The 1-loop anomalous dimension of massless quark field

[DaniV]

Homework Statement

Considering SU(N) gauge theory with nf massless quarks

I want to find the anomalous dimension to order of 1-loop of the massless quark field, that defined by: [ tex ]\gamma_q(g^{(R)})=\frac{1}{2Z_q}\mu\frac{\partial Z_q}{\partial \mu} [ /tex ]
when [ tex ]\mu [ /tex ] is energy scale, q is symbolize quark field, g is the strong coupling and $Z_q$ is the renormalization parameter such that: [ tex ]q^{(R)}=\frac{1}{\sqrt{Z_q}}q ,\bar q^{(R)}=\frac{1}{\sqrt{Z_q}}\bar q [ /tex ]

Relevant Equations

[ tex ]\gamma_q(g^{(R)})=\frac{1}{2Z_q}\mu\frac{\partial Z_q}{\partial \mu} [ /tex ]
[ tex ]q^{(R)}=\frac{1}{\sqrt{Z_q}}q ,\bar q^{(R)}=\frac{1}{\sqrt{Z_q}}\bar q [ /tex ]

I tried as first step to find $Z_q$ the renormalization parameter, to do so I did the same procedure to find the renormalization parameter of the gauge field of the gluon $A^a_\mu$ when $a$ is representation index $a \in {1,2,...,N^2-1}$ such that $$A^{a{(R)}}_{\mu}=\frac{1}{\sqrt{Z_A}}A^{a}_{\mu}$$. the procedure is to find 1-loop correction to the 2 point function of the field A, as a result we find the 1-loop beta function of the coupling $g$: $$\beta(g^{(R)})=-\frac{(g^{(R)})^2}{16\pi^2}\cdot \left(\frac{11N-2n_f}{3}\right)$$
I try to find the 1-loop correction to the two point function of the quark field, to do so I have to take into account and calculate the following diagram (putting quark's field mass to zero):

eventually I get the following effective action:
$$\Gamma=-\int d^dk \bar q^P(k) q^P(-k) \left[1-\frac{g^2}{32\pi^2} \cdot \left(\frac{1}{\epsilon}-\ln(k^2)+C \right) \right]k_\mu\gamma^\mu$$ when k is the momentum, and by dimensional regularization $2\epsilon=4-d$, C is just a numerical value, and $P\in 1,2,...,n_f$ is the flavour index.

according to this effective action we define:
$$Z^{-1}_q=1-\frac{g^2}{32\pi^2} \cdot \left(\frac{1}{\epsilon}-\ln(\mu^2)+C \right)$$

So the calculation of the anomalous dimension is depended on $\beta(g^{(R)})$ but omitted because it have a coefficient with higher order value of g (taking all $O(g^3)$ to zero because it higher then 1-loop approximation).
the 1-loop anomalous dimension that I got is: $$\gamma_q(g^{(R)})=-\frac{(g^{(R)})^2}{32\pi^2}$$

But I wonder if the answer have to be:
$$\gamma_q(g^{(R)})=-n_f\frac{(g^{(R)})^2}{32\pi^2}$$
I claim that my previous answer is right because given a flavor P I can contract each flavor to itself and cannot contract with other flavors in order to build the loop correction we calculate earlier so we don`t need to multiply by $n_f$ the answer, moreover the definition of the anomalous dimension came from the Callan Symanzik equation for the n- correlator when the correlator for my opinion have to include n quarks with the same flavor (we can't contract different flavors) such that:
$$\left(n\gamma_q(g^{(R)})+\beta(g^{(R)})\frac{\partial}{\partial g^{(R)}}+\mu \frac{\partial}{\partial \mu}\right)G^{(R)}_{g^{R}(\mu)}(x_1,...,x_n)=0$$
when: $$G^{(R)}_{g^{R}(\mu)}(x_1,...,x_n)=\langle T[\bar q_P^{(R)}(x_1),q_P^{(R)}(x_2),..., \bar q_P^{(R)}(x_{n-1}),q_P^{(R)}(x_n)]\rangle=\mathcal N \int Dq D \bar q \bar D A q_P^{(R)}(x_1),q_P^{(R)}(x_2),..., \bar q_P^{(R)}(x_{n-1}),q_P^{(R)}(x_n)e^{iS[g^{(R)},A,q]}$$

Am I right?

 

[mark726]

I cannot definitively say whether your answer is right or not without further context and understanding of the specific theory and calculations you are working with. However, based on the information provided, it seems that your reasoning and approach are sound. It is important to carefully consider the relevant symmetry and constraints when calculating quantities in a theory, and it appears that you have done so in your derivation of the anomalous dimension. Additionally, the Callan-Symanzik equation does involve multiple quark fields with the same flavor, as you have correctly pointed out.
If you have any doubts about your answer, it may be helpful to consult with your colleagues or supervisor to discuss and verify your calculations. It is also important to carefully check any assumptions or approximations made in the calculation.