## Abstract

I give a self-contained review on radiative -broadening and radiative energy loss of high-energy partons in QCD matter. The typical of high-energy partons receives a double logarithmic correction due to the recoiling effect of medium-induced gluon radiation. Such a double logarithmic term, averaged over the path length of the partons, can be taken as the radiative correction to the jet quenching parameter and hence contributes to radiative energy loss. This has also been confirmed by detailed calculations of energy loss by radiating two gluons.

###### keywords:
QCD matter, jet quenching, double logarithmic corrections
1

00 \journalnameNuclear Physics B Proceedings Supplement \runauth \jidnppp \jnltitlelogoNuclear Physics B Proceedings Supplement

## 1 Introduction

High energy partons serve as hard probes to bulk QCD matter created in ultra-relativistic heavy-ion collisions at RHIC and the LHC. The motion of such hard probes, determined by the properties of bulk matter, can be theoretically studied by first-principle (pQCD) calculations. The nature of such a medium can be deciphered only by confronting theoretical calculations with experimental data. Calculations beyond leading order (LO) in pQCD are of great importance to testing the reliability of theoretical studies and hence help improves our understanding of the properties of bulk matter. Significant progress has recently been made on this topic. And I shall only limit myself to a self-contained summary on the leading-logarithmic correction proportional to obtained in a slightly generalized BDMPS-Z formalismBaier:1998 (); Zakharov:1996:1997 (). Here, is the length of the QCD medium and is the size of its constituents. Such a correction is universally present in radiative -broadeningWu:2011 (); Liou:Mueller:Wu:2013 (), the transport coefficient Blaizot:2014 (); Iancu:2014 () and radiative energy lossBlaizot:2014 (); Wu:2014 (); Arnold:2015 (). Let us start with the discussion about the typical and averaged transverse momentum broadening at LO in Baier:1996 (); Arnold:2009 ().

### 2.1 Typical p⊥-broadening at LO

is defined as the typical transverse momentum squared per unit length transferred from the medium to a high energy parton in color representation Baier:1996 (). In terms of the gluon distribution and the number density of the medium constituents, it can be written in the following formBaier:1996 ()

 ^qR≡ρ∫dq2⊥q2⊥dσRdq2⊥=4π2αsCRN2c−1ρxG(x,^qRL) (1)

with the differential cross section for single scattering. In a hot quark-gluon plasma (QGP) the gluon distribution in the wave function of each plasma particle is given by with for (anti-)quarks and for gluons and the (total) number density of each species is with the dimension of color representation . Here, () is the Debye mass and, in terms of the phase-space distribution ,

 ni≡∫d3p(2π)3fi(1+ϵifi)  with ϵA/F=±1. (2)

The following discussions shall be applicable to both cold and hot matter in terms of the corresponding .

A high-energy parton traversing a medium of length picks up a typical equal to , in which only the contribution from multiple soft scattering is includedBaier:1996 (). This can be easily understood in terms of the mean free path of the high-energy parton (with a typical momentum transfer to the medium)

 1λR≡σRρxG∼αsCRq2⊥ρxG∼^qRq2⊥. (3)

By comparing with , one can distinguish typical multiple soft scattering with in each individual scattering from rare single hard scattering with Arnold:2009 (). The distribution of transverse momenta of the parton takes the form

 dNd2p⊥=∫d2x⊥(2π)2eρL∫d2q⊥dσRd2q⊥(eip⊥⋅x⊥−1)S(x). (4)

Including only typical multiple scattering gives, in the logarithmic approximation,

 dNd2p⊥≈1π^qRLe−p2⊥^qRL. (5)

And it is modified by rare hard single scattering only at in which case one has

 dNd2p⊥≈ρLdσRd2p⊥∝p−4⊥. (6)

Quantities such as the average do receive large corrections from such rare scattering at high energies. However, the distribution carries more information and its shape is mainly characterized by the typical . This motivates us to focus only on multiple soft scattering and typical -broadening shall be simply denoted by in all the following discussions.

### 2.2 Double logarithmic correction to p⊥-broadening

In QCD matter the high-energy parton undergoes multiple scattering and radiates gluons. It hence generates a recoiling transverse momentum. Such a radiative correction to has been studied in Wu:2011 (); Liou:Mueller:Wu:2013 () using a formalism as an extension of that by BDMPS-ZBaier:1998 (); Zakharov:1996:1997 (). The complete analytic evaluation of the contribution from one-gluon emission is complicated by the presence of multiple scattering but double logarithmic terms, , and single logarithmic terms, , can be evaluated analytically.

The radiated gluon and the high-energy parton are in a coherent state within the formation time with the gluon’s energy and its transverse momentum. The evolution of the coherent pair is governed by a Schrödinger-type evolution equation with a potentialWu:2011 ()

 V≈−iNc^qRB2⊥4CR  for B2⊥≃1k2⊥≳1^qRL. (7)

The inverse of is the typical time scale for the pair of partons to undergo one individual collision and hence referred to as the mean free path of the coherent state

 λc≡4CRk2⊥Nc^qR⇒{λc≳t:single % scattering,λc≲t:multiple scattering. (8)

By taking one can see that becomes larger than that of a single parton when .

The double logarithmic contribution only comes from a single scattering. As discussed above, the single scattering phase space can be singled out by requiring the formation time of the gluon, radiated inside of the medium, is not larger than the mean free path of the coherent pair, that is,

 L≳λc≃k2⊥^q≳t  and  L≳t≳l0. (9)

Within this region, the double logarithmic correction can be easily reproduced from the differential cross section for the emission process in single scattering

 dσR→Rgdωdk2⊥d2p⊥=αsNcπ2q2⊥ωk2⊥p2⊥dσRd2q⊥∣∣ ∣∣→q⊥=→k⊥+→pR⊥ (10)

with the limits of the integral given by (9). And one hasLiou:Mueller:Wu:2013 ()

 Δp2⊥(L)≡αsNcπ^qRL∫Ll0dtt∫^qRL^qRtdk2⊥k2⊥ =^qRLΔ(L)  with Δ(L)≡αsNc2πln2(Ll0). (11)

Subleading single logarithmic terms have also been evaluated by crossing different boundaries between different regions in the phase space of the radiated gluon.

A resummation of leading double logarithmic terms from radiating an arbitrary number of gluons induced by a single scattering can be easily carried out by repeating the calculation in (2.2)Liou:Mueller:Wu:2013 (). Let us start with two-gluon emission. As illustrated in Fig. 1, the term proportional to comes only from the phase space in which the second gluon is radiated within the formation time of the first one and the transverse momentum of the second gluon is smaller than that of the first one. Accordingly, the next term in the leading double logarithmic series is given by

 ⟨p2⊥⟩2=(αsNcπ^qRL)2∫Ll0dt2t2∫^qRL^qRt2dk22⊥k22⊥ ×∫t1l0dt1t1∫k22⊥^qRt1dk21⊥k21⊥=(^qRL)23!2![2Δ(L)]2. (12)

By induction, one can show that the leading double logarithmic contribution from -gluon emission is given by

 ⟨p2⊥⟩n=(^qRL)n(n+1)!n![2Δ(L)]n (13)

and hence obtain Liou:Mueller:Wu:2013 ().

 ⟨p2⊥⟩ = ^qRL[2Δ(L)]−12I1([8Δ(L)]12). (14)

The -broadening in (2.2) is the typical and, after being averaged over the path length, can be taken as an effective (renormalized) Blaizot:2014 (); Iancu:2014 ()

 ^qrenR(L)≈^qR[1+Δ(L)]. (15)

Even though the correction includes contributions from radiating gluons with a life time , it can be still taken as a correction to the (local) transport coefficient as a consequence of the property of logarithmic integrationsBlaizot:2014 (). Let us only focus on leading double logarithmic terms from radiating gluons without overlapping formation times. They can be evaluated by repeating the calculation in (2.2) for each radiated gluon. Specifically, receives a correction of the form

 ΔSn(x⊥,L)=−αsNc4πn^qRLx2⊥∫Ll0dlntln(Lt) × ΔSn−1(x⊥,L−t) (16)

with

 ΔS1(x⊥,L)≈−14^qRLΔ(L)x2⊥e−14^qRLx2⊥. (17)

Each gluon has a formation time larger than and hence vanishes if . This simply indicates that the double logarithmic approximation breaks down when . Therefore, we have to restrict ourselves to the case . In this case leading logarithmic approximation applies and one has

 (18)

From this, one can justify the exponentiation of the double logarithmic integral in (2.2)

 S(x⊥)=e−14^qRLx2⊥∑n≲L/l0ΔSn≈e−14(^qRL+Δp2⊥)x2⊥, (19)

if satisfies

 (14Δp2⊥x2⊥)L/l0(L/l0)!

At it asymptotically gives and, in this case, plugging (19) into (4) gives

 dNdp2⊥≈e−p2⊥^qRL(1+Δ(L))^qRL(1+Δ(L)). (21)

This justifies that the leading logarithmic result in (2.2) is the typical value of .

## 3 Radiative parton energy loss

At high energies, the parton loses its energy mainly due to medium-induced gluon radiation (the LPM effect). Since the formation time of a radiated gluon grows with its energy, the average energy loss within any period of time is dominated by radiating one gluon with the formation time comparable with . Within , the gluon picks up a transverse momentum broadening . Accordingly, one has and the maximum energy that the gluon may carry away is given by . From this, one can get parametrically two well-known results: the LPM spectrumBaier:1998 (); Zakharov:1996:1997 ()

 ωdIdω∼αsNcLt(ω)∼αsNc√^qLω (22)

and the average energy loss per unit length Baier:1996 ()

 −dEdz∼αsNcωc(L)L=αsNc^qL. (23)

Will these LO results be modified by the radiative correction? The answer is: in double logarithmic accuracy the only thing one needs to do is replace by in (15), as justified in Blaizot:2014 (); Iancu:2014 (). This has been further confirmed by a detailed calculation of average energy loss from radiating two and more gluonsWu:2014 () and a more elaborated and careful analysis of two-gluon emission beyond the double logarithmic approximationArnold:2015 (). In the following I shall only take average energy loss for example to illustrate the underlying physical picture.

The energy loss due to two-gluon emission is greatly simplified in the double logarithmic region discussed in the previous section. The first gluon with energy , similar to that in the case with one-gluon emission, typically has a formation time , and . In the double logarithmic region, the second gluon with energy typically has

 k22⊥≲k21⊥,  ω2≲ω1,  t2≲√ω2^qA≲√ω1^qA≃t1. (24)

Therefore, the second gluon plays the same role as that in the calculation of the radiative -broadening in (2.2). After integrating out the contribution from the second gluon, we haveWu:2014 ()

 ΔE2≈αsNc12^qL2Δ(L), (25)

which has exactly the same prefactor as the LO result (for the parton approaching the medium from outside) inBaier:1998 (), i.e., .

The resummation of the double logarithmic correction in eq. (25) can be carried out in exactly the same way as that in the calculation of . For -gluon emission, besides the most long-lived gluon, the other gluons undergoes one single scattering. Using the same orderings in the formation time and transverse momenta of these gluons that lead to (13) gives a contribution of the form

 ΔEn+1=αsNc12L^qLn!(n+1)![αsNcπln2Ll0]n. (26)

Therefore, the total energy loss is given byWu:2014 ()

 ΔE=∞∑n=0ΔEn+1=αsNc12L⟨p2⊥⟩ (27)

with given by (14), including all the leading double logarithmic contributions from radiating arbitrary number of gluons.

## 4 Discussions

Our discussions so far have been focused on the medium-induced radiative corrections. Rather, in the above results the limit has been subtracted outBaier:1998 (); Zakharov:1996:1997 (). One effect neglected from such a subtraction is the running of the QCD coupling constant. We have also discussed this effect in the radiative correction to from one-gluon emission and find a factor 2 larger than in the fixed coupling caseLiou:Mueller:Wu:2013 (). How the inclusion of the running coupling modifies the evolution of has also been discussed in more detail in Iancu:Trian:2014 ().

At the end, I briefly comment on the differences between our double logarithmic radiative correction to and some previous results beyond LO in CaronHuot:2008 (); CasalderreySolana:Wang:2007 (). In CaronHuot:2008 (), the NLO () correction to transverse momentum broadening has been obtained analytically in a hot () quark-gluon plasma using a perturbative-kinetic approach. This result was calculated from the soft contribution to the two-body collisional kernel from soft collisions with . In contrast our result comes from radiation, which is of and would be present at next-next-to-leading order in that approachGhiglieri:2015 ().

The authors of Ref. CasalderreySolana:Wang:2007 () (see Kang:2015 () for a recent progress) have also calculated the radiative correction to -broadening in a QGP. The effect of multiple (soft) scattering was not considered in this formalism. As a result, the regime of double logarithmic approximation (Sec. V) has been extended into and . This is different from ours in (9) in which the phase space includes only single scattering inside the medium (multiple scattering sets the boundary at ). It will be intriguing to see how to formulate multiple scattering in this formalism.

## References

1. volume:

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