Transverse momentum broadening and gauge invariance

# Transverse momentum broadening and gauge invariance

## Abstract

In the framework of the soft-collinear effective theory, we present a gauge invariant definition of the transverse momentum broadening probability of a highly-energetic collinear quark in a medium and consequently of the jet quenching parameter .

Jet Quenching, SCET
###### :
12.38.-t,12.38.Mh,13.87.-a
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6x9

address=Physik-Department, Technische Universität München, James-Franck-Str. 1, 85748 Garching, Germany

## 1 Introduction

Jet quenching occurs when in a heavy-ion collision an energetic parton propagating in one light-cone direction loses sufficient energy that few high momentum hadrons are seen in the final state, where in the vacuum there would be a jet. In this context, a parton is considered highly energetic when its momentum is much larger than any other energy scale, including those characterizing the medium. Jet quenching, which has been observed at RHIC [1] and at LHC [2], manifests itself in many ways. In particular, the hard partons produced in the collision lose energy and change direction of their momenta. This last phenomenon goes under the name of transverse momentum broadening.

A way to describe the transverse momentum broadening is by means of the probability, , that after propagating through the medium for a distance () the hard parton acquires a transverse momentum (see Fig. 1): . A related quantity is the jet quenching parameter, , which is the mean square transverse momentum picked up by the hard parton per unit distance traveled:

 ^q=1L∫d2k⊥(2π)2k2⊥P(k⊥). (1)

In the following, we will review the derivation of a gauge invariant expression for in the case of the propagation of a highly energetic quark. The original detailed derivation can be found in [3].

## 2 Scales and effective field theory

We consider a highly energetic quark of momentum propagating along one light-cone direction . The light-cone momentum coordinates are , , with , and , which is the momentum component that is transverse with respect to the light-cone directions and , see Fig. 2. If the quark propagates in a medium whose energy scales are much smaller than , then we can define a parameter , which is the ratio of the energy scale characterizing the medium and . This small parameter may serve to classify the different modes of the propagating quark and interacting gluons.

We assume that the quark, after traveling along the medium, undergoes a transverse momentum broadening of order . If the virtuality of the quark is small, i.e. of order , then the parton has momentum and is called collinear. We set up to describe the propagation of a single collinear quark in the medium. A collinear quark may scatter in the medium with ultrasoft gluons, whose momenta scale like , with Glauber gluons , whose momenta scale like or or with soft gluons scaling like through the emission of virtual hard-collinear quarks scaling like . The relevant degrees of freedom are shown in Fig. 3.

The effective field theory that describes the propagation of a collinear quark in the light-cone direction is the soft-collinear effective theory (SCET) [4] coupled to Glauber gluons [5]. After rescaling the quark field by , the Lagrangian may be organized as an expansion in :

 L¯n=¯ξ¯nin/¯n⋅Dξ¯n+¯ξ¯nD2⊥2Qn/ξ¯n+¯ξ¯nigFμν⊥4Qγμγνn/ξ¯n+…, (2)

where and is the gluon field strength. The fragmentation of the collinear quark into collinear partons is not taken into account by the above Lagrangian; a preliminary study of this effect can be found in [6].

## 3 Momentum broadening in covariant gauges

Collinear and hard-collinear quark fields, , scale in the same way. The operators and scale like and respectively when acting on a collinear field , and both scale like when acting on a hard-collinear field . Soft gluon fields scale like and ultrasoft gluon fields scale like , for they are homogeneous in the soft and ultrasoft scale respectively. In contrast, the power counting of Glauber gluons depends on the gauge. The equations of motion require to scale like . In a covariant gauge, if the gluon field is coupled to a homogeneous soft source, this also implies that . The leading order Lagrangian in is then

 Missing dimension or its units for \hskip (3)

Because ultrasoft gluons decouple at lowest order from collinear quarks trough the field redefinition , where P stands for the path ordering operator, only one relevant vertex involving either Glauber or soft gluons has to be taken into account:

The transverse momentum broadening probability is then given by the imaginary part of the differential scattering amplitude

b
taken for and normalized by the number of collinear quarks in the medium. The scattering amplitude has the form (evaluated on a background of gluon fields)

 ∫∏id4qi(2π)4⋯iQ2Qq+2−q22⊥+iϵ¯n/A+(q2−q1)n/iQ2Qq+1−q21⊥+iϵ¯n/A+(q1−q0)n/ξ¯n(q0),

where the Dirac spinor satisfies and is normalized as . For Glauber gluons, the free propagator may be approximated by (e.g. in Feynman gauge)

 Dμν(k)=D(k2)gμν≈D(k2⊥)gμν, (4)

which implies that the scattering amplitude in coordinate space is at leading order

 ∫dy+d2y⊥∏idy−i⋯θ(y−3−y−2)A+(y+,y−2,y⊥)θ(y−2−y−1)A+(y+,y−1,y⊥)ξ¯n(q0). (5)

The same result also holds when considering the case of (hard-)collinear quarks interacting with soft gluons.

Because (5) is just a term in the expansion of the Wilson line for , the transverse momentum broadening probability of a quark in covariant gauges is given by

 Missing or unrecognized delimiter for \right (6)

where denotes a field average. The Wilson lines of (6) are shown in Fig. 4. The relation between jet quenching and Wilson lines oriented along one of the light-cone directions was derived within different approaches in [7] and within SCET in [8]. Clearly the above expression is, in general, not gauge invariant (e.g. in the light-cone gauge , one would have ).

## 4 Momentum broadening in light-cone gauge

In the light-cone gauge , the free gluon propagator reads

 Dμν(k)=D(k2)(gμν−kμ¯nν+kν¯nμ[k+]). (7)

For Glauber gluons , which leads to on enhancement of order in the singular part of the propagator. Moreover, because of the singularity, one can write [9] , where contributes to the non-singular part of the propagator and vanishes at , while with . The field does not vanish at infinity where it becomes pure gauge, for the field tensor does (the energy of the gauge field is finite).

In the gauge, the scaling of the Glauber fields appearing in the Lagrangian changes to and . The leading order Lagrangian in is then

 L¯n=¯ξ¯nin/¯n⋅∂ξ¯n+¯ξ¯n(∇⊥+igAsin⊥)22Qn/ξ¯n, (8)

where gluons are just Glauber gluons. The relevant vertices are now two

b
From the vertices one constructs the scattering amplitude (on the left of the cut)

b
The function is a convolution of , which involves only fields at and , which involves only fields at :

 Gn(k−,k⊥)=n∑j=0∫d4q(2π)4G+n−j(k−,k⊥,q)iQ¯n/2Qq+−q2⊥+iϵG−j(q). (9)

The computation is done by solving recursively the equation (analogously for )

 G−n(q)=∫d4q′(2π)4G−n−1(q′)\raisebox−15.0pt\includegraphics[[width=56.905512pt]]./vlc1bis.eps+∫d4q′′(2π)4G−n−2(q′′)\raisebox−10.0pt\includegraphics[[width=56.905512pt]]./vlc2bis.eps, (10)

writing the differential amplitude as

 1L3√2Q∫dk+2π∫dk−2π2πQδ(2Qk+−k2⊥)¯ξ¯n(q0)G†m(k−,k⊥)¯n/Gn(k−,k⊥)ξ¯n(q0),

and eventually summing over all and . The expression of the transverse momentum broadening probability in light-cone gauge then reads

 P(k⊥)=∫d2x⊥eik⊥⋅x⊥1Nc⟨Tr{T†[0,−∞,x⊥]T[0,∞,x⊥]T†[0,∞,0]T[0,−∞,0]}⟩, (11)

where (for the definition of see also [10]). The transverse vector is arbitrary. The Wilson lines of (11) are shown in Fig. 5.

## 5 Gauge invariant momentum broadening

Combining the results in covariant and light-cone gauge for , we obtain a gauge invariant expression for , which reads

 P(k⊥) = ∫d2x⊥eik⊥⋅x⊥1Nc⟨Tr{T†[0,−∞,x⊥]W†[0,x⊥]T[0,∞,x⊥] (12) ×T†[0,∞,0]W[0,0]T[0,−∞,0]}⟩.

The Wilson lines of (12) are shown in Fig. 6. Note that the fields are path ordered but not time ordered as in usual Wilson loops [11]. This difference should not be surprising since it reflects the fact that describes the propagation of a single particle, while usual Wilson loops describe the propagation of a particle-antiparticle pair.

The expression of may be simplified into

 P(k⊥)=∫d2x⊥eik⊥⋅x⊥1Nc⟨Tr{[0,x⊥]−W†[0,x⊥][x⊥,0]+W[0,0]}⟩, (13)

where , because contiguous adjoint lines cancel, fields separated by space-like intervals commute and because of the cyclicity of the trace. The Wilson lines of (13) are shown in Fig. 7.

The obtained expression for does not depend on . It is also gauge invariant. In fact, under a gauge transformation , transforms to , which is equal to the original expression, , after noticing that the fields in commute with all the others (because of space-like separations) and after using the cyclicity of the trace.

## 6 Conclusion

Having derived the transverse momentum broadening probability, , we are in the position to write the jet quenching parameter in a manifestly gauge invariant fashion:

 ^q = ∫d2k⊥(2π)2d2x⊥dx−eik⊥⋅x⊥√2Nc⟨Tr{[0,x⊥]−U†x⊥[x−,−∞]gF+i⊥(0,x−,x⊥) (14) ×U†x⊥[∞,x−][x⊥,0]+U0⊥[∞,0]gF+i⊥(0,0,0)U0⊥[0,−∞]}⟩,

where the fields come from the derivatives, , acting on the Wilson lines, and . We recall that the above expression holds when the integral over has an ultraviolet cut-off of order , which is the size of the transverse momentum broadening that we have been considering.

###### Acknowledgements.
I thank Michael Benzke, Nora Brambilla and Miguel Angel Escobedo for collaboration on the work presented here. I acknowledge financial support from the DFG cluster of excellence “Origin and structure of the universe” (http://www.universe-cluster.de).

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