1-loop matching of a thermal Lorentz force

Published for SISSA by Springer Received: April 5, 2021 Revised: June 4, 2021 Accepted: June 4, 2021 Published: June 23, 2021 M. Laine AEC, Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, Bern CH-3012, Switzerland E-mail: Abstract: Studying the diffusion and kinetic equilibration of heavy quarks within a hot QCD medium profits from the knowledge of a coloured Lorentz force that acts on them. Starting from the spatial components of the vector current, and carrying out two matching computations, one for the heavy quark mass scale (M ) and another for thermal scales √ ( M T , T ), we determine 1-loop matching coefficients for the electric and magnetic parts of a Lorentz force. The magnetic part has a non-zero anomalous dimension, which agrees with that extracted from two other considerations, one thermal and the other in vacuum. The matching coefficient could enable a lattice study of a colour-magnetic 2-point correlator. Keywords: Thermal Field Theory, Heavy Quark Physics, Quark-Gluon Plasma, Lattice QCD ArXiv ePrint: 2103.14270 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP06(2021)139 JHEP06(2021)139 1-loop matching of a thermal Lorentz force Contents 1 2 Outline of a procedure 2 3 QCD vacuum contribution 4 4 QCD thermal contribution 7 5 Non-relativistic determination of the thermal contribution 10 6 Infrared side of the matching 14 7 Result and discussion 16 1 Introduction The motion of heavy probe particles is a classic tool for extracting information about the microscopic properties of an interacting statistical system. In heavy ion collision experiments, one manifestation of this philosophy is to inspect how efficiently heavy flavours (charm and bottom quarks) participate in hydrodynamic flow (cf., e.g., ref. [1]). In cosmology, assuming that dark matter is made of weakly interacting massive particles, it would be important to know for how long they stay in kinetic equilibrium with the other particles, as this may affect, amongst others, structure formation (cf., e.g., ref. [2]). To be concrete, consider a particle whose mass M is much larger than the temperature T . Given that the average (equilibrium) velocity is below unity, v 2 ∼ 3T /M  1, and  3/2 the (equilibrium) density is exponentially suppressed, n ∼ M2πT e−M/T , we find ourselves in a non-relativistic dilute regime. Thinking of a single such particle, and assuming that it carries the gauge charge g, the classical Lorentz force acting on it reads dpµ = gF µν vν , dt (1.1) where pµ is the four-momentum and v µ ≡ (1, v) is the velocity. The Lorentz force contains an electric part (∼ gE) and a magnetic one (∼ gv × B). It has thus been argued that at zeroth order in v, heavy quarks are affected by colour-electric forces [3, 4], whereas at first order in v, corrections originate from colour-magnetic ones [5]. For dark matter, we could similarly consider the forces originating from the weak gauge group. Being a classical description, eq. (1.1) is guaranteed to hold only at large time scales where phase decoherence has taken place, t  1/(α2 T ), where α = g 2 /(4π). Due to their large inertia, the time scale associated with the kinetic equilibration of heavy particles is –1– JHEP06(2021)139 1 Introduction 2 Outline of a procedure Let us consider the vector current, JµQCD = ψ̄γµ ψ, associated with one heavy flavour in R QCD.2 The spatial integral over the zeroth component, x J0QCD , measures the net number of this species (particles minus antiparticles), and is conserved in the absence of weak inR teractions. In contrast, the spatial components, x JiQCD , are not conserved. They measure velocities, and velocities can be changed by elastic reactions. Following eq. (1.1), our focus here is on time derivatives of velocities, i.e. accelerations. R The QCD operator that we are interested in can formally be expressed as ∂0 x JiQCD . In a vacuum setting, we could take matrix elements of this operator in the presence of a background gauge field Ā(Q) [14], where Q = (q0 , q) is a four-momentum. As we are aiming at an infrared (IR) description, Q is considered small compared with other energy scales. Schematically, then, we could consider matrix elements like *  Z p1 ∂0 QCD x Ji  Ā(Q) Z p1 +  QCD x J0 p2  p2 ' δ (3) (p2 + q − p1 ) AQCD [Ā(Q)] + O(q02 , q2 , v2 ) , i (2.1) ' δ (3) (p2 − p1 ) N0QCD + O(v2 ) , (2.2) where the precise way to extract the external states will be discussed presently, and v is the heavy-quark velocity in the medium rest frame. The matrix elements in eqs. (2.1) and (2.2) are subject to wave function renormalization, which drops out in the ratio aQCD ≡ i AQCD i . N0QCD 1 (2.3) There is a famous history of quantum-mechanical derivations of the Lorentz force, cf. e.g. ref. [7]. We do not elaborate on the overall factors ±i, ±1 of the various operators, on one hand because these play no role in the end, on the other because we work in Euclidean spacetime, with Euclidean Dirac matrices, and then additional factors may originate from the time coordinate, temporal gauge field components, spatial Dirac matrices, and raising/lowering of indices. It would be a distraction to discuss all of them. 2 –2– JHEP06(2021)139 ∼ M/(α2 T 2 ) [6]. For M  T , there should thus be a broad range of time scales for which eq. (1.1) is valid. At the same time, thermal effects break Lorentz invariance and distinguish between electric and magnetic fields, modifying the respective couplings (cf. eq. (2.5)). In fact, we recover an unmodified eq. (1.1) only in vacuum,1 where the decoherence argument does not apply, but M  ΛMS still provides for a hierarchy of time scales (cf. eq. (3.22)). Given that colour interactions are strong in QCD, their effects should be investigated up to the non-perturbative level. For colour-electric forces, large-scale lattice simulations have indeed been carried out in recent years [8–13], whereas for the colour-magnetic corrections, the challenge lies ahead of us. In preparation for this task, the goal of the current study is to clarify the renormalization of the colour-magnetic part of eq. (1.1). Specifically, we show how a divergence found in ref. [5], cf. eq. (7.6), gets cancelled after the inclusion of the proper matching coefficient. It is for the cause of such an acceleration, multiplied by a (thermally corrected) pole mass M , that we would like to find an operator reminiscent of the Lorentz force. Before proceeding, we note that for the thermal effects that we are mostly concerned with, the notion of matrix elements such as eqs. (2.1) and (2.2) is ambiguous. Therefore, we generalize the definitions to certain “partition functions”, defined in configuration space. Let the Euclidean time coordinate be τ and a generic spatially averagedoperator  O(τ ). β β The time direction is compact and is chosen to lie in the interval τ ∈ − 2 , 2 , where β ≡ T1 is the (...truncated)