# Coulomb corrections to two-particle interaction in artificial traps

Peng Guo<EMAIL_ADDRESS>Department of Physics and Engineering, California
State University, Bakersfield, CA 93311, USA  Kavli Institute for Theoretical
Physics, University of California, Santa Barbara, CA 93106, USA

###### Abstract

In present work, we discuss the effect of Coulomb interaction to the dynamics
of two-particle system bound in various traps. The strategy of including
Coulomb interaction into the quantization condition of trapped system is
discussed in a general and non-perturbative manner. In most cases, Coulomb
corrections to quantization condition largely rely on numerical approach or
perturbation expansion. Only for some special cases, such as the spherical
hard wall trap, a closed-form of quantization condition with all orders of
Coulomb corrections can be obtained.

##  I Introduction

Recent advances in lattice quantum Chromodynamics (LQCD), ab initio nuclear
many-body theory and developments in computer technology have now made it
possible for the high precision computation of hadron and nuclei systems from
the first principle. However, most of these computations are performed in
various traps, for instance, harmonic oscillator trap in nuclear physics and
periodic cubic box in LQCD. The typical observables from these ab initio
computations are discrete energy spectrum of trapped systems. Therefore,
extracting particle interactions from discrete energy spectrum in the trap and
building connection between trapped dynamics and infinite volume dynamics have
became an important subject in both LQCD and nuclear physics communities in
recent years. In elastic two-particle sector, such a connection between
trapped system and infinite volume system can be formulated in a closed form,
such as Lüscher formula Lüscher (1991) in a periodic cubic box in LCQD and
BERW formula Busch et al. (1998) in a harmonic oscillator trap in nuclear
physics community. Since then, Lüscher and BERW formula have been quickly
extended into both coupled-channel and few-body sectors, see e.g. Refs.
Rummukainen and Gottlieb (1995); Christ et al. (2005); Bernard et al. (2008);
He et al. (2005); Lage et al. (2009); Döring et al. (2011); Guo et al. (2013);
Guo (2013); Kreuzer and Hammer (2009); Polejaeva and Rusetsky (2012); Hansen
and Sharpe (2014); Mai and Döring (2017, 2019); Döring et al. (2018); Guo
(2017); Guo and Gasparian (2017, 2018); Guo and Morris (2019); Mai et al.
(2019); Guo et al. (2018); Guo (2020a); Guo and Döring (2020); Guo (2020b);
Guo and Long (2020a); Guo (2020c); Guo and Long (2020b); Guo (2020d); Guo and
Gasparian (2021); Guo and Long (2021). Both Lüscher and BERW formula have the
form of

$\det\left[\cot\delta(E)-\mathcal{M}(E)\right]=0\,,$ (1)

where $\delta(E)$ refers to the diagonal matrix of scattering phase shifts,
and the analytic matrix function $\mathcal{M}(E)$ is associated to the
geometry and dynamics of trap itself. Lüscher and BERW formula as the matter
of fact is the result of the presence of two well separated physical scales:
(1) short-range interaction between two particles and (2) size of trap. Hence
the short-range dynamics that is described by scattering phase shift and long-
range correlation effect due to the trap can be factorized.

The aim of present work is to extend such a relation to include long-range
Coulomb interaction between charged particles. Coulomb interaction becomes
dominant for charged particles interactions at low energy Kong and Ravndal
(2000), including Coulomb interaction may be crucial for charged system
interaction in a trap, see e.g. charge hadron system in LQCD Beane et al.
(2020). In fact, some early works on including Coulomb corrections in finite
volume has already been presented in Refs. Beane and Savage (2014); Stellin
and Meißner (2021). The discussion in Refs. Beane and Savage (2014); Stellin
and Meißner (2021) was primarily based on effective perturbation field theory
approach. It has been well known fact that both incoming plane wave and
scattered spherical wave are distorted by long-range Coulomb interaction
Messiah (1999),

$\displaystyle\psi^{(\infty)}_{l}(r,q)$ $\displaystyle\stackrel{{\scriptstyle
r\rightarrow\infty}}{{\sim}}\frac{\sin(qr-\frac{\pi}{2}l+\frac{Z\mu}{q}\ln
2qr)}{qr}$
$\displaystyle+t_{l}(q)\frac{e^{i(qr-\frac{\pi}{2}l+\frac{Z\mu}{q}\ln
2qr)}}{qr},$ (2)

where $Z=-Z_{1}Z_{2}e^{2}$ is Coulomb interaction strength, and $\mu$ and $q$
refers to the effective mass and incoming momentum of two-particle system.
$t_{l}(q)$ is the partial wave scattering amplitude. Hence perturbation breaks
down and Coulomb corrections must be dealt with non-perturbatively, see e.g.
Kong and Ravndal (2000). When it comes to formulating Lüscher and BERW formula
in the presence of long-range Coulomb force, Coulomb propagator must be used
instead of free particle propagator. In this work, we offer a general
perspective for the formulating Lüscher and BERW formula in presence of long-
range Coulomb force. All the discussion are based on Lippmann-Schwinger (LS)
equation approach, hence, the discussion can be made general in non-
perturbative way for various types of trap. However, except the hard-sphere
wall trap, the analytic form of Green’s function in a trap is usually not
available, Dyson equation must be solved either numerically or in terms of
perturbation expansion.

The paper is organized as follows. The derivation of the Coulomb corrections
to the quantization condition of trapped system is presented in Sec. II. The
discussions and summary are given in Sec. III.

##  II Connecting bound states in a trap to infinite volume scattering state
with Coulomb force

In this section, we present a general formalism on the topic of bridging
discrete bound state energy spectrum in a trap and infinite volume scattering
dynamics in the presence of Coulomb force. The commonly used traps are
periodic finite box in LQCD Beane et al. (2020), harmonic potential in nuclear
physics Rotureau et al. (2010, 2012); Luu et al. (2010); Zhang et al. (2020),
and spherical hard wall in some of the lattice implementations of chiral
effective field theory Elhatisari et al. (2016); Rokash et al. (2015). The
brief discussion of formal scattering in presence of both a short-range
interaction and a long-range Coulomb interaction is given in Appendix A.
Before the technical presentation of detailed derivation of quantization
conditions, our notations for describing the dynamics of two-particle
interaction in trap and in infinite volume are established as follows:

##### Dynamics in a trap:

the relative motion of two charged spinless particles interacting with both
Coulomb and short-range interactions in a trap is described by Schrödinger
equation

$\left[\varepsilon-\hat{H}_{t}-V_{C}(r)\right]\psi^{(t)}_{\varepsilon}(\mathbf{r})=\int_{trap}d\mathbf{r}^{\prime}V_{S}(\mathbf{r},\mathbf{r}^{\prime})\psi^{(t)}_{\varepsilon}(\mathbf{r}^{\prime}),$
(3)

where $\int_{trap}d\mathbf{r}^{\prime}$ refers to the integral over space of
unit cell of the trap, e.g.

$\int_{trap}d\mathbf{r}^{\prime}=\int^{\frac{L}{2}}_{-\frac{L}{2}}dx^{\prime}dy^{\prime}dz^{\prime}$

in a periodic box with the size of $L$. The Hamiltonian operator of the trap
is given by

$\hat{H}_{t}=\hat{H}_{0}+V_{trap}(\mathbf{r}),$ (4)

with

$\hat{H}_{0}=-\frac{\nabla_{\mathbf{r}}^{2}}{2\mu}$ (5)

and $V_{trap}(\mathbf{r})$ representing the free particle Hamiltonian operator
and trap potential respectively. $\mu$ stands for the reduced mass of two
particles, and $\varepsilon$ is energy of trapped particles associated with
relative motion.

$V_{C}(r)=-\frac{Z}{r}$ (6)

and $V_{S}(\mathbf{r},\mathbf{r}^{\prime})$ denote the Coulomb and short-range
interactions between particles respectively.

##### Dynamics in infinite volume:

the dynamics of two charged interacting particles through the same short-range
interaction $V_{S}(\mathbf{r},\mathbf{r}^{\prime})$ in infinite volume is
given by

$\left[\varepsilon_{\infty}-\hat{H}_{0}-V_{C}(r)\right]\psi^{(\infty)}_{\varepsilon_{\infty}}(\mathbf{r})=\int_{-\infty}^{\infty}d\mathbf{r}^{\prime}V_{S}(\mathbf{r},\mathbf{r}^{\prime})\psi^{(\infty)}_{\varepsilon_{\infty}}(\mathbf{r}^{\prime}),$
(7)

where $\varepsilon_{\infty}$ stands for the relative motion energy of
particles in infinite volume. $\varepsilon_{\infty}$ is related to
$\varepsilon$ in the trap by total energy conservation,

$\varepsilon_{\infty}+\frac{\mathbf{P}^{2}}{2M}=\varepsilon+E^{(t)}_{CM}=E,$
(8)

where $\frac{\mathbf{P}^{2}}{2M}$ and $E^{(t)}_{CM}$ are the center of mass
(CM) energy of system in infinite volume and in the trap respectively.

##### Short-range interaction:

the separable zero-range potential is assumed in follows for the derivation of
quantization condition. In coordinate space, it has the form of, see Refs.Guo
and Gasparian (2021); Guo and Long (2021),

$V_{S}(\mathbf{r},\mathbf{r}^{\prime})=\frac{\delta(r)\delta(r^{\prime})}{(rr^{\prime})^{2}}\sum_{lm}\frac{V^{(S)}_{l}}{(rr^{\prime})^{l}}Y_{lm}(\mathbf{\hat{r}})Y^{*}_{lm}(\mathbf{\hat{r}}^{\prime}).$
(9)

We emphasis that the assumption of separable zero-range potential is not
essential for obtaining the Lüscher or BERW formula type quantizations
condition, due to the fact that Lüscher or BERW formula type quantization
conditions are model independent asymptotic result when the size of trap is
much larger than the range of nuclear interactions, see e.g. discussion in
Refs.Guo and Gasparian (2021); Guo and Long (2021). However, separable zero-
range potential does serve as a convenient tool for the derivation of
quantization condition.

Next, the dynamical equations of charged particles interaction in a trap is
presented in II.1, and quantization condition of trapped particles system that
connects the strength of short-range interaction and the Coulomb Green’s
function in a trap is also derived and given in II.1. Then, under the same
assumption of separable short-range interaction, the scattering solutions of
charged particles in infinite volume are given in details in II.2, the similar
relation that connecting the strength of short-range interaction, infinite
volume Coulomb Green’s function and scattering phase shift is also obtained.
At last, by combining dynamical equations in a trap and infinite volume
together, the Lüscher or BERW formula type of quantization condition is
obtained and given in II.3.

###  II.1 Coulomb force modified dynamical equations in a trap

In the trap, the integral representation of Eq.(3) is given by

$\displaystyle\psi^{(t)}_{\varepsilon}(\mathbf{r})$
$\displaystyle=\int_{trap}d\mathbf{r}^{\prime\prime}G^{(C,t)}(\mathbf{r},\mathbf{r}^{\prime\prime};\varepsilon)$
$\displaystyle\times\int_{trap}d\mathbf{r}^{\prime}V_{S}(\mathbf{r}^{\prime\prime},\mathbf{r}^{\prime})\psi^{(t)}_{\varepsilon}(\mathbf{r}^{\prime}),$
(10)

where

$G^{(C,t)}(\mathbf{r},\mathbf{r}^{\prime\prime};\varepsilon)=\langle\mathbf{r}|\frac{1}{\varepsilon-\hat{H}_{t}-\hat{V}_{C}}|\mathbf{r}^{\prime\prime}\rangle$
(11)

stands for the Coulomb Green’s function in a trap. The Coulomb Green’s
function $G^{(C,t)}$ satisfies Dyson equation,

$\displaystyle
G^{(C,t)}(\mathbf{r},\mathbf{r}^{\prime\prime};\varepsilon)=G^{(t)}(\mathbf{r},\mathbf{r}^{\prime\prime};\varepsilon)$
$\displaystyle+\int_{trap}d\mathbf{r}^{\prime}G^{(t)}(\mathbf{r},\mathbf{r}^{\prime};\varepsilon)V_{C}(r^{\prime})G^{(C,t)}(\mathbf{r}^{\prime},\mathbf{r}^{\prime\prime};\varepsilon),$
(12)

where

$G^{(t)}(\mathbf{r},\mathbf{r}^{\prime\prime};\varepsilon)=\langle\mathbf{r}|\frac{1}{\varepsilon-\hat{H}_{t}}|\mathbf{r}^{\prime\prime}\rangle$
(13)

is particle Green’s function in a trap. The partial wave expansions

$\psi^{(t)}_{\varepsilon}(\mathbf{r})=\sum_{lm}\psi^{(t)}_{lm}(r)Y_{lm}(\mathbf{\hat{r}})$
(14)

and

$\displaystyle G^{(C,t)}(\mathbf{r},\mathbf{r}^{\prime\prime};\varepsilon)$
$\displaystyle=\sum_{lm,l^{\prime\prime}m^{\prime\prime}}Y_{lm}(\mathbf{\hat{r}})G_{lm,l^{\prime\prime}m^{\prime\prime}}^{(C,t)}(r,r^{\prime\prime};\varepsilon)Y^{*}_{l^{\prime\prime}m^{\prime\prime}}(\mathbf{\hat{r}}^{\prime\prime})$
(15)

yields

$\displaystyle\psi^{(t)}_{lm}(r)$
$\displaystyle=\sum_{l^{\prime}m^{\prime}}\int_{trap}{r^{\prime\prime}}^{2}dr^{\prime\prime}G_{lm,l^{\prime}m^{\prime}}^{(C,t)}(r,r^{\prime\prime};\varepsilon)$
$\displaystyle\times\int_{trap}{r^{\prime}}^{2}dr^{\prime}V^{(S)}_{l^{\prime}}(r^{\prime\prime},r^{\prime})\psi^{(t)}_{l^{\prime}m^{\prime}}(r^{\prime}).$
(16)

With separable potential given in Eq.(9), the quantization condition that
determines the discrete bound state energy spectrum of trapped system is thus
given by

$\det\left[\delta_{lm,l^{\prime}m^{\prime}}\frac{1}{V^{(S)}_{l}}-\frac{G_{lm,l^{\prime}m^{\prime}}^{(C,t)}(r,r^{\prime};\varepsilon)}{r^{l}{r^{\prime}}^{l^{\prime}}}|_{r,r^{\prime}\rightarrow
0}\right]=0.$ (17)

The Coulomb Green’s function, $G^{(C,t)}$, that describes the propagation of
charged particles in a trap is an essential ingredient of quantization
condition, and must be solved first. Specifically, only commonly used traps in
lattice and nuclear physics communities are considered in this work:

####  II.1.1 Harmonic oscillator trap

In the harmonic oscillator (HO) trap with trap potential:

$V_{trap}(r)=\frac{1}{2}\mu\omega^{2}r^{2},$ (18)

the rotational symmetry is still preserved, thus only diagonal elements of
partial wave Green’s function contribute. The partial wave Dyson equation for
harmonic trap Green’s function in presence of Coulomb force is given by

$\displaystyle
G_{l}^{(C,\omega)}(r,r^{\prime};\varepsilon)=G^{(\omega)}_{l}(r,r^{\prime};\varepsilon)$
$\displaystyle-\int_{0}^{\infty}{r^{\prime\prime}}^{2}dr^{\prime\prime}G_{l}^{(\omega)}(r,r^{\prime\prime};\varepsilon)\frac{Z}{r^{\prime\prime}}G_{l}^{(C,\omega)}(r^{\prime\prime},r^{\prime};\varepsilon),$
(19)

where $G^{(\omega)}_{l}$ is the partial-wave HO Green’s function and is given
in Refs. Blinder (1984); Guo and Long (2021) by

$\displaystyle
G^{(\omega)}_{l}(r,r^{\prime};\varepsilon)=-\frac{1}{\omega(rr^{\prime})^{\frac{3}{2}}}\frac{\Gamma(\frac{l}{2}+\frac{3}{4}-\frac{\varepsilon}{2\omega})}{\Gamma(l+\frac{3}{2})}$
$\displaystyle\times\mathcal{M}_{\frac{\varepsilon}{2\omega},\frac{l}{2}+\frac{1}{4}}(\mu\omega
r^{2}_{<})\mathcal{W}_{\frac{\varepsilon}{2\omega},\frac{l}{2}+\frac{1}{4}}(\mu\omega
r^{2}_{>}).$ (20)

$\mathcal{M}_{a,b}(z)$ and $\mathcal{W}_{a,b}(z)$ are the Whittaker functions
as defined in Ref. DLMF , and $r_{<}$ and $r_{>}$ represent the lesser and
greater of $(r,r^{\prime})$ respectively.

####  II.1.2 Periodic cubic box

In finite volume, the trap potential is replaced by periodic boundary
condition. The rotational symmetry is broken, and angular orbital momenta are
no longer good quantum numbers. In addition, the periodic boundary condition
is not satisfied by infinite volume Coulomb potential:
$V_{C}(r)=-\frac{Z}{r}$. The infinite volume Coulomb potential is usually
replaced by infrared singularity regularized periodic Coulomb potential, see
Refs. Beane and Savage (2014); Stellin and Meißner (2021),

$V_{C}^{(L)}(\mathbf{r})=-\frac{1}{L^{3}}\sum_{\mathbf{p}=\frac{2\pi\mathbf{n}}{L},\mathbf{n}\in\mathbb{Z}^{3},\mathbf{n}\neq\mathbf{0}}\frac{4\pi
Z}{|\mathbf{p}|^{2}}e^{i\mathbf{p}\cdot\mathbf{r}},$ (21)

where $L$ is size of cubic box, and

$V_{C}^{(L)}(\mathbf{r}+\mathbf{n}L)=V_{C}^{(L)}(\mathbf{r}),\ \ \ \
\mathbf{n}\in\mathbb{Z}^{3}.$ (22)

In momentum space, Dyson equation in finite volume is given by

$\displaystyle\widetilde{G}^{(C,L)}(\mathbf{p},\mathbf{p}^{\prime};\varepsilon)=\frac{L^{3}\delta_{\mathbf{p},\mathbf{p}^{\prime}}}{\varepsilon-\frac{\mathbf{p}^{2}}{2\mu}}$
$\displaystyle-\frac{1}{\varepsilon-\frac{\mathbf{p}^{2}}{2\mu}}\frac{1}{L^{3}}\sum_{\mathbf{p}^{\prime\prime}=\frac{2\pi\mathbf{n}}{L},\mathbf{n}\in\mathbb{Z}^{3}}^{\mathbf{p}^{\prime\prime}\neq\mathbf{p}}\frac{4\pi
Z}{|\mathbf{p}-\mathbf{p}^{\prime\prime}|^{2}}\widetilde{G}^{(C,L)}(\mathbf{p}^{\prime\prime},\mathbf{p}^{\prime};\varepsilon).$
(23)

The finite volume Coulomb force modified Green’s function in coordinate space
is thus given by finite volume Fourier transform

$G^{(C,L)}(\mathbf{r},\mathbf{r}^{\prime};\varepsilon)=\frac{1}{L^{6}}\sum_{\mathbf{p},\mathbf{p}^{\prime}\in\frac{2\pi\mathbf{n}}{L}}^{\mathbf{n}\in\mathbb{Z}^{3}}e^{i\mathbf{p}\cdot\mathbf{r}}\widetilde{G}^{(C,L)}(\mathbf{p},\mathbf{p}^{\prime};\varepsilon)e^{-i\mathbf{p}^{\prime}\cdot\mathbf{r}^{\prime}}.$
(24)

####  II.1.3 Spherical hard wall

The hard-sphere boundary condition is accomplished by the trap potential

$V_{trap}(r)=\begin{cases}0,&r<R\,,\\\ \infty,&r>R\,,\end{cases}$ (25)

where $R$ is the radius of the sphere. Hence, inside of spherical hard wall:
$|\mathbf{r}|<R$, Coulomb force modified Green’s function satisfies

$\left[\varepsilon-\hat{H}_{0}-\hat{V}_{C}\right]G^{(C,h.s.)}(\mathbf{r},\mathbf{r}^{\prime};\varepsilon)=\delta(\mathbf{r}-\mathbf{r}^{\prime}),$
(26)

which is just regular differential equation for Coulomb Green’s function
except boundary condition.

###  II.2 Coulomb force modified infinite volume dynamical equations

In infinite volume, the scattering solution of two charged interacting
particles in presence of Coulomb interaction is described by inhomogeneous LS
equation,

$\displaystyle\psi^{(\infty)}_{\varepsilon_{\infty}}(\mathbf{r},\mathbf{q})=\psi^{(C,\infty)}_{\varepsilon_{\infty}}(\mathbf{r},\mathbf{q})$
$\displaystyle+\int_{-\infty}^{\infty}d\mathbf{r}^{\prime\prime}G^{(C,\infty)}(\mathbf{r},\mathbf{r}^{\prime\prime};q)\int_{-\infty}^{\infty}d\mathbf{r}^{\prime}V_{S}(\mathbf{r}^{\prime\prime},\mathbf{r}^{\prime})\psi^{(\infty)}_{\varepsilon_{\infty}}(\mathbf{r}^{\prime},\mathbf{q}),$
(27)

where $\mathbf{q}$ is on-shell incoming momentum:

$q=\sqrt{2\mu\varepsilon_{\infty}}.$ (28)

$\psi^{(C,\infty)}_{\varepsilon_{\infty}}$ and $G^{(C,\infty)}$ are Coulomb
wave function and Coulomb Green’s function respectively. The partial wave
expansion

$\displaystyle\psi^{(\infty)}_{\varepsilon_{\infty}}(\mathbf{r},\mathbf{q})=\sum_{lm}Y^{*}_{lm}(\mathbf{\hat{q}})\psi^{(\infty)}_{l}(r,q)Y_{lm}(\mathbf{\hat{r}}),$
$\displaystyle
G^{(C,\infty)}(\mathbf{r},\mathbf{r}^{\prime\prime};q)=\sum_{lm}Y_{lm}(\mathbf{\hat{r}})G_{l}^{(C,\infty)}(r,r^{\prime\prime};q)Y^{*}_{lm}(\mathbf{\hat{r}}^{\prime\prime}),$
(29)

and separable potential in Eq.(9) yield an algebra equation

$\displaystyle\frac{\psi^{(\infty)}_{l}(r,q)}{r^{l}}=\frac{\psi^{(C,\infty)}_{l}(r,q)}{r^{l}}$
$\displaystyle+V^{(S)}_{l}\frac{G_{l}^{(C,\infty)}(r,r^{\prime\prime};q)}{(rr^{\prime\prime})^{l}}\frac{\psi^{(\infty)}_{l}(r^{\prime},q)}{{r^{\prime}}^{l}}|_{r^{\prime},r^{\prime\prime}\rightarrow
0}.$ (30)

####  II.2.1 Coulomb wave function and Coulomb Green’s function

The analytic expression of $\psi^{(C,\infty)}_{l}$ and $G_{l}^{(C,\infty)}$
are given in Refs. Messiah (1999); Hostler (1964) respectively by

$\displaystyle\psi^{(C,\infty)}_{l}(r,q)=4\pi\frac{\Gamma(l+1+i\gamma)}{(2l+1)!}e^{-\frac{\pi}{2}\gamma}$
$\displaystyle\times(2iqr)^{l}e^{iqr}M(l+1+i\gamma,2L+2,-2iqr),$ (31)

and

$\displaystyle
G_{l}^{(C,\infty)}(r,r^{\prime\prime};q)=2\mu(2iq)\frac{\Gamma(l+1+i\gamma)}{(2l+1)!}$
$\displaystyle\times(-2iqr_{<})^{l}e^{iqr_{<}}M(l+1+i\gamma,2l+2,-2iqr_{<})$
$\displaystyle\times(-2iqr_{>})^{l}e^{iqr_{>}}U(l+1+i\gamma,2l+2,-2iqr_{>}),$
(32)

where $M(a,b,z)$ and $U(a,b,z)$ are two linearly independent Kummer functions
DLMF , and

$\gamma=-\frac{Z\mu}{q}.$ (33)

For the convenience, let’s introduce two real functions:

$\displaystyle
j_{l}^{(C)}(\gamma,qr)=C_{l}(\gamma)(qr)^{l}e^{iqr}M(l+1+i\gamma,2l+2,-2iqr),$
(34)

and

$\displaystyle n_{l}^{(C)}(\gamma,qr)$
$\displaystyle=i(-2qr)^{l}e^{\frac{\pi}{2}\gamma}e^{iqr}U(l+1+i\gamma,2l+2,-2iqr)e^{i\delta_{l}^{(C)}}$
$\displaystyle-i(-2qr)^{l}e^{\frac{\pi}{2}\gamma}e^{-iqr}U(l+1-i\gamma,2l+2,2iqr)e^{-i\delta_{l}^{(C)}},$
(35)

where the Sommerfeld factor and Coulomb phase shift are defined in Messiah
(1999) by

$C_{l}(\gamma)=2^{l}\frac{|\Gamma(l+1+i\gamma)|}{(2l+1)!}e^{-\frac{\pi}{2}\gamma},$
(36)

and

$e^{2i\delta_{l}^{(C)}}=\frac{\Gamma(l+1+i\gamma)}{\Gamma(l+1-i\gamma)}.$ (37)

At the limit of $\gamma\rightarrow 0$, $j_{l}^{(C)}(\gamma,qr)$ and
$n_{l}^{(C)}(\gamma,qr)$ are reduced to the regular spherical Bessel
functions,

$\left(j_{l}^{(C)}(\gamma,qr),n_{l}^{(C)}(\gamma,qr)\right)\stackrel{{\scriptstyle\gamma\rightarrow
0}}{{\rightarrow}}\left(j_{l}(qr),n_{l}(qr)\right).$ (38)

Also using identity

$\displaystyle M(l+1+i\gamma,2l+2,-2iqr)$
$\displaystyle=-(-1)^{l}\frac{(2l+1)!}{\Gamma(l+1-i\gamma)}e^{\pi\gamma}U(l+1+i\gamma,2l+2,-2iqr)$
$\displaystyle-(-1)^{l}\frac{(2l+1)!}{\Gamma(l+1+i\gamma)}e^{\pi\gamma}e^{-2iqr}U(l+1-i\gamma,2l+2,2iqr),$
(39)

the partial wave Coulomb wave function and Coulomb Green’s function can thus
be rewritten as

$\psi^{(C,\infty)}_{l}(r,q)=4\pi
i^{l}j_{l}^{(C)}(\gamma,qr)e^{i\delta_{l}^{(C)}},$ (40)

and

$G_{l}^{(C,\infty)}(r,r^{\prime\prime};q)=-i2\mu
qj_{l}^{(C)}(\gamma,qr_{<})h_{l}^{(C,+)}(\gamma,qr_{>}),$ (41)

where

$\displaystyle h_{l}^{(C,\pm)}(\gamma,qr)=j_{l}^{(C)}(\gamma,qr)\pm
in_{l}^{(C)}(\gamma,qr)$
$\displaystyle=-2(-2qr)^{l}e^{\frac{\pi}{2}\gamma}e^{\pm iqr}U(l+1\pm
i\gamma,2l+2,\mp 2iqr)e^{\pm i\delta_{l}^{(C)}}.$ (42)

The Coulomb Green’s function in Eq.(41) thus resemble the free particle
Green’s function,

$G_{l}^{(0,\infty)}(r,r^{\prime\prime};q)=-i2\mu
qj_{l}(qr_{<})h_{l}^{(+)}(qr_{>}),$ (43)

where $j_{l}$ and $h_{l}^{(+)}$ are regular spherical Bessel and Hankel
functions.

####  II.2.2 Coulomb force modified scattering amplitudes

In presence of Coulomb force, the total scattering amplitude now is composed
of two components: (1) the short-range interaction scattering amplitude
modified by Coulomb interaction and (2) the pure Coulomb scattering amplitude.

##### Coulomb force modified short-range interaction scattering amplitude:

the short-range interaction scattering amplitude can be defined by the
solution of Eq.(30),

$\displaystyle\psi^{(\infty)}_{l}(r,q)$ $\displaystyle=4\pi
i^{l}\bigg{[}j^{(C)}_{l}(\gamma,qr)e^{i\delta_{l}^{(C)}}$
$\displaystyle+it^{(SC)}_{l}(q)h_{l}^{(C,+)}(\gamma,qr)e^{-i\delta_{l}^{(C)}}\bigg{]},$
(44)

where $t^{(SC)}_{l}(q)$ is the Coulomb force modified short-range interaction
scattering amplitude and is given by

$t^{(SC)}_{l}(q)=-\frac{2\mu
q\left(\frac{j_{l}^{(C)}(\gamma,qr)}{r^{L}}|_{r\rightarrow
0}\right)^{2}}{\frac{1}{V^{(S)}_{l}}-\frac{G_{l}^{(C,\infty)}(r^{\prime},r^{\prime\prime};q)}{(r^{\prime}r^{\prime\prime})^{l}}|_{r^{\prime},r^{\prime\prime}\rightarrow
0}}e^{2i\delta_{l}^{(C)}}.$ (45)

The $t^{(SC)}_{l}(q)$ is typically parameterized by both Coulomb phase shift
$\delta_{l}^{(C)}$ and a short-range scattering phase shift
$\delta_{l}^{(S)}$,

$t^{(SC)}_{l}(q)=\frac{1}{\cot\delta_{l}^{(S)}-i}e^{2i\delta_{l}^{(C)}}.$ (46)

Using the asymptotic form:

$\displaystyle\frac{j_{l}^{(C)}(\gamma,qr)}{r^{l}}|_{r\rightarrow 0}$
$\displaystyle=q^{l}C_{l}(\gamma),$
$\displaystyle\Im\left[\frac{G_{l}^{(C,\infty)}(r^{\prime},r^{\prime\prime};q)}{(r^{\prime}r^{\prime\prime})^{l}}|_{r^{\prime},r^{\prime\prime}\rightarrow
0}\right]$ $\displaystyle=-2\mu q^{2l+1}C_{l}^{2}(\gamma),$ (47)

and Eq.(45) and Eq.(46), one thus find

$\displaystyle\frac{1}{V^{(S)}_{l}}$ $\displaystyle=-2\mu
q^{2l+1}C^{2}_{l}(\gamma)\cot\delta_{l}^{(S)}(q)$
$\displaystyle+\Re\left[\frac{G_{l}^{(C,\infty)}(r^{\prime},r^{\prime\prime};q)}{(r^{\prime}r^{\prime\prime})^{l}}|_{r^{\prime},r^{\prime\prime}\rightarrow
0}\right].$ (48)

##### Pure Coulomb scattering amplitude:

the pure Coulomb scattering amplitude is defined by Coulomb wave function. By
introducing

$h_{l}^{(C,\pm)}(\gamma,qr)=e^{\pm
i\delta_{l}^{(C)}}H_{l}^{(C,\pm)}(\gamma,qr),$ (49)

where

$\displaystyle H_{l}^{(C,\pm)}(\gamma,qr)=J_{l}^{(C)}(\gamma,qr)\pm
iN_{l}^{(C)}(\gamma,qr)$
$\displaystyle=-2(-2qr)^{l}e^{\frac{\pi}{2}\gamma}e^{\pm iqr}U(l+1\pm
i\gamma,2l+2,\mp 2iqr),$ (50)

we can thus rewrite the Coulomb wave function in Eq.(40) to

$\psi^{(C,\infty)}_{l}(r,q)=4\pi
i^{l}\left[J_{l}^{(C)}(\gamma,qr)+it_{l}^{(C)}(q)H_{l}^{(C,+)}(\gamma,qr)\right],$
(51)

where $t_{l}^{(C)}(q)$ is the pure Coulomb scattering amplitude:

$t_{l}^{(C)}(q)=\frac{e^{2i\delta_{l}^{(C)}}-1}{2i}.$ (52)

##### Total scattering amplitude:

the total wave function in Eq.(44) is now also given by

$\displaystyle\psi^{(\infty)}_{l}(r,q)$ $\displaystyle=4\pi
i^{l}\bigg{[}J^{(C)}_{l}(\gamma,qr)+it_{l}(q)H_{l}^{(C,+)}(\gamma,qr)\bigg{]},$
(53)

where

$\displaystyle
t_{l}(q)=t^{(C)}_{l}(q)+t^{(SC)}_{l}(q)=\frac{e^{2i\delta_{l}^{(C)}}e^{2i\delta_{l}^{(S)}}-1}{2i}.$
(54)

##### Asymptotic forms of wave functions:

using asymptotic form of $H_{l}^{(C,\pm)}$ functions,

$H_{l}^{(C,\pm)}(\gamma,qr)\stackrel{{\scriptstyle
r\rightarrow\infty}}{{\rightarrow}}h^{(\pm)}_{l}(qr)e^{\mp i\gamma\ln 2qr},$
(55)

one can easily illustrate that

$\displaystyle\psi^{(C,\infty)}_{l}(r,q)$
$\displaystyle\stackrel{{\scriptstyle r\rightarrow\infty}}{{\rightarrow}}4\pi
i^{l}\bigg{[}\frac{\sin(qr-\frac{\pi}{2}l-\gamma\ln 2qr)}{qr}$
$\displaystyle+it_{l}^{(C)}(q)h^{(+)}_{l}(qr)e^{-i\gamma\ln 2qr}\bigg{]},$
(56)

and

$\displaystyle\psi^{(\infty)}_{l}(r,q)$ $\displaystyle\stackrel{{\scriptstyle
r\rightarrow\infty}}{{\rightarrow}}4\pi
i^{l}\bigg{[}\frac{\sin(qr-\frac{\pi}{2}l-\gamma\ln 2qr)}{qr}$
$\displaystyle+it_{l}(q)h^{(+)}_{l}(qr)e^{-i\gamma\ln 2qr}\bigg{]},$ (57)

where the factor $\gamma\ln 2qr$ represents the long-range Coulomb distortion
effect to the incoming plane wave and outgoing spherical wave.

###  II.3 Quantization condition in a trap in presence of Coulomb interaction

Combining Eq.(17) and Eq.(48) by eliminating $V^{(S)}_{l}$, one thus find
Coulomb force modified Lüscher formula-like relation,

$\det\left[\delta_{lm,l^{\prime}m^{\prime}}\cot\delta^{(S)}_{l}(q)-\mathcal{M}^{(C,t)}_{lm,l^{\prime}m^{\prime}}(\varepsilon)\right]=0,$
(58)

where $\mathcal{M}^{(C,t)}$ is generalized zeta function in presence of
Coulomb force,

$\displaystyle\mathcal{M}^{(C,t)}_{lm,l^{\prime}m^{\prime}}(\varepsilon)=-\frac{1}{2\mu
q^{2l+1}C^{2}_{l}(\gamma)}\frac{G_{lm,l^{\prime}m^{\prime}}^{(C,t)}(r,r^{\prime};\varepsilon)}{r^{l}{r^{\prime}}^{l^{\prime}}}|_{r,r^{\prime}\rightarrow
0}$ $\displaystyle+\delta_{lm,l^{\prime}m^{\prime}}\frac{1}{2\mu
q^{2l+1}C^{2}_{l}(\gamma)}\frac{\Re\left(G_{l}^{(C,\infty)}(r,r^{\prime};q)\right)}{(rr^{\prime})^{l}}|_{r,r^{\prime}\rightarrow
0}.$ (59)

Both $G_{lm,l^{\prime}m^{\prime}}^{(C,t)}$ and $G_{l}^{(C,\infty)}$ are
ultraviolet divergent, and after cancellation between two terms, generalized
zeta function is finite and well-defined. At the limit of $\gamma\rightarrow
0$,

$C_{l}(\gamma)\stackrel{{\scriptstyle\gamma\rightarrow
0}}{{\rightarrow}}C_{l}(0)=\frac{\sqrt{\pi}}{2^{l+1}\Gamma(l+\frac{3}{2})}=\frac{j_{l}(qr)}{(qr)^{l}}|_{r\rightarrow
0},$ (60)

and

$\frac{\Re\left(G_{l}^{(C,\infty)}(r,r^{\prime\prime};q)\right)}{(rr^{\prime\prime})^{l}}|_{r,r^{\prime\prime}\rightarrow
0}\stackrel{{\scriptstyle\gamma\rightarrow 0}}{{\rightarrow}}2\mu
q\frac{j_{l}(qr)n_{l}(qr)}{r^{2l}}|_{r\rightarrow 0},$ (61)

hence, Eq.(59) is reduced to Eq.(B32) in Ref. Guo and Gasparian (2021).

##  III Discussion and summary

###  III.1 Perturbation expansion

The key element of generalized zeta function in Eq.(59) is Coulomb interaction
modified Green’s function in a trap, which is given by Dyson equation,
Eq.(12). Solving Dyson equation in most cases is not an easy task, great
effort must to be made to deal with both ultraviolet divergence (UV) and
infrared divergence (IR) caused by Coulomb interaction. Therefore the
perturbation expansion may be more practical in general, see discussion in
Beane and Savage (2014); Stellin and Meißner (2021). Symbolically, the Coulomb
force modified zeta function is a real function and given by

$\mathcal{\hat{M}}_{C,t}\sim\frac{1}{C^{2}(\gamma)}\left[\hat{G}_{C,t}-\Re\left(\hat{G}_{C,\infty}\right)\right],$
(62)

where the solution of $\hat{G}_{C,t}$ is given by

$\hat{G}_{C,t}=\frac{\hat{G}_{t}}{1-\hat{V}_{C}\hat{G}_{t}}=\sum_{n=0}^{\infty}\hat{G}_{t}\left(\hat{V}_{C}\hat{G}_{t}\right)^{n},$
(63)

and $\hat{G}_{t}$ denotes Green’s function in a trap,

$\hat{G}_{t}(E)=\frac{1}{E-\hat{H}_{t}}.$ (64)

Although the analytic expression of infinite volume Coulomb Green’s function
$\hat{G}_{C,\infty}$ is known already, in order to make sure the UV and IR
divergences cancelled out properly order by order, $\hat{G}_{C,\infty}$ can be
expanded in terms of perturbation theory as well

$\hat{G}_{C,\infty}=\frac{\hat{G}_{0,\infty}}{1-\hat{V}_{C}\hat{G}_{0,\infty}}=\sum_{n=0}^{\infty}\hat{G}_{0,\infty}\left(\hat{V}_{C}\hat{G}_{0,\infty}\right)^{n},$
(65)

where

$\hat{G}_{0,\infty}(E)=\frac{1}{E-\hat{H}_{0}}.$ (66)

Hence, Coulomb corrected zeta function may be computed by perturbation
expansion systematically,

$\displaystyle C^{2}(\gamma)\mathcal{\hat{M}}_{C}$
$\displaystyle\sim\sum_{n=0}^{\infty}\left[\hat{G}_{t}\left(\hat{V}_{C}\hat{G}_{t}\right)^{n}-\Re\left(\hat{G}_{0,\infty}\left(\hat{V}_{C}\hat{G}_{0,\infty}\right)^{n}\right)\right].$
(67)

Perturbation expansion may be well applied to both HO trap and periodic cubic
box, also see Beane and Savage (2014); Stellin and Meißner (2021) for the
discussion in finite volume from effective field theory perspective.

##### HO trap:

for the harmonic oscillator trap, iterating Dyson equation in Eq.(19) once,
the leading order and first order perturbation result can be written down
formally by,

$\displaystyle\frac{C^{2}_{l}(\gamma)}{C^{2}_{l}(0)}\mathcal{M}^{(C,0th)}_{l}(\varepsilon)=(-1)^{l+1}\left(\frac{4\mu\omega}{q^{2}}\right)^{l+\frac{1}{2}}\frac{\Gamma(\frac{l}{2}+\frac{3}{4}-\frac{\varepsilon}{2\omega})}{\Gamma(\frac{1}{4}-\frac{l}{2}-\frac{\varepsilon}{2\omega})},$
(68)

and

$\displaystyle\frac{C^{2}_{l}(\gamma)}{C^{2}_{l}(0)}\mathcal{M}^{(C,1st)}_{l}(\varepsilon)$
$\displaystyle=-\frac{2^{2l+2}\Gamma^{2}(l+\frac{3}{2})}{2\mu
q^{2l+1}\pi}\frac{\triangle
G_{l}^{(C,1st)}(r,r^{\prime};\varepsilon)}{(rr^{\prime})^{l}}|_{r,r^{\prime}\rightarrow
0},$ (69)

where

$\displaystyle\triangle G_{l}^{(C,1st)}(r,r^{\prime};\varepsilon)$
$\displaystyle=-\int_{0}^{\infty}{r^{\prime\prime}}^{2}dr^{\prime\prime}G_{l}^{(\omega)}(r,r^{\prime\prime};\varepsilon)\frac{Z}{r^{\prime\prime}}G_{l}^{(\omega)}(r^{\prime\prime},r^{\prime};\varepsilon)$
$\displaystyle+\Re\int_{0}^{\infty}{r^{\prime\prime}}^{2}dr^{\prime\prime}G_{l}^{(0,\infty)}(r,r^{\prime\prime};q)\frac{Z}{r^{\prime\prime}}G_{l}^{(0,\infty)}(r^{\prime\prime},r^{\prime};q).$
(70)

##### Periodic cubic box:

similarly, in finite volume, the leading order and first order perturbation
result are given by

$\displaystyle\frac{C^{2}_{l}(\gamma)}{C^{2}_{l}(0)}\mathcal{M}^{(C,0th)}_{lm,l^{\prime}m^{\prime}}(\varepsilon)$
$\displaystyle=-\frac{1}{L^{3}}\sum_{\mathbf{p}\in\frac{2\pi\mathbf{n}}{L}}^{\mathbf{n}\in\mathbb{Z}^{3}}\frac{p^{l+l^{\prime}}}{q^{2l+1}}\frac{Y_{lm}(\mathbf{\hat{p}})Y^{*}_{l^{\prime}m^{\prime}}(\mathbf{\hat{p}})}{2\mu\varepsilon-\mathbf{p}^{2}}$
$\displaystyle-\delta_{lm,l^{\prime}m^{\prime}}\frac{2^{2l+1}\Gamma(l+\frac{1}{2})\Gamma(l+\frac{3}{2})}{\pi}\frac{1}{(qr)^{2l+1}}|_{r\rightarrow
0},$ (71)

and

$\displaystyle\frac{C^{2}_{l}(\gamma)}{C^{2}_{l}(0)}\mathcal{M}^{(C,1st)}_{lm,l^{\prime}m^{\prime}}(\varepsilon)$
$\displaystyle=\frac{1}{2\mu
q^{2l+1}}\frac{1}{L^{6}}\sum_{\mathbf{p},\mathbf{p}^{\prime}\in\frac{2\pi\mathbf{n}}{L}}^{\mathbf{n}\in\mathbb{Z}^{3}}\frac{p^{l}Y_{lm}(\mathbf{\hat{p}})}{\varepsilon-\frac{\mathbf{p}^{2}}{2\mu}}\frac{4\pi
Z}{|\mathbf{p}-\mathbf{p}^{\prime}|^{2}}\frac{{p^{\prime}}^{l^{\prime}}Y^{*}_{l^{\prime}m^{\prime}}(\mathbf{\hat{p}}^{\prime})}{\varepsilon-\frac{\mathbf{p^{\prime}}^{2}}{2\mu}}$
$\displaystyle-\frac{\delta_{lm,l^{\prime}m^{\prime}}}{2\mu
q^{2l+1}}\Re\int\frac{d\mathbf{p}d\mathbf{p}^{\prime}}{(2\pi)^{6}}\frac{p^{l}Y_{lm}(\mathbf{\hat{p}})}{\varepsilon-\frac{\mathbf{p}^{2}}{2\mu}}\frac{4\pi
Z}{|\mathbf{p}-\mathbf{p}^{\prime}|^{2}}\frac{{p^{\prime}}^{l}Y^{*}_{lm}(\mathbf{\hat{p}}^{\prime})}{\varepsilon-\frac{\mathbf{p^{\prime}}^{2}}{2\mu}}.$
(72)

###  III.2 Analytic solutions in a spherical hard wall trap

The rotational symmetry inside of a hard-sphere trap is also well-preserved,
so angular orbital momentum is still a good quantum number, only diagonal
elements of Coulomb Green’s function contribute. The partial wave Coulomb
Green’s function inside hard-sphere must be the combination of regular and
irregular functions of Coulomb differential equation: $j_{l}^{(C)}(\gamma,qr)$
and $n_{l}^{(C)}(\gamma,qr)$ defined in Eq.(34) and Eq.(35) respectively.
Similar to the hard-sphere trap without Coulomb interaction, see Eq.(54) in
Ref.Guo and Long (2021), the closed form of Coulomb force modified Green’s
function inside hard-sphere is given by

$\displaystyle G_{l}^{(C,h.s.)}$
$\displaystyle(r,r^{\prime\prime};\varepsilon)=-2\mu
qj_{l}^{(C)}(\gamma,qr_{<})j_{l}^{(C)}(\gamma,qr_{>})$
$\displaystyle\times\left[\frac{n_{l}^{(C)}(\gamma,qR)}{j_{l}^{(C)}(\gamma,qR)}-\frac{n_{l}^{(C)}(\gamma,qr_{>})}{j_{l}^{(C)}(\gamma,qr_{>})}\right].$
(73)

The real part of Coulomb Green’s function in infinite volume is given by

$\Re\left(G_{l}^{(C,\infty)}(r,r^{\prime\prime};\varepsilon)\right)=2\mu
qj_{l}^{(C)}(\gamma,qr_{<})n_{l}^{(C)}(\gamma,qr_{>}).$ (74)

Hence, after UV cancellation in Eq.(59), the analytic expression of Coulomb
force modified generalized zeta function for hard-sphere trap is obtained

$\mathcal{M}^{(C,h.s.)}_{lm,l^{\prime}m^{\prime}}(\varepsilon)=\delta_{lm,l^{\prime}m^{\prime}}\frac{C^{2}_{l}(0)}{C^{2}_{l}(\gamma)}\frac{n_{l}^{(C)}(\gamma,qR)}{j_{l}^{(C)}(\gamma,qR)}.$
(75)

The quantization condition in a hard-sphere trap in presence of Coulomb
interaction is given in a closed-form:

$\cot\delta^{(S)}_{l}(q)=\frac{C^{2}_{l}(0)}{C^{2}_{l}(\gamma)}\frac{n_{l}^{(C)}(\gamma,qR)}{j_{l}^{(C)}(\gamma,qR)},$
(76)

where $j_{l}^{(C)}(\gamma,qr)$ and $n_{l}^{(C)}(\gamma,qr)$ are defined in
Eq.(34) and Eq.(35) respectively.

###  III.3 Summary

In summary, we present a general discussion on the topic of formulating
quantization condition of trapped systems by including long-range Coulomb
interaction. Although all the discussion are based on non-perturbative LS
equation approach, in most cases, the Coulomb force modified Green’s function
in a trap must be solved either numerically or by perturbation expansion. In
special cases, such as the spherical hard wall trap, the closed-form of
quantization condition is obtained and given in Eq.(76).

###### Acknowledgements.

We thank fruitful discussion with Bingwei Long. P.G. also acknowledges support
from the Department of Physics and Engineering, California State University,
Bakersfield, CA. The work was supported in part by the National Science
Foundation under Grant No. NSF PHY-1748958.

##  Appendix A Formal scattering theory with short-range and Coulomb
interactions

In this section, the formal scattering theory in presence of both a short-
range interaction and a long-range Coulomb interaction is briefly discussed,
the complete discussion can be found in Refs. Mott and Massey (1985);
Goldberger and Watson (1964). The connection to trapped system is also briefly
discussed symbolically.

###  A.1 Coulomb force modified scattering amplitude in infinite volume

The infinite volume scattering amplitude in the presence of both Coulomb and
short-range nuclear interactions is defined by

$T_{\infty}=-\langle\Psi_{0}|(\hat{V}_{C}+\hat{V}_{S})|\Psi^{(+)}\rangle,$
(77)

where $|\Psi_{0}\rangle$ stands for plane wave. The $|\Psi^{(+)}\rangle$ is
defined by LS equation,

$|\Psi^{(\pm)}\rangle=|\Psi_{0}\rangle+\hat{G}_{0}(E\pm
i0)(\hat{V}_{C}+\hat{V}_{S})|\Psi^{(\pm)}\rangle,$ (78)

where

$\hat{G}_{0}(E\pm i0)=\frac{1}{E-\hat{H}_{0}\pm i0}.$ (79)

Using Eq.(78) and also LS equation for pure Coulomb interaction,

$\langle\Psi_{0}|=\langle\Psi^{(-)}_{C}|-\langle\Psi^{(-)}_{C}|\hat{V}_{C}\hat{G}_{0}(E+i0),$
(80)

the total infinite volume scattering amplitude $T_{\infty}$ can be rewritten
as, also see Refs. Mott and Massey (1985); Goldberger and Watson (1964),

$T_{\infty}=-\langle\Psi^{(-)}_{C}|\hat{V}_{C}|\Psi_{0}\rangle-\langle\Psi^{(-)}_{C}|\hat{V}_{S}|\Psi^{(+)}\rangle.$
(81)

The first term in Eq.(81) is identified as pure Coulomb interaction scattering
amplitude,

$T_{C,\infty}=-\langle\Psi^{(-)}_{C}|\hat{V}_{C}|\Psi_{0}\rangle=-\langle\Psi_{0}|\hat{V}_{C}|\Psi^{(+)}_{C}\rangle.$
(82)

The partial wave coulomb amplitude is parameterized by Coulomb phase shifts,

$T^{(C,\infty)}_{l}\propto\frac{e^{2i\delta^{(C)}_{l}}-1}{2i},$ (83)

where the Coulomb phase shift $\delta_{l}^{(C)}$ is defined in Eq.(37).

The second term in Eq.(81) is the result of short-range interaction in the
presence of Coulomb interaction, using LS equation

$|\Psi^{(+)}\rangle=|\Psi^{(+)}_{C}\rangle+\hat{G}_{C}(E+i0)V_{S}|\Psi^{(+)}\rangle,$
(84)

where

$\hat{G}_{C}(E\pm i0)=\frac{1}{E-\hat{H}_{0}-\hat{V}_{C}\pm i0},$ (85)

it can be shown rather straight-forwardly that second term satisfies equation

$\displaystyle-\langle\Psi^{(-)}_{C}|\hat{V}_{S}|\Psi^{(+)}\rangle$
$\displaystyle=-\langle\Psi^{(-)}_{C}|\hat{V}_{S}|\Psi^{(+)}_{C}\rangle-\langle\Psi^{(-)}_{C}|\hat{V}_{S}\hat{G}_{C}(E+i0)\hat{V}_{S}|\Psi^{(+)}\rangle.$
(86)

Hence, it may be useful and more convenient to define a Coulomb force modified
scattering operator

$\hat{T}_{SC,\infty}|\Psi_{C}^{(+)}\rangle=-\hat{V}_{S}|\Psi^{(+)}\rangle,$
(87)

thus, the second term in Eq.(81) now can be written as

$-\langle\Psi^{(-)}_{C}|\hat{V}_{S}|\Psi^{(+)}\rangle=\langle\Psi^{(-)}_{C}|\hat{T}_{SC,\infty}|\Psi^{(+)}_{C}\rangle.$
(88)

According to Eq.(86), $\hat{T}_{SC,\infty}$ satisfies operator equation

$\hat{T}_{SC,\infty}=-\hat{V}_{S}+\hat{V}_{S}\hat{G}_{C}(E+i0)\hat{T}_{SC,\infty}.$
(89)

The total scattering amplitude is now given by

$T_{\infty}=T_{C,\infty}+\langle\Psi^{(-)}_{C}|\hat{T}_{SC,\infty}|\Psi^{(+)}_{C}\rangle.$
(90)

Given the fact that the symbolic solution of $\hat{T}_{SC,\infty}$ operator is
given by

$\hat{T}^{-1}_{SC,\infty}=-\hat{V}_{S}^{-1}+\hat{G}_{C}(E+i0),$ (91)

and

$\langle\Psi^{(-)}_{C}|\Psi^{(+)}_{C}\rangle=1+2iT_{C,\infty}=S_{C,\infty}\propto
e^{2i\delta^{(C)}_{l}}$ (92)

is the pure Coulomb interaction $S$-matrix, the partial wave expansion of
$\langle\Psi^{(-)}_{C}|\hat{T}_{SC,\infty}|\Psi^{(+)}_{C}\rangle$ is
conventionally parameterized by both Coulomb phase shift $\delta_{l}^{(C)}$
and the short-range interaction phase shift, $\delta^{(S)}_{l}$, see Refs.
Mott and Massey (1985); Goldberger and Watson (1964),

$\langle\Psi^{(-)}_{C}|\hat{T}_{SC,\infty}|\Psi^{(+)}_{C}\rangle\propto
e^{2i\delta^{(C)}_{l}}\frac{e^{2i\delta^{(S)}_{l}}-1}{2i}.$ (93)

Therefore, the partial wave total infinite volume scattering amplitude is thus
defined by a total phase shift,

$\delta_{l}=\delta^{(S)}_{l}+\delta^{(C)}_{l},$ (94)

and

$T^{(\infty)}_{l}\propto\frac{e^{2i\delta_{l}}-1}{2i}=\frac{e^{2i\delta^{(C)}_{l}}-1}{2i}+e^{2i\delta^{(C)}_{l}}\frac{e^{2i\delta^{(S)}_{l}}-1}{2i}.$
(95)

###  A.2 Charged particles in a trap in presence of Coulomb force

In the trap, the Eq.(89) is now modified to

$\hat{T}_{SC,t}=-\hat{V}_{S}+\hat{V}_{S}\hat{G}_{C,t}(E+i0)\hat{T}_{SC,t}\,,$
(96)

where

$\hat{G}_{C,t}(E\pm i0)=\frac{1}{E-\hat{H}_{t}-\hat{V}_{C}\pm i0},$ (97)

is Coulomb Green’s function in the trap, and

$\hat{H}_{t}=\hat{H}_{0}+\hat{V}_{t}$

is trap Hamiltonian operator. The quantization condition including Coulomb
interaction thus is given by

$\det\left[V_{S}^{-1}-G_{C,t}(E+i0)\right]=0.$ (98)

###  A.3 Quantization condition including Coulomb interaction

Eliminating $V^{-1}_{S}$ from Eq.(91) and Eq.(98), the quantization condition
Eq.(98) thus can be rewritten as

$\det\left[T_{SC,\infty}^{-1}+G_{C,t}(E+i0)-G_{C}(E+i0)\right]=0.$ (99)

In general, Coulomb Green’s function in the trap is obtained by Dyson
equation,

$\hat{G}_{C,t}(E)=\hat{G}_{t}(E)+\hat{G}_{t}(E)\hat{V}_{C}\hat{G}_{C,t}(E),$
(100)

where

$\hat{G}_{t}(E\pm i0)=\frac{1}{E-\hat{H}_{t}\pm i0}.$ (101)

In practice, Coulomb effect may be treated as perturbation by summing over all
the ladder diagrams generated by Coulomb exchanges,

$\hat{G}_{C,t}(E)=\hat{G}_{t}(E)\sum_{n=0}^{\infty}\left(\hat{V}_{C}\hat{G}_{t}(E)\right)^{n}.$
(102)

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[a]Enrico Scomparin

# Hard and electromagnetic probes:
plans for future measurements at the CERN SPS

###### Abstract

The CERN SuperProtoSynchrotron (SPS) represents an ideal facility for fixed-
target heavy-ion experiments exploring the phase diagram of strongly
interacting matter in the region $200\leq\mu_{\rm B}\leq 500$ MeV. It can
deliver high-intensity beams ($>10^{6}$ Pb/s), allowing a study of rare probes
of the Quark-Gluon Plasma, including electromagnetic and hard processes. The
NA61/SHINE experiment is currently active and plans to perform a first direct
measurement of open charm production in Pb–Pb collisions at top SPS energy and
possibly at lower energies. The project of a new experiment, NA60+, based on a
muon spectrometer coupled to a vertex spectrometer is currently being
developed, for the study of dimuon and heavy quark production, and a Letter of
Intent was recently submitted. In this contribution the physics motivation for
the studies of rare probes, the existing and planned experimental set-ups and
their expected physics performance will be discussed.

##  1 Introduction

Studies of electromagnetic and hard probes have represented one of the most
important sources of information on the early stages of the strongly
interacting partonic system created in ultra-relativistic heavy-ion
collisions. From 1986 to 2003, historic first results in this domain were
obtained by experiments taking data at the CERN SPS. Among those, one can
mention the first evidence for a modification of the $\rho$-meson spectral
function by the CERES/NA45 experiment [1], the discovery of the anomalous
suppression of the J/$\psi$ by NA50 [2] and the first direct measurement of a
temperature exceeding $T_{\rm c}$, via thermal dimuons by NA60 [3]. However,
all of these measurements were performed at top SPS energy ($\sqrt{s_{\rm
NN}}=17.3$ GeV), with the extension to lower energies made difficult by the
quickly vanishing production cross sections. Among the hard probes of QGP,
open charm production was also marginally explored, with an indirect result
from NA60 obtained in the study of the dimuon spectrum in the region around
$m_{\rm\mu\mu}\sim 2$ GeV/$c^{2}$ [3] and an upper limit on D0 production [4]
from NA49.

An extension of all these measurements to lower energies presents a great
physics interest. Concerning thermal dileptons, their emission takes place all
along the collision history but the yield is dominated by the high-temperature
phase. The $T_{\rm slope}$ parameter of the spectral shape of the dilepton
invariant mass spectrum can be considered as a space-time average of the
thermal temperature $T$ over the fireball evolution. Measurements carried out
by the HADES [5] and NA60 [3] experiments, respectively in Au–Au collisions at
$\sqrt{s_{\rm NN}}=2.42$ GeV and In–In at $\sqrt{s_{\rm NN}}=17.3$ GeV,
allowed experimental estimates of the medium temperature. Values increasing
from $T_{\rm slope}=71.8\pm 2.1$ MeV to $205\pm 12$ MeV were obtained, showing
that the crossing of the pseudo-critical temperature should be reached in the
collision energy interval defined by the two existing measurements. A precise
measurement of the dilepton spectrum below top SPS energy is therefore quite
interesting and a precision study of its $\sqrt{s_{\rm NN}}$-dependence might
also be sensitive to the presence of a first order phase transition [6].
Measurements of lepton pairs also allow a study of modifications of the
spectral function of the $\rho_{0}$ and of its chiral partner $a_{\rm 1}$,
related to chiral symmetry restoration. The latter resonance is not directly
coupled to the dilepton channel, but its modification leads to an enhancement
of the continuum in the region $1<m_{\rm ll}<1.4$ GeV/c2 [6].

Concerning hard probes of the QGP, quarkonium production can also be
investigated by a dilepton measurement. Studying the J/$\psi$ suppression, and
possibly that of higher-mass states ($\psi({\rm 2S})$, $\chi_{\rm c}$) is of
utmost importance, seen its relation with deconfinement, studied in great
detail from top SPS [7] to RHIC [8] and LHC energy [9]. A superposition of a
suppression mechanism, driven by color screening and gluodissociation
processes, and regeneration at hadronization is able to explain the wealth of
data at collider energies. Extending measurements towards lower energies would
allow the identification of a threshold for the suppression, which is the
dominant mechanism at low energy, and its correlation with the temperature
measured via thermal dileptons. In addition to hidden charm, open charm
production represents a crucial measurement to get information on the
transport coefficients of the medium, as well as on its hadronization
mechanisms [10]. At low energy, the system spends comparatively more time in
the hadronic phase than in the QGP, and it is not clear to which degree heavy
quark would thermalize in the medium. Accurate results on $R_{\rm AA}$ and
$v_{\rm 2}$ of charm hadrons would test an environment very different from
that accessible to collider experiments and represent a stringent testbench
for models. Furthermore, separate measurements of charmed baryon over meson
and strange over nonstrange hadron ratios would allow a study of coalescence
effects around the threshold for QGP production. Clearly, accurate
measurements of open charm hadrons can only be performed by accessing their
hadronic decays, requiring the capability of tracking/identifying charged
particles in a high-multiplicity environment.

The physics program sketched in these paragraphs can be addressed by a new SPS
experiment, NA60+ [11], based on a muon spectrometer coupled to a vertex
spectrometer, and inspired to the former NA60 experiment. Specific items, and
in particular a first direct measurement of open charm production, can be
addressed by the existing NA61/SHINE [12] experiment at the same facility. In
the next sections details will be given on the foreseen measurements by NA61
and on the progress of the design and R&D phase of the NA60+ experiment. For
the latter, the current estimates of the physics performance and the planning
of the data taking phase will be given.

##  2 Open charm measurements at NA61/SHINE

NA61/SHINE, active since 2009, is a multi-purpose experiment investigating
hadron production in nuclear collisions. It has performed systematic studies
with various ion species at different energies, investigating signals of the
onset of deconfinement. Among the main physics goals for 2022-2025, the
experiment is planning a measurement of open charm, also thanks to an upgrade
of the vertex detector to stand a 1 kHz interaction rate. The expected
statistics is $5\cdot 10^{8}$ minimum bias events at $E_{\rm lab}=150A$ GeV
and $2.5\cdot 10^{8}$ at $E_{\rm lab}=40A$ GeV. The data should provide, at
high energy, the D-meson rapidity and transverse momentum spectra for central
collisions (tens of thousands reconstructed decays) and establish the
centrality dependence of mean D-meson yield in the acceptance (thousands of
reconstructed decays). Together with the low-energy data, the first
measurement of the collision energy dependence of the mean yield should become
possible. A low-statistics invariant mass spectrum is shown in Fig. 1.

Figure 1: Invariant mass distribution of D0 \+ $\bar{\rm D}^{0}$ candidates in
central Pb–Pb collisions, from NA61/SHINE, after background suppression cuts
[13].

##  3 The NA60+ experiment

A sketch of the NA60+ experiment can be seen in Fig. 2. A muon spectrometer is
separated from the target region by means of a thick hadron absorber, made of
BeO and graphite that filters out the hadrons. Candidate muons are tracked by
means of six tracking stations. The two upstream stations are followed by a
large-aperture toroidal magnet and by two further tracking stations. A “muon
wall” acts as a final filter to remove hadrons escaping the absorber as well
as low-momentum muons, and is followed by the two final tracking stations. In
the target region a system of five closely spaced targets is followed by a
vertex spectrometer which includes from five up to ten tracking MAPS planes.
The vertex spectrometer is embedded in the gap of a dipole magnet. By matching
muon tracks in the muon and vertex spectrometers, in the coordinate and
momentum space, a measurement of the muon kinematics not affected by multiple
scattering and energy loss in the hadron absorber becomes possible. The muon
spectrometer is mounted on a system of rails that allows keeping a constant
rapidity coverage when changing the collision energy. When moving the
spectrometer downstream along the beam line (high energy collisions) the
thickness of the absorber will be increased by adding further graphite
elements.

Figure 2: The foreseen set-up of the NA60+ experiment at the CERN SPS.

In Fig. 3 (left) a possible arrangement of the vertex spectrometer is shown.
Each plane contains four large MAPS sensors ($15\times 15$ cm2 each). The R&D
phase is in progress in the frame of a common development of the ALICE (ITS3
project [14]) and NA60+ collaborations. For NA60+ each sensor is based on 25
mm long units, replicated several times through a stitching procedure. For
each stitched long sensor, the control logic to steer the priority encoders,
the interfaces for the configuration of the chip and serial data transmitters
are all located at the periphery of the sensor, outside the detector
acceptance. The thickness of the sensor will be $<0.1$% radiation length and
the spatial resolution will be $\leq 5$ $\mu$m. From the mechanical point of
view the sensors will be mounted on a light frame, as shown in Fig. 3(right).
Studies of cooling options show that a mixed fluid (0.5 l/s at T=18 oC) + air
($\sim$ 1 m/s) system should allow an efficient control of the temperature
over all the surface of the detection plane. A test set-up is being built,
with thin graphite frames, in order to perform tests of positioning and gluing
of sensors (dummy Si sensors are used for these studies). It will also be
possible to check the chosen cooling option using a flex circuit with resistor
arrays, glued on the sensor, to simulate the expected power dissipation. Flex
cables for wire-bonding tests and high-speed data transmission will also be
designed.

Figure 3: Possible layout of the target system and the vertex spectrometer
(left); sketch of the mechanical support of a MAPS plane (right).

For the muon tracking, due to the relatively low rates after the hadron
absorber ($\sim 2$ kHz, at the expected Pb beam rate of $10^{6}$ s-1) GEMs or
MWPCs options are considered. For the latter, two prototypes were built and
tested in the laboratory with cosmic rays. They will be exposed to an SPS test
beam at the end of 2023. The tracking stations will be built by replicating
the prototype detector in such a way to obtain the desired geometry. The two
upstream stations will include 12 trapezoid detectors each, while those
downstream of the magnet and the muon wall will have 36 and 84 detectors
respectively. Figure 4 shows an MWPC prototype unit (left) and the foreseen
arrangement for the largest stations (right). The GEM option is also under
study, based on the design of a triple GEM detector for the MOLLER experiment
at JLab [15]. Also this detector will be tested on an SPS beam this year.

Figure 4: First prototype of a MWPC unit for the NA60+ muon spectrometer
(left); layout of the largest muon tracking stations, downstream of the Muon
wall (right).

The toroidal magnet represents a major element of the set-up. A warm pulsed
magnet is foreseen, producing 0.5 T of magnetic field magnitude over a volume
of 120 m3. The toroid is made from eight sectors, each coil has 12 turns and
the conductor has a square copper section with 50 mm side length and a
circular cooling channel in the centre. The current is 190 kA, and the total
power is $\sim$3 MW. A demonstrator (scale 1:5) was constructed and tested,
allowing a cross-check of various aspects of the design [16]. Measurements of
the magnetic field in the prototype were found to be in agreement with
simulations within 3%. In Fig. 5 (left) the results of the field calculations
for the toroidal magnet are reported, while on the right panel a photo of the
demonstrator can be seen. Finally, for the dipole magnet the plan is to use
the MEP48 magnet, currently stored at CERN. It can deliver a 1.5 T field, over
a 40 cm gap.

Figure 5: Magnetic field calculations for the NA60+ toroidal magnet (left);
the scale 1:5 demonstrator with its support mechanics (right).

After integration studies, the PPE138 hall on the H8 beam line of the SPS was
chosen as installation site for the NA60+ set-up [17]. Radio-protection
calculations have led to the design of an iron/concrete shielding, to keep the
dose levels below 3 $\mu$Sv/h externally to the experimental hall for the
intended 106 s-1 Pb beam intensity. Specific beam optics studies have shown
that a sub-millimetric beam can be delivered to the experiment from $E_{\rm
lab}=30$ to 150 AGeV. A first check of these optics was carried out, making
use of a telescope of ALPIDE Si sensors, in the fall of 2022 at $E=150$ GeV.
Values as small as $\sigma_{\rm x}=280$ $\mu$m and $\sigma_{\rm y}=250$ $\mu$m
were measured.

Accurate studies of the expected physics performance of the NA60+ experiment
were carried out. They are extensively reported in [11] and a selection of the
results is reported here. With the expected Pb beam intensity, 1011 minimum
bias collision events should be collected in the typical 1-month period
dedicated to heavy-ion running at CERN each year. For the measurement of the
$T_{\rm slope}$ parameter, in the mass region $1.5<m_{\mu\mu}<2.5$
GeV/$c^{2}$, uncertainties as low as $\sim 3\%$ can be reached. The result on
the measurement of the energy dependence of $T_{\rm slope}$ is shown in Fig. 6
(left) together with existing measurements from HADES and NA60 and
corresponding studies for the CBM experiment at GSI, which explores a lower,
complementary energy range. It will be possible to explore the region close to
the pseudocritical temperature and detect possible signals of a first-order
phase transition that can be expected at these energies. The expected effect
of a full chiral mixing on the dilepton spectrum is shown in Fig. 6 (right).
An increase of the dimuon yield by up to $\sim$30% in the region around
$m_{\mu\mu}=1$ GeV/$c^{2}$ is visible and should be detectable within the
expected uncertainties.

Figure 6: The energy dependence of the $T_{\rm slope}$ parameter (left), the
blue triangles show the expected NA60+ performance; comparison of the thermal
dimuon spectrum at $\sqrt{s_{\rm NN}}=8.8$ GeV in case of no chiral mixing and
full chiral mixing (black line) (right). The ratio between the two spectra is
also shown.

Open charm measurements will be performed starting from the tracks
reconstructed in the vertex spectrometer. Charm hadron candidates are built by
combining pairs or triplets of tracks with the proper charge signs. The huge
combinatorial background can be reduced via geometrical selections and
exploiting the fact that the mean proper decay lengths $c\tau$ are of about
60–310 $\mu$m, and therefore their decay vertices are typically displaced by a
few hundred $\mu$m from the primary interaction vertex. In Fig. 7 the expected
signals from the ${\rm D}^{\rm 0}\rightarrow{\rm K}\pi$ decay, for $E=60$ AGeV
incident energy and from the 3-body decay $\Lambda_{\rm c}\rightarrow{\rm
pK}\pi$, for $E=158$ AGeV are shown. Measurements of the energy dependence of
${\rm D}^{\rm 0}$ production, as well as of its nuclear modification factor
and $v_{\rm 2}$ at top SPS energy are within reach. Also studies of charmed
baryon/meson ratios, sensitive to the hadronization mechanisms, will be
possible.

Figure 7: Invariant mass distribution of candidates for the $D^{\rm
0}\rightarrow{\rm K}\pi$ decay at $E=60$ AGeV (left) and the $\Lambda_{\rm
c}\rightarrow{\rm pK}\pi$ decay at $E=158$ AGeV (right).

Measurements of hidden charm production will also be performed. The expected
statistics range from $10^{4}$ to $10^{5}$ J/$\psi$ from low to high SPS
energy, allowing a precise study of the suppression signal, that is expected
to vanish at low energy. Corresponding measurements in p-A collisions will
also be carried out. The latter are crucial in order to establish the
contribution of cold nuclear matter effects that is expected to become
stronger at low energy. In Fig. 8 (left) the number of ${\rm
J}/\psi\rightarrow\mu\mu$ decays that can be detected as a function of the
integrated luminosity, at various energies, is shown. With one month of data
taking for Pb–Pb collisions one has $L_{\rm int}\sim 25$ nb-1. In Fig. 8
(right) a possible centrality dependence of the J/$\psi$ nuclear modification
factor, at $E_{\rm lab}=50$ AGeV, is shown. The cold nuclear matter reference
is obtained by means of an extrapolation, using the Glauber model, of results
on J/$\psi$ production in p–A collisions, obtained with $5\cdot 10^{13}$
protons on a target system made of various nuclear species. As an hypothesis,
a 20% suppression effect in Pb–Pb, beyond cold nuclear matter effects and
starting at $N_{\rm part}\sim 100$, has been assumed. Although with larger
uncertainties, corresponding measurements for the $\psi(2S)$ meson should be
feasible down to $E_{\rm lab}\sim$ 100 GeV.

Figure 8: The expected number of detected ${\rm J}/\psi\rightarrow\mu\mu$
decays, as a function of the integrated luminosity (left); the centrality
dependence of the nuclear modification factor for the J/$\psi$ in Pb–Pb
collisions at $E_{\rm lab}=50$ AGeV (right), superimposed to the corresponding
values for cold nuclear matter effects, extrapolated from the results of p–A
collisions (see text for the assumptions).

Finally, in Fig. 9 (left), the invariant mass spectrum from the
$\Omega^{-}\rightarrow\Lambda^{0}{\rm K}^{-}$ decay (+ charge conjugate), is
shown, as an example of the NA60+ capabilities for the measurement of hyperons
in nuclear collisions. A realistic background was simulated, starting from the
parametrization of hadron production from NA49 results, topological selections
were then applied and Boosted Decision Trees were employed to enhance the
significance of the signals. In Fig. 9 (right) an invariant mass spectrum for
the decay ${}^{5}_{\Lambda}{\rm He}\rightarrow^{4}{\rm He}\,{\rm p}\,\pi^{-}$
is shown, demonstrating the possibility for NA60+ to detect hypernuclei. The
separation of heavily ionising particles from ordinary hadrons, essential for
the identification of the decay nucleus, is performed by means of a study of
the cluster size associated with the track in the vertex spectrometer.

Figure 9: Invariant mass spectrum from the
$\Omega^{-}\rightarrow\Lambda^{0}{\rm K}^{-}$ (+ charge conjugate) decay, at
$E_{\rm lab}=40$ AGeV (left), and for the ${}^{5}_{\Lambda}{\rm He}$
hypernucleus (right).

In summary, the NA60+ experiment should be able to access for the first time
e.m. and hard probes from low to top SPS energy, via dimuon measurements and
reconstruction of hadronic decays of open charm hadrons. Also measurements in
the strangeness sector will be feasible. After the submission of the Letter of
Intent at the end of 2022, a technical proposal should be completed by 2024,
with the plan of taking data from 2029. Together with the first measurements
of open charm planned by NA61/SHINE in the immediate future, these two
experiments will be able to advance our knowledge of the features of
high-$\mu_{\rm B}$ QGP, complementing efforts that are or will be undertaken
at other facilities (FAIR, RHIC, NICA, J-PARC).

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