republic

# The effect of thermal photons on exceptional points in coupled resonators

Grzegorz Chimczak<EMAIL_ADDRESS>Anna Kowalewska-Kudłaszyk  Ewelina
Lange  Karol Bartkiewicz  [  Institute of Spintronics and Quantum Information,
Faculty of Physics, Adam Mickiewicz University, 61-614 Poznań, Poland  Jan
Peřina Jr  RCPTM, Joint Laboratory of Optics of Palacký University and
Institute of Physics of Czech Academy of Sciences, 17. listopadu 12, 771 46
Olomouc, Czech Republic

###### Abstract

We analyse two quantum systems with hidden parity-time (${\cal{PT}}$)
symmetry: one is an optical device, whereas another is a superconducting
microwave-frequency device. To investigate their symmetry, we introduce an
equilibrium frame, in which loss and gain terms for a given Hamiltonian are
balanced. We show that the non-Hermitian Hamiltonians of both systems can be
tuned to reach an exceptional point (EP), i.e., the point in parameter space
at which a transition from broken to unbroken hidden ${\cal{PT}}$ symmetry
takes place. We calculate a degeneracy of a Liouvillian superoperator, which
is called the Liouvillian exceptional point (LEP), and show that, in the
optical domain, LEP is equivalent to EP obtained from the non-Hermitian
Hamiltonian (HEP). We also report breaking the equivalence between LEP and HEP
by a non-zero number of thermal photons for the microwave-frequency system.

PT symmetry, thermal environment, exceptional points

††preprint: APS/123-QED

##  I Introduction

In recent years, there has been increasing interest in exploring non-Hermitian
systems as a source of novel physical effects [for examples see Refs. 1, 2, 3,
4]. It has been shown that the special group of non-Hermitian Hamiltonians,
which is parity-time (${\cal{PT}}$) symmetric, exhibits entirely real spectra,
like Hermitian Hamiltonians, in the region of a parametric space where this
symmetry is in unbroken phase [5, 6, 7]. From both a theoretical and an
experimental points of view, much more interesting than non-Hermitian
Hamiltonians having entirely real spectra are degeneracies of these
Hamiltonians, which are placed in points of the parameter space, where a phase
transition occurs from an unbroken to a broken ${\cal{PT}}$ symmetry. Such
degeneracies, known as exceptional points (EPs), are the points of the
parametric space where the eigenvalues and the corresponding eigenvectors of a
Hamiltonian coincide [8]. Only non-Hermitian Hamiltonians can display EPs [9],
and therefore, only in non-Hermitian systems, all the interesting physics
associated with these degeneracies can be observed. These nontrivial phenomena
include enhancement of weak signal sensing [10], enhancement of spontaneous
emission [11], asymmetric light propagation [12, 13], single-mode laser [14],
electromagnetically induced transparency [15] just to name the few. The EPs
are usually studied in the semiclassical regime, where optical and photonic
systems are driven with strong classical external fields. Recently, these
studies have been extended to the fully quantum regime [16, 17, 18]. These
studies do not use the Schrödinger equation with a non-Hermitian Hamiltonian,
but are based on two different fully quantum descriptions of the open system
dynamics, namely the master equation with a Liouvillian superoperator and the
Heisenberg-Langevin equations. Since the matrix form of a Liouvillian
superoperator is a non-Hermitian matrix, it can display degeneracies just like
non-Hermitian Hamiltonians in the semiclassical regime. These degeneracies,
known as Liouvillian EPs (LEPs), and their influence on features of the
quantum system are attracting increasing attention [19, 20]. The master
equation approach allows investigating fully quantum exceptional points, i.e.,
LEPs. It also helps to design a quantum system associated with a given non-
Hermitian Hamiltonian. The quantum dynamics included in the master equation
can be decomposed to give quantum trajectories [21, 22]. In the quantum
trajectory method, the evolution of an open quantum system during the time
intervals without quantum jumps is governed by a non-Hermitian Hamiltonian.
Therefore, it is possible to realise non-Hermitian Hamiltonians in the fully
quantum system using postselection. In this way, EP of a non-Hermitian
Hamiltonian (HEP) has recently been observed in an experimental
superconducting system [23]. The important difference between LEP and HEP is
in accounting for quantum jumps. The former includes quantum jumps, whereas
the latter assumes their absence. Therefore, in general, LEPs are different
from HEPs. Shortly after the first observation of HEP in a fully quantum
system, several papers were published comparing LEPs with HEPs [24, 25, 26,
27, 28]. It was shown that in some systems LEP can be equivalent to HEP in the
sense that the position of both in the parameter space is the same. In one of
the mentioned papers, Arkhipov _et al._ [24] have investigated a quantum
system composed of two coupled cavities, where one cavity experiences
incoherent gain, while another only damping, and have found such equivalence
of LEP and HEP.

In the present work, we study a quantum system consisting of two laser-driven
coupled optical cavities, from which a field leaks out to the reservoir. The
non-Hermitian Hamiltonian describing this system is not
${\cal{PT}}$-symmetric, because it does not include an incoherent gain term.
Nevertheless, we find the position of the HEP by revealing the ${\cal{PT}}$
symmetry hidden in this Hamiltonian and the point where a phase transition
occurs. To this end, we introduce the idea of the equilibrium frame (EF) — a
frame, where the hidden ${\cal{PT}}$ symmetry is clearly seen. We demonstrate
that in a system with coherent gain HEP is equivalent to LEP. A similar
observation was reported for a system with incoherent gain in Ref. 24.
Finally, we consider a superconducting circuit realised in the microwave
domain, which is described by the same master equation as the optical system
in the case when the thermal photon number in a thermal environment is
negligible. We report breaking the equivalence between LEP and HEP by a non-
zero number of thermal photons.

##  II Results

###  II.1 Equilibrium frame

The main idea of the transformation to equilibrium frame (EF) is based on a
rotating frame transformation, frequently used in quantum optics. We assume
that the total Hamiltonian can be written as a sum of two terms $H^{\cal{PT}}$
and $H_{0}$. The Schrödinger equation is thus given by ($\hbar=1$)

$\displaystyle i\partial_{t}|\psi\rangle$ $\displaystyle=$
$\displaystyle(H^{\cal{PT}}+H_{0})|\psi\rangle\,.$ (1)

Now we make the substitution $|\psi\rangle=S\,|\widetilde{\psi}\rangle$, where
$S$ and $|\widetilde{\psi}\rangle$ are time-dependent. If we set
$S=\exp(-i\,H_{0}\,t)$ then the Schrödinger equation reduces to

$\displaystyle i\partial_{t}|\widetilde{\psi}\rangle$ $\displaystyle=$
$\displaystyle\widetilde{H}|\widetilde{\psi}\rangle\,,$ (2)

where $\widetilde{H}=S^{-1}H^{\cal{PT}}S$. In the case of the transformation
to a rotating frame, $S$ is unitary, because $H_{0}$ is Hermitian. However, in
the transformation to EF the operator $S$ is not a unitary one, because
$H_{0}$ is not Hermitian. In both cases $\widetilde{H}$ and $H^{\cal{PT}}$
have the same eigenvalues. In order to obtain a ${\cal{PT}}$-symmetric
Hamiltonian in EF, we restrict ourselves to the cases, where
$[H^{\cal{PT}},H_{0}]=0$. Using the Baker–Hausdorff lemma

$\displaystyle e^{Y}Xe^{-Y}$ $\displaystyle=$ $\displaystyle
X+[Y,X]+(1/2!)\big{[}Y,[Y,X]\big{]}+\dots$ (3)

one can easily prove that $\widetilde{H}=H^{\cal{PT}}$ for these cases.

For $[H^{\cal{PT}},H_{0}]=0$, both Hamiltonians have the same set of
eigenstates, and then we may relate the eigenvalues of the Hamiltonian given
in the initial frame (IF) to those in EF. Therefore, an $i$-th eigenvalue of
the total Hamiltonian in IF

$\displaystyle H|\phi_{i}\rangle$ $\displaystyle=$ $\displaystyle
H^{\cal{PT}}|\phi_{i}\rangle+H_{0}|\phi_{i}\rangle=\widetilde{H}|\phi_{i}\rangle+H_{0}|\phi_{i}\rangle$
$\displaystyle E_{i}|\phi_{i}\rangle$ $\displaystyle=$
$\displaystyle\widetilde{E}_{i}|\phi_{i}\rangle+E^{(0)}_{i}|\phi_{i}\rangle$
(4)

is equal to the sum of the $i$-th eigenvalue of $\widetilde{H}$ and the
corresponding eigenvalue of $H_{0}$. This fact is important when one is
looking for Hamiltonians displaying EPs, i.e., points in the parameter space,
where two (or more) eigenvalues have the same value. If $H^{\cal{PT}}$ is PT-
symmetric, then the Hamiltonian in EF, i.e., $\widetilde{H}$, can display EPs.
In such a case, at least two eigenvalues of $\widetilde{H}$ coincide (
$\widetilde{E}_{i}=\widetilde{E}_{j}$ for some $i$ and $j$). Therefore, one
may state that the Hamiltonian given in IF, i.e., $H=H^{\cal{PT}}+H_{0}$,
being not a PT-symmetric one, can also display EP if
$E^{(0)}_{i}=E^{(0)}_{j}$. The second condition for EP is also fulfilled
because $H^{\cal{PT}}$ and $H$ have the same set of eigenstates. Therefore, in
this point the eigenvectors of $H$ also coincide. Thus, we can say that EF
reveals the hidden symmetry of $H$.

The existence of degenerate eigenvalues of $H_{0}$ is a necessary condition
for $H$ displaying EP. Moreover, the eigenvalues suggest the convenient frame.
If the eigenvalues of $H_{0}$ are real, then we transform to a rotating frame.
If they are imaginary, we deal with a transformation to a frame in which the
length of the eigenstates scales with time. In the case when a system is in an
unbroken, ${\cal{PT}}$-symmetric phase, i.e., $H^{\cal{PT}}$ has a real
spectrum and the eigenvalues of $H_{0}$ are imaginary, we can associate these
two parts of $H$ with the observable energy of the system ($H^{\cal{PT}}$) and
the metric describing the geometric nature of the Hilbert space ($H_{0}$) [29,
30, 31].

It should be noted that condition $[H^{\cal{PT}},H_{0}]=0$ does not mean that
$H_{0}$ is a constant of motion, since $H^{\cal{PT}}$ is not Hermitian. The
conserved quantities for ${\cal{PT}}$-symmetric Hamiltonian are given by
intertwining operators [32, 33].

It is also worth mentioning that the equilibrium frame can also be useful to
reveal hidden pseudo-Hermiticity of non-Hermitian Hamiltonians. It is known
that the ${\cal{PT}}$ symmetry is a special case of pseudo-Hermiticity [34,
35, 36]. If the total Hamiltonian can be written as a sum of commuting parts,
i.e., a pseudo-Hermitian Hamiltonian and $H_{0}$, then one can expect that the
eigenvalues of the Hamiltonian given in IF are related to those in EF.

###  II.2 Hidden ${\cal{PT}}$ Symmetry of Passive Optical System with Coherent
Gain

Let us apply the idea of EF to investigate the symmetry and EP of a non-
Hermitian Hamiltonian of a physical system described by the master equation
($\hbar=1$)

$\displaystyle\dot{\rho}=-i[H,\rho]+\frac{1}{2}\sum_{i}\left(2C_{i}\rho
C^{+}_{i}-C^{+}_{i}C_{i}\rho-\rho C^{+}_{i}C_{i}\right)\,.$ (5)

The Hamiltonian of the system is given by

$\displaystyle H$ $\displaystyle=$ $\displaystyle
g(a^{\dagger}b+b^{\dagger}a)+i\varepsilon(a-a^{\dagger})+i\varepsilon(b-b^{\dagger})\,,$
(6)

and collapse operators are given by

$\displaystyle C_{1}=\sqrt{2\gamma_{a}}\,a\,,$ $\displaystyle
C_{2}=\sqrt{2\gamma_{b}}\,b\,.$ (7)

Here, $g$ is a coupling strength, $a$ and $b$ denote the annihilation
operators, $\gamma_{a}$, and $\gamma_{b}$ are the field damping rates of both
modes. For simplicity, we assume that $g$ is real and positive. The above
master equation describes a quantum system composed of two coupled optical
cavities, which are both driven by a classical field, and from both of them a
field leaks out to the reservoir, as shown in Fig. 1.

Figure 1: Schematic representation of the optical setup, in which the hidden
${\cal{PT}}$ symmetry is present.

Therefore, this system contains both incoherent loss and coherent gain. This
is a necessary condition to obtain a steady state solution different from the
vacuum state. Such a system can serve as a source of light and it should be
possible to implement it experimentally.

The master equation (5) can be rewritten to the form

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle-i(H_{\rm{{}_{nH}}}\rho-\rho
H_{\rm{{}_{nH}}}^{\dagger})+\sum_{i}C_{i}\rho C_{i}^{\dagger}\,,$ (8)

where

$\displaystyle H_{\rm{{}_{nH}}}=H-\frac{i}{2}\sum_{i}C_{i}^{\dagger}C_{i}\,.$
(9)

Therefore, the non-Hermitian Hamiltonian is given by

$\displaystyle H_{\rm{{}_{nH}}}$ $\displaystyle=$ $\displaystyle
g(a^{\dagger}b+b^{\dagger}a)+i\varepsilon(a-a^{\dagger})+i\varepsilon(b-b^{\dagger})$
(10) $\displaystyle-i\gamma_{a}a^{\dagger}a-i\gamma_{b}b^{\dagger}b\,.$

First, let us express Eq. (10) in terms of the new bosonic operators defined
by

$\displaystyle c=a+\varepsilon\alpha\hat{I}\,,$ $\displaystyle
c^{+}=a^{+}+\varepsilon\beta\hat{I}\,,$ $\displaystyle
d=b+\varepsilon\delta\hat{I}\,,$ $\displaystyle
d^{+}=b^{+}+\varepsilon\theta\hat{I}\,,$ (11)

where $\alpha=(\gamma_{b}-ig)/\xi$, $\beta=-(\gamma_{b}-ig)/\xi$,
$\delta=(\gamma_{a}-ig)/\xi$, $\theta=-(\gamma_{a}-ig)/\xi$ and
$\xi=g^{2}+\gamma_{a}\gamma_{b}$. Note that $c^{+}$($d^{+}$) is not Hermitian
conjugation of $c$($d$). Hence, we have used the symbol ”$+$” instead of
”$\dagger$”. Nevertheless, the operators $c$ and $d$ commute with each other
and satisfy $[c,c^{+}]=1$ and $[d,d^{+}]=1$, so actually they satisfy
commutation relations of independent oscillators. In terms of these new
bosonic operators, the Hamiltonian takes the form

$\displaystyle H_{\rm{{}_{nH}}}$ $\displaystyle=$ $\displaystyle
g(c^{+}d+d^{+}c)-i\gamma_{a}c^{+}c-i\gamma_{b}d^{+}d-\chi\hat{I}\,,$ (12)

where $\chi=i2\varepsilon^{2}\gamma/(g^{2}+\gamma^{2}-\kappa^{2})$ and
$\gamma=(\gamma_{a}+\gamma_{b})/2$. We have dropped the real part of $\chi$,
because it contributes only an overall irrelevant phase factor. After
introducing $\kappa=(\gamma_{a}-\gamma_{b})/2$, this Hamiltonian can be
rewritten as a sum of two parts

$\displaystyle H_{\rm{{}_{nH}}}$ $\displaystyle=$
$\displaystyle\underbrace{g(c^{+}d+d^{+}c)-i\kappa c^{+}c+i\kappa
d^{+}d}_{=H^{\cal{PT}}}$ (13)
$\displaystyle\underbrace{-i\gamma(c^{+}c+d^{+}d)-\chi\hat{I}}_{=H_{0}}\,.$

It can be verified that $H^{\cal{PT}}$ is ${\cal{PT}}$-symmetric using the
spatial reflection defined by

$\displaystyle{\cal{P}}$ $\displaystyle=$ $\displaystyle
P_{\rm{S}}\exp[i\pi(c^{\dagger}c+d^{\dagger}d)]\,,$ (14)

where $P_{\rm{S}}$ is the exchange operator [37], which spatially interchanges
the modes (i.e. $c\leftrightarrow d$). A matrix representation of $P_{\rm{S}}$
is given by a perfect shuffle [38]. We define the time-reversal operator
${\cal{T}}$ just as the complex conjugation operator
(${\cal{T}}i{\cal{T}}=-i$). Note that ${\cal{P}}$ given by Eq. (14) is a
reflection operator (i.e., ${\cal{P}}={\cal{P}}^{-1}$) and
$[{\cal{P}},{\cal{T}}]=0$. Using it and formulas: $\exp(\alpha
c^{\dagger}c)c\exp(-\alpha c^{\dagger}c)=\exp(-\alpha)c$ and $\exp(\alpha
c^{\dagger}c)c^{\dagger}\exp(-\alpha c^{\dagger}c)=\exp(\alpha)c^{\dagger}$,
one can easily check that $({\cal{PT}})c({\cal{PT}})=-d$,
$({\cal{PT}})c^{\dagger}({\cal{PT}})=-d^{\dagger}$,
$({\cal{PT}})d({\cal{PT}})=-c$,
$({\cal{PT}})d^{\dagger}({\cal{PT}})=-c^{\dagger}$ and
$({\cal{PT}})i({\cal{PT}})=-i$.

To find the eigenvalues, we use bosonic algebra combined with Fock space
representation of (12) [39]. To this end, we introduce the operators
$[e,f]^{\rm{T}}=\bm{R}\,[c,d]^{\rm{T}}$ and
$[e^{+},f^{+}]^{\rm{T}}=\bm{R}\,[c^{\dagger},d^{\dagger}]^{\rm{T}}$, where

$\bm{R}\equiv\begin{bmatrix}\cos\frac{\alpha}{2}&\sin\frac{\alpha}{2}\\\
-\sin\frac{\alpha}{2}&\cos\frac{\alpha}{2}\end{bmatrix}\,,$ (15)

$\sin{(\alpha/2)}=\sqrt{(\Omega+i\kappa)/(2\Omega)}$,
$\Omega=\sqrt{g^{2}-\kappa^{2}}$ and
$\cos{(\alpha/2)}=\sqrt{(\Omega-i\kappa)/(2\Omega)}$. The new operators
satisfy the following commutation relations $[e,e^{+}]=1$, $[f,f^{+}]=1$,
$[e,f^{+}]=0$, $[f,e^{+}]=0$, $[e,f]=0$ and $[f^{+},e^{+}]=0$, and therefore,
can be considered as annihilation and creation operators [39]. In terms of
these operators, the Hamiltonian takes the form

$\displaystyle H_{\rm{{}_{nH}}}$ $\displaystyle=$
$\displaystyle\underbrace{\Omega\,(e^{+}e-f^{+}f)}_{=H^{\cal{PT}}}\underbrace{-i\gamma(e^{+}e+f^{+}f)-\chi\hat{I}}_{=H_{0}}\,.$
(16)

Now, it is also easy to check that $[H^{\cal{PT}},H_{0}]=0$, so we can expect
that $H_{\rm{{}_{nH}}}$ has a hidden ${\cal{PT}}$ symmetry. The geometric part
of the Hamiltonian corresponds to its imaginary part [29], and therefore,
assuming an unbroken ${\cal{PT}}$-symmetric phase, it is given just by
$H_{0}$. This part is important because the rate at which each of the
eigenstates scales in the EF is determined by the eigenvalues of $H_{0}$. Note
that the geometric part is not just the operator of the total number of
photons in both modes.

The eigenvalues of $H^{\cal{PT}}$ (EF) and $H_{\rm{{}_{nH}}}$ (IF) are given
by

$\displaystyle\lambda^{PT}$ $\displaystyle=$
$\displaystyle\Omega\,(N_{e}-N_{f})\,,$ (17) $\displaystyle\lambda^{\rm{nH}}$
$\displaystyle=$
$\displaystyle\Omega\,(N_{e}-N_{f})-i\gamma\,(N_{e}+N_{f})-\chi\hat{I}\,,$
(18)

respectively. We have denoted excitation numbers in the supermodes $e$ and $f$
by $N_{e}$ and $N_{f}$. From these formulas, it is evident that all
eigenvalues have the same value for $\kappa=g$ in EF. In IF all eigenvalues
have the same real part but imaginary parts can be different.

Figure 2: Eigenvalues of the Hamiltonian presented in the equilibrium frame
(EF) given by Eq. (17) [panels (a) and (b)] and in the initial frame (IF)
given by Eqs. (18) [panels (c) and (d)] as functions of the gain/loss
coefficient $\kappa$, for $(\gamma,\varepsilon)/g=(2,1)$. Real parts are the
same in both frames, but imaginary parts are different in these frames.
Nevertheless, in both frames, the exceptional point exists and is placed at
the same point of the parameter space, i.e., $g=\kappa$.

To illustrate it, we have compared in Fig. 2 eigenvalues corresponding to the
following four eigenstates: $|\psi_{1}\rangle=|1\rangle_{e}|0\rangle_{f}$,
$|\psi_{2}\rangle=|0\rangle_{e}|1\rangle_{f}$,
$|\psi_{3}\rangle=|2\rangle_{e}|0\rangle_{f}$, and
$|\psi_{4}\rangle=|0\rangle_{e}|2\rangle_{f}$ in both frames. As expected for
${\cal{PT}}$-symmetric theory, it can be seen in panels (a) and (b) that in
the EF all eigenvalues are real when a system is in unbroken,
${\cal{PT}}$-symmetric phase and complex-conjugate pairs of eigenvalues appear
when ${\cal{PT}}$ symmetry is broken. It is also seen that a transition from
broken to unbroken PT symmetry occurs at the point $\kappa=g$. This is EP, in
which all four eigenvalues have the same real and imaginary parts. In this
case, all eigenvalues at the EP are equal to zero. We can conclude from panels
(a) and (b) that the investigated system is ${\cal{PT}}$-symmetric in EF and
eigenvalues of the Hamiltonian describing this system behave exactly as
predicted by the ${\cal{PT}}$-symmetric theory.

Fig. 2 in panels (c) and (d) show real and imaginary parts of the Hamiltonian
(12), which describes the system investigated in IF. In this frame all
eigenvalues have always non-zero imaginary parts. Nevertheless, it is clearly
seen that there is a correspondence between IF and EF. According to Eq. (18),
EP is also present in the point $\kappa=g$. In this case, however, the same
real and imaginary parts have only such eigenvalues, which correspond to
states with the same excitation number $N=N_{e}+N_{f}$. Therefore, there are
two pairs of coalescing eigenvalues in panels (c) and (d):
{$\lambda^{\rm{nH}}_{1}$, $\lambda^{\rm{nH}}_{2}$} and
{$\lambda^{\rm{nH}}_{3}$, $\lambda^{\rm{nH}}_{4}$}. The first pair corresponds
to $N=1$ and the second to $N=2$. It is also seen that for $\kappa<g$
eigenvalues corresponding to the same excitations number $N$ have different
real parts and equal imaginary parts. For $\kappa>g$, these eigenvalues have
different imaginary parts and equal real parts. Due to the similarities
between EF and IF, we can say that $H_{\rm{{}_{nH}}}$ has a hidden
${\cal{PT}}$ symmetry despite the fact that $H_{\rm{{}_{nH}}}$ is not
${\cal{PT}}$-symmetric. Similarly, we can say that $\kappa<g$ is the region of
unbroken ${\cal{PT}}$ hidden symmetry, and $\kappa>g$ is the region of broken
${\cal{PT}}$ hidden symmetry.

It is worth comparing the results presented in Fig. 2 with the results
presented in [40]. The real part of the eigenvalues in [40] diverge, whereas
these seen in Fig. 2 are finite and continuous. This is because the
Hamiltonian (10) describes a real physical system and the
${\cal{PT}}$-symmetric Hamiltonian considered in [40] is just a mathematical
model. If we chose $\kappa>\gamma$, i.e., if we assumed an incoherent gain,
then a divergence would also appear here.

Let us discuss the effect of the laser driving $\varepsilon$ on the
eigenvalues. One can see that the laser driving changes only $\chi$, which is
multiplied by an identity operator in the Hamiltonian. Since $\chi$ is purely
imaginary, it increases equally the decay rates of all eigenstates. This leads
to an increase in the probability of a collapse occurring during the observed
time interval. Hence, driving the system by an external laser field increases
the average rate at which photons are emitted from the system to the
environment.

We have shown that the non-Hermitian Hamiltonian $H_{\rm{{}_{nH}}}$ given by
Eq. (10) has a hidden ${\cal{PT}}$ symmetry and displays EP. Moreover, we know
that $H_{\rm{{}_{nH}}}$ is appropriate for describing a real quantum system
provided that the conditional evolution of the system is assumed. According to
quantum trajectory theory [21, 22] the non-Hermitian Hamiltonian
$H_{\rm{{}_{nH}}}$ describes the conditional time evolution of an open system
when the system’s interaction with the environment is monitored by perfect
detectors. During the time intervals when no photon decay is detected, the
evolution is governed by $H_{\rm{{}_{nH}}}$. This evolution is interrupted by
collapses corresponding to the action of the collapse operators (7). The
quantum trajectory theory makes it possible to describe a state evolution
conditioned on a sequence of detected collapses. The master equation approach
cannot describe the state evolution conditioned on a particular detection
record, because the master equation evolves all possible trajectories in time
as a single package. In other words, the master equation approach does not
assume any knowledge of the detection events.

The non-Hermitian Hamiltonian $H_{\rm{{}_{nH}}}$ corresponding to the master
equation (5) gives us the opportunity to compare EP calculated from the non-
Hermitian Hamiltonian (HEP) with EP calculated from a Liouvillian
superoperator (LEP). In general, LEPs can be different from HEPs, because we
calculate LEPs taking into account also the last term of the master equation
(10), i.e., $\sum_{i}C_{i}\rho C_{i}^{\dagger}$, which describes quantum
jumps. $H_{\rm{{}_{nH}}}$ governs the evolution of the system in the absence
of quantum jumps, and thus quantum jumps are not taken into account here. The
best way to calculate LEP in the case of this infinite-dimensional system is
to use the Heisenberg-Langevin equations averaged over the reservoir [24].
This approach is equivalent to calculating LEP from the Liouvillian
superoperator as shown in Ref. 41. Knowing the Hamiltonian (6) and the
collapse operators (7) we can obtain the Heisenberg-Langevin equation without
noise terms for an operator $A$ using the following formula [42]:

$\displaystyle\dot{A}=i[H,A]+\sum_{k}\frac{1}{2}\big{(}2C_{k}^{\dagger}AC_{k}-C_{k}^{\dagger}C_{k}A-AC_{k}^{\dagger}C_{k}\big{)}\,.$
(19)

In this way we obtain the closed set of differential equations for the fields’
operator moments

$\displaystyle\langle\dot{a}\rangle$ $\displaystyle=$
$\displaystyle-\gamma_{a}\langle a\rangle-ig\langle b\rangle-\varepsilon\,,$
$\displaystyle\langle\dot{b}\rangle$ $\displaystyle=$ $\displaystyle-ig\langle
a\rangle-\gamma_{b}\langle b\rangle-\varepsilon\,.$ (20)

The matrix form of this set of linear equations is given by

$\displaystyle\bm{\dot{v}}$ $\displaystyle=$
$\displaystyle-i\bm{M}\,\bm{v}-\bm{v_{0}}\,,$ (21)

where

$\bm{M}=\left(\begin{array}[]{cc}-i\gamma_{a}&g\\\ g&-i\gamma_{b}\\\
\end{array}\right)\,,$ (22)

$\bm{v}=[\langle a\rangle,\langle b\rangle]^{T}$ and
$\bm{v_{0}}=[\varepsilon,\varepsilon]^{\mathrm{T}}$. The diagonalisation of
$\bm{M}$ leads to formulas for the eigenvalues
$\lambda_{\pm}=\pm\Omega-i\gamma$ and the corresponding eigenvectors
$\bm{v_{\pm}}=[\pm\Omega-i\kappa,\,g]^{T}$, where
$\Omega=\sqrt{g^{2}-\kappa^{2}}.$ It is evident that the point $\kappa=g$ is
LEP, because at this point both eigenvalues coincide and the corresponding
eigenvectors coalesce. One can see that in the case of this optical system LEP
is equivalent to HEP.

###  II.3 The Effect of Thermal Photons in the Reservoir on HEP in Circuit QED

Let us now consider another physical system — two coupled superconducting
resonators driven by an external electromagnetic field. This physical system
should also be experimentally feasible [43]. The circuit diagram is shown in
Fig. 3.

Figure 3: Schematic diagram of a superconducting circuit realised in the
microwave domain, which is equivalent to the optical setup shown in Fig. 1.
The difference between these setups is that the frequency of the optical
cavity is four orders of magnitude higher than that of the superconducting
resonator LC. Therefore, thermal photons present in the environment can be
neglected in the optical case, but should be taken into account in the
microwave case.

These two LC resonators are inductively coupled to each other via mutual
inductance $M$ and capacitively coupled to sources of loss via $\rm{C}_{a}$
and $\rm{C}_{b}$. These two capacitors, i.e., $\rm{C}_{a}$ and $\rm{C}_{b}$
also allow for driving the LC resonators. It is evident that this electrical
circuit system can be also described by the Hamiltonian (6). However, the
master equation has to be modified because of thermal effects. In the case of
optical systems, which interact with a thermal environment, the number of
thermal photons is negligibly small. However, in the case of superconducting
microwave resonators, even at liquid helium temperatures, a few thermal
photons are present in the thermal bath [21]. It is easy to take into account
the number of thermal photons by substituting the following collapse operators
into the master equation (5)

$\displaystyle C_{1}=\sqrt{2\gamma_{a}(n+1)}a\,,$ $\displaystyle
C_{2}=\sqrt{2\gamma_{a}n}a^{\dagger}\,,$ $\displaystyle
C_{3}=\sqrt{2\gamma_{b}(n+1)}b\,,$ $\displaystyle
C_{4}=\sqrt{2\gamma_{b}n}b^{\dagger}\,.$ (23)

For the sake of simplicity, we assume that both resonators are coupled to
baths with the same number of thermal photons $n$, i.e., to baths with the
same temperatures. Using Eq. (9), we obtain the non-Hermitian Hamiltonian

$\displaystyle H_{\rm{{}_{nH}}}$ $\displaystyle=$ $\displaystyle
H-i\gamma_{a}(n+1)a^{\dagger}a-i\gamma_{a}naa^{\dagger}-i\gamma_{b}(n+1)b^{\dagger}b$
(24) $\displaystyle-i\gamma_{b}nbb^{\dagger}\,,$

which governs the evolution of the system between quantum jumps. Using the
bosonic commutation relations we can rewrite the non-Hermitian Hamiltonian to
the form

$\displaystyle H_{\rm{{}_{nH}}}$ $\displaystyle=$ $\displaystyle
g(a^{\dagger}b+b^{\dagger}a)+i\varepsilon(a-a^{\dagger})+i\varepsilon(b-b^{\dagger})$
(25)
$\displaystyle-i\gamma^{\prime}_{a}a^{\dagger}a-i\gamma^{\prime}_{b}b^{\dagger}b-\chi_{t}\hat{I}\,,$

where $\gamma^{\prime}_{a}=\gamma_{a}(2n+1)$,
$\gamma^{\prime}_{b}=\gamma_{b}(2n+1)$ and
$\chi_{t}=i\,n(\gamma_{a}+\gamma_{b})$. Note that the Hamiltonians given by
Eq. (25) and Eq. (10) have a very similar form. Hamiltonian (25) has a
different damping constants and an extra term — an identity operator
multiplied by a constant. Therefore, this microwave system also has hidden
${\cal{PT}}$ symmetry and the point in which a transition from broken to
unbroken hidden ${\cal{PT}}$ symmetry takes place. Due to this similarity, we
can diagonalise non-Hermitian Hamiltonian (25) in exactly the same way as
Hamiltonian (10), which yields

$\displaystyle H_{\rm{{}_{nH}}}$ $\displaystyle=$
$\displaystyle\Omega^{\prime}\,(e^{+}e-f^{+}f)-i\gamma^{\prime}(e^{+}e+f^{+}f)-\chi^{\prime}\hat{I}\,,$
(26)

where $\Omega^{\prime}=\sqrt{g^{2}-\kappa^{\prime\,2}}$,
$\gamma^{\prime}=(2n+1)\gamma$, $\kappa^{\prime}=(2n+1)\kappa$ and

$\displaystyle\chi^{\prime}$ $\displaystyle=$ $\displaystyle
i\,2\gamma\Big{[}n+\frac{\varepsilon^{2}(2n+1)}{g^{2}+(\gamma^{2}-\kappa^{2})(2n+1)^{2}}\Big{]}\,.$
(27)

The eigenvalues of this microwave system are then given by

$\displaystyle\lambda^{\rm{nH}}$ $\displaystyle=$
$\displaystyle\Omega^{\prime}\,(N_{e}-N_{f})-i\gamma^{\prime}\,(N_{e}+N_{f})-\chi^{\prime}\hat{I}\,.$
(28)

Figure 4: Real and imaginary parts of eigenvalues given by Eq. (28) as
functions of $\kappa/g$ for different number of thermal photons. Parameters:
$\gamma$ = 2, $g$ = 1, $\varepsilon$ = 1. Panels (a) and (d): $n$ = 0, panels
(b) and (e): $n$ = 0.1, panels (c) and (f): $n$ = 0.2.

In Fig. 4 we have plotted for different values of $n$ four eigenvalues
corresponding to the four eigenstates, which were considered in the case of
the optical system, i.e., $|\psi_{1}\rangle$, $|\psi_{2}\rangle$,
$|\psi_{3}\rangle$, and $|\psi_{4}\rangle$. It can be seen that the system has
a hidden ${\cal{PT}}$ symmetry and displays HEP between the unbroken and
broken phases. It can also be seen that the position of HEP is dependent on
the number of thermal photons $n$. From Eq. (28) we can infer that the
eigenvalues corresponding to the eigenstates with the same number of
excitations are equal for $g$ satisfying the condition:

$\displaystyle g_{{}_{\rm{HEP}}}$ $\displaystyle=$
$\displaystyle(2n+1)\kappa\,.$ (29)

Therefore, the effect of thermal photons in the reservoir on the HEP is to
move it in such a direction that the parametric region of the unbroken hidden
${\cal{PT}}$ symmetry is reduced.

Let us now calculate the eigenvalues of Liouvillian using Heisenberg-Langevin
equations averaged over the reservoir to obtain the formula for LEP. In order
to generate equations for the fields’ operator moments we substitute the
Hamiltonian (6) and collapse operators (II.3) into Eq. (19). We obtain the
same set of differential equations as in the case of the optical setup, i.e.,
the set given by Eqs. (II.2). So, we obtain also the same eigenfrequencies.
Thus, the system displays LEP for g satisfying the following condition:

$\displaystyle g_{{}_{\rm LEP}}$ $\displaystyle=$ $\displaystyle\kappa\,.$
(30)

It should be noted that $g_{{}_{\rm LEP}}$ is independent of $n$ in contrast
to $g_{{}_{\rm HEP}}$. Therefore, the effect of thermal photons is that LEP is
not equivalent to HEP in this circuit QED system. This effect of thermal
photons on EPs has not been observed yet, to the best of our knowledge.

This circuit QED system should be experimentally feasible, so it should also
be possible to observe the spectrum of the field radiated by this system.
Therefore, the question arises whether this circuit QED system displays EP in
the point $g_{{}_{\rm LEP}}$ or $g_{{}_{\rm HEP}}$. As mentioned earlier, a
non-Hermitian Hamiltonian in the quantum trajectory method describes a
conditional evolution of a system, which is monitored by perfect detectors,
whereas the master equation method describes an evolution of an open system,
for which a sequence of jumps events is not known. We note that HEPs are
observed in experiments that include conditional measurements [26].

Finally, let us explain the difference in the positions of EPs revealed by
considering the non-Hermitian Hamiltonian $H_{\rm nH}$ (24) and by analysing
the dynamical matrix of the equations (II.2) for mean values [41]. The
Hamiltonian $H_{\rm nH}$ in Eq. (24) describes two bosonic modes interacting
with two independent reservoirs at finite temperature, i.e., with non-zero
mean reservoir photon numbers $n$. Without the loss of generality, let us
concentrate our attention to one bosonic mode. The form of the non-Hermitian
Hamiltonian $H_{\rm nH}$ corresponds to the following master equation for the
mode statistical operator $\rho$ describing its interaction with the reservoir
[44]:

$\displaystyle\dot{\rho}$ $\displaystyle=$
$\displaystyle\gamma_{a}(n+1)\left([a\rho,a^{\dagger}]+[a,\rho
a^{\dagger}]\right)$ (31)
$\displaystyle+\gamma_{a}n\left([a^{\dagger}\rho,a]+[a^{\dagger},\rho
a]\right).$

To identify the drift and diffusion terms in the evolution of the mode as
described by the master equation (31), let us rewrite it for the quasi-
distribution function $\Phi_{\cal N}$ introduced in the Glauber-Sudarshan
representation of the statistical operator $\rho$ in the basis of coherent
states $|\alpha\rangle$ [44]:

$\rho=\int d^{2}\alpha\,\Phi_{\cal
N}(\alpha,\alpha^{*})|\alpha\rangle\langle\alpha|.$ (32)

Using the properties of coherent states, we arrive at the following Fokker-
Planck equation [45]:

$\displaystyle\frac{d\Phi_{\cal
N}}{dt}=\gamma_{a}\frac{\partial}{\partial\alpha}(\alpha\Phi_{\cal
N})+\gamma_{a}\frac{\partial}{\partial\alpha^{*}}(\alpha^{*}\Phi_{\cal N}).$
(33)

According to Eq. (33), the drift terms correspond to the following Heisenberg-
Langevin equations for the operators $a$ and $a^{\dagger}$,

$\dot{a}=-\gamma_{a}a+L,\hskip
14.22636pt\dot{a}^{\dagger}=-\gamma_{a}a^{\dagger}+L^{\dagger},$ (34)

and the stochastic operator forces $L$ and $L^{\dagger}$ serve to describe the
influence of the diffusion term. The form of Eq. (34) corresponds to the
following non-Hermitian Hamiltonian $H_{\rm nH}^{\rm drift}$,

$H_{\rm nH}^{\rm drift}=-i\gamma_{a}a^{\dagger}a,$ (35)

that completely describes the mode evolution caused by the drift terms.

The consideration of non-Hermitian Hamiltonian (24) in the form of Eq. (35),
i.e.

$H_{\rm nH}^{\rm drift}=H-i\gamma_{a}a^{\dagger}a-i\gamma_{b}b^{\dagger}b$
(36)

then leads to the EPs according to the condition (30) that identifies LEPs.

The approach that gives the non-Hermitian Hamiltonian of Eq. (24) incorporates
the drift terms only partly, which results in shifted temperature-dependent
positions of EPs. Subsequent inclusion of quantum jumps [26] then has to
correct for both the dynamics of the drift terms and the diffusion terms. Once
this correction is done, the positions of EPs (HEPs) shift to those identified
from the analysis of the whole Liouvillian of the system (LEPs).

##  III Discussion

We have introduced the concept of the equilibrium frame (EF) — a frame, which
scale is time dependent. This frame makes it possible to reveal ${\cal{PT}}$
symmetry hidden in passive non-Hermitian Hamiltonians, which do not have an
incoherent gain term. More specifically, we have shown that if a
non-${\cal{PT}}$-symmetric non-Hermitian Hamiltonian can be expressed as a sum
of two terms (a ${\cal{PT}}$-symmetric term and a second term commuting with
the first one), then such Hamiltonian has a hidden ${\cal{PT}}$ symmetry.
Using the EF method we have proved that the non-Hermitian Hamiltonian
describing a quantum system composed of two coupled optical cavities, which
are both driven by a classical field, and from both of them a field leaks out
to the reservoir, has a hidden ${\cal{PT}}$ symmetry. Although systems
composed of two coupled optical resonators have been investigated many times
in the context of ${\cal{PT}}$ symmetry [39, 24, 40], to our knowledge, the
non-Hermitian Hamiltonian (10) with hidden ${\cal{PT}}$ symmetry and coherent
gain has been never presented. The presence of the coherent gain is important
because it allows for using this system as a source of light. Thanks to the EF
method, we have identified the region of the parameter space where the hidden
${\cal{PT}}$ symmetry is unbroken and the region of the broken phase. In this
way, we have also found the position of the exceptional point (EP), where a
transition from the broken phase to the unbroken phase takes place.

The non-Hermitian Hamiltonian (10) describes a conditional evolution of a
feasible optical open system interacting with its environment. Therefore, it
is possible to write the master equation for the considered system. We have
written the master equation that describes the evolution of this optical
system and we have calculated EP from a Liouvillian superoperator. We have
found that in the case of this optical system, EP obtained from the
Liouvillian superoperator (LEP) is equivalent to EP determined from the non-
Hermitian Hamiltonian (HEP). Next, we have investigated a circuit QED system,
which is described by the same Hermitian Hamiltonian as the optical system.
However, in this microwave system, the effect of thermal photons present in
the environment cannot be neglected, so the master equation is different than
in the optical case. We have found that in the case of the circuit QED system
LEPs differ from HEPs as a consequence of non-zero number of thermal photons.
This means breaking the equivalence between HEPs and LEPs.

###### Acknowledgements.

This work was supported by the Polish National Science Centre (NCN) under the
Maestro Grant No. DEC-2019/34/A/ST2/00081. KB also acknowledges financial
support from the Czech Science Foundation under project No. 19-19002S and
project No. CZ.1.05/2.1.00/19.0377 of the Ministry of Education, Youth and
Sports of the Czech Republic. JP acknowledges the support from MŠMT ČR
projects No. CZ.02.2.69/0.0/0.0/18_053/0016919.

## References

  * Özdemir  _et al._ [2019] Ş. K. Özdemir, S. Rotter, F. Nori, and L. Yang, Parity–time symmetry and exceptional points in photonics, Nat. Mater. 18, 783 (2019).
  * El-Ganainy  _et al._ [2018] R. El-Ganainy, K. G. Makris, M. Khajavikhan, Z. H. Musslimani, S. Rotter, and D. N. Christodoulides, Non-hermitian physics and pt symmetry, Nature Physics 14, 11 (2018).
  * Miri and Alù [2019] M.-A. Miri and A. Alù, Exceptional points in optics and photonics, Science 363, eaar7709 (2019).
  * Ashida  _et al._ [2020] Y. Ashida, Z. Gong, and M. Ueda, Non-hermitian physics, Advances in Physics 69, 249 (2020).
  * Bender and Boettcher [1998] C. M. Bender and S. Boettcher, Real spectra in non-hermitian hamiltonians having $\mathcal{PT}$ symmetry, Phys. Rev. Lett. 80, 5243 (1998).
  * Bender [2007] C. M. Bender, Making sense of non-Hermitian Hamiltonians, Rep. Prog. Phys. 70, 947 (2007).
  * Bender  _et al._ [2019] C. M. Bender, P. E. Dorey, C. Dunning, A. Fring, D. W. Hook, H. F. Jones, S. Kuzhel, G. Lévai, and R. Tateo,  _PT Symmetry_ (World Scientific (Europe), 2019).
  * Kato [1966] T. Kato,  _Perturbation theory for linear operators_ (Springer Berlin, Heidelberg, 1966).
  * Mostafazadeh [2015] A. Mostafazadeh, Physics of spectral singularities, in  _Geometric Methods in Physics_, edited by P. Kielanowski, P. Bieliavsky, A. Odzijewicz, M. Schlichenmaier, and T. Voronov (Springer Berlin, 2015) pp. 145–165.
  * Chen  _et al._ [2017] W. Chen, Ş. Kaya Özdemir, G. Zhao, J. Wiersig, and L. Yang, Exceptional points enhance sensing in an optical microcavity, Nature 548, 192 (2017).
  * Lin  _et al._ [2016] Z. Lin, A. Pick, M. Lončar, and A. W. Rodriguez, Enhanced spontaneous emission at third-order dirac exceptional points in inverse-designed photonic crystals, Phys. Rev. Lett. 117, 107402 (2016).
  * Makris  _et al._ [2008] K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, Beam dynamics in $\mathcal{P}\mathcal{T}$ symmetric optical lattices, Phys. Rev. Lett. 100, 103904 (2008).
  * Jin  _et al._ [2017] L. Jin, P. Wang, and Z. Song, One-way light transport controlled by synthetic magnetic fluxes and $\mathcal{PT}$-symmetric resonators, New J. Phys. 19, 015010 (2017).
  * Feng  _et al._ [2014] L. Feng, Z. J. Wong, R.-M. Ma, Y. Wang, and X. Zhang, Single-mode laser by parity-time symmetry breaking, Science 346, 972 (2014).
  * Wang  _et al._ [2020] C. Wang, X. Jiang, G. Zhao, M. Zhang, C. W. Hsu, B. Peng, A. D. Stone, L. Jiang, and L. Yang, Electromagnetically induced transparency at a chiral exceptional point, Nat. Phys. 16, 334 (2020).
  * Hatano [2019] N. Hatano, Exceptional points of the lindblad operator of a two-level system, Molecular Physics 117, 2121 (2019).
  * Peřina Jr.  _et al._ [2019] J. Peřina Jr., A. Lukš, J. Kalaga, W. Leoński, and A. Miranowicz, Nonclassical light at exceptional points of a quantum $\mathcal{PT}$-symmetric two-mode system, Phys. Rev. A 100, 053820 (2019).
  * Minganti  _et al._ [2019] F. Minganti, A. Miranowicz, R. W. Chhajlany, and F. Nori, Quantum exceptional points of non-hermitian hamiltonians and liouvillians: The effects of quantum jumps, Phys. Rev. A 100, 062131 (2019).
  * Minganti  _et al._ [2018] F. Minganti, A. Biella, N. Bartolo, and C. Ciuti, Spectral theory of liouvillians for dissipative phase transitions, Phys. Rev. A 98, 042118 (2018).
  * Arkhipov  _et al._ [2020a] I. I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, Liouvillian exceptional points of any order in dissipative linear bosonic systems: Coherence functions and switching between $\mathcal{PT}$ and anti-$\mathcal{PT}$ symmetries, Phys. Rev. A 102, 033715 (2020a).
  * Carmichael [1993] H. Carmichael,  _An Open Systems Approach to Quantum Optics_ (Springer Berlin, Heidelberg, 1993).
  * Plenio and Knight [1998] M. B. Plenio and P. L. Knight, The quantum-jump approach to dissipative dynamics in quantum optics, Rev. Mod. Phys. 70, 101 (1998).
  * Naghiloo  _et al._ [2019] M. Naghiloo, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Quantum state tomography across the exceptional point in a single dissipative qubit, Nature Physics 15, 1232 (2019).
  * Arkhipov  _et al._ [2020b] I. I. Arkhipov, A. Miranowicz, F. Minganti, and F. Nori, Quantum and semiclassical exceptional points of a linear system of coupled cavities with losses and gain within the scully-lamb laser theory, Phys. Rev. A 101, 013812 (2020b).
  * Wiersig [2020] J. Wiersig, Robustness of exceptional-point-based sensors against parametric noise: The role of hamiltonian and liouvillian degeneracies, Phys. Rev. A 101, 053846 (2020).
  * Minganti  _et al._ [2020] F. Minganti, A. Miranowicz, R. W. Chhajlany, I. I. Arkhipov, and F. Nori, Hybrid-Liouvillian formalism connecting exceptional points of non-Hermitian Hamiltonians and Liouvillians via postselection of quantum trajectories, Phys. Rev. A 101, 062112 (2020).
  * Ou  _et al._ [2021] X. Ou, J. Huang, and C. Lee, Quantum exceptional points of non-hermitian hamiltonian and liouvillian in dissipative quantum rabi model, Chinese Physics B 30, 110309 (2021).
  * Chen  _et al._ [2021] W. Chen, M. Abbasi, Y. N. Joglekar, and K. W. Murch, Quantum jumps in the non-Hermitian dynamics of a superconducting qubit, Phys. Rev. Lett. 127, 140504 (2021).
  * Zhang  _et al._ [2019a] D.-J. Zhang, Q.-h. Wang, and J. Gong, Time-dependent $\mathcal{PT}$-symmetric quantum mechanics in generic non-hermitian systems, Phys. Rev. A 100, 062121 (2019a).
  * Mostafazadeh [2018] A. Mostafazadeh, Energy observable for a quantum system with a dynamical hilbert space and a global geometric extension of quantum theory, Phys. Rev. D 98, 046022 (2018).
  * Zhang  _et al._ [2019b] D.-J. Zhang, Q.-h. Wang, and J. Gong, Quantum geometric tensor in $\mathcal{PT}$-symmetric quantum mechanics, Phys. Rev. A 99, 042104 (2019b).
  * Bian  _et al._ [2020] Z. Bian, L. Xiao, K. Wang, X. Zhan, F. A. Onanga, F. Ruzicka, W. Yi, Y. N. Joglekar, and P. Xue, Conserved quantities in parity-time symmetric systems, Phys. Rev. Research 2, 022039 (2020).
  * Mostafazadeh [2010] A. Mostafazadeh, Pseudo-Hermitian representation of quantum mechanics, Int. J. Geom. Methods Mod. Phys. 07, 1191 (2010).
  * Mostafazadeh [2002a] A. Mostafazadeh, Pseudo-hermiticity versus pt symmetry: The necessary condition for the reality of the spectrum of a non-hermitian hamiltonian, Journal of Mathematical Physics 43, 205 (2002a).
  * Mostafazadeh [2002b] A. Mostafazadeh, Pseudo-hermiticity versus pt-symmetry. ii. a complete characterization of non-hermitian hamiltonians with a real spectrum, Journal of Mathematical Physics 43, 2814 (2002b).
  * Mostafazadeh [2002c] A. Mostafazadeh, Pseudo-hermiticity versus pt-symmetry iii: Equivalence of pseudo-hermiticity and the presence of antilinear symmetries, Journal of Mathematical Physics 43, 3944 (2002c).
  * Horodecki and Ekert [2002] P. Horodecki and A. Ekert, Method for direct detection of quantum entanglement, Phys. Rev. Lett. 89, 127902 (2002).
  * Loan [2000] C. F. Loan, The ubiquitous Kronecker product, J. Comput. Appl. Math. 123, 85 (2000).
  * Teimourpour  _et al._ [2018] M. H. Teimourpour, Q. Zhong, M. Khajavikhan, and R. El-Ganainy, Higher order exceptional points in discrete photonics platforms, in  _Parity-time Symmetry and Its Applications_, edited by D. Christodoulides and J. Yang (Springer Singapore, Singapore, 2018) pp. 261–275.
  * Lange  _et al._ [2020] E. Lange, G. Chimczak, A. Kowalewska-Kudłaszyk, and K. Bartkiewicz, Rotation-time symmetry in bosonic systems and the existence of exceptional points in the absence of $\mathcal{P}\mathcal{T}$ symmetry, Sci. Rep. 10, 19906 (2020).
  * Perina Jr  _et al._ [2022] J. Perina Jr, A. Miranowicz, G. Chimczak, and A. Kowalewska-Kudłaszyk, Quantum Liouvillian exceptional and diabolical points for bosonic fields with quadratic Hamiltonians: The Heisenberg-Langevin equation approach, Quantum 6, 883 (2022).
  * Plankensteiner  _et al._ [2022] D. Plankensteiner, C. Hotter, and H. Ritsch, QuantumCumulants.jl: A Julia framework for generalized mean-field equations in open quantum systems, Quantum 6, 617 (2022).
  * Purkayastha  _et al._ [2020] A. Purkayastha, M. Kulkarni, and Y. N. Joglekar, Emergent $\mathcal{PT}$ symmetry in a double-quantum-dot circuit qed setup, Phys. Rev. Research 2, 043075 (2020).
  * Peřina [1991] J. Peřina,  _Quantum Statistics of Linear and Nonlinear Optical Phenomena_ (Kluwer, Dordrecht, 1991).
  * Riksen [1996] H. Riksen,  _The Fokker-Planck equation_ (Springer, Berlin, 1996).


#  Improved Superscaling in Quasielastic Electron Scattering with Relativistic
Effective Mass

P.R. Casale  Departamento de Física Atómica, Molecular y Nuclear
and Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada, E-18071 Granada, Spain.  J.E. Amaro<EMAIL_ADDRESS>Departamento de Física Atómica, Molecular y Nuclear
and Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada, E-18071 Granada, Spain.  V.L. Martinez-Consentino
<EMAIL_ADDRESS>Departamento de Física Atómica, Molecular y Nuclear
and Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada, E-18071 Granada, Spain.  I. Ruiz Simo<EMAIL_ADDRESS>Departamento de Física Atómica, Molecular y Nuclear
and Instituto Carlos I de Física Teórica y Computacional
Universidad de Granada, E-18071 Granada, Spain.

###### Abstract

Superscaling in electron scattering from nuclei is re-examined paying special
attention to the definition of the averaged single-nucleon responses. The
validity of the extrapolation of nucleon responses in the Fermi gas has been
examined, which previously lacked a theoretical foundation. To address this
issue, we introduce new averaged responses with a momentum distribution
smeared around the Fermi surface, allowing for momenta above the Fermi
momentum. This approach solves the problem of negativity in the extrapolation
away from the scaling region and, at the same time, validates its use in the
scaling analysis. This work has important implications for the interpretation
of scaling data and contributes to the development of a more complete
understanding of the scaling approach.

quasielastic electron scattering; superscaling analysis; relativistic mean
field; relativistic effective mass

##  I Introduction

In the field of nuclear physics, understanding the behavior of atomic nuclei
under various conditions is of utmost importance. One such phenomenon is the
electromagnetic response of nuclei in electron scattering experiments Wes75 ;
Sar93 ; Ben08 ; Bof96 ; Wal04 . More recently, neutrino experiments with
accelerators have increased the interest and the need to describe the electro-
weak response of the atomic nucleus Alv18 ; Mos16 ; Kat17 ; Alv14 . Electron
and neutrino scattering processes are closely related, as the electromagnetic
current is linked to the weak isovector current Ama20 ; Ank22 ; Ama06 .
Therefore, it is of central importance to describe first the electromagnetic
response, as there is an abundance of experimental data on these reactions
archive ; archive2 .

In this article, we focus on the nuclear quasielastic response in electron
scattering, and more specifically, on the superscaling model Don99 ; Don99a ,
whose basic theoretical foundations we aim to examine. One-nucleon emission is
the most important contribution to the inclusive cross section in the
quasielastic region, centered around $\omega=|Q^{2}|/2m^{*}_{N}$, where
$\omega$ is the energy transfer, $Q^{2}=\omega^{2}-q^{2}<0$, and $q$ is the
momentum transfer to a nucleon with relativistic effective mass $m_{N}^{*}$
Ros80 ; Ser86 ; Dre89 ; Weh93 .

The widely used model of superscaling assumes factorization of the electron-
nucleus scattering cross section, which is proportional to the average
electron-nucleon scattering probability times a phenomenological scaling
function that incorporates the nuclear structure information Alb88 ; Cen97 .
Despite their limitations, such as neglecting the effects of final-state
interactions and meson-exchange currents, this approach has the potential to
provide an accurate description of the quasielastc electron and neutrino
scattering data in the quasielastic peak region with only a few parameters:
the Fermi momentum, $k_{F}$, and the relativistic effective mass, $m_{N}^{*}$
—or the nucleon separation energy depending on the particular approach to
superscaling Ama21 — as well as the phenomenological scaling function. Several
methods have been employed to extract the phenomenological scaling function
from experimental data. The SuSA (superscaling approach) model utilizes
longitudinal response data Ama04 , additionally the SuSA-v2 uses theoretical
input to construct a scaling function in the transverse channel Meg16b , while
the more recent SuSAM* (superscaling approach with relativistic effective
mass) model extract the scaling function directly from cross section data and
incorporates medium corrections through the effective mass of the nucleon
Mar17 ; Ama18 ; Rui18 . Attempts to extend the formalism to the inelastic
region have also been made Ama04 ; Mai09 ; Meg16b .

Despite the success of the phenomenological SuSA and SuSAM* models in the
quasielastic peak, one aspect of the theory that remains unverified is the
choice of the nuclear average of the single-nucleon response. In most works,
the single-nucleon response was averaged over the relativistic Fermi gas Alb88
, and then extrapolated by analytic continuation to the energy transfer region
that is prohibited in the Fermi gas due to the Pauli blocking effect Ama04 ;
Meg16b ; Ama18 . While this approach yield good results, extrapolating a
function outside the range of validity is dangerous and needs a physical
justification.

In this article, we investigate the behavior of the single-nucleon responses
when averaged over a Fermi gas and extrapolated outside of the kinematic range
allowed by the Pauli blocking effect. We show that as we move further away
from the scaling region, the extrapolation loses its physical meaning and
yields negative results for the response, which should be positive. On the
other hand, we demonstrate that using extrapolation in the scaling region is
appropriate because it produces results similar to those obtained by averaging
the response over a nuclear momentum distribution, which does not suffer from
this issue.

Our proposed framework involves a new definition of the single-nucleon
response averaged over momentum space, with a momentum distribution where the
Fermi surface is smeared out instead of using the sharp Fermi gas
distribution. This average therefore has a theoretical justification, in
contrast to the extrapolation approach Ama04 ; Meg16b ; Ama18 , and produces
results that are similar to those of the traditional superscaling models. With
this approach, we have a solid argument that justifies the choice of the
single-nucleon response and does not suffer from the previous issues. While we
will show that the use of the new averaged single-nucleon or the extrapolated
one is indifferent in the scaling region, this work improves the superscaling
formalism from the theoretical point of view by providing a physical
justification for its use, which strengthens the applicability of such
phenomenological models. Our findings have implications for the SuSA and
SuSAM* model, as well as for other phenomenological models in nuclear physics.

The scheme of the paper is as follows. In Sect. 2 we present a brief review of
superscaling formalism in the context of the SuSAM* approach. In sect. 3 we
analyze the averaged single-nucleon responses, discuss the problems of the
extrapolation, and propose a new definition. We give some details on the
calculation of the single-nucleon responses in Appendix A. In sect 4 we
present results of the single-nucleon cross sections and perform an updated
scaling analysis of the 12C data using the new definition. In sect. 5 we
discuss the results and finally in sect 6 we present our conclusions.

##  II Review of superscaling formalism

In this section we will briefly review the theory of the relativistic Fermi
gas (RFG) response function and its connection with the theory of
superscaling. The scaling variable $\psi$ was first introduced in ref. Alb88 .
The scaling formalism was refined in subsequent works Cen97 ; Don99 ; Don99a
until reaching the most up-to-date version of the SuSA-v2 model Ama20 .

The formalism in this work is an extension of the SuSA to the SuSAM* approach
—based on the equations of nuclear matter interacting with a relativistic mean
field (RMF) Ros80 ; Ser86 ; Dre89 ; Weh93 . The RMF model differs from the RFG
mainly in that the nucleons acquire a relativistic effective mass $m_{N}^{*}$.
The on-shell energy with effective mass is defined as

$E=\sqrt{p^{2}+(m_{N}^{*})^{2}}.$ (1)

In the RMF this is not the total energy of the nucleon, but rather, the
nucleons acquire an additional positive vector energy that partly cancels the
(negative) attraction energy of the scalar field. However in this work we deal
with the one-particle one-hole (1p1h) response functions where the vector
energy of particles and holes cancel. So the response only depends on the
effective mass and the Fermi momentum.

###  II.1 Electromagnetic response functions

We consider the inclusive electron scattering process where an incident
electron with energy $\epsilon$ scatters off a nucleus with scattering angle
$\theta$. The final electron energy is $\epsilon^{\prime}$. The momentum
transfer is $q$ and the energy transfer is $\omega$, and
$Q^{2}=\omega^{2}-q^{2}<0$. The cross section in plane-wave Born approximation
with one photon-exchange is written

$\frac{d\sigma}{d\Omega d\epsilon^{\prime}}=\sigma_{\rm
Mott}(v_{L}R_{L}(q,\omega)+v_{T}R_{T}(q,\omega)),$ (2)

were $\Omega$ is the final electron solid angle, $\sigma_{\rm Mott}$ is the
Mott cross section,

$\sigma_{\rm
Mott}=\left(\frac{\alpha\cos\theta/2}{2\epsilon\sin^{2}\theta/2}\right)^{2},$
(3)

$v_{L}$ and $v_{T}$ are the kinematic factors

$v_{L}=\frac{Q^{4}}{q^{4}},\kern
28.45274ptv_{T}=\tan^{2}\frac{\theta}{2}-\frac{Q^{2}}{2q^{2}},$ (4)

and finally, $R_{K}(q,\omega)$, $K=L,T$, are the longitudinal and transverse
response functions defined below.

We focus on the description of the nuclear response functions resulting from
the interaction of the electron with the one-body electromagnetic current,
giving rise to 1p1h excitation of the Fermi gas. They are defined in a similar
way to the usual RFG formalism Ama20 , with the difference that in our case
the nucleons have an effective mass $m^{*}_{N}<m_{N}$. The hole momentum is
${\bf h}$ with $h<k_{F}$ and on-shell energy $E=\sqrt{h^{2}+(m_{N}^{*})^{2}}$.
By momentum conservation, the particle momentum is ${\bf p}^{\prime}={\bf
h}+{\bf q}$ with on-shell energy
$E^{\prime}=\sqrt{p^{\prime}{}^{2}+(m_{N}^{*})^{2}}$. Pauli blocking implies
$p^{\prime}>k_{F}$. The nuclear response functions are then given by

$R_{K}^{QE}(q,\omega)=\frac{V}{(2\pi)^{3}}\int
d^{3}h\frac{(m^{*}_{N})^{2}}{EE^{\prime}}2w_{K}\,\theta(p^{\prime}-k_{F})\theta(k_{F}-h)\delta(E^{\prime}-E-\omega),$
(5)

where $w_{K}$ are the single-nucleon responses for the 1p1h excitation

$\displaystyle w_{L}=w^{00},\kern 28.45274ptw_{T}=w^{11}+w^{22},$ (6)

corresponding to the single-nucleon hadronic tensor

$w^{\mu\nu}=\frac{1}{2}\sum_{ss^{\prime}}(J^{\mu}_{s^{\prime}s})^{*}J^{\nu}_{s^{\prime}s}$
(7)

and $J^{\mu}$ is the electromagnetic current matrix element

$J^{\mu}_{s^{\prime}s}=\overline{u}_{s^{\prime}}({\bf
p}^{\prime})\left[F_{1}\gamma^{\mu}+i\frac{F_{2}}{2m_{N}}\sigma^{\mu\nu}Q_{\nu}\right]u_{s}({\bf
h}),$ (8)

where $F_{1}$ and $F_{2}$, are the Dirac and Pauli form factors of the
nucleon. Note that we use the current operator in the vacuum, but the spinors
correspond to nucleons with effective mass $m_{N}^{*}$.

To compute the integral (5), we use the variables $E,E^{\prime},\phi$, with
Jacobian $h^{2}dhd\cos\theta=(EE^{\prime}/q)dEdE^{\prime}$. Then the integral
over $E^{\prime}$ is made using the Dirac delta. This fixes the angle between
${\bf q}$ and ${\bf h}$ to the value

$\cos\theta_{h}=\frac{2E\omega+Q^{2}}{2hq},$ (9)

and the integration over the azimuth angle $\phi$ gives $2\pi$ by symmetry of
the responses when ${\bf q}$ is on the $z$-axis Ama20 . We are left with an
integral over the initial nucleon energy

$R_{K}^{QE}(q,\omega)=\frac{V}{(2\pi)^{3}}\frac{2\pi
m_{N}^{*3}}{q}\int_{\epsilon_{0}}^{\infty}d\epsilon\,n(\epsilon)\,2w_{K}(\epsilon,q,\omega),$
(10)

where $\epsilon=E/m^{*}_{N}$ is the initial nucleon energy in units of
$m_{N}^{*}$, and $\epsilon_{F}=E_{F}/m_{N}^{*}$ is the (relativistic) Fermi
energy in the same units. Moreover we have introduced the energy distribution
of the Fermi gas $n(\epsilon)=\theta(\epsilon_{F}-\epsilon)$. The lower limit,
$\epsilon_{0}$ of the integral in Eq. (10) corresponds to the minimum energy
for a initial nucleon that absorbs energy $\omega$ and momentum $q$. It can be
written as (see Appendix C of ref. Ama20 )

$\epsilon_{0}={\rm
Max}\left\\{\kappa\sqrt{1+\frac{1}{\tau}}-\lambda,\epsilon_{F}-2\lambda\right\\},$
(11)

where we have introduced the dimensionless variables

$\displaystyle\lambda=\omega/2m_{N}^{*}$ $\displaystyle\kappa=q/2m_{N}^{*}$
$\displaystyle\tau=\kappa^{2}-\lambda^{2}.$ (12)

Figure 1: Integration path in momentum space of the initial nucleon
corresponding to the integral (13) for different values of the energy transfer
$\omega$ (indicated in MeV in the key for each panel) and for three values of
the momentum transfer

###  II.2 Geometrical interpretation

For a fixed value of $\phi,q,\omega$, the integral over energy $\epsilon$ in
Eq. (10) corresponds to integrating the single nucleon response over a path in
the momentum space of the hole ${\bf h}$, weighted with the momentum
distribution. This curve is easily obtained from Eq. (9), giving the angle
$\theta_{h}$ as a function of the hole energy. Some examples are shown in Fig.
1 for three values of $q$. For each $q$ we plot the integration trajectories
in the $(h_{x},h_{z})$-plane for several values of $\omega$. The semicircles
indicate the moment distribution for $k_{F}=250$ MeV. The nuclear response
function, $R_{K}(q,\omega)$, therefore correspond to the sum (the integral) of
the single-nucleon responses along one path. The minimum momentum $h_{0}$, and
therefore the minimum energy $\epsilon_{0}$, correspond to the intersection of
each curve with the $h_{z}$ axis. The curves for different values of $\omega$
do not intersect. The case $h_{0}=0$ only occurs for a certain value of
$\omega$, which is precisely the position of the quasielastic peak; this
corresponds also to $\epsilon_{0}=1$ (or $\psi^{*}=0$ for the scaling
variable, see below). For very large or very small $\omega$-values, the curves
lie in the region where the momentum distribution is zero, and therefore the
corresponding response function is also zero.

Now we define a mean value of the single-nucleon responses by averaging with
the energy distribution $n(\epsilon)$

$\displaystyle\overline{w}_{K}(q,\omega)=\frac{\int^{\infty}_{\epsilon_{0}}d\epsilon\,n(\epsilon)w_{K}(\epsilon,q,\omega)}{\int_{\epsilon_{0}}^{\infty}d\epsilon\,n(\epsilon)}.$
(13)

This corresponds to the average of the single-nucleon response
$w_{K}(\epsilon,q,\omega)$ over one of the paths in Fig. 1. Using these
averaged single-nucleon responses we can rewrite Eq. (10) in the form

$\displaystyle R_{K}^{QE}(q,\omega)$ $\displaystyle=$
$\displaystyle\frac{V}{(2\pi)^{3}}\frac{2\pi
m_{N}^{*3}}{q}2\overline{w}_{K}(q,\omega)\int_{\epsilon_{0}}^{\infty}d\epsilon\,n(\epsilon).$
(14)

This last integral depends on the variable $\epsilon_{0}$, which in turn
depends on $(q,\omega)$.

###  II.3 Scaling

In the super-scaling approach the $\psi^{*}$-scaling variable is used instead
of the minimum energy of the nucleon, $\epsilon_{0}$. This energy is
transformed by a change of variable into the scaling variable, $\psi^{*}$,
defined as

$\psi^{*}=\sqrt{\frac{\epsilon_{0}-1}{\epsilon_{F}-1}}{\rm
sgn}(\lambda-\tau),$ (15)

where $\psi^{*}$ is negative (positive) for $\lambda<\tau$ ($\lambda>\tau$).

The superscaling function is defined as

$\frac{4}{3}(\epsilon_{F}-1)f^{*}(\psi^{*})=\int_{\epsilon_{0}}^{\infty}n(\epsilon)d\epsilon,$
(16)

where $\epsilon_{F}-1\ll 1$ is the kinetic Fermi energy in units of
$m_{N}^{*}$. The definition (16) is, except for a factor, similar to that of
the $y$-scaling function $f(y)$ Sar93 ; Wes75 , where the scaling variable $y$
was the minimum moment of the initial nucleon.

In RFG and nuclear matter with RMF Eq. (16) is easily evaluated (remember that
the RFG is recovered as the particular case $M^{*}=1$) as

$\int_{\epsilon_{0}}^{\infty}\theta(\epsilon_{F}-\epsilon)d\epsilon=\theta(\epsilon_{F}-\epsilon_{0})(\epsilon_{F}-\epsilon_{0})=(\epsilon_{F}-1)(1-\psi^{*}{}^{2})\theta(1-\psi^{*}{}^{2}).$
(17)

Therefore the scaling function of nuclear matter is

$f^{*}(\psi^{*})=\frac{3}{4}(1-\psi^{*}{}^{2})\theta(1-\psi^{*}{}^{2}).$ (18)

Note that the scaling function of nuclear matter is zero for
$\epsilon_{0}>\epsilon_{F}$, and this is equivalent to $|\psi^{*}|>1$. This is
a consequence of the maximum momentum $k_{F}$ for the nucleons in nuclear
matter, which implies that $\epsilon_{0}<\epsilon_{F}$.

Using $V/(2\pi)^{3}=N/(\frac{8}{3}\pi k_{F}^{3})$ for nuclear matter we can
write the response functions (14) as

$R^{QE}_{K}(q,\omega)=\frac{\epsilon_{F}-1}{m_{N}^{*}\eta_{F}^{3}\kappa}(Z\overline{w}^{p}_{K}(q,\omega)+N\overline{w}^{n}_{K}(q,\omega))f^{*}(\psi^{*}),$
(19)

where we have added the contribution of $Z$ protons and $N$ neutrons to the
response functions, and $\eta_{F}=k_{F}/m_{N}^{*}$.

Figure 2:  Super scaling analysis with relativistic effective mass (SuSAM*) of
12C data. Top panel: experimental scaling data $f^{*}_{exp}$ plotted against
$\psi^{*}$. Middle panel: data surviving after cleanup of non-quasielastic
sparse points. The red curve is Gaussian fit made in this work,
$f^{*}_{QE}(\psi^{*})$. In the bottom panel we compare the two scaling
functions obtained with two different definitions of the averaged single-
nucleon responses: using the extrapolated Fermi gas responses and performing
the average with a Fermi distribution defined in Sect 3.

###  II.4 SuSAM*

The SuSAM* approach extends the formula (19) by replacing $f^{*}(\psi^{*})$ by
a phenomenological scaling function obtained from experimental data of
$(e,e^{\prime})$. In a real, finite nucleus the momentum is not limited by
$k_{F}$ (in particular correlated nucleons can greatly exceed the Fermi
momentum). This has the effect that the phenomenological superscaling function
is not zero for $|\psi^{*}|>1$, and therefore takes into account that the
nucleons are not limited by a maximum Fermi momentum.

Several approaches have been used in the past to obtain a phenomenological
scaling function. In the original SuSA model, based on the RFG without
effective mass, the scaling function was obtained from the longitudinal
response data. In the SuSAv2 model, a scaling function for the transverse
response was also introduced by means of a RMF theoretical model in finite
nuclei. In this paper we will focus on the SuSAM* model with effective mass
where the phenomenological scaling function is obtained directly from the
quasielastic data of the inclusive cross section. Different scaling models
with effective mass and without effective mass provide different scaling
functions, but all may reproduce the quasielastic cross section reasonably
well, since they have been fitted to experimental data.

In the procedure followed in ref Mar17 ; Ama18 ; Rui18  the inclusive cross
section data are divided by the contribution of the single nucleon.

$f_{exp}^{*}=\frac{\displaystyle\left(\frac{d\sigma}{d\Omega
d\omega}\right)_{exp}}{\sigma_{M}(v_{L}r_{L}+v_{T}r_{T})},$ (20)

where

$r_{K}=\frac{\epsilon_{F}-1}{m_{N}^{*}\eta_{F}^{3}\kappa}(Z\overline{w}^{p}_{K}(q,\omega)+N\overline{w}^{n}_{K}(q,\omega)).$
(21)

In Fig. 2 these experimental data, $f^{*}_{exp}$, are plotted against
$\psi^{*}$ in the interval $-2<\psi^{*}<2$, which we call the quasielastic
scaling region in this work. It is observed that about half of them roughly
collapse forming a thin band around the quasielastic peak. This band
constitutes the set of selected data that can be considered QE and we reject
the rest, which mainly contribute to inelastic processes. The selected
quasielastic data are well parameterized with a sum of two Gaussians, thus
obtaining the phenomenological quasielastic function $f^{*}_{QE}$, shown also
in Fig. 2.

The SuSAM* model was extended in refs. Mar21 ; Mar21b , by subtracting the
theoretical contribution of the meson-exchange currents (MEC) in the 2p2h
channel from the experimental data before dividing by the single nucleon, that
is

$f_{exp}^{*}=\frac{\displaystyle\left(\frac{d\sigma}{d\Omega
d\omega}\right)_{exp}-\left(\frac{d\sigma}{d\Omega
d\omega}\right)_{MEC}}{\sigma_{M}(v_{L}r_{L}+v_{T}r_{T})}.$ (22)

The resulting SuSAM*+MEC model provided a somewhat smaller scaling function.
However in this work we use the scaling function (20) without subtraction of
MEC, since our focus will be on the average single nucleon responses. Both
models give similar results for the quasielastic cross section and we do not
want to complicate the calculation by introducing the 2p2h contribution, that
is not relevant for our further discussion.

##  III Averaged single-nucleon response functions

One of the most confusing aspects in the superscaling formalism is the
definition and meaning of the averaged single-nucleon response functions for
$|\psi^{*}|>1$ or, equivalently, $\epsilon_{0}>\epsilon_{F}$, i.e., outside
the allowed $\omega$-range of the Fermi gas. One of the goals of this paper is
to shed light on this matter. Traditionally an extrapolation of the Fermi gas
formula has often been used. In this section we expose the intrinsic
theoretical problems of the Fermi gas extrapolation, and propose an
alternative definition that is more satisfactory from the theoretical point of
view.

###  III.1 RFG extrapolation

In the traditional superscaling approach, first the averaged single-nucleon
responses $\overline{w}_{K}$ are calculated for $\epsilon_{0}<\epsilon_{F}$
(or $|\psi^{*}|<1$) using the Fermi gas momentum distribution,

$\displaystyle\overline{w}_{K}(q,\omega)$ $\displaystyle=$
$\displaystyle\frac{\int_{\epsilon_{0}}^{\infty}w_{K}(\epsilon,q,\omega)\theta(\epsilon_{F}-\epsilon)d\epsilon}{\int_{\epsilon_{0}}^{\infty}\theta(\epsilon_{F}-\epsilon)d\epsilon}$
(23) $\displaystyle=$
$\displaystyle\frac{\theta(\epsilon_{F}-\epsilon_{0})\int_{\epsilon_{0}}^{\epsilon_{F}}w_{K}(\epsilon,q,\omega)}{\theta(\epsilon_{F}-\epsilon_{0})\int_{\epsilon_{0}}^{\epsilon_{F}}d\epsilon}.$

Note that this expression is only defined for $\epsilon_{0}<\epsilon_{F}$, in
which case the step functions cancel and we obtain

$\overline{w}_{K}(q,\omega)=\frac{1}{\epsilon_{F}-\epsilon_{0}}\int_{\epsilon_{0}}^{\epsilon_{F}}w_{K}(\epsilon,q,\omega)d\epsilon,\kern
28.45274pt(\epsilon_{0}<\epsilon_{F}).$ (24)

The function $w_{K}(\epsilon,q,\omega)$ inside the integral is well defined
and positive only if $\epsilon>\epsilon_{0}$, because it corresponds to the
response of a single nucleon with energy $\epsilon$, that absorbs momentum $q$
and energy $\omega$. In the traditional SuSA and SuSAM* approaches the
function (24) is extended analytically for $\epsilon_{0}>\epsilon_{F}$ in the
obvious way. This is called in this work the extrapolated single nucleon
response function, and it can be written equivalently in the way

$\overline{w}_{K}(q,\omega)=\frac{1}{\epsilon_{0}-\epsilon_{F}}\int_{\epsilon_{F}}^{\epsilon_{0}}w_{K}(\epsilon,q,\omega)d\epsilon.$
(25)

From this expression it is clear that,for $\epsilon_{0}>\epsilon_{F}$, the
function $w_{K}(\epsilon,q,\omega)$ inside the integral must be evaluated for
$\epsilon<\epsilon_{0}$. But this is not possible for a nucleon on-shell that
absorbs $(q,\omega)$, because its minimum energy is $\epsilon_{0}$. Therefore
it is not guaranteed that the function $w_{K}(\epsilon,q,\omega)$ inside the
integral is positive if is evaluated for $\epsilon<\epsilon_{0}$. This is a
fundamental problem of the single nucleon extrapolation. Next we will study
some particular cases where the extrapolated responses are explicitly negative
for $\epsilon_{0}>\epsilon_{F}$, that is, for $|\psi^{*}|>1$.

###  III.2 Longitudinal single-nucleon response

We use the analytical formulas of the single nucleon responses from Appendix
A.

$w_{L}=\frac{(G_{M}^{*})^{2}}{1+\tau}[\tau(\epsilon+\lambda)^{2}-(1+\tau)\kappa^{2}]+\frac{(G_{E}^{*})^{2}}{1+\tau}(\epsilon+\lambda)^{2}.$
(26)

To better understand the kinematic dependence of this response function it is
convenient to express it in terms of the minimal nucleon energy $\epsilon_{0}$
using

$\epsilon_{0}+\lambda=\kappa\sqrt{\frac{1+\tau}{\tau}}\Longrightarrow\kappa^{2}(1+\tau)=\tau(\epsilon_{0}+\lambda)^{2}.$
(27)

in the regime without Pauli blocking. Then Eq. (42) becomes

$w_{L}=\frac{(G_{M}^{*})^{2}\tau}{1+\tau}[(\epsilon+\lambda)^{2}-(\epsilon_{0}+\lambda)^{2}]+\frac{(G_{E}^{*})^{2}}{1+\tau}(\epsilon+\lambda)^{2}.$
(28)

In this equation it is evident that the electric term is always positive.
However the magnetic term is positive only for $\epsilon>\epsilon_{0}$. For
this reason, if $w_{L}$ is calculated using the Fermi gas momentum
distribution and then extrapolated to values $\epsilon_{0}>\epsilon_{F}$ (or
$\psi^{*}>1$), the magnetic term becomes negative. This does not make physical
sense because the longitudinal response must be positive, by definition,
regardless of the value of the form factors. In fact if we artificially turn
off the electric contribution, a negative averaged response $\overline{w_{L}}$
is obtained for $\epsilon_{0}>\epsilon_{F}$. Let suppose for simplicity that
$G_{E}^{*}=0$. Then the extrapolated single-nucleon longitudinal response
would be

$\overline{w}_{L}=\frac{(G_{M}^{*})^{2}}{\epsilon_{0}-\epsilon_{F}}\frac{\tau}{1+\tau}\int_{\epsilon_{F}}^{\epsilon_{0}}[(\epsilon+\lambda)^{2}-(\epsilon_{0}+\lambda)^{2}]d\epsilon,$
(29)

that is negative for $\epsilon_{0}>\epsilon_{F}$.

###  III.3 Transverse single-nucleon response

We find a similar situation in the case of the transverse response from Eq.
(47) in the Appendix A

$w_{T}=2\tau(G_{M}^{*})^{2}+\frac{(G_{E}^{*})^{2}+\tau(G_{M}^{*})^{2}}{1+\tau}\frac{\tau}{\kappa^{2}}\left[(\epsilon+\lambda)^{2}-\kappa^{2}\frac{1+\tau}{\tau}\right].$
(30)

Again we can rewrite this response as a function of the minimum nucleon
energy, $\epsilon_{0}$, using
$\kappa^{2}(1+\tau)/\tau=(\epsilon_{0}+\lambda)^{2}$

$w_{T}=2\tau(G_{M}^{*})^{2}+\left[(G_{E}^{*})^{2}+\tau(G_{M}^{*})^{2}\right]\left[\left(\frac{\epsilon+\lambda}{\epsilon_{0}+\lambda}\right)^{2}-1\right].$
(31)

Rearranging terms containing $G_{E}^{*}$ and $G_{M}^{*}$ the single-nucleon
transverse response becomes finally

$w_{T}=(G_{E}^{*})^{2}\left[\left(\frac{\epsilon+\lambda}{\epsilon_{0}+\lambda}\right)^{2}-1\right]+\tau(G_{M}^{*})^{2}\left[\left(\frac{\epsilon+\lambda}{\epsilon_{0}+\lambda}\right)^{2}+1\right].$
(32)

Written in this way, it is evident that the magnetic contribution of $w_{T}$
is always positive. While the electrical term is positive only for
$\epsilon>\epsilon_{0}$. The situation is similar to what we found with the
longitudinal response, but in the transverse response it is the electrical
term that becomes negative in the extrapolation to
$\epsilon_{0}>\epsilon_{F}$. We now turn off the magnetic contribution and
suppose that $G_{M}^{*}=0$. Then the averaged T response in RFG would be, with
analogy to Eq. (29)

$\overline{w}_{T}=\frac{(G_{E}^{*})^{2}}{\epsilon_{0}-\epsilon_{F}}\int_{\epsilon_{F}}^{\epsilon_{0}}\left[\left(\frac{\epsilon+\lambda}{\epsilon_{0}+\lambda}\right)^{2}-1\right]d\epsilon.$
(33)

From this expression it is clear that the extrapolated $\overline{w}_{T}$ is
negative for $\epsilon_{0}>\epsilon_{F}$ because the function inside the
integral is negative, which is not physically acceptable: the transverse
response should be positive by definition regardless of the form factors
values. In other words, the electrical contribution to the transverse
response, although samll, cannot be negative.

###  III.4 Alternative to the extrapolated single-nucleon responses

In this work we propose an alternative definition of the averaged single-
nucleon responses that solves the extrapolation problem in the superscaling
model. As we have seen, the problem is a consequence of the fact that in the
Fermi gas there is a maximum momentum for the nucleons. If this momentum is
exceeded by extrapolation, i.e. $\epsilon_{0}>\epsilon_{F}$, mathematically
this is equivalent to assuming nucleons with energy less than $\epsilon_{0}$,
which is impossible in the Fermi gas because nucleons are on-shell. Hence
results without physical sense, such as negative responses, are obtained if
the extrapolated formula is applied.

The proposed solution involves using equation (13) for the averaged single-
nucleon responses, but introducing a momentum distribution without a maximum
momentum, and that at the same time does not differ much from the Fermi gas
distribution, for $h<k_{F}$. An appropriate function is a distribution of
Fermi type

$n(h)=\frac{a}{1+e^{(h-k_{F})/b}}.$ (34)

Where $b$ is a smearing parameter for the Fermi surface, which is no longer
restricted to a sphere as in figure 1. Then the integrals by averaging in Eq.
(13) extend to infinity and therefore there is no longer an upper limit for
$\epsilon_{0}$, which can take any value up to infinity. The single-nucleon
responses of the integrand always are evaluated for $\epsilon>\epsilon_{0}$
and they are therefore positive definite (see eqs. (28,32)).

Figure 3: Averaged and extrapolated longitudinal and transverse response
functions for proton plus neutron, as a function of $\omega$ and of the
scaling variable $\psi^{*}$, for three values of the momentum transfer.

Besides, for $\epsilon<\epsilon_{F}$, the momentum distribution is similar to
the Fermi Gas distribution, $\theta(k_{F}-h)$, and then it is expected that
the averaged single-nucleon be similar to that of the RFG (see Fig.1). Now,
for $\epsilon>\epsilon_{F}$ the integration (13) extends in momentum space
along one of the paths outside the Fermi sphere of Fig. 1. Then the average
has the physical sense of coming from regions above the Fermi sphere, that is
to say, from the high momentum zone that the Fermi gas cannot describe. This
is in accordance with the meaning attached to the experimental scaling
function for $|\psi^{*}|>1$, which comes mainly from high momentum nucleons.

##  IV Results

In this section we present results for the averaged nucleon responses and for
the total nuclear responses in the SuSAM* model. The calculations are made for
electron scattering off the nucleus 12C with Fermi momentum $k_{F}=225$ MeV/c
and effective mass $m_{N}^{*}=0.8m_{N}$. These values were fitted to the
quasielastic data of $f^{*}_{exp}$ to obtain the best possible scaling Mar17 ;
Ama18 . We evaluate the validity of the scaling model when using the Fermi gas
extrapolation for the nucleon response function. Specifically, the results
obtained by averaging the single-nucleon response function over a smeared
Fermi momentum distribution, Eq. (34) are compared with the extrapolated
response function obtained from the Fermi gas model.

In Fig. 3 we compare the averaged nucleon responses with the extrapolated
ones. The sum of proton plus neutron is shown. The averaged responses have
been calculated with a Fermi distribution using a smearing parameter $b=50$
MeV/c. The responses do not depend much on the precise value of this parameter
for small variations. We see that the averaged responses are practically the
same as the extrapolated responses of the Fermi gas in the quasielastic
scaling region, $-2<\psi^{*}<2$,. But both results start to diverge for large
$\omega$ or $\psi^{*}>2$. The extrapolated transverse response becomes
negative for $\psi^{*}>4$, 5 and 7, for $q=500$, 700, and 1000 MeV/c,
respectively, very close to the photon line. This is easily explained because
in Eq. (32) the magnetic term is multiplied by $\tau$. Therefore the $w_{T}$
response is dominated by the electric term for $\tau\rightarrow 0$, that is,
for large $\omega$, and in Eq. (33) we have seen that this term is negative
when extrapolated to $\epsilon_{0}>\epsilon_{F}$.

More details can be seen in Fig. 4 where we show the averaged and extrapolated
response functions separated for protons and neutrons, as a function of the
scaling variable. The extrapolated and averaged responses start to differ in
the region $\psi^{*}>2$ and the discrepancy increases with $\psi^{*}$. The
extrapolated longitudinal response of neutrons is negative for $\psi^{*}>2$.
This agrees with what was seen analytically in the previous section, because
the extrapolation of the longitudinal magnetic response is negative and the
electric form factor of the neutron is negligible. This does not affect the
results of the SuSAM* model in the scaling region because the longitudinal
response of the neutron is much smaller than that of the proton.

Figure 4: Averaged and extrapolated longitudinal and transverse response
functions for protons and neutrons, as a function of the scaling variable and
for three values of the momentum transfer.

In fig. 4 we also can see see that the averaged proton transverse response is
very similar to the extrapolation in the scaling region and differ for
$\psi^{*}>2$. They also start to differ in the $\psi^{*}$-negative region for
$\psi^{*}<-2$. The extrapolated transverse response of protons is negative
from $\psi^{*}\sim 4$–6 depending on the value of $q$. Again this is because
the electrical term of the proton dominates this response for large $\omega$
since the magnetic term carries a factor $\tau$, which tends to zero for
$\omega\rightarrow q$. In contrast the averaged proton transverse responses
are always positive.

The averaged transverse neutron response shown in Fig. 4 is similar in shape
to the Fermi gas extrapolation in the scaling region. But again they differ
for $|\psi^{*}|>2$, where the averaged one is the largest, and the difference
between the two increases with the momentum transfer.

Figure 5: Averaged and extrapolated transverse response functions for protons
and neutrons, for $G_{M}^{*}=0$, as a function of the scaling variable and for
three values of the momentum transfer. Averaged and extrapolated longitudinal
response functions for protons and neutrons, for $G_{E}^{*}=0$, as a function
of the scaling variable and for three values of the momentum transfer.

We have seen in the extrapolation formulas, Eqs. (29,33), that the magnetic
contribution to the longitudinal response and the electrical contribution to
the transverse response become both negative for $\epsilon_{0}>\epsilon_{F}$,
This can be explicitly seen in the results in Fig. 5, where we plot the
longitudinal responses computed for $G^{*}_{E}=0$ and the transverse responses
computed for $G^{*}_{M}=0$, for protons and neutrons. In fact, in all cases of
Fig. 5 the extrapolated responses are negative for $|\psi^{*}|>1$. On the
contrary the averaged responses are always positive.

Figure 6: Longitudinal and transverse response functions separated for protons
and neutrons in the SuSAM* model using the averaged and extrapolated single
nucleon responses for $q=500$ (two left panels) and 1000 MeV/c (two right
panels).

In fig 6 we use the superscaling model to investigate the nuclear responses
under various inputs for the single-nucleon. The nuclear response is computed
from the product of the averaged nucleon-responses and a phenomenological
scaling function obtained from the data, using Eq. (19).

The results in Fig. 6 demonstrate that there are no significant differences in
the separate responses of protons and neutrons when computed with the averaged
single-nucleon compared to the extrapolation. The only difference is seen in
the longitudinal neutron response for high $\omega$, which becomes negative in
the extrapolated model. However this is not relevant for the total nuclear
response, as the neutron contribution is negligible in the longitudinal
response as compared to the proton one.

This is verified in the results of Fig. 7 for the total responses. Both the
averaged and the extrapolated single-nucleon responses give essentially the
same result. The results obtained have two important implications. Firstly,
they provide support for the validity of using the single-nucleon response
extrapolated from the Fermi gas, as this approach yields the same results as
using a response averaged with a nuclear momentum distribution that does not
have a maximum momentum. Secondly, they justify the use of the averaged
response as a means of avoiding the potential issues that we have identified
with the extrapolation method.

Finally we have conducted a new scaling analysis of the ${}^{12}C$ data using
the single-nucleon response averaged with the Fermi distribution. The results,
as shown in Figure 2, demonstrate that the scaling function obtained using
this approach is virtually indistinguishable from the one obtained through
extrapolation. These findings highlight the robustness of the scaling approach
and suggest that using the averaged response may be a viable alternative to
extrapolation in certain cases. Furthermore, in Figures 8 and 9, we compare
the cross-section of 12C using the SuSAM* model and the RMF model of nuclear
matter for a selected set of kinematics. The SuSAM* model still proves to be
an excellent method to parameterize the quasielastic cross-section through a
single scaling function.

Figure 7: Longitudinal and transverse response functions in the SuSAM* model
using the averaged and extrapolated single nucleon responses

##  V Discussion

The findings of the results section demonstrate the robustness and versatility
of the superscaling models with respect to the choice of the averaged single-
responses, and its potential applications in a variety of situations in
electron and neutrino scattering. The updated single-nucleon responses provide
a well-defined theoretical basis for the scaling function that is compatible
with the traditional extrapolation in the scaling region. This reinforces the
universality of the scaling function because it is independent of the way in
which the average response of the nucleon is defined. This means that the
scaling function can be used to describe the electromagnetic response of
nucleons in different types of nuclei, regardless of their size or
composition.

Figure 8: Quasielastic $(e,e^{\prime})$ cross section of ${}^{12}C$ as a
function of $\omega$ for several values of the electron energy, $\epsilon$,
and scattering angles $\theta$, computed with the present SuSAM* model (black
lines) compared to the RFG with effective mass (blue lines). Experimental data
are from refs. archive ; archive2

The averaged single-nucleon model has promising applications in other
situations outside the scaling region for high-energy transfer. For instance
in two-particle emission reactions, two-particle two-hole (2p2h) excitation
can be produced by the one-body current due to nuclear short-range
correlations. The electromagnetic interaction with a nucleon belonging to a
correlated pair can result in the emission of both nucleons because the
correlated nucleons acquire high-momentum components that allow the overlap of
the wave function with states above the Fermi momentum. A simple model of
emission of two correlated nucleons has been proposed in ref. Mar22  to
explain phenomenologically the tail of the scaling function at high energies.
The probability of emission of a proton-neutron pair is approximated by a
factorized model, similar to the scaling approach. One factor is the sum of
the averaged proton and neutron responses considered in this work. The other
factor is the probability of emitting two particles while conserving energy
and momentum, assumed to be proportional to the phase space of two particles
in the Fermi gas. The total response is assumed to be the product of these two
factors with an additional correlation factor $c_{pn}(q)$ that accounts for
the average probability of the high momentum proton-neutron correlated pair.
The factor $c_{pn}(q)$ is obtained phenomenologically by fitting the tail of
the scaling function. In such 2p2h correlation model the contribution of the
single nucleon for high $\omega$ outside of the scaling region $\psi^{*}>2$
plays an important role, and the extrapolation of the Fermi gas single-nucleon
model is not appropriate.

Figure 9: Quasielastic $(e,e^{\prime})$ cross section of ${}^{12}C$ as a
function of $\omega$ for several values of the electron energy, $\epsilon$,
and scattering angles $\theta$, computed with the present SuSAM* model (black
lines) compared to the RFG with effective mass (blue lines). Experimental data
are from refs. archive ; archive2

Another direct application of this method concerns the calculation of the
contribution of meson-exchange currents (MEC) to the quasielastic 1p1h
response in the superscaling model. This calculation was performed in the RFG
for instance in Refs. Ama01 ; Ama03  and involves computing an effective one-
body current as the sum of one-body plus MEC, 1p1h matrix elements. The
traditional scaling model with extrapolation is not trivial to apply in this
case, as the single-nucleon responses of the MEC must be computed numerically.
However, the averaged single-nucleon responses of the OB+MEC operator can be
directly computed as we have done in this work.

##  VI Conclusions

In this work, we have re-examined the scaling formalism from a theoretical
standpoint, with a particular emphasis on the definition of the averaged
electron-nucleon responses, which are assumed to factorize in the model.
Within the SuSAM* model, which takes into account the relativistic mean field
through the effective mass of the nucleon, we have investigated the validity
of the traditional approach of extrapolating to $|\psi^{*}|>1$ the single-
nucleon responses averaged over the Fermi gas. A detailed analysis shows that
that, for $|\psi^{*}|>1$, the extrapolation formulas produce nonphysical
negative results for the responses in some particular cases, which contradict
the physical expectation that the response functions should always be
positive. Specifically, the magnetic contribution of the longitudinal response
and the electrical contribution of the transverse response become negative for
$|\psi^{*}|>1$. This is propagated to the total responses, resulting in the
extrapolated single-nucleon transverse response becoming negative for very
high values of $\omega$.

Therefore we have proposed a different definition for the averaged single-
nucleon responses with a smeared momentum distribution around the Fermi
surface. This approach does not suffer from the problems associated with the
extrapolation method, and on the other hand produces results that are similar
to those of the extrapolated SuSAM* model. Our proposed approach, which takes
into account the high-momentum nucleons to a certain extent, does not depend
significantly on the fine details of the nuclear density due to the averaging
procedure.

Despite the theoretical problems with extrapolation, in this work, we have
shown that the extrapolated model produces results similar to the correctly
averaged model within the scaling region $-2<\psi^{*}<2$. In conclusion, we
have provided a solid basis for the traditional superscaling model in the
quasielastic peak region. The new physically motivated definition of the
averaged single-nucleon responses strengthens the physical interpretation of
the superscaling model for understanding the response of atomic nuclei in
electron and neutrino scattering experiments. The new method has other
applications as well, for example, it allows for the inclusion of the MEC
effect in the superscaling model. A calculation of this type will be presented
in the future, demonstrating the versatility and potential of this new
approach.

##  VII Acknowledgments

Work supported by: Grant PID2020-114767GB-I00 funded by MCIN/ AEI/ 10.13039/
501100011033; FEDER/Junta de Andalucia-Consejeria de Transformacion Economica,
Industria, Conocimiento y Universidades/A-FQM-390-UGR20; and Junta de
Andalucia (Grant No. FQM-225).

##  Appendix A Single nucleon responses

The single-nucleon hadronic tensor is computed performing the spin traces (7)
with the current matrix elements (8), and can be written as

$w^{\mu\nu}=-w_{1}\left(g^{\mu\nu}-\frac{Q^{\mu}Q^{\nu}}{Q^{2}}\right)+w_{2}V^{\mu}V^{\nu},$
(35)

where we have defined the four-vector $V^{\mu}=(H^{\mu}+Q^{\mu}/2)/m_{N}^{*}$,
and $H^{\mu}=(E,{\bf h})$ is the initial nucleon four-momentum with effective
mass $m_{N}^{*}$. The four-momentum of the final nucleon is
$P^{\mu}=H^{\mu}+Q^{\mu}$. The nucleon structure functions are given by

$\displaystyle w_{1}(Q^{2})$ $\displaystyle=$
$\displaystyle\tau(G_{M}^{*})^{2}>0,$ (36) $\displaystyle w_{2}(Q^{2})$
$\displaystyle=$
$\displaystyle\frac{(G_{E}^{*})^{2}+\tau(G_{M}^{*})^{2}}{1+\tau}>0,$ (37)

where the electric and magnetic form factors for nucleons with effective mass
are Ama15

$G_{E}^{*}=F_{1}-\tau\frac{m^{*}_{N}}{m_{N}}F_{2},\kern
28.45274ptG_{M}^{*}=F_{1}+\frac{m_{N}^{*}}{m_{N}}F_{2}.$ (38)

For the $F_{i}$ form factors of the nucleon, we use the Galster
parametrizations Gal71 .

Note that $w_{1}$ and $w_{2}$ are positive and depend only on $Q^{2}$. Here we
compute the longitudinal and transverse components of the hadronic tensor,
$w_{L}=w^{00}$ and $w_{T}=w^{11}+w^{22}$ respectively, appearing in inclusive
electron scattering.

#### Longitudinal single-nucleon response

We use the following results for the time components of the basic tensors and
vectors in terms of adimensional variables, $\kappa,\lambda,\tau$

$g^{00}-\frac{Q^{0}Q^{0}}{Q^{2}}=-\frac{q^{2}}{Q^{2}}=\frac{\kappa^{2}}{\tau},$
(39) $V^{0}=\frac{E+\omega/2}{m_{N}^{*}}=\epsilon+\lambda.$ (40)

Substituting the values of these time components and of the structure
functions in the hadronic tensor (35), the longitudinal single-nucleon
response function becomes

$w_{L}=-\kappa^{2}(G_{M}^{*})^{2}+\frac{(G_{E}^{*})^{2}+\tau(G_{M}^{*})^{2}}{1+\tau}(\epsilon+\lambda)^{2}.$
(41)

Rearranging terms containing $G_{E}^{*}$ and $G_{M}^{*}$ this becomes

$w_{L}=\frac{(G_{M}^{*})^{2}}{1+\tau}[\tau(\epsilon+\lambda)^{2}-(1+\tau)\kappa^{2}]+\frac{(G_{E}^{*})^{2}}{1+\tau}(\epsilon+\lambda)^{2}.$
(42)

#### Transverse single-nucleon response

In the case of the transverse response $g^{ii}=-1$ and
$V^{i}=h_{i}/m_{N}^{*}=\eta_{i}$, for $i=1,2$, where we have defined the
three-vector $\mbox{\boldmath$\eta$}={\bf h}/m_{N}^{*}$. Then the T response
is

$w_{T}=w^{11}+w^{22}=2w_{1}+w_{2}(\eta_{1}^{2}+\eta_{2}^{2}).$ (43)

Note that
$\eta_{1}^{2}+\eta_{2}^{2}=\eta^{2}-\eta_{3}^{2}=\epsilon^{2}-1-\eta_{3}^{2}$.
The value of $\eta_{3}^{2}$ is the projection of the vector $\eta$ over the
${\bf q}$ direction, which is determined by energy-momentum conservation. In
fact, using Eq (9)

$\eta_{3}=\frac{h\cos\theta}{m_{N}^{*}}=\frac{E\omega+Q^{2}}{m_{N}^{*}q}=\frac{\epsilon\lambda-\tau}{\kappa}.$
(44)

Then we have

$\eta_{1}^{2}+\eta_{2}^{2}=\epsilon^{2}-1-\left(\frac{\epsilon\lambda-\tau}{\kappa}\right)^{2}.$
(45)

Expanding the square and using $\kappa^{2}-\lambda^{2}=\tau$, this gives gives

$\eta_{1}^{2}+\eta_{2}^{2}=\frac{\tau}{\kappa^{2}}\left[\epsilon^{2}-\frac{\kappa^{2}}{\tau}-\tau+2\epsilon\lambda\right]=\frac{\tau}{\kappa^{2}}\left[(\epsilon+\lambda)^{2}-\kappa^{2}\frac{1+\tau}{\tau}\right].$
(46)

Inserting this result in Eq. (43) and using the values of $w_{i}$ from Eqs.
(36,37), the transverse response becomes

$w_{T}=2\tau(G_{M}^{*})^{2}+\frac{(G_{E}^{*})^{2}+\tau(G_{M}^{*})^{2}}{1+\tau}\frac{\tau}{\kappa^{2}}\left[(\epsilon+\lambda)^{2}-\kappa^{2}\frac{1+\tau}{\tau}\right].$
(47)

## References

  * (1) G. B. West, Phys. Rept. 18, 263-323 (1975)
  * (2) A. M. Saruis, Phys. Rept. 235, 57-188 (1993)
  * (3) O. Benhar, D. Day, and I. Sick, Rev. Mod. Phys. 80, 189 (2008).
  * (4) ] S. Boffi, C. Giusti, F. d. Pacati, and M. Radici, Electromagnetic Response of Atomic Nuclei, Oxford Studies in Nuclear Physics, Vol. 20. 20 (Clarendon Press, Oxford UK, 1996).
  * (5) J. D. Walecka, Theoretical Nuclear And Subnuclear Physics (Editorial World Scientific Pub. Co. Inc., 2004).
  * (6) L. Alvarez-Ruso et al., Progress in Particle and Nuclear Physics 100, 1 (2018).
  * (7) U. Mosel, Ann. Rev. Nuc. Part. Sci. 66 (2016), 171.
  * (8) T. Katori and M. Martini, J. Phys. G 45 (2018) no.1, 013001.
  * (9) L. Alvarez-Ruso, Y. Hayato, J. Nieves, New J. Phys. 16 (2014) 075015.
  * (10) J. E. Amaro, M. B. Barbaro, J. A. Caballero, R. González-Jiménez, G. D. Megias and I. Ruiz Simo, J. Phys. G 47 (2020) no.12, 124001
  * (11) A. M. Ankowski, A. Ashkenazi, S. Bacca, J. L. Barrow, M. Betancourt, A. Bodek, M. E. Christy, L. D. S. Dytman, A. Friedland and O. Hen, et al. [arXiv:2203.06853 [hep-ex]].
  * (12) J. E. Amaro, M. B. Barbaro, J. A. Caballero and T. W. Donnelly, Phys. Rev. Lett. 98, 242501 (2007)
  * (13) O. Benhar, D. Day and I. Sick, arXiv:nucl-ex/0603032.
  * (14) O. Benhar, D. Day, and I. Sick, http://faculty.virginia.edu/qes-archive/
  * (15) T. W. Donnelly and I. Sick, Phys. Rev. Lett. 82, 3212-3215 (1999)
  * (16) T. W. Donnelly and I. Sick, Phys. Rev. C 60, 065502 (1999)
  * (17) R. Rosenfelder, Ann. Phys. (N.Y.) 128, 188 (1980).
  * (18) B.D. Serot, and J.D. Walecka, Adv. Nucl. Phys. 16 (1986) 1.
  * (19) D Drechselt and M M Giannini, Rep. Prog. Phys. 52 (1989) 1083.
  * (20) K. Wehrberger, Phys. Rep. 225 (1993) 273.
  * (21) W. M. Alberico, A. Molinari, T. W. Donnelly, E. L. Kronenberg and J. W. Van Orden, Phys. Rev. C 38, 1801-1810 (1988)
  * (22) R. Cenni, T. W. Donnelly and A. Molinari, Phys. Rev. C 56, 276-291 (1997)
  * (23) J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, R. Gonzalez-Jimenez, G. D. Megias and I. R. Simo, Eur. Phys. J. ST 230, no.24, 4321-4338 (2021)
  * (24) J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly, A. Molinari and I. Sick, Phys. Rev. C 71, 015501 (2005)
  * (25) G. D. Megias, J. E. Amaro, M. B. Barbaro, J. A. Caballero and T. W. Donnelly, Phys. Rev. D 94, 013012 (2016).
  * (26) V. L. Martinez-Consentino, I. Ruiz Simo, J. E. Amaro and E. Ruiz Arriola, Phys. Rev. C 96, no. 6, 064612 (2017).
  * (27) J. E. Amaro, V. L. Martinez-Consentino, E. Ruiz Arriola and I. Ruiz Simo, Phys. Rev. C 98 (2018) 024627
  * (28) I. Ruiz Simo, V. L. Martinez-Consentino, J. E. Amaro and E. Ruiz Arriola, Phys. Rev. D 97, 116006 (2018).
  * (29) C. Maieron, J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly and C. F. Williamson, Phys. Rev. C 80, 035504 (2009)
  * (30) V. L. Martinez-Consentino, I. R. Simo and J. E. Amaro, Phys. Rev. C 104, no.2, 025501 (2021)
  * (31) V. L. Martinez-Consentino, J. E. Amaro and I. Ruiz Simo, Phys. Rev. D 104, no.11, 113006 (2021)
  * (32) J. E. Amaro, E. Ruiz Arriola and I. Ruiz Simo, Phys. Rev. C 92, no.5, 054607 (2015) [erratum: Phys. Rev. C 100, no.1, 019904 (2019)]
  * (33) S. Galster, H. Klein, J. Moritz, K. H. Schmidt, D. Wegener and J. Bleckwenn, Nucl. Phys. B 32, 221 (1971).
  * (34) V. L. Martinez-Consentino, J. E. Amaro, P. R. Casale and I. Ruiz Simo, [arXiv:2210.09982 [nucl-th]].
  * (35) J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly and A. Molinari, Nucl. Phys. A 697, 388-428 (2002)
  * (36) J. E. Amaro, M. B. Barbaro, J. A. Caballero, T. W. Donnelly and A. Molinari, Nucl. Phys. A 723, 181-204 (2003)