# How to derive the effective action or Vertex generating functional of complex scalar field?

I am studying-self QFT. Recently, I am studying and trying to follow the calculation of the Ginzburg-Landau free energy functional of superconductor in this paper:http://arxiv.org/abs/hep-ph/0108256. But I meet the problem in deriving the coefficient of 4th order term of the order parameter.Anyone knew it,please help me.

In this paper,the author seem the Ginzburg-Landau functional as the vertex generating functional or effective action of order parameter complex field. So, the coefficient of the 2nd and 4th order term of G-L functional is the 2-point and 4-point vertex function,respectively.In part.V, for the quartic term coefficient, I had derived the connected diagram generating functional Eq.5.2 as $$W(J,{J}^{\dagger})=-\frac{{k}_{B}T}{2}\sum_{P}{\frac{tr[{J}^{\dagger}(P)J(P)]}{ {\upsilon}_{n}^{2}+{(p-\mu)}^{2}}}+\frac{{k}_{B}T}{4}\sum_{P}{\frac {{tr[{J}^{\dagger}(P)J(P)]}^{2}}{{({\upsilon}_{n}^{2}+{(p-\mu)}^{2})}^{2}}}$$ with $k_B$ is Boltzmann constant, $T$ is temperature, $J$ and $J^{\dagger}$ are the source of order parameter $B^{\dagger}$ and $B$. $J$ and $J^{\dagger}$ are matrix, so there is a trace. $\nu_{n}$ and $p$ is matsubara frequency and momentum, respectively.

The $W(J,{J}^{\dagger})$ have two sources because the field is complex. But I don't know How to take the Legendre transform to effective action with two source, I only found out the explicit process for real scalar field with one source. So, I gauss that it should be satisfy conditions $$\frac{\delta W}{\delta J }={B}^{\dagger}$$ and $$\frac{\delta W}{\delta J^{\dagger}}={B}$$ same time. This form a system of equations and lead to two groups solutions of $J$ and $J^{\dagger}$ in terms of $B$ and $B^{\dagger}$(used The Mathematica). Then I substituted the $J$ and $J^{\dagger}$ with these solutions and form the effective action(Eq.2.6 in this paper) as $$\Gamma(B,B^{\dagger})=W(J,{J}^{\dagger})-\sum_{P}{{J}^{\dagger}(P)B+{B}^{\dagger}J(P)}$$

But, unfortunately, I can not get the result in Eq.5.3 when I take 4-order derivative of the resulting $\Gamma(B,B^{\dagger})$. I want know what is the correct process in this case and where can I find the explicit calculating process of the functional Legendre transform to the effective action with two sources.

PS: I also want to know Is there the technique to calculate the vertex function without calculating effective action and where to find the explicit process?

Thanks

• why nobody give me some recommend？ – alxandernashzhang Sep 16 '15 at 23:59
• I only want to know to how take the Legendre transform when we have two sources – alxandernashzhang Sep 17 '15 at 0:13