Computation of non scalar loop integrals

[CAF123]

Consider the following integral that comes out of a loop calculation along with some fermionic propagators (e.g virtual one loop correction to a $p \gamma^* \rightarrow p'$ process such as in DIS):
$$ \int \frac{\text{d}^d l}{l^2 (l-p)^2 (p+q-l)^2} \text{Tr}(\not p \gamma^{\nu} (\not p + \not q) \gamma^{\sigma} (\not p + \not q - \not l) \gamma^{\mu} (\not p - \not l) \gamma^{\rho})$$

What is the general methodology to go about solving such integrals? I think I can obtain a closed form solution for the trace but not sure how to proceed from there.

 

[CAF123]

Ok, so I proceeded and obtained the integral $$\int \frac{d^d l}{l^2 (p+q-l)^2 (p-l)^2} ((2p\cdot q - l \cdot p + q^2 - l \cdot q)(l \cdot p) + \dots$$ Now, without the numerator, application of feynman paramaters allows me to make progress. But, in this case, I want to express the numerator in terms of propagators that would allow me to cancel terms in denominator. I can write $$2p\cdot q - l \cdot p + q^2 - l \cdot q = (p+q-l)^2 - l^2 + l \cdot (p+q)$$ How to deal with the last term given it is not a propagator?

Thanks to anyone who can help :)

 

[vanhees71]

Just use the standard formulae of dim. reg. You can find an introduction in my lecture notes on QFT:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

 

[CAF123]

Just use the standard formulae of dim. reg. You can find an introduction in my lecture notes on QFT:

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

Are those the formulae on p.298 in the appendix? I'd rather not just quote some formulae - is there a way to proceed from my last display in my last post?
I was thinking once I have the last term expressed as a propagator I can just apply feynman paramaters since then I have no non trivial numerators to deal with.
Thanks!

 

[vanhees71]

But you can use Feynman parameters right away and then use the equations in App. C. Of course, to simplify the integral before, can be crucial to get a result in closed form, particularly to extract the divergent piece (poles in $1/(4-d)$), which can get hidden in the Feynman-parameter integrals and then are not so simple to isolate.

For a non-trivial example see the calculation of the sunset diagram in $\phi^4$ theory on p. 151ff.

For a one-loop integral with a non-trivial numerator, see the calculation of the gluon self-energy (polarization) on p. 250ff.

 

[CAF123]

Ok I see. So, just to check, in going from eqn (7.122) to eqn (7.123) on p.251, you evaluated the trace and rewrote the denominator by using feynman parameters? Then you used the formulae in Appendix C.2 to get a closed form solution?

Thanks!

 

[CAF123]

I have reduced
$$ \int \frac{\text{d}^D l}{l^2 (l-p)^2 (p+q-l)^2} \text{Tr}(\not p \gamma^{\nu} (\not p + \not q) \gamma^{\sigma} (\not p + \not q - \not l) \gamma^{\mu} (\not p - \not l) \gamma^{\rho})$$ down to $$ 2 \int d^D l \int_0^1 dx \int_0^{1-x} dy \frac{1}{(l^2 - \mathcal M^2)^3} \text{Tr}(\not p \gamma^{\nu} (\not p + \not q) \gamma^{\sigma} (\not p + \not q - \not l) \gamma^{\mu} (\not p - \not l) \gamma^{\rho})$$ where $\mathcal M$ is some function of $p,q$ and the feynman parameters which is not too complicated: $-\mathcal M^2 = q^2 x + 2p \cdot q x - ((q+p)x+py)^2$.

Now, in the case where there is no trace on the numerator, a Wick rotation allows use of Schwinger parameters and then the remaining integrals can be done by Gaussian integration. But there is $l$ dependence in the numerator because of the trace so how would you suggest I proceed?

Thanks!

 

[vanhees71]

Stay tuned. I'm right now in the process of evaluating the one-loop corrections for QED. I'll put complete steps of the calculation into the manuscript, but you can just use the forumlae in appendix C. Just write out, e.g., $\not{l}=l_{\mu} \gamma^{\mu}$. The $\gamma^{\mu}$ are just momentum-independent matrices which you can take out of the integral (if you don't want to evaluate the trace for some reason or if you calculate, e.g., radiation corrections to the the fermion-fermion-gauge-boson vertex which contains a germ with an explicit untraced Dirac matrix) and then use the standard formulae of Appendix C.

 

[CAF123]

Thanks! Ok, I think I can continue but just to make sure I understand the notation in the results of Appendix C - is $\omega = D/2$, where D is the dimensionality in which we compute the integral? And also for the denominator there, $(m^2 - p^2 - 2pq - i\eta)$ - does this correspond to my $(l-A)^2 - \mathcal M^2 = l^2 - 2l \cdot A + A^2 - \mathcal M^2$ with your $q$ taking the place of my $A$ and your $m^2 = A^2 - \mathcal M^2$?

I also thought about shifting all the momenta in the integral $$\int d^D l \frac{1}{l^2 (p+q-l)^2(p-l)^2} ((2p \cdot q - l \cdot p + q^2 - l\cdot q)(-l \cdot p) + \dots)$$ by $l$ so that e.g $l \rightarrow l'=l+l$ and similarly for the other two terms: $(p+q-l) \rightarrow (p+q)$, $p-l \rightarrow p$ and in that case I have only even powers of $l$ in the denominator and odd powers in the numerator for the first and third terms. So for these terms I have an odd integral over all $l$ so it must be zero. Is that a correct argument?

 

[vanhees71]

Yes. I use the convention for dim. reg. (I think it's from the book by Ramond) that $d=2 \omega=4-2 \epsilon$. It's a pain that different authors use different conventions :-)).

Also your argument is fine, because your integral is dimensionally regularized, i.e., you are allowed to shift integration variables (also differently in different terms of the integral) without changing the result. Then your argument is fine.

Note, however, the important exception of linearly divergent integrals in the case of theories with chiral fermions, where you cannot naively use dim. reg. That's related to the issue of anomalies, where you have to carefully define your renormalization conditions from other considerations like gauge invariance, which determines uniquely, how to regularize the integrals in such a case. The problem with dim. reg. are specifically four-dimensional objects like $\gamma^5=\mathrm{i} \gamma^{0} \gamma^1 \gamma^2 \gamma^3$ or the Levi-Civita tensor $\epsilon^{\mu \nu \rho \sigma}$.

For the usual gauge theories like QCD, where the gauge group is not chiral you have to obey gauge invariance and thus have to anomalously break the axial-vector-current conservation and must keep that of the vector current. Here you have to define $\gamma^5$ such that it anti-commutes with the $\gamma0$ to $\gamma^{3}$ but commute with all other $\gamma$-matrices (the socalled 't Hooft-Veltman prescription).

For chiral gauge theories like the Glashow-Salam-Weinberg theory of the weak and electromagnetic interaction (QFD), you must make sure that there are no anomalies, because here the gauge coupling is through currents of the type "vector minus axialvector".

 

[CAF123]

Thanks for the detailed response.

Also your argument is fine, because your integral is dimensionally regularized, i.e., you are allowed to shift integration variables (also differently in different terms of the integral) without changing the result. Then your argument is fine.

I noticed that I was shifting all my momenta by $l$ but is this permissible since I am integrating over $l$?

Just a few more comments: If I use feynman parameters, complete the square etc... I can put my integral into the form $$\int d^d l \frac{1}{(l^2 - \mathcal M^2)^3} ((2 q \cdot p -l \cdot p + q^2 - l \cdot q)(-l \cdot p) + \dots )$$ so, given this rearrangement, $l$ is now even in the denominator so the only non vanishing contribution in the above sum would be the second and last term there involving two l's $$p_{\mu} p_{\nu} \int d^d l \frac{l^{\mu} l^{\nu}}{(l^2 - \mathcal M^2)^3}$$ since the rest yield a single power of l on the numerator which vanishes in the symmetric integration. I can use, in principle, the dim reg formulae to evaluate the remaining pieces.

I was just wondering, if that is correct, if you could say a few words about how in practice one goes about obtaining such dim reg formulae that for example you found in your notes or in end of QFT books? I am guessing the numerator is rewritten in such a way that we can cancel some terms in the denominator thereby yielding a trivial numerator etc...

Thanks!