# Effect of non-eikonal corrections on azimuthal asymmetries in the color glass condensate

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## Abstract

We analyse the azimuthal structure of two gluon correlations in the color glass condensate including those effects that result from relaxing the shockwave approximation for the target. Working in the Glasma graph approach suitable for collisions between dilute systems, we compute numerically the azimuthal distributions and show that both even and odd harmonics appear. We study their dependence on model parameters, energy of the collision, pseudorapidity and transverse momentum of the produced particles, and length of the target. While the contribution from non-eikonal corrections vanishes with increasing collision energy and becomes negligible at the energies of the Large Hadron Collider, it is found to be sizeable up to top energies at the Relativistic Heavy Ion Collider.

## 1 Introduction

The existence of azimuthal asymmetries in particle production stretched for a long pseudorapidity interval – named the ridge – has been observed in small collision systems, proton–proton and proton–nucleus, at the Large Hadron Collider (LHC) at CERN [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14] and the Relativistic Heavy Ion Collider (RHIC) at BNL [15, 16, 17, 18, 19]. The corresponding observation in nucleus–nucleus collisions finds a standard explanation in final state interactions that lead to a macroscopic description in terms of relativistic hydrodynamics. But for small systems such explanation is a matter of active debate, see e.g. the recent works [20, 21, 22], and initial state dynamics have also been invoked.

Concerning initial state explanations, those based on the effective theory for high-energy Quantum Chromodynamics [23] named the Color Glass Condensate (CGC) [24, 25, 26], have been explored intensively in recent years. For the collision of dilute objects like proton–proton, the “Glasma graph” approximation [27, 28], that encodes both Bose enhancement and Hanbury–Brown–Twiss (HBT) effects [29, 30, 31, 32], has been developed and used to describe experimental data [33, 34, 35, 36]. Besides two gluon correlations, those among three and four [37, 38] have also been studied, and also those between two quarks [39, 40]. The extension to dilute–dense (proton–nucleus) collisions was later done numerically [41] and analytically [42, 43, 44], including three gluon correlations [43], and applied to describe data [45, 46]. Complementary explanations in terms of density gradients [47] have also been considered to explain the observed azimuthal structure.

In this framework, the two remaining key problems are the analytical extension to dense–dense collisions, and the absence of odd azimuthal harmonics in standard calculations. To overcome the latter, several alternatives have been essayed: density corrections in the projectile [48, 49, 50] (implemented to attempt a description of data in [51, 52, 53]), quark correlations [45, 54, 55, 56] and a more involved description of the target [57, 58] than the one provided by the commonly used McLerran–Venugopalan (MV) model [59, 60].

In this work we explore a different direction. Usual calculations in the CGC employ the eikonal approximation: the process of propagation of an energetic parton from the projectile through the target, considered as a background field, is computed in the light cone gauge neglecting its transverse components and considering it as infinitely time dilated and Lorentz contracted – a shockwave. Also terms subleading in energy (among them, spin flip ones) are disregarded. This is to be contrasted to the calculations of elastic and radiative energy loss of energetic partons traversing a medium composed of coloured scattering centers – jet quenching. Here, the shockwave approximation is relaxed and the target is considered to have a finite length, see e.g. the reviews [61, 62].^{1}

Some years ago, a systematic expansion of the gluon propagator in non-eikonal terms stemming from the relaxation of the shockwave approximation was performed in [73, 74] and applied to particle production in the CGC in [75]. Using those ideas, in a recent paper [76] we have computed single, double and triple gluon production in the CGC including those non-eikonal corrections within the Glasma graph approximation – thus suitable for collisions of two dilute objects. It was anticipated there that an asymmetry between the near and away side ridges appeared for certain kinematic regions, which would lead to odd azimuthal harmonics. Restricted to two gluon correlations, it is the goal of the present work to study numerically the impact of these non-eikonal corrections on even and odd harmonics, and their dependence on model parameters, energy of the collision, pseudorapidity and transverse momentum of the produced particles, and length of the target.

As discussed in this introduction, non-eikonal corrections are not the only source of odd harmonics, others being density corrections or a more sophisticated treatment of the target beyond the MV model. Besides, they vanish with increasing energy, a trend that is not observed for the odd harmonics in experimental data. Therefore, here we make no attempt to compare with experimental data but only address the existence and size of the non-eikonal effects on the azimuthal structure.

The plan of the paper is as follows: in Sect. 2 we present the formulae for two-gluon correlations in a form derived from that in [76] but more suitable for a numerical implementation, and present the details of the model. In Sect. 3 we show the results for azimuthal harmonics. Finally, in Sect. 4 we provide our conclusions and outlook.

## 2 Non-eikonal double gluon production

*i*denote transverse coordinates, we use the shorthand notation \(\underline{k} \equiv (k^+,\mathbf k )\) for the three-momenta of the produced gluons, \(\int _\mathbf{q }\equiv \int d^2\mathbf q /(2\pi )^2\), \(N_c\) is the number of colors, \(\alpha _s=g^2/(4 \pi )\) the strong coupling constant, and the non-eikonal correction functions coming from the finite extension of the target in the \(+\) lightcone direction \(L^+\) read

- 1.We assume a Gaussian distribution of the colour sources inside the projectile, the MV model [59, 60], such thatwhere \(\mu \) is the width of the Gaussian and has units of mass squared.$$\begin{aligned} \mu ^2(\mathbf k ,\mathbf q )=\mu ^2 (2 \pi )^2 \delta ^{(2)}(\mathbf k +\mathbf q ), \end{aligned}$$(8)
- 2.We choose a Yukawa-type potential generated by the colour sources inside the target:where \(\mu _T\) is an infrared regulator analogous to a Debye mass.$$\begin{aligned} |a(\mathbf q )|^2=\frac{\mu _T^2}{(q^2+\mu _T^2)^2}, \end{aligned}$$(9)

## 3 Azimuthal harmonics

- (i)
\(k_1=p_T^{ref}\)

*and*\(k_2=p_T\):One way of defining the \(p_T\) dependence of the Fourier coefficients is by fixing one of the momenta, say \(k_1\), to a some reference momentum \(p_T^{ref}\) and letting the other momentum as the free variable, that is, \(k_2=p_T\). With this choice, the azimuthal harmonics are defined as (see e.g. [13])$$\begin{aligned} v_n(p_T)=\frac{V_{n\Delta }(p_T,p_T^{ref})}{\sqrt{V_{n\Delta }(p_T^{ref},p_T^{ref})}}. \end{aligned}$$(18) - (ii)
\(k_1=k_2=p_T\):

Another way of fixing the \(p_T\) dependence is by setting \(k_1=k_2=p_T^{ref}=p_T\) and therefore$$\begin{aligned} v_n(p_T)=\sqrt{V_{n\Delta }(p_T,p_T)}. \end{aligned}$$(19) - (iii)
*Integrating over*\(k_1\)*and*\(k_2=p_T\):Following we can define \(v_n(p_T)\) by integrating over \(k_1\) and letting \(k_2\) free as in [56], that is,$$\begin{aligned} 2v_n^2(p_T)&=\frac{\int _{0}^{\infty }k_1 dk_1 a_n(k_1,p_T)}{\int _{0}^{\infty }k_1 dk_1 a_0(k_1,p_T)}\nonumber \\&=2 \frac{\int _{0}^{\infty }k_1 dk_1\int _{0}^{\pi } N(k_1,p_T,\Delta \phi ) \cos (n \Delta \phi ) d\Delta \phi }{\int _{0}^{\infty }k_1 dk_1\int _{0}^{\pi } N(k_1,p_T,\Delta \phi ) d\Delta \phi }\, . \end{aligned}$$(20)

### 3.1 Numerical results

If *L* is the size of the target in its rest frame, then we have that \(L^+=\frac{1}{\gamma \sqrt{2}}L\approx 2 A^{1/3}/\gamma \ \text {fm}\approx 10A^{1/3}/\gamma \ \text {GeV}^{-1}\), where *A* is the mass number of the nucleus and \(\gamma \simeq \sqrt{s_\mathrm{NN}}/(2 m_N)\) accounts for the Lorentz contraction in the center of mass frame (therefore, our pseudorapidities will be considered in this frame). Furthermore, for the numerics we take the gluonic size of the projectile to be \(B_p = 4 \ \text {GeV}^{-2}\) [77], \(S_\perp =2 \pi B_p \approx 9.8\) mb, \(L=12\) fm (Pb nucleus) unless otherwise stated^{2} and \(N_c=3\). We also take \(\lambda ^+=0\) in Eq. (5) – note that this factor is irrelevant for the azimuthal harmonics using definitions in Eqs. (18) and (19) and gives a very small contribution using Eq. (20).

^{3}As mentioned, we are omitting in these plots a peak around \(p_T=p_T^{ref}\) which comes from the HBT contribution.

*x*or high energies where our formalism can be safely applied.

We would like to also mention that in Fig. 2, it is apparent that both even and odd harmonics peak around \(p_T\sim p_T^{ref}\). This is due to the fact that \(p_T^{ref}\) is chosen to be \(k_1\) and \(p_T\) is defined as \(k_2\), and the values of both even and odd harmonics maximise when the momenta of both produced gluons (\(k_1\) and \(k_2\) in our notation) are close to each other.

On the other hand, the unrealistic peaked shape of the HBT contribution is due to the fact we have used a simplistic approach, \(\mu ^2(\mathbf k ,\mathbf q )\propto \delta ^{(2)}(\mathbf k +\mathbf q )\). A more realistic approach would employ some function \(F[(\mathbf k +\mathbf q ) \sqrt{B_p}]\) (with \(B_p\) being the gluonic size of the projectile), which is peaked around \(\mathbf k +\mathbf q =0\), e.g. a Gaussian, in which case we should obtain a bell shape with smaller values for the harmonics when \(p_T=p_T^{ref}\). We will show results using a Gaussian distribution below.

One interesting behaviour of the odd azimuthal harmonics, stemming from the non-eikonal effects, that we observe in Fig. 2 is that at any fixed energy the value of the odd harmonics decreases with increasing value of rapidity \(\eta \). This behaviour is completely natural since the size of the odd harmonics is directly related the non-eikonal corrections in our framework. When expressed in terms of the rapidity, the eikonal expansion parameter can be written as \(p_TL^+e^{-\eta }\). With increasing value of the rapidity, non-eikonal corrections (and therefore odd harmonics) get smaller and vanish completely in the strict eikonal limit.

Another interesting feature of our result is that odd harmonics depend strongly of the size of the target while even ones are almost independent. Furthermore, all odd harmonics and all even harmonics show a good scaling with \(L^+\), as can be seen in Fig. 3. There we plot \(v_n\), using the definition Eq. (18), divided by its value for \(L^+=1.5\) fm. While the dependence with centrality and multiplicity would demand a detailed study and the variation of parameters in the model, see e.g. [52, 78], the increase of \(L^+\) with increasing centrality should be one of the ingredients in such dependence and this finding resembles qualitatively that in [52].

For the sake of completeness, we also compute the azimuthal harmonics using prescriptions Eqs. (19) and (20). Now, since we are integrating over variable \(k_1\), we have to regulate the \(1/k^2\) term that arises in Eq. (7). In order to do so, we just substitute \(1/k^2\rightarrow 1/(k^2+\mu _g^2)\) and we choose \(\mu _g=0.4\) GeV.^{4} The results are shown in Figs. 4 and 5 where we have used \(\mu _T=0.4 \ \text {GeV}\), \(\mu _P=0.2 \ \text {GeV}\) and \(\eta _1=\eta _2=1.5\). The dashed lines are our results for a Dirac delta in \(\mu ^2(\mathbf k _1,\mathbf k _2)\), and we observe that the shape of \(v_n(p_T)\) is very abrupt and unrealistic for small \(p_T\). This is what we should expect since \(\mu ^2(\mathbf k _1,\mathbf k _2) \propto (2\pi )^2 \delta ^{(2)} (\mathbf k _1-\mathbf k _2) \) comes from assuming translational invariance and this is only valid for large \(|\mathbf k _1-\mathbf k _2|\) or \(B_p\) but, in our case, we are using small values for both \(|\mathbf k _1-\mathbf k _2|\) and \(B_p\). In order to deal with this problem we make the substitution \((2\pi )^2 \delta ^{(2)} (\mathbf k _1-\mathbf k _2)\rightarrow 2 \pi B_p \exp {(-\frac{B_p}{2}(\mathbf k _1-\mathbf k _2)^2)}\) in the HBT term Eq. (A.10) since this is the dominant contribution. The corresponding results can be seen in the continuous lines of Figs. 4 and 5 and they are smoother.

In Fig. 6, we have plotted the ratio of the non-eikonal cross section with respect to the eikonal one both in the forward (\(\Delta \phi =0\)) and the backward (\(\Delta \phi =\pi \)) peaks and for \(\eta _1=0\), \(k_1=1\) GeV and \(k_2=1.2\) GeV (we set \(k_1\ne k_2\) to not include the HBT contribution). We can see that there is a sizeable difference between the peaks up to \(1.5-2\) units in rapidity for \(\sqrt{s_{\mathrm{NN}}}=20\) and 60 GeV, and that the difference becomes negligible for higher energies, \(\sqrt{s_\mathrm{NN}}=200\) GeV, as expected.

## 4 Conclusions

In this manuscript we have analyzed the effect on the non-eikonal corrections stemming from relaxing the shockwave approximation for the target which, therefore, acquires a finite length, on the two gluon inclusive cross section in the CGC. We work in the Glasma graph approximation suitable for collisions between dilute objects (pp). While the corresponding expressions were derived in a previous publication [76], here we focus on the numerical implementation, for which several model assumptions are made. We make no attempt to compare with experimental data but only address the existence and size of the non-eikonal effects on the azimuthal structure.

We explore how the non-eikonal corrections break the accidental forward–backward symmetry present in usual CGC calculations, and thus lead to sizeable odd harmonics. We discuss the different contributions: Bose enhancement of the projectile and target wave functions and HBT, and check the stability of the qualitative behavior of the results against variations in the functional forms and parameters in the model assumptions. We find a good scaling of all even and all odd harmonics with respect to the length of the target, with even harmonics being constant and odd ones growing with increasing length. The non-eikonal corrections vanish with increasing energy of the collision, being sizeable up to the top energies at RHIC but negligible for those at the LHC. Furthermore, they turn to be significant for pseudorapidity differences between the produced gluons up to about 2.5 units. Therefore, we conclude that non-eikonal effects cannot be the dominant source of odd harmonics at the highest energies but they can be relevant for those at RHIC.

The outlook of this work is its extension to dilute–dense (pA) collisions that will be the subject of a forthcoming publication, and a comparison to experimental data.

## Footnotes

- 1.
- 2.
As our aim is not to describe experimental data but to discuss the effect of the considered non-eikonal corrections, we will apply the calculation for proton–nucleus collisions though, as indicated above, the Glasma graph approach is only valid for collisions between dilute objects. See [41] for a comparison of the results of the Glasma graph approximation with a full dilute–dense numerical computation.

- 3.
While these values lie close to \(\Lambda _\mathrm{QCD}\), it is difficult to say how realistic they can be considered.

- 4.
The effect of changing this value to 0.2 GeV affects the azimuthal harmonics for \(p_T< 0.5\) GeV when the delta function form of the HBT term is used. For a Gaussian, see below, no sizeable effect of this change of \(\mu _g\) is observed.

- 5.
Were the projectile a dense object, the natural infrared regulator would be its saturation scale.

## Notes

### Acknowledgements

PA and NA are supported by Ministerio de Ciencia e Innovación of Spain under projects FPA2017-83814-P and Unidad de Excelencia María de Maetzu under project MDM-2016-0692, by Xunta de Galicia under project ED431C 2017/07, and by FEDER. The work of TA is supported by Grant no. 2018/31/D/ST2/00666 (SONATA 14 – National Science Centre, Poland). This work has been performed in the framework of COST Action CA15213 “Theory of hot matter and relativistic heavy-ion collisions” (THOR), MSCA RISE 823947 “Heavy ion collisions: collectivity and precision in saturation physics” (HIEIC) and has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant agreement no. 824093.

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