# TMD parton showers for associated $\gamma$ $+$ jet production in electron-
proton collisions at high energies

A.V. Lipatov1,2, M.A. Malyshev1

1Skobeltsyn Institute of Nuclear Physics, Lomonosov Moscow State University,
119991, Moscow, Russia
2Joint Institute for Nuclear Research, 141980, Dubna, Moscow region, Russia

Abstract

An earlier developed $k_{T}$-factorization framework to calculate the
associated prompt photon and hadronic jets production cross sections at high
energies is extended now to the electron-proton deep inelastic scattering. The
proposed method is based on joint usage of the pegasus and cascade Monte-Carlo
event generators, which deal with the transverse momentum dependent (TMD)
gluon and quark densities in a proton. First of them, pegasus, is applied to
produce off-shell photon-gluon fusion events. Then, to properly simulate the
kinematics of the produced jets, the TMD parton shower algorithm implemented
into the cascade is used. We demonstrate basic features of the approach
considering latest H1 and ZEUS data on prompt photon plus jet production at
low $Q^{2}$ collected at HERA. We achieve a good description of the
measurements and point out that a correct simulation of parton showers is
essential for studying non-inclusive processes at high energies within the
TMD-based approaches.

Keywords: deep inelastic scattering, high energy factorization, CCFM
evolution, TMD gluon density, prompt photon production

##  1 Introduction

Inclusive prompt photon111Photons are considered as prompt if they are
produced in the hard interaction subprocess rather than in hadron decays or
via fragmentation. production at high energies is an important subject of
investigation at modern colliders, such as LHC. Being quite clean in sense of
final state effects, these processes can serve as an important source of
information about the proton structure (expressed in terms of parton density
functions) and represent an important background to many processes involving
photons in the final state, including Higgs boson production. Also, a rather
clear and robust way to study parton content of a proton can be provided by
the prompt photon photoproduction in deep inelastic $ep$ scattering (DIS),
where the exchanged photon is of small virtuality $Q^{2}\lesssim 1$ GeV2[1, 2,
3]. In fact, the corresponding cross section at leading order (LO) of
perturbative QCD is mainly governed by the direct $\gamma+q\to\gamma+q$ and
several resolved photon subprocesses, where the photon emitted by the electron
fluctuates into a hadronic state. A gluon and/or a quark of this hadronic
fluctuation takes part then in hard interactions222Beyound the LO, the
difference between these two mechanisms disappears.. Including into the
consideration accompanying final state jet(s) gives an opportunity to study
jet observables, different photon-jet correlations, thus allowing to test the
production dynamics in many additional aspects. Investigation of prompt photon
production will be an important part of the physical program at future
electron-proton or electron-ion colliders, such as LHeC[4], FCC-eh[5], EiC[6]
and EiCC[7]. Thus, relevant theoretical studies and developing corresponding
Monte-Carlo event generators are rather important tasks at present.

Usually, a theoretical description of prompt photon photoproduction relies on
conventional (collinear) QCD factorization, which is based on well-known
Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) quark and gluon evolution
equations[8]. Within this formalism, next-to-leading order (NLO) calculations
are available[9, 10, 11]. The prompt photon production events can be produced
also with some Monte-Carlo generators, such as pythia[12] or herwig[13].
Alternative calculations can be done in the framework of high-energy
factorization[14] or $k_{T}$-factorization approach[15] based on Balitsky-
Fadin-Kuraev-Lipatov (BFKL)[16] or Catani-Ciafaloni-Fiorani-Marchesini
(CCFM)[17] gluon evolution equations. The $k_{T}$-factorization QCD approach
has certain technical advantages in the ease of including higher-order pQCD
radiative corrections corresponding to real initial-state gluon emissions
(namely, terms, proportional to $\alpha_{s}^{n}\ln^{n}s/\Lambda_{\rm
QCD}^{2}\sim\alpha_{s}^{n}\ln^{n}1/x$, important at high energies $s$, or
small $x$) in the form of transverse momentum dependent (TMD, or unintegrated)
gluon density (see review[18] for more information).

Studies of prompt photon production within the $k_{T}$-factorization approach
have their own long history (see, for example,[19, 20, 21, 22, 23, 24, 25, 26,
27, 28] and references therein). Moreover, such processes have been also
investigated[29, 30, 31, 32, 33, 34, 35, 36, 37] in the framework of Color
Glass Condensate (CGC) formalism[38, 39, 40]. In particular, a good
description of experimental data for inclusive production events was obtained
for both electron-proton[19, 20, 21] and proton-proton collisions[26, 27, 28].
A simple model[41] to reconstruct the jet kinematics in associated
$\gamma+{\rm jet}$ production was applied[25] and necessity for correct
reconstruction of this kinematics was pointed out. The latter is a special
task for any calculations where the non-collinear QCD evolution of the initial
parton (gluon) cascade is used. In the case of CCFM evolution, a method based
on numerical simulation of TMD parton showers was proposed[42] and
successfully applied to associated $\gamma+{\rm jet}$ production at LHC[27].
In fact, earlier description of the experimental data was significantly
improved. Here we extend the consideration to electron-proton collisions and
analyse latest HERA data[1, 2, 3] on prompt photon and associated jet
photoproduction taken by the H1 and ZEUS Collaborations at $\sqrt{s}=319$ GeV.

The paper is organized as follows. In Section 2 we briefly describe our
approach. In Section 3 we present our numerical results and compare them with
available HERA data. Section 4 contains our conclusions.

##  2 Theoretical framework

Let us shortly describe the main calculation steps. In the photoproduction
regime the colliding electron emits a quasi-real ($Q^{2}\sim 0$) photon, so
one deals then with photon-proton interaction:

$\displaystyle d\sigma(ep\to e^{\prime}\gamma X)=\int
f_{\gamma/e}(y)d\sigma(\gamma p\to\gamma X)dy,$ (1)

where $y$ is the fraction of initial electron energy carried by the photon in
the laboratory frame. We use the well-known Weizacker-Williams approximation
for bremsstrahlung photon distribution from the electron:

$\displaystyle
f_{\gamma/e}(y)=\frac{\alpha}{2\pi}\left(\frac{1+(1-y)^{2}}{y}\ln\frac{Q^{2}_{\text{max}}}{Q^{2}_{\text{min}}}+2m^{2}_{e}y\left(\frac{1}{Q^{2}_{\text{max}}}-\frac{1}{Q^{2}_{\text{min}}}\right)\right),$
(2)

where $m_{e}$ is the electron mass, $Q^{2}_{\rm min}=m_{e}^{2}y^{2}/(1-y)^{2}$
and $Q^{2}_{\rm max}=1$ GeV2, which is a typical value for photoproduction
measurements at the HERA.

As it was already mentioned above, direct contribution to the prompt photon
photoproduction comes from the ${\cal O}(\alpha^{2})$ deep inelastic Compton
scattering subprocess, $\gamma+q\to\gamma+q$. In the consideration below we
decompose the quark content of a proton into the sea and valence parts and
exploit the idea that the sea quarks all appear from a perturbative chain as a
result of the QCD evolution of gluon density333A similar treatment has been
used recently in the prompt photon hadroproduction case[28].. Then, appending
an explicit gluon splitting vertex, we come to the ${\cal
O}(\alpha^{2}\alpha_{s})$ photon-gluon fusion subprocess

$\displaystyle\gamma(k_{1})+g^{*}(k_{2})\to\gamma(p)+q(p_{1})+\bar{q}(p_{2}).$
(3)

Here $q$ denotes any quark flavor and the particles four-momenta are given in
parentheses. The subprocess (3) covers also the main part of LO resolved
photon contributions. We employ the $k_{T}$-factorization approach, thus
effectively taking into account a large piece of higher-order pQCD corrections
via CCFM-evolved TMD gluon density in a proton. In this way, we keep the non-
zero virtuality of the incoming gluon, $k_{2}^{2}\simeq
k_{2T}^{2}=-{\mathbf{k}}_{2T}^{2}\neq 0$. The corresponding off-shell
production amplitude was calculated in [21] and is implemented now in the
Monte-Carlo generator pegasus[43]. We note that the subprocess (3), being
supplemented with the final state parton radiation, covers (at least partly)
many other off-shell subprocesses — for example, $\gamma+q^{*}\to\gamma+q+g$,
which has been taken into account in our previous calculations[21] separately.
The effects of initial and final state parton emissions (ISR and FSR) can be
simulated using the standard parton showering algorithms, such as implemented
into cascade or pythia. The remaining contributions from valence quarks,
important at large $x$, are taken into account via simple ${\cal
O}(\alpha^{2})$ deep inelastic Compton scattering subprocess. Of course, here
one can safely neglect the off-shellness of valence quarks and employ the
conventional (collinear) QCD factorization. Additionally we will take into
account ${\cal O}(\alpha^{2}\alpha_{s}^{2})$ box contribution:

$\displaystyle\gamma(k_{1})+g^{*}(k_{2})\to\gamma(p_{1})+g(p_{2}),$ (4)

since it is known to be sizeable due to high gluon luminosity at small $x$.
Here we improve the previous consideration[21] by taking into account non-zero
transverse momentum of initial gluon and use analytical expressions[24] for
off-shell production amplitude444We would like to note that the expression
given in the Appendix of[24] is a bit misleading. Namely, the denominator of
(A5) from[24] must be taken universal for the helicity amplitudes (A1) — (A3);
alternatively, one can multiply (A9) by $(u+t_{1})/(s+t_{1})$., where the one-
loop integrals were evaluated with qcdloop[44].

To calculate the photon-proton interaction cross section in the
$k_{T}$-factorization QCD approach one has to convolute the corresponding
partonic cross section with the TMD gluon density in a proton
$f_{g}(x,\mathbf{k}_{T}^{2},\mu^{2})$:

$\displaystyle d\sigma(\gamma p\to\gamma
X)=\int\frac{dx_{2}}{x_{2}}\frac{d\phi_{2}}{2\pi}d\mathbf{k}_{2T}^{2}f_{a}(x_{2},\mathbf{k}_{2T}^{2},\mu^{2})d\hat{\sigma}(\gamma
g^{*}\to\gamma X),$ (5)

where $x_{2}$ and $\phi_{2}$ are the longitudinal momentum fraction of the
colliding proton carried by incoming off-shell gluon and its azimuthal angle,
respectively. To avoid the collinear divergencies appearing in (3) when the
final photon becomes collinear to the outgoing quark, we follow an
approach[23] and split the cross section into two pieces:

$\displaystyle\sigma=\sigma_{\rm pert}(\mu^{2}_{\rm reg})+\sigma_{\rm non-
pert}(\mu^{2}_{\rm reg}),$ (6)

where $\sigma_{\rm pert}(\mu^{2}_{\rm reg})$ is the perturbative contribution
calculated under usual condition when the wavelength of produced photon (in
the emitting quark rest frame) becomes larger than the typical hadronic scale,
${\cal O}(1\,{\rm GeV}^{-1})$. Below this scale, different non-perturbative
effects, including photon fragmentation, have to be taken into account. To
separate these two pieces the regularization scale $\mu_{\rm reg}$ is used.
Following[23], we restrict $\sigma_{\rm pert}(\mu^{2}_{\rm reg})$ to the
region $M\geq\mu_{\rm reg}$, where $M$ is the invariant mass of the photon +
quark subsystem and $\mu_{\rm reg}\sim 1$ GeV is the hadronic scale. Under
this condition, the contribution $\sigma_{\rm pert}(\mu^{2}_{\rm reg})$ is
free from collinear divergences. Moreover, a special photon isolation
criterion (so-called cone isolation), introduced in experimental analyses[1,
2, 3], in our calculations is used as a tool555Another photon isolation
criterion was also proposed [45]. to remove the non-perturbative part
$\sigma_{\rm non-pert}(\mu^{2}_{\rm reg})$, where the final photon is radiated
close to the quark (inside the isolation cone). The sensitivity of the
numerical results to $\mu_{\rm reg}$ is reasonably soft if the photon
isolation condition is applied (see[23] for more information). We note also
that under this condition the fragmentation contributions become negligible[1,
2, 3], so we can safely omit it in the consideration below.

Concerning the TMD gluon density, in the present paper we have tested two
latest sets obtained from the numerical solution of CCFM evolution equation.
First of them is the JH’2013 set 2[46], which is widely used in
phenomenological applications (see, for example,[28, 27, 47] and references
therein). The parameters of the corresponding (rather empirical) initial gluon
density were derived from a fit to precision HERA data on the proton structure
functions $F_{2}(x,Q^{2})$ and $F_{2}^{c}(x,Q^{2})$ at $x<5\cdot 10^{-3}$ and
$Q^{2}>3.5$ GeV2. Also we have tried a very recent TMD gluon distribution in a
proton, namely, LLM’2022[48]. In contrast with JH’2013 set 2, the analytical
expression for the starting distribution was chosen in a more physically
motivated way — namely, from the best description of the LHC data on charged
hadron production at low transverse momenta $p_{T}\sim 1$ GeV in the framework
of modified soft quark-gluon string model (QGSM)[49, 50]. Moreover, the gluon
saturation effects, which are important at low scales, were taken into
account. Some phenomenological parameters were determined from the LHC and
HERA data on several hard QCD processes (see[48, 51] for more information).
Both these TMD gluon densities are available in the Monte-Carlo event
generator pegasus and popular tmdlib package[52], which is a C$++$ library
providing a framework and an interface to the different parametrizations.

A last important point of our calculations is connected with the proper
determination of the associated jet momentum. In fact, the quarks and/or
gluons produced in the hard subprocesses described above can form final state
hadronic jets. However, in addition to that, the produced photon is
accompanied by a number of gluons radiated in the course of the non-collinear
evolution, which also give rise to final jets. This is a distinct and
remarkable feature of the CCFM-based approach used. From all of the hadronic
jets we choose the one, carrying the largest transverse momentum and
satisfying the experimental cuts (leading jet) and then compute the cross
section of $\gamma+{\rm jet}$ production. Technically, we produce a Les Houche
Event file[53] in the parton level calculations performed using the pegasus
and then process the file with the TMD shower routine implemented into the
Monte-Carlo event generator cascade[54]. This routine only recently became
available for DIS processes, so our work presents the first application of
this procedure to photoproduction processes at HERA, to the best of our
knowledge. Thus, we fully reconstruct the CCFM evolution ladder. The hadronic
jets are reconstructed with the $k_{T}$-algorithm, implemented in the fastjet
tool[55]. This method gives us the possibility to take into account the
contributions from initial state parton showers in a consistent way. Moreover,
it was successfully applied already to investigate prompt photon (or Higgs
boson) and associated jet production at the LHC (see[27, 47]). Note that
additionally we performed final state hadronization procedure, which is
neccesary to fully reproduce the experimental setup[1, 2, 3].

##  3 Numerical results

Now we are in a position to present the numerical results. First we describe
our input and kinematical conditions. So, as it is often done, we set the
renormalization scale $\mu_{R}$ equal to the produced photon transverse energy
$E_{T}^{\gamma}$. The factorization scale is defined as
$\mu_{F}^{2}=\hat{s}+{\mathbf{Q}}^{2}$ with $\hat{s}$ and ${\mathbf{Q}}_{T}$
being the subprocess invariant energy and net transverse momentum of the
initial state, respectively. Note that the definition of $\mu_{F}$ is dictated
mainly by the CCFM evolution algorithm (see[17]). Then, we use the massless
limit for light ($u$, $d$ and $s$) quarks and set the charm and beauty masses
to $m_{c}=1.28$ GeV and $m_{b}=4.75$ GeV. Also, we apply the two-loop formula
for the strong coupling constant $\alpha_{s}$ with $N_{f}=4$ quark flavours at
$\Lambda_{\rm QCD}=200$ MeV, as it was originally done in the fits[46, 48].

The experimental data for associated prompt photon and jet photoproduction at
HERA were taken by both the H1 and ZEUS collaborations. The H1 data[1] were
obtained in the following kinematical region666Here and in the following all
kinematic quantities are given in the laboratory frame with positive $OZ$ axis
direction given by the proton beam.: $6<E_{T}^{\gamma}<15$ GeV, $E_{T}^{\rm
jet}>4.5$ GeV, $-1.0<\eta^{\gamma}<2.4$, $-1.3<\eta^{\rm jet}<2.3$. The
fraction $y$ of the electron energy transferred to the photon is restricted to
the range $0.1<y<0.7$. More recent ZEUS measurements[2, 3] refer to the region
defined by $6<E_{T}^{\gamma}<15$ GeV, $4<E_{T}^{\rm jet}<35$ GeV,
$-0.7<\eta^{\gamma}<0.9$, $-1.5<\eta^{\rm jet}<1.8$ and $0.2<y<0.7$. The
data[1, 2, 3] were obtained with the electron energy $E_{e}=27.6$ GeV and
proton energy $E_{p}=920$ GeV.

Our results are compared with experimental data in Figs. 1 —8. So, the
transverse energy $E_{T}^{\gamma}$, $E_{T}^{\rm jet}$ and pseudorapidity
$\eta^{\gamma}$, $\eta^{\rm jet}$ distributions of the produced photon and jet
are shown in Figs. 1 and 2. We find that the calculations based on recently
proposed LLM’2022 gluon density (represented by the green histograms) are
consistent with the H1 data in most bins, although some discrepancies are
present. In particular, these predictions tend to underestimate the H1
measurements at low $E_{T}^{\rm jet}$ and rear photon pseudorapidity
$\eta^{\gamma}$, although coincide with the data within the theoretical and
experimental uncertainties. Note that theoretical uncertainties of our
calculations are estimated in a traditional way, varying the renormalization
scale777In the CCFM-based approach, the factorization scale $\mu_{F}$ is
related with the evolution variable and therefore should not be varied.
See[17] for more details. around its default value as $\mu_{R}\to\xi\mu_{R}$
with $\xi=1/2$ or $2$. One can see that the obtained results reproduce
behavior of the measured $\eta^{\rm jet}$ spectrum (shifted towards positive
pseudorapidities), which could not be achieved in earlier calculations[21,
24]. This is a direct consequense of the applied method of proper
determination of jet kinematics based on the TMD shower algorithm. Similar
conclusions were already done in [27, 47] where production of prompt photons
or Higgs bosons associated with hadronic jet(s) in $pp$ collisions at the LHC
was studied. Overall agreement of our predictions with more recent ZEUS data
is a bit worse, but still rather reasonable (see Fig. 2). Although there is
some overestimation of the data at low $E_{T}^{\gamma}$, large $E_{T}^{\rm
jet}$ and low separation in azimuthal angle between the produced photon and
jet, $\Delta\phi$, the shapes of $\eta^{\rm jet}$ and
$\Delta\eta=\eta^{\gamma}-\eta^{\rm jet}$ spectra are reproduced well. These
observables are sensitive to the proper determination of jet kinematics.

As it has been mentioned above, to investigate the dependence of our results
on the TMD gluon density in a proton, we have repeated the calculations using
another set, JH’2013 set 2. In contrast with LLM’2022, it leads to systematic
overestimation of the HERA data, that coincides with the observation of [51].
It was argued[48, 51] that better agreement achieved with the LLM’2022 is an
immediate consequence of using a physically motivated expression for the
corresponding starting distribution. Thus, our calculations demonstrate that
the HERA data on associated $\gamma+{\rm jet}$ production in DIS are sensitive
to the TMD gluon densities and could help to clearly distinguish the latter.
It could be important for experiments at future electron-proton or electron-
ion colliders, such as LHeC, FCC-eh, EiC and EiCC. Additionally, we find that
contribution from the valence quarks is negligible in the considered
kinematical region.

For comparison we also show results of conventional NLO pQCD calculations
taken from[1, 2, 3]. These predictions are slightly below the LLM’2022 ones
but rather close to them for most of observables within the uncertainties. The
NLO pQCD calculations also tend to underestimate the H1 data. However, they
agree well with both the H1 and ZEUS measurements at large $E_{T}^{\rm jet}$
and small $\Delta\phi<90^{\circ}$. The JH’2013 set 2 gluon density
significantly overshoots the NLO pQCD results.

Other important variables are the longitudinal momenta fractions carried by
the colliding partons. The momentum fractions of the initial photon and proton
are introduced in the ZEUS analyses[2, 3] as the following:

$x_{\gamma}^{\rm obs}=\frac{E^{\gamma}_{T}e^{-\eta^{\gamma}}+E_{T}^{\rm
jet}e^{-\eta^{\rm jet}}}{2yE_{e}},\quad x_{p}^{\rm
obs}=\frac{E^{\gamma}_{T}e^{\eta^{\gamma}}+E_{T}^{\rm jet}e^{\eta^{\rm
jet}}}{2E_{p}}.$ (7)

At $x_{\gamma}^{\rm obs}>0.8$ the cross section is believed to be dominated by
the ”direct photon” contributions, whereas at lower $x_{\gamma}^{\rm obs}$ the
”resolved photon” contributions play the main role. The H1 Collaboration
refers to $x_{\gamma}^{\rm LO}$ and $x_{p}^{\rm LO}$ estimators given by[1]:

$x_{\gamma}^{\rm LO}=\frac{E_{T}^{\gamma}(e^{-\eta^{\gamma}}+e^{-\eta^{\rm
jet}})}{2yE_{e}},\quad x_{p}^{\rm
LO}=\frac{E_{T}^{\gamma}(e^{-\eta^{\gamma}}+e^{\eta^{\rm jet}})}{2E_{p}}.$ (8)

The $x_{\gamma}^{\rm LO}$ and $x_{p}^{\rm LO}$ variables explicitly use only
of the photon energy, which is better measured than the jet energy. Our
predictions for these observables are shown in Fig. 3. We find that results
obtained with LLM’2022 gluon density agree well with the H1 data. In contrast
with earlier calculations[21], they show more smeared cross sections for the
$x_{\gamma}^{\rm obs}$ and $x_{\gamma}^{\rm LO}$ spectra, which is in a better
agreement with the data. Again, this is due to more accurate determination of
the jet kinematics compared to the previous considerations. However, there is
still some overestimation of the ZEUS data at small $x_{\gamma}^{\rm obs}$ and
large $x_{p}^{\rm obs}$.

Although we cannot distinguish between the ”resolved photon” and ”direct
photon” contributions, the dedicated study of cross sections measured at low
and high $x_{\gamma}^{\rm obs}$ can provide an additional test of theoretical
calculations. In fact, in our simulations different interplay between the
subprocesses (3) and (4) can result then in different cross sections in these
two regions and thus there might be a deviation from the data. To investigate
it in more details, we have performed the calculations at low $x_{\gamma}^{\rm
obs}(x_{\gamma}^{\rm LO})<0.8$ and high $x_{\gamma}^{\rm obs}(x_{\gamma}^{\rm
LO})>0.8$ values and compared our results with the H1[1] and ZEUS[3]
measurements, see Figs. 4 — 8. One can see that transverse energy
$E_{T}^{\gamma}$, $E_{T}^{\rm jet}$ and pseudorapidity $\eta^{\gamma}$,
$\eta^{\rm jet}$ spectra predicted by the LLM’2022 gluon density are in a
reasonable agreement with the ZEUS data[3] in the both these kinematical
regions within the uncertainties, except only last $E_{T}^{\rm jet}$ and
forward $\eta^{\rm jet}$ bins at $x_{\gamma}^{\rm obs}<0.8$. However, there is
some overestimation of the ZEUS data at large $x_{p}^{\rm obs}\geq 0.015$,
that is clearly visible at $x_{\gamma}^{\rm obs}<0.8$. It comes from the
events where the produced photon and jet are close to each other, as one can
see from $\Delta\phi$ distributions shown in Fig. 7. Note that our predictions
overshoot the ZEUS data on $\Delta\phi$ spectra at $\Delta\phi<150^{\circ}$
and $x_{\gamma}^{\rm obs}>0.8$, but agree well with the H1 measurements which
were performed in the similar kinematical region. In contrast, at low
$x_{\gamma}^{\rm obs}<0.8$ the H1 data for $\Delta\phi$ distributions are
clearly underestimated everywhere while corresponding ZEUS data are reasonably
well described. Therefore, at this point it could be some contradiction
between the H1 and ZEUS measurements. In any case, special studies of
transverse correlations between the final state particles are known to be
useful to investigate the production dynamics (see, for example,[19, 20, 21,
25] and references therein). So, the H1 Collaboration has investigated the
distribution on the component of the photon’s momentum perpendicular to the
jet direction in the transverse plane,
$p_{\perp}=|{\mathbf{p}}_{T}^{\gamma}\times{\mathbf{p}}_{T}^{\rm
jet}|/|{\mathbf{p}}_{T}^{\rm jet}|=E_{T}^{\gamma}\sin\Delta\phi$. Similar to
$\Delta\phi$ spectra, the distribution over $p_{\perp}$ is partucularly
sensitive to the higher-order pQCD corrections888In the conventional
(collinear) LO approximation, it must be simply a delta function since the
produced photon and the jet are back-to-back in the transverse plane., which
are taken into account in the form of CCFM-evolved TMD gluon densities in our
calculations. The predictions for $p_{\perp}$ spectra are confronted with the
H1 data in Fig. 8. We find that $k_{T}$-factorization results agree well with
the H1 data in the ”direct photon” region, although LLM’2022 gluon slightly
underestimates the data at low $p_{\perp}$ and JH’2013 set 2 tends to
overshoots the latter. The conventional NLO pQCD calculations do not reproduce
the H1 data both in the normalization and shape. At the same time we find that
none of the calculations is able to describe the overall normalization of the
data at $x_{\gamma}^{\rm LO}<0.8$, although the shape of measured distribution
is reproduced well by all of them. Thus, further investigation of such an
observable could be important to discriminate between the different
approaches.

Finally, we can conclude that the LLM’2022 predictions reproduce well the
latest HERA data for most of the observables within the theoretical and
experimental uncertainties. It is due to more accurate determination of jet
kinematics in our consideration compared to earlier analyses performed within
the TMD-based approaches. These calculations are rather close to the NLO pQCD
ones at $x_{\gamma}^{\rm obs}>0.8$ and tend to lie above the latter at
$x_{\gamma}^{\rm obs}<0.8$. The NLO pQCD provides a better description of the
jet transverse energy spectra, but unable to reproduce well the data on most
of correlation variables (such as $p_{\perp}$ ones). The JH’2013 set 2 gluon
density overshoots the HERA data, although at $x_{\gamma}^{\rm obs}<0.8$ the
difference between these predictions and LLM’2022 ones become smaller. So, the
cross sections of associated $\gamma+{\rm jet}$ production in DIS events are
sensitive to the TMD gluon density in a proton and can be used to constrain
it.

##  4 Conclusion

We have considered associated production of prompt photon and hadronic jets in
photoproduction regime of deep inelastic electron-proton scattering at high
energies. The calculations were performed in the framework of
$k_{T}$-factorization QCD approach and mainly based on two off-shell photon-
gluon fusion subprocesses, $\gamma+g^{*}\to\gamma+q+\bar{q}$ and
$\gamma+g^{*}\to\gamma+g$, implemented now into the Monte-Carlo generator
pegasus. First of them, being supplemented with effects of the final-state
parton radiation, covers many other subprocesses, including resolved photon
contributions. An additional valence quark-induced subprocess $\gamma+q_{\rm
val}\to\gamma+q$, which can be important at large $x$ region, has been taken
into account in the conventional (collinear) QCD factorization. In the
numerical calculations we have tested two CCFM-evolved gluon distributions in
a proton, namely, JH’2013 set 2 and very recent LLM’2022 gluon, both available
in pegasus and tmdlib packages. The LLM’2022 gluon density is based on
simultaneous fit to the number of HERA and LHC processes sensitive to the
gluon content of the proton. To reconstruct correctly the kinematics of the
hadronic jets the TMD parton shower implemented into the Monte-Carlo generator
cascade has been applied for the first time for the DIS process.

We have achieved reasonably good agreement between our predictions obtained
with the recent LLM’2022 gluon density in a proton and latest H1 and ZEUS
experimental data, thus demonstrating again the importance of initial state
TMD parton showers for jet determination in the TMD-based approaches. The
previously developed framework to calculate the jet associated processes is
extended now to the electron-proton deep inelastic scattering. It is important
for forthcoming studies at future electron-proton and electron-ion colliders,
such as LHeC, FCC-he, EiC and EiCC.

The next version of Monte-Carlo event generator pegasus (1.08), which is
extended now to the considered DIS process, will be released shortly.

## Acknowledgements

We thank S.P. Baranov and G.I. Lykasov for their important comments and
remarks. We are also grateful to H. Jung for help with implementation of
cascade in DIS. Framework setting, matrix elements adaptation and the analysis
of the results were supported by the grant of the Foundation for the
Advancement of Theoretical Physics and Mathematics “BASIS” 20-1-3-11-1. Work
on implementation of parton showers and hadronization for photoproduction
processes were supported by the Russian Science Foundation under grant
22-22-00119.

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Figure 1: The associated prompt photon and jet photoproduction cross section
as functions of photon and jet transverse energies and pseudorapidities. The
green histograms and shaded bands correspond to the predictions obtained with
LLM’2022 gluon density and estimated scale uncertainties of these
calculations. The yellow histograms represent the JH’2013 set 2 predictions.
Separately shown the contributions from $\gamma+q_{\rm val}\to\gamma+q$
subprocess and conventional NLO pQCD results (taken from[1]). The experimental
data are from H1[1].

Figure 2: The associated prompt photon and jet photoproduction cross section
as functions of photon and jet transverse energies, pseudorapidities,
differences in their azimuthal angles and pseudorapidities. The notations are
the same as in Fig. 1. The experimental data are from ZEUS[2, 3].

Figure 3: The associated prompt photon and jet photoproduction cross section
as functions of $x_{\gamma}^{\rm LO}$, $x_{\gamma}^{\rm obs}$, $x_{p}^{\rm
LO}$ and $x_{p}^{\rm obs}$ variables. The notations are the same as in Fig. 1.
The experimental data are from H1[1] and ZEUS[2].

Figure 4: The associated prompt photon and jet photoproduction cross section
as functions of photon and jet transverse energies $E_{T}^{\gamma}$ and
$E_{T}^{\rm jet}$ at $x_{\gamma}^{\rm obs}>0.8$ (left panels) and
$x_{\gamma}^{\rm obs}<0.8$ (right panels). The notations are the same as in
Fig. 1. The experimental data are from ZEUS [3].

Figure 5: The associated prompt photon and jet photoproduction cross section
as functions of photon and jet pseudo-rapidities $\eta^{\gamma}$ and
$\eta^{\rm jet}$ at $x_{\gamma}^{\rm obs}>0.8$ (left panels) and
$x_{\gamma}^{\rm obs}<0.8$ (right panels). The notations are the same as in
Fig. 1. The experimental data are from ZEUS [3].

Figure 6: The associated prompt photon and jet photoproduction cross section
as functions of $x_{p}^{\rm obs}$ at $x_{\gamma}^{\rm obs}>0.8$ (left panels)
and $x_{\gamma}^{\rm obs}<0.8$ (right panels). The notations are the same as
in Fig. 1. The experimental data are from ZEUS [3].

Figure 7: The associated prompt photon and jet photoproduction cross section
as functions of the prompt photon and jet azimuthal angle and pseudorapidity
differences $\Delta\phi$ and $\Delta\eta$ at $x_{\gamma}^{\rm obs}>0.8$ (left
panels) and $x_{\gamma}^{\rm obs}<0.8$ (right panels). The notations are the
same as in Fig. 1. The experimental data are from ZEUS [3].

Figure 8: The associated prompt photon and jet photoproduction cross section
as functions of photon and jet azimuthal angle difference $\Delta\phi$ and
photon momentum perpendicular to the jet direction $p_{\perp}$ at
$x_{\gamma}^{\rm LO}>0.8$ (left panels) and $x_{\gamma}^{\rm LO}<0.8$ (right
panels). The notations are the same as in Fig. 1. The experimental data are
from H1 [1].


Bogumił Pilecki Cepheids with giant companions

19 2023 376 2024 10.1017/S1743921323004258

Proceedings IAU Symposium

# Cepheids with giant companions: A new, abundant source of Cepheid
astrophysics

Bogumił Pilecki<EMAIL_ADDRESS>Centrum Astronomiczne im. Mikołaja
Kopernika, PAN, Bartycka 18, 00-716 Warsaw, Poland

###### Abstract

We present a progress report of our project aiming to increase the number of
known Cepheids in double-lined binary (SB2) systems from six to 100 or more.
This will allow us, among other goals, to accurately measure masses for a
large sample of Cepheids. Currently, only six accurate Cepheid masses are
available, which hinders our understanding of their physical properties and
renders the Cepheid mass–luminosity relation poorly constrained. At the same
time, Cepheids are widely used for essential measurements (e.g., extragalactic
distances, the Hubble constant). To examine Cepheid period–luminosity
relations, we selected as binary candidates Cepheids that are too bright for
their periods. To date, we have confirmed 56 SB2 systems, including the
detection of significant orbital motions of the components for 32. We
identified systems with orbital periods up to five times shorter than the
shortest reported period to date, as well as systems with mass ratios
significantly different from unity (suggesting past merger events). Both
features are essential to understand how multiplicity affects the formation
and destruction of Cepheid progenitors and what effect this has on global
Cepheid properties. We also present eight new systems composed of two Cepheids
(only one such system was known before). Among confirmed SB2 Cepheids, there
are also several wide-orbit systems. In the future, these may facilitate
independent accurate geometric distance measurements to the Large and Small
Magellanic Clouds.

###### keywords:

Cepheids, binary systems, physical properties, P-L relation

††editors: Richard de Grijs, Patricia Whitelock and Márcio Catelan, eds.

##  1 Introduction

Classical Cepheids are perhaps the most important objects in astrophysics,
given that they are crucial to gain improved physical insights in various
fields, including as regards stellar oscillations and the evolution of
intermediate-mass and massive stars. They wield enormous influence on modern
cosmology. Since the discovery of the relationship between their pulsation
period and their luminosity more than a century ago (the Leavitt Law; Leavitt
and Pickering, 1912), Cepheids have been used extensively to measure distances
both within and outside of our Galaxy. The recent local Hubble constant
determination accurate to 1.8% (Riess et al.,, 2022), which shows a
significant discrepancy with the value inferred from Planck data, depends
sensitively on this period–luminosity (P–L) relation.

Although theoretical studies of Cepheids are quite advanced (e.g., Bono et
al., 1999, 2005; Valle et al., 2009), our empirical knowledge is still
limited. Theory predicts masses of Cepheids in the range of 3–11 M⊙, but their
measured masses clump between 3.6 M⊙ and 5 M⊙, with only one higher but
uncertain value of 6 M⊙ (Pilecki et al.,, 2013, 2018; Evans et al.,, 2018;
Gallenne et al., 2019a, ). This makes the Cepheid mass–luminosity relation
very poorly constrained (Anderson et al.,, 2016). Nevertheless, it is crucial
for our theoretical understanding of the P–L relation and basic stellar
physics regarding, e.g., convection, mass loss, and rotation. Moreover, the
blue loops predicted by current evolution theory are too short to explain the
existence of low-mass, short-period Cepheids (see, e.g., Espinoza-Arancibia
and Pilecki, these proceedings).

Masses covering a wider range would be of the utmost importance to resolve
these issues, but we in practice can measure them only for Cepheids in
spectroscopic double-lined binaries (SB2), for which lines of companions are
easily detectable. Unfortunately, such systems are very rare. Most binary
Cepheids, and all found in the Milky Way, have an early-type main-sequence
companion (exhibiting few and broad lines) which is typically 2–5 mag fainter
in the $V$ band, making it extremely hard to determine its velocity and thus
the mass of the Cepheid.

One good source for more binary Cepheids would be to look for eclipses, but
this solution has three serious limitations. First, such systems are not
numerous, because a very specific orbital orientiation is needed for eclipses
to occur. Second, according to current statistics, only about half of
eclipsing systems involving Cepheids are double-lined, and third, the best
places to look for them (the Galactic disk, Magellanic Clouds) have already
been examined, and we do not expect many more such binaries to be found there
any time soon. Therefore, we decided to look for another possible source that
had not yet been considered and remained unexplored.

In the first paper of the series (Pilecki et al.,, 2021) we described such a
source, highlighting its potential, and we presented the first results for the
Large Magellanic Cloud (LMC). Here, we present the current stage of the
project, providing updates based on new observations and expansion of the
sample to the Small Magellanic Cloud (SMC) and the Milky Way (MW).

##  2 The project

The basic idea of the project was to look for Cepheids in binary systems, for
which lines of both components are present in their spectra. To meet these
conditions, one has to find Cepheids accompanied by stars of similar
luminosity and preferentially of late spectral types, i.e., subgiants or later
stages of stellar evolution. To identify such candidates, we considered three
observable features caused by such companions. Compared with single Cepheids
(or Cepheids with significantly fainter companions), for Cepheids with giant
companions we expect:

  * •

their total observed brightness to increase significantly;

  * •

their photometric pulsation amplitude to decrease; and

  * •

their color to be either similar or redder.

Figure 1: P–L relation based on the reddening-free Wesenheit index. Filled red
circles show known eclipsing binaries involving Cepheids, while empty circles
mark their brightness without any companion light. Our candidates for binary
Cepheids are $\geq$50% brighter than an average Cepheid for a given period.
Note that their distribution parallel to the P–L relation advocates strongly
against blending by random stars.

Looking at the P–L diagram for LMC Cepheids (Figure 1), we noticed that all
previously confirmed eclipsing giant–giant SB2 systems including Cepheids
(Pilecki et al.,, 2018) lie significantly above the corresponding P–L
relation, being at least 50% (0.44 mag) brighter than a typical Cepheid for
its period. At the same time, their amplitudes are about half the typical
ones, and their colors are typical or redder. From the number of known
eclipsing binary Cepheids we also estimated how many similar binary Cepheids
may exist but have not yet been detected in photometric studies due to a lack
of eclipses. Assuming random inclinations, we expected on the order of 50 SB2
systems composed of a Cepheid and a giant companion exhibiting a range of
orbital periods from 1 to $\sim$5 years, i.e., similar to the periods for
eclipsing Cepheids.

Indeed, when we investigated the P–L diagram for all Cepheids from the OGLE-3
catalog (Soszyński et al.,, 2008), we identified 41 additional Cepheids that
lie 0.44 mag (4.7$\sigma$) above the P–L relation, but for which eclipses were
not detected (Figure 1). Virtually all Cepheids selected this way have low
amplitudes and colors that are on average redder than typical by 0.12 mag in
the $(V-I)$ index. As stated above, these three features together clearly
indicate that the overbright Cepheids have luminous, late-type giant
companions. This, in turn, made them perfect candidates for SB2 systems
composed of giants, for which lines of both components could be easily
detected and their radial velocities (RVs) measured.

The Cepheids in our sample are distributed along the full P–L relation, with
periods from 0.26 to 10.5 days. Among them, 20 pulsate in the fundamental (F)
mode and 21 in the first-overtone (1O) mode. Three of the latter are actually
double-mode Cepheids pulsating in the second overtone (2O) as well. In Figure
1 the periods of 1O Cepheids ($P_{1O}$) were fundamentalized to match the
periods of F-mode Cepheids ($P_{F}$) using the prescription of Pilecki et al.,
(2021).

Meanwhile, the project has been expanded by inclusion of new candidates in the
LMC from the OGLE-4 catalog (Soszyński et al.,, 2017) and a dozen candidates
in the SMC (Soszyński et al.,, 2010). In the samples selected as described
above we found all seven known double Cepheids (i.e., objects for which
pulsation of two Cepheids was detected at the same coordinates), which were
all selected as candidates. This statistical result suggests that they share
the same properties as other P–L-overbright Cepheids. This allows us to add to
our SB2 candidate list double Cepheids from the MW, where the P–L relation
method of detecting SB2 Cepheids cannot be applied directly (individual
distances to the objects are needed to do so).

##  3 Data

Spectroscopic monitoring of our first SB2 candidates started in October 2020,
with new objects added subsequently. Observations were performed using three
of the world’s best instruments mounted on telescopes located at observatories
in Chile. The brighest targets ($V\leq 15.7$ mag) were observed with the High
Accuracy Radial velocity Planet Searcher (HARPS) instrument mounted on the 3.6
meter telescope at the La Silla observatory. Most of the acquired spectra were
obtained with the Magellan Inamori Kyocera Echelle (MIKE) spectrograph mounted
on the 6.5 m Magellan Clay telescope at the Las Campanas observatory. A
significant fraction of data were also acquired with the European Southern
Observatory’s UV–Visual Echelle Spectrograph (UVES) on the 8.2 m Very Large
Telescope (VLT) at Paranal Observatory. HARPS and MIKE were used in visitor
mode, so that the observations were confined to the specific nights of the
corresponding runs. UVES observations were obtained in service mode and could
thus cover the orbital cycles more uniformly, while several-day-long HARPS
runs were useful for covering pulsation phases of bright, long-period
Cepheids.

##  4 Results

Figure 2: O–C diagram for one of the SB2 Cepheid candidates. A clear LTTE
modulation due to binary motion is present. The red line is the best fit to
the Cepheid’s orbit.

In a preliminary study, we analyzed photometric data from the OGLE catalogs
for our initial sample of 41 Cepheids. The study of their O–C diagrams led to
detection of the light travel-time effect (LTTE) due to binary motion for four
candidates. One example, with an orbital period of about 2500 days, is shown
in Figure 2.

###  4.1 Double-lined binary Cepheids

During the pilot program, we found 16 of 18 ($\sim$90%) of our brightest
candidate Cepheids to be components of SB2 systems, showing that the majority
of these overbright pulsators are normal Cepheids with luminous giant
companions. Additional observations of fainter candidates led to the discovery
of a Cepheid in a binary system with an orbital period of only 59 days, five
times shorter than any measured before for a firmly confirmed binary Cepheid.
For a more detailed description of the system, please refer to Pilecki et al.,
(2022). Updated RV curves for this object are shown in Figure 3.

Figure 3: Updated RV curves for OGLE-LMC-CEP-1347. The scatter in the Cepheid
RVs is due to unaccounted-for 2O-mode pulsations.

Figure 4: Preliminary orbits for two example double-lined binary Cepheids with
best-covered orbital cycles. More data are still needed to cover the pulsation
curves and improve the precision of the derived orbital parameters but no
major change is expected to the solution.

The observations have continued and, to date, 56 candidate Cepheids in the
LMC, SMC, and MW have been confirmed as members of SB2 systems. For 32 of
them, the anticorrelated orbital motions of the Cepheid and its companion were
detected, which represents the ultimate proof that they are gravitationally
bound. For 23, preliminary orbital solutions could be obtained, although in
general the data coverage is not yet satisfactory due to long orbital periods
and the complexity of the variabilities involved. For a few cases we managed
to cover the orbital cycles well, and the solutions are not expected to change
significantly (see Figure 4). Moreover, eight of the confirmed systems are
composed of two Cepheids (see the next subsection).

Figure 5: Example of a binary Cepheid with a preliminary mass ratio that is
significantly different from unity.

For systems with preliminary orbital solutions we calculated the mass ratios
of the components and the minimum Cepheid masses, $M_{\rm Cep}\sin^{3}(i)$.
The minimum masses obtained for several systems are close to the masses
derived for eclipsing Cepheids, showing that the inclinations of these systems
are high and that they are good targets for accurate and precise mass
measurements. Mass ratios obtained for some of the systems are different from
unity. As $q=M_{2}/M_{1}\sim 1$ is expected for components at similar
evolutionary stages and the same age, which means that they could have passed
through binary interactions in the past, either by means of a merger event or
via mass transfer. One example of such a system is shown in Figure 5.

Figure 6: Another binary Cepheid with an orbital period below the 200-day
limit.

Among the binary Cepheids with orbital solutions, a new system was found with
an orbital period below 200 days, which is the lower limit obtained by Neilson
et al., (2015) for binary Cepheids that passed through their evolution on the
RGB. As systems with periods below this limit would be destroyed at that
evolutionary stage, their components are probably younger, and the Cepheids
are likely on their first crossing through the instability strip (in the
Hertzsprung gap). Our newly discovered system has a period of only 163 days,
which is the second shortest period ever measured for a binary Cepheid (Figure
6).

Figure 7: Example of a wide-orbit binary Cepheid with a long orbital period.
The current solution suggests that its period may reach 10 years or even
longer.

The longest orbital period measured so far for an extragalactic binary system
is $P=1550$ days ($\sim$4 yr; Gieren et al., 2015). Meanwhile, preliminary
solutions for some of our new systems suggest orbital periods longer than 4
years, even up to about 10 years or more (see Figure 7). These binary systems,
for which we can obtain spectroscopic orbits, are currently the best
candidates for direct geometric distance measurements using interferometric
methods, offering the best possible accuracy. And although at the moment we do
not expect high precision for extragalactic measurements, with upcoming
improvements to instrumentation, such systems may eventually become our best
and most direct tools for distance measurements to nearby galaxies, surpassing
previously used methods (e.g., Pietrzyński et al., 2019).

###  4.2 Binary double Cepheids

An interesting subsample of double-lined binary Cepheids form those binary
systems composed of two Cepheid components. As part of our project we observed
nine candidates for such systems. Binary double (BIND) Cepheids are unique
laboratories for pulsation and evolution studies. However, only one system of
this kind was known before. To date, we have confirmed orbital motions for
eight additional systems of a similar nature. Preliminary orbits for two are
shown in Figure 8. Altough the orbital solutions may change once a full
orbital cycle has been covered, the anti-correlated orbital motion of both
components is already clear, proving binarity of these double Cepheids.

Interestingly, some preliminary solutions suggest mass ratios significantly
different from unity, while for others the pulsation periods are very
different (after fundamentalization of the 1O periods, for comparison). Both
features may indicate past binary interaction events, which would affect the
physical properties of the same-age components. Another interesting feature of
the new BIND Cepheids is their long orbital periods. For most, periods longer
than five years are expected according to the current models (based on only
partially covered cycles). Although long periods make the observations more
challenging, in the future these systems will also offer an opportunity to
measure direct distances to their host galaxies, as described in the previous
section. More recent results for BIND Cepheids can be found in Pilecki et al.,
(2024).

Figure 8: Preliminary orbits for two example binary double Cepheids show a
clear change of the components’ orbital velocities. However, as the orbital
cycle has not yet been fully covered, the final, exact solutions (including
the orbital periods) may change significantly.

##  5 Conclusions

Analysis of spectroscopic data we have collected so far clearly shows that the
Cepheids selected as described in Section 2 are indeed components of double-
lined binary systems. This has important implications for the interpretation
of period–luminosity relations and for our general knowledge of Cepheids and
their evolution. We know now that overbright Cepheids, which are often
rejected as P–L relation outliers, are in the great majority Cepheids with
red, luminous giant companions. These conclusions can also likely be extended
to other pulsating stars with well-defined P–L relations, e.g., Type II
Cepheids or RR Lyrae stars.

With 56 confirmed candidates, we have already ten-fold increased the number of
Cepheids in spectroscopic double-lined binaries. This includes over 90% of all
candidates in the LMC. In Figure 1 one can see that many of them have periods
longer than those of known SB2 Cepheids, reaching up to 10.5 days.
Extrapolation of the number of confirmed cases to the SMC suggests that about
50 new SMC Cepheids in SB2 systems can be expected once all SMC objects have
been studied. This would increase the total number of known SB2 Cepheids by a
further 80% (i.e., 20-fold compared with the state before we commenced the
project).

Eventually, our study will yield firm mass estimates for a large sample of
Cepheids, including long-period, high-mass Cepheids for which the lack of data
is the most severe and for those from the uncertain low-mass end. Similar
evolutionary stages for both components imply mass ratios close to unity. Any
significant deviation from unity may indicate a probable past merger event.
This is the only way merged Cepheids can be unambiguously detected (as
compared with, e.g., dubious chemical composition peculiarities) and
characterized. From the simulations of Sana et al., (2012) we can estimate
that up to 30% of Cepheids may be merger products. In our study we will be
able to estimate this fraction observationally and compare it with
simulations.

Geometric methods provide the most accurate and direct distances to
astronomical objects. One such method is based on the determination of both
astrometric (angular) and spectroscopic (absolute) orbits of binary systems,
which combined give directly the distance (see, e.g., Gallenne et al., 2019b
). Unfortunately, at increasingly large distances, it becomes commensurately
harder to measure orbital angular sizes, as one needs systems of increasingly
higher luminosity and larger component separations. Using traditional methods,
such systems are extremely hard to identify in other galaxies, as one needs
decades of monitoring to find just a few. However, our method of detection of
binary Cepheids is independent of the orbital size, and several systems with
very wide orbits have already been identified. They are currently our best
candidates for direct geometric distance determinations to the LMC and SMC,
which can eventually lead to the ultimate calibration of the first rung of the
cosmic distance ladder.

###### Acknowledgements.

The research leading to these results received funding from the Polish
National Science Center (grant SONATA BIS 2020/38/E/ST9/00486). This work is
based on observations collected at the Las Campanas Observatory and the
European Southern Observatory, ESO (ESO programs: 106.21GB, 108.229Z,
108.22BS, 110.2434, and 110.2436). We thank the Carnegie Institution and the
CNTAC for the allocation of observing time for this project. This research has
made use of NASA’s Astrophysics Data System Service.

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