QCD Shear Viscosity at (almost) NLO
Jacopo Ghiglieri Guy D. Moore Derek Teaney









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    #1Eq. (<ref>)𝒞#1Appendix <ref>#1Section <ref>#1Ref. <cit.>#1Fig. <ref>1]Jacopo Ghiglieri,2]Guy D. Moore,3]and Derek Teaney[1]Theoretical Physics Department, CERN, Geneva, Switzerland[2]Institut für Kernphysik, Technische Universität Darmstadt
    
    Schlossgartenstraße 2, D-64289 Darmstadt, Germany[3]Department of Physics and Astronomy, Stony Brook University,
    
    Stony Brook, New York 11794-3800, United Statesjacopo.ghiglieri@cern.chguymoore@theorie.ikp.physik.tu-darmstadt.dederek.teaney@stonybrook.eduWe compute the shear viscosity 
      of QCD with matter, including almost all
      next-to-leading order corrections – that is, corrections suppressed
      by one power of g relative to leading order.  We argue that the
      still missing terms are small.  The next-to-leading order
      corrections are large and
      bring η/s down by more than a factor of 3 at physically
      relevant couplings.  The perturbative expansion is problematic
      even at T ≃ 100 GeV.  The largest next-to-leading order 
      correction to η/s arises from modifications to the q̂ parameter, 
      which determines the rate of transverse momentum diffusion. 
      We also explore quark number diffusion,
      and shear viscosity in pure-glue QCD and in QED.CERN-TH-2018-009QCD Shear Viscosity at (almost) NLO
        [
        
    ===================================
    
    
    
    § INTRODUCTION
    
    
    The original idea of the Quark-Gluon Plasma phase
    <cit.> was that it would
    consist of weakly-interacting, nearly-free quarks and gluons
    (this assumption is implicit, for instance, in treatments of the
    cosmological QCD phase transition <cit.>).
    This picture was naive, since the QCD coupling varies only
    logarithmically with scale <cit.>, so the
    coupling is in fact quite large at any achievable temperature.
    Although thermodynamical quantities approach the expected
    weak-coupling values rather quickly
    <cit.>,
    this does not necessarily indicate weak coupling; even in the limit of
    infinite coupling, analogue theories display 3/4 of the free theory
    value for the pressure, for instance
    <cit.>.
    
    Weak coupling would imply large transport coefficients, characterized
    for instance by a large ratio of the shear viscosity to the entropy
    density, η/s ≫ 1.  In fact, leading-order (LO) perturbative
    calculations of η<cit.>
    find η/s ∼ 0.5 for coupling values of physical relevance for
    achievable temperatures.  Unfortunately, the presentation in
    Ref. <cit.> has led to frequent misinterpretation of the
    results, such as using the next-to-leading-log
    (NLL) pocket formulae in regimes where the paper cautions that they
    are not applicable.
    
    But in any case, experimental results at RHIC
    <cit.>
    and the LHC
    <cit.>
    indicate that the shear viscosity is even
    smaller:  numerous authors have found that the experimental data on
    angular correlations and other experimental measurables are fit very
    well by relativistic, viscous hydrodynamics, but only if the shear
    viscosity to entropy ratio is quite small,
    η/s ∼ 0.1–0.3
    (see <cit.> for reviews).
    This would indicate that η/s is quite close to the value in
    extremely strongly coupled theories with holographic duals
    <cit.>. First attempts
    at non-perturbative QCD determinations from the lattice, which require
    a highly non-trivial analytical continuation, also point towards
    small values
    <cit.>.
    So do FRG analyses, which also require analytical continuation and
    other truncations <cit.>.
    
    So how should we understand this discrepancy with the weak-coupling
    calculations?  The best way to address this question is to compute the
    next-to-leading order (NLO) corrections to the shear viscosity.  We finally
    have the technology to do so.  One key breakthrough, due to
    Caron-Huot, was the development of a technique to understand how
    particles are “kicked” transversely as they move through the plasma,
    at next-to-leading order <cit.>.
    Then there was the development of sum-rule tools for next-to-leading
    order longitudinal momentum diffusion, identity change, and collinear emission,
    developed to study photon production
    <cit.>
    and recently extended to treat jet energy modification at
    subleading order
    <cit.>.
    With rather modest modifications, we can apply this technology to
    perform an “almost” next-to-leading order 
    calculation of η, and of quark number
    diffusion D_q, in a hot QCD plasma.
    
    In the following sections, we will give a rather detailed explanation
    of how one computes η/s perturbatively in QCD, and of what is and
    is not included in our “almost” NLO calculation.  But for the
    impatient reader, we will give a short summary of the procedure, of
    what is included and what is missing, why we think the remaining
    “missing” parts should give only a small correction, and of our
    final results.
    
    Shear viscosity describes the persistence of any anisotropy in the
    stress tensor T_μν.  When a fluid flows in a nonuniform way,
    such anisotropy constantly develops from the fluid flow, and
    constantly disappears due to dissipative physics.  Shear viscosity
    measures the inefficiency of that dissipation.  It can also be studied
    by using random thermal fluctuations, through which T_μν
    accidentally becomes anisotropic.  The fluctuation-dissipation theorem
    says that the persistence of these fluctuations also determines the
    shear viscosity.  These concepts are well defined in any theory with
    well defined thermodynamics, whether or not the stress tensor can be
    understood in terms of some “particle” degrees of freedom.
    
    The perturbative picture is that the plasma is made up primarily of
    quasiparticle excitations with momenta of the order of the temperature,
     and these are responsible for carrying the
    stress tensor T_μν of the plasma.  An anisotropic T_μν
    arises when the quasiparticles are distributed anisotropically in
    momentum space.  Their scattering relaxes
    T_μν towards its equilibrium value.  This description is
    sufficient at both leading order and (g) NLO order.  The
    challenge is to determine the exact form of the collision operator
    which relaxes the particles towards equilibrium.  The LO
    calculation <cit.> requires two sorts of scattering
    process, the  scatterings with all hard ((T)) external
    particles and  effective splitting processes between hard
    participants.  There are two features in the calculation.  First,
    there is the momentum a particle carries into a scattering and the
    momentum it carries out.  Second, there is the effect of the momentum
    which it “dumps” into the other particle in the scattering process.
    While the first effect always makes the momentum distribution more
    isotropic, the second effect can make it more or less isotropic,
    depending on the relative angles of the participants.
    
    To treat the problem at NLO, we need to find all new scattering
    processes, and corrections to the already-considered processes, which
    are suppressed by a single power of g.  No other corrections are
    needed because the quasiparticle picture first needs amending at
    (g^2) or higher.  As we shall show in detail, there are only a
    few such (g) subleading effects.  First, the rate of soft
     scattering is modified; this can be described as an
    additional momentum-diffusion coefficient δ.  This
    modification, and an (g) correction to the medium corrections to
    dispersion, also provide an (g) shift in the  splitting
    rate.  Next, the  splitting rate must be corrected wherever
    one participant becomes “soft” (p ∼ gT) or when the opening
    angle becomes less collinear.  And finally, the numerical
    implementation of the LO scattering kernel <cit.>
    already resums a small amount of these NLO effects, requiring a
    subtraction (or “counterterm”) to δ and
    δ (longitudinal momentum diffusion).
    
    We are able to give a relatively simple determination of these effects
    by the use of light-cone techniques.  Unfortunately, these methods
    typically keep track of the incoming and outgoing momentum of a
    particle, but lose track of the momentum which it transfers to the
    other participants.  This momentum transfer also affects the departure
    from equilibrium of the other particle or particles which receive the
    momentum, an effect which we will fail to account for at NLO.
    Therefore our treatment is only “almost” NLO.  However we compute
    the importance of this effect in the leading-order  case and
    use it to make an estimate for this incomplete treatment.  The
    associated errors turn out to be small, much smaller than the
    difference between LO and NLO, and therefore presumably smaller than
    still-uncomputed NNLO effects.
    
    
    
    Our main results are presented in Section <ref>, but we
    will present one “summary” result right away in Figure
    <ref>.  The figure shows the ratio of the shear viscosity to
    the entropy density, computed at LO and NLO.  The temperature enters
    in the choice of renormalized coupling and the number of quark species
    (there are slight discontinuities where we cross quark-number
    “thresholds”).  The solid thinner band represents our “best
    estimate” based on 2-loop renormalization group flow from the
    Z-pole and the coupling fixed via the EQCD choice of Laine and
    Schröder <cit.>. The renormalization
    uncertainty is estimated by varying the scale μ_EQCD over
    the range μ_EQCD=(2.7↔4π) T.
    The wider bands represent fixing g^2 from the
    scale π T to 4π T with the standard 
    approach, to indicate the importance of the
    renormalization uncertainty.
    The plot shows that next-to-leading order corrections lower the shear
    viscosity by a factor of two at high temperatures T∼ 1000  GeV, and
    by a factor of four for physically relevant temperatures,
    T ∼ 250  MeV.  This large change
    is suggestive that the true value of η/s is smaller than the
    leading-order perturbative estimate, but it also signals severe
    convergence problems in the perturbative expansion, even for
    surprisingly large temperatures or, equivalently, small values
    of g.  The figure also shows the analogous result for the (light) quark
    diffusion coefficient, which displays very similar coupling dependence.
    We present more results and discussion in later sections, but we point
    out now that the largest NLO correction arises from NLO modifications
    of q̂– see fig_visc_md. 
    Accurate fits of or NLO results for η/s as a 
    function of coupling are provided in an appendix.
    
    Having finished a quick summary of the problem, our approach, and our
    main conclusions, we now summarize the content
    of the remainder of the paper.  In Section <ref> we
    review the definition of transport coefficients and their calculation
    within the kinetic theory of Arnold, Moore, and Yaffe
    <cit.>.  In Section <ref> we show how to
    interpret parts of the leading-order calculation in terms of
    transverse and longitudinal momentum diffusion and of
    identity-changing processes.  The NLO effects take the form of these
    three effects, plus a shift in the rate of  splittings, and
    can therefore be efficiently included once we express the problem in
    terms of these pieces.  We do this in Section <ref>, with
    special attention to the “overlap” regions between these processes.
    With all pieces available, we present the main results in Section
    <ref>.  We also decompose the NLO correction into the
    respective pieces to see which are most influential.  Some technical
    details, together with fits for our NLO
    results as a function of the coupling, are postponed to the  appendices.
    
    
    
    
    § INGREDIENTS
    
    
    
    Let us start by briefly summarizing how the transport coefficients
    we investigate are defined and how they have been computed to leading
    order in the Effective Kinetic Theory (EKT). Transport coefficients
    characterize a system's response to weak, slowly varying
    inhomogeneities or external forces.  In the case of the viscosity, if
    the flow velocity of the plasma is nonuniform,  then the
    stress-energy tensor (which defines the flux of momentum density)
    departs from its perfect fluid form.
    In the local (Landau-Lifshitz) fluid rest frame at a point x,
    the stress tensor, to first order in the velocity gradient,
    has the form
    
        ⟨ T_ij(x) ⟩
            =
            δ_ij ⟨⟩
            - ησ_ij
            - ζ δ_ij ∇^l   u_l  ,
        	  σ_ij≡∇_i   u_j + ∇_j   u_i
        		- 2/3 δ_ij ∇^l   u_l ,
    
    where the metric is the “mostly-plus” one,
      is the equilibrium pressure associated with
    the energy density ⟨ T_00(x) ⟩=ϵ,
    and the coefficients η and ζ are known as
    the shear and bulk viscosities, respectively.
    The flow velocity $̆ equals the momentum density divided by the
    enthalpy densityϵ+=sT.
    We will only be concerned with the shear viscosity in this paper;
    the bulk viscosity requires a more complicated analysis, which has been
    carried out at leading order in <cit.> for
    a scalar theory and in <cit.> for a
    gauge theory with massless quarks. The charm
    contribution has been computed in <cit.>.
    Additional coefficients such asτ_πwould appear at higher order in the gradient expansion
    <cit.>,
    but we leave their evaluation for a future investigation.
    
    In the presence of further conserved global charges beyond four-momentum,
    such as baryon or lepton number, the associated charge densityn ≡ j^0and current densityȷsatisfy a diffusion equation,
    
        ⟨ȷ⟩
            = -D ⟨ n ⟩ ,
    
    in the local (Landau-Lifshitz) rest frame of the medium.
    The coefficientDis called the diffusion constant.
    
    When (some of) the diffusing species of excitations carry electric charge,
    as is the case for baryon and lepton number, the diffusion constants for these
    charged species determine
    the electric conductivityσthrough an Einstein relation
    (see Refs. <cit.>).
    If the net number of each species of charge carriers is conserved, then
    
        σ = ∑_a   e_a^2   D_a  ∂ n_a/∂μ_a ,
    
    where the sum runs over the different species or flavors of excitations
    withe_a,D_aandμ_athe corresponding electric charge, diffusion constant
    and chemical potential, respectively.
    
    
    
    These transport coefficients
     all find a field-theoretical definition through Kubo-type formulae
    relating them to the zero-frequency (transport) limit of the spectral functions
    of two-point correlators of the appropriate operators (the stress-energy tensor
    or other conserved currents). However, for a leading- and next-to-leading-order
     perturbative evaluation, the diagrammatic approach that would result from a direct
     application of these Kubo formulae would require cumbersome resummations
     to all orders
     of many classes of sub-diagrams. It is thus more convenient to use the
     linearized version <cit.>
    of the EKT developed in <cit.>.
    Solving the linearized theory automatically accounts for
    the needed resummations. The leading-order equivalence between the diagrammatic and
    kinetic approaches has been proven in <cit.> for a scalar theory and
    in <cit.>
    for gauge theories.
    
    The leading-order
    EKT introduced in <cit.> is given by this Boltzmann equation
    
        [
        	∂/∂ t
        	+
        	_̌·∂/∂
        	+
        	^a ·∂/∂]
            f^a(,,t)
            =
            -C_a[f]=-(C_a^[f] + C_a^[f] )   ,
    
    wheref^a(,,t)=dN^a/d^3 d^3is the phase space distribution function for
    the excitation (gluon, quark, antiquark ) of indexa. The
    leading-order collision operator encodes the contribution of tree-levelscattering processes, with Hard-Loop resummed
    propagators in the soft-sensitive channels,
    as well as collinear, effectiveprocesses resumming the effect
    of an infinite number of soft scatterings. Both processes
    contribute to orderg^4Tto the collision operator; a subset ofC_a^[f]is logarithmically enhanced,g^4Tln(1/g), due to the aforementioned sensitivity
    to the soft scalegT.C_a^[f]andC_a^[f]are described in detail
    in <cit.>.
    
    
    We now proceed to the gradient expansion of eq:Boltz,
    following the notation of <cit.>.
    Schematically,f^a=f_0^a+f_0(1± f_0)[f_1^a+f_2^a+…], wheref_0^ais the equilibrium
    distribution[
       We use capital letters for four-vectors, bold lowercase ones
    for three-vectors and italic lowercase for the modulus of the
    latter. We work in the “mostly plus” metric, so that
    P^2=-p_0^2+p^2. The upper sign is for bosons, and the lower sign is
    for fermions.  The full collision operator is C_a; the collision
    operator linearized in the departure from equilibrium is _a.],f_0^a=(exp(-β u^μ P_μ - q^a_αβμ)∓ 1)^-1,
    which  is determined by the Boltzmann equation at zeroth order in
    the gradients,C_a[f_0]=0.
    The inverse
    temperatureβ, the chemical potentialμ,  and the flow velocityu^μare
    functions of(t,)and obey the equations of ideal hydrodynamics.
    We will consideru^i(t,)andμ(t,)to be small perturbations
    on top of an approximately homogeneous background. The charge
    of speciesaunder conserved chargeQ_αisq_α^a, whereαis a label for the
    flavor symmetry of interest (i.e. quark number in our case).
    
    Substitutingf_0^ainto the lefthand side of the Boltzmann
    equation, eq:Boltz,
    yields a source 
    for the first dissipative correction,
    which (after using the hydrodynamic equations of motion)
    is proportional to  the strains <cit.>
        X_ĳ≡ ∇_i  μ_α ,    1/√(6)( ∇_i u_j + ∇_j u_i
        	- 2/3 δ_ij∇·)  ,
    
    depending on whether we are considering chemical
    potential fluctuations (diffusion) or velocity fluctuations (shear viscosity).
    The source takes the form
    
        ^a()≡   β^2 ^a_ĳ() X_ĳ , 
        
           =    - β q^a f_0^a(p)[1±f_0^a(p)]   I_ĳ() X_ĳ  .
    
    Hereq^adenotes the relevant conserved charge
    carried by speciesaassociated with
    the transport coefficient of interest,
    
        q^a ≡
        
        	q^a_α  ,    
        	||   ,I_ĳis the uniqueℓ=1orℓ=2rotationally covariant
    tensor depending only on,
    
        I_ĳ()
            ≡ p̂_i  ,    ℓ = 1 √(3/2)  (p̂_i p̂_j - 1/3δ_ij)  ,
        	   ℓ = 2
    
    The normalization onI_ĳwas chosen so that
    
        I_ĳ()   I_ĳ() = 1,
    
    and more generally,
    
        I_ĳ()   I_ĳ(̨̂) = P_ℓ^(·̨̂) ,
    
    whereP_ℓ(x)is theℓ'th Legendre polynomial.
    
    The linearized kinetic equation may be written compactly as
    
        ^a() = ( f_1)^a()  ,
    
    whereis a linearized collision operator defined  below.
    To linear order the first dissipative correction
    must be proportional to the driving termX, allowing us to define
    the proportionality coefficientχ_ĳ(),
    
        f_1^a() ≡β^2 X_ĳ χ_ĳ^a()
          ≡β^2 X_ĳ  I_ĳ()  χ^a(p)  ,
    
    where we have also used rotational invariance of the collision
    operator (in the rest frame) to define a scalar proportionality
    coefficientχ(p), which describes how the departure from
    equilibrium varies as a function of the magnitude of the momentum.
    
    The first-order transport coefficients are then obtained
    from the kinetic-theory expressions forT^μνand
    for the conserved currentJ^μ_αassociated to the conserved
    chargeQ_α, i.e.,
    
        T^μν(,t)=∫d^3/(2π)^3p^μ p^ν/p∑_aν_a
        	f^a(,,t),   J_α^μ(,t)=∫d^3/(2π)^3p^μ/p∑_aν_a q^a_α f^a(,,t),
    
    whereν_ais the spin and color degeneracy of the excitationa(2for quarks and antiquarks,2(^2-1)for gluons).
    Upon inserting the first-order
    deviationf^1in these equations, the first-order coefficients
    are easily recovered.
    Hence, the solution of the first-order, linear eq:Boltz1
    yieldsηandD.
    
    As in <cit.>,
    we will solve eq:Boltz1 and its NLO extension by means
    of a variational method. To this end, we introduce the inner
    product
    
        ( f,g ) ≡β^3 ∑_a  _a ∫  f^a()   g^a()  .
    
    The linearized collision operatoris symmetric with respect to
    this inner product, and is given by variation  of the  quadratic form
    
       ( f_1)^a() = (2π)^3/β^3 ν_aδ /δ f_1^a()  (f_1,  f_1)   .is a positive semi-definite operator, and is strictly positive
    definite in theℓ= 1andℓ=2channels relevant
    for diffusion or shear viscosity. As we will show, some
    NLO contributions are negative, so some care will be needed
    in defining a positive definite𝒞_. Once that
    is taken care of,
    the linearized Boltzmann equation (<ref>) at LO and
    NLO is precisely the condition for maximizing the functional
    
        β^4 𝒬[χ]
            ≡( f_1, )
            -  ( f_1,   f_1 ).
    
    Note that the maximized𝒬determines the rate per volume at which work is
    dissipated into heat;β𝒬then gives the rate per
    volume of entropy production.
    This structure is valid at LO and NLO, so we
    have not explicitly labeledandf_1in that respect.
    The strainsX_ĳmay be pulled out of the integrals,  and then
    rotational invariance of the measure and collision operator
    guarantees that
    
        T^4 ( f_1,  f_1 )  =    X^2/2ℓ + 1 ( χ_ĳ,  χ_ĳ) , 
        
          T^4 ( f_1, ) =    X^2/2ℓ + 1 ( χ_ĳ, S_ĳ) ,
    
    whereX^2 = X_ĳ  X_ĳ.
    
    The explicit forms of the source and LO collision parts of this quadratic
    functional are
    
        T^4 ( f_1, )
               =
             X^2/2ℓ +1∑_a β^2 ν_a
            ∫
        	f_0(p)   [1± f_0(p)] 
           q^a  χ^a(p)  ,
        
        and( f_1, _  f_1 )
               =
            ( f_1,   f_1 ) +
            ( f_1,   f_1 )  .
    
    Thecontribution reads, after symmetrization
    of the departures from equilibrium <cit.>
        ( f_1, ^  f_1 )
            ≡    β^3/8∑_abcd∫_''̨| ^ab_cd(,;̨','̨) |^2 /(2p) (2k) (2p') (2k')  (2π)^4 δ^(4)(P+K-P'-K')
           ×	
        f^a_0(p)   f^b_0(k)   [1±f^c_0(p')]   [1±f^d_0(k')]
        	 [
        	    f_1^a() + f_1^b()̨ -
        	    f_1^c(') - f_1^d('̨)
        	]^2_ ,
    
    where∫_is shorthand for∫ d^3/(2π)^3and| ^ab_cd(,;̨','̨) |^2 is the matrix element
    squared for theab cdprocess, summed over all
    spins polarizations and colors and Hard Thermal Loop (HTL) resummed
    <cit.>
    in the IR-sensitive cases. A complete
    list of these matrix elements appears in <cit.>.
    Thecontribution reads instead <cit.>
        ( f_1, ^  f_1 )
            ≡    β^3/2∑_abc∫ dΩ_∫_0^∞ dp∫_0^p dk  γ^a_bc(p;p-k,k)  
           ×
                  f_0^a(p)   [1 ± f_0^b(p-k)]   [1 ± f_0^c(k)] [
              f_1^a(p ) - f_1^b((p-k)) - f_1^c(k)
        	]^2  .
    
    The splitting rate is given by
    
        γ^a_bc(p;p-k,k)=g^2d_R_bC_R_b/64π^4{[ p^4+k^4+(p-k)^4/p^3k^3(p-k)^3                         g↔ gg;     p^2+(p-k)^2/p^2(p-k)^2k^3                        q↔ q g;     (p-k)^2+k^2/(p-k)^2k^2p^3                        g↔ qq̅ ].
        ∫d^2h/(2π)^2  2· _b() ,
    
    where=×$̨ is a transverse, two-dimensional vector
    related to the transverse momentum picked up during the splitting
    process. d_R_b and C_R_b are the dimension and quadratic Casimir
    operator of the representation R of the particle b.
     A complete leading-order treatment of collinear radiation must consistently
    resum the effect of the many soft, transverse collisions
    to account for the Landau-Pomeranchuk-Migdal
    (LPM) effect
    <cit.>.
    This is achieved through the following integral equation for  _b():
    
        2   =   iδ E()_b()+∫d^2/(2π)^2C̅()
        {(C_R_b-/2)[_b()-_b(-k)]
              +/2[
        _b()-_b(+p)]+/2[
        _b()-_b(-(p-k))]}.
    
    For the case of
    g→ qq̅, C_R_b-/2=-/2 multiplies the term with _b(-p) rather than
    _b(-k). The equation depends on two inputs,
    C̅()  and δ E(,p,k). The former is the leading-order
     transverse scattering kernel C() in units of the Casimir factor,
    <cit.>
        C̅()≡C_R()/C_R=g^2 T
        	^2/^2(^2+^2).
    δ E is the energy difference between the initial and final collinear
    particles. It reads
    
        δ E()=h^2/2pk(p-k)+m^2_∞ c/2k
        +m^2_∞ b/2(p-k)
        -m^2_∞ a/2p,
    
    where m_∞ a^2 is the asymptotic mass of the particle a. For gluons
    m_∞ g^2≡=^2/2 (with ^2=g^2T^2/3(T_A+T_FN_f)),
     for quarks m_∞ q^2≡= g^2T^2/4.
    
    The maximization of the functional (<ref>) is then
    carried out using a variational Ansatz of form
    
        χ(p) = ∑_m=1^K c_m ϕ^(m)(p)  ,     ϕ^(m)(p) = p^ℓ-1 (p/T)^m-1/(1+p/T)^K-3/2 ,
    
    which, when substituted into eq:Q1, transforms its extremization
    into a matrix algebra problem <cit.>.  Our choice of functional basis
    is motivated by the need <cit.> to allow infrared
    behavior of form ∝ p^ℓ-1 and ultraviolet behavior of form
    ∝ p^ℓ-1/2.  The
    variational procedure is only guaranteed to converge to the right
    answer as the functional basis becomes complete, but in practice we
    see good convergence above 4 functions.  Later, in our numerical
    results, we will use 6 basis functions, but our results shift by less
    than 10^-4 when we make the basis still larger.
    
    
    
    § REORGANIZATION OF THE LO QUADRATIC FUNCTIONAL
    
    
    The effective kinetic theory introduced in <cit.> has
    been extended to next-to-leading order in <cit.>
     for the case where one follows
    the evolution of a dilute set of high-energy particles of typical energy E
    interacting with an
    equilibrated medium at a temperature T such that exp(-E/T)≪ 1,
    which is a sensible approximation for the evolution of the
    leading partons in a jet. There, we found that a reorganization
    of the form of the LO collision operator was necessary
    to systematically compute
    NLO corrections. Specifically, (g) NLO corrections can only occur
    where one or more lines carry a soft (gT) momentum, because only
    there do statistical functions give rise to a 1/g enhancement of
    loop level effects.  But transport coefficients are only sensitive to
    hard (T) momenta.  So NLO corrections only occur when there is a
    momentum hierarchy within a diagram.  In such cases one can always
    re-express the diagram as an effective process.  When the soft
    particle is a gluon and does not change particle identity, the process
    can be understood as giving rise to momentum diffusion; when
    the exchanged particle is a quark and therefore changes quantum
    numbers, it is a conversion process.  One can already isolate
    such processes at the leading order.  Doing so will make it easier to
    see how to incorporate NLO processes.
    
    In the diffusion case, the action of the soft gluon exchange is to randomize (diffuse)
    the momentum of the hard particles by small, (g) amounts that can
    be described by a Fokker-Planck equation. The drag, longitudinal and
    transverse momentum diffusion coefficients appearing in the Fokker-Planck equation can be defined
    field-theoretically in terms of Wilson-line operators supported on light fronts,
    which can in turn be evaluated in analytical form using the light-cone techniques
    mentioned in the introduction. In the conversion
    case, the soft quark exchange converts a hard quark (gluon) into a gluon (quark)
    of the same momentum, up to (gT). Again, a light-front Wilson line definition
    for the conversion rate was introduced in <cit.>, leading
    to a simple closed form expression. At NLO, the diffusion and conversion rates
    receive (g) corrections, which were computed in
    <cit.>. These,
    together with corrections to the collinear  rate and a new,
    semi-collinear process (which only contributes starting from NLO),
    constitute the entirety of the NLO corrections to the EKT in the “dilute-hard”
    approximation appropriate for energy loss.
    
    For computing the transport coefficients we will first show (again) how the effect
    of soft-gluon exchange can be  reorganized
    into a Fokker-Planck equation in sec:softglue.  However, in order to conserve
    energy and momentum, the Fokker-Planck equation must be supplemented by
    gain terms, which describe precisely how the momentum lost by a parton
    in the bath is redistributed. This redistribution
    of energy and momentum is unimportant for determining the energy loss, but plays
    an essential role in determining the transport coefficients.
    The computation of these gain terms is not amenable to an evaluation using light-cone techniques
     since more than one light-like particle is involved, and therefore computing the gain terms constitutes a major obstacle to
    computing transport coefficients at NLO.  We will use the LO analysis
    in this section to motivate a NLO Ansatz for the gain terms in sec_nlo2.
    The treatment of soft fermion exchange and conversions is analogous and
    will be discussed in sec:softquark.
    
    
    
     §.§ Soft gluon exchange
    
    
    We will now analyze soft gluon exchange shown in fig_softgluon.
