# Production of the $\eta_{1}(1855)$ and its possible $C$-parity partner via
kaon induced reactions

Xiao-Yun Wang<EMAIL_ADDRESS>Department of physics, Lanzhou University of
Technology, Lanzhou 730050, China  Lanzhou Center for Theoretical Physics, Key
Laboratory of Theoretical Physics of Gansu Province, Lanzhou University,
Lanzhou, Gansu 730000, China  Fan-Cong Zeng  Department of physics, Lanzhou
University of Technology, Lanzhou 730050, China  Xiang Liu
<EMAIL_ADDRESS>School of Physical Science and Technology, Lanzhou
University, Lanzhou 730000, China  Research Center for Hadron and CSR Physics,
Lanzhou University and Institute of Modern Physics of CAS, Lanzhou 730000,
China  Lanzhou Center for Theoretical Physics, Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou, Gansu 730000, China
Frontier Science Center for Rare Isotopes, Lanzhou University, Lanzhou 730000,
China

###### Abstract

By the reaction of kaon interacting with a proton, we investigate the
production of the newly observed $\eta_{1}(1855)$ and its $C$-parity partner
$\eta_{1}^{\prime}(1855)$ predicted in the picture of the $K\bar{K}_{1}(1400)$
molecular state. The total and differential cross sections of the concrete
$K^{-}p\to\eta_{1}(1855)\Lambda$ and $\eta_{1}^{\prime}(1855)\Lambda$
reactions are obtained. At the center of mass energies about 3.4$\sim$3.6 GeV,
the minimum cross-section of $\eta_{1}(1855)$ and and
$\eta_{1}^{\prime}(1855)$ production via the $K^{-}p$ reaction are about 0.59
$nb$ and 19.2 $nb$, respectively. The differential cross sections for both
reactions at the different center of mass energies are also available.
Furthermore, we present the Dalitz processes of $2\rightarrow 3$ and
$2\rightarrow 4$, and initially discuss the feasibility of finding out the
$\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$ in experiments like J-PARC.

##  I Introduction

Until now, the BESIII experiment has collected the world largest data sample
of $J/\psi$, which provide good platform to carry out the study of light
hadron spectrum BESIII:2020nme . Recently, by analyzing the partial wave of
the $J/\psi\rightarrow\gamma\eta\eta^{\prime}$ decay, the BESIII Collaboration
BESIII:2022qzu ; BESIII:2022riz  reported the observation of the
$\eta_{1}(1855)$ in the $\eta\eta^{\prime}$ invariant mass spectrum with a
significance 19$\sigma$, which has quantum number
$I^{G}(J^{PC})=0^{+}(1^{-+})$. And its mass and width are

$\displaystyle m=1855\pm 9_{-1}^{+6}\text{ MeV},\quad\Gamma=188\pm
18_{-1}^{+6}\text{ MeV,}$ (1)

respectively. Obviously, the observed $\eta_{1}(1855)$ cannot be grouped into
the conventional hadron, which is a good candidate of exotic states as
indicated by some theoretical groups, where the $\eta_{1}(1855)$ was
interpreted as a hybrid state Chen:2022qpd ; Qiu:2022ktc ; Shastry:2022mhk , a
molecular state Dong:2022cuw ; Yang:2022lwq , and a tetraquark state
Wan:2022xkx .

At present, the $\eta_{1}(1855)$ was only observed in the radiative decay
$J/\psi\rightarrow\gamma\eta\eta^{\prime}$, confirming this observation by
others processes is an interesting experimental topic, which will be helpful
to establish this exotic state in experiment and may provide some valuable
information to reflect its inner structure. Under different assignments to the
$\eta_{1}(1855)$ Chen:2022qpd ; Dong:2022cuw , the strong decays of the
$\eta_{1}(1855)$ were studied, where the $\eta_{1}(1855)$ has strong
interaction with the $K^{(*)}\bar{K}^{(*)}$ channels. This result inspires us
to propose that the reaction of Kaon and proton can be applied to study the
production of the $\eta_{1}(1855)$.

If checking the Particle Data Group pdg2021 , we may find that the reaction of
kaon and proton is an ideal process to explore light hadron spectrum. Taking
the $\phi(1020)$ as example, the $\phi(1020)$ was firstly found in the
$K^{-}p\to\Lambda K\bar{K}$ reaction Schlein:1963zz . When investigating the
production problem of the $\eta_{1}(1855)$, we naturally pay attention to the
$Kp$ reaction, which is an effective approach among these possible production
processes. There are available experiments at OKA@U-70 Obraztsov:2016lhp  and
SPS@CERN Velghe:2016jjw , and J-PARC Nagae:2008zz ; jparc . For answering
whether the $\eta_{1}(1855)$ can be accessible at these facilities, we should
have a realistic study of the production of the $\eta_{1}(1855)$ via the
$K^{-}p$ reaction.

In this work, we calculate the production of $K^{-}p\to\eta_{1}(1855)\Lambda$
with a $t$-channel $K/K^{\ast}$ exchange, where the effective Lagrangian
approach and Regge model are adopted. In the following section, the details
will be given. Besides studying the production of the $\eta_{1}(1855)$, we
also explore the production of the predicted $\eta_{1}^{\prime}(1855)$ through
$K^{-}p\to\eta_{1}^{\prime}(1855)\Lambda$. In the picture of the
$K\bar{K}_{1}(1400)$ molecular state Dong:2022cuw , as the $C$-parity partner
of the $\eta_{1}(1855)$, the $\eta_{1}^{\prime}(1855)$ with $J^{PC}=1^{--}$
was predicted. Thus, searching for the predicted $\eta_{1}^{\prime}(1855)$ can
be as a test of molecule explanation to the $\eta_{1}(1855)$ Dong:2022cuw .

This paper is organized as follows. After introduction, we present formalism
including Lagrangians and amplitudes of the $\eta_{1}(1855)$ and
$\eta_{1}^{\prime}(1855)$ production in Sec. II. The numerical results are
discussed in Sec. III, followed by a brief summary in Sec. IV.

##  II The production of the $\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$
via the $Kp$ reaction

The exotic states $\eta_{1}(1855)$ $(\equiv\eta_{1})$ and
$\eta_{1}^{\prime}(1855)$ $(\equiv\eta_{1}^{\prime})$ can be produced via the
$Kp$ reaction, where the diagram is shown in Fig. 1 when the $t$-channel
exchange of the $K/K^{\ast}$ mesons is considered. In the present work, the
contribution from $s$-channel with nucleon pole is not considered since such
contribution is negligibly small Wang:2017qcw . In addition, since the
coupling strength of $\eta_{1}^{(\prime)}NN$ is unknown, and the Reggeized
$u$-channel with nucleon exchange usually has a small contribution to the
cross section Wang:2017qcw , therefore, the contributions from the $u$-channel
with nucleon exchange is also not considered in this work.

Figure 1: Schematic diagram for the $t$-channel contribution to the
$K^{-}p\rightarrow\eta_{1}^{(\prime)}\Lambda$ reaction.

For kaon induced production of the $\eta_{1}(1855)$ and its $C$-parity partner
$\eta_{1}^{\prime}(1855)$, the relevant Lagrangians for the $t$-channel read
as below Wan:2015gsl ; Ryu:2012tw ; Liu:2008qx ; Xiang:2020phx

$\displaystyle\mathcal{L}_{\eta_{1}^{(\prime)}KK}$ $\displaystyle=$
$\displaystyle-
ig_{\eta_{1}^{(\prime)}KK}[(\partial^{\mu}K)\bar{K}-(\partial^{\mu}\bar{K})K]\eta_{1\mu}^{(\prime)},$
(2) $\displaystyle\mathcal{L}_{KN\Lambda}$ $\displaystyle=$ $\displaystyle
ig_{KN\Lambda}\bar{N}\gamma_{5}\Lambda K{\ +}\text{ H.c.},$ (3)
$\displaystyle\mathcal{L}_{\eta_{1}^{(\prime)}K^{\ast}K}$ $\displaystyle=$
$\displaystyle\frac{g_{\eta_{1}^{(\prime)}K^{\ast}K}}{m_{\eta_{1}^{(\prime)}}}\epsilon_{\alpha\beta\mu\nu}\partial^{\beta}\eta_{1}^{(\prime)\alpha}\partial^{\nu}K^{\ast\mu}K,$
(4) $\displaystyle\mathcal{L}_{K^{\ast}N\Lambda}$ $\displaystyle=$
$\displaystyle-g_{K^{\ast}N\Lambda}\bar{N}\left(\hbox
to0.0pt{$/$\hss}K^{\ast}-\frac{\kappa_{K^{\ast}N\Lambda}}{2m_{N}}\sigma_{\mu\nu}\partial^{\nu}K^{\ast\mu}\right)\Lambda+\text{H.c.}~{},$
(5)

where $\epsilon_{\alpha\beta\mu\nu}$ is the Levi-Civita tensor. $\eta_{1}$,
$\eta_{1}^{\prime}$, ${K}$, $K^{\ast}$, $N$ and $\Lambda$ stand for the fields
of the $\eta_{1}(1855)$, $\eta_{1}^{\prime}(1855)$, $K$, $K^{\ast}$ meson,
nucleon and $\Lambda$, respectively. The coupling constant
$g_{KN\Lambda}=-13.24$ can be determined Wan:2015gsl  by the SU(3) flavor
symmetry relation Oh:2006hm ; Oh:2006in . Moreover, in the Nijmegen potential
Stoks:1999bz , the calculated results show that the values of the coupling
constants $g_{K^{\ast}N\Lambda}$ and $\kappa_{K^{\ast}N\Lambda}$ are -4.26 and
2.66, respectively. The partial widths of the $\eta_{1}(1855)$ and
$\eta_{1}^{\prime}(1855)$ decays to $KK$ and $KK^{\ast}$ are given in Ref.
Dong:2022cuw , by which one can calculate the corresponding coupling constants
$g_{\eta_{1}KK}\simeq 0$, $g_{\eta_{1}^{\prime}KK}\simeq 0.26$,
$g_{\eta_{1}K^{\ast}K}\simeq 0.73$ and $g_{\eta_{1}^{\prime}K^{\ast}K}\simeq
0.77$.

Table 1: The partial decay widths of the $\eta_{1}(1855)$ under the hybrid Chen:2022qpd  and molecular Dong:2022cuw  pictures, and the partial decay widths of the predicted $\eta_{1}^{\prime}(1855)$ in the molecular explanation Dong:2022cuw  . Here, the unit is MeV for the listed partial widths. Channels |  $\eta_{1}\,(J^{PC}=1^{-+})$ |  $\eta_{1}^{\prime}\,(J^{PC}=1^{--})$ |  |  |
---|---|---|---|---|---
Hybrid Chen:2022qpd  |  Molecule Dong:2022cuw  |  Molecule Dong:2022cuw  |  |  |
$K^{*}\bar{K}^{*}$ | — | 26.3 | 38.1 |  |  |
$K\bar{K}$ | — | 0 | 0.5 |  |  |
$K\bar{K}^{*}$ | $98.1$ | 0.9 | 1.0 |  |  |
$a_{1}\pi$ | — | 9.2 | 0 |  |  |
$f_{1}\eta$ | — | 0.2 | 0 |  |  |
$\eta\eta^{\prime}$ | — | 26.9 | 0 |  |  |
$\sigma\omega$ | — | 0 | 0.2 |  |  |
$\rho\rho$ | — | 0.04 | 0 |  |  |
$\pi\rho$ | — | 0 | 6.4 |  |  |
$\eta\omega$ | — | 0 | 0.4 |  |  |
$\omega\omega$ | — | 0.01 | 0 |  |  |
$\omega\phi$ | — | 0.4 | 0 |  |  |
$K\bar{K}^{*}\pi$ | — | 105.0 | 130.0 |  |  |

With the above Lagrangians, the amplitude of the $\eta_{1}(1855)$ and
$\eta_{1}^{\prime}(1855)$ production via $t$-channel $K$ or $K^{\ast}$
exchange in the $K^{-}p$ scatterings can be written as

$\displaystyle i\mathcal{M}_{K}$ $\displaystyle=$ $\displaystyle
g_{\eta_{1}^{(\prime)}KK}g_{KN\Lambda}F(q^{2})\epsilon_{\eta_{1}^{(\prime)}}^{\mu}(k_{2})\bar{u}_{N}(p_{2})\gamma_{5}\frac{1}{t-m_{K}^{2}}$
(6) $\displaystyle\times(k_{1\mu}+q_{\mu})u_{\Lambda}(p_{1}),$ $\displaystyle
i\mathcal{M}_{K^{\ast}}$ $\displaystyle=$ $\displaystyle
i\frac{g_{\eta_{1}^{(\prime)}K^{\ast}K}}{m_{\eta_{1}^{(\prime)}}}g_{K^{\ast}N\Lambda}F(q^{2})\bar{u}_{N}(p_{2})\left(\gamma_{\nu}-\frac{\kappa_{K^{\ast}N\Lambda}}{2m_{N}}\gamma_{\nu}\hbox
to0.0pt{$/$\hss}q_{K^{\ast}}\right)$ (7)
$\displaystyle\times\epsilon_{\alpha\beta\mu\xi}\frac{\mathcal{P}^{\mu\nu}}{t-m_{K^{\ast}}^{2}}k_{2}^{\beta}\epsilon_{\eta_{1}^{(\prime)}}^{\alpha}(k_{1}-k_{2})^{\xi}$

with

$\mathcal{P}^{\mu\nu}=i\left(g^{\mu\nu}+q_{K^{\ast}}^{\mu}q_{K^{\ast}}^{\nu}/m_{K^{\ast}}^{2}\right),$
(8)

where $\epsilon_{\eta_{1}^{(\prime)}}$ is the polarization vector of the
$\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$, and $\bar{u}_{N}$ or
$u_{\Lambda}$ is the Dirac spinor of nucleon or $\Lambda$ baryon. For the $t$
channel meson exchange Liu:2008qx , the form factor
$F(q^{2})=(\Lambda_{t}^{2}-m^{2})/(\Lambda_{t}^{2}-q^{2})$ is adopted. Here,
$t=q^{2}=(k_{1}-k_{2})^{2}$ is the Mandelstam variables. The cutoff
$\Lambda_{t}$ in the form factor is the only free parameter. In Sec. III, we
discuss how to fix it when presenting the results.

Usually, the Regge trajectory model is successfully applied to analyze the
hadron production at high energy Wan:2015gsl ; Wang:2015xwa ;
Haberzettl:2015exa ; Wang:2015hfm ; Ozaki:2010wp ; Wang:2017qcw ; Wang:2017plf
. The Reggeization can be done by replacing the $t$-channel propagator in the
Feynman amplitudes (Eqs. (6)-(7)) with the Regge propagator

$\displaystyle\frac{1}{t-m_{K}^{2}}$ $\displaystyle\rightarrow$
$\displaystyle\left(\frac{s}{s_{scale}}\right)^{\alpha_{K}(t)}\frac{\pi\alpha_{K}^{\prime}}{\Gamma[1+\alpha_{K}(t)]\sin[\pi\alpha_{K}(t)]},$
(9) $\displaystyle\frac{1}{t-m_{K^{\ast}}^{2}}$ $\displaystyle\rightarrow$
$\displaystyle\left(\frac{s}{s_{scale}}\right)^{\alpha_{K^{\ast}}(t)-1}\frac{\pi\alpha_{K^{\ast}}^{\prime}}{\Gamma[\alpha_{K^{\ast}}(t)]\sin[\pi\alpha_{K^{\ast}}(t)]}.$
(10)

The scale factor $s_{scale}$ is fixed at 1 GeV. In addition, the Regge
trajectories $\alpha_{K}(t)$ and $\alpha_{K^{\ast}}(t)$ read as Ozaki:2010wp

$\alpha_{K}(t)=0.70(t-m_{K}^{2}),\
\alpha_{K^{\ast}}(t)=1+0.85(t-m_{K^{\ast}}^{2}).\quad\ \ $ (11)

It is necessary to note that no additional parameter is introduced after
applying the Reggeized treatment .

##  III Numerical results

###  III.1 Cross section

With the preparation shown in the previous section, the cross section of the
$K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ and
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda$ reactions can be calculated.
The differential cross section in the center of mass (c.m.) frame is written
as

$\frac{d\sigma}{d\cos\theta}=\frac{1}{32\pi
s}\frac{\left|\vec{k}_{2}^{{~{}\mathrm{c.m.}}}\right|}{\left|\vec{k}_{1}^{{~{}\mathrm{c.m.}}}\right|}\left(\frac{1}{2}\sum\limits_{\lambda}\left|\mathcal{M}\right|^{2}\right),$
(12)

where $s=(k_{1}+p_{1})^{2}$ is defined, and $\theta$ denotes the angle of the
outgoing $\eta_{1}^{(\prime)}$ meson relative to the $K$ beam direction in the
c.m. frame. $\vec{k}_{1}^{{~{}\mathrm{c.m.}}}$ and
$\vec{k}_{2}^{{~{}\mathrm{c.m.}}}$ are the three-momenta of initial $K$ beam
and final $\eta_{1}^{(\prime)}$, respectively.

Since there does not exist the experimental data for the
$K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ and
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda$ reaction, here we give the
prediction of the cross section of these two reactions as presented in Figures
2-3. In these calculations, the cutoff parameter involved in the form factor
is the only free parameter. In Ref. Xiang:2020phx , by fitting the
experimental data of the $\pi^{-}p\rightarrow K^{\ast}\Sigma^{\ast}$ process,
the $\Lambda_{t}=1.67\pm 0.04$ GeV corresponding to the contributions of
Reggeized $t$-channel $K^{(\ast)}$ exchange is obtained. Moreover, in Ref.
Ozaki:2010wp , for the Reggeized $t$ channel with the $K$ and $K^{\ast}$
exchanges, the experimental data can be reproduced well by taking cutoff
$\Lambda_{t}=1.55$ GeV. Also in Ref. Wang:2018mjz , the result indicate that
$\Lambda_{t}=1.60$ GeV is a reasonable value for the $K^{-}p\rightarrow
f_{1}(1420)\Lambda$ process through Reggeized $t$-channel $K^{\ast}$ exchange.
Thus, in this work one intends to take the value of cutoff for the $t$-channel
$K^{(\ast)}$ exchange to be $1.6\pm 0.3$ GeV.

Figure 2: (color online). The energy dependence of the total cross section for
production of the $\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$ through $t$
channel with cutoff $\Lambda_{t}=1.6\pm 0.3$ GeV. The Full (red) and dashed
(blue) lines are for the $K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ and
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda$ reactions, respectively. The
bands stand for the error bar of the cutoff $\Lambda_{t}.$

In Fig. 2, we present the total cross section of the
$K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ and
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda$ reactions within the Regge
trajectory model by taking $\Lambda_{t}=1.6\pm 0.3$ GeV, respectively. It is
found that the line shape of the total cross sections of the
$K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ and
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda$ processes goes up very
rapidly and has a peak around $W=3.4\sim 3.6$ GeV. The peak of the total cross
section of the $\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$ production in
the $K$ induced reaction can reach up to about 2.5 $nb$ and 35 $nb$ at
$W=3.4\sim 3.6$ GeV, respectively, which indicates that the $W\in(3.4\,{\rm
GeV},3.6\,{\rm GeV})$ is the best energy window for searching for the
$\eta_{1}(1855)$ or $\eta_{1}^{\prime}(1855)$ via the kaon induced reaction.
Even if we take into account the error of the cutoff value, at the best energy
window, the minimum cross section of the $\eta_{1}(1855)$ or
$\eta_{1}^{\prime}(1855)$ production via the $K^{-}p$ reaction are about 0.59
$nb$ and 19.2 $nb$, respectively. In addition, the partial width of the decay
of $\eta_{1}$ to $KK^{\ast}$ given in the Ref. Chen:2022qpd  is 98.1 MeV, and
the coupling constant $g_{\eta_{1}K^{\ast}K}\simeq 7.6$ is determined by this
partial width, which is about an order of magnitude higher than the coupling
constant obtained by using the partial width in the Ref. Dong:2022cuw . The
corresponding cross section is also an order of magnitude higher than the
current cross section of $K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ in Fig. 2.
Of course, we still try to consider the small cross section value in the
subsequent discussion, which is helpful for us to examine the feasibility of
the experimental measurement.

In Fig. 3, one presents the prediction of the differential cross section of
the $K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ and
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda$ reactions within the Regge
trajectory model by taking a cutoff $\Lambda_{t}=1.6\pm 0.3$ GeV. It can be
seen that the differential cross sections of these two reactions are very
sensitive to the $\theta$ angle and show strong forward-scattering
enhancements especially at higher energies. Thus, the measurement at forward
angles is suggested, which can be used to check the validity of the Reggeized
treatment.

Figure 3: (color online). The differential cross section $d\sigma/d\cos\theta$
of the $\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$ production at different
c.m. energies $W=3.5\,{\rm GeV},\,3.6\,{\rm GeV},\,4,6\,{\rm GeV}$ and $8$
GeV. Here, the notations are the same as in Fig. 2. Figure 4: (color online).
The $t$-distribution for the $K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ and
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda$ reactions at different c.m.
energies $W=3.5{\rm GeV},\,3.6\,{\rm GeV}$ and $4$ GeV. Here, the notations
are as in Fig. 2.  Figure 5: (color online). Same as Fig. 4 except that the
c.m. energy is expanded to 6 GeV, 8 GeV and 10 GeV. Figure 6: (color online).
The limiting value of $|t|_{\min}$ as a function of $W$.

In Fig. 4 and Fig. 5, the $t$-distributions for the
$K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ and
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda$ at low and high energies are
given, respectively. As can be seen from both figures, in the small momentum
transfer $t$ region, there are some differences in the shape of the $t$
distributions of the $K^{-}p\rightarrow\eta_{1}(1855)\Lambda$ and
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda$ reaction, which may help us
distinguish the production mechanism of the $\eta_{1}(1855)$ and its possible
$C$-parity partner $\eta_{1}^{\prime}(1855)$. And through comparison, it is
found that the difference of this shape is more obvious at high energy, which
is mainly caused by the limitation of $\left|t_{\min}\right|$. Figure 6 shows
the change of $\left|t_{\min}\right|$ value with energy. When the center of
mass energy is 3.5 GeV and 6 GeV, the corresponding $\left|t_{\min}\right|$
values are 0.298 GeV2 and 0.046 GeV2, respectively. It means that at low
energies, neither experiment nor theory can give values of $t$ distributions
in the region of $t\sim 0$.

###  III.2 Dalitz process

Note that both $\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$ are the
intermediate states usually reconstructed by their decays. And from Table 1,
the decay widths of the $\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$ to
different final states are different. For example, the partial widths of
$\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$ decays to the $\rho\pi$ final
state are 0 MeV and 6.4 MeV, respectively. Thus, one should consider, for
example, the Dalitz process of
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda\rightarrow\rho\pi\Lambda$ and
the ratio of
$\sigma(K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda\rightarrow\rho\pi\Lambda)/\sigma(K^{-}p\rightarrow\rho\pi\Lambda)$,
which may provide a useful reference for future experiments. Usually, the
invariant mass distribution for the Dalitz process
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda\rightarrow\rho\pi\Lambda$ can
be defined with the two-body process Kim:2017nxg

$\frac{d\sigma_{K^{-}p\rightarrow\eta_{1}^{\prime}\Lambda\rightarrow\rho\pi\Lambda}}{dM_{\rho\pi}}\approx\frac{2M_{\eta_{1}^{\prime}}M_{\rho\pi}}{\pi}\frac{\sigma_{K^{-}p\rightarrow\eta_{1}^{\prime}\Lambda}\Gamma_{\eta_{1}^{\prime}\rightarrow\rho\pi}}{(M_{\rho\pi}^{2}-M_{\eta_{1}^{\prime}}^{2})^{2}+M_{\eta_{1}^{\prime}}^{2}\Gamma_{\eta_{1}^{\prime}}^{2}}.$

Simple to
$K^{-}p\rightarrow\eta_{1}^{\prime}(1855)\Lambda\rightarrow\rho\pi\Lambda$
process, the Dalitz process
$K^{-}p\rightarrow\eta_{1}(1855)\Lambda\rightarrow\eta(548)\eta^{\prime}(958)\Lambda$
can be wirtten as Kim:2017nxg

$\frac{d\sigma_{K^{-}p\rightarrow\eta_{1}\Lambda\rightarrow\eta\eta^{\prime}\Lambda}}{dM_{\eta\eta^{\prime}}}\approx\frac{2M_{\eta_{1}}M_{\eta\eta^{\prime}}}{\pi}\frac{\sigma_{K^{-}p\rightarrow\eta_{1}\Lambda}\Gamma_{\eta_{1}\rightarrow\eta\eta^{\prime}}}{(M_{\eta\eta^{\prime}}^{2}-M_{\eta_{1}}^{2})^{2}+M_{\eta_{1}}^{2}\Gamma_{\eta_{1}}^{2}},$

where the full width $\Gamma_{\eta_{1}^{\prime}}$ and $\Gamma_{\eta_{1}}$ are
both taken as $188$ MeV. For the partial width
$\Gamma_{\eta_{1}\rightarrow\eta\eta^{\prime}}$ and
$\Gamma_{\eta_{1}^{\prime}\rightarrow\rho\pi}$, one take the theoretical
prediction value predicted in Ref. Dong:2022cuw  as
$\Gamma_{\eta_{1}\rightarrow\eta\eta^{\prime}}$ and
$\Gamma_{\eta_{1}^{\prime}\rightarrow\rho\pi}$ is taken as 26.9 MeV and 6.4
MeV, respectively, as listed in Table 1. Thus the invariant-mass distribution
$d\sigma_{K^{-}p\rightarrow\eta_{1}\Lambda\rightarrow\eta\eta^{\prime}\Lambda}/dM_{\eta\eta^{\prime}}$
and
$d\sigma_{K^{-}p\rightarrow\eta_{1}^{\prime}\Lambda\rightarrow\rho\pi\Lambda}/dM_{\rho\pi}$
for $W=3.5-8$ GeV are calculated, as shown in Fig. 7 and Fig. 8. It is seen
that there exists an obvious peak at $M_{\rho\pi}$ and $M_{\eta\eta^{\prime}}$
near $1.86$ GeV. These results will inform future experimental measurements.

Figure 7: The invariant mass distribution
$d\sigma_{K^{-}p\rightarrow\eta_{1}\Lambda\rightarrow\eta\eta^{\prime}\Lambda}/dM_{\eta\eta^{\prime}}$
reactions at different c.m. energies $W=3.5\,{\rm GeV},\,3.6\,{\rm
GeV},\,4\,{\rm GeV},\,6\,{\rm GeV}$ and $8$ GeV. Here, the cutoff
$\Lambda_{t}$ is taken as 1.6 GeV. Figure 8: The invariant mass distribution
$d\sigma_{K^{-}p\rightarrow\eta_{1}^{\prime}\Lambda\rightarrow\rho\pi\Lambda}/dM_{\rho\pi}$
reactions at different c.m. energies $W=3.5\,{\rm GeV},\,3.6\,{\rm
GeV},\,4\,{\rm GeV},\,6\,{\rm GeV}$ and $8$ GeV. Here, the cutoff
$\Lambda_{t}$ is taken as 1.6 GeV.

As mentioned above, the minimum cross section of $\eta_{1}^{\prime}(1855)$
production via the $K^{-}p$ reaction is above 19.2 $nb$ at $W=3.5$ GeV. With
the branching ratio $BR(\eta_{1}^{\prime}\rightarrow
K\bar{K}^{*}\pi)=(\Gamma_{\eta_{1}^{\prime}\rightarrow
K\bar{K}^{*}\pi}/\Gamma_{\eta_{1}^{\prime}})\times 100\%\approx 69.1\%$, one
obtains the total cross section
$\sigma_{K^{-}p\rightarrow\eta_{1}^{\prime}\Lambda\rightarrow
K\bar{K}^{*}\pi\Lambda}\approx 13.3$ $nb$ at $W=3.5$ GeV. There are several
experimental data available for the cross section of the
$K^{-}p\rightarrow\Lambda\pi KK^{*}$ reaction data , which are listed in Table
2.

Table 2:  The cross section for the $K^{-}p\to\Lambda\pi KK^{*}$ reaction at different beam momentum from several experiments data .  Reaction |  $W$ (GeV) |  Cross-section ($\mu b$)
---|---|---
$K^{-}p\to\Lambda\pi^{+}K^{0}K^{*-}$ | 3.52 | $13\pm 6$
$K^{-}p\to\Lambda\pi^{0}K^{0}K^{*0}$ | 3.52 | $12\pm 9$
$K^{-}p\to\Lambda\pi^{-}K^{+}K^{*0}$ | 3.52 | $16.5\pm 7.5$

In the $K^{-}p\rightarrow\Lambda\pi KK^{*}$ process, we estimate the total
cross section is 13.8 $\mu b$ near the c.m. energy $W=3.5$ GeV according to
the average cross-section of these energies. Therefore, we get the the ratio
at $W=3.5$ GeV as

$\frac{\sigma(K^{-}p\rightarrow\eta_{1}^{\prime}\Lambda\rightarrow
K\bar{K}^{*}\pi\Lambda)}{\sigma(K^{-}p\rightarrow
K\bar{K}^{*}\pi\Lambda)}\simeq 0.096\%.$ (13)

According to the minimum cross section of the $\eta_{1}(1855)$ production is
above 0.59 $nb$ near the $W=3.5$ GeV and the branching ratio
$BR(\eta_{1}\rightarrow K\bar{K}^{*}\pi)=55.9\%$, we get the the cross-section
ratio at $W=3.5$ GeV as

$\frac{\sigma(K^{-}p\rightarrow\eta_{1}\Lambda\rightarrow
K\bar{K}^{*}\pi\Lambda)}{\sigma(K^{-}p\rightarrow
K\bar{K}^{*}\pi\Lambda)}\simeq 0.0024\%.$ (14)

When checking the present experimental status, we notice that the J-PARC
experiment can provide Kaon meson beams Nagae:2008zz ; jparc , which an ideal
facility for probing the $\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$
production by the $Kp$ reaction. Assuming that the acceptance of the
$K^{-}p\rightarrow K\bar{K}^{*}\pi\Lambda$ reaction on J-PARC can reach $50\%$
and taking the branching ratios of the $K^{*-}$ decay chain to be as 0.67, it
can be expected to detect about $0.9\times 10^{5}$ events for the
$K\bar{K}^{*}\pi\Lambda$ production per day, in which about dozens of events
are related to the $\eta_{1}(1855)$ and $\eta_{1}^{\prime}(1855)$.

##  IV Summary

Recently, the BESIII Collaboration observed $\eta_{1}(1855)$ in the
$J/\psi\to\gamma\eta\eta^{\prime}$ decay BESIII:2022riz . Since the
$\eta_{1}(1855)$ is an isoscalar resonance with exotic $J^{PC}=1^{-+}$ quantum
numbers, different explanations of exotic states like hybrid Chen:2022qpd ;
Qiu:2022ktc ; Shastry:2022mhk , the $K\bar{K}_{1}(1400)$ molecular state
Dong:2022cuw ; Yang:2022lwq , and tetraquark state Wan:2022xkx  were proposed,
which shows that this observation attracted theorist’s attention in the past
months. Besides decoding the nature of the observed $\eta_{1}(1855)$,
establishing the $\eta_{1}(1855)$ in experiment is also crucial. How to
confirm the observation of the $\eta_{1}(1855)$ by other experiments becomes a
central issue.

In this work, we propose that the $Kp$ reaction can be as one way to further
explore the $\eta_{1}(1855)$, which is stimulated by the study of its strong
decays Dong:2022cuw . Our calculation provides the information of the total
and differential cross sections of the $K^{-}p\to\eta_{1}(1855)\Lambda$
reaction. We also notice the possible $C$-parity partner
$\eta_{1}^{\prime}(1855)$, which was predicted in the picture of the
$K\bar{K}_{1}(1400)$ molecular state Dong:2022cuw . Thus, in this work, the
production of the predicted $\eta_{1}^{\prime}(1855)$ via the $Kp$ reaction is
discussed, where the $K^{-}p\to\eta_{1}(1855)^{\prime}\Lambda$ process is
focused. For telling our experimental colleague more information, we also
further analyze the relevant Dalitz processes, which is also valuable to
future experimental search for them.

In summary, we suggest these available experiments at OKA@U-70
Obraztsov:2016lhp  and SPS@CERN Velghe:2016jjw , and J-PARC Nagae:2008zz ;
jparc  to explore the newly reported $\eta_{1}(1855)$ and its possible partner
$\eta_{1}^{\prime}(1855)$ due to considerable events as estimated in the
present work.

##  V Acknowledgments

This project is supported by the National Natural Science Foundation of China
under Grant Nos. 12065014 and 12047501, and by the West Light Foundation of
The Chinese Academy of Sciences, Grant No. 21JR7RA201. X.L. is also supported
by the China National Funds for Distinguished Young Scientists under Grant No.
11825503, National Key Research and Development Program of China under
Contract No. 2020YFA0406400, and the 111 Project under Grant No. B20063, and
by the Fundamental Research Funds for the Central Universities.

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Goutam Das

# Automated Calculation of Beam Functions at NNLO

Guido Bell  Kevin Brune  Marcel Wald

###### Abstract

We present an automated framework for the calculation of beam functions that
describe collinear initial-state radiation at hadron colliders at next-to-
next-to leading order (NNLO) in perturbation theory. By exploiting the
infrared behaviour of the collinear matrix elements, we factorise the phase-
space singularities with suitable observable-independent parametrisations. Our
numerical approach applies to a large class of collider observables, and as a
check of its validity, we compute the quark beam functions for transverse-
momentum resummation and N-jettiness, which are known analytically at this
order, finding excellent agreement.

##  1 Introduction

Beam functions constitute a key ingredient in factorisation theorems at hadron
colliders. For the measurement of a global observable $\omega$ that is
sensitive to soft and collinear QCD radiation, the differential cross section
typically takes the following schematic form,

$\displaystyle\frac{{\rm d}\sigma}{{\rm
d}\omega}=H(Q)\cdot\prod_{i}B_{i/h}(\omega)\otimes\prod_{j}J_{j}(\omega)\otimes
S(\omega)\,.$ (1)

The hard function $H(Q)$ describes the virtual corrections to the Born
process, and it depends only on the hard scale $Q$ of the process, while it is
independent of the specific measurement $\omega$. On the other hand, the beam
functions $B_{i/h}(\omega)$, the jet functions $J_{j}(\omega)$ and the soft
function $S(\omega)$ describe initial-state collinear, final-state collinear
and soft emissions, respectively, and they are observable-dependent. One thus
needs to compute these functions on a case-by-case basis for each observable,
which for the beam functions has been achieved either analytically [1, 2, 3,
4, 5, 6, 7, 8, 9, 10, 11] or semi-analytically [12, 13] in some cases relevant
e.g. for transverse-momentum or jet-veto resummation. An automated approach
for the calculation of beam functions at next-to-next-to-leading order (NNLO)
in perturbation theory has been initiated only recently [14, 15], and in this
article we report on the status of these developments.

An automated framework for the calculation of soft functions at NNLO is
already available through the public package SoftSERVE [16, 17, 18]. This has
been achieved by introducing suitable phase-space parametrisations to
factorise the divergences of the soft matrix elements at the second order of
the strong coupling. The singularity structure of the collinear matrix
elements is, on the other hand, significantly more complicated. The beam
functions are, moreover, defined as proton matrix elements of collinear field
operators, and they need to be matched onto the standard parton distribution
functions to extract the relevant perturbative information. By employing
suitable phase-space parametrisations, sector-decomposition techniques and
non-linear transformations, we recently set up a similar automated framework
for the calculation of the beam-function matching kernels [14]. As a first
application of our approach, we computed the quark beam function for jet-veto
resummation [15], and in this work we present our results for transverse-
momentum resummation and the event-shape variable N-jettiness. While these
beam functions are known analytically at the considered NNLO for quite some
time [2, 6], they provide important reference observables for our setup. In
particular, they allow us to test the numerical accuracy of our predictions
for different classes of observables, which are known as SCET-1 and SCET-2
beam functions.

##  2 Quark beam functions

We are concerned with quark beam functions that are defined via

$\displaystyle\frac{1}{2}\left[\frac{{\not{n}}}{2}\right]_{\beta\alpha}{\cal
B}_{q/h}(x,\tau,\mu)=$
$\displaystyle\sum_{X}\,\delta\Big{(}(1-x)P^{-}-\sum_{i}k_{i}^{-}\Big{)}\,{\cal
M}(\tau;\\{k_{i}\\})\bra{h(P)}\bar{\chi}_{\alpha}\ket{X}\bra{X}\chi_{\beta}\ket{h(P)},$
(2)

where $\chi=W^{\dagger}_{\bar{n}}\frac{{\not{n}}{\not{\bar{n}}}}{4}\psi$ is
the collinear quark field, and we used light-cone coordinates with
$k_{i}^{-}=\bar{n}\cdot k_{i}$, $k_{i}^{+}=n\cdot k_{i}$ and a transverse
component $k_{i}^{\perp,\mu}$ that satisfies $n\cdot
k_{i}^{\perp}=\bar{n}\cdot k_{i}^{\perp}=0$, along with $n^{2}=\bar{n}^{2}=0$
and $n\cdot\bar{n}=2$. The sum over $X$ represents the phase space of the
collinear emissions with momenta $\\{k_{i}\\}$, while the external state
$\ket{h(P)}$ refers to a hadronic state with momentum
$P^{\mu}=P^{-}n^{\mu}/2$. The function ${\cal M}(\tau;\\{k_{i}\\})$
furthermore specifies the observable, and in order to avoid distribution-
valued expressions, we assume that it is given in Laplace space, with $\tau$
being the corresponding Laplace variable (see also [16, 17, 18]).

Unlike soft and jet functions, beam functions are intrinsically non-
perturbative objects, but as long as the relevant scale of the collinear
emissions is perturbative, i.e. $\tau\ll 1/\Lambda_{\rm QCD}$, they can be
matched onto the usual parton distribution functions (pdf). The matching
relation is most conveniently expressed in Mellin space,
${\cal\widehat{B}}_{q/h}(N,\tau,\mu)=\int_{0}^{1}dx\;x^{N-1}\;{\cal
B}_{q/h}(x,\tau,\mu)$, where it becomes

$\displaystyle\mathcal{\widehat{B}}_{q/h}(N,\tau,\mu)$
$\displaystyle=\sum_{k}\,{\cal\widehat{I}}_{q\leftarrow
k}(N,\tau,\mu)\leavevmode\nobreak\ \widehat{f}_{k/h}(N,\mu)\,,$ (3)

and the sum runs over all partonic channels. The development of an automated
framework to compute the matching kernels ${\cal\widehat{I}}_{q\leftarrow
k}(N,\tau,\mu)$ to NNLO accuracy is the goal of the current project. The
matching kernels can, in fact, be extracted from partonic rather than hadronic
beam functions, i.e. the external states $\ket{h(P)}$ in (2) can be
interpreted as partonic states for this purpose. If the matching is performed
on-shell in dimensional regularisation, the partonic pdf evaluate to
$\widehat{f}_{k/j}(N,\mu)=\delta_{kj}$ to all orders in perturbation theory,
and the extraction of the matching kernels boils down to the calculation of
the bare partonic beam functions.

Depending on the observable, dimensional regularisation may not be sufficient
to resolve all phase-space singularities that appear in the beam-function
calculation. It is well-known that one needs an additional prescription to
regularise rapidity divergences for transverse-momentum dependent SCET-2
observables. In our approach, we regularise these divergences with a symmetric
version of the phase-space regulator proposed in [19]. We thus introduce the
following phase-space factor for each emission with momentum $k_{i}^{\mu}$,

$\int
d^{d}k_{i}\;\left(\frac{\nu}{k_{i}^{-}+k_{i}^{+}}\right)^{\alpha}\;\delta(k_{i}^{2})\,\theta(k_{i}^{0})\,,$
(4)

where $\alpha$ is the rapidity regulator and $\nu$ the corresponding rapidity
scale. The choice of a symmetric regulator under $n\leftrightarrow\bar{n}$
exchange enables us to derive the anti-collinear beam function directly from
the collinear one. For consistency one then also has to calculate the soft
function in the same regularisation scheme, for which we make use of
SoftSERVE.

We finally expand the bare matching kernels in the renormalised strong
coupling $\alpha_{S}$ as

$\displaystyle{\cal\widehat{I}}_{q\leftarrow
k}^{0}(N,\tau,\nu)=\delta_{qk}+\bigg{(}\frac{Z_{\alpha}\alpha_{s}}{4\pi}\bigg{)}\,\big{(}\mu^{2}\bar{\tau}^{2}\big{)}^{\epsilon}\;\bigg{(}\frac{\nu}{q_{-}}\bigg{)}^{\alpha}\,{\cal\widehat{I}}_{q\leftarrow
k}^{R}(N,\epsilon,\alpha)$ (5)
$\displaystyle\quad+\bigg{(}\frac{Z_{\alpha}\alpha_{s}}{4\pi}\bigg{)}^{2}\,\big{(}\mu^{2}\bar{\tau}^{2}\big{)}^{2\epsilon}\;\Bigg{\\{}\bigg{(}\frac{\nu}{q_{-}}\bigg{)}^{\alpha}\,{\cal\widehat{I}}_{q\leftarrow
k}^{RV}(N,\epsilon,\alpha)+\bigg{(}\frac{\nu}{q_{-}}\bigg{)}^{2\alpha}\,{\cal\widehat{I}}_{q\leftarrow
k}^{RR}(N,\epsilon,\alpha)\Bigg{\\}}+\mathcal{O}(\alpha_{s}^{3})\,,$

where $\epsilon=(4-d)/2$ is the dimensional regulator, $\bar{\tau}=\tau
e^{\gamma_{E}}$, and $Z_{\alpha}=1-\beta_{0}\alpha_{s}/(4\pi\epsilon)$ is the
coupling renormalisation factor in the $\overline{\text{MS}}$-scheme. We
furthermore traded the large component $P_{-}$ in the beam-function definition
by the component $q_{-}=xP_{-}$ that enters the hard interaction, before
performing the Mellin transformation. We note that the rapidity regulator
$\alpha$ is only required for SCET-2 observables and it can thus be set to
zero in the SCET-1 case.

##  3 NLO calculation

As purely virtual corrections are scaleless and vanish in our setup, only the
real-emission process contributes at NLO. Denoting the momentum of the emitted
parton by $k^{\mu}$, the on-shell condition along with the delta function in
(2) fixes two light-cone components to $k^{+}=|\vec{k}^{\perp}|^{2}/k^{-}$ and
$k^{-}=(1-x)P^{-}$. The remaining components are then parametrised as

$\displaystyle k_{T}=|\vec{k}^{\perp}|\,,\qquad\qquad
t_{k}=\frac{1-\cos\theta_{k}}{2}\,,$ (6)

where $\vec{k}^{\perp}$ is the transverse part of $k^{\mu}$ in light-cone
coordinates, and $\theta_{k}$ is its angle with respect to a reference vector
$\vec{v}^{\perp}$ in the transverse plane.

In terms of these variables, we write the one-emission measurement function in
the form (see also [16, 17, 18]),

$\displaystyle{\cal M}_{1}(\tau;k)$ $\displaystyle=\exp\left[-\tau
k_{T}\left(\frac{k_{T}}{(1-x)P^{-}}\right)^{n}f(t_{k})\right].$ (7)

The observable is thus characterised by the parameter $n$, which controls the
scaling of the observable in the soft-collinear limit [17], and its azimuthal
dependence is described by the function $f(t_{k})$. With this ansatz one can
derive a master formula for the calculation of the NLO matching kernels,

$\displaystyle{\cal\widehat{I}}_{q\leftarrow q}^{R}(N,\epsilon,\alpha)=$
$\displaystyle\left(\tau
q_{-}\right)^{\frac{-2n\epsilon}{1+n}}\frac{8e^{-\gamma_{E}\epsilon}}{(1+n)\sqrt{\pi}}\,\frac{\Gamma\left(-\frac{2\epsilon}{1+n}\right)}{\Gamma\left(\frac{1}{2}-\epsilon\right)}$
(8)
$\displaystyle\times\;\int_{0}^{1}\\!dx\;\,x^{N+\frac{2n\epsilon}{1+n}+\alpha}\;\bar{x}^{-1-\frac{2n\epsilon}{1+n}-\alpha}\;\left[\bar{x}\,\mathbb{P}_{q\to
q^{*}g}^{(0)}(x)\right]\;\int_{0}^{1}\\!dt_{k}\;(4t_{k}\bar{t}_{k})^{-\frac{1}{2}-\epsilon}\;f(t_{k})^{\frac{2\epsilon}{1+n}}\,,$

where we introduced the notation $\bar{x}=1-x$, etc. The off-diagonal kernel
${\cal\widehat{I}}_{q\leftarrow g}^{R}(N,\epsilon,\alpha)$ takes a similar
form with $\mathbb{P}_{q\to q^{*}g}^{(0)}(x)$ replaced by $\mathbb{P}_{g\to
q^{*}\bar{q}}^{(0)}(x)$. The latter quantities are related to the well-known
(crossed) splitting functions, and they are given by

$\displaystyle\mathbb{P}_{q\to q^{*}g}^{(0)}(x)$
$\displaystyle=\frac{C_{F}}{x\bar{x}}\left[1+x^{2}-\epsilon\bar{x}^{2}\right]\,,\qquad\mathbb{P}_{g\to
q^{*}\bar{q}}^{(0)}(x)=\frac{T_{F}}{(1-\epsilon)x}\left[x^{2}+\bar{x}^{2}-\epsilon\right].$
(9)

From the master formula in (8) we read off that the rapidity regulator
$\alpha$ is only required for the diagonal channel to control the divergence
in the limit $x\to 1$ as long as $n=0$, which corresponds to the SCET-2 case.

##  4 NNLO calculation

At NNLO there are two different types of contributions, viz. the real-virtual
(RV) and the real-real (RR) contribution. In the former case, the calculation
follows along the lines outlined above, with additional explicit divergences
coming from the one-loop corrections to the splitting functions [20, 21, 22].
In particular, the phase-space divergences can still be exposed with the
parametrisation given in (6), and one can derive a similar master formula as
for NLO contribution.

The RR case, on the other hand, is more complicated. It involves two emissions
with momenta $k^{\mu}$ and $l^{\mu}$, which we parametrise according to

$\displaystyle a=\frac{k^{-}l_{T}}{l^{-}k_{T}},\qquad
b=\frac{k_{T}}{l_{T}},\qquad z=\frac{k^{-}+l^{-}}{P^{-}},\qquad
q_{T}=\sqrt{(k^{-}+l^{-})(k^{+}+l^{+})}\,,$ (10)

where again $k_{T}=|\vec{k}^{\perp}|$ and $l_{T}=|\vec{l}^{\perp}|$. In
physical terms, the variable $a$ represents a measure of the rapidity
difference of the two emissions, $b$ is the ratio of their transverse
components, and $z$ is the splitting variable of the joint system composed out
of the two emitted particles. Notice that $q_{T}$ is the only dimensionful
variable in this parametrisation, whereas there are further angular variables
that we parametrise similar to the NLO case from above. Two of these angular
variables ($t_{k},t_{l}$) are defined with respect to the reference vector
$\vec{v}^{\perp}$, whereas the third one $(t_{kl})$ represents the relative
angle between the two emitted partons in the transverse plane.

The matrix element of the RR contribution is proportional to the leading-order
triple-collinear splitting functions [23, 24], which contain overlapping
divergences that cannot be disentangled with a single parametrisation. We then
start from the following ansatz for the two-emission measurement function,

$\displaystyle{\cal M}_{2}(\tau;k,l)$ $\displaystyle=\exp\left[-\tau
q_{T}\left(\frac{q_{T}}{(1-x)P^{-}}\right)^{n}{\cal
F}(a,b,z,t_{k},t_{l},t_{kl})\right],$ (11)

in which the observable is again described by the parameter $n$ and a function
${\cal F}(a,b,z,t_{k},t_{l},t_{kl})$, whose explicit form can become quite
lengthy. Even worse, this function may vanish in the singular limits of the
matrix element, and in this case one needs to apply sector-decomposition steps
to correctly extract the associated divergences.

With the above form of the measurement function, one can easily follow the
dependence on the variable $q_{T}$, which we integrate out analytically. In
order to factorise the remaining phase-space divergences, we apply a mixed
strategy that consists of sector-decomposition steps, non-linear
transformations and selector functions. This allows us to bring all
divergences into monomial form, and we finally perform a Laurent expansion in
the two regulators. For the numerical integration of the coefficients in this
double expansion, we rely on pySecDec [25] and its Cuba implementation [26].

##  5 Renormalisation

While the computation of the bare matching kernels can be performed in a
universal framework for SCET-1 and SCET-2 observables, their renormalisation
aspects are different, and we discuss them one by one in this section.

###  5.1 SCET-1 observables

In the combined Mellin-Laplace space the renormalisation of the beam-function
matching kernels takes a multiplicative form, ${\cal\widehat{I}}_{q\leftarrow
j}=Z_{q}^{B}\sum_{k}{\cal\widehat{I}}_{q\leftarrow
k}^{0}\,\widehat{Z}_{k\leftarrow j}^{f}$, where $Z_{q}^{B}$ subtracts the UV
divergences of the beam function, whereas $\widehat{Z}_{k\leftarrow j}^{f}$
captures the IR divergences that match the UV divergences of the pdf. The
renormalised matching kernels fulfil the renormalisation-group equation (RGE)

$\displaystyle\frac{d}{d\ln\mu}\;{\cal\widehat{I}}_{i\leftarrow
j}(N,\tau,\mu)=$ $\displaystyle\left[2g(n)\,\Gamma_{\rm
cusp}^{i}(\alpha_{S})\,L+\gamma^{B}(\alpha_{S})\right]{\cal\widehat{I}}_{i\leftarrow
j}(N,\tau,\mu)$ $\displaystyle-2\sum_{k}\,{\cal\widehat{I}}_{i\leftarrow
k}(N,\tau,\mu)\,\widehat{P}_{k\leftarrow j}(N,\alpha_{S})\,,$ (12)

where $g(n)=(n+1)/n$,
$L=\ln\big(\mu\bar{\tau}/\left(q_{-}\bar{\tau}\right)^{1/g(n)}\big{missing})$,
$\Gamma_{\rm cusp}^{i}(\alpha_{S})$ is the cusp anomalous dimensions in the
representation of the parton $i$, $\gamma^{B}(\alpha_{S})$ is the non-cusp
anomalous dimension, and $\widehat{P}_{k\leftarrow j}(N,\alpha_{S})$ are the
DGLAP splitting functions in Mellin space. Expanding the anomalous dimensions
in the form
$G(\alpha_{s})=\sum_{m=0}^{\infty}G_{m}\,(\frac{\alpha_{s}}{4\pi})^{m+1}$, the
two-loop solution of the RGE becomes,

$\displaystyle{\cal\widehat{I}}_{i\leftarrow
j}(N,\tau,\mu)=\delta_{ij}+\left(\frac{\alpha_{s}}{4\pi}\right)\bigg{\\{}\Big{(}g(n)\,\Gamma_{0}^{i}\,L^{2}+\gamma_{0}^{B}\,L\Big{)}\delta_{ij}-2L\,\widehat{P}_{i\leftarrow
j}^{(0)}(N)+\widehat{I}_{i\leftarrow j}^{(1)}(N)\bigg{\\}}$ (13)
$\displaystyle\;+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\bigg{\\{}\bigg{(}\frac{g(n)^{2}\,(\Gamma_{0}^{i})^{2}}{2}L^{4}+g(n)\,\Gamma_{0}^{i}\Big{(}\gamma_{0}^{B}+\frac{2\beta_{0}}{3}\Big{)}L^{3}+\Big{(}g(n)\,\Gamma_{1}^{i}+\frac{1}{2}(\gamma_{0}^{B})^{2}+\beta_{0}\gamma_{0}^{B}\Big{)}L^{2}+\gamma_{1}^{B}L\bigg{)}\delta_{ij}$
$\displaystyle\qquad\qquad\quad-2\Big{(}g(n)\,\Gamma_{0}^{i}L^{3}+\big{(}\beta_{0}+\gamma_{0}^{B}\big{)}L^{2}\Big{)}\widehat{P}_{i\leftarrow
j}^{(0)}(N)+\Big{(}g(n)\,\Gamma_{0}^{i}L^{2}+(\gamma_{0}^{B}+2\beta_{0})L\Big{)}\,\widehat{I}_{i\leftarrow
j}^{(1)}(N)$
$\displaystyle\qquad\qquad\quad+2\sum_{k}\bigg{(}L^{2}\,\widehat{P}_{i\leftarrow
k}^{(0)}(N)-L\,\widehat{I}_{i\leftarrow
k}^{(1)}(N)\bigg{)}\widehat{P}_{k\leftarrow
j}^{(0)}(N)-2L\,\widehat{P}_{i\leftarrow j}^{(1)}(N)+\widehat{I}_{i\leftarrow
j}^{(2)}(N)\bigg{\\}}\,.$

The renormalisation constants $Z_{q}^{B}$ and $\widehat{Z}_{k\leftarrow
j}^{f}$ fulfil similar RGE that are controlled by the first line or the second
line of (12), respectively. Their explicit form up to two loop-order can be
found e.g. in [14] for $Z_{q}^{B}$ and in [15] for $\widehat{Z}_{k\leftarrow
j}^{f}$. From the pole terms of the bare matching kernels we then extract the
non-cusp anomalous dimension $\gamma_{m}^{B}$, and from the finite terms we
obtain the non-logarithmic coefficients $\widehat{I}_{i\leftarrow
j}^{(m)}(N)$, which we sample for different values of the Mellin parameter
$N$.

###  5.2 SCET-2 observables

In the SCET-2 case we follow the collinear-anomaly approach [27, 28], which
states that the product of the soft, collinear and anti-collinear functions
can be refactorised in the form

$\displaystyle\left[\widehat{\cal I}_{q\leftarrow
i}(N_{1},\tau,\mu,\nu)\;\widehat{\cal I}_{\bar{q}\leftarrow
j}(N_{2},\tau,\mu,\nu)\;{\cal S}_{q\bar{q}}(\tau,\mu,\nu)\right]_{Q}$
$\displaystyle\qquad=\left(Q\bar{\tau}\right)^{-2F_{q\bar{q}}(\tau,\mu)}\;\widehat{I}_{q\leftarrow
i}(N_{1},\tau,\mu)\;\widehat{I}_{\bar{q}\leftarrow j}(N_{2},\tau,\mu)\,,$ (14)

where $Q^{2}=q_{+}q_{-}$, and the quantities on the right-hand side are
referred to as the collinear-anomaly exponent $F_{q\bar{q}}(\tau,\mu)$ and the
refactorised matching kernels $\widehat{I}_{i\leftarrow j}(N,\tau,\mu)$. The
former renormalises additively,
$F_{q\bar{q}}^{0}=F_{q\bar{q}}+Z_{q\bar{q}}^{F}$, and it obyes the RGE

$\displaystyle\frac{d}{d\ln\mu}\;F_{q\bar{q}}(\tau,\mu)=2\,\Gamma_{\rm
cusp}^{q}(\alpha_{S})\,,$ (15)

which is solved by

$\displaystyle F_{q\bar{q}}(\tau,\mu)$
$\displaystyle=\left(\frac{\alpha_{s}}{4\pi}\right)\Big{\\{}2\Gamma_{0}^{q}\,L+d_{1}\Big{\\}}+\left(\frac{\alpha_{s}}{4\pi}\right)^{2}\Big{\\{}2\beta_{0}\Gamma_{0}^{q}\,L^{2}+2\left(\Gamma_{1}^{q}+\beta_{0}d_{1}\right)L+d_{2}\Big{\\}},$
(16)

where now $L=\ln(\mu\bar{\tau})$. The two-loop expression for the anomaly
counterterm $Z_{q\bar{q}}^{F}$ can be found e.g. in [15].

The refactorised matching kernels, on the other hand, obey an RGE that is
structually of the same form as the one discussed in the previous section with
$g(n)\to 1$ and $L\to\ln(\mu\bar{\tau})$, while the non-cusp anomalous
dimension is usually expressed in terms of the collinear quark and gluon
anomalous dimensions in this case, i.e.
$\gamma^{B}(\alpha_{S})\to-2\gamma^{i}(\alpha_{S})$ with $i\in\\{q,g\\}$. As
the latter are observable-independent, we use them to cross-check our
calculation, similar to the cusp anomalous dimension $\Gamma_{\rm
cusp}^{i}(\alpha_{S})$. In essence we thus extract the non-logarithmic terms
$d_{m}$ of the anomaly exponent and the coefficients $\widehat{I}_{i\leftarrow
j}^{(m)}(N)$ of the refactorised matching kernels for SCET-2 observables.

##  6 Results

With this setup, we computed the quark beam functions for transverse-momentum
resummation and N-jettiness. As the NLO case is trivial, we focus here on the
NNLO numbers, which we present in the form
$\gamma_{1}^{B}=\gamma_{1}^{C_{F}}C_{F}^{2}+\gamma_{1}^{C_{A}}C_{F}C_{A}+\gamma_{1}^{n_{f}}C_{F}T_{F}n_{f}$,
and similarly for $d_{2}$. For the non-logarithmic terms of the matching
kernels we use the decomposition

$\displaystyle\widehat{I}_{q\leftarrow q}^{(2)}(N)$
$\displaystyle=C_{F}^{2}\;\widehat{I}_{q\leftarrow
q}^{(2,C_{F})}(N)+C_{F}C_{A}\;\widehat{I}_{q\leftarrow
q}^{(2,C_{A})}(N)+C_{F}T_{F}n_{f}\;\widehat{I}_{q\leftarrow
q}^{(2,n_{f})}(N)+C_{F}T_{F}\;\widehat{I}_{q\leftarrow q}^{(2,T_{F})}(N)\,,$
$\displaystyle\widehat{I}_{q\leftarrow g}^{(2)}(N)$
$\displaystyle=C_{F}T_{F}\;\widehat{I}_{q\leftarrow
g}^{(2,C_{F})}(N)+C_{A}T_{F}\;\widehat{I}_{q\leftarrow g}^{(2,C_{A})}(N)\,,$
$\displaystyle\widehat{I}_{q\leftarrow\bar{q}}^{(2)}(N)$
$\displaystyle=C_{F}(C_{A}-2C_{F})\;\widehat{I}_{q\leftarrow\bar{q}}^{(2,C_{AF})}(N)+C_{F}T_{F}\;\widehat{I}_{q\leftarrow
q}^{(2,T_{F})}(N)\,,$ $\displaystyle\widehat{I}_{q\leftarrow
q^{\prime}}^{(2)}(N)$
$\displaystyle=\widehat{I}_{q\leftarrow\bar{q}^{\prime}}^{(2)}(N)=C_{F}T_{F}\;\widehat{I}_{q\leftarrow
q}^{(2,T_{F})}(N)\,.$ (17)

There are thus seven independent coefficients to describe quark beam functions
at NNLO, which we evaluate for ten values of the Mellin parameter
$N\in\\{2,4,6,8,10,12,14,16,18,20\\}$.

$p_{T}$ |  Analytic |  This work
---|---|---
$d_{2}^{n_{f}}$ | $-8.296$ | $-8.293(4)$
$d_{2}^{C_{A}}$ | $-3.732$ | $-3.705(40)$
$d_{2}^{C_{F}}$ | $0$ | $0.044(74)$

N-jet |  Analytic |  This work
---|---|---
$\gamma_{1}^{n_{f}}$ | $-26.699$ | $-26.700(34)$
$\gamma_{1}^{C_{A}}$ | $-6.520$ | $-6.549(324)$
$\gamma_{1}^{C_{F}}$ | $21.220$ | $21.223(518)$

Table 1: Two-loop anomaly coefficient for transverse-momentum resummation
(left) and two-loop non-cusp anomalous dimension for N-jettiness (right). The
analytic results have been extracted from [2, 6].

Figure 1:  Two-loop non-logarithmic matching kernels for transverse-momentum
resummation (upper line) and N-jettiness (lower line). Left: The contributions
to the diagonal kernels $\widehat{I}_{q\leftarrow q}^{(2)}(N)$. Right: The
same for the off-diagonal kernels $\widehat{I}_{q\leftarrow g}^{(2)}(N)$ and
$\widehat{I}_{q\leftarrow\bar{q}}^{(2)}(N)$. The solid lines represent the
analytical results from [2, 6], and the dots show the numbers of our automated
approach. The numerical uncertainties of the latter are too small to be
visible in the plots.

We start with transverse-momentum resummation, which is a SCET-2 observable
and hence corresponds to the case $n=0$. According to the collinear-anomaly
relation (14), we need to combine the bare beam-function matching kernels with
the corresponding soft function that we obtain from SoftSERVE. Our results for
the collinear-anomaly exponent $d_{2}$ are shown in Table 1, and the non-
logarithmic terms of the Mellin-space matching kernels are displayed in the
upper panels of Figure 1. The plots also show the analytic results from [2],
and we observe a very good agreement with the numerical predictions of our
novel automated approach.

We next consider the N-jettiness beam function, which is a SCET-1 observable
with $n=1$. In this case we extract the two-loop non-cusp anomalous dimension
$\gamma_{1}^{B}$ and the non-logarithmic terms of the matching kernels
$\widehat{I}_{i\leftarrow j}^{(2)}(N)$. The former are given in Table 1, and
the latter are displayed in the lower panels of Figure 1. Our numbers are
again in excellent agreement with the known analytic results [6].

##  7 Conclusion

We have presented a novel framework to calculate beam functions at two-loop
order for a broad class of observables. Our approach is based on suitable
phase-space parametrisations and a systematic application of sector
decomposition and non-linear transformations, which allows us to expose all
phase-space singularities in an observable-independent fashion. In order to
avoid distributions, we furthermore chose to work in the Mellin-Laplace space,
but we emphasise that our approach is not limited to this assumption, and we
in fact already started to explore a direct calculation in momentum space. Our
method has been implemented in the public package pySecDec, and it has been
validated against known results in the literature for transverse-momentum
resummation and N-jettiness. As a further application, we recently calculated
the quark beam function for jet-veto resummation [15]. In the future we plan
to extend our setup to gluon beam functions as well as to publish an automated
standalone C++ code in the spirit of SoftSERVE.

### Acknowledgement

This work was supported by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under grant 396021762 - TRR 257 (_“Particle Physics
Phenomenology after the Higgs Discovery”_).

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