High-energy theory

Setting the small mass to zero

About 20 min if you have seen this material before, 55 min or so if it is new to you.

Background

Assumed background: integrating a rational function, and the natural logarithm. A quantity that often shows up when calculating a process in particle physics is an integral over the momentum carried by a rather than observed. That momentum is not measured, so every value it could take is integrated over, from zero up to infinity, and every mass in the theory appears somewhere in the integrand. Integrals like this can look as though they do not converge. This problem is about one particular way that happens, called an infrared divergence: an infinity that comes from the low-momentum end of the range. The integral below is a real one with the easy parts already done, so what is left is a single integration over the momentum's magnitude

k

. It was chosen because it can also be evaluated exactly, which is unusual.

I(m, M) = \int_0^\infty \frac{k\,dk}{(k^2+m^2)(k^2+M^2)}

The two masses are the electron and the W boson:

m = 0.511

MeV,

M = 80400

MeV. Every energy below is in MeV.

The small mass is five orders of magnitude smaller than the large one, so one might be tempted to set it to zero. The parts below work out whether that is a fine approximation, and at which step in the calculation it can actually be made.

(a)

With

m

set to zero in the integrand, the integral diverges. Below what value of

k

does the

m = 0

integrand exceed the true one by more than

10\%

(b)

Do the integral exactly, keeping both masses. Give

M^2 I

, which is dimensionless.

(c)

Most real integrals of this kind cannot be evaluated exactly in terms of elementary functions, so here is the method used when they cannot. Pick a momentum

\mu

somewhere between the two masses,

m \ll \mu \ll M

, and write

I

as a sum of two integrals: one from

0

\mu

, the other from

\mu

to infinity. In the first,

k

never gets near

M

, so replace

k^2 + M^2

M^2

. In the second,

k

is always far above

m

, so replace

k^2 + m^2

k^2

. Both integrals are then elementary. Evaluate them, add them, and compare the total with (b). Each replacement introduces a relative error, and the two errors move in opposite directions as

\mu

is varied. Give the

\mu

at which they are the same size.

(d)

By either route, what fraction of

I

comes from

k

below

1

GeV? Give a percentage.

Past 1.4 hr on this? That usually means a missing prerequisite rather than a lack of effort — worth reading up before pushing further, starting with Method of matched asymptotic expansions (Wikipedia).

Want to go deeper?