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On p104 of Srednicki's QFT, he does an integral in closed form, equations 14.43 and 14.44. I just ran the calculations for this in Mathematica, and I get his answer exactly except for my constants [tex] c_1=4-\pi\sqrt{3} [/tex] and [tex] c_2=4-2\pi\sqrt{3} [/tex].

The mathematica code I used to generate this was:

Then collecting the terms in k^2 and m^2, you find the constants I posted above, rather than the very similar but different Srednicki ones.In[3]:= d[x_] := x*(1 - x)*k^2 + m^2

In[4]:= d0[x_] := (1 - x*(1 - x))*m^2

In[5]:= p[x_] = (1/2)*a*d[x]*Log[d[x]/d0[x]]

Out[5]= 1/2 a (m^2 + k^2 (1 - x) x) Log[(m^2 + k^2 (1 - x) x)/(

m^2 (1 - (1 - x) x))]

In[12]:= Integrate[p[x], {x, 0, 1},

Assumptions -> {Element[m, Reals], Element[k, Reals], m > 0, k > 0}]

Out[12]= (1/(12 k Sqrt[

k^2 + 4 m^2]))a (k Sqrt[

k^2 + 4 m^2] (4 (k^2 + m^2) - Sqrt[3] (k^2 + 2 m^2) \[Pi]) +

2 (k^2 + 4 m^2)^2 ArcTanh[k/Sqrt[k^2 + 4 m^2]])

It's not listed on his errata page if this is an error, perhaps I am missing something? just seems very close, to not be correct.