# Beta-function for the Gross-Neveu model

#### Manu_

Problem Statement
In the Peskin & Schroeder textbook, the $\beta$-function for the Gross-Neveu model is discussed in problem 12.2. After computing it, I have tried checking my results with some solutions found online. My problem is that they all disagree among each other (something quite recurrent for this book actually).
Relevant Equations
$$\mathcal{L} = \bar{\Psi}_{i}(i\not{\partial} )\Psi_{i}-g\sigma \bar{\Psi}_i\Psi_i-\frac{1}{2}\sigma^2$$

$$\delta_{Z_{\Psi}}=(ig)^2\int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i\not{k}}{k^2}\frac{i}{(k+p)^2}]=0$$

$$\delta_{Z_{\sigma}}=-N(ig)^2\int \frac{d^{d}k}{(2\pi)^d}\frac{i\not{k}}{k^2} \frac{i(\not{k}+\not{p})}{(k+p)^2} =-Ng^2 \int \frac{d^{d}k}{(2\pi)^d}\frac{1}{(k+p)^2} =-Ng^2\frac{i}{4\pi\epsilon}+finite$$

$$\delta_{g}=(ig)^3 \int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i(\not{k}+\not{p_1})}{(k+p_1)^2} \frac{i(\not{k}+\not{p_2})}{(k+p_2)^2}] = ig^3 \int \frac{d^{d}k}{(2\pi)^d}\frac{-2k^2}{(k+p_1)^2 (k+p_2)^2}+...=-2ig^3\frac{i}{4\pi\epsilon}$$
In the Peskin & Schroeder textbook, the $\beta$ function for the Gross-Neveu model is discussed in problem 12.2. After computing it, I have tried checking my results with some solutions found online. My problem is that they all disagree among each other (something quite recurrent for this book actually). I have basically followed the original paper from Gross and Neveu:

- We start from the Lagrangian $$\mathcal{L} = \bar{\Psi}_{i}(i\not{\partial} )\Psi_{i}-g\sigma \bar{\Psi}_i\Psi_i-\frac{1}{2}\sigma^2.$$
- We compute the field strength renormalisation

For $\Psi:$ $$\delta_{Z_{\Psi}}=(ig)^2\int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i\not{k}}{k^2}\frac{i}{(k+p)^2}]=0$$

For $\sigma:$ $$\delta_{Z_{\sigma}}=-N(ig)^2\int \frac{d^{d}k}{(2\pi)^d}\frac{i\not{k}}{k^2} \frac{i(\not{k}+\not{p})}{(k+p)^2} =-Ng^2 \int \frac{d^{d}k}{(2\pi)^d}\frac{1}{(k+p)^2} =-Ng^2\frac{i}{4\pi\epsilon}+finite$$

- We compute the coupling renormalisation (though in the Gross-Neveu original paper, the authors argue the vertex need not be renormalised. That is a point I don't get)

$$\delta_{g}=(ig)^3 \int \frac{d^{d}k}{(2\pi)^d} Tr[\frac{i(\not{k}+\not{p_1})}{(k+p_1)^2} \frac{i(\not{k}+\not{p_2})}{(k+p_2)^2}] = ig^3 \int \frac{d^{d}k}{(2\pi)^d}\frac{-2k^2}{(k+p_1)^2 (k+p_2)^2}+...=-2ig^3\frac{i}{4\pi\epsilon}$$

- We use relation (12.53)

$$\beta (g) = M\frac{\partial}{\partial M}\left(-\delta_g +\frac{1}{2}g\sum_i \delta_{Z_i} \right)=-\frac{2g^3}{4\pi}+g\frac{-Ng^2}{4\pi}=-\frac{g^3}{4\pi}(N+2)$$

It is asymptotically free, but I do fully trust my result. Any comment welcome!

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#### MathematicalPhysicist

Gold Member
Can you provide the references you found for this calculation?

Thanks!

#### Manu_

I compared the results described in:

And the result found by Gross and Neveu in their paper:

The factor of $g^2$ found by Gross and Neveu seems to be because they are using a different definition from P&S, which would amount to $\beta/g$.

#### king vitamin

Gold Member
You should find that the beta function vanishes for $N = 1$ (where $N$ is the number of two-component Dirac fermions, as in P&S's notation). This is because of a well-known equivalence between that model and that of a free massless boson. So I'm biased towards the Xianyu solutions being correct, unless somehow Murayama's definition of $N$ is given by twice the definition in P&S (which some authors like to do, since that definition of $N$ is the total number of components*flavors).

Also, after converting the difference in notations, Zinn-Justin's textbook also gives the same beta function as Xianyu's solution (he uses a different regularization scheme, but it is well-known that this doesn't change the one-loop contribution to the beta function).

I will take a look at your computations when I have time, but in the meantime I would say Xianyu's solutions are likely trustworthy.

#### Manu_

Thanks King Vitamin, I will take a look in that textbook.
Actually, there is a mistake in my first post, where I used (ig) instead of (-ig) for the coupling, and this changes the final result to (N-2) instead of (N+2). But I still have to find out how to edit my post...

"Beta-function for the Gross-Neveu model"

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