A relation between screening masses and real-time rates

Thermal screening masses related to the conserved vector current are determined for the case that the current carries a non-zero Matsubara frequency, both in a weak-coupling approach and through lattice QCD. We point out that such screening masses are sensitive to the same infrared physics as light-cone real-time rates. In particular, on the perturbative side, the inhomogeneous Schrodinger equation determining screening correlators is shown to have the same general form as the equation implementing LPM resummation for the soft-dilepton and photon production rates from a hot QCD plasma. The static potential appearing in the equation is identical to that whose soft part has been determined up to NLO and on the lattice in the context of jet quenching. Numerical results based on this potential suggest that screening masses overshoot the free results (multiples of 2piT) more strongly than at zero Matsubara frequency. Four-dimensional lattice simulations in two-flavour QCD at temperatures of 250 and 340 MeV confirm the non-static screening masses at the 10% level. Overall our results lend support to studies of jet quenching based on the same potential at T>250 MeV.


Introduction
Even though an asymptotically free gauge theory at a temperature (T ) much higher than the confinement scale is sometimes called weakly coupled, its dynamics is non-trivial. Denoting the gauge coupling by g = √ 4πα s , such a theory possesses three parametrically different momentum scales [1]: πT , gT , and g 2 T /π, with by assumption πT ≫ gT ≫ g 2 T /π. The structure of any physical observable can be viewed in various ways: (i) In a strict weak-coupling expansion, observables are computed in a power series in g.
It is a consequence of the momentum scales as mentioned above that odd powers and logarithms of g appear [2,3] and that some of the coefficients are non-perturbative [4]. It is also commonly believed that the series converges slowly unless g is extremely small, a problem often associated with the dynamics of the intermediate scale gT .
(ii) In an effective theory approach [5,6], only the "hardest" scale is treated perturbatively. It is "integrated out" in order to derive an effective low-energy description for the "soft" scales gT and g 2 T /π. The dynamics of the low-energy modes is solved nonperturbatively, often with the help of "dimensionally reduced" lattice simulations.
(iii) In principle the most precise level is a fully non-perturbative solution of a given problem, with methods of four-dimensional lattice QCD. A major practical limitation of this approach is that the simulations are carried out in the imaginary-time formalism. If real-time observables are to be considered, an analytic continuation is required, which in practice is ill-controlled (for a review, see ref. [7]).
There are many phenomenologically interesting observables in thermal QCD, notably screening masses and real-time rates such as the photon and dilepton production rates from the plasma, or the rate of "jet quenching" of energetic probes passing through the plasma, which are dominantly determined by the soft scale gT . Given the systematic uncertainties of the third approach, it is suggestive to also follow the second approach for the study of these observables. For screening masses related to flavour-singlet (gluonic) states, this approach leads to a good description of thermal QCD down to temperatures of a few hundred MeV [8].
Recently it has been proposed to apply the same approach to jet quenching [9], and indeed first simulation results exist already [10].
Nevertheless, it may be questioned with every observable how accurate the effective theory approach really is; certainly it breaks down at temperatures very close to the confinement scale, which it does not capture. The purpose of this paper is to elaborate on a non-trivial if indirect crosscheck: we point out that there is a class of Euclidean observables, namely flavour non-singlet (mesonic) screening masses at non-zero Matsubara frequency, which are sensitive to the same infrared physics as is relevant for jet quenching or photon and dilepton production. By measuring these observables on a 4-dimensional lattice and comparing with results based on the effective theory approach, we can lend credibility to the latter.
The screening masses related to mesonic operators are at leading order multiples of 2πT , because of the boundary conditions imposed on quarks across the time direction. Corrections originate from a "potential" V (r) ∼ (g 2 T /π) φ(gT r, g 2 T r/π). The potential balances against a kinetic energy ∼ (1/πT )∂ 2 r , so that the typical momentum scale probed is 1/r ∼ g 2 T 2 ∼ gT . Therefore it would be helpful to determine the function φ without recourse to any expansion, and this is what can be achieved with the second approach.
The plan of this paper is the following. After defining the correlators in sec. 2, we compute them in non-interacting QCD in sec. 3. In sec. 4 we show that the QCD results can be reproduced through an effective theory. The parameters of the effective theory are determined through matching computations in sec. 5, and in sec. 6 we recall how the solution of the problem within the effective theory reduces to a 2-dimensional Schrödinger equation. Numerical estimates following from this equation are displayed in sec. 7. A lattice calculation in two-flavor QCD is presented in sec. 8, where we also compare with the predictions following from the effective Schrödinger equation. An outlook and conclusions are offered in sec. 9.

Basic definitions
Letting γ µ denote Euclidean Dirac matrices, with {γ µ , γ ν } = 2δ µν and γ † µ = γ µ , we consider the quark-connected (or flavour non-singlet) vector current correlator where k n ≡ 2πnT is a bosonic Matsubara frequency, T is the temperature, and x ≡ (x 1 , x 2 ) denotes a 2-dimensional vector in a "transverse" plane. A corresponding Fourier transform is formally defined as It is also convenient to define a "spectral function" as µν (ω) is odd in its argument. Then G (kn) µν (z) can be represented as a Laplace transform: The low-lying spectrum of ρ (kn) µν (ω) is discrete; the corresponding energies, leading to an exponential falloff of G which plays a role in dilepton production, lead to It is therefore sufficient to compute G (kn) 00 , whose analysis turns out to be simpler than that of G (kn) 33 (cf. ref. [11]). Apart from G (kn) 00 , we also consider the transverse part which is not constrained by Ward identities. Given that we have chosen a particular direction (z) in which to measure the correlators, it is convenient to choose a representation of the Dirac matrices which is commensurate with this choice. Starting with the standard (Euclidean) representation, this can be achieved through a transformation γ µ → U γ µ U −1 , with a matrix U given in ref. [12]. After this transformation, the matrices γ 0 γ µ relevant for the "non-relativistic" effective description (cf. e.g. eq. (5.11)) read 7) where the blocks are 2 × 2-matrices, σ j are Pauli matrices, and ǫ 12 = 1. Unless stated otherwise, latin indices take values labelling the transverse directions, i, j ∈ {1, 2}.
In a previous study [12], the screening masses of G (kn) T at k n = 0, as well as similar results for scalar and pseudoscalar densities and the axial current, were determined up to next-to-leading order (NLO). All of the screening masses are equal in this approximation: m = 2πT + cg 2 N c T /(2π), where c is a small positive coefficient whose value depends on the number of dynamical fermions. Numerical measurements (cf. refs. [13,14,15] and references therein) have detected discrepancies with respect to this prediction, particularly for the scalar and pseudoscalar channels where the results are clearly below 2πT . Here we extend the study to k n = 0, whereby the coefficient c and the quality of the comparison both change.

Leading-order computation in full QCD
Before considering NLO corrections, we work out the leading-order (LO) predictions. It turns out that analytic results can be given for the case that no average over the transverse directions is taken in eq. (2.1). Let us denote such correlators by The correlators can be computed with the mixed coordinate space-momentum space techniques introduced in ref. [16]. In coordinate space, spatial propagators have the form Subsequently one is faced with sums of the type where {p n } denotes a fermionic Matsubara frequency and P α is a polynomial of degree α ∈ {0, 1, 2}. The sums can be carried out in analytic form, cf. e.g. ref. [17]. Denotinḡ Structures with sinhr in the denominator are exponentially suppressed atr ≫ 1; however they are relevant for n = 0 in which case the other terms disappear. (For n = 0 a similar expression for the pseudoscalar correlator was given in ref. [18]. NLO corrections could be worked out with the techniques introduced in ref. [19].) Let us now take the transverse averages x . The powerlike terms can be integrated in terms of the exponential integral The equations simplify greatly for |n| = 1 (we also assume z > 0 here): In order to gain an intuitive understanding, eqs. (3.10), (3.11) can be represented by spectral functions like in eq. (2.3). We obtain, for ω > 0, These results are reproduced below from a "low-energy description", valid for the regime |ω − k 1 | ≪ k 1 , but it is already clear that the physics corresponds to a 2-particle threshold, with a discontinuous (ρ (k 1 ) 00 ) or continuous (ρ (k 1 ) T ) spectral function. We note that the asymptotic behaviours of eqs. (3.10), (3.11) contain a power-law in addition to an exponential decay. Physically, this corresponds to an approximation in which two free heavy particles are generated with a continuous spectrum; the extra suppression in eq. (3.11) compared with eq. (3.10) is due to the fact that the latter is a P -channel correlator. After interactions are taken into account, the particles are bound together, and the spectrum is discrete, ρ(ω) ∼ n c n δ(ω − ω n ); therefore we expect that in the full theory there is no power correction to the exponential decay.
In the "static" sector, k n = 0, the roles of the two channels are interchanged. The spatially averaged correlators become 14) 15) and the corresponding spectral functions read

Effective description
We now build an effective theory which allows us to describe the physics of the correlators considered around the threshold ω ∼ max(k 1 , k n ) (we restrict to k n ≥ 0 without loss of generality). We start with a tree-level construction, and promote it to loop level in sec. 5. The correlator of eq. (2.1) can be re-written as In order to represent these operators within an effective theory, it is convenient to introduce an abelian source field B µ which couples to eq. (4.2). This can be achieved by adding S B ≡ 1/T 0 dτ x,zψ γ µ B µ ψ to the original QCD action, with B µ expressed in Matsubara modes as The full action is S ≡ S QCD + S B , where S QCD is the part without B µ . The vector currents and their correlators can then be derived from the identity The idea of the effective approach is dimensional reduction, i.e. keeping only the Matsubara zero modes of the SU(3) gauge fields in the covariant derivatives D µ = ∂ µ −igA µ (cf. ref. [20]). At tree-level, this means that we replace the original action through (4.5) Making use of the representation of Dirac matrices in eq. (2.7) and denoting we thereby get From S 0 it is observed that free propagators, (4.9) are proportional to θ(z 1 − z 2 ) for χ pn>0 and φ pn<0 ; and to θ(z 2 − z 1 ) for χ pn<0 and φ pn>0 . For any p n one of the fields is thus "non-propagating" or "short-range" and can be integrated out. Given that fermionic fields appear quadratically, the integration out can equivalently be achieved by solving equations of motion. This yields the simplified representation Given that χ pn<0 , φ pn>0 are non-propagating (we consider z 1 > z 2 ), forward-propagating mesons are of the types φ † pn χ p ′ n and φ † pn φ −p ′ n with p n , p ′ n > 0. It is seen from eq. (4.8) that B couple to operators of this type for 0 < p n < k n . The transverse source B (kn) i couples to ǫ ij φ † pn σ j χ pn−kn − χ † pn σ j φ pn−kn which is non-propagating for 0 < p n < k n . 1 However, by making use of equations of motion for the non-propagating modes χ pn−kn and χ † pn , there is still a 1/p n or 1/(k n − p n )-suppressed projection to a forward-propagating mode: and total derivatives were omitted. Therefore, the correlator G (kn) T is also non-zero; it is simply power-suppressed with respect to G (kn) 00 .
Whereas the operators are of the type φ † pn φ −p ′ n in the non-static sector, they are of the type φ † pn χ pn in the static sector (i.e. for k n = 0). For V (0) i this is immediately visible from eq. (4.8), whereas for V (0) 0 the elimination of non-propagating modes (separately for p n > 0 and p n < 0) yields This is clearly a P -channel operator.

Mass and vertex corrections
In the discussion of the previous section, only Matsubara zero modes of gauge fields appeared. In full QCD, there are obviously also non-zero Matsubara modes. The description of eqs. (4.8), (4.10) should be viewed as a low-energy effective theory from which the non-zero Matsubara modes have been integrated out. The effect of the integration out is to modify the parameters of the low-energy description, and this is the topic of the present section.
Before proceeding, let us discuss the kinematic regime relevant for the problem. As became clear in sec. 3, for k n = 0 the long-distance screening concerns a distance scale z ∼ 1/k n and is therefore determined by the kinematic regime K 2 = k 2 n + k 2 3 ∼ 0. As was discussed in sec. 4 (cf. e.g. eq. (4.9)), the quark Matsubara modes are close to on-shell, with P 2 = p 2 n +p 2 3 ∼ 0. In a typical case (as discussed in more detail below) the two "constituents" have the Matsubara modes p n = k n /2. Therefore, even though we are considering a Euclidean problem, the kinematics is formally similar to that of collinear splitting, in which a nearly on-shell photon with Minkowskian four-momentum K = (k 0 , k) splits into two fermions with four-momenta P = K/2. This formal similarity suggests a relationship of the current problem to that of photon (K 2 = 0) or soft-dilepton (K 2 ∼ g 2 T 2 ) production from a QCD plasma.
The similarity turns out to extend into practical computations, notably the determination of effects from non-zero Matsubara modes. Indeed, the mass and vertex corrections induced by the non-zero modes can be extracted from computations which are essentially equivalent to the derivation of the Hard Thermal Loop (HTL) effective action [21,22]. The reason is that the assumptions needed in the computations are K 2 ≪ (πT ) 2 , P 2 ≪ (πT ) 2 , (K − P ) 2 ≪ (πT ) 2 , which as we have argued are true in our situation as well. The graphs to be considered are shown in fig. 1, in which a 4-quark operator has been included as well (cf. ref. [20]).
After computing the graphs and expanding to leading order in K 2 /(πT ) 2 , P 2 /(πT ) 2 , (K − P ) 2 /(πT ) 2 , the results can be expressed as corrections to the actions in eqs.  Using for the moment the original fermion fields, the free part of S 0 becomes 2 and v is the integral over directions of a unit vector (|v| = 1), normalized as v 1 = 1. The "asymptotic mass" parameter reads The coupling to the vector current is It might be expected that the correction here is suppressed by O(m 2 ∞ /p 2 n ) ∼ O(α s ), but this is not the case, because parts of the velocity integral give terms of O(m 2 ∞ /P 2 ) ∼ O(1). Let us define an "on-shell" spinor u satisfying Consider the dispersion relation following from eq. (5.4). It is known that in Minkowskian space-time the dispersion relation of the "particle branch" reads p 0 = p+m 2 ∞ /(2p)+..., where p = |p| [23]. Continuing the frequency to imaginary time, this corresponds to p 2 From here a "rest mass" can be identified and subsequently used as a matching coefficient, This agrees with the effective mass derived from an explicit matching computation in ref. [12]; a derivation through HTL expressions like above was previously presented in ref. [24]. The computation of the vertex correction is more cumbersome; however, the task can be simplified by carrying out the matching with the special kinematics 3 R = −P = −K/2, in an "on-shell" configuration. Consider the matrix element The velocity integrals appearing here are all doable. In particular, it can be shown that the transverse part of the current, i.e. the part coupling to ; this correction will be neglected in the following. As far as the longitudinal parts are concerned, we focus on the component coupling to B 0 like before (cf. eq. (2 .5)). An explicit computation yields Rewriting this with 2-component spinors like in eq. (4.8), the HTL-corrected vertex for the operator to which B 0 couples reads However, for the on-shell configuration of φ † pn , p 3 = −ip n + O(α s ) (cf. eq. (4.7)). Similarly, for on-shell χ † pn , p 3 = ip n + O(α s ). Therefore the prefactors of the operators in eq. (5.10) equal unity. Thus the end result is that for the zero component of the current, we can simply use naive vertices as read off from eq. (4.8). 4 The non-zero Matsubara modes also induce higher-dimensional operators. In particular, as pointed out in ref. [20], they generate 4-quark operators which can be represented as (5.11) Here the matrices γ 0 γ µ are as given in eq. (2.7), and T a are Hermitean generators of SU(3), normalized as Tr [T a T b ] = δ ab /2. The role of these operators is that they cause mixings; for instance, a state ∼ φ † pn−ωn φ qn can be transferred to ∼ φ † pn φ qn+ωn , both of which have the same screening mass p n − q n − ω n at tree level, but a different "decomposition". This implies that all decompositions decay with the same screening mass when δS 0 is included.

Schrödinger equation
In this section we recall how the computation of the spatial correlators within the effective theory of secs. 4, 5 reduces to the solution of a two-dimensional Schrödinger equation. In particular, we show that in the free limit eqs. (3.12), (3.13), (3.16), (3.17) can be reproduced this way; and that, going to NLO, the equation to be solved is closely related to that for soft-dilepton and photon production in ref. [11]. The theory is the same as in eq. (4.10), with the modification p n → M n as discussed around eq. (5.6).

Charge density correlator (G 00 )
The charge density can be expressed in terms of low-energy fields as (cf. eqs. (4.4), (4.8)) The fields φ † pn and φ pn−kn = φ −|kn−pn| are forward-propagating and contribute in eq. (4.1) if z > 0. We now rewrite eq. (4.1) with an auxiliary point-splitting in the operator: where (for z > 0) and W y,z is a transverse Wilson line. Computing the correlator to leading order in the weakcoupling expansion and taking already the limit y ′ → 0, a straightforward analysis yields Here (6.6) Two things can be learned from eq. (6.5). First, w LO can equivalently be represented as a solution of a first order differential equation with a particular boundary condition, w LO (0, y) = δ (2) (y) . (6.8) Second, the point-split spectral function corresponding to eq. (6.5) can be determined, The original spectral function thereby becomes, combining eqs. (6.4) and (6.9), Setting n = 1 and considering the leading order (i.e. m 2 ∞ → 0), we have M cm = k 1 and M r = k 1 /4. Then eq. (6.10) agrees with eq. (3.12) when the latter is expanded to leading non-trivial order in ω − k 1 (the case of general k n is discussed in appendix A).
Consider now NLO corrections to eq. (6.2). Keeping y, y ′ = 0, the computation can be carried out by omitting the transverse motion suppressed by 1/(2M r ), whereby the quark propagators are straight Wilson lines. Sending z → ∞ and suppressing y ′ , we obtain where C F = (N 2 c − 1)/(2N c ); g 2 E = g 2 T is the gauge coupling of the dimensionally reduced theory; m 2 E = ( Nc 3 + N f 6 )g 2 T 2 is the Debye mass parameter appearing in the static propagator of A 0 ; and K 0 is a modified Bessel function.
We finally combine eqs. (6.7), (6.11). If we set w LO ∼ O(1), than according to eq. (6.11), w NLO ∼ O(α s ). Moreover, in the kinematic regime ∇ ∼ gT of relevance to us, −∇ 2 /M r ∼ O(α s ). It follows that, up to a perturbative error of ∼ O(α 2 s ), we can write where w = w LO + w NLO + . . . and we have denoted 14) The initial condition remains that same as in eq. (6.8), up to corrections of O(α s ). The Schrödinger equation and the initial condition can be combined into a single equation by taking a Fourier transform. The system ∂ z w(z, y) = −sign(z)Ĥ + w(z, y) , w(0, y) = δ (2) (y) (6.15) can formally be solved as w(z, y) = e −Ĥ + |z| w(0, y). Its Fourier transform (cf. eq. (2.2)) reads The spectral function follows from the cut. Defining an auxiliary function g(ω, y) as the solution of a z-independent inhomogeneous equation we obtain (for ω > 0 and assuming a positive spectrum) It may be noted that eqs. (6.17), (6.18) bear a close resemblance to the corresponding equations appearing in the LPM resummation of longitudinal modes for dilepton production, cf. eqs. (22), (24) of ref. [11]. The overall normalizations of g + , as determined by the coefficient of the inhomogeneous term, as well as of the parameters appearing do differ, but this is a matter of conventions. In addition some imaginary parts appear differently, 5 but this is related to the Minkowskian versus Euclidean nature of the observable considered. The functional form of the potential appearing inĤ + is identical, as well as the fact that we are looking for a scalar (S-wave) solution, as determined by the inhomogeneous term.

Transverse current correlator (G T )
Let us repeat the analysis for the transverse components of the current, cf. eq. (2.6). We again introduce an auxiliary point-splitting into the currents: where, following eq. (4.11), Like with the charge density, the LO solution can be represented as a differential equation, Also, a point-split spectral function corresponding to eq. (6.22) can be determined, The original spectral function thereby becomes (cf. eq. (6.21)) ρ (kn) The parameters appearing here are defined in eq. (6.6). For n = 1 eq. (6.26) agrees with eq. (3.13) when the latter is expanded to leading non-trivial order in ω − k 1 (the case n > 1 is discussed in appendix A). The inclusion of interactions proceeds like for the charge density, with the only difference that the "wave function" is now a vector. In particular, introducing a z-independent inhomogeneous Schrödinger equation the cut of the solution (denoted by Im) yields the spectral function, and ρ (kn) Im ∇ · f + (ω, y) .
(6.28) Equations (6.27), (6.28) again have the same general form as the ones in the LPM resummation of the photon or dilepton production rate, cf. eqs. (22), (24) of ref. [11], with differences originating from the chosen normalization and parameters and from differences of the signatures. The functional form of the potential is the same, as is the fact that we are looking for a vector-valued (P -wave) solution, as dictated by the inhomogeneous term in eq. (6.27).

Static sector
The static case k n = 0 differs qualitatively from k n = 0. We go here beyond the previous discussion of ref. [12] by including the charge density correlator and by giving the inhomogeneous Schrödinger equations determining the absolute values of the correlators. Starting with the transverse case (which for n = 0 corresponds to the S-wave), we write where (cf. eq. (4.8)) The subsequent steps go as in sec. 6.1, with the difference that the fields appearing are φ † pn χ pn rather than φ † pn φ −p ′ n , which leads to a different potential [12]: With this potential the Hamiltonian readŝ The inhomogeneous Schrödinger equation becomes 33) and the spectral function is Im g − (ω, y) . (6.34) Here the modes p n > 0 and p n < 0 have been summed together even though, when 4-quark operators are included, their degeneracy may be lifted. In the free limit the result can be extracted from eq. (6.10), and agrees with the threshold expansion of eq. (3.17). The charge density correlator (which for n = 0 corresponds to the P -wave) reads where from eq. (4.12), The inhomogeneous Schrödinger equation reads 37) and the spectral function is Im ∇ · f − (ω, y) . (6.38) Again the modes p n > 0 and p n < 0 have been summed together. In the free limit, the result can be extracted from eq. (6.26), and agrees with the threshold expansion of eq. (3.16).

Non-perturbative potential and numerical predictions
The potential given in eq. (6.12) can be defined more generally from point-split correlators such as eq. (6.2). It can be defined in the "infinite-mass" limit (∇ 2 /M r ≪ M cm ), whereby the propagators are just straight lines. In this situation we can set y ′ = y and x = 0 in eq. (6.2). More specifically, let us define a loop (z > 0) where we inserted eq. (6.3), restricted to a contribution from a single p n , and defined the "longitudinal" Wilson lines as (q n ≡ k n − p n ) Here ... A denotes a fermion propagator in a fixed gauge field background. The Wilson lines satisfy the equations of motion which can be integrated in terms of path-ordered exponentials as usual. Note that L(y, z) is z-independent at y = 0. The potential is subsequently extracted from The lowest "S-wave" eigenvalue obtained with the LO (eq. (6.12)), NLO [9], and EQCD potential V + [10]. For EQCD the "P -wave" result is shown as well. For the LO case the leading-log asymptotics readÊ 0 ≈ 1 2 ln 1 ρ for ρ ≪ 1 andÊ 0 ≈ ( ln ρ 2ρ ) 1 2 for ρ ≫ 1. Right: The "amplitudes" corresponding to the lowest eigenmodes, as defined in the text (cf. eqs. (7.11), (7.14)).
The potential of eq. (7.6) agrees with the one derived and computed up to O(α 3/2 s ) in ref. [9]. A coordinate space expression was given in ref. [25]. The potential has been measured within a dimensionally reduced effective field theory (EQCD) in ref. [10]; we make use of the "cold" (T ≈ 400 MeV) β = 16 data set. 6 At short distances, a polynomial interpolation is employed; for estimating the potential at distances larger than those for which measurements exist, we fit the 5 largest distances (yg 2 E > 2.5) to the confining form σy + µ + γ y which describes the asymptotics well [27]. 7 We express the screening masses as "energies", and define the dimensionless quantities where M r is the reduced mass (cf. eqs. (6.6), (6.32)). The radial homogeneous part of the Schrödinger equation to be solved (cf. eqs. (6.17), (6.27), (6.33), (6.37)) reads where l = 0, 1, 2, ... denotes the angular quantum number. Assuming R l finite atȳ = 0 and integrable atȳ → ∞, the eigenvaluesÊ (l) are easily determined numerically. Results for the ground state (Ê (l) 0 ) for V + are shown in fig. 2(left) and for V − in fig. 3(left). Apart from the energies, the magnitudes of the correlators are also of interest. These can be obtained by solving eqs. (6.17), (6.27) in a spectral representation. Assuming a discrete spectrum and letting ψ i be wave functions normalized as d 2 y ψ * i (y)ψ j (y) = δ ij , the solution of eq. (6.17) and subsequently (6.18) reads where the sum 0<pn<kn has been suppressed for notational simplicity. Inserting this into eq. (2.4), we obtain the long-distance asymptotics For the P -wave case, eqs. (6.27), (6.28) lead similarly to 13) and the configuration space correlator reads (for |z| ≫ 1/[E 14) The "amplitudes" A + 0 , A + 1 are illustrated for the various potentials in fig. 2(right). In the static sector, we similarly get from eqs. (6.34) and (6.38) where the eigenvalues and eigenfunctions are solved withĤ − , and the suppressed sum now reads pn>0 . The "amplitudes" A − 0 , A − 1 are illustrated in fig. 3(right). There is one more comment to make about the energies in eq. (7.7). For the non-static case, eq. (6.6) implies Here we have re-expressed the parameter m 2 ∞ of eq. (5.2) in terms of the gauge coupling g 2 E . It should be noted however that m 2 ∞ is only known at 1-loop level whereas the parameters m 2 E and g 2 E are known at 2-loop level. Within the same approximation, the ground state energies of the static sector are of the form

Basic setup
For a non-perturbative crosscheck we make use of lattice simulations in two-flavour QCD, with physical parameters corresponding to Λ MS = 310 (20) MeV and m π ≈ 270 MeV [28]. Lattices of spatial size N 3 s = 64 3 and lattice spacing a = 0.0486(4)(5) fm are considered. The thermal ensembles have temporal extents N τ = 16, 12, corresponding to the temperatures T = 254(4) MeV and T = 338(5) MeV, respectively. (In terms of the pseudocritical temperature of the QCD crossover these amount to T /T c ≃ 1.2 and T /T c ≃ 1.6 at N f = 2 [29].) Further details concerning the lattice setup and measurements are given in appendix B.
For a comparison with the results of the previous sections, the parameters of the effective theory need to be estimated. The 2-loop values for these, as well as for the parameter ρ defined in eq. (7.8), are given in table 1. The subsequent predictions for the four-dimensional physical observables are shown in table 2. The energies come from eqs. (7.16) and (7.17); the amplitudes from eqs. (7.11), (7.14) and (7.15). (5) 3.2(2) 3.5 (4) 0.61(4) 0.92(9) 1.09 (7) Table 2: Weak-coupling (for n = 0, marked with an asterisk) and EQCD (for n > 0) predictions for screening masses and "amplitudes", with the latter defined as G ≡ −A e −E|z| at large |z|. For the amplitudes all states that are degenerate at the current level of precision have been summed together. The errors, based on those in table 1, should be considered as underestimates.
To the order we are working at, the ground state is degenerate in several channels (cf. table 2). We expect this degeneracy to be lifted at higher orders, in particular by the effect of the term δS 0 in the action (cf. eq. (5.11)). In the practical lattice analysis we see no indications of closely lying states. Therefore, in the following, only single states are discussed on both sides.

Fitting strategy
In order to extract the screening masses and amplitudes from the non-perturbative lattice correlators, a fitting ansatz needs to be chosen. The discussion below refers to the form but we have also considered purely exponential fits of the form G exp (z) ≡ 2 n=1 A n e −Mnz . The right edge of the fitting range is set to L z /2, and the fits are repeated for all possible positions of the left edge. The results are extracted from uncorrelated fits with errors originating from a jackknife procedure. To decide which of the resulting parameters is the best we impose a stability criterion with respect to the position of the left edge. To this end we compute the adjacent and next-to-adjacent edge-position parameter values and demand that T /MeV = 254 (4) T /MeV = 338(5) n = 0 n = 1 n = 2 n = 0 n = 1 n = 2 E 00 /T 7.87(10) 7.45 (6) 13.6(4) * 7.69(23) 7.252 (11)   the difference of their average and the current parameters be smaller than some tolerance. In the next step we reduce this tolerance to the point where only a single parameter set fulfills the stability criterion, and quote this number as our final result.
The qualities of such fits can be illustrated by defining "effective masses" and "effective amplitudes". Effective masses are defined by the implicit equation In order to define effective amplitudes we divide the data by a function with the fitted mass value (M fit ) inserted into eq. (8.1): The results are shown for the lower temperature T = 254(4) MeV in fig. 4. Like before, we refer to the screening masses in the following as "energies" (M fit → E).
For n = 0 a non-trivial crosscheck on the overall procedure can be obtained through Ward identities. Given the complex nature of the fitting procedure and the fact that the lattice correlator measured is of a local-conserved type (cf. appendix B), the Ward identities are not trivially fulfilled. They assert that E

Results and comparisons with perturbative predictions
Our final results for the screening masses are shown in fig. 5, where they are also compared with the perturbative ones from table 2. The evolution of the perturbative results in the non-static sector when going from LO to EQCD results is illustrated in fig. 6. The final lattice results for the screening masses and amplitudes are collected in table 3.
The following observations can be made: • On a rough level, the free-theory predictions E (n=0) = k 1 , E (n>0) = k n can be recognized in the full lattice data, with deviations that are < ∼ 50% (cf. fig. 5).
• More quantitatively, for n = 0 the lattice and perturbative screening masses differ in general by less than 10% (cf. fig. 5). A fairly good agreement is also observed for the P -wave screening mass in the static sector (E (n=0) 00 ).
• The S-wave screening masses in the static sector (E ) differ by about 15% at the lower temperature, by 10% at the higher one. Although not large per se, such discrepancies are beyond the estimated systematic errors of the effective description. We recall, however, that in the static sector only the LO potential is known, so that an additional approximation has been made: in the language of the introduction these predictions are of type (i) rather than (ii).
• The splitting between the two S-wave masses (E compared with the corresponding weak-coupling or effective-theory results from table 2. Note that discretization effects are expected to be larger at the higher temperature (i.e. N τ = 12), because then the distance scale 1/(4πT ) = aN τ /(4π) is close to the lattice spacing. particularly at the higher temperature. It may be noted that at LO, the splitting is entirely due to the difference in the sign of the force from the A 0 exchange, represented by the function K 0 (m E y) in eqs. (6.12) and (6.31).
• The "amplitudes" do not compare well with each other: for the S-wave cases (A ) the difference is ∼ 30%, but for the P -wave cases (A ). The amplitudes may however be expected to suffer from larger systematic uncertainties than the screening masses. On the perturbative side, we have determined the correlators only to LO as far as the amplitudes are concerned, not to NLO like for the screening masses. Moreover, the LO result arises from a numerical solution of the wave function and is thereby sensitive to soft scales, yet for heavy states the effective theory description of the soft dynamics is likely to be less accurate than for the ground state. On the lattice side, the amplitudes could be overestimated by misjudging where the plateau starts (put another way, they could include contributions from almost degenerate excited states). In addition, discretization effects have not been estimated, given that we only consider a single lattice spacing. For all of these reasons we think that the ∼ 30% discrepancy in the S-wave cases at n = 0, 1 is a reasonable reflection of the systematic uncertainties of a perturbative LO potential V + is from eq. (6.12), ref. [9], or ref. [10], respectively) and lattice (LAT) results for the non-static screening masses at T = 254 MeV. Right: The same for the corresponding amplitudes.
Going from LO to EQCD yields in most cases an improvement, particularly for the amplitudes.
computation at the temperatures considered, whereas little can be deduced from the amplitudes related to the heavy states (P -wave or n ≥ 2).
• The ∼ 30% resolution in the S-wave cases at n = 1 (A (n=1) 00 ) could only be reached thanks to the availability of the non-perturbative EQCD potential. Had only the LO potential been available, the discrepancy would have been ∼ 75%, cf. fig. 6(right).
• Finally, we point out that in principle there is a "two-meson" threshold in each sector of fixed n. This physics has not been included in the current effective-theory analysis, however we believe that the states we have measured are light enough not to be affected.

Conclusions
We have considered mesonic screening masses related to the vector current both in a "static" (zero Matsubara frequency) and "non-static" (non-zero Matsubara frequency) sector. It turns out that even though in the weak-coupling limit both sectors probe physics at the momentum scale gT and can be represented by "non-relativistic" low-energy effective theories in 2 + 1 dimensions, the physical significances of the sectors are different. Indeed only the non-static sector has a clear relation to real-time physics: we have shown that the potential appearing in the effective description (cf. eq. (6.12)) is identical to that previously considered in the context of jet quenching and photon and dilepton production in thermal QCD (cf. e.g. refs. [9]- [11]). On a general level, this observation is consistent with the statement that only non-zero Matsubara modes play a role in analytic continuation from imaginary to real time [32].
Apart from computing screening masses with an effective theory, we have also measured them with large-scale lattice Monte Carlo simulations in two-flavour QCD at temperatures of about 250 MeV and 340 MeV. We find a remarkably good agreement in the non-static sector, and also in the static sector for P -channel screening masses (cf. fig. 5). This adds confidence to the applicability of effective theory methods for the study of phenomenologically interesting observables in the temperature range relevant for heavy ion collision experiments. (For "amplitudes" the agreement is poorer than for screening masses, but they also suffer from larger systematic errors both on the perturbative and on the lattice side, as has been discussed around the end of sec. 8.) A general lesson, based on fig. 2, is that higher-order corrections from the momentum scales gT and g 2 T /π, even though formally suppressed by powers of g, are numerically of order 100% compared with results from the LO potential. In the case of the non-static sector we can at least partly account for these corrections, thanks to NLO computations and lattice simulations carried out in the context of jet quenching [9,10]. Including these corrections in the solution of the Schrödinger equation indeed improves the overall agreement with lattice data (cf. fig. 6).
In the case of the static sector, in contrast, the potential is different and no NLO or nonperturbative results exist for the moment. It seems conceivable, however, that computing such corrections might permit to reduce discrepancies between weak-coupling predictions and S-channel lattice data in the static sector. (The situation in the vector channel, illustrated in fig. 5, is not too bad, however the discrepancy is larger in the scalar and pseudoscalar channels, cf. refs. [14,15] and references therein.) Part of the discrepancy may be due to non-potential effects and spin-dependent terms, but it should be possible to incorporate these in the low-energy description as well.
In the present paper, we have demonstrated the existence of a relation between screening masses and real-time rates through perturbative arguments, employing four-dimensional lattice simulations only as a crosscheck for the accuracy of the perturbative description in the temperature range considered. It would be very interesting if similar relations could be established on a non-perturbative level. As a modest step in this direction, we may note that measuring the screening mass related to the operator 1/T 0 dτ e iknτψ (τ, y 2 , z) γ 0 W y,z ψ τ, − y 2 , z directly in four dimensions, for k n large, and cancelling the free behaviour through an exponential factor exp(k n z) like in eq. (7.1), would allow us to define fully non-perturbative variants of the potential V + of eq. (7.6). The local (non-conserved) vector current J l µ requires a finite renormalization factor Z V (cf. e.g. ref. [40]). Correspondingly the bare vector correlators were renormalized using with the non-perturbative value Z V (g 2 0 ) = 0.768(5) at 6/g 2 0 = 5.50 [41]. We have not included O(a) contributions from the improvement term proportional to the derivative of the antisymmetric tensor operator [42,43], nor a quark-mass dependent improvement of the form 1 + b V (g 2 0 )am q [43]. Both should be included to ensure a smooth scaling behaviour as the continuum limit is taken, however our present study concerns a single (fine) lattice spacing.
The renormalized lattice data as well as the fitted correlation functions for the ground states are shown as the coloured shaded bands in fig. 7. The error estimates were obtained via a jackknife procedure.