Thermal radiation and inclusive production in the CGC/saturation approach at high energies
Abstract
In this paper, we discuss the inclusive production of hadrons in the framework of the CGC/saturation approach. We argue, that gluon jet inclusive production stems from the vicinity of the saturation momentum, even for small values of the transverse momenta \(p_T\). Since in this region, we theoretically, know the scattering amplitude, we claim that we can provide reliable estimates for this process. We demonstrate, that in a widely accepted model for confinement, we require a thermal radiation term to describe the experimental data. In this model the parton (quark or gluon) with the transverse momenta of the order of \(Q_s\) decays into hadrons with the given fragmentation functions, and the production of the hadron with small transverse momenta is suppressed by the mass of the gluon jet. In addition we show that other approaches for confinement, also describe the data, without the need for thermal emission.
1 Introduction
In this paper we discuss the dynamics of generating multihadron processes at high energy in the framework of the Color Glass Condensate(CGC)/saturation approach (see Ref. [1] for the review). These processes occur at long distances and therefore to treat them theoretically, we need to develop a nonperturbative QCD approach. This is a very difficult and challenging problem, which is far from being solved. The CGC/saturation approach, being an effective QCD theory at high energies, states that the new phase of QCD: the dense system of partons (gluons and quarks) is produced in collisions with a new characteristic scale: saturation momentum \(Q_s(W)\), which increases as a function of energy W [2, 3, 4, 5, 6, 7]. However, the transition from this system of partons to the measured state of hadrons is still an unsolved problem. At the moment, we need to use pure phenomenological input for the long distance nonperturbative physics, due to our lack of theoretical understanding of the confinement of quarks and gluons. In particular, we wish to use phenomenological fragmentation functions. Hence, our model for confinement is that the parton (quark or gluon) with the transverse momenta of the order of \(Q_s\) decays into hadrons with the given fragmentation functions. The experimental data confirm this model of hadronization, which is the foundation of all Monte Carlo simulation programs, and leads to descriptions of the transverse momenta distribution of the hadrons at the LHC energies. As an example, we refer to Ref. [8], which shows that the nexttoleading order QCD calculations with formation of the hadrons in accord with the fragmentation functions ([9, 10]), is able to describe the transverse momentum spectra for the LHC range of energies. However, such a description is only successful for large values of \(p_T\, >\, 3\, \,\mathrm{{GeV}}\) [9] or \(p_T\, >\, 5 \,\mathrm{{GeV}}\) [10], while we assume that one can use these fragmentation functions in the region of small \(p_T\) as well. In a sense, at present, this model is the best that we can propose to describe multi  hadron production.
For protonproton scattering in the lab. frame, the parton configuration in QCD is formed long before the interaction at distances 1 / (mx), where m  denotes the proton mass, and x the fraction of longitudinal momentum carried by parton which interacts with the target. However, before the collision, the wave function of this partonic fluctuaction is the eigenfunction of the Hamiltonian and, therefore, the system has zero entropy. The interaction with the target of size R destroys the coherence of the parton wave function of the projectile. The typical time, which is needed for this, is of the order of \(\Delta t \propto R\), and is much smaller than the lifetime of all faster partons in the fluctuation. Hence, this interaction can be viewed as a rapid quench of the entangled partonic state [26] with substantial entanglement entropy. After this rapid quench, the interaction of the gluons change the Hamiltonian. In the CGC approach, all partons with rapidity larger than that of a particular gluon \(y_i\), live longer than this parton. They can be considered as the source of the classical field that emits this gluon. It was shown that after the quench, the fast gluons create the longitudinal chromoelectrical background field, which leads to the thermal distribution of the produced gluons.
The goal of the paper is to revisit inclusive production in the CGC/saturation approach for a more thorough consideration, and to show that the thermal term with the temperature given by Eq. (3), is needed for describing the experimental data at high energies. It should be noted that in the first attempt [29] to compare the CGC prediction with the experiment at \(W\,=\,7 \,TeV\), the thermal term was not required.
The paper is organized as follows. In the next section we discuss the general procedure for the calculation of the gluon inclusive production in CGC/saturation approach. In Sect. 3, we consider the evolution equation for the theory with a simplified BFKL kernel. We show that the solution to this equation confirms our key idea, that the main contribution to the inclusive production stems from the kinematic region in the vicinity of the saturation scale. Since theoretically we know the scattering amplitude in this region, we demonstrate that we are able to provide reliable estimates for this process. In Sect. 4 we develop the saturation model which we need to use due to the long standing unsolved problem i.e. the behaviour of the scattering amplitudes at large impact parameter. In Sect. 5 we compare our estimates with the experimental data, and demonstrate that within our model for confinement: the fragmentation function for the gluon jets, needs to have a thermal radiation term with temperature, which is proportional to the saturation scale \(Q_s\). In the Conclusions we summarize our results.
2 Inclusive production in CGC/saturation approach: generalities
We note that Eq. (8) as well as Eq. (9) leads to a cross section which is proportional to \(1/p^2_T\). This behaviour results in a logarithmic divergency of the integral over \(p_T\), or in other words gives an infinite number of produced pions at fixed rapidity.This divergency also indicates that we need to reformulate our assumptions about confinement, since using the fragmentation functions does not suppress the divergency at low \(p_T\). We believe that the reason for this divergency, is the fact that we neglected the mass of the jet of hadrons that stem from the decay of the gluon. The simple estimates [35] give for a gluon with the value of the transverse momentum \(p_T\), the mass of the jet \(m^2_{\mathrm{jet}}\,=\,2 p_T\, m_{\mathrm{eff}}\), where \( m_{\mathrm{eff}} = \sqrt{m^2 + k_T^2 + k^2_L}  k_L\), m is the mass of the lightest hadron in the jet, \(k_T\) is it’s transverse momentum and \(k_L \approx \,k_T\) is the longitudinal momentum of this hadron. Since most pions stem from the decay of \(\rho \)resonances we expect that \( m_{\mathrm{eff}} \approx m_\rho \).
3 Nonlinear evolution for the leading twist BFKL kernel
3.1 Equations
At first sight, we need to solve this equation to obtain the inclusive cross section. However, we need to know the dependence of the dipole amplitude on the impact parameter, which cannot be found from Eq. (12). The failure to reproduce the correct large b behaviour of the scattering amplitude, is a long standing problem of nonperturbative QCD contributions [45, 46] to the BK equation. As a consequence we are doomed to use a phenomenological input in addition to Eq. (12). Hence, we suggest the following strategy: to use a simplified form of Eq. (12) and to study in this approach the main features of the inclusive production. After such an investigation, we will select the model which satisfies both the BK equation, and reproduces the correct behaviour at large b.

\({\tau \,\,=\,\, r\,Q_s \ll \,1}\)

\( {\tau \,\,=\,\, r\,Q_s \gg \,1}\)
The advantage of the simplified kernel of Eq. (14) is that, in the Double Log Approximation (DLA) for \(\tau < 1\), it provides a matching with the DGLAP evolution equation [50, 51, 52, 53, 54].
3.2 Solutions
3.2.1 Perturbative QCD: linear equation
3.2.2 Solution in the region \(\tau \,<\,1\)
3.2.3 Solution in the region \(\tau \,>\,1\)
3.2.4 Inclusive production
Taking \({\bar{\alpha }}_{S}= 0.2\) we find that integration over negative z gives 85% of the total contribution.
We wish to emphasize that the contribution to \(\mathcal{I}\left( \tilde{p}_T\right) \) at sufficiently short distances does not lead to the suppression of this function at \(\tilde{p}_T \,=\, 0\). Therefore, the cross section is still divergent at \(p_T \rightarrow 0\), which results in the large production of soft gluons in the framework of CGC/saturation approach, and is suppressed by the process of hadronization.
4 Impactparameter dependent CGC dipole model
Fitted parameters of the model [28], which we use in our estimates
\(\bar{\gamma }\)  \(N_0\)  \(\lambda \)  \(x_0\)  \(B_{CGC}\) (\(\,\mathrm{{GeV}}^{2}\)) 

0.6599 ± 0.0003  0.3358± 0.0004  0.2063 ± 0.0004  \(0.00105\pm 1.13 10^{5}\)  5.5 
The fact that the impact parameter behaviour of the saturation momentum determines the bdependence of the scattering amplitude comes from theory, while the particular form and result of the integration over b, stems from the model for \(S\left( b\right) \).
For \(\tau \,\ge \,1\) we have discussed the form of Eq. (48) in the previous section, and have given strong arguments for such an expression. The b integration is performed with the phenomenological \(S\left( b \right) \).
In Fig. 4 we plot the numerical result for \(\mathcal{I}\left( \tilde{p}_T\right) \) with functions given by Eq. (48) and the analytical expression of Eq. (50). Note that for \(\tilde{p}_T\, \ge \,2\) (or \( p_T \,\ge \,2 \,Q_s(x)\) ) both functions coincide.
For \(\tilde{p}_T=0\), or more generally for \(p_T \ll Q_s\), the typical distances turns out to be \(r =1/(2 Q_s(x))\), and for large \(\tilde{p}_T\) they are of the order of \(1/(4 p_T)\). Hence, we suggest to use in Eq. (53) the calculated \(\langle \tau \rangle \left( \tilde{p}_T\right) \) for \(\tilde{p} \le 4\) and \(1/(4 \tilde{p}_T\)) for \(\tau \ge 4\).
5 Comparison with the experimental data
As we have discussed in the introduction, our goal is to answer the question. Do we need the thermal radiation term to describe the experimental data in the framework of the CGC/saturation approach? Our answer is yes.
We have partly explained that \(\gamma _{\mathrm{eff}}\) from Fig. 6 is not able to describe the shape of \(p_T\) distribution at high \(p_T\). On the other hand, we predict the value of the cross section, while in Eq. (1) this value was a fitted parameter. We will discuss below the dependence of the cross section on the value of \(m_{\mathrm{eff}}\).
\(T_{\mathrm{th}} \,\propto \,Q_S\) was taken from Eq. (3) , however, it turns out that the experimental data can be described with \(c = 2.3\) which is almost twice larger than estimated in Ref. [25].
\(R\,\,=\,\,d^2\sigma /d^2p_T (\text{ thermal } \text{ radiation })\Big / d^2\sigma /d^2p_T (\text{ sum })\) versus the values of energies and the value of \(m_{\mathrm{eff}}\)
W (TeV)  \(m_{\mathrm{eff}}\) = 0.5 GeV)  \(m_{\mathrm{eff}}\) = 0.12 GeV 

13  70%  46% 
7  70%  43% 
2.76  59%  13% 
0.9  53%  7% 
We see that at small values of the effective mass, we can describe the experimental data without the thermal radiation term. It should be stressed that we do not need the so called K factor, to include the nexttoleading order corrections. Even for the multiplicity distribution at W = 13 TeV we are able to describe the data using \(\sigma _{in} \,\,=\,\,\sigma _{tot}  \sigma _{el}  \sigma _{diff}\) from Refs. [61, 62]. We recall that the simple formula for \(m_\mathrm{eff}= \sqrt{\mu ^2 + k^2_T +k^2_L}  k_L\) leads to \(m_{\mathrm{eff}} = 0.5\,\,\mathrm{{GeV}}\) if \(\mu \) is equal to the mass of \(\rho \)resonance since the value of \(k_T = k_L = 0.45\,\,\mathrm{{GeV}}\) (see Ref. [63] for the measurement and Ref. [64]) and reference therein for theoretical discussions). For the minimal mass of \(\mu = m_\pi = 0.14 \,\,\mathrm{{GeV}}\) we obtain \(m_{\mathrm{eff}}\,=0.2\,\,\mathrm{{GeV}}\).
Discussing hadron production we have to construct a model for the process of hadronization. Our model is the production of the gluon jets with the hadronization, which is given by the fragmentation functions. We showed that in this model for confinement, we obtained a reasonable description of the experimental data, with the thermal radiation and with the temperature of Eq. (3), which is predicted in the CGC approach. It is possible that our hadronization model is too primitive, and for the gluon with the transverse momenta of the order of \(\Lambda _{\mathrm{QCD}}\), we should not apply the CGC formulae which are based on the perturbative QCD approach. If we cut our gluon spectra at \(p_T =\Lambda _{\mathrm{QCD}}\), we obtain a good description of the experimental data, without the thermal radiation. An alternate picture could be the following: The propagator of the gluon with transverse momentum \(p_T\) in the CGC medium with the temperature \(T_{\mathrm{th}}\), acquires a mass \(m_g\,\,\propto \,\, T_{\mathrm{th}}\) [65] and the propagator acquires the form \(1/(p^2_T + m^2_g)\). This mass provides the infrared cutoff in the gluon spectrum. Therefore, the same rescatterings in the produced CGC medium, which generates the thermal spectrum can be a reason for blocking the small gluon \(p_T \approx T_{\mathrm{th}} \approx 0.120.14 \,\,\mathrm{{GeV}}\). In this picture we will not see any thermal emission in the spectrum of hadrons. Note, that \(p_T \sim 0.12 0.14\, \,\mathrm{{GeV}} \) corresponds to the \(m_{\mathrm{eff}} \approx 0.06\,\,\mathrm{{GeV}}\) and to the distribution of Fig. 10b.
In Fig. 11 we present the estimates with the gluon propagator \(1/(p^2_T + m^2)\) with \(m = T_{\mathrm{th}}\). One can see that we are able to successfully describe the data without the thermal radiation term. Such a description should only be considered with a grain of salt, since \( m = g T\) with small g in Ref. [65], and realistic estimates will overshoot the data.
6 Conclusions
The main result of the paper is, that we show the need for thermal emission within a particular model for confinement: the parton (quark or gluon) with the transverse momenta of the order of \(Q_s\) decays into hadrons with the given fragmentation functions. The temperature of this emission turns out to be equal to \(2.3/(2 \,\pi )\,Q_s\), as was expected in the CGC/saturation approach. Note, that the coefficient c in Eq. (3) turns out to be in almost two times larger than predicted in Ref. [25].
We develop the formalism for the calculation of the transverse momenta spectra in CGC/saturation approach, which is based on the observation that even for small values of \(p_T\) the main contribution stems from the kinematic region in vicinity of the saturation momentum, where theoretically, we know the scattering amplitude. In other words, it means that we do not need to introduce the nonperturbative corrections due to the unknown physics at long distances (see Refs. [66, 67] for example) in the dipole scattering amplitude. The nonperturbative corrections have to be included to describe the process of hadronization, which we discuss in the model. This model incorporates the decay of the gluon jet with the effective mass \(m^2_{\mathrm{eff}} = 2 Q_s \mu _{\mathrm{soft}}\) where \(\mu _\mathrm{soft}\) is the soft scale, and with the fragmentation functions of Eq. (10) at all values of the transverse momenta.
It should be emphasized that we reproduce the experimental data without any Kfactor, which is used for accounting for higher order corrections. We wish also to mention, that we have calculated the inclusive production taking \({\bar{\alpha }}_{S}= 0.25\). This value is less that \({\bar{\alpha }}_{S}\left( Q_s\right) \,\,=\,\,0.3\) which appears more natural in Eq. (8). For \({\bar{\alpha }}_{S}= {\bar{\alpha }}_{S}\left( Q_s\right) \), we need to introduce a Kfactor of about 1.3–1.5.
The value of the thermal radiation term contribution depends on the value of the effective mass \(m_{\mathrm{eff}}\), however, in the region of possible values for this mass 0.12–0.5 GeV we need to account for the thermal emission to describe the spectrum at low \(p_T\).
We show that a different mechanism of confinement that blocks the emission of gluons with \(p_T \,\le \Lambda _{\mathrm{QCD}} \) or/and that generates the gluon mass \(m = T_{\mathrm{th}}\), is able to describe the experimental data without the thermal radiation.
Hence, we state that the existence of the thermal term in the \(p_T\) spectrum of produced hadrons, depends crucially on the model for hadronization.
In our approach we are able to evaluate the kinematic region that we can use the formalism of the deep inelastic structure functions. The structure function is related to the scattering amplitude being \(\int d^2 b N\left( Y, r,b\right) \). However, as we have discussed the inclusive production is determined by function \(N_G\) [see Eq. (6)]. In the region where we can neglect the \(N^2\) term in \(N_G\), we can safely perform the integration over b, and obtain the expression for the inclusive production through the structure functions. In Fig. 13 we show Eq. (50) and the first term of this equation which is the gluon structure function in the vicinity of the saturation scale. One can conclude that for \(p_T \,>\,2 \,Q_s\) we can safely use the gluon structure function which has been measured for DIS at HERA. On the other hand, the thermal emission comes from the region \(p_T \,<\,2 \,Q_s\) and, therefore, the existence of this phenomenon depends completely on the inclusive prodiction in CGC/saturation approach in this region.
Footnotes
 1.
The DIS data has also been described by the saturation model of Ref. [57], as this model does not produce the theoretically correct behaviour deep in the saturation region, we do not consider it in this paper.
 2.
Note that we introduce the extra factor \(\frac{1}{2}\) in the definition of the saturation scale since we use \(\tau = r\,Q_s\), while in Ref. [28] \(\tau \) is defined as \(\tau = r Q_s/2\).
Notes
Acknowledgements
We thank our colleagues at Tel Aviv university and UTFSM for encouraging discussions. Our special thanks go to Keith Baker and Dmitry Kharzeev for fruitful discussions on the subject which prompted the appearance of this paper.
This research was supported by Proyecto Basal FB 0821(Chile), Fondecyt (Chile) grant 1180118 and by CONICYT grant PIA ACT1406.
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