I am currently trying to understand a really old paper of Jackiw and Coleman: "Why dilatation generators do not generate dilatations".

There, at some point they arrive at the following integral from a Feynman propagator (wth loops) and regulator masses:

\begin{equation} \lim_{M_f \rightarrow \infty} g^2 M_f Tr \int \frac{d^d k}{(2 \pi)^d} \gamma_5 \frac{1}{{\not}{k} - M_f} \gamma_5 \frac{1}{{\not}{k} - {\not}p - M_f} \frac{1}{{\not}{k} - {\not} q - M_f} \end{equation} where ${\not} k, {\not}p $ and ${\not}q$ are the dirac slashed $k_{\mu} \gamma^{\mu},p_{\mu} \gamma^{\mu},q_{\mu} \gamma^{\mu}$ momenta.

They argue that the integral is convergent and my first question is how can I see it?

I only fully understand the DR regularization way and never actually seen regularization via auxilliary fields and masses. From what I can see the integral will give a k-order divergence and a $M_f^2$ on the denominator. Is it enough to say that the limit $\lim_{M_f \rightarrow \infty} \frac{k}{M_f^2} =0$ ?

Secondly, they argue that if we expand the integral with respect to $p$ and $q$, the only terms that will survive are of order $p^2$ and $q^2$.

So my question is, what do I expand and how since I have gamma matrices on the denominator. Do I expand around $\not k - \not p$?


1 Answer 1


Ok, here's my guess as to what's happening here. I'm not an expert in this so hopefully, someone more knowledgable can correct me. In the paper it is stated that

For these values of the masses, the integral (3.26) has, for any fixed $p$ and $q$, a convergent power series expansion in $p$ and $q$ for sufficiently large regulator mass.

I am reading this as "the power series converges in $p$ and $q$, but we haven't said anything about the loop momentum integral". So apart from worrying about the divergence arising from the loop momentum $k$, the authors want to make sure that the power series expansion in the external momenta is also well-defined.

The type of regularization that they are using is called Pauli-Villars regularization. Each one of the integrals is still divergent but taking differences between integrals of different masses can parametrize the divergence in terms of the fictitious regulator mass $M_f$. This is achieved by the following replacement of the propagator

$$\frac{1}{p^2+i \epsilon} \rightarrow\frac{1}{p^2+i \epsilon}- \frac{1}{p^2+M_f^2+i \epsilon}.$$

Before we can show why the integral is divergent, let's first address the gamma matrices in the denominator. When one performs fermion loops, it is standard to use the identity:

$$\frac{1}{\not k -m}=\frac{\not k +m}{k^2 -m^2}, \tag{1} $$

since $(\not k-m)(\not k+ m) = k^2 -m^2$. After such a manipulation, the integral becomes

\begin{equation} \lim_{M_f \rightarrow \infty} g^2 M_f Tr \int \frac{d^4 k}{(2 \pi)^4} \gamma_5 \frac{{\not}{k} + M_f}{k^2 - M_f^2} \gamma_5 \frac{{\not}{k} - {\not}p + M_f}{(k-p)^2 - M_f^2} \frac{{\not}{k} - {\not} q + M_f}{(k-q)^2 - M_f^2}. \end{equation}

After performing the trace, we can see that the highest order term will look something like $$\sim \int \frac{d^4k}{(2\pi)^4} \frac{k^2}{(k^2- \Delta)^3}\sim \int \frac{k^5}{k^6} \rightarrow \infty,$$ where $\Delta$ is some algebraic combination of the regulator masses, Feynman parameters etc... The divergence is logarithmic, not linear since the trace of an odd number of $\gamma$ matrices is always $0$ in $4$ dimensions. This doesn't change the main message.

The point is that the loop momentum integral diverges, as it should in a QFT. This is why the authors compute the quantities $A(p,q)$, which are the regularized versions of this integral. The convergence they are referring to in that paragraph most likely has to do with a convergence in the sense of justifying their use of Taylor series.

Finally, to address the possibility of performing a Taylor series, I hope the identity (1) answers this question.


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