A surprisingly easy string theory problem

The above, I imagine, seems to be a direct paradox.  I’ve been perusing O.C.W. again, as per usual – I’m currently examining the second problem set featured in 8.821.

Whilst it’s exceedingly difficult for me to get through these problems, components of them are possible to approach with the correct tidbits of knowledge; noting that M.I.T. professors’ material is nearly always meticulously organized, it becomes easy to retrieve what one needs.

That said, I’m currently on the third problem, which reads:

1.  Show that the one-loop β-function for N = 4 SYM vanishes.

Let’s begin by defining a beta function in the context of theoretical physics.  Basely, a beta function β(g) is a way of expressing the dependence of a coupling parameter g on a given energy scale μ.  The aforementioned coupling constant is just a number that rates the strength of some interaction.  This is the relation:

\beta(g) = \mu\, \frac{\partial g}{\partial \mu}.

However, N = 4 SYM has been mentioned; this is a gauge theory, and, more specifically, a supersymmetric Yang-Mills theory.  We’re looking solely at it.

At this point, I’d like to derive the necessary beta function, but I’m attempting to showcase the easy aspect of this problem.  Additionally, as I’ll mention later, I’ve no idea as to how to do it.  So, as was permitted in the problem set, I will proceed to cheat, looking up the function for a gauge theory with Nf Weyl fermions and Ns complex scalars; that will be the below:

\mu\, \frac{\partial g}{\partial \mu} = -\frac{b}{16\pi^{2}} + O(g^{5})

where

b = \frac{11}{6}T(adj) - \frac{1}{3}\displaystyle\sum_aT(r_a) - \frac{1}{6}\displaystyle\sum_nT(r_n)

Now, we just have to realize that a) everything is in the adjoint, b) N = 4 SYM has four Weyl fermions, and c) N = 4 SYM has six real scalars, and, consequently, three complex ones.  This yields a nicer formula that you can find in the solutions to the second problem set; I will not bother reproducing it here, as the fun lies in obtaining it.  I’ve theoretically handed it to you, given the above three notes.  You should be able to show that the function vanishes in two steps, one of which is very simple.

I’ve a few questions, however – firstly, how would I derive the aforementioned beta function?  Are Feynman diagrams needed?  Secondly, erm, is it possible to write a path-integral for N=4 SYM in analytic superspace?  I’m new to examining this theory, so please excuse any questions that may harbor perfectly obvious answers.  I also apologize for any fails where LaTeX is concerned – it seems that I’m having some sizing issues; I will remedy them tomorrow.

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