Isolated photon production in $pp$ collisions at forward rapidities and high multiplicity events

TL;DR

This paper investigates isolated photon production in high multiplicity proton-proton collisions at forward rapidities using the Color Glass Condensate formalism, providing predictions for photon yields and their dependence on event multiplicity.

Contribution

It introduces a novel analysis of isolated photon production in high multiplicity events within the CGC framework, considering multiple solutions of the BK equation.

Findings

Photon yield increases with multiplicity

Yield slope decreases at larger rapidities

Provides predictions for future experimental tests

Abstract

The production of isolated photons in high multiplicity events is investigated considering the Color Glass Condensate (CGC) formalism. The associated cross-section for proton - proton collisions is estimated considering three distinct solutions of the Balitsky - Kovchegov (BK) equation and predictions for the normalized photon yield as a function of the multiplicities of co - produced charged particles are presented. We predict the increasing of the yield with the multiplicity, with the slope being smaller for larger rapidities. As the isolated photon production is not affected by the fragmentation process, a future experimental investigation of this process in current high energy hadronic colliders is ideal to test the treatment of high multiplicity events using the CGC formalism, previously applied only for the production of hadronic final states.

Equations

$$\frac{dN^{pp\to\gamma X}}{d^{2}\mathbf{k}_{T}dy}=\sum_{q}\frac{e^{2}_{q}\alpha_{em}}{\pi(2\pi)^{3}}\int d^{2}\mathbf{l}_{T}\int_{x_{min}}dx_{p}z^{2}[1+(1-z)^{2}]\frac{q(x_{p},\mu^{2})}{\mathbf{k}^{2}_{T}}\frac{(\mathbf{k}_{T}+\mathbf{l}_{T})^{2}}{[z\mathbf{l}_{T}-(1-z)\mathbf{k}_{T}]^{2}}\int d^{2}\mbox{\boldmath$b$}S(\mathbf{k}_{T}+\mathbf{l}_{T},\mbox{\boldmath$b$},x_{g})\,,$$

$$x_g=\frac{|{\mathbf{k}}_T|e^{-y}+|{\mathbf{l}}_T|e^{-y_q}}{\sqrt{s}},y_q=\log \left(\frac{-e^y|{\mathbf{k}}_T|+x_p\sqrt{s}}{|{\mathbf{l}}_T|}\right),z=\frac{|{\mathbf{k}}_T|}{x_p\sqrt{s}{e}^y}.$$

$$\displaystyle\int d^{2}\mbox{\boldmath$b$}\,{\cal{N}}(x_{g},\mbox{\boldmath$r$},\mbox{\boldmath$b$})=\frac{\sigma_{0}}{2}\,N(x_{g},\mbox{\boldmath$r$})\,\,,$$

Full text

Photon production, Color Glass Condensate framework, High multiplicity events

The production of isolated photons in hadronic collisions at high energies is one the cleanest probes of the strong interactions and the structure of hadrons (For recent reviews see e.g. Refs. Blau:2023bvi ; David:2019wpt ). Over the last decades, this process has been largely studied, mainly motivated by the fact that, at leading order (LO), the process is dominated by the Compton scattering $q+g\rightarrow q+\gamma$, which implies that the isolated photon production is sensitive to the gluon distribution at small values of the Bjorken - $x$ variable Aurenche:1988vi ; Vogelsang:1995bg ; BrennerMariotto:2007yf ; BrennerMariotto:2008st ; Arleo:2011gc ; dEnterria:2012kvo ; Helenius:2014qla ; Klasen:2017dsy ; Goharipour:2018sip and to the description of the QCD dynamics at high energies Kopeliovich:1998nw ; Gelis:2002ki ; Kopeliovich:2007yva ; Kopeliovich:2009yw ; Machado:2008nz ; amir ; Basso:2015pba ; Ducloue:2017kkq ; Benic:2016uku ; Benic:2017znu ; Benic:2018hvb ; Goncalves:2020tvh ; SampaiodosSantos:2020lte ; Golec-Biernat:2020cah ; Benic:2022ixp . Another important aspect, is that contribution from photons generated by the collinear fragmentation of final - state partons is significantly reduced, suppressing the impact of final - state interactions and making the isolated photon production an excellent probe of the wave function of incoming particles.

At high energies (or at very small $x$), the hadronic wave functions are characterized by a large number of gluons hdqcd . Such dense system can be described by the Color Glass Condensate (CGC) effective theory CGC , with the evolution of the gluon distribution being given, in the mean - field approximation, by the Balitsky - Kovchegov (BK) equation BAL ; KOVCHEGOV . This framework implies the limitation on the maximum phase-space parton density that can be reached in the hadron wave function (parton saturation) and the emergence of a semihard saturation scale $Q_{s}(x)$, which is energy and atomic number dependent. Such scale characterizes the transition between the linear and non - linear regimes of the QCD dynamics and defines the main aspects of the particle production in minimum bias events. Moreover, the CGC framework also predicts rare parton configurations in the hadronic wave function, which are characterized by larger values of $Q_{s}$ and generate high multiplicity events. As this formalism is a promising approach for the particle production in low and high multiplicity events at the LHC, several authors Ma:2018bax ; Levin:2019fvb ; Kopeliovich:2019phc ; Gotsman:2020ubn ; Siddikov:2020lnq ; Siddikov:2021cgd ; Stebel:2021bbn ; Salazar:2021mpv ; Lima:2022mol have applied it to describe the event activity dependence of $J/\Psi$, $D$, $K_{S}^{0}$ and $\Lambda$ yields measured in proton - proton ($pp$) collisions, which are observed to grow rapidly as a function of the multiplicities of co - produced charged particles ALICE:2015ikl ; ALICE:2017wet ; STAR:2018smh ; ALICE:2020msa ; ALICE:2020eji ; ALICECollaboration:2020 ; ALICE:2021zkd . These studies demonstrate that the CGC framework provides a satisfactory description of the current data for low and medium multiplicities, with the predictions overshooting the data when the multiplicity becomes very high, indicating that new ingredients in the formalism are needed and/or new dynamical effects become relevant in these rare events. One possibility is the modification of the hadronization process in high multiplicity events, which can suppress the yields of hadronic final states. However, its modelling is not an easy task due to dominance of non - perturbative contributions and eventual medium effects that may be present. In addition, it is important to have a clear picture of what are the limitations of the current treatment of high multiplicity events using the CGC framework. Both aspects motivate the study of the isolated photon production in these rare events. As discussed above, the associated cross section is not affected by fragmentation processes while being sensitive to the non - linear effects in the hadronic wave function. Therefore, a future experimental analysis of this process will be ideal to disentangle the contribution of initial and final state effects in high multiplicity events and will help us to establish the necessary improvements in the theory. In what follows, we will perform, for the first time, the analysis of the isolated photon production in high multiplicity events using the CGC framework and present predictions for the normalized photon yield for different rapidities and distinct ranges of the photon transverse momentum. The results will be shown for three different solutions for the BK equation, making it possible to investigate different descriptions of the initial hadronic wave function. Our goal in this letter is twofold. Provide the CGC predictions for the isolated photon production at high multiplicity events and motivate the experimental analysis of this process in a near future.

Initially, let’s present a brief review of the CGC formalism for the isolated photon production. In this approach, the photon production is considered as a $\gamma$ Bremsstrahlung off a fast projectile quark propagating through the low-$x$ color field of the target Kopeliovich:1998nw ; Gelis:2002ki , with the photon radiation occurring either after or before the quark scatters off the target. Due to the interference between the corresponding amplitudes, the photon Bremsstrahlung process can be viewed as scattering of a $q\bar{q}$ dipole with a given transverse separation. This formalism predicts that in the coordinate space the associated differential cross section can be expressed in terms of the light-cone (LC) wave function $\Psi_{q\gamma}$, describing the real photon radiation off the projectile quark, and the dipole $S$ matrix, S(x_{g},\mbox{\boldmathr},\mbox{\boldmathb}), which describes the dipole - target scattering interaction for a dipole separation $r$ and impact parameter $b$. Such quantity can be expressed in terms of the dipole - target scattering amplitude {\cal{N}}(x_{g},\mbox{\boldmathr},\mbox{\boldmathb}), $S=1-{\cal{N}}$, with the evolution in $x_{g}$ being obtained by solving the Balitsky - Kovchegov (BK) equation. One has that in the transverse momentum space and in the massless quark limit, the inclusive photon yield in a $pp$ collision is given at leading order (LO) by Ducloue:2017kkq

[TABLE]

where $\mathbf{k}_{T}$ and $y$ are the photon transverse momentum and rapidity, respectively, $z$ represents the longitudinal momentum fraction of the quark carried by the photon, $x_{min}=k_{T}e^{y}/\sqrt{s}$, $q(x_{p},\mu^{2})$ is the collinear quark distribution function for a hard scale $\mu^{2}$ and the $S$ matrix in the momentum space is the Fourier transform of S(x_{g},\mbox{\boldmathr},\mbox{\boldmathb}). Moreover, one has that

[TABLE]

In order to reduce the fragmentation component for the photon production, we will enforce an isolation cut by multiplying the integrand of Eq. (1) by $\theta[\sqrt{(y-y_{q})^{2}+\Delta\phi}-R]$, where $\Delta\phi$ is the azimuthal angle between the scattering quark and the photon and $R$ is a chosen isolation cone radius. As in Ref. Ducloue:2017kkq , we will use the leading order CTEQ6 parton distribution functions cteq to describe the quark content of the projectile and assume that the impact parameter in the $S$ matrix can be factorized, such that

[TABLE]

where $\sigma_{0}$ is a constant determined by fitting the HERA data for the reduced cross section and N(x_{g},\mbox{\boldmathr}) is obtained by solving the running coupling BK equation Albacete:2007yr ; Albacete:2009fh ; Albacete:2010sy . In our analysis we will consider three different solutions for the rcBK equation derived assuming different parametrizations for the initial condition N(x_{g}=0.01,\mbox{\boldmathr}) and that provide similar values of $\chi/d.o.f$ for the HERA data Albacete:2012xq . In particular, we will consider the solution derived assuming the GBW parameterization, N(x_{g}=0.01,\mbox{\boldmathr})=1-\exp[-\frac{\mbox{\boldmathr}^{2}Q_{s,0}^{2}}{4}] with $Q_{s,0}^{2}=0.24$ GeV2, and two solutions obtained assuming a MV - type initial condition N(x_{g}=0.01,\mbox{\boldmathr})=1-\exp[-\frac{(\mbox{\boldmathr}^{2}Q_{s,0}^{2})^{\gamma}}{4}\ln(\frac{1}{\Lambda r}+e)], characterized by ($Q_{0}^{2}=0.157$ GeV2, $\gamma=1.101$) and ($Q_{0}^{2}=0.1597$ GeV2, $\gamma=1.118$). In what follows, the predictions derived with these distinct input will be denoted, respectively, by g1 (GBW), g1.101 (MV) and g1.118 (MV).

In Fig. 1 (left panel) we present our predictions for the transverse momentum distribution integrated over distinct ranges of rapidity considering $pp$ collisions at $\sqrt{s}=8$ TeV and varying the factorization scale in the range $0.25Q^{2}\leq\mu^{2}\leq 4Q^{2}$, with $Q^{2}=\mbox{max}\{\mathbf{l}_{T}^{2},\mathbf{k}_{T}^{2}\}$ Ducloue:2017kkq . Moreover, we have assumed $R=0.4$ and considered the distinct solutions of the BK equation for the dipole - proton scattering amplitude. One has that the current ATLAS data ATLAS:2016 for the isolated photon production are satisfactorily described by our predictions. Our results for the transverse momentum dependence of the invariant photon yield for $pp$ collisions at $\sqrt{s}=13$ TeV are presented in Fig. 1 (right panel) assuming distinct values for the photon rapidity. One has that the predictions differ mainly at large values of $k_{T}$. The comparison of these predictions with future experimental data will be useful to check the validity of the formalism, especially at forward rapidities where we expect a larger impact of the non - linear effects on the QCD dynamics.

In what follows we will focus on the isolated photon production in high multiplicity events at the LHC. Our goal is to estimate the ratio $dN_{\gamma}/dy/\langle dN_{\gamma}/dy\rangle$ as a function of the multiplicity, where $dN_{\gamma}/dy$ is the rapidity distribution for a given multiplicity, obtained from Eq. (1) by integrating it over a given range of $k_{T}$, and $\langle dN_{\gamma}/dy\rangle$ is its minimum bias value. As in previous studies Ma:2018bax ; Levin:2019fvb ; Kopeliovich:2019phc ; Gotsman:2020ubn ; Siddikov:2020lnq ; Siddikov:2021cgd ; Stebel:2021bbn ; Salazar:2021mpv ; Lima:2022mol , we will assume that the particle production mechanism is the same for low and high - multiplicity events, with the main difference being the saturation scale present in these two classes of events. Such assumption implies that Eq. (1) is assumed to be valid for both classes, and that the high multiplicity configurations can be approximated by increasing the value of $Q_{s,0}^{2}$ in the initial condition of the BK equation as follows $Q_{s,0}^{2}(n)=n\cdot Q_{s,0}^{2}$. Our results for the dependence on $n$ of the normalized yield are presented in Fig. 2 considering the production of an isolated photon with $y=0$ in $pp$ collisions at $\sqrt{s}=13$ TeV and assuming distinct solutions of the BK equation. In the top and middle panels, we present results derived by integrating the transverse momentum of the photon in the range $1.0\leq k_{T}\leq 12$ GeV. The dependence on these values is analyzed in the bottom panel. Initially, in the upper panel, we analyze the dependence of our predictions on the value of the isolation cone radius $R$. It is important to note that the predictions derived using the distinct solutions of the BK equation have been rescaled by different constant factors to improve visualization. One has that the predictions are almost identical for the two values of $R$ considered, which is expected since we are estimating the normalized yield. As a consequence, in the next calculations we only will present predictions derived assuming $R=0.1$. The dependence on the factorization scale $\mu^{2}$ is presented in the middle panel for the g1(GBW) solution. One has that the high $n$ behavior is sensitive to the choice for the hard scale, with larger values of $\mu^{2}$ implying a reduction of the normalized photon yield. A similar behavior is verified for the other solutions of the BK equation. Finally, in the lower panel, we present the results derived by integrating over distinct $k_{T}$ ranges. One has that the predictions are sensitive to the $k_{T}$ range and BK solution considered in the calculation. In particular, the increasing of the normalized photon yield with $n$ is steeper when the minimum value of $k_{T}$ ($k_{T}^{\min}$) is larger. A similar behavior already been observed for other final states Ma:2018bax ; Levin:2019fvb ; Kopeliovich:2019phc ; Gotsman:2020ubn ; Siddikov:2020lnq ; Siddikov:2021cgd ; Stebel:2021bbn ; Salazar:2021mpv ; Lima:2022mol . Moreover, the predictions also become more dependent on the BK solution used in the calculation for a larger $k_{T}^{\min}$. Such behaviour is directly associated to the fact that the main difference between the BK solutions occurs for larger values of the transverse momentum and this difference is amplified when larger values of initial saturation scale are assumed.

In recent years, the dependence on the multiplicity of a process has been analyzed by studying the correlation between the normalized yield for the considered final state and for charged hadrons. The latter is described by $dN_{ch}/d\eta/\langle dN_{ch}/d\eta\rangle$ and is predicted to be proportional to $n$ when the non - linear effects are taken into account (See e.g. Ref. Lappi:2011gu ). In order to study this correlation for the isolated photon case, we will estimate the normalized yield for charged particles as in Ref. Lima:2022mol , taking into account the contribution of charged pions, baryons and strange mesons and assuming the distinct solutions for the BK equation. As in Ref. ALICE:2021zkd , the charged hadron yield will be estimated in all cases assuming that the particles are produced at central rapidities. One has verified that the corresponding predictions describe the current data for the inclusive hadron production at central rapidities in $pp$ collisions at the LHC. Our predictions for the correlation are presented in Fig. 3 for different values of the photon rapidity $y$ and distinct solutions of the BK equation. The results have been obtained by integrating the photon momentum in the range $4.0\leq k_{T}\leq 12.0$ GeV, the same range considered by the ALICE Collaboration in its measurement of the $K_{S}^{0}$ meson ALICECollaboration:2020 . Predictions for other $k_{T}$ ranges can be provided upon request. The solid line in Fig. 3 indicates the expected result for a linear correlation between the yields. One has that the increasing of the isolated photon yield with the multiplicity is strongly dependent on the rapidity, becoming weaker as $y$ increases. In particular, for very forward rapidities, we predict an almost linear dependence of the multiplicity, with the results obtained by employing different solutions of the BK equation being similar. Such a result is expected, since for large values of $y$ one has large values of the saturation scale, implying that both the isolated photon and charged hadron yields will be impacted by the non - linear effects in the QCD dynamics in a similar way. Moreover, our results indicate that the predictions for large multiplicities and central rapidities are sensitive to the description of the QCD dynamics, with the g1.101 (MV) solution predicting a larger enhancement. A future comparison of these predictions with the experimental data will be an useful test of the CGC formalism as well of the main assumptions present in the treatment of the high multiplicity events.

As a summary, in this letter we have investigated the isolated photon production in high multiplicity $pp$ collisions at the LHC considering the CGC formalism, which provides an unified approach for the treatment of low and high multiplicity events as well for the description of the hadron and photon production at high energies. Our study has been motivated by the fact that the isolated photon yield is not expected to be affected by final state interactions and hadronization effects, which can modify the predictions for the hadron production in rare events. Therefore, a future comparison between the predictions presented in this letter with experimental data will be a clean probe of the CGC formalism and an important test of the assumptions assumed in the modelling of the high multiplicity events. We strongly motivate the experimental analysis of the isolated photon production at distinct multiplicities and different rapidities in the forthcoming years.

Acknowledgements

This work was partially supported by INCT-FNA (Process No. 464898/2014-5). V.P.G. was partially supported by CNPq, CAPES and FAPERGS. Y.N.L. was partially financed by CAPES (process 001). A.V.G. has been partially supported by CNPq.