Alternative Feynman Diagram Calculation of Axial Anomaly

[Ian Lovejoy]

Homework Statement

I'm working my way through Quantum Field Theory in a Nutshell by A. Zee. I'm religiously doing the exercises but since I'm doing it on my own (I'm not in school) I have no one to ask when I get stuck. Any hints would be appreciated.

The problem is IV.7.5, on page 253 of the book. The desire is to derive the anomaly using the triangle diagram. However this is not to be done using the standard shift of integration variable treatment (that is developed in the chapter itself). Instead, one is to start with the massive Fermion case, and use Lorentz invariance, Bose statistics, and vector current conservation. All of that part I get. But in the end, one still has to evaluate a Feynman integral, and that is where I am stuck.

Homework Equations

The equation one gets from the triangle diagram is:

$$\Delta^{\lambda\mu\nu}(k_1,k_2) = (-1)i^3\int\frac{d^4p}{(2\pi)^4} tr(\gamma^\lambda\gamma^5\frac{1}{p\!\!\!/ - q\!\!\!/ - m}\gamma^\nu\frac{1}{p\!\!\!/ - k\!\!\!/_1 - m}\gamma^\mu\frac{1}{p\!\!\!/ - m}) + \{\mu, k_1 \leftrightarrow \nu, k_2\}$$

Also one can argue by Lorentz invariance that:

$$\Delta^{\lambda\mu\nu}(k_1,k_2) = \epsilon^{\lambda\mu\nu\sigma}k_{1\sigma}A_1 + \epsilon^{\lambda\mu\nu\sigma}k_{2\sigma}A_2 + \epsilon^{\lambda\nu\sigma\tau}k_{1\sigma}k_{2\tau}k^{1\mu}A_3 + \epsilon^{\lambda\mu\sigma\tau}k_{1\sigma}k_{2\tau}k^{1\nu}A_4 + \epsilon^{\mu\nu\sigma\tau}k_{1\sigma}k_{2\tau}k^{1\lambda}A_5$$
$$+ \epsilon^{\lambda\nu\sigma\tau}k_{1\sigma}k_{2\tau}k^{2\mu}A_6 + \epsilon^{\lambda\mu\sigma\tau}k_{1\sigma}k_{2\tau}k^{2\nu}A_7 + \epsilon^{\mu\nu\sigma\tau}k_{1\sigma}k_{2\tau}k^{2\lambda}A_8$$

Where the $$A_i$$ are functions of the Lorentz scalars. Using Bose statistics one can show:

$$A_2(k_1, k_2) = -A_1(k_2, k_1)$$
$$A_6(k_1, k_2) = -A_4(k_2, k_1)$$
$$A_7(k_1, k_2) = -A_3(k_2, k_1)$$
$$A_8(k_1, k_2) = A_5(k_2, k_1)$$

And using vector current conservation you can show:

$$A_2 = {k_1}^2A_3 + k_1\cdot k_2 A_6$$
$$A_1 = k_1\cdot k_2 A_4 + {k_2}^2A_7$$

(continued in reply, having trouble with the length of this post)

 

[Ian Lovejoy]

(continuation of original post)

The Attempt at a Solution

That is actually the main point of the exercise, that $$A_1$$ and $$A_2$$ are divergent but because of these relations one need only calculate $$A_3$$, $$A_4$$, and $$A_5$$.

Fair enough. But in trying to actually calculate any of these, rationalizing the denominators and focusing on only the relevant terms, I always end up with an integral that looks roughly like (some constant times) this:

$$\int\frac{d^4p}{(2\pi)^4} \frac{p_\tau}{[(p - q)^2 - m^2][(p - k_1)^2 - m^2][p^2 - m^2]}$$

OK, so I know to introduce Feynman parameters $$\alpha$$ and $$\beta$$ to combine the factors in the denominator, at which point one can complete the square, then substitute $$l=p - (\alpha q + \beta k_1)$$. The factor of $$l$$ in the numerator cancels out by symmetry, and what is left is a standard Minkowski space integral. I'm left with:

$$\int d\alpha d\beta \frac{-i}{16\pi^2} \frac{\alpha q_\tau + \beta k_{1\tau}}{(\alpha q + \beta k_1)^2 + m^2 - \alpha q^2 - \beta {k_1}^2}$$

Where the integral is to be done over the triangle in the alpha, beta plane bounded by alpha = 0, beta = 0, and alpha + beta=1.

And that is where I'm stuck. From here I attempt to integrate on alpha or beta first, and the resulting equations are so complicated that I can't do the other integral. I suspect I need to do some clever substitution, or combine terms somehow, or symmetrize, or use some approximation (especially approximating m as small since we presumably want to set it equal to zero in the end). I haven't been able to get any of that to work. Even if I make special assumptions such as k_1^2 = k_2^2 = 0, the integral looks too complicated to be right, especially since in the end I'm expecting a relatively simple answer, the standard anomaly equation.

The hint in the back of the book refers the reader to the lectures given by S. Adler in the 1970 Brandeis Summer School, but I can't find the lecture notes online.

Any help or hints appreciated.

 

[Ian Lovejoy]

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[Ian Lovejoy]

Anyone? Bueller?