Considering $SU(N)$ gauge theory with $n_f$ massless quarks

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

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): enter image description here

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 ommited 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?

  • $\begingroup$ Hi! I have changed the < and > to \langle and \rangle. If this isn't what you were intending, please change it back – both =< and >= seemed unfamiliar to me though. $\endgroup$
    – jng224
    Aug 4, 2021 at 21:23
  • $\begingroup$ Great thanks!! Aprreciate $\endgroup$ Aug 5, 2021 at 11:09


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