aainstitutetext: Department of Physics and Center for Field Theory and
Particle Physics, Fudan University, Shanghai, Chinabbinstitutetext: Key
Laboratory of Nuclear Physics and Ion-beam Application (MOE), Fudan
University, Shanghai, Chinaccinstitutetext: Shanghai Qi Zhi Institute,
Shanghai 200030, China

# Azimuthal decorrelation for photon induced dijet production in ultra-
peripheral collisions of heavy ions

Cheng Zhang a , Qian-Shun Dai a,b,c,1 and Ding Yu Shao111Corresponding author.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the azimuthal angular decorrelation of the dijet production via
photon fusion in ultra-peripheral heavy ion collisions. The impact parameter
dependent cross section of quark-antiquark pairs production is derived using
the equivalent photon approximation, and the contribution from final-state QCD
radiations to the azimuthal angular distribution are calculated within Soft-
Collinear Effective Theory. We carry out the QCD resummation of large
logarithms of the azimuthal angle as well as the jet radius at next-to-leading
logarithmic accuracy. In the end we present the normalized differential cross
section for azimuthal decorrelation of the dijet pair and find that our
results are consistent with the measurements reported by the ATLAS
collaboration.

##  1 Introduction

Ultraperipheral collisions (UPCs) of high-energy nuclei are processes that the
impact parameter is large enough so that only electromagnetic interactions
happen with hadronic interactions being excluded. UPCs provide unique
opportunities to study the strong electromagnetic processes in relativistic
heavy-ion collisions Baur:1998ay ; Baur:2003ar ; Bertulani:2005ru ;
Baltz:2007kq ; Zhao:2022dac . Besides, UPCs involve two-photon fusions and
photonuclear interactions, which provide an opportunity to study the quasi-
real photons emitted by one of the colliding nuclei and probe the nuclear
parton distribution functions (PDFs). Typical photonuclear processes
ZEUS:2002wfj ; PHENIX:2009xtn ; Contreras:2013oan ; Schmidke:2016ccw ;
CMS:2016itn ; ALICE:2019tqa ; ALICE:2021tyx ; LHCb:2021bfl ; STAR:2021wwq ;
STAR:2022wfe  involve the photoproduction of vector mesons via photon-gluon
fusions, which is a clean probe to study the gluon distributions inside the
energetic nucleus Brodsky:1994kf ; Klein:1999qj ; Kowalski:2006hc ;
Rebyakova:2011vf ; Lappi:2013am ; Guzey:2013qza ; Xie:2016ino ; Xing:2020hwh ;
Zha:2020cst ; Brandenburg:2022jgr ; Mantysaari:2022sux . Typical two-photon
interactions involve dilepton productions ATLAS:2022cbd ; STAR:2004bzo ;
STAR:2018ldd ; STAR:2018xaj ; ATLAS:2018pfw ; ALICE:2018ael ; ATLAS:2020epq
or diphoton productions from light-by-light scatterings ATLAS:2017fur . The
dilepton production process can serve to probe the strong electromagnetic
fields generated from the initial state relativistic heavy-ion Vidovic:1992ik
; Hencken:1994my ; Krauss:1997vr ; Zha:2018tlq ; Klein:2018fmp ;
Brandenburg:2021lnj ; Wang:2021kxm , as well as the polarizations of these
quasi-real photons Li:2019sin ; Li:2019yzy ; Wang:2022gkd , or to study the
final-state Quantum Electrodynamics (QED) radiations from the produced charged
leptons.

The photon flux coherently emitted from the charged nucleus is highly enhanced
by a factor of the squared nuclear charge $Z^{2}$, and these photons are
nearly real with a small virtuality $k^{2}\lesssim(\hbar/R_{A})^{2}$, where
$R_{A}$ is the nuclear radius. The quasi-real photon flux can be well
described by the well-known equivalent photon approximation (EPA) introduced
by Fermi, von Weizs$\ddot{\textrm{a}}$cker and Williams Fermi:1924tc ;
vonWeizsacker:1934nji ; Williams:1935dka , where the strong photon flux is
treated as a highly Lorentz-contracted classical electromagnetic field emitted
from the fast-moving nucleus. Within the EPA method, the photon spectrum is
the Fourier transform of the nuclear charge spatial distribution.

Recently, in ultraperipheral $Pb+Pb$ collisions Quantum Chromodynamics (QCD)
jets production in $0n0n$ events was first observed by the ATLAS collaboration
ATLAS:2022cbd , and they find Pythia8 Monte Carlo event generator
Bierlich:2022pfr  underestimates the total cross section by about an order of
magnitude. Usually, dijet and multi-jet production in UPCs with no nuclear
breakup involves both photon fusion and diffractive photo-production
processes. This discrepancy in magnitude may result from the missing processes
of the diffractive photo-production in Pythia8, and these processes has been
calculated in Guzey:2016tek , where the results are obtained by using the
next-to-leading order (NLO) collinear factorization formalism of perturbative
QCD (pQCD). Besides, in ATLAS:2022cbd  the ATLAS collaboration also present
the experimental results of the azimuthal angle $\Delta\phi_{jj}$ between the
two jets with the highest transverse momenta. In order to directly compare the
shape of the measured distribution with the Pythia results, they have rescaled
the Pythia curve to match the number of measured events. As a result, they
find the measured distribution is obviously wider than that from the Pythia8
event generator. Therefore, it is significant to have a theoretical
calculation for the $\Delta\phi_{jj}$-distribution.

Starting with the pioneering papers Banfi:2008qs ; Hautmann:2008vd , all-order
resummation of azimuthal decorrelation for QCD jets has been performed in
various processes Sun:2014gfa ; Sun:2015doa ; Chen:2018fqu ; Sun:2018icb ;
Liu:2018trl ; Chien:2019gyf ; Liu:2020dct ; Chien:2020hzh ; Abdulhamid:2021xtt
; Chien:2022wiq ; Bouaziz:2022tik ; Yang:2022qgk ; Martinez:2022dux . Usually,
fixed-order calculations in pQCD can be used to systematically improve the
description of hadronic radiation. However, in nearly back-to-back region
$\Delta\phi_{jj}\sim\pi$ the fixed-order expansion of pQCD diverges due to the
existence of logarithms $\alpha_{s}^{n}\ln^{m}(\pi-\Delta\phi_{jj})$ (with
$0<m\leq 2n$) at each perturbative order $n$, so that an all-order resummation
is necessary for the validity of theoretical predictions. Besides, as pointed
in Banfi:2008qs ; Chien:2022wiq  the all-order resummation formula strongly
depends on the recombination schemes in sequential jet clustering algorithms.
Especially when the jets are reconstructed with the $p_{T}^{n}$-weighted
recombination scheme, the resummation is straightforward since it does not
require any non-trivial treatment of non-global logarithms (NGLs)
Dasgupta:2001sh ; Banfi:2002hw .

In this paper we perform a detailed theoretical study for the azimuthal
decorrelation of dijet productions from photon-photon fusion in
ultraperipheral heavy ion collisions, shown in figure 1. Specifically, we
consider the $\Delta\phi_{jj}$-distribution of the leading dijet pair produced
nearly back to back in the transverse plane to the beam. In order to describe
such configuration we need to consider nonzero transverse momenta from both
incoming and outgoing partons. First of all, since the initial photons can
have nonzero transverse momentum, one needs to consider the transverse
momentum dependent photon distribution. This effect can be captured by the
impact parameter dependent cross section in EPA, and the corresponding
calculations have been extensively studied in Vidovic:1992ik ; Krauss:1997vr ;
Zha:2018tlq ; Li:2019yzy ; Li:2019sin ; Zha:2020cst ; Brandenburg:2021lnj ;
Wang:2021kxm ; Wang:2022gkd . Moreover, final-state radiations could also
introduce azimuthal angular decorrelation. We apply Soft-Collinear Effective
Theory (SCET) Bauer:2000yr ; Bauer:2001ct ; Bauer:2001yt ; Bauer:2002nz ;
Beneke:2002ph  to develop a factorization and resummation formalism for
studying the azimuthal angular distribution in back-to-back dijet production
in ultraperipheral heavy ion collisions, where large logarithms of the
azimuthal angle and the jet radius are resumed at next-to-leading logarithmic
(NLL) accuracy.

Figure 1: Photon-induced dijet production in UPCs.

The rest of this paper is organized as follows. In section 2 we discuss the
impact parameter dependent cross section of quark-antiquark pairs production
via photon-photon fusions in UPCs. In section 3 we give the factorization and
resummation formula of $\gamma\gamma\to 2$ jets within SCET. We present the
numerical results using the theoretical formula, enumerate all theoretical
uncertainties and compare our predictions with the ATLAS experimental data in
section 4. We conclude in section 5. The details of the one-loop calculations,
resummation, and anomalous dimensions are provided in the appendix.

##  2 Quark-antiquark pairs production from $\gamma\gamma$ fusions

In this section, we give the impact parameter dependent cross section of
quark-antiquark pairs production from photon-photon fusions in UPCs at the
lowest QED order, which provides the initial state of the dijet production.
Associating that with the final-state QCD emissions, we arrive at the dijet
production from two-photon interactions in UPCs. The two-photon fusion process
reads,

$\gamma(\omega_{1},\bm{k}_{1T})+\gamma(\omega_{2},\bm{k}_{2T})\rightarrow
q(y_{1},\bm{p}_{1T})+\bar{q}(y_{2},\bm{p}_{2T}).$ (1)

After applying the Weizsäcker-Williams equivalent photon approximation (EPA)
Fermi:1924tc ; vonWeizsacker:1934nji ; Williams:1935dka , one can drive the
unpolarized UPC differential cross section as Krauss:1997vr ; Li:2019yzy ;
Li:2019sin

$\displaystyle\frac{\mathrm{d}^{5}\sigma_{0}}{\mathrm{d}^{2}\bm{q}_{T}\mathrm{d}p_{T}\mathrm{d}y_{1}\mathrm{d}y_{2}}=$
$\displaystyle\,N_{c}\sum_{q}e_{q}^{4}\frac{Z^{4}\alpha_{{\rm
em}}^{4}}{\pi^{5}M^{4}}p_{T}\int\mathrm{d}^{2}\bm{b}_{T}\mathrm{d}^{2}\bm{k}_{1T}\mathrm{d}^{2}\bm{k}_{2T}\mathrm{d}^{2}\bm{k}_{1T}^{\prime}\mathrm{d}^{2}\bm{k}_{2T}^{\prime}$
$\displaystyle\times\delta^{(2)}(\bm{k}_{1T}+\bm{k}_{2T}-\bm{q}_{T})\,\delta^{(2)}(\bm{k}_{1T}^{\prime}+\bm{k}_{2T}^{\prime}-\bm{q}_{T})\,e^{i\,(\bm{k}_{1T}-\bm{k}_{1T}^{\prime})\cdot\bm{b}_{T}}$
$\displaystyle\times
k_{1T}\frac{F(-k_{1}^{2})}{-k_{1}^{2}}k_{2T}\frac{F(-k_{2}^{2})}{-k_{2}^{2}}k_{1T}^{\prime}\frac{F(-k_{1}^{\prime
2})}{-k_{1}^{\prime 2}}k_{2T}^{\prime}\frac{F(-k_{2}^{\prime
2})}{-k_{2}^{\prime 2}}\frac{M^{2}-2p_{T}^{2}}{p_{T}^{2}}$
$\displaystyle\times\cos\left(\phi_{\bm{k}_{1T}}-\phi_{\bm{k}_{1T}^{\prime}}+\phi_{\bm{k}_{2T}}-\phi_{\bm{k}_{2T}^{\prime}}\right),$
(2)

with the nuclear electric charge $Z$ ($79$ for $Au$, $82$ for $Pb$), the fine-
structure constant $\alpha_{{\rm em}}=1/137$, the number of colors $N_{c}=3$,
the quark fractional electric charge $e_{q}$, and the quark flavors $q$. Since
we only consider QCD light flavor jet production, the cross section is a sum
over the light quark flavors $q=u,d,s$. Here we define the average transverse
momentum $\bm{p}_{T}$ of the two jets and the transverse momentum imbalance
$\bm{q}_{T}$ as follows

$\displaystyle\bm{p}_{T}\equiv\left(\bm{p}_{1T}-\bm{p}_{2T}\right)/2,~{}~{}~{}\bm{q}_{T}\equiv\bm{p}_{1T}+\bm{p}_{2T}.$
(3)

The quark-antiquark pairs are produced nearly back-to-back in the transverse
plane with respect to the beam direction, so the transverse momentum imbalance
$q_{T}\equiv|\bm{q}_{T}|$ is small. In the limit of $q_{T}\ll p_{T}\sim
p_{1T}\sim p_{2T}$, the invariant mass of quark-antiquark pairs reads

$\displaystyle M=p_{T}\sqrt{2+2\cosh(y_{1}-y_{2})},$ (4)

with quark rapidities $y_{1,2}$. The impact parameter $b_{T}$ should be
integrated over from $2R_{A}$ to $\infty$ to exclude the central collisions,
where $R_{A}$ is the nuclear radius. The photon possesses different transverse
momentum $\bm{k}_{T}$ (with the azimuthal angle $\phi_{\bm{k}_{T}}$) and
$\bm{k}_{T}^{\prime}$ inside the amplitude and the conjugate amplitude
respectively, as we have kept the impact parameter dependence of two colliding
photon fluxes. The quasi-real photon flux with a small virtuality
$k^{2}=-\omega^{2}/\gamma^{2}-\bm{k}_{T}^{2}$ (with the Lorentz boost factor
$\gamma=2676$ at a centre-of-mass energy per nucleon pair of
$\sqrt{s_{\mathrm{NN}}}=5.02$ TeV at the CERN LHC) and photon energy
$\omega_{1,2}=p_{T}(e^{\pm y_{1}}+e^{\pm y_{2}})/2$ can be described by the
EPA, which depicts the photon momentum $k$ distribution via the nuclear charge
form factor $F(-k^{2})$. This form factor can be obtained after performing
Fourier transformation from the nuclear charge distribution $\rho(r)$ as

$F(\bm{k}^{2})=\int\mathrm{d}^{3}\bm{r}e^{i\bm{r}\cdot\bm{k}}\rho(r),$ (5)

where the charge distribution inside the heavy nucleus is well described by
the Woods-Saxon distribution,

$\rho_{\mathrm{WS}}(r)=\frac{1}{1+e^{(r-R_{A})/d}}\Bigg{/}\\!\\!\int\mathrm{d}^{3}\bm{r}\frac{1}{1+e^{(r-R_{A})/d}},$
(6)

with the nuclear radius $R_{A}\simeq 1.2\,A^{1/3}$ (where $A$ is the nucleon
number) and the nuclear skin depth $d$ ($d=0.546$ fm for $Pb$ and $d=0.535$ fm
for $Au$). In the left panel of figure 2 we plot the nuclear charge spatial
distribution of lead (red solid curve) and gold (blue dashed curve) nuclei
within the Woods-Saxon model. One can see that the lead nuclei possesses a
slightly wider spatial broadening and a less steep boundary compared to the
gold nuclei. For simplicity, in the latter calculation, we alternatively
choose the approximate Woods-Saxon form factor used in the STARlight Monte
Carlo event generator Klein:2016yzr ,

$F_{\rm AWS}(\kappa^{2})=\frac{3[\sin(\kappa R_{A})-\kappa R_{A}\cos(\kappa
R_{A})]}{(\kappa R_{A})^{3}(a^{2}\kappa^{2}+1)},$ (7)

with $a=0.7$ fm. The corresponding nuclear charge form factors are shown in
the right panel of figure 2, and we find the difference between these two
models is negligible. From the plot we find that the typical transverse
momentum of the photon is about $50$ MeV for $Pb$, and then the corresponding
azimuthal angular is too small and can not explain the results observed by the
ATLAS collaboration. Therefore we also need to systematically calculate
higher-order QCD and QED corrections in order to describe data at the LHC. In
the following section, we will consider the all-order resummation corrections
from final-state QCD radiations, which play a dominant role in jet production,
and we leave a full QCD and QED showers study for the future work.

Figure 2: Left: The nuclear charge spatial distribution $\rho(r)$ within the
Woods-Saxon model of lead (red solid curve) and gold nuclei (blue dashed
curve), both of which have been normalized to unity
$\int\mathrm{d}^{3}\bm{r}\rho(r)=1$; Right: The lead nuclear charge form
factor $F(\kappa^{2})$ of the Woods-Saxon type (solid red curve) and from
STARlight Monte Carlo event generator (dashed blue curve).

##  3 Factorization and resummation for final-state radiations

In this section we will present the resummation formula describing final-state
QCD emissions from the dijet system, which contributes to the azimuthal
decorrelation of the dijet pair. In order to obtain such formalism, we first
apply SCET to write down a factorized formula, and express the differential
cross section as the product of single-scale key ingredients at leading power.
Based on the factorization formula and requirements of the factorization scale
independence for the cross section, we obtain the all-order resummation
formula via solving corresponding renormalization group (RG) equations.

Before introducing our methodology, we first briefly describe two methods on
the market used to perform resummation of large logarithms of the azimuthal
angle for jets Banfi:2008qs ; Hautmann:2008vd ; Sun:2014gfa ; Sun:2015doa ;
Chen:2018fqu ; Sun:2018icb ; Liu:2018trl ; Chien:2019gyf ; Liu:2020dct ;
Chien:2020hzh ; Abdulhamid:2021xtt ; Chien:2022wiq ; Bouaziz:2022tik ;
Yang:2022qgk ; Martinez:2022dux . The first method is deriving an all-order
resummation formula for the transverse momentum imbalance $\bm{q}_{T}$, and
then constructing the azimuthal decorrelation $\Delta\phi$ distribution from
the $\bm{q}_{T}$ distribution, which can be named as the “indirect method”.
The second method is calculating the resumed results of the azimuthal angular
distribution directly, so we name it as the “direct method”. The relation
between the two methods is obvious for Drell-Yan-like processes, but for the
scattering processes involving jet production it is not easy to establish the
relation, since one also needs to carry out the resummation of large
logarithms from final-state QCD radiations. As shown in Buffing:2018ggv ;
Chien:2019gyf ; delCastillo:2020omr , in the indirect method the single
logarithmic anomalous dimensions of the soft and collinear-soft function with
the narrow cone approximation both depend on the azimuthal angle of
$\bm{q}_{T}$ (or the azimuthal angle of $\bm{b}$ in the Fourier conjugate
transverse coordinate space). After taking into account QCD evolution, this
azimuthal integral is divergent in some phase space regions. In order to
regularize such divergences, different schemes have been proposed in
Buffing:2018ggv ; Chien:2019gyf ; delCastillo:2021znl . In this paper we
consider QCD resummation of large logarithms of azimuthal angle and jet radius
at the NLL accuracy. Therefore, in order to avoid such difficulties, we will
apply the direct method in the latter calculations.

###  3.1 QCD factorization formalism in SCET

A factorization formula for the boson-jet azimuthal decorrelation has been
comprehensively derived by one of us in Chien:2022wiq  within SCET, for the
case where jets are defined using the anti-$k_{T}$ algorithm Cacciari:2008gp
with the $p_{T}^{n}$-weighted recombination scheme. Although in this paper we
consider the jet definition using the standard recombination schemes, which is
different from $p_{T}^{n}$-weighted scheme, the factorization formula still
shares many similar properties. Therefore, we only present the main features
of the factorized structure, and a more detailed discussion on the
factorization analysis within SCET can be found in Chien:2022wiq .

At the first step, we use scaling arguments to identify the regions of phase
space that contribute to the factorization formula at the leading power. In
the back-to-back limit where $\Delta\phi_{jj}\to\pi$, the relevant low energy
modes for the factorized expression in SCET are given by

$\displaystyle{\color[rgb]{0,0.5,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.5,1}\pgfsys@color@cmyk@stroke{1}{0.50}{0}{0}\pgfsys@color<EMAIL_ADDRESS>collinear}}:$ $\displaystyle~{}~{}p_{c_{i}}^{\mu}\sim
p_{T}\,(R^{2},1,R)_{n_{i}\bar{n}_{i}},$ (8)
$\displaystyle{\color[rgb]{0,0.88,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0.88,0}\pgfsys@color@cmyk@stroke{0.91}{0}{0.88}{0.12}\pgfsys@color<EMAIL_ADDRESS>collinear-soft}}:$
$\displaystyle~{}~{}p_{cs_{i}}^{\mu}\sim\frac{p_{T}\,\delta\phi}{R}(R^{2},1,R)_{n_{i}\bar{n}_{i}},$
(9)
$\displaystyle{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\textbf{soft}}:$
$\displaystyle~{}~{}p_{s}^{\mu}\sim
p_{T}\,(\delta\phi,\delta\phi,\delta\phi),$ (10)

with $\delta\phi\equiv\pi-\Delta\phi_{jj}$. Here all the momenta
$p^{\mu}\equiv(n_{i}\cdot p,\bar{n}_{i}\cdot
p,p_{n_{i}\perp})_{n_{i}\bar{n}_{i}}$ are expressed using light-cone
coordinates, where $n_{1}^{\mu}$ and $n_{2}^{\mu}$ are light-like vectors
along leading and sub-leading jets, separately, and $\bar{n}_{i}^{\mu}$ is the
direction backwards to the jet. In the above analysis we have considered the
limits $R\ll 1$, therefore the narrow cone approximation as well. As a result,
the $n_{i}$ collinear mode (8) describes energetic emissions inside the jet
with radius $R$, which only contributes to the normalization of the
$\Delta\phi_{jj}$-distribution, not to its shape. While both $n_{i}$
collinear-soft and soft modes contribute to $\Delta\phi_{jj}$-distribution,
where $p_{cs_{i}}^{\mu}$ is sensitive to the jet boundary but $p_{s}^{\mu}$ is
not. Besides, the soft mode describes large-angle radiations without any
direction preference, so in (10) the subscript of $p_{s}^{\mu}$ in light-cone
coordinates can be ignored.

With the relevant scaling identified, we can write down a factorized form for
the differential cross section at the leading power of $\delta\phi$

$\displaystyle\frac{\mathrm{d}^{4}\sigma}{\mathrm{d}q_{x}\mathrm{d}p_{T}\mathrm{d}y_{1}\mathrm{d}y_{2}}=\,$
$\displaystyle\int\mathrm{d}k_{x}\,\mathrm{d}\lambda_{x}\,\mathrm{d}l_{1,x}\,\mathrm{d}l_{2,x}\delta(q_{x}+\lambda_{x}+l_{1,x}+l_{2,x}-k_{x})B(k_{x},p_{T},y_{1},y_{2})$
$\displaystyle\times H(p_{T},\Delta
y,\mu)S(\lambda_{x},y_{1},y_{2},\mu,\nu)U_{1}(l_{1,x},R,y_{1},\mu,\nu)J_{1}(p_{T},R,\mu)$
$\displaystyle\times U_{2}(l_{2,x},R,y_{2},\mu,\nu)J_{2}(p_{T},R,\mu),$ (11)

with $|q_{x}|\approx p_{T}\sin(\pi-\Delta\phi_{jj})=p_{T}\sin\delta\phi$ as
shown in the left panel of figure 3, where we take the leading jet’s
transverse momentum as reference for $-y$ direction,

$\displaystyle\bm{p}_{1T}\approx(0,-p_{T}),~{}~{}~{}\bm{p}_{2T}\approx
p_{T}(\sin\delta\phi,\cos\delta\phi).$ (12)

In (3.1) the Dirac delta function term in the first line indicates the
momentum conservation of different ingredients along the $x$-axis, neglecting
power corrections. Explicitly, the pieces of the factorized formula (3.1) are:

Figure 3: Left: The definition of the azimuthal angle $\Delta\phi_{jj}$
between leading $j_{1}$ and subleading jet $j_{2}$ in $x$-$y$ plane, and the
relation between $\Delta\phi_{jj}$ and momentum $q_{x}$ in the factorized
formula (3.1), where $\bm{q}_{T}$ is the vector sum of the transverse momenta
for the dijet, and $q_{x}$ is the projection of $\bm{q}_{T}$ in the $x$-axis.
Right: Sample one-loop diagrams of the hard, jet, soft as well as collinear-
soft functions in (3.1).

  * •

$B(k_{x},p_{T},y_{1},y_{2})$ is the Born cross section of the process
$\gamma\gamma\to q\bar{q}$ calculated in section 2, and $k_{x}$ indicates $x$
component of the transverse momentum from incoming photon beams. Explicitly,
we obtain the function $B(k_{x},p_{T},y_{1},y_{2})$ from (2) as

$\displaystyle
B(k_{x},p_{T},y_{1},y_{2})=\int\mathrm{d}k_{y}\frac{\mathrm{d}^{5}\sigma_{0}}{\mathrm{d}^{2}\bm{k}_{T}\mathrm{d}p_{T}\mathrm{d}y_{1}\mathrm{d}y_{2}},~{}~{}~{}{\rm
with}~{}~{}~{}\bm{k}_{T}=(k_{x},k_{y}).$ (13)

  * •

$H(p_{T},\Delta y)$ is the hard function that can be determined order-by-order
in perturbation theory by a matching calculation in QCD and in SCET at the
hard scale $\mu_{h}$. Here $\Delta y\equiv y_{1}-y_{2}$, is defined as the
rapidity difference between leading and sub-leading jets.

  * •

$J_{i}(p_{T},R)$ is the jet function describing the emission of collinear
radiations inside the anti-$k_{T}$ jet with radius $R$, and its mode has
momenta that scale as $p_{c_{i}}^{\mu}$.

  * •

$U_{i}(l_{i,x},R,y_{i})$ is the collinear-soft function describing the soft
radiation along the direction of the jet, and it is sensitive to the jet
direction and the jet boundary. In addition, the collinear-soft mode
$p_{cs_{i}}^{\mu}$ can also resolve any possible collinear constituents of the
jet, which can give the so-called NGLs, as discussed in the following
paragraph.

  * •

$S(\lambda_{x},y_{1},y_{2})$ is the transverse momentum dependent soft
function that integrates the radiation from the leading and sub-leading jets,
so it depends on the rapidity of jets explicitly.

Their one-loop sample diagrams in SCET are given in the right panel of figure
3, and in the appendix A we present the explicit calculations of the soft and
collinear-soft functions at one loop. In addition to the factorization scale
$\mu$ dependence, we also present the soft $S$ and collinear-soft $U_{i}$
functions with explicit rapidity scale $\nu$ dependence, which stems from
rapidity divergence and the corresponding regulator. In the following
subsection we will apply the collinear anomaly formalism in Becher:2010tm ;
Becher:2011xn  to resum corresponding rapidity logarithms. Alternatively, it
can be dealt with using the rapidity renormalization group method Chiu:2011qc
; Chiu:2012ir . Finally, it should be noted that the above factorized
expression (3.1) has ignored the structure from non-global logarithms (NGLs),
which start contributing at two-loop order Dasgupta:2001sh . The TMD
factorization formula including those effects has been discussed in
Liu:2018trl ; Chien:2019gyf ; Kang:2020xez , and one finds that NGLs can be
resumed via a fitting function given in Dasgupta:2001sh  at NLL level. In our
phenomenology, we have included their contributions in the resummation
formula.

After performing Fourier transform for (3.1), we obtain the factorized formula
in the coordinate space as follows

$\displaystyle\frac{\mathrm{d}^{4}\sigma}{\mathrm{d}q_{x}\mathrm{d}p_{T}\mathrm{d}y_{1}\mathrm{d}y_{2}}=$
$\displaystyle\int_{-\infty}^{+\infty}\frac{\mathrm{d}b_{x}}{2\pi}e^{iq_{x}b_{x}}\tilde{B}(b_{x},p_{T},y_{1},y_{2})H(p_{T},\Delta
y,\mu)\tilde{S}(b_{x},y_{1},y_{2},\mu,\nu)$
$\displaystyle\times\tilde{U}_{1}(b_{x},R,y_{1},\mu,\nu)J_{1}(p_{T},R,\mu)\tilde{U}_{2}(b_{x},R,y_{2},\mu,\nu)J_{2}(p_{T},R,\mu),$
(14)

where $\tilde{B}$, $\tilde{S}$ and $\tilde{U}_{i}$ are the Fourier transform
of $B$, $S$ and $U_{i}$ in (3.1), respectively. Except for the Born cross
section $\tilde{B}$, all other ingredients are normalized to $1$ at the
leading order. Accordingly, as a check one can easily see that at the leading
order the above formula (3.1) degenerate (2) for the production of the quark-
antiquark pair without final-state radiations.

###  3.2 QCD resummation formalism of final-state radiation

In this section, we present the RG equations for the factorization scale
dependent ingredients in (3.1), including the hard function $H$, the jet
function $J_{i}$, the soft function $\tilde{S}$, and collinear-soft function
$\tilde{U}_{i}$. After presenting their RG evolution equations, we check the
RG consistency at one loop. In the end, we present the all-order QCD
resummation formula for the azimuthal angular distribution.

For the hard scattering process $\gamma\gamma\to q\bar{q}$, the QCD
corrections have been perturbatively calculated at three loop Caola:2020dfu ,
and the corresponding RG equation of the hard function reads

$\displaystyle\frac{\mathrm{d}}{\mathrm{d\ln\mu}}H(p_{T},\Delta
y,\mu)=\underbrace{\left[-2C_{F}\gamma_{\rm
cusp}(\alpha_{s})\ln\frac{\mu^{2}}{M^{2}}+4\gamma_{q}(\alpha_{s})\right]}_{\begin{subarray}{c}\equiv\Gamma_{H}(\alpha_{s})\end{subarray}}H(p_{T},\Delta
y,\mu),$ (15)

where we have defined the hard anomalous dimension $\Gamma_{H}(\alpha_{s})$,
and $M$ is the invariant mass of dijet pair defined in (4). In the appendix B
we present the perturbative expression of all relevant anomalous dimensions
for the NLL resummation.

The one-loop quark jet function for the anti-$k_{T}$ algorithm with radius $R$
is calculated in Ellis:2010rwa , and it satisfies the RG evolution equations

$\displaystyle\frac{\mathrm{d}}{\mathrm{d\ln\mu}}J_{i}(p_{T},R,\mu)$
$\displaystyle=\underbrace{\left[-C_{F}\gamma_{\rm
cusp}(\alpha_{s})\ln\frac{p_{T}^{2}R^{2}}{\mu^{2}}-2\gamma_{q}(\alpha_{s})\right]}_{\begin{subarray}{c}\equiv\Gamma_{J}(\alpha_{s})\end{subarray}}J_{i}(p_{T},R,\mu).$
(16)

From (15) and (16) one can see that the characteristic scales related to the
hard and jet functions would be $\mu_{h}\sim M$ and $\mu_{j}\sim p_{T}R$,
respectively.

Due to rapidity divergences, the calculation of soft and collinear-soft
functions involves extra complication which is not seen in the calculation of
the hard and jet functions. In appendix A we present their one-loop
calculations in (A) and (A), where we introduce a rapidity regulator by
modifying the phase-space integrals Bell:2018oqa .

$\displaystyle\int\mathrm{d}^{d}k\to\int\mathrm{d}^{d}k\left(\frac{\nu}{2k^{0}}\right)^{\eta}.$
(17)

Such divergences are artificial because the product of the soft and collinear-
soft functions is independent on the scale $\nu$. However, the rapidity
divergence introduces additional logarithmic dependence of the jet radius $R$,
so (3.1) does not achieve complete factorization. By refactorizing out
$R$-dependence terms within the collinear anomaly framework Becher:2010tm ;
Becher:2011xn , the product of soft and collinear-soft functions should be
expressed as

$\displaystyle\tilde{U}_{1}(b,R,y_{1},\mu,\nu)\tilde{U}_{2}(b,R,y_{2},\mu,\nu)\tilde{S}(b,y_{1},y_{2},\mu,\nu)=R^{2F_{q\bar{q}}(b,\,\mu)}W(b,\Delta
y,\mu),$ (18)

where the reminder function $W(b,\Delta y,\mu)$ no longer contains large
logarithms of the jet radius, and the relevant $R$-dependent logarithms are
resummed by the help of the anomaly exponent $F_{q\bar{q}}(b,\mu)$.
Explicitly, their one-loop expressions are given by

$\displaystyle F_{q\bar{q}}(b,\mu)$
$\displaystyle=\frac{\alpha_{s}}{4\pi}C_{F}\gamma_{0}^{\rm
cusp}\ln\frac{b^{2}\mu^{2}}{b_{0}^{2}}+\mathcal{O}(\alpha_{s}^{2}),$ (19)
$\displaystyle W(b,\Delta y,\mu)$
$\displaystyle=1-\frac{\alpha_{s}}{4\pi}C_{F}\left[\gamma_{0}^{\rm
cusp}\ln(2+2\cosh\Delta
y)\ln\frac{b^{2}\mu^{2}}{b_{0}^{2}}\right]+\mathcal{O}(\alpha_{s}^{2}),$ (20)

where the reminder function $W$ is normalized to $1$ at the leading order, and
they satisfy the following RG equations:

$\displaystyle\frac{\mathrm{d}}{\mathrm{d\ln\mu}}F_{q\bar{q}}(b,\mu)$
$\displaystyle=2C_{F}\gamma_{\rm cusp}(\alpha_{s}),$ (21)
$\displaystyle\frac{\mathrm{d}}{\mathrm{d\ln\mu}}W(b,\Delta y,\mu)$
$\displaystyle=\Big{[}\\!-2C_{F}\gamma_{\rm
cusp}(\alpha_{s})\ln(2+2\cosh\Delta y)\Big{]}W(b,\Delta y,\mu).$ (22)

In appendix A we present the one-loop calculation of the soft and collinear-
soft functions, and one can easily verify that the rapidity poles are canceled
in the product of $\tilde{S}$ and $\tilde{U}_{i}$, and also reproduce the
refactorization formula (18).

With the anomalous dimensions presented for all the ingredients, one can
easily verify that the factorized formula given in (3.1) satisfies the
consistency relations for the RG evolution

$\displaystyle\frac{\mathrm{d}}{\mathrm{d}\ln\mu}\left[R^{2F_{q\bar{q}}(b,\,\mu)}W(b,\Delta
y,\mu)H(p_{T},\Delta y,\mu)J_{1}(p_{T},R,\mu)J_{2}(p_{T},R,\mu)\right]=0.$
(23)

In the end all large logarithms can be resummed by evolving the hard and jet
functions from their intrinsic scales $\mu_{h}$ and $\mu_{j}$ to the scale of
the reminder function and the anomaly exponent at $\mu_{b}$ separately, and at
NLL accuracy we have

$\displaystyle\frac{\mathrm{d}^{4}\sigma^{\rm
NLL}}{\mathrm{d}q_{x}\mathrm{d}p_{T}\mathrm{d}y_{1}\mathrm{d}y_{2}}=$
$\displaystyle\int_{0}^{\infty}\frac{\mathrm{d}b_{x}}{\pi}\cos(q_{x}b_{x})\tilde{B}(b_{x},p_{T},y_{1},y_{2})$
$\displaystyle\times\exp\left[\int_{\mu_{h}}^{\mu_{b}}\frac{\mathrm{d}\mu}{\mu}\,\Gamma_{H}(\alpha_{s})+2\int_{\mu_{j}}^{\mu_{b}}\frac{\mathrm{d}\mu}{\mu}\,\Gamma_{J}(\alpha_{s})\right]U_{\rm
NG}^{2}(\mu_{b},\mu_{j}),$ (24)

where we have incorporated NGLs resummation effects included by the function
$U_{\rm NG}$. As shown in Liu:2018trl ; Chien:2019gyf ; Kang:2020xez  the
fitting function given in Dasgupta:2001sh  can be used to capture leading-
logarithmic NGLs after choosing proper initial and final evolution scales. In
our process, the resummation of NGLs comes from a non-linear RG evolution
between the jet and the collinear-soft function Chien:2019gyf , so we choose
the jet scale $\mu_{j}$ and $\mu_{b}$ for the reminder function in the
function $U_{\rm NG}$ that is given by

$\displaystyle
U_{\mathrm{NG}}\left(\mu_{b},\mu_{j}\right)=\exp\left[-C_{A}C_{F}\frac{\pi^{2}}{3}u^{2}\frac{1+(au)^{2}}{1+(bu)^{c}}\right],$
(25)

with $a=0.85\,C_{A}$, $b=0.86\,C_{A}$, $c=1.33$ and
$u=\ln[\alpha_{s}(\mu_{b})/\alpha_{s}(\mu_{j})]/\beta_{0}$. Since there are
two jet functions in the factorized formula (3.1), we need to include the
square of $U_{\rm NG}$ to take into account the NGL resummation associated
with each jet.

##  4 Numerical results

In this section we will present the numerical results using the resummation
formula (3.2), which captures both non-zero transverse momentum from incoming
photons and QCD evolution of final-state radiations at NLL accuracy.

Figure 4: Left: The scale $\mu_{b}$ in (26) depending on the functional form
chosen for $b_{*}$: the red solid line is obtained with the square-root form,
while the blue dashed line with the exponential form; see (27) and (28).
Right: Uncertainties of the differential cross section estimated by varying
the functional forms in the $b_{*}$-prescription. In both cases we choose
$b_{\rm max}=1.123\,{\rm GeV}^{-1}$.

We see that, after the change of integration variable from $q_{x}$ to
$\Delta\phi_{jj}$ in (3.2), the theoretical results for
$\Delta\phi_{jj}$-distribution can be directly obtained where we choose the
intrinsic scales in the resummation formula as

$\displaystyle\mu_{h}=M,~{}~{}~{}\mu_{j}=p_{T}R,~{}~{}~{}\mu_{b}=\frac{b_{0}}{b_{*}(b_{x})}.$
(26)

In the above expressions, we introduce a function $b_{*}(b_{x})$ to freeze the
value of $\mu_{b}$ in the non-perturbative region as $b_{x}\to\infty$, since
pQCD calculations will hit the Landau pole. The standard “square-root form”
Collins:1984kg  of this function is defined as

$\displaystyle b_{*}(b)=\frac{b}{\sqrt{1+b^{2}/b_{\rm max}^{2}}}.$ (27)

It is noted that in the original Collins-Soper-Sterman (CSS) resummation,
$b_{*}$-prescription was introduced with additional non-perturbative factors,
which are proposed in terms of different functional forms and obtained from
fitting experimental data Collins:2014jpa ; Aidala:2014hva ; Sun:2014dqm ;
Landry:2002ix ; Konychev:2005iy ; Bacchetta:2017gcc ; Bacchetta:2022awv .
However, in our process there are no same double logarithmic terms that from
initial-state QCD radiations, so that the nonperturbative contribution would
be different from the original ones in the CSS resummation. Therefore, we do
not introduce such non-perturbative factors in the numerical results.

In order to estimate theoretical uncertainties from different function forms
of $b_{*}(b_{x})$, we also present the result using the “exponential form”
Bacchetta:2017gcc

$\displaystyle b_{*}(b)=b_{\rm max}\left[1-\exp\left(-b^{4}/b_{\rm
max}^{4}\right)\right]^{1/4}.$ (28)

to avoid the Landau pole. By introducing the function $b_{*}(b_{x})$, one has
$\mu_{b}\sim b_{0}/b_{x}$ when $b_{x}\ll b_{\rm max}$, and $\mu_{b}\sim
b_{0}/b_{\rm max}$ when $b_{x}\gg b_{\rm max}$, as is shown in the left panel
of figure 4, where the solid red and dashed blue curves are the scales
$\mu_{b}$ with square-root and exponential form, respectively. In both cases,
we choose the scale $\mu_{b}$ freeze at $1$ GeV as $b_{x}\to\infty$. As
expected, these two forms have the same asymptotic behaviors in both large
$b_{x}$ and small $b_{x}$ regions. The right panel of figure 4 shows that
these two forms lead to similar predictions for the differential cross section
in (3.2). From the figure, we find that as $\Delta\phi_{jj}\sim\pi$ the
sensitivity of our predictions to this function form is about $10\%$, and when
$\Delta\phi_{jj}<2.9$ it has a negligible effect on numerics, where we choose
$p_{T}=20$ GeV and $\Delta y=$1\. Since the uncertainty from different
function forms is smaller than that from the scale variation, in the rest of
this paper we will use the standard “square-root form” in (27).

Figure 5: Comparison between theoretical calculations of the azimuthal
decorrelation with the preliminary ATLAS data ATLAS:2022cbd , where
$\Delta\phi_{jj}$ is the azimuthal angle between two jets with highest
transverse momenta. The theoretical and experimental distributions are
normalized such that the area under the curve is equal to $1$. The black dots
are the ATLAS results, and the uncertainties of the data are smaller than the
symbol size used in the plot. The solid red curve is the result with the scale
choice in (26), and the theoretical uncertainty for the red band is obtained
from the variation of scales in the resummation formula.

In the end we will compare our theoretical calculation of the azimuthal
angular decorrelation $\Delta\phi_{jj}$ with the experimental results in
ATLAS:2022cbd , where we only consider the kinematic region where the jet pair
is produced nearly back to back. In this region the contribution from
multijets production is power suppressed, and we leave fixed-order pQCD
calculations of multijets production for future studies.222Very recently, a
new phenomenological code for automated fixed-order calculations of the
photon-induced processes in UPCs is presented in Shao:2022cly . In order to
make a comparison, we calculate (3.2) with the same kinematic cuts as in
ATLAS:2022cbd

$\displaystyle p_{T}>15\,{\rm GeV},~{}~{}~{}|y_{1,2}|<4.4,~{}~{}~{}M>35\,{\rm
GeV},$ (29)

and the jet radius $R=0.4$. As is shown in figure 5 the red curve is the
theoretical predictions with the scale choice in (26), and the error bands are
shown as the shaded regions where we consider the theoretical uncertainty from
the scales. Specifically, we perform a variation of $\mu_{h}$, $\mu_{j}$, and
$\mu_{b}$ scales by a factor of $2$ about the central values, and then combine
these uncertainties by taking the envelope. The black dots are the ATLAS
results, where we find experimental errors are small and can hardly be seen in
the plot. The theoretical and experimental distributions are normalized such
that the area under the curve is equal to $1$. From the plot it is clear to
see that our result is consistent with the experimental data.

##  5 Conclusion

In this paper we study the azimuthal decorrelation of dijet productions from
photon-photon fusions in UPCs, where the dijet is produced nearly back-to-back
in the transverse plane with respect to the beam direction. We calculate the
key measurement, coplanarity distribution, which is the distribution of the
azimuthal angle $\Delta\phi_{jj}$ between the leading and the subleading jet.
The azimuthal decorrelation is produced via both the non-trivial transverse
momentum distribution of the incoming photon flux and QCD radiations from the
outgoing partons, which leads to the factorized formula (3.1), namely the
convolution of the initial state and the final state. The initial quark-
antiquark pairs production is performed by the hard scattering of the diphoton
associated with the EPA photon spectrum. In addition, the final-state
radiations are described by the factorization and resummation formula at NLL
accuracy within the SCET framework.

We vary the scales in resummation formula and obtain a theoretical band of the
normalized coplanarity distribution. Figure 5 shows a good agreement with the
ATLAS experimental data in the nearly back-to-back region, where we note that
the Pythia8 Monte Carlo event generator simulates a noticeably narrower
$\Delta\phi_{jj}$ distribution compared to the data ATLAS:2022cbd . However,
both our theoretical calculations and the Pythia8 simulation severely
underpredict the total cross section of the experiment (roughly by a factor of
$10$), which implies some missing progresses (e.g. diffractive dijet
photoproductions) in these UPC $0n0n$ events to be studied in the future. We
argue that these additional dijet photoproductions may enhance the dijet
production yields, but should barely change the azimuthal angular distribution
shape since it is dominated by the final-state radiations.

###### Acknowledgements.

C.Z. is supported by the National Science Foundations of China under Grant No.
12147125. D.Y.S. is supported by the National Science Foundations of China
under Grant No. 12275052 and the Shanghai Natural Science Foundation under
Grant No. 21ZR1406100.

##  Appendix A Soft, collinear-soft functions and the collinear anomaly

The operator definition of the soft function can be found in Chien:2022wiq ,
and here we summarize its one-loop results. The NLO soft function reads

$\displaystyle\tilde{S}($ $\displaystyle
b_{x},y_{1},y_{2},\mu,\nu)=1+2\,C_{F}g_{s}^{2}\tilde{\mu}^{2\epsilon}\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d-1}}\delta(k^{2})\theta(k^{0})\left(\frac{\nu}{2k^{0}}\right)^{\eta}\frac{n_{1}\cdot
n_{2}}{n_{1}\cdot k\,k\cdot n_{2}}e^{-ib_{x}k_{x}}$
$\displaystyle=1+\frac{\alpha_{s}}{4\pi}C_{F}\left[-\left(\frac{2}{\eta}+\ln\frac{\nu^{2}n_{1}\cdot
n_{2}}{2\mu^{2}}\right)\left(\frac{4}{\epsilon}+4\ln\frac{\mu^{2}b_{x}^{2}}{b_{0}^{2}}\right)+\frac{4}{\epsilon^{2}}-2\ln^{2}\frac{\mu^{2}b_{x}^{2}}{b_{0}^{2}}-\frac{\pi^{2}}{3}\right],$
(30)

with $\tilde{\mu}^{2}=\mu^{2}e^{\gamma_{E}}/(4\pi)$ and
$b_{0}=2e^{-\gamma_{E}}$. Here the rapidity regulator is given in (17). In
order to evaluate the one-loop collinear-soft function $U_{i}$, it is
convenient to expand the integrated momentum $k^{\mu}$ along the light-like
reference vector $n_{i}^{\mu}$. Explicitly, we have

$\displaystyle k^{\mu}=\frac{n_{i}^{\mu}}{2}\bar{n}_{i}\cdot
k+\frac{\bar{n}_{i}^{\mu}}{2}n_{i}\cdot k+k_{\perp}^{\mu}.$ (31)

Then the one-loop collinear-soft function is written as

$\displaystyle\tilde{U}_{i}$
$\displaystyle(b_{\perp},y_{i},\mu,\nu)=1+2\,C_{F}g_{s}^{2}\tilde{\mu}^{2\epsilon}\int\frac{\mathrm{d}^{d}k}{(2\pi)^{d-1}}\delta(k^{2})\theta(k^{0})\left(\frac{\nu}{\bar{n}_{i}\cdot
k}\right)^{\eta}\frac{n_{i}\cdot\bar{n}_{i}}{n_{i}\cdot k\,k\cdot\bar{n}_{i}}$
$\displaystyle\times\left\\{e^{-i\bm{b}_{\perp}\cdot\bm{k}_{\perp}}\theta\left[\frac{n_{i}\cdot
k}{\bar{n}_{i}\cdot k}-\left(\frac{R}{2\cosh
y_{i}}\right)^{2}\right]+\theta\left[\left(\frac{R}{2\cosh
y_{i}}\right)^{2}-\frac{n_{i}\cdot k}{\bar{n}_{i}\cdot k}\right]\right\\}$
$\displaystyle=1+\frac{\alpha_{s}}{4\pi}C_{F}\left[\left(\frac{2}{\eta}+\ln\frac{\nu^{2}R^{2}}{4\mu^{2}\cosh^{2}y_{i}}\right)\left(\frac{2}{\epsilon}+2\ln\frac{\mu^{2}b_{\perp}^{2}}{b_{0}^{2}}\right)-\frac{2}{\epsilon^{2}}+\ln^{2}\frac{\mu^{2}b_{\perp}^{2}}{b_{0}^{2}}+\frac{\pi^{2}}{6}\right],$
(32)

where the rapidity regulator is also expanded at the leading power as
suggested in (9). The collinear-soft mode describes low energy radiations
emitted from the collinear partons in the jet at an angle $\theta\sim R$, and
the step functions in the second line indicate that only emissions outside the
jet contribute to transverse momentum imbalance at one loop.

As shown in (18), after combing soft and collinear-soft functions the jet
radius dependence would be factorized out. At order $\alpha_{s}$, using (A)
and (A), we find

$\displaystyle\tilde{S}(b_{x},y_{1},y_{2},\mu,\nu)$
$\displaystyle\tilde{U}_{1}(b_{x},y_{1},\mu,\nu)\tilde{U}_{2}(b_{x},y_{2},\mu,\nu)$
$\displaystyle=1+C_{F}\frac{\alpha_{s}}{\pi}\bigg{[}\ln
R^{2}-\ln(2+2\cosh\Delta
y)\bigg{]}\left(\frac{1}{\epsilon}+\ln\frac{b_{x}^{2}\mu^{2}}{b_{0}^{2}}\right)+\mathcal{O}(\alpha_{s}^{2}),$
(33)

where the poles in $\eta$ and scale $\nu$ dependence in the NLO terms cancel
out. Since we consider $R\ll 1$ in (3.1), the logarithm of $R$ indicates that
a complete separation of scales was not achieved. In order to carry out the
resummation of the logarithmic of the jet radius $R$, one obtain the
refactorized formula in (18). At one-loop order we find $\overline{\rm MS}$
renormalized result of (A) agrees with (18) using the exponent $F_{q\bar{q}}$
in (19) and the reminder function $W$ in (20).

In the above calculation, we evaluate both the one-loop soft and collinear-
soft functions in the lab frame where rapidities $y_{1,2}$ of dijet are
arbitrary, and obtain general expansions of them. However, since the product
of the soft and the collinear-soft functions are boost invariant along the
beam axis as given in (3.1), they can also be simultaneously calculated in the
center-of-mass frame of two incoming photons, which would give the same result
as in (A) after combining them together.

##  Appendix B Anomalous dimension

The QCD $\beta$-function and the cusp and non-cusp anomalous dimensions in the
$\overline{\textrm{MS}}$ renormalization scheme are expanded as

$\beta(\alpha_{s})=-2\alpha_{s}\sum_{n=0}^{\infty}\beta_{n}\left(\frac{\alpha_{s}}{4\pi}\right)^{n+1},\quad\gamma(\alpha_{s})=\sum_{n=0}^{\infty}\gamma_{n}\left(\frac{\alpha_{s}}{4\pi}\right)^{n+1}.$
(34)

The two-loop coefficients of the $\beta$-function and the cusp anomalous
dimensions, and the one-loop coefficient of the non-cusp anomalous dimensions
read,

$\beta_{0}=\frac{11}{3}C_{A}-\frac{4}{3}T_{F}n_{f},\quad\beta_{1}=\frac{34}{3}C_{A}^{2}-\frac{20}{3}T_{F}C_{A}n_{f}-4T_{F}C_{F}n_{f},$
(35) $\gamma_{0}^{\text{cusp }}=4,\quad\gamma_{1}^{\text{cusp
}}=\left(\frac{268}{9}-\frac{4\pi^{2}}{3}\right)C_{A}-\frac{80}{9}T_{F}n_{f},\quad\gamma_{0}^{q}=-3C_{F},$
(36)

with $T_{F}=1/2,\>C_{A}=3,\>C_{F}=4/3,\>n_{f}=5.$

## References

  * (1) G. Baur, K. Hencken and D. Trautmann, _Photon-photon physics in very peripheral collisions of relativistic heavy ions_ , _J. Phys. G_ 24 (1998) 1657–1692, [hep-ph/9804348].
  * (2) G. Baur, K. Hencken, A. Aste, D. Trautmann and S. R. Klein, _Multiphoton exchange processes in ultraperipheral relativistic heavy ion collisions_ , _Nucl. Phys. A_ 729 (2003) 787–808, [nucl-th/0307031].
  * (3) C. A. Bertulani, S. R. Klein and J. Nystrand, _Physics of ultra-peripheral nuclear collisions_ , _Ann. Rev. Nucl. Part. Sci._ 55 (2005) 271–310, [nucl-ex/0502005].
  * (4) A. J. Baltz, _The Physics of Ultraperipheral Collisions at the LHC_ , _Phys. Rept._ 458 (2008) 1–171, [0706.3356].
  * (5) J. Zhao, J. Chen, X.-G. Huang and Y.-G. Ma, _Electromagnetic fields in ultra-peripheral relativistic heavy-ion collisions_ , 2211.03968.
  * (6) ZEUS collaboration, S. Chekanov et al., _Exclusive photoproduction of $J/\psi$ mesons at HERA_, _Eur. Phys. J. C_ 24 (2002) 345–360, [hep-ex/0201043].
  * (7) PHENIX collaboration, S. Afanasiev et al., _Photoproduction of $J/\psi$ and of high mass $e^{+}e^{-}$ in ultra-peripheral Au+Au collisions at $\sqrt{s}=200$GeV_, _Phys. Lett. B_ 679 (2009) 321–329, [0903.2041].
  * (8) ALICE collaboration, J. G. Contreras, _Photoproduction of $J/\psi$ in Pb-Pb and p-Pb collisions at the LHC with the ALICE detector_, _EPJ Web Conf._ 60 (2013) 13007.
  * (9) STAR collaboration, W. Schmidke, _$J/\psi$ Production in Ultra-Peripheral Collisions at STAR_, _PoS_ DIS2016 (2016) 189.
  * (10) CMS collaboration, V. Khachatryan et al., _Coherent $J/\psi$ photoproduction in ultra-peripheral PbPb collisions at $\sqrt{s_{\rm NN}}=$ 2.76 TeV with the CMS experiment_, _Phys. Lett. B_ 772 (2017) 489–511, [1605.06966].
  * (11) ALICE collaboration, S. Acharya et al., _Coherent J/ $\psi$ photoproduction at forward rapidity in ultra-peripheral Pb-Pb collisions at $\sqrt{s_{\rm{NN}}}=5.02$ TeV_, _Phys. Lett. B_ 798 (2019) 134926, [1904.06272].
  * (12) ALICE collaboration, S. Acharya et al., _First measurement of the $|t|$-dependence of coherent $J/\psi$ photonuclear production_, _Phys. Lett. B_ 817 (2021) 136280, [2101.04623].
  * (13) LHCb collaboration, R. Aaij et al., _Study of coherent $J/\psi$ production in lead-lead collisions at $\sqrt{{\mathrm{s}}_{\mathrm{NN}}}$ = 5 TeV_, _JHEP_ 07 (2022) 117, [2107.03223].
  * (14) STAR collaboration, M. Abdallah et al., _Probing the Gluonic Structure of the Deuteron with $J/\psi$ Photoproduction in d+Au Ultraperipheral Collisions_, _Phys. Rev. Lett._ 128 (2022) 122303, [2109.07625].
  * (15) STAR collaboration, M. Abdallah et al., _Tomography of Ultra-relativistic Nuclei with Polarized Photon-gluon Collisions_ , 2204.01625.
  * (16) S. J. Brodsky, L. Frankfurt, J. F. Gunion, A. H. Mueller and M. Strikman, _Diffractive leptoproduction of vector mesons in QCD_ , _Phys. Rev. D_ 50 (1994) 3134–3144, [hep-ph/9402283].
  * (17) S. Klein and J. Nystrand, _Exclusive vector meson production in relativistic heavy ion collisions_ , _Phys. Rev. C_ 60 (1999) 014903, [hep-ph/9902259].
  * (18) H. Kowalski, L. Motyka and G. Watt, _Exclusive diffractive processes at HERA within the dipole picture_ , _Phys. Rev. D_ 74 (2006) 074016, [hep-ph/0606272].
  * (19) V. Rebyakova, M. Strikman and M. Zhalov, _Coherent $\rho$ and J/$\psi$ photoproduction in ultraperipheral processes with electromagnetic dissociation of heavy ions at RHIC and LHC_, _Phys. Lett. B_ 710 (2012) 647–653, [1109.0737].
  * (20) T. Lappi and H. Mantysaari, _$J/\psi$ production in ultraperipheral Pb+Pb and $p$+Pb collisions at energies available at the CERN Large Hadron Collider_, _Phys. Rev. C_ 87 (2013) 032201, [1301.4095].
  * (21) V. Guzey and M. Zhalov, _Exclusive $J/{\psi}$ production in ultraperipheral collisions at the LHC: constrains on the gluon distributions in the proton and nuclei_, _JHEP_ 10 (2013) 207, [1307.4526].
  * (22) Y.-p. Xie and X. Chen, _The coherent cross section of vector mesons in ultraperipheral PbPb collisions at the LHC_ , _Eur. Phys. J. C_ 76 (2016) 316, [1602.00937].
  * (23) H. Xing, C. Zhang, J. Zhou and Y.-J. Zhou, _The cos 2 $\phi$ azimuthal asymmetry in $\rho^{0}$ meson production in ultraperipheral heavy ion collisions_, _JHEP_ 10 (2020) 064, [2006.06206].
  * (24) W. Zha, J. D. Brandenburg, L. Ruan, Z. Tang and Z. Xu, _Exploring the double-slit interference with linearly polarized photons_ , _Phys. Rev. D_ 103 (2021) 033007, [2006.12099].
  * (25) J. D. Brandenburg, Z. Xu, W. Zha, C. Zhang, J. Zhou and Y. Zhou, _Exploring gluon tomography with polarization dependent diffractive J/ $\psi$ production_, 2207.02478.
  * (26) H. Mäntysaari, F. Salazar and B. Schenke, _Nuclear geometry at high energy from exclusive vector meson production_ , 2207.03712.
  * (27) ATLAS collaboration, _Photo-nuclear jet production in ultra-peripheral Pb+Pb collisions at $\sqrt{s_{\rm{NN}}}$ = 5.02 TeV with the ATLAS detector_, .
  * (28) STAR collaboration, J. Adams et al., _Production of $e^{+}e^{-}$ pairs accompanied by nuclear dissociation in ultra-peripheral heavy ion collision_, _Phys. Rev. C_ 70 (2004) 031902, [nucl-ex/0404012].
  * (29) STAR collaboration, J. Adam et al., _Low- $p_{T}$ $e^{+}e^{-}$ pair production in Au$+$Au collisions at $\sqrt{s_{\rm NN}}$ = 200 GeV and U$+$U collisions at $\sqrt{s_{\rm NN}}$ = 193 GeV at STAR_, _Phys. Rev. Lett._ 121 (2018) 132301, [1806.02295].
  * (30) STAR collaboration, J. Adam et al., _Measurements of Dielectron Production in Au $+$Au Collisions at $\sqrt{s_{\rm NN}}$= 27, 39, and 62.4 GeV from the STAR Experiment_, 1810.10159.
  * (31) ATLAS collaboration, M. Aaboud et al., _Observation of centrality-dependent acoplanarity for muon pairs produced via two-photon scattering in Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV with the ATLAS detector_, _Phys. Rev. Lett._ 121 (2018) 212301, [1806.08708].
  * (32) ALICE collaboration, S. Acharya et al., _Measurement of dielectron production in central Pb-Pb collisions at $\sqrt{{\textit{s}}_{\mathrm{NN}}}$ = 2.76 TeV_, _Phys. Rev. C_ 99 (2019) 024002, [1807.00923].
  * (33) ATLAS collaboration, G. Aad et al., _Exclusive dimuon production in ultraperipheral Pb+Pb collisions at $\sqrt{s_{\mathrm{NN}}}=5.02$ TeV with ATLAS_, _Phys. Rev. C_ 104 (2021) 024906, [2011.12211].
  * (34) ATLAS collaboration, M. Aaboud et al., _Evidence for light-by-light scattering in heavy-ion collisions with the ATLAS detector at the LHC_ , _Nature Phys._ 13 (2017) 852–858, [1702.01625].
  * (35) M. Vidovic, M. Greiner, C. Best and G. Soff, _Impact parameter dependence of the electromagnetic particle production in ultrarelativistic heavy ion collisions_ , _Phys. Rev. C_ 47 (1993) 2308–2319.
  * (36) K. Hencken, D. Trautmann and G. Baur, _Impact parameter dependence of the total probability for the electromagnetic electron - positron pair production in relativistic heavy ion collisions_ , _Phys. Rev. A_ 51 (1995) 1874–1882, [nucl-th/9410014].
  * (37) F. Krauss, M. Greiner and G. Soff, _Photon and gluon induced processes in relativistic heavy ion collisions_ , _Prog. Part. Nucl. Phys._ 39 (1997) 503–564.
  * (38) W. Zha, J. D. Brandenburg, Z. Tang and Z. Xu, _Initial transverse-momentum broadening of Breit-Wheeler process in relativistic heavy-ion collisions_ , _Phys. Lett. B_ 800 (2020) 135089, [1812.02820].
  * (39) S. Klein, A. H. Mueller, B.-W. Xiao and F. Yuan, _Acoplanarity of a Lepton Pair to Probe the Electromagnetic Property of Quark Matter_ , _Phys. Rev. Lett._ 122 (2019) 132301, [1811.05519].
  * (40) J. D. Brandenburg, W. Zha and Z. Xu, _Mapping the electromagnetic fields of heavy-ion collisions with the Breit-Wheeler process_ , _Eur. Phys. J. A_ 57 (2021) 299, [2103.16623].
  * (41) R.-j. Wang, S. Pu and Q. Wang, _Lepton pair production in ultraperipheral collisions_ , _Phys. Rev. D_ 104 (2021) 056011, [2106.05462].
  * (42) C. Li, J. Zhou and Y.-J. Zhou, _Impact parameter dependence of the azimuthal asymmetry in lepton pair production in heavy ion collisions_ , _Phys. Rev. D_ 101 (2020) 034015, [1911.00237].
  * (43) C. Li, J. Zhou and Y.-J. Zhou, _Probing the linear polarization of photons in ultraperipheral heavy ion collisions_ , _Phys. Lett. B_ 795 (2019) 576–580, [1903.10084].
  * (44) R.-j. Wang, S. Lin, S. Pu, Y.-f. Zhang and Q. Wang, _Lepton pair photoproduction in peripheral relativistic heavy-ion collisions_ , 2204.02761.
  * (45) E. Fermi, _On the Theory of the impact between atoms and electrically charged particles_ , _Z. Phys._ 29 (1924) 315–327.
  * (46) C. F. von Weizsacker, _Radiation emitted in collisions of very fast electrons_ , _Z. Phys._ 88 (1934) 612–625.
  * (47) E. J. Williams, _Correlation of certain collision problems with radiation theory_ , _Kong. Dan. Vid. Sel. Mat. Fys. Med._ 13N4 (1935) 1–50.
  * (48) C. Bierlich et al., _A comprehensive guide to the physics and usage of PYTHIA 8.3_ , 2203.11601.
  * (49) V. Guzey and M. Klasen, _Diffractive dijet photoproduction in ultraperipheral collisions at the LHC in next-to-leading order QCD_ , _JHEP_ 04 (2016) 158, [1603.06055].
  * (50) A. Banfi, M. Dasgupta and Y. Delenda, _Azimuthal decorrelations between QCD jets at all orders_ , _Phys. Lett. B_ 665 (2008) 86–91, [0804.3786].
  * (51) F. Hautmann and H. Jung, _Angular correlations in multi-jet final states from $k_{\perp}$-dependent parton showers_, _JHEP_ 10 (2008) 113, [0805.1049].
  * (52) P. Sun, C. P. Yuan and F. Yuan, _Soft Gluon Resummations in Dijet Azimuthal Angular Correlations in Hadronic Collisions_ , _Phys. Rev. Lett._ 113 (2014) 232001, [1405.1105].
  * (53) P. Sun, C. P. Yuan and F. Yuan, _Transverse Momentum Resummation for Dijet Correlation in Hadronic Collisions_ , _Phys. Rev. D_ 92 (2015) 094007, [1506.06170].
  * (54) L. Chen, G.-Y. Qin, L. Wang, S.-Y. Wei, B.-W. Xiao, H.-Z. Zhang et al., _Study of Isolated-photon and Jet Momentum Imbalance in $pp$ and $PbPb$ collisions_, _Nucl. Phys._ B933 (2018) 306–319, [1803.10533].
  * (55) P. Sun, B. Yan, C. P. Yuan and F. Yuan, _Resummation of High Order Corrections in $Z$ Boson Plus Jet Production at the LHC_, 1810.03804.
  * (56) X. Liu, F. Ringer, W. Vogelsang and F. Yuan, _Lepton-jet Correlations in Deep Inelastic Scattering at the Electron-Ion Collider_ , _Phys. Rev. Lett._ 122 (2019) 192003, [1812.08077].
  * (57) Y.-T. Chien, D. Y. Shao and B. Wu, _Resummation of Boson-Jet Correlation at Hadron Colliders_ , _JHEP_ 11 (2019) 025, [1905.01335].
  * (58) X. Liu, F. Ringer, W. Vogelsang and F. Yuan, _Lepton-jet Correlation in Deep Inelastic Scattering_ , _Phys. Rev. D_ 102 (2020) 094022, [2007.12866].
  * (59) Y.-T. Chien, R. Rahn, S. Schrijnder van Velzen, D. Y. Shao, W. J. Waalewijn and B. Wu, _Azimuthal angle for boson-jet production in the back-to-back limit_ , 2005.12279.
  * (60) M. I. Abdulhamid et al., _Azimuthal correlations of high transverse momentum jets at next-to-leading order in the parton branching method_ , _Eur. Phys. J. C_ 82 (2022) 36, [2112.10465].
  * (61) Y.-T. Chien, R. Rahn, D. Y. Shao, W. J. Waalewijn and B. Wu, _Precision boson-jet azimuthal decorrelation at hadron colliders_ , 2205.05104.
  * (62) H. Bouaziz, Y. Delenda and K. Khelifa-Kerfa, _Azimuthal decorrelation between a jet and a Z boson at hadron colliders_ , 2207.10147.
  * (63) H. Yang et al., _Back-to-back azimuthal correlations in $\mathrm{Z}+$jet events at high transverse momentum in the TMD parton branching method at next-to-leading order_, _Eur. Phys. J. C_ 82 (2022) 755, [2204.01528].
  * (64) A. B. Martinez and F. Hautmann, _Azimuthal di-jet correlations with parton branching TMD distributions_ , in _29th International Workshop on Deep-Inelastic Scattering and Related Subjects_ , 8, 2022, 2208.08446.
  * (65) M. Dasgupta and G. P. Salam, _Resummation of nonglobal QCD observables_ , _Phys. Lett. B_ 512 (2001) 323–330, [hep-ph/0104277].
  * (66) A. Banfi, G. Marchesini and G. Smye, _Away from jet energy flow_ , _JHEP_ 08 (2002) 006, [hep-ph/0206076].
  * (67) C. W. Bauer, S. Fleming, D. Pirjol and I. W. Stewart, _An Effective field theory for collinear and soft gluons: Heavy to light decays_ , _Phys. Rev._ D63 (2001) 114020, [hep-ph/0011336].
  * (68) C. W. Bauer and I. W. Stewart, _Invariant operators in collinear effective theory_ , _Phys. Lett._ B516 (2001) 134–142, [hep-ph/0107001].
  * (69) C. W. Bauer, D. Pirjol and I. W. Stewart, _Soft collinear factorization in effective field theory_ , _Phys. Rev._ D65 (2002) 054022, [hep-ph/0109045].
  * (70) C. W. Bauer, S. Fleming, D. Pirjol, I. Z. Rothstein and I. W. Stewart, _Hard scattering factorization from effective field theory_ , _Phys. Rev._ D66 (2002) 014017, [hep-ph/0202088].
  * (71) M. Beneke, A. P. Chapovsky, M. Diehl and T. Feldmann, _Soft collinear effective theory and heavy to light currents beyond leading power_ , _Nucl. Phys._ B643 (2002) 431–476, [hep-ph/0206152].
  * (72) S. R. Klein, J. Nystrand, J. Seger, Y. Gorbunov and J. Butterworth, _STARlight: A Monte Carlo simulation program for ultra-peripheral collisions of relativistic ions_ , _Comput. Phys. Commun._ 212 (2017) 258–268, [1607.03838].
  * (73) M. G. A. Buffing, Z.-B. Kang, K. Lee and X. Liu, _A transverse momentum dependent framework for back-to-back photon+jet production_ , 1812.07549.
  * (74) R. F. del Castillo, M. G. Echevarria, Y. Makris and I. Scimemi, _TMD factorization for di-jet and heavy meson pair in DIS_ , 2008.07531.
  * (75) R. F. del Castillo, M. G. Echevarria, Y. Makris and I. Scimemi, _Transverse momentum dependent distributions in dijet and heavy hadron pair production at EIC_ , _JHEP_ 03 (2022) 047, [2111.03703].
  * (76) M. Cacciari, G. P. Salam and G. Soyez, _The anti- $k_{t}$ jet clustering algorithm_, _JHEP_ 04 (2008) 063, [0802.1189].
  * (77) T. Becher and M. Neubert, _Drell-Yan Production at Small $q_{T}$, Transverse Parton Distributions and the Collinear Anomaly_, _Eur. Phys. J._ C71 (2011) 1665, [1007.4005].
  * (78) T. Becher, M. Neubert and D. Wilhelm, _Electroweak Gauge-Boson Production at Small $q_{T}$: Infrared Safety from the Collinear Anomaly_, _JHEP_ 02 (2012) 124, [1109.6027].
  * (79) J.-y. Chiu, A. Jain, D. Neill and I. Z. Rothstein, _The Rapidity Renormalization Group_ , _Phys. Rev. Lett._ 108 (2012) 151601, [1104.0881].
  * (80) J.-Y. Chiu, A. Jain, D. Neill and I. Z. Rothstein, _A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory_ , _JHEP_ 05 (2012) 084, [1202.0814].
  * (81) Z.-B. Kang, K. Lee, D. Y. Shao and J. Terry, _The Sivers Asymmetry in Hadronic Dijet Production_ , _JHEP_ 02 (2021) 066, [2008.05470].
  * (82) F. Caola, A. Von Manteuffel and L. Tancredi, _Diphoton Amplitudes in Three-Loop Quantum Chromodynamics_ , _Phys. Rev. Lett._ 126 (2021) 112004, [2011.13946].
  * (83) S. D. Ellis, C. K. Vermilion, J. R. Walsh, A. Hornig and C. Lee, _Jet Shapes and Jet Algorithms in SCET_ , _JHEP_ 11 (2010) 101, [1001.0014].
  * (84) G. Bell, R. Rahn and J. Talbert, _Generic dijet soft functions at two-loop order: correlated emissions_ , _JHEP_ 07 (2019) 101, [1812.08690].
  * (85) J. C. Collins, D. E. Soper and G. F. Sterman, _Transverse Momentum Distribution in Drell-Yan Pair and W and Z Boson Production_ , _Nucl. Phys. B_ 250 (1985) 199–224.
  * (86) J. Collins and T. Rogers, _Understanding the large-distance behavior of transverse-momentum-dependent parton densities and the Collins-Soper evolution kernel_ , _Phys.Rev._ D91 (2015) 074020, [1412.3820].
  * (87) C. Aidala, B. Field, L. Gamberg and T. Rogers, _Limits on TMD Evolution From Semi-Inclusive Deep Inelastic Scattering at Moderate $Q$_, _Phys.Rev._ D89 (2014) 094002, [1401.2654].
  * (88) P. Sun, J. Isaacson, C. P. Yuan and F. Yuan, _Nonperturbative functions for SIDIS and Drell–Yan processes_ , _Int. J. Mod. Phys. A_ 33 (2018) 1841006, [1406.3073].
  * (89) F. Landry, R. Brock, P. M. Nadolsky and C. P. Yuan, _Tevatron Run-1 Z boson data and Collins-Soper-Sterman resummation formalism_ , _Phys. Rev._ D67 (2003) 073016, [hep-ph/0212159].
  * (90) A. V. Konychev and P. M. Nadolsky, _Universality of the Collins-Soper-Sterman nonperturbative function in gauge boson production_ , _Phys.Lett._ B633 (2006) 710–714, [hep-ph/0506225].
  * (91) A. Bacchetta, F. Delcarro, C. Pisano, M. Radici and A. Signori, _Extraction of partonic transverse momentum distributions from semi-inclusive deep-inelastic scattering, Drell-Yan and Z-boson production_ , _JHEP_ 06 (2017) 081, [1703.10157].
  * (92) MAP collaboration, A. Bacchetta, V. Bertone, C. Bissolotti, G. Bozzi, M. Cerutti, F. Piacenza et al., _Unpolarized transverse momentum distributions from a global fit of Drell-Yan and semi-inclusive deep-inelastic scattering data_ , _JHEP_ 10 (2022) 127, [2206.07598].
  * (93) H.-S. Shao and D. d’Enterria, _gamma-UPC: Automated generation of exclusive photon-photon processes in ultraperipheral proton and nuclear collisions with varying form factors_ , 2207.03012.