# Nonperturbative study of dynamical SUSY breaking in ${\cal N}=(2,2)$ Yang-
Mills

Simon Catterall<EMAIL_ADDRESS>Department of Physics, Syracuse University,
Syracuse, NY 13244, USA  Raghav G. Jha<EMAIL_ADDRESS>Department of Physics,
Syracuse University, Syracuse, NY 13244, USA  Anosh Joseph
<EMAIL_ADDRESS>International Centre for Theoretical Sciences (ICTS-
TIFR),
Tata Institute of Fundamental Research,
Bangalore, 560089 INDIA

###### Abstract

We examine the possibility of dynamical supersymmetry breaking in two-
dimensional ${\cal N}=(2,2)$ supersymmetric Yang-Mills theory. The theory is
discretized on a Euclidean spacetime lattice using a supersymmetric lattice
action. We compute the vacuum energy of the theory at finite temperature and
take the zero temperature limit. Supersymmetry will be spontaneously broken in
this theory if the measured ground state energy is non-zero. By performing
simulations on a range of lattices up to $96\times 96$ we are able to perform
a careful extrapolation to the continuum limit for a wide range of
temperatures. Subsequent extrapolations to the zero temperature limit yield an
upper bound on the ground state energy density. We find the energy density to
be statistically consistent with zero in agreement with the absence of
dynamical supersymmetry breaking in this theory.

##  I Introduction

The investigations of supersymmetric gauge theories on a spacetime lattice are
important for understanding the non-perturbative structure of such theories
and in particular they can address the question of whether dynamical
supersymmetry breaking takes place in such theories. This is a crucial
question for efforts to construct supersymmetric theories which go beyond the
Standard Model since the low energy world is clearly not supersymmetric while
non-renormalization theorems typically ensure that supersymmetry cannot break
in perturbation theory Grisaru:1979wc .

Unfortunately, there are a plethora of problems to overcome for lattice
formulations of supersymmetric theories. Supersymmetry is a spacetime
symmetry, which is generically broken by the lattice regularization procedure.
Hence, the effective action of the lattice theory typically contains relevant
supersymmetry breaking interactions. To achieve a supersymmetric continuum
limit it is necessary to fine tune the lattice couplings to these terms as the
lattice spacing is reduced. Since generically there are very many such terms
this is in practice impossible. Some exceptions to this are - ${\cal N}=1$
super Yang-Mills where only a single coupling, the gluino mass, must be tuned.
In addition, it has also been shown that fine-tuning to a supersymmetric
continuum limit is also possible for ${\cal N}=(2,2)$ in two dimensions. Using
Wilson fermions, the only relevant parameter that has to be fine-tuned is the
scalar mass since the bare gluino mass is an irrelevant parameter. The
continuum value for the critical scalar mass is known up to one-loop order in
lattice perturbation theory and that has already been employed in the
numerical simulations. See Ref. Montvay:2001aj ; Suzuki:2005dx ;
August:2018esp  for discussions and references therein.

The attempt to formulate supersymmetric theories on the lattice has a long
history starting in Refs. Elitzur:1982vh ; Banks:1982ut ; Sakai:1983dg ;
Kostelecky:1983qu ; Aratyn:1984bc ; Scott:1983ha . Recent approaches to this
problem have focused on preserving a subalgebra of the full supersymmetry
algebra which can protect the theory from some of these dangerous
supersymmetry violating terms - for a review, see Ref. Catterall:2009it . For
supersymmetric theories with extended supersymmetry various supersymmetric
lattice formulations exist. One approach that was pioneered by Cohen, Kaplan,
Katz and Ünsal in Refs. Cohen:2003xe ; Cohen:2003qw ; Kaplan:2005ta  is based
on orbifolding and deconstruction of a supersymmetric matrix model. A second
approach uses the idea of topological twisting to isolate appropriate
nilpotent scalar supersymmetries that can be transferred to the lattice. Two
independent discretization schemes have been proposed in this approach - that
proposed by Sugino in Refs. Sugino:2003yb ; Sugino:2004qd  where the fermions
are associated with sites and a geometrical approach in which fermions are
generically associated with links Catterall:2003wd 111Yet another construction
was formulated by D’Adda, Kanamori, Kawamoto and Nagata, DAdda:2005rcd  but
was later shown to be equivalent to the orbifolding constructions when
restricted to a sector containing a scalar supercharge Damgaard:2007eh .. In
four spacetime dimensions, the geometrical approach has been used to construct
a supersymmetric lattice action for ${\cal N}=4$ SYM Catterall:2005fd ;
Catterall:2014vka  and has been shown to be identical to the orbifolding
constructions in Ref. Unsal:2006qp ; Catterall:2007kn . For an elaborate
discussion on the relation between all these constructions, see Ref.
Takimi:2007nn .

In this paper, we will study ${\cal N}=(2,2)$ super Yang-Mills (SYM) theory
using the geometrical discretization scheme. It is the simplest two-
dimensional supersymmetric theory that can be studied on the lattice. This
theory is a particularly interesting theory in the continuum because of its
exotic phases as discussed by Witten in Ref. Witten:1993yc . This theory is
conjectured to flow in the infrared (IR) to a conformal field theory. For
recent developments, see Ref. Park:2016dpb . The goal of this paper is to
calculate the vacuum energy density accurately for this theory and hence
determine whether supersymmetry breaking occurs. It is well known
Witten:1981nf  that the vacuum energy can be thought of as an order parameter
for SUSY breaking. The spontaneous breaking of supersymmetry in this two-
dimensional theory has been considered theoretically in Ref. Hori:2006dk  and
numerically in Refs. Kanamori:2009dk ; Kanamori:2007yx . In Hori:2006dk  it
was conjectured that in fact supersymmetry may break in this theory. Related
work for ${\cal N}=(2,2)$ super QCD on the lattice was described in
Catterall:2015tta . In the context of orbifold lattice theories, it was shown
in Ref. Matsuura:2007ec  that the vacuum energy of these theories does not
receive any quantum corrections in perturbation theory leaving only non-
perturbative mechanisms to drive supersymmetry breaking.

In this four supercharge theory, unlike the sixteen supercharge case in two
dimensions, the thermal instabilities at low temperatures are less severe and
we can access relatively small temperatures without truncating the $U(1)$
degree of freedom as done in our recent work Catterall:2017lub ; Jha:2017zad .
However, we have to use a small mass term to control the classical flat
directions associated with the scalars. This small mass term was also
implemented while exploring the phase structure at large $N$ using Sugino’s
lattice construction in Ref. Hanada:2009hq .

The plan of this paper will be as follows. In Sec. II we review the lattice
construction for ${\cal N}=(2,2)$ SYM on a two-dimensional square lattice.
Then in Sec. III we mention results on the phase of the pfaffian, discuss our
procedure of extracting the ground state energy and comment on the ${\cal
O}(a)$ improved action we use for the analysis. We end the paper with
conclusions and brief discussion in Sec. IV.

##  II Two-dimensional ${\cal N}=(2,2)$ Lattice SYM

The two-dimensional ${\cal N}=(2,2)$ SYM theory is the simplest supersymmetric
gauge theory which admits topological twisting Witten:1988ze  and thus
satisfies the requirements for a supersymmetric lattice construction following
the prescription given in Refs. Catterall:2004np ; Catterall:2006jw , where
the first numerical simulations of this construction were performed. The
theory has global symmetry group $G=SO(2)_{E}\times SO(2)_{R_{1}}\times
U(1)_{R_{2}}$, where $SO(2)_{E}$ is the two-dimensional Euclidean Lorentz
rotation symmetry, $SO(2)_{R_{1}}$ is the symmetry due to reduced directions
and $U(1)_{R_{2}}$ is the R-symmetry of the parent four-dimensional ${\cal
N}=1$ SYM theory. This theory can be twisted in two inequivalent ways (the
A-model and B-model twists) depending on how we embed $SO(2)_{E}$ group into
$SO(2)_{R_{1}}\times SO(2)_{R_{2}}$ the internal symmetry group.

We are interested in the B-model twist, which gives rise to a strictly
nilpotent twisted supersymmetry charge. After twisting, the fields and
supersymmetries are expressed as representations of the the twisted Euclidean
Lorentz group

$SO(2)^{\prime}={\rm diag}\Big{(}SO(2)_{E}\times SO(2)_{R_{1}}\Big{)}.$ (1)

The action of continuum ${\cal N}=(2,2)$ SYM takes the following ${\cal
Q}$-exact form after twisting

$S=\frac{N}{2\lambda}{\cal Q}\int d^{2}x\,\Psi,$ (2)

where

$\Psi={\rm Tr\;}\left(\chi_{ab}{\cal F}_{ab}+\eta[{\overline{\cal
D}}_{a},{\cal D}_{b}]-\frac{1}{2}\eta d\right),$ (3)

and $\lambda=g^{2}N$ is the ’t Hooft coupling. We use an anti-hermitian basis
for the generators of the gauge group with ${\rm
Tr\;}(T_{a}T_{b})=-\delta_{ab}$.

The four degrees of freedom appearing in the above action are just the twisted
fermions $(\eta,\psi_{a},\chi_{ab})$ and a complexified gauge field ${\cal
A}_{a}$. The complexified field is constructed from the usual gauge field
$A_{a}$ and the two scalars $B_{a}$ present in the untwisted theory: ${\cal
A}_{a}=A_{a}+iB_{a}$. The twisted theory is naturally written in terms of the
complexified covariant derivatives

${\cal D}_{a}=\partial_{a}+{\cal A}_{a},\quad\quad{\overline{\cal
D}}_{a}=\partial_{a}+{\overline{\cal A}}_{a},$ (4)

and complexified field strengths

${\cal F}_{ab}=[{\cal D}_{a},{\cal D}_{b}],\quad\quad{\overline{\cal
F}}_{ab}=[{\overline{\cal D}}_{a},{\overline{\cal D}}_{b}].$ (5)

The nilpotent supersymmetry transformations associated with the scalar
supercharge ${\cal Q}$ are given by

$\displaystyle{\cal Q}\;{\cal A}_{a}$ $\displaystyle=$
$\displaystyle\psi_{a},$ $\displaystyle{\cal Q}\;\psi_{a}$ $\displaystyle=$
$\displaystyle 0,$ $\displaystyle{\cal Q}\;{\overline{\cal A}}_{a}$
$\displaystyle=$ $\displaystyle 0,$ $\displaystyle{\cal Q}\;\chi_{ab}$
$\displaystyle=$ $\displaystyle-{\overline{\cal F}}_{ab},$ $\displaystyle{\cal
Q}\;\eta$ $\displaystyle=$ $\displaystyle d,$ $\displaystyle{\cal Q}\;d$
$\displaystyle=$ $\displaystyle 0.$ (6)

Performing the ${\cal Q}$-variation on $\Psi$ and integrating out the
auxiliary field $d$ yields

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{N}{2\lambda}\int{\rm
Tr\;}\bigg{(}-{\overline{\cal F}}_{ab}{\cal
F}_{ab}+\frac{1}{2}[{\overline{\cal D}}_{a},{\cal D}_{a}]^{2}$ (7)
$\displaystyle-$ $\displaystyle\chi_{ab}{\cal
D}_{\left[a\right.}\psi_{\left.b\right]}-\eta{\overline{\cal
D}}_{a}\psi_{a}\bigg{)}.$

The prescription for discretization is straightforward. The complexified gauge
fields are mapped to complexified Wilson links

${\cal A}_{a}(x)\rightarrow{\cal U}_{a}({\bf n}),$ (8)

living on the links of a square lattice with integer-valued basis vectors
along two directions,

$\widehat{\bm{\mu}}_{1}=(1,0),\quad\quad\widehat{\bm{\mu}}_{2}=(0,1).$ (9)

They transform in the appropriate way under $U(N)$ lattice gauge
transformations

${\cal U}_{a}({\bf n})\to G({\bf n}){\cal U}_{a}({\bf n})G^{\dagger}({\bf
n}+\widehat{\bm{\mu}}_{a}).$ (10)

Supersymmetry invariance then implies that $\psi_{a}({\bf n})$ live on the
same links and transform identically. The scalar fermion $\eta({\bf n})$ is
associated with a site and transforms the following way under gauge
transformations

$\eta({\bf n})\to G({\bf n})\eta({\bf n})G^{\dagger}({\bf n}).$ (11)

The field $\chi_{ab}({\bf n})$, as a 2-form, should be associated with a
plaquette. In practice, we introduce diagonal links running through the center
of the plaquette and choose $\chi_{ab}({\bf n})$ to lie with opposite
orientation along those diagonal links. This orientation ensures gauge
invariance. Fig. (1) shows the unit cell of the lattice theory with field
orientation assignments.

Figure 1: The unit cell and field orientations of the two-dimensional ${\cal
N}=(2,2)$ lattice SYM theory.

The continuum covariant derivatives are replaced by covariant difference
operators and they act on the twisted fields the following way

$\displaystyle{\overline{\cal D}}^{(-)}_{a}f_{a}({\bf n})$ $\displaystyle=$
$\displaystyle f_{a}({\bf n}){\overline{\cal U}}_{a}({\bf n})-{\overline{\cal
U}}_{a}({\bf n}-\widehat{\bm{\mu}}_{a})f_{a}({\bf n}-\widehat{\bm{\mu}}_{a}),$
$\displaystyle{\cal D}^{(+)}_{a}f_{b}({\bf n})$ $\displaystyle=$
$\displaystyle{\cal U}_{a}({\bf n})f_{b}({\bf
n}+\widehat{\bm{\mu}}_{a})-f_{b}({\bf n}){\cal U}_{a}({\bf
n}+\widehat{\bm{\mu}}_{b}).$

The lattice field strength is given by ${\cal F}_{ab}({\bf n})={\cal
D}^{(+)}_{a}{\cal U}_{b}({\bf n})$, and is anti-symmetric. It transforms like
a lattice 2-form and yields a gauge invariant loop on the lattice when
contracted with $\chi_{ab}({\bf n})$. Similarly, the term involving the
covariant backward difference operator, ${\overline{\cal D}}^{(-)}_{a}{\cal
U}_{a}({\bf n})$, transforms as a 0-form or site field and hence can be
contracted with the site field $\eta({\bf n})$ to yield a gauge invariant
expression.

The lattice action is ${\cal Q}$-exact

$\displaystyle S$ $\displaystyle=$ $\displaystyle\frac{N}{2\lambda}\sum_{{\bf
n}}{\rm Tr\;}{\cal Q}\Big{(}\chi_{ab}({\bf n}){\cal D}_{a}^{(+)}{\cal
U}_{b}({\bf n})$ (12) $\displaystyle+$ $\displaystyle\eta({\bf
n}){\overline{\cal D}}_{a}^{(-)}{\cal U}_{a}({\bf n})-\frac{1}{2}\eta({\bf
n})d({\bf n})\Big{)}.$

Applying the ${\cal Q}$ transformation on the lattice fields and integrating
out the auxiliary field $d$, we obtain the gauge invariant and ${\cal Q}$
supersymmetric lattice action

$\displaystyle S$ $\displaystyle=$ $\displaystyle S_{B}+S_{F},$ (13)

where the bosonic action is

$\displaystyle S_{B}$ $\displaystyle=$
$\displaystyle\frac{N}{2\lambda}\sum_{{\bf n}}{\rm Tr\;}\Big{(}{\cal
F}_{ab}^{\dagger}({\bf n}){\cal F}_{ab}({\bf
n})+\frac{1}{2}\Big{(}{\overline{\cal D}}_{a}^{(-)}{\cal U}_{a}({\bf
n})\Big{)}^{2}\Big{)},$

and the fermionic piece

$\displaystyle S_{F}$ $\displaystyle=$
$\displaystyle\frac{N}{2\lambda}\sum_{{\bf n}}{\rm Tr\;}\Big{(}-\chi_{ab}({\bf
n}){\cal D}^{(+)}_{[a}\psi_{b]}({\bf n})-\eta({\bf n}){\overline{\cal
D}}^{(-)}_{a}\psi_{a}({\bf n})\Big{)}.$

It was correctly noted in Ref. Kanamori:2008bk  that for simulation purposes,
we need to add a small supersymmetry breaking scalar potential to stabilize
the $SU(N)$ flat directions of the theory. We add a single trace deformation
term to the action in Eq. (13) as,

$S_{\rm soft}=\frac{N}{2\lambda}\mu^{2}\sum_{{\bf n},\ a}{\rm
Tr\;}\bigg{(}{\overline{\cal U}}_{a}({\bf n}){\cal U}_{a}({\bf
n})-\mathbb{I}_{N}\bigg{)}^{2},$ (14)

with a tunable parameter $\mu$. Exact supersymmetry at $\mu=0$ ensures that
all ${\cal Q}$-breaking terms vanish as some (positive) power of $\mu$.

##  III Lattice Simulations

We simulate the theory on a square lattice with anti-periodic boundary
conditions (aPBC) for fermions in the temporal direction. The physical size of
the lattice is $\beta\times L$, where $\beta$ is the dimensionful temporal
extent and $L$ the dimensionful spatial extent. We denote the lattice spacing
as $a$ while $N_{t}$ is the number of lattice sites along the temporal
direction and $N_{x}$ number of sites along the spatial direction. Thus the
dimensionful quantities are $\beta=aN_{t}$ and $L=aN_{x}$. In our case the
lattice is symmetric: $N_{t}=N_{x}$.

In two dimensions, the ’t Hooft coupling $\lambda$ is dimensionful and we can
construct the dimensionless temporal circle size,

$r_{\tau}=\sqrt{\lambda}\beta.$ (15)

The quantity $r_{\tau}$ also serves as the effective coupling. Its inverse is
the dimensionless temperature $t$. Since we have only considered symmetric
lattices, the spatial circle size is the same as the temporal circle size,
$r_{x}=r_{\tau}$. As discussed above we use a small mass parameter
$\mu=\zeta\frac{r_{\tau}}{N_{t}}=\zeta\sqrt{\lambda}a$ to regulate potential
divergences associated with the flat directions. As for case of sixteen
supercharge theory in two dimensions Catterall:2017lub ; Jha:2017zad , we
extrapolate all our results to $\mu=0$.

To examine the question of supersymmetry breaking we consider the system at
non-zero temperature and subsequently take the temperature to zero after
taking the limits $\zeta\to 0$ and $a\to 0$. A non-zero value of the vacuum
energy would indicate supersymmetry breaking. Notice that if supersymmetry is
intact in a finite volume, it is unbroken even in infinite volume
Witten:1982df .

We compute the ground state energy density in two-dimensional ${\cal N}=(2,2)$
SYM using the publicly available code presented in Ref. Schaich:2014pda . In
the four-supercharge case, the expression for the _effective_ bosonic action,
which is related to the dimensionless energy density we measure, was first
given in Ref. Catterall:2008dv .

We can have two different definitions for the ground state energy based on
whether we take the massless (scalar mass) limit followed by the continuum
limit or vice versa. In both cases, the zero temperature limit is taken at the
end. Thus, we have

$\frac{\mathcal{E}_{\mathrm{VAC}}^{\prime}}{N^{2}\lambda}=\lim_{\beta\to\infty}\leavevmode\nobreak\
\leavevmode\nobreak\ \lim_{\mathrm{a}\to 0}\leavevmode\nobreak\
\leavevmode\nobreak\ \lim_{\mu\to
0}\Bigg{\langle}\mathrm{VAC}\Bigg{|}\left(\frac{-2{\bar{S}}}{N^{2}\lambda}\right)\Bigg{|}\mathrm{VAC}\Bigg{\rangle},$
(16)

and

$\frac{\mathcal{E}_{\mathrm{VAC}}}{N^{2}\lambda}=\lim_{\beta\to\infty}\leavevmode\nobreak\
\leavevmode\nobreak\ \lim_{\mu\to 0}\leavevmode\nobreak\ \leavevmode\nobreak\
\lim_{\mathrm{a}\to
0}\Bigg{\langle}\mathrm{VAC}\Bigg{|}\left(\frac{-2{\bar{S}}}{N^{2}\lambda}\right)\Bigg{|}\mathrm{VAC}\Bigg{\rangle},$
(17)

where,

${\bar{S}}=\frac{1}{L\beta}\left(S_{B}-\frac{3}{2}N^{2}N_{x}N_{t}\right).$
(18)

Figure 2: The $\zeta^{2}\to 0$ extrapolation of the ground state energy
density for $U(3)$, ${r_{\tau}}=9$. Figure 3: The $(a/L)^{2}\to 0$
extrapolation of the ground state energy density for $U(3)$, ${r_{\tau}}=9$.

We provide the simulation data in Tables (2) and (3). It is clear from the
tables that the order of taking these _different_ limits is consistent within
errors and we will quote results only for
$\frac{\mathcal{E}_{\mathrm{VAC}}}{N^{2}\lambda}$.

Figure 4: Pfaffian phase fluctuations, $1-\left\langle\cos\phi\right\rangle$,
for some U(3) ensembles used in this work. We have measured the phase for
three couplings used in this work. We keep the mass parameter, $\zeta=0.50$
for all couplings. Note that at sufficiently weak couplings, large lattices
are not needed to control sign problem.

We integrate out the fermions to produce a Pfaffian, which in turn is
represented by square root of a determinant. The fermion determinant with a
fractional power can be simulated using Rational Hybrid Monte Carlo (RHMC)
algorithm Clark:2004cp . In the simulations we used the absolute value of the
Pfaffian. The phase of the Pfaffian may be incorporated back in the
expectation values of observables by re-weighting although as will be seen in
the next section the measured Pfaffian phase is always small in our
simulations.

###  III.1 Phase of the Pfaffian

The phase of the Pfaffian was studied in Ref. Kanamori:2007ye  for two
different lattice constructions. Soon after, the phase of Pfaffian for the
construction we use here was calculated in Ref. Catterall:2011aa  and it was
observed that it vanishes as one approaches the continuum limit. It was
correctly noted in Ref. Hanada:2010qg  that the absence of the sign is a
property of the correct continuum limit. In this paper, we will study the
phase of the Pfaffian at stronger couplings than have been explored before and
on much larger lattices using the parallel code developed in Ref.
Schaich:2014pda . We show that the phase fluctuations become small and vanish
as we take the continuum limit. This is true for all couplings we have
considered. However, on a fixed lattice volume, the magnitude of the phase
fluctuations grows with the coupling. This implies that accessing stronger
couplings ($t\leq 1/9$) requires the use of larger lattices if we are to avoid
a sign problem. We show these results in Fig. 4.

###  III.2 Ground State Energy

We now present our simulation results on the ground state energy of the
theory. We would like to extrapolate the lattice data for ground state energy
density $\frac{\mathcal{E}_{\mathrm{VAC}}}{N^{2}\lambda}$ to zero temperature
after taking the continuum ($a\to 0$) and massless ($\mu\to 0$) limits. A
representative example of the mass extrapolations and continuum extrapolations
are shown in Fig. 2 and Fig. 3, respectively. At the end, we perform three
types of extrapolations in temperature - using power law, exponential, and
constant fits.

We show the vacuum energy density vs inverse temperature for $U(2)$ in Fig. 5.
Extrapolating ${r_{\tau}}\to\infty$ using the range ${r_{\tau}}\in[6,9]$

$\frac{\mathcal{E}_{\mathrm{VAC}}}{N^{2}\lambda}=\left\\{\begin{array}[]{ll}0.06(4),\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{$\chi^{2}/\text{d.o.f.}$}=0.40&:\text{power law
fit}\\\ 0.06(2),\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{$\chi^{2}/\text{d.o.f.}$}=1.26&:\text{exponential fit}\\\
0.08(2),\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{$\chi^{2}/\text{d.o.f.}$}=0.63&:\text{constant fit}\\\
\end{array}\right.$ (19)

In Fig. 6 we show the vacuum energy density vs inverse temperature for gauge
group $U(3)$. Extrapolating ${r_{\tau}}\to\infty$ using the range
${r_{\tau}}\in[6,9]$

$\frac{\mathcal{E}_{\mathrm{VAC}}}{N^{2}\lambda}=\left\\{\begin{array}[]{ll}0.05(2),\leavevmode\nobreak\
\leavevmode\nobreak\ \mbox{$\chi^{2}/\text{d.o.f.}$}=0.11&:\text{power law
fit}\\\ 0.04(4),\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{$\chi^{2}/\text{d.o.f.}$}=0.11&:\text{exponential fit}\\\
0.05(2),\leavevmode\nobreak\ \leavevmode\nobreak\
\mbox{$\chi^{2}/\text{d.o.f.}$}=0.06&:\text{constant fit}\\\
\end{array}\right.$ (20)

Figure 5: The $\beta\to\infty$ extrapolation of the ground state energy for
$U(2)$ gauge group. The inset zooms in to show the low-temperature regime. For
details, see Table (2). Figure 6: The $\beta\to\infty$ extrapolation of the
ground state energy for $U(3)$ gauge group. The inset zooms in to show the
low-temperature regime. For details, see Table (3).

We note that the errors in our results do not allow us to make conclusive
statements about the exact form of the energy dependence on the temperature.
Both power, exponential and constant fitting functions yield comparable
results consistent with vanishing ground state energy. Our calculation puts an
upper bound on the dimensionless energy density using the constant fit at
$\frac{\mathcal{E}_{\mathrm{VAC}}}{N^{2}\lambda}=0.08(2)$ for U(2) and
$\frac{\mathcal{E}_{\mathrm{VAC}}}{N^{2}\lambda}=0.05(2)$ for U(3).

While this work was in progress results were presented on the tree-level
${\cal O}(a)$ improvement of the Sugino’s lattice action for two-dimensional
${\cal N}=(2,2)$ SYM Hanada:2017gqc . We note that our lattice formulation
already possesses this improvement which we see in Fig. 3 and in Table 1.

$\zeta$ |  $\propto(a/L)^{p}$ |  $\propto(a/L)^{p}+c$ |
---|---|---|---
0.40 | 1.86(9) | 1.76(22) |
0.50 | 1.76(6) | 1.60(15) |
0.55 | 1.79(5) | 1.90(11) |
0.60 | 1.74(4) | 1.70(11) |

$\zeta$ |  $\propto(a/L)^{p}$ |  $\propto(a/L)^{p}+c$ |
---|---|---|---
0.40 | 1.73(10) | 1.58(24) |
0.50 | 1.71(7) | 1.74(17) |
0.55 | 1.69(6) | 1.57(14) |
0.60 | 1.78(5) | 1.98(12) |

Table 1: Numerical results showing that our action is effectively ${\cal
O}(a)$ improved. We measure the deviation of the bosonic action/site from its
supersymmetric value of $\frac{3}{2}N^{2}$ and fit it to power law. The first
column shows the soft-mass parameter, $\zeta$, we use to regulate the flat
directions. The second column is the obtained value of the power, $p$,
constraining vanishing intercept, the third is the obtained value of the
power, $p$, _without_ constraining the intercept. We quote results from one of
the couplings used in this work, ${r_{\tau}}$ = 6. On the top, we show the
results with $U(3)$ and with $U(2)$ at the bottom. The fits are very good with
maximum $\chi^{2}/\text{d.o.f.}$= 2.80.

##  IV Conclusions

In this paper we have examined the possibility of dynamical supersymmetry
breaking in two-dimensional ${\cal N}=(2,2)$ SYM through lattice simulations.
The lattice theory is exact supersymmetric, gauge invariant, local, and
doubler free. We find an upper bound on the vacuum energy density of
$\frac{\mathcal{E}_{\mathrm{VAC}}}{N^{2}\lambda}=0.08(2)$ and
$\frac{\mathcal{E}_{\mathrm{VAC}}}{N^{2}\lambda}=0.05(2)$ for U(2) and U(3)
respectively. The energy density is statistically consistent with zero and
hence with the absence of dynamical supersymmetry breaking. It would be
interesting to examine the spectrum in future work to confirm the absence of
spontaneous supersymmetry breaking perhaps by searching for signals of a
Goldstino as was done in Catterall:2015tta . We have also measured the phase
of the Pfaffian on all our ensembles and find that while the average phase
grows with coupling it decreases as we take the continuum limit in agreement
with theoretical expectations. In practice, it is numerically small for all
our ensembles. The question of supersymmetry breaking in this model was
addressed before in Kanamori:2009dk . Our current work, in addition to using a
different lattice action, has employed stronger couplings (and hence lower
temperatures) and much smaller lattice spacings. For example, the lowest
temperature used in the earlier work was $t=1/6$ as compared to $t=1/9$ in
this work while the largest lattice used here is $96\times 96$ as compared to
$30\times 12$ in the earlier study.

## ACKNOWLEDGEMENTS

SC and RGJ were supported by the US Department of Energy (DOE), Office of
Science, Office of High Energy Physics, under Award Number DE-SC0009998. AJ
gratefully acknowledges support from the International Centre for Theoretical
Sciences (ICTS-TIFR), the Infosys Foundation and the Indo-French Centre for
the Promotion of Advanced Research (IFCPAR/CEFIPRA). Numerical calculations
were carried out on the DOE-funded USQCD facilities at Fermilab, and on the
Mowgli cluster at ICTS-TIFR. AJ thanks Srinivasa R. and Mohammad Irshad for
providing timely technical support with the usage of the computing facility at
ICTS-TIFR. RGJ would like to thank Issaku Kanamori for discussions during a
workshop at YITP, Kyoto in April 2017.

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##  Appendix A Data Tables for $U(2)$ and $U(3)$

${r_{\tau}}$ | $N_{x}\times N_{t}$ | $\left.-{\bar{S}}/N^{2}\lambda\right|_{\zeta=0.4}$ | $\left.-{\bar{S}}/N^{2}\lambda\right|_{\zeta=0.5}$ | $\left.-{\bar{S}}/N^{2}\lambda\right|_{\zeta=0.55}$ | $\left.-{\bar{S}}/N^{2}\lambda\right|_{\zeta=0.6}$ |
---|---|---|---|---|---|---
1.0 | $24\times 24$ | 1.14(33) | — | — | — |
2.0 | $24\times 24$ | 0.53(10) | — | — | — |
4.0 | $24\times 24$ | 0.253(21) | 0.332(23) | 0.440(20) | 0.502(21) |
| $32\times 32$ | 0.272(34) | 0.355(35) | 0.407(35) | 0.501(33) |
| $48\times 48$ | 0.354(43) | 0.378(48) | 0.531(45) | 0.538(45) |
| $96\times 96$ | 0.26(10) | 0.40(10) | 0.48(10) | 0.63(10) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$ , then $\lim_{\mu\to 0}=0.151(65)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.148(65)$ |  |
6.0 | $24\times 24$ | 0.20(1) | 0.30(1) | 0.35(1) | 0.43(1) |
| $32\times 32$ | 0.22(1) | 0.32(2) | 0.39(2) | 0.44(1) |
| $48\times 48$ | 0.25(2) | 0.37(2) | 0.44(2) | 0.48(2) |
| $96\times 96$ | 0.27(5) | 0.45(5) | 0.48(2) | 0.62(5) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.083(37)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.079(38)$ |  |
7.0 | $24\times 24$ | 0.20(1) | 0.28(1) | 0.33(1) | 0.38(1) |
| $32\times 32$ | 0.22(1) | 0.30(1) | 0.38(1) | 0.43(1) |
| $48\times 48$ | 0.25(1) | 0.34(2) | 0.41(2) | 0.47(2) |
| $96\times 96$ | 0.27(4) | 0.38(3) | 0.45(3) | 0.56(4) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.056(28)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.055(28)$ |  |
7.5 | $24\times 24$ | 0.19(1) | 0.29(1) | 0.33(5) | 0.38(1) |
| $32\times 32$ | 0.19(1) | 0.31(1) | 0.36(1) | 0.42(1) |
| $48\times 48$ | 0.23(1) | 0.34(2) | 0.39(1) | 0.47(2) |
| $96\times 96$ | 0.24(3) | 0.40(3) | 0.45(3) | 0.48(3) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.035(23)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.033(24)$ |  |
8.0 | $24\times 24$ | 0.20(1) | 0.28(1) | 0.33(1) | 0.36(1) |
| $32\times 32$ | 0.20(1) | 0.29(1) | 0.35(1) | 0.40(1) |
| $48\times 48$ | 0.22(1) | 0.34(1) | 0.37(1) | 0.44(1) |
| $96\times 96$ | 0.27(2) | 0.36(3) | 0.44(3) | 0.54(3) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.023(19)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.022(20)$ |  |
9.0 | $24\times 24$ | 0.184(5) | 0.260(4) | 0.304(4) | 0.361(4) |
| $32\times 32$ | 0.20(1) | 0.30(1) | 0.35(1) | 0.39(1) |
| $48\times 48$ | 0.22(1) | 0.32(1) | 0.36(1) | 0.45(1) |
| $96\times 96$ | 0.254(22) | 0.364(17) | 0.435(20) | 0.534(22) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.040(17)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.039(17)$ |  |

Table 2:  The action density, $-{\bar{S}}/N^{2}\lambda$, which is related to
the ground state energy density using Eq. (16) or Eq. (17) with gauge group
$U(2)$ for different lattices, mass parameters and couplings used in this
work. The results are obtained through blocked jackknife analyses. We have
considered at least 5000 thermalized molecular dynamics time units (MDTU) in
each case.

${r_{\tau}}$ | $N_{x}\times N_{t}$ | $\left.-{\bar{S}}/N^{2}\lambda\right|_{\zeta=0.4}$ | $\left.-{\bar{S}}/N^{2}\lambda\right|_{\zeta=0.5}$ | $\left.-{\bar{S}}/N^{2}\lambda\right|_{\zeta=0.55}$ | $\left.-{\bar{S}}/N^{2}\lambda\right|_{\zeta=0.6}$ |
---|---|---|---|---|---|---
1.0 | $24\times 24$ | 1.05(27) | — | — | — |
2.0 | $24\times 24$ | 0.50(9) | — | — | — |
4.0 | $24\times 24$ | 0.228(17) | 0.329(18) | 0.405(19) | 0.443(18) |
| $32\times 32$ | 0.235(26) | 0.370(27) | 0.445(22) | 0.512(25) |
| $48\times 48$ | 0.302(41) | 0.460(37) | 0.484(37) | 0.531(32) |
| $96\times 96$ | 0.164(60) | 0.423(78) | 0.665(78) | 0.631(78) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$ , then $\lim_{\mu\to 0}=0.097(77)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.074(54)$ |  |
6.0 | $24\times 24$ | 0.205(8) | 0.297(10) | 0.363(8) | 0.412(8) |
| $32\times 32$ | 0.228(11) | 0.332(10) | 0.391(11) | 0.428(12) |
| $48\times 48$ | 0.220(24) | 0.366(21) | 0.397(17) | 0.515(20) |
| $96\times 96$ | 0.246(35) | 0.399(35) | 0.511(35) | 0.580(31) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.028(34)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.034(34)$ |  |
7.0 | $24\times 24$ | 0.206(7) | 0.285(5) | 0.335(5) | 0.394(6) |
| $32\times 32$ | 0.195(9) | 0.309(9) | 0.351(10) | 0.406(11) |
| $48\times 48$ | 0.231(13) | 0.327(13) | 0.387(13) | 0.473(11) |
| $96\times 96$ | 0.262(28) | 0.463(34) | 0.440(25) | 0.550(28) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.020(24)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.026(24)$ |  |
7.5 | $24\times 24$ | 0.196(5) | 0.275(5) | 0.330(5) | 0.376(5) |
| $32\times 32$ | 0.214(7) | 0.299(9) | 0.358(8) | 0.418(9) |
| $48\times 48$ | 0.23(1) | 0.33(1) | 0.40(1) | 0.46(1) |
| $96\times 96$ | 0.24(2) | 0.39(2) | 0.49(2) | 0.55(3) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.031(19)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.030(20)$ |  |
8.0 | $24\times 24$ | 0.192(5) | 0.273(5) | 0.324(5) | 0.367(3) |
| $32\times 32$ | 0.210(6) | 0.300(6) | 0.347(6) | 0.413(7) |
| $48\times 48$ | 0.212(9) | 0.340(9) | 0.408(13) | 0.463(7) |
| $96\times 96$ | 0.261(15) | 0.387(19) | 0.471(17) | 0.529(15) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.025(15)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.026(16)$ |  |
9.0 | $24\times 24$ | 0.182(4) | 0.259(4) | 0.312(3) | 0.351(4) |
| $32\times 32$ | 0.199(6) | 0.288(6) | 0.331(6) | 0.394(5) |
| $48\times 48$ | 0.207(7) | 0.314(7) | 0.380(7) | 0.446(9) |
| $96\times 96$ | 0.246(17) | 0.358(12) | 0.444(17) | 0.521(17) |
|  |  |  |  $\lim_{\mathrm{a}\to 0}$, then $\lim_{\mu\to 0}=0.021(14)$ |  |
|  |  |  |  $\lim_{\mu\to 0}$ , then $\lim_{\mathrm{a}\to 0}=0.016(14)$ |  |

Table 3:  The action density, $-{\bar{S}}/N^{2}\lambda$, which is related to
the ground state energy density using Eq. (16) or Eq. (17) with gauge group
$U(3)$ for different lattices, mass parameters and couplings used in this
work. The results are obtained through blocked jackknife analyses. We have
considered at least 4000 thermalized molecular dynamics time units (MDTU) in
each case.


# Light nuclei photoproduction in relativistic heavy ion ultraperipheral
collisions

Jin-Yu Hu<EMAIL_ADDRESS>Department of Modern Physics,
University of Science and Technology of China, Anhui 230026, China  Shuo Lin
<EMAIL_ADDRESS>Department of Modern Physics, University of Science
and Technology of China, Anhui 230026, China  Shi Pu<EMAIL_ADDRESS>Department of Modern Physics, University of Science and Technology of China,
Anhui 230026, China  Southern Center for Nuclear-Science Theory (SCNT),
Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000,
Guangdong Province, China  Qun Wang<EMAIL_ADDRESS>Department of Modern
Physics, University of Science and Technology of China, Anhui 230026, China
School of Mechanics and Physics, Anhui University of Science and Technology,
Huainan,Anhui 232001, China

###### Abstract

We have investigated light nuclei pair photoproduction in relativistic heavy
ion ultraperipheral collisions. As a first attempt, we employ our previously
developed quantum electrodynamics model, which incorporates a wave-packet
description of initial nuclei, to compute the cross section for proton-
antiproton pair photoproduction. The effective vertex for the photon and
proton interaction is chosen based on studies of two-photon exchange effects
in hadron physics. We present the transverse momentum, invariant mass, and
azimuthal angle distributions of proton-antiproton pairs at
$\sqrt{s_{NN}}=200$ GeV in Au+Au ultraperipheral collisions. We observe a
$\cos(2\phi)$ modulation and an almost negligible $\cos(4\phi)$ modulation in
the azimuthal angle distribution. Our studies helps us better understand the
matter generated by light.

##  I Introduction

In relativistic heavy ion collisions, two heavy nuclei are accelerated to near
the speed of light, collide, and generate extremely strong electromagnetic
fields on the order of $10^{17}-10^{18}$ Gauss (Bzdak:2011yy, ; Deng:2012pc, ;
Roy:2015coa, ). Although these fields decay rapidly in a vacuum, they can
persist for a long time after interacting with the medium generated by the
collisions (Roy:2015kma, ; Pu:2016ayh, ; Pu:2016bxy, ; Siddique:2019gqh, ;
Inghirami:2016iru, ; Zhang:2022lje, ). Such strong fields provide a novel
platform to study quantum transport phenomena (see Refs. (Kharzeev:2020jxw, ;
Gao:2020vbh, ; Hidaka:2022dmn, ) for recent reviews), the Schwinger mechanism
(Schwinger:1951nm, ; Copinger:2018ftr, ; Copinger:2020nyx, ; Copinger:2022jgg,
), and photon-nuclear and photon-photon interactions. In the early 20th
century, Fermi (Fermi:1924tc, ) first suggested, and Williams
(Williams:1934ad, ) and Weizs$\ddot{a}$cker (vonWeizsacker:1934nji, ) later
improved, the idea that strong electromagnetic fields can be regarded as a
flux of quasi-real photons. In relativistic heavy ion collisions, photon
fluxes are significantly enhanced due to the large nuclear charge number.
Additionally, the virtuality of photons is highly suppressed by the Lorentz
factor, meaning the photons generated by the moving nuclei can be considered
real. Therefore, relativistic heavy ion collisions offer a new opportunity to
investigate photon-nuclear and photon-photon interactions.

Photon-nuclear interactions in high-energy scenarios, known as the
photoproduction of vector mesons, have been recently measured in relativistic
heavy ion ultraperipheral collisions (UPCs), e.g. see the measurements from
RHIC-STAR (STAR:2019wlg, ; STAR:2021wwq, ; STAR:2022wfe, ) and LHC-ALICE
experiments (ALICE:2019tqa, ; ALICE:2020ugp, ; ALICE:2021gpt, ). In the
photoproduction of vector mesons, either nucleus can emit a photon while the
other acts as a target, and vice versa. These two possibilities combine at the
amplitude level and produce an interference effect, akin to a high-energy
version of the double-slit interference phenomenon. This interference
phenomenon was recently observed in UPCs (STAR:2022wfe, ) as a significant
breakthrough. Theoretical studies have discussed this phenomenon using the
color glass condensate effective theory (Xing:2020hwh, ; Hagiwara:2020juc, ;
Hagiwara:2021xkf, ) and the vector meson dominance model (Zha:2020cst, ). This
phenomenon has also been found to detect nuclear deformation effects in
isobaric collisions (Mantysaari:2023prg, ; Lin:2024mnj, ). Another interesting
photon-nuclear interaction is the photoproduction of di-jets, which has been
measured at LHC-CMS (CMS:2022lbi, ) and discussed using Wigner distributions
(Hagiwara:2017fye, ) or diffractive transverse momentum-dependent parton
distribution functions of gluons (Iancu:2023lel, ).

The two key processes for photon-photon interactions in UPCs are light-by-
light scattering, measured by LHC-ATLAS (ATLAS:2017fur, ), and lepton pair
photoproduction, measured by RHIC-STAR (STAR:2018ldd, ; STAR:2019wlg, ), LHC-
ATLAS (ATLAS:2018pfw, ) and LHC-ALICE experiments (ALICE:2022hvk, ). Since the
photon generated by the moving nuclei is nearly real, as mentioned, lepton
pair photoproduction in relativistic heavy ion collisions can also be
considered as the Breit-Wheeler processes (Breit:1934zz, ). Although early
pioneering works proposed the equivalent photon approximation (Klein:2016yzr,
; Bertulani:1987tz, ; Bertulani:2005ru, ; Baltz:2007kq, ), it has been found
inadequate to accurately describe lepton pair photoproduction in UPCs due to
its lack of dependence on impact parameters and the transverse momentum of the
initial photons. Several theoretical methods have been developed to address
photon-photon interactions in UPCs, including the generalized equivalent
photon approximation and quantum electrodynamics (QED) in the background field
approach (Wang:2022ihj, ; Vidovic:1992ik, ; Hencken:1994my, ; Hencken:2004td,
; Zha:2018tlq, ; Zha:2018ywo, ; Brandenburg:2020ozx, ; Brandenburg:2021lnj, ;
Li:2019sin, ; Luo:2023syp, ), methods based on the factorization theorem
(Klein:2018fmp, ; Klein:2020jom, ; Li:2019yzy, ; Xiao:2020ddm, ; Shi:2024gex,
), and QED models incorporating a wave-packet description of nuclei
(Wang:2021kxm, ; Wang:2022gkd, ; Lin:2022flv, ). Higher-order effects, such as
Coulomb corrections (Ivanov:1998ka, ; Eichmann:1998eh, ; Segev:1997yz, ;
Baltz:1998zb, ; Baltz:2001dp, ; Baltz:2003dy, ; Baltz:2009jk, ; Zha:2021jhf, ;
Klein:2020jom, ; Sun:2020ygb, ) and Sudakov effects (Li:2019sin, ; Li:2019yzy,
; Klein:2018fmp, ; Klein:2020jom, ; Shao:2023zge, ; Shao:2022stc, ), have also
been considered. For further discussion, see the recent review (Shi:2023nko, )
and references therein.

The successful discovery of the Breit-Wheeler process in UPCs naturally raises
the question: could we observe the photoproduction of other particle-
antiparticle pairs in UPCs? In this work, we study light nuclei pair
photoproduction in UPCs using a QED model that incorporates a wave-packet
description of nuclei (Wang:2021kxm, ; Wang:2022gkd, ; Lin:2022flv, ). As a
first attempt, we compute the cross section for proton-antiproton ($p\bar{p}$)
pairs, the lightest nuclei, at the Born level. The effective photon-proton
vertex is chosen following Refs. (Guttmann:2011ze, ; Zhou:2009xb, ). We
present results for the transverse momentum, invariant mass, and azimuthal
angle distributions of $p\bar{p}$ pairs. Recently, we also notice the related
work on $\gamma+\gamma\rightarrow p+\bar{p}$ through a different schemes
(Zhang:2024mql, ) and $\gamma+\gamma\rightarrow\pi^{+}+\pi^{-}$ in
$e^{+}+e^{-}$ collisions (Jia:2024xzx, ).

The paper is organized as follows. In Sec. II, we briefly review our
theoretical formula for the differential cross section in UPCs. In Sec. III,
we present the transverse momentum, invariant mass, and azimuthal angle
distributions for $p\bar{p}$ pairs photoproduction at $\sqrt{s_{NN}}=$200 GeV
in UPCs. We summarize our work and discuss the results in Sec. IV.

Figure 1: The Feynman diagram for the particle and anti-particle pairs in two
photon fusion processes at Born level.

Figure 2: Differential cross sections as functions of (a) transverse momentum
$\mathbf{P}_{{\rm T}}^{p\overline{p}}$, (b) invariant mass $M^{p\bar{p}}$ for
$p\bar{p}$ pairs. Different lines represent results computed with different
$\mathcal{P}(b_{T})$. The black solid lines use $\mathcal{P}(b_{T})$ from Eq.
(10) with $w$ from Eq. (12). The orange dashed lines use $\mathcal{P}(b_{T})$
from Eq. (11) with $w$ from Eq. (12). The blue dotted lines use
$\mathcal{P}(b_{T})$ from Eq. (11) with $w$ from Eq. (12).

##  II Theoretical framework

We first review the theoretical framework for calculating the differential
cross section of particle and antiparticle pair photoproduction. The framework
follows Refs. (Wang:2021kxm, ; Wang:2022gkd, ), in which some of us derived
the differential cross section of lepton pair photoproduction using the
classical field approximation with a wave packet description of nuclei.
Consider two identical nuclei $A_{1}$ and $A_{2}$ move in $\pm z$ direction
with the velocity $u_{1,2}^{\mu}=\gamma(1,0,0,\pm v)$, where
$\gamma=1/\sqrt{1-v^{2}}$ is the Lorentz factor. We can use a wave packet
description for the initial nuclei. The sub-process involves two photons from
the colliding nuclei producing a particle-antiparticle pair, described by

$\gamma(p_{1})+\gamma(p_{2})\rightarrow H(k_{1})+\overline{H}(k_{2}),$ (1)

where $p_{1}^{\mu}$ and $p_{2}^{\mu}$ are four-momenta of photons, and
$k_{1}^{\mu}=(E_{k1},\mathbf{k}_{1})$ and
$k_{2}^{\mu}=(E_{k2},\mathbf{k}_{2})$ are on-shell four-momenta for particle
$H$ and anti-particle $\overline{H}$, respectively. The Born-level total cross
section can be expressed in a compact form (Wang:2021kxm, ; Wang:2022gkd, ),

$\displaystyle\sigma$ $\displaystyle=$
$\displaystyle\frac{Z^{4}e^{4}}{2\gamma^{4}v^{3}}\int
d^{2}\mathbf{b}_{T}d^{2}\mathbf{b}_{1T}d^{2}\mathbf{b}_{2T}\delta^{(2)}(\mathbf{b}_{T}-\mathbf{b}_{1T}+\mathbf{b}_{2T})$
(2) $\displaystyle\times$
$\displaystyle\int\frac{d\omega_{1}d^{2}\mathbf{p}_{1T}}{(2\pi)^{3}}\frac{d\omega_{2}d^{2}\mathbf{p}_{2T}}{(2\pi)^{3}}$
$\displaystyle\times$
$\displaystyle\int\frac{d^{2}\mathbf{p}_{1T}^{\prime}}{(2\pi)^{2}}e^{-i\mathbf{b}_{1T}\cdot(\mathbf{p}_{1T}^{\prime}-\mathbf{p}_{1T})}\frac{F^{*}(-\overline{p}_{1}^{\prime
2})}{-\overline{p}_{1}^{\prime
2}}\frac{F(-\overline{p}_{1}^{2})}{-\overline{p}_{1}^{2}}$
$\displaystyle\times$
$\displaystyle\int\frac{d^{2}\mathbf{p}_{2T}^{\prime}}{(2\pi)^{2}}e^{-i\mathbf{b}_{2T}\cdot(\mathbf{p}_{2T}^{\prime}-\mathbf{p}_{2T})}\frac{F^{*}(-\overline{p}_{2}^{\prime
2})}{-\overline{p}_{2}^{\prime
2}}\frac{F(-\overline{p}_{2}^{2})}{-\overline{p}_{2}^{2}}$
$\displaystyle\times$
$\displaystyle\int\frac{d^{3}k_{1}}{(2\pi)^{3}2E_{k1}}\frac{d^{3}k_{2}}{(2\pi)^{3}2E_{k2}}(2\pi)^{4}\delta^{(4)}(\overline{p}_{1}+\overline{p}_{2}-k_{1}-k_{2})$
$\displaystyle\times$ $\displaystyle\sum_{\textrm{spin of
}l,\overline{l}}\left[u_{1\mu}u_{2\nu}L^{\mu\nu}(\overline{p}_{1},\overline{p}_{2};k_{1},k_{2})\right]$
$\displaystyle\times$
$\displaystyle\left[u_{1\sigma}u_{2\rho}L^{\sigma\rho*}(\overline{p}_{1}^{\prime},\overline{p}_{2}^{\prime};k_{1},k_{2})\right],$

where $Z$ is the proton number of the nuclei, $\mathbf{b}_{iT}$ is the
transverse position of the photon emission in nucleus $A_{i}$,
$\mathbf{b}_{T}$ is the impact parameter of colliding nuclei,
$\overline{p}_{i}$ and $\overline{p}_{i}^{\prime}$ are photon momenta in the
classical field approximation and defined as ($i=1,2$ correspond to $+$ and
$-$),

$\displaystyle\overline{p}_{i}^{\mu}$ $\displaystyle=$
$\displaystyle\left(\omega_{i},\boldsymbol{p}_{iT},\pm\frac{\omega_{i}}{v}\right),\quad\overline{p}_{i}^{\prime\mu}=\left(\omega_{i},\boldsymbol{p}_{iT}^{\prime},\pm\frac{\omega_{i}}{v}\right),$
(3)

respectively. The form factor $F(-p^{2})$ is the Fourier transform of the
nuclear charge density distribution. In this work, we choose the form factor
as in Ref. (Klein:2016yzr, )

$\displaystyle F(p)$ $\displaystyle=$
$\displaystyle\frac{4\pi\rho^{0}}{p^{3}}[\sin(pR_{A})-pR_{A}\cos(pR_{A})]\frac{1}{a^{2}p^{2}+1},$
(4)

where $a=0.7$ fm, $\rho^{0}=3A/(4\pi R_{A}^{3})$ is the normalization factor,
$A$ is the number of nucleons in the nucleus, $R_{A}=1.1A^{1/3}$ fm is the
nucleus radius.

The tensor $L^{\mu\nu}$ in Eq. (2) depends on the final states generated
through photoproduction. The typical Feynman diagrams for particle-
antiparticle pairs in two-photon fusion processes at the Born level are
presented in Fig. 1. If the particle $H$ is a fermion, then the effective
$L^{\mu\nu}$ can be written in a compact form,

$\displaystyle L^{\mu\nu}(p_{1},p_{2};k_{1},k_{2})$ $\displaystyle=$
$\displaystyle-\overline{u}(k_{1})[\Gamma^{\mu}S_{F}(k_{1},p_{1})\Gamma^{\nu}+\Gamma^{\nu}S_{F}(p_{1},k_{2})\Gamma^{\mu}]v(k_{2}),$

where $S_{F}$ is the effective propagator and $\Gamma^{\mu}$ is the effective
vertex, which will reduce to $\gamma^{\mu}$ in lepton pair photoproduction
(Wang:2021kxm, ; Wang:2022gkd, ).

Now, let us consider light nuclei photoproduction. As a first attempt, we
consider the photoproduction of the lightest nuclei and anti-nuclei pair,
i.e., $p\bar{p}$ pairs. Since light nuclei are not elementary particles, the
effective vertex $\Gamma^{\mu}$ and the effective propagator $S_{F}$ must be
chosen based on phenomenological models. The photon-proton interactions are
widely discussed in the context of two-photon exchange effects in
$e^{-}+p\rightarrow e^{-}+p$ and $e^{+}+e^{-}\rightarrow p+\overline{p}$
(Guttmann:2011ze, ; Zhou:2009xb, ; Blunden:2003sp, ; Guichon:2003qm, ), also
see recent review (Carlson:2007sp, ; Arrington:2011dn, ) and references
therein. Several methods are also proposed to detect the two two-photon
exchange effects in experiments (PANDA:2009yku, ; Nikolenko:2010zz, ;
Kohl:2011zz, ).

Therefore, here, we adopt the fermionic propagator $S_{F}(k,p)$ in the lowest
order of perturbative theory,

$-iS_{F}(k,p)=\frac{\gamma\cdot(k-p)+m_{p}}{(k-p)^{2}-m_{p}^{2}+i\varepsilon},$
(6)

with $m_{p}$ being the mass of proton and we follow Ref.(Blunden:2003sp, ) to
adopt,

$\displaystyle\Gamma^{\mu}(q)$ $\displaystyle=$ $\displaystyle
F_{1}(q^{2})\gamma^{\mu}+i\frac{F_{2}(q^{2})}{2m_{p}}\sigma^{\mu\nu}q_{\nu},$
(7)

where $\sigma^{\mu\nu}=i[\gamma^{\mu},\gamma^{\nu}]/2$ and $F_{1}$ and $F_{2}$
are related to the Sachs form factors $G_{E}$ and $G_{M}$,

$\displaystyle F_{1}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{G_{E}(q^{2})+\tau G_{M}(q^{2})}{1+\tau},$ $\displaystyle
F_{2}(q^{2})$ $\displaystyle=$
$\displaystyle\frac{G_{M}(q^{2})-G_{E}(q^{2})}{1+\tau},$ (8)

with $\tau=-q^{2}/4m_{p}^{2}$. The form factors $G_{E}$ and $G_{M}$ are
commonly parameterized as,

$G_{E}(q^{2})=\frac{G_{M}(q^{2})}{\mu_{p}}=-\frac{\Lambda^{2}}{q^{2}-\Lambda^{2}},$
(9)

with $\mu_{p}$ being the magnetic moment of proton and $\Lambda=0.84$ GeV
(Blunden:2003sp, ). In general, particle-antiparticle pairs can also be
generated through the $s$-channel. However, this contribution may strongly
depend on the effective coupling constant and is usually found to be smaller
than that of the current scheme (also see Ref. (Zhang:2024mql, ) for related
discussions).

In RHIC-STAR experiments, the UPC events are triggered by detecting the
neutrons emitted from nuclei-photon interactions. To compare with experimental
data, we further need to add an extra factor $\mathcal{P}^{2}(b_{\perp})$ in
the cross sections. The $\mathcal{P}(b_{\perp})$ is the probability of
emitting neutrons from the excited nucleus and can usually be parameterized as
(Bertulani:1987tz, ),

$\mathcal{P}(b_{T})=1-\exp(-w),$ (10)

where $w$ is a function of $b_{T}$. When $w$ is small, $\mathcal{P}(b_{T})$ in
Eq. (10) is also approximately given by (Hencken:2004td, ),

$\mathcal{P}(b_{\perp})\simeq w.$ (11)

One popular way to estimate $w$ is to use the Giant Dipole Resonance model
(Bertulani:1987tz, ; Baur:1998ay, ; Hencken:2004td, ; Baltz:2009jk, ),

$w=5.45\times 10^{-5}Z^{3}(A-Z)/(A^{2/3}b_{T}^{2}).$ (12)

Since $w$ in Eq. (12) only depends on $A,Z$ and $b_{T}$, the details of the
nuclear mass distribution or deformation effects are missing in this model.
Another way to derive $w$ is to use the equivalent photon approximation
(Brandenburg:2020ozx, ; Baltz:1998ex, ; Baltz:2002pp, ; Broz:2019kpl, ;
Baltz:2009jk, ; Pshenichnov:2001qd, ),

$w_{Xn}(b_{T})=\int d\omega n(\omega,b_{T})\sigma_{\gamma+A\rightarrow
A^{\prime}+Xn}(\omega),$ (13)

where the $X\geq 1$ stands for the number of neutrons emitted by a nucleus and
$n(\omega,b_{T})$ is the photon flux (Jackson:1998nia, ; Vidovic:1992ik, ;
Wang:2021kxm, ), and the photon-nucleus cross section
$\sigma_{\gamma+A\rightarrow A^{\prime}+Xn}$ is given by fixed-target
experiments (Veyssiere:1970ztg, ; Berman:1987zz, ; Baltz:1998ex, ). In this
work, we consider three kinds of $\mathcal{P}(b_{T})$, (i)
$\mathcal{P}(b_{T})$ from Eq. (10) with $w$ from Eq. (12), labeled as
“$\mathcal{P}(b_{T})=1-\exp(-w)$”, (ii) $\mathcal{P}(b_{T})$ from Eq. (11)
with $w$ from Eq. (12), labeled as “$\mathcal{P}(b_{T})=w$” , (iii)
$\mathcal{P}(b_{T})$ from Eq. (10) with $w$ from Eq. (13), labeled as “XnXn”.
These labels are used in the following figures.

Table 1: The total cross section for $p\bar{p}$ pair photoproduction with different choices of $\mathcal{P}(b_{T})$ at $\sqrt{s_{NN}}=200$ GeV and $54$ GeV in Au+Au UPCs. We have chosen $\mathbf{k}_{{\rm 1T}},\mathbf{k}_{{\rm 2T}}\geq 200$ MeV, $|y^{p\bar{p}}|\in[0.05,0.5]$ and $\eta^{p\bar{p}}\in[-0.9,-0.9]$. The errors from the numerical integration are estimated by ZMCintegral package. $\sqrt{s_{NN}}$ | $\mathcal{P}(b_{T})=1-e^{-w}$ | $\mathcal{P}(b_{T})=w$ |  $\mathcal{P}(b_{T})$ for XnXn
---|---|---|---
$200$ GeV | $4.5\pm 0.1\mu\text{b}$ | $6.3\pm 0.1\mu\text{b}$ | $15.7\pm 0.3\mu\text{b}$
$54$ GeV | $51\pm 3\text{nb}$ | $78\pm 4\text{nb}$ | $170\pm 1\text{nb}$

##  III Transverse momentum, invariant mass spectrum and azimuthal angle
distributions

In this section, we present numerical results for the transverse momentum,
invariant mass, and azimuthal angle distributions of $p\bar{p}$ pair
photoproduction in Au+Au UPCs. We define $\mathbf{P}_{{\rm
T}}^{p\overline{p}}=\mathbf{k}_{{\rm 1T}}+\mathbf{k}_{{\rm 2T}}$ as the
transverse momentum of the $p\bar{p}$ pair, where $\mathbf{k}_{{\rm 1T}}$ and
$\mathbf{k}_{{\rm 2T}}$ are the transverse momentum for $p$ and $\bar{p}$,
respectively, as introduced in Eq. (2). Additionally, we also define
$M^{p\overline{p}},\phi,$ and $y^{p\bar{p}}$ as the invariant mass of the
$p\bar{p}$ pair, the angle between $\mathbf{P}_{{\rm
T}}^{\mathrm{p\overline{p}}}$ and $\frac{1}{2}(\mathbf{k}_{{\rm
2T}}-\mathbf{k}_{{\rm 1T}})$, the rapidity of $p\bar{p}$ pair. We use
$\eta^{p(\bar{p})}$ for pseudo-rapidity of single $p$ or $\bar{p}$. We set
$\mathbf{k}_{{\rm 1T}},\mathbf{k}_{{\rm 2T}}\geq 200$ MeV,
$|y^{p\bar{p}}|\in[0.05,0.5]$ and $\eta^{p(\bar{p})}\in[-0.9,0.9]$. The high
dimensional integration in Eq. (2) is computed by ZMCintegral package
(Wu:2019tsf, ; Zhang:2019nhd, ).

We first present the total cross section for $p\bar{p}$ pairs photoproduction
at the $\sqrt{s_{NN}}=200$ GeV and $54$ GeV in Au+Au UPCs with the different
choice of $\mathcal{P}(b_{T})$ in Table 1. We observe that the total cross
section are consistently in the range of $\sim 4-10\mu\textrm{b}$ at
$\sqrt{s_{NN}}=200$ GeV and $\sim 50-170$ nb at $\sqrt{s_{NN}}=54$ GeV,
respectively.

The differential cross section as a function of transverse momentum
$\mathbf{P}_{{\rm T}}^{\mathrm{p\overline{p}}}$ for $p\bar{p}$ pairs is shown
in Fig. 2(a). We observe that the $d\sigma/d\mathbf{P}_{{\rm
T}}^{p\overline{p}}$ ranges from $0.1$ to $0.3$ $\textrm{mb}/\textrm{GeV}$,
with peaks around $0.035-0.040$ GeV. The differential cross section with
$\mathcal{P}(b_{T})$ from XnXn is slightly higher than those from the other
two forms of $\mathcal{P}(b_{T})$.

We also plot the invariant mass $M^{p\overline{p}}$ distribution of $p\bar{p}$
pairs in Fig. 2(b). Interestingly, the differential cross section grows
smoothly when $M^{p\overline{p}}<2.4$ GeV, which contrasts with the rapid rise
near the threshold seen in lepton pair photoproduction. This difference arises
from the non-trivial effective electromagnetic form factors for protons in Eq.
(7). For $M^{p\overline{p}}>2.4$ GeV, the differential cross section decreases
slowly. Therefore, further experimental measurements of $p\bar{p}$ pairs
photoproduction can provide additional constraints on the effective vertex
$\Gamma^{\mu}$.

Figure 3: Differential cross sections as a function of azimuthal angle $\phi$.
The different lines have the same meanings as in Fig. 2.

Finally, we plot the azimuthal angle distribution for $p\bar{p}$ pairs in Fig.
3. The azimuthal angle distribution for lepton pair production exhibits
$\cos(4\phi)$ and $\cos(2\phi)$ modulation due to the linear polarization of
initial photons (Li:2019sin, ; Li:2019yzy, ). We fit the differential cross
section $d\sigma/d\phi$ using the function
$\mathcal{A}_{0}+\mathcal{A}_{2}\cos(2\phi)+\mathcal{A}_{4}\cos(4\phi)$. The
coefficients $\mathcal{A}_{0},\mathcal{A}_{2}$ and $\mathcal{A}_{4}$ are shown
in Table 2. In previous studies of lepton pair photoproduction (Li:2019sin, ;
Li:2019yzy, ; Wang:2022gkd, ), it was found that the $\cos(2\phi)$ modulation
is proportional to the mass squared of leptons. It was observed that dimuon
pair photoproduction has a much larger $\cos(2\phi)$ modulation than
dielectron production. Therefore, it is not surprising to observe
$\cos(2\phi)$ modulation for $p\bar{p}$ pairs photoproduction, despite the
significant differences in the vertex $\Gamma^{\mu}$ for protons in Eq. (7)
compared to leptons. Interestingly, we do not observe significant
$\cos(4\phi)$ modulation in Fig. 3, and $\mathcal{A}_{4}$ is always an order
of magnitude smaller than $\mathcal{A}_{2}$ in Table 2. Therefore,
measurements of the azimuthal angle can also impose constraints on the vertex
$\Gamma^{\mu}$.

Table 2: We fit the $d\sigma/d\phi$ using the function $\mathcal{A}_{0}+\mathcal{A}_{2}\cos(2\phi)+\mathcal{A}_{4}\cos(4\phi)$. The coefficients $\mathcal{A}_{0},\mathcal{A}_{2}$ and $\mathcal{A}_{4}$ for different $\mathcal{P}(b_{T})$.  | $\quad\mathcal{A}_{0}(\mu\textrm{b})\quad$ | $\quad\mathcal{A}_{2}(\mu\textrm{b})\quad$ | $\quad\mathcal{A}_{4}(\mu\textrm{b})\quad$
---|---|---|---
$\mathcal{P}(b_{T})=1-e^{-w}$ | $0.73$ | $-0.08$ | $\sim 10^{-3}$
$\mathcal{P}(b_{T})=w$ | $1.01$ | $-0.11$ | $\sim 10^{-3}$
$\mathcal{P}(b_{T})$ for XnXn | $2.52$ | $-0.28$ | $0.01$

##  IV Conclusion and discussion

In this work, we studied light nuclei photoproduction in UPCs using our QED
model that incorporates a wave-packet description of nuclei. As a first
attempt, we considered the photoproduction of the lightest nuclei pairs,
$p\bar{p}$ pairs, at the Born level in UPCs. The effective vertex between
photons and protons, $\Gamma^{\mu}$ , was chosen based on studies of two-
photon exchange effects, including information on the electromagnetic form
factors of the proton. We then present the transverse momentum and invariant
mass distributions of $p\bar{p}$ pairs at $\sqrt{s_{NN}}=200$ GeV in UPCs in
2(a) and (b), respectively. Interestingly, we observe a significant
$\cos(2\phi)$ modulation and an almost vanishing $\cos(4\phi)$ modulation in
the azimuthal angle distribution of $p\bar{p}$ pairs in Fig. 3. Our study is
closely linked to two-photon exchange effects in hadron physics. Therefore,
further measurements in UPCs can also provide constraints on these effects.

Light nuclei photoproduction opens a new window to generate light nuclei and
helps us better understand the matter generated by light. Following the
similar strategy, one can also consider deuterium or helium nuclei
photoproduction within our theoretical framework. For deuterium nuclei, which
are spin-$1$, the propagator in tensor $L^{\mu\nu}$ needs to be replaced
accordingly, and the effective vertex may include contributions from the
electric quadrupole moment. For helium nuclei, more complex nuclear
information may be involved. Further discussions along this line will be
presented elsewhere.

###### Acknowledgements.

We would like to thank Wang-Mei Zha, Xin Wu and Cheng Zhang for helpful
discussion. This work is supported in part by the National Key Research and
Development Program of China under Contract No. 2022YFA1605500, by the Chinese
Academy of Sciences (CAS) under Grants No. YSBR-088 and by National Nature
Science Foundation of China (NSFC) under Grants No. 12075235, No. 12135011,
No. 12275082, No. 12035006, and No. 12075085.

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