Finite angle effects in jet quenching

. The medium that forms in a heavy-ion collision modiﬁes the properties of jets traversing it. These modiﬁcations give substantial information about the nature of the medium and, therefore, they are one of the main focuses of the heavy-ion program at LHC. The inﬂuence of the medium into highly energetic partons depends on correlators of Wilson lines, which have been studied in perturbation theory and in several phenomenological models. Here, we focus on the antenna conﬁguration, the study of the evolution of the large energy partons resulting from a collinear splitting inside of the medium. In the eikonal limit, the interaction of each daughter parton with the medium can be parametrized by a light-like Wilson line in a given direction. We consider the case in which the angle between these two directions θ is small but such that L θ is of the order of the medium resolution scale, where L is the size of the medium. We ﬁnd that ﬁnite angle e ﬀ ects give sub-leading corrections to jet broadening. The physical e ﬀ ect in the antenna conﬁguration is to make the decoherence e ﬀ ect larger when ﬁnite angle corrections are taken into account.


Introduction
A consequence of asymptotic freedom is that at very large temperatures and densities quarks and gluons are not confined within hadrons anymore.This new state of matter is called the quark-gluon plasma (QGP).The only way that we can create this state, that was present in the early universe, on earth is by colliding relativistic heavy ions.However, it is very challenging to learn about the properties of the QGP by observing the particles produced in these collisions.The reason is that the QGP exists only during a time much shorter than what it takes the particles to hit the detectors.Therefore, we can only measure the indirect influence of the QGP.
We need well calibrated probes to be able to characterize the QGP.A particularly useful type of probes are those known as hard probes.These are particles whose production need a very large energy and, therefore, they can only be produced at the beginning of the collision in a hard process before the particles have time to thermalize.However, they are affected in a substantial way by the QGP that is formed afterwards.Then, by comparing the results in heavy ion collisions for these probes with the naive extrapolation of proton-proton data we can gain a lot of information about the medium.
In this work we are going to discuss a phenomenon called jet quenching (see [1] for a review).A jet is a collimated set of particles with a large energy and a small opening angle.
They are useful in the context of Quantum Chromodynamics (QCD) because the are defined in such a way that they are minimally sensitive to infrared non-perturbative physics.Jets are quenched (lose energy) when traversing a hot medium.Therefore, measuring jet quenching we can characterize the opacity of the medium and, from this, we can infer its properties.They are many models that can be used to describe jet quenching.However, in most of them it exists a parameter, called q, that controls the strength of the medium-jet interaction.
In recent years, there is an increasing interest in jet substructure, both in proton-proton and in heavy-ion collisions.Several techniques appeared that allow to re-cluster jets within a jet.In the context of heavy-ion collisions, this is interesting because it allows to test the interaction with the medium at different scales.However, to make the most of these techniques we need to understand how jet quenching depends on the opening angle of the jet constituents.
The angular structure of a jet is very well understood in the absence of a medium.We find the so-called angular ordering, meaning that each subsequent splitting has a smaller opening angle.This implies a high degree of coherence within the different splittings in the vacuum.The question then is what happens when the high energy partons split inside of a medium.A setting widely used in the literature to address these issues is the antenna configuration [2][3][4][5][6][7].It consists of a pair of quark-antiquark pairs that are created inside of the medium as a result of the splitting of a high energy particle.It has been found that, when the splitting angle is small, the medium can not resolve the products of the splitting and it interacts with jet as if the splitting never happened.For example, for a case of a photon splitting into a quark-antiquark pair with a small opening angle, medium interaction is highly suppressed.The opposite happens when the angle is large, from the point of view of the medium the two daughter partons are very far away from each other and they radiate energy as if they were uncorrelated particles.Note that this implies that there is no angular-ordering in this case.
Previously discussed results were obtained as a leading-order computation in the small angle limit.In this work, we want to study what happens when the angle is a little bit larger.What we mean by this is the following.In most studies, it is assumed that the daughter partons that form the antenna travel approximately among the same light-like direction as the mother parton.In our case, we are going to consider that each daughter parton follows a different light-like direction.Then, we are going to compute the leading-order corrections that this introduces.This work is organized as follows, in section 2 we discuss the general setting of the problem.Then, in section 3 we give some details about the computation.Finally, the conclusions are given in section 4.More details will be given in an upcoming publication [8].

Setting of the problem
We focus on the simple scenario of a pair of quark-antiquark created from a photon that radiate a high energy gluon after traversing the medium.This process is represented in fig. 1.It is a simple setting but it is already sensitive to the coherence effects that we wish to highlight.To discuss this problem, we use approximations that are common in the literature: • The large N c limit.
• We take the classical limit.In this context, we mean by classical that the gauge fields are equal in the amplitude and in the complex-conjugated amplitude.This is a good approximation for soft fields at finite temperature and on-shell particles at tree level.• The eikonal approximation.This means that collisions of the high energy partons with medium particles do not deviate them from their original trajectory along a light-like line.Blue lines represent the original amplitude while red elements comes from the folding of the complex-conjugated amplitude itself vanishes outside of the medium.However, it does imply that the field is a pure gauge outside of the QCD brick.
First, let us focus on the diagram on the left-hand side of fig. 1, that we call the self-energy diagram.We recall that in the eikonal limit a high energy quark (gluon) can be substituted by a light-like fundamental (adjoin) Wilson line.In this sense, in the large N c limit we can substitute the gluon by the exchange of a quark-antiquark pair separated by a very small distance and multiply the diagram by an overall 1  2 factor.This is equivalent to applying the Fierz identity to leading order in the large N c counting.Moreover, note that in the classical limit a quark line in the complex conjugate amplitude can be folded into an antiquark line in the amplitude such that we get a cancellation.The combined effect of the large N c plus the classical approximation for the self-energy diagram is illustrated in fig. 2. A blue line cancels with an equivalent red line with the arrows pointing out to the opposite direction.Therefore, all Wilson lines cancel in this process and we can conclude that there are no thermal effects.
The situation is completely different for the right-hand side diagram of fig. 1, that we call the interference diagram.Applying the same approximations (large N c and classical limits) and using the property that the field is a pure gauge outside of the medium, we can see that  all the thermal effects are equivalently included in the modulus square of the diagram of fig. 3.In summary, the contribution from the interference term is proportional to Δ med and that contains all medium effects.
where the contour is shown in fig. 3. The quark (antiquark) goes around a light-cone direction that we call n 1 (n 2 ).This has to be compared with previous results in which the role of Δ med was played Δ old med with where n is the light-cone direction of the parent photon.Although Δ med and Δ old med are equal at leading order in perturbation theory, they have important differences.First, we now take into account components of the gauge field perpendicular to n both in the quark and the antiquark lines.Second, there is an additional vertical lines that goes along the border of the medium.We can trace back the origin of this Wilson line to the effect that the gauge field have on the large energy particles once they escape the medium.We remind that these fields are a pure gauge outside of the medium, therefore, we could regard this effect as similar to the Aharonov-Bohm effect.Finally, Δ med is manifestly gauge invariant.
Let us now discuss some aspects related to the eikonal approximation.We assume that the quark (antiquark) moves eikonally with direction n 1 (n 2 ).They are related by the formula where θ 1.It is useful to define and ,  then W LO  2 can be read from eq. ( 7).We did not arrive to a simple analytic expression for W NLO   2   .However, we were able to compute it numerically and to understand its behavior for small and large angles.For large θ and for small θ where δn = θ/ √ 2 and L + = √ 2L.The numerical results can be found in fig. 5.More details about the computation can be found in a future publication [8].

Conclusions
In this work we have computed corrections that need to be considered when the opening angle of the antenna configuration is small but we are interested in sub-leading corrections.At leading order, the high energy partons only couple directly to space-like (off-shell) gluons.In other words, the interaction with medium particles can be encoded in a potential.At nextto-leading order this is no longer the case.As we have seen, we need to consider corrections coming from on-shell gluons.This implies, that at this level of accuracy we can no longer regard the medium as a collection of static scattering centers.
Our results are compatible with the previous literature as long as α s LT 1, which is true for nowadays heavy ion collisions.The leading corrections come from the influence of onshell gluons.They have less suppression in α s compared to off-shell gluons (because they get In the left panel we compare with the large LT θ limit, while in the right panel we compare with the opposite limit. thermal corrections already at tree level) but they do not have any phase space enhancement that results in an enhancement due to the size of the medium (LT power).
We note that the interaction with the gauge fields after the medium ends up giving the vertical Wilson line, which can not be ignored at next-to-leading order.
In conclusion, we can consistently add higher order corrections, that matter when θ is not so small, in a manifestly gauge invariant way.
I am grateful for the collaboration of Carlos Salgado and Manoel Rodríguez in this project.My work was funded by Maria de Maetzu excellence program under project CEX2020-001035-M; by Spanish Research State Agency under project PID2020-119632GB-I00; and by Xunta de Galicia (Centro singular de investigación de Galicia accreditation 2019-2022), by European Union ERDF.

Figure 1 .L 1 2 Figure 2 .
Figure1.Representation of the contributions to the cross-section of the process studied in this work.Only high energy particles are represented.The green-shaded area represents the extent of the medium.On the left (right) of the dashed line we represent the amplitude (complex-conjugated amplitude).

EPJ Web of Conferences 274 XV 0 LFigure 3 .
Figure 3. Diagram whose square contains all the relevant thermal modifications that are needed to describe the right-hand diagram of fig. 1

Figure 4 .
Figure 4. Relevant diagrams in the one-gluon-exchange approximation.The blue wavy line represents a soft thermal gluon exchange.

,Figure 5 .
Figure 5. Numerical evaluation of W NLO 2 .In the left panel we compare with the large LT θ limit, while in the right panel we compare with the opposite limit.

•
In this work, or main interest is the medium influence of partons traversing a static high temperature QCD brick.For this reason, we consider that chromoelectric and chromomagnetic fields outside of the medium vanish.Note that this does not imply that the gauge field