finiteT.py

This module provides the functions for the one-loop finite temperature corrections to a potential in QFT. The two basic functions are:

Jb(x) = int[0->inf] dy +y^2 log( 1 - exp(-sqrt(x^2 + y^2)) )

Jf(x) = int[0->inf] dy -y^2 log( 1 + exp(-sqrt(x^2 + y^2)) )

Call them by:

Jb(x, approx=’high’, deriv=0, n = 8)

Here, approx can either be ‘exact’, ‘spline’, ‘high’, or ‘low’. Exact calculates the integral numerically, while high and low calculate the high and low x expansions of J to order n. Specify the derivative with the ‘deriv’ parameter.

Jf_exact(x)

Jf calculated directly from the integral.

Jf_exact2(theta)

Jf calculated directly form the integral; input is theta = x^2.

Jb_exact(x)

Jb calculated directly from the integral.

Jb_exact2(theta)

Jb calculated directly form the integral; input is theta = x^2.

dJf_exact(x)

dJf/dx calculated directly from the integral.

dJb_exact(x)

dJb/dx calculated directly from the integral.

Jf_spline(X, n=0)

Jf interpolated from a saved spline. Input is (m/T)^2.

Jb_spline(X, n=0)

Jb interpolated from a saved spline. Input is (m/T)^2.

Jb_low(x, n=20)

Jb calculated using the low-x (high-T) expansion.

Jf_low(x, n=20)

Jf calculated using the low-x (high-T) expansion.

Jb_high(x, deriv=0, n=8)

Jb calculated using the high-x (low-T) expansion.

Jf_high(x, deriv=0, n=8)

Jf calculated using the high-x (low-T) expansion.

Jb(x, approx='high', deriv=0, n=8)

A shorthand for calling one of the Jb functions above.

Parameters:

• approx (str, optional) – One of ‘exact’, ‘high’, ‘low’, or ‘spline’.
• deriv (int, optional) – The order of the derivative (0 for no derivative). Must be <= (1, 3, 0, 3) for approx = (exact, high, low, spline).
• n (int, optional) – Number of terms to use in the low and high-T approximations.

Jf(x, approx='high', deriv=0, n=8)

A shorthand for calling one of the Jf functions above.

Parameters:

• approx (str, optional) – One of ‘exact’, ‘high’, ‘low’, or ‘spline’.
• deriv (int, optional) – The order of the derivative (0 for no derivative). Must be <= (1, 3, 0, 3) for approx = (exact, high, low, spline).
• n (int, optional) – Number of terms to use in the low and high-T approximations.