NLO spectral functions

Based on: 1910.07552 (PRD,2019).

Here I provide functions to compute the imaginary part of two-loop diagrams, like the one shown below. These are thermal self energies, functions of an external four momentum K=(k₀,k) and dependent on the statistical configuration.

Powers of the internal propagators for P, Q, R, L and V respectively are used to classify the masters by a string `abcde'. Here the powers are all either 0, 1 or 2. (Propagators in the figure will be absent if the power is 0.) The s_i are statistical factors: +1 for bosons and -1 for fermions. By "cutting" the diagram, some momenta are put on-shell which allows the master to be computed as a 2-dimensional integral over p and q (corresponding to the 3-momenta of the loop variables above).

See below for a list of the included functions. (They are defined in inc/core.hh)

Physical applications

Two specific examples are supplied.

QGP dilepton rate

See: 1910.09567

The combination of master integrals which is required to compute the dilpton emission rate from a hot QCD plasma is demonstrated in examples/dileptons.cpp (see functions in file for more information). This source file can be compiled with:

make dileptons

This file was used to compute the rates implemented in hydrodynamic simulations in 2311.06675 and 2311.06951. Note that these spectral functions are valid for finite quark chemical potential.

Neutrino interaction rate

See: 2312.07015

The QED corrections to the neutrino interaction rate at MeV temperatures requires an integration over the electron-positron spectral function. We supply code for evaluating this rate in examples/neutrinos.cpp (see functions in file, accuracy may be adjusted to suit). This source file can be compiled with:

make neutrinos

Usage

The program handles a master integral that is read from the config file in the working directory. An example, which demonstrates the correct formatting, reads:

11020
++-
1
0

When processed, this input sets up a master integral with no propagators for R and V. Those with P and Q are single propagators and L is repeated. Here s₀=s₁=+1 and s₂=-1 summarise the statistical configuration: (s₀,s₁,s₂). Here m=1 and n=0 are powers of energies p₀ and q₀ respectively.

The main project can be built using make in the same directory as this readme, which puts the executable in bin/. It can then be run, for example using

./bin/rho -k 1.0

which will set k/T=1.0 and then perform a sweep of energies from k₀/T=10^-2 to 10⁺² using 500 points on a logarithmic scale (this can be changed by modifying main.cpp and recompiling).

Use the flag -h to see more options. Note that any changes to main.cpp will require a clean to be picked up by the makefile.

Data files

Output from the main function is stored under out/data. (If the file is empty, it will be created by the makefile.) Naming convention is

diag.[type]{k=[k/T]}.([s0][s1][s2]).[m][n].dat

where [...] takes the value specified within the square brackets. Each file tabulates the energy master integrals, as a function of energy. There are four columns: (1) energy k₀/T, (2) rho scaled with T=1, (3) the leading OPE and (4) the next-to-leading OPE.

These can be plotted with the help of the gnuplot scripts in the out directory.

Abusage

You can manipulate the master integrals by first declaring a pointer to a Master object, and then constructing the type with energy power and exponents:

Master *rho;
rho = _11020(m,n,s);

The s pointer is to an int [3] array of entries +1 (boson) or -1 (fermion).

To get the value of the function, one can simply call:

(*rho)(k0,k)

Some examples are provided.

NB, Instances of the master functions return a quantity in units of the temperature (i.e. T=1).

Requirements

• GNU Science Library

• C++ compiler (v11 or higher)

List of included integrals

Type	I	II	III	IV	V	VI
Propagators	01020, 00120	11010, 10110, 11020, 10120	11011	11100	11110, Star	11111