Phase Transition In $SU(N)\times U(1)$ Gauge Theory With Many Fundamental Bosons

TL;DR

This paper investigates the phase transition behavior of an $SU(N) imes U(1)$ gauge theory with many fundamental bosons by analyzing the renormalization group flow and identifying a new stable fixed point indicating a second order phase transition.

Contribution

It introduces a new stable fixed point in the RG flow for the gauge theory with many bosons, expanding the understanding of phase transitions in such models.

Findings

Discovery of a new stable fixed point for $M > M_{crit}$

Identification of a second order phase transition

Calculation of critical exponents in $\e$ and large-$M$ expansions

Abstract

Here we study the Renormalization group flow of $SU(N)\times U(1)$ gauge theory with $M$-fundamental bosons in $4-\epsilon$ dimension by calculating the beta functions. We found a new stable fixed point in the zero mass plane for $M>M_\text{crit}$ by expanding upto $O(\epsilon)$. This indicates a second order phase transition. We also calculated the critical exponents in both $\epsilon$ expansion and also in the large-$M$ expansion.

Figures

Equations

$$\mathcal{H}=\frac{J}{M}\sum _{⟨i,j⟩}{{\hat{S}}^{\beta }}_{\alpha }(i){{\hat{S}}^{\alpha }}_{\beta }(j)$$

$$b_{\alpha a}^{†}(i)b^{\alpha b}(i)={\delta }_a^b{n}_c,\text{ no sum on }i$$

$${\bar{b}}^{\alpha a†}(j){\bar{b}}_{\alpha b}(j)={\delta }_a^b{n}_c,\text{ no sum on }j$$

$${\hat{S}}_{\beta }^{\alpha }(i)=\sum _{a=1}^N{b}_{\beta a}^{†}(i)b^{\alpha a}(i),i\in A\text{ sublattice}$$

$${\hat{S}}_{\alpha }^{\beta }(j)=-\sum _{a=1}^N{\bar{b}}^{\beta a†}(j){\bar{b}}_{\alpha a}(j),j\in B\text{ sublattice}$$

$$\mathcal{Z}=\int \mathcal{D}Q\mathcal{D}b\mathcal{D}\bar{b}\mathcal{D}\lambda \exp \left[-{\int }_0^{\beta }\mathcal{L}d\tau \right]$$

$$\begin{gathered} \mathcal{L}=\sum _{i\in A}\left[b_{\alpha a}^{†}(i)\left({\delta }_b^a\frac{d}{d\tau }+i{\lambda }_b^a(i)\right)b^{\alpha b}(i)-i{\lambda }_a^a(i)n_c\right] \\ +\sum _{j\in B}\left[{\bar{b}}^{\alpha a†(j)}\left({\delta }_a^b\frac{d}{d\tau }+i{\lambda }_a^b(j)\right){\bar{b}}_{\beta b}(j)-i{\lambda }_a^a(j)n_c\right] \\ +\sum _{i,\hat{n}}\left[\frac{M}{J}\text{absolutevalue}{{Q_b^a}_{i,i+\hat{n}}}^2-\left({{Q^{\ast b}}_a}_{i,i+\hat{n}}{b}^{\alpha a}(i){\bar{b}}_{\alpha b}(i+\hat{n})+H.c.\right)\right] \end{gathered}$$

$$Q_{bi,i+\hat{n}}^a=\left[\bar{Q}{\delta }_c^a+q_c^a(i)\right]\exp {\left[i\hat{n}\cdot B(i)\right]}_b^c$$

$$i{\lambda }_b^a(i)=\bar{\lambda }{\delta }_b^a+i{B_b^a}_{\tau }(i),\text{for }i\in A$$

$$i{\lambda }_b^a(j)=\bar{\lambda }{\delta }_b^a-i{B_b^a}_{\tau }(j),\text{for }j\in B$$

$$S_{\text{eff}}=\int d^{d}r{\int }_0^{c\beta }d\tilde{\tau }\frac{a^{1-d}}{2\sqrt{d}}\left[\text{absolutevalue}({\partial }_{\mu }{\delta }_b^a-i{B}_{\mu b}^a{z}^{\alpha b})+\frac{{\Delta }^2}{c^2}\text{absolutevalue}{z^{\alpha a}}^2\right]$$

$$B_{b\mu }^a={\delta }_b^a{A}_{\mu }+W_{b\mu }^a$$

$$S[\psi,\vec{B},\vec{W}^{a}]=\int d^{3}x\bigg{[}\absolutevalue{(\partial_{\mu}-iyA_{\mu}-igT^{a}W_{\mu}^{a})\psi}^{2}\\ +\frac{1}{4}F^{\mu\nu}F_{\mu\nu}+\frac{1}{4}G^{a\mu\nu}G^{a}_{\mu\nu}+a\absolutevalue{\psi}^{2}+\frac{b}{2}\absolutevalue{\psi}^{4}\bigg{]}$$

$$F_{\mu \nu }$$

$$G_{\mu \nu }^a$$

$$a=\frac{a^{\prime }(T-T_c)}{T_c}$$

$$\text{absolutevalue}\psi =\pm \sqrt{-\frac{a}{b}}$$

$$\frac{T-T_c}{T_c}<\frac{1}{8{\pi }^2}\frac{b^2{T}_c^2}{a^{\prime }}.$$

$$T^a=\frac{1}{2}{\sigma }^a.$$

$$A_{\mu }$$

$$Z_{\mu }$$

$$\exp (-S(\psi )/T)=\int \mathcal{D}A\mathcal{D}Z\mathcal{D}{W}^i\exp [-S[\psi ,\vec{B},{\vec{W}}^a]/T].$$

$$\frac{dS}{dv}=2(vol)av+2(vol)b{v}^3+(g^2/4)⟨{W_{\mu }^i}^2⟩v+(g^2/4+y^2)⟨Z_{\mu }^2⟩v.$$

$$⟨{W_{\mu }^i}^2⟩=\frac{2(vol)}{{\pi }^2}\left[\Lambda -\frac{m_i\pi }{2}\right]$$

$$⟨{Z_{\mu }}^2⟩=\frac{2(vol)}{{\pi }^2}\left[\Lambda -\frac{m_Z\pi }{2}\right]$$

$$\frac{S}{vol}=\left[\left(a+\frac{3\Lambda }{2{\pi }^2}\right)v^2+\frac{b}{2}{v}^4-3\left(2g+\sqrt{g^2+4y^2}\frac{v^3}{4\sqrt{2}\pi }\right)\right]$$

$${\beta }_{{\alpha }_1}$$

$${\beta }_{{\alpha }_2}$$

$${\beta }_a$$

$$\begin{gathered} {\beta }_b=ϵb-\frac{b^2(NM+4)}{8{\pi }^2}-\frac{3{\alpha }_1^2}{4{\pi }^2}-\frac{3{\alpha }_2^2(N^3+N^2-4N+2)}{32{\pi }^2{N}^2} \\ -\frac{3{\alpha }_1{\alpha }_2}{{\pi }^{2}N}(N-1)+\frac{3b{\alpha }_1}{4{\pi }^2}+\frac{3b{\alpha }_2}{4{\pi }^2}\left(\frac{N^2-1}{2N}\right). \end{gathered}$$

$$\frac{1}{\nu }$$

$$\eta$$

$$M_{\text{crit}}=607.765+174.594N+106.058N^2$$

$$\eta$$

$$\nu$$

$$\begin{gathered} diagrama1 \\ =b\int \frac{d^{d}l}{(2\pi {)}^d}\frac{{\delta }_{kl}}{l^2+a}({\delta }_{ik}{\delta }_{jl}+{\delta }_{il}{\delta }_{jk}) \\ =b(N+1)delt{a}_{ij}\int \frac{d^{d}l}{(2\pi {)}^d}\frac{1}{l^2+a} \\ =\frac{b(N+1){\delta }_{ij}}{(4\pi {)}^{2-ϵ/2}}\frac{\Gamma (1-d/2)}{\Gamma (1)}{\left(\frac{1}{a}\right)}^{1-d/2} \\ =-\frac{ab(N+1){\delta }_{ij}}{8{\pi }^2}\frac{1}{ϵ}+O(ϵ) \end{gathered}$$

Full text

I Introduction

Phase transitions in gauge theories are very interesting because gauge theories appear as effective theories in many physical problems. Historically, in particle physics gauge theories have been studied in detail because of their potential application to phenomenology. More recently, there are several examples of emergent gauge degrees of freedom in condensed matter physicssachdev ; senthil ; senthil2 ; lammert ; lee ; huo ; blasone . Phase transitions in those theories hold very rich physics. We will be concerned solely with continuous gauge symmetries.

The simplest example of a phase transition in a continuous gauge theory is in $U(1)$ gauge theory with a single boson. This is the Ginzburg-Landau theory of superconductor-insulator transitiongor . Fluctuations around mean field were first studied by Coleman and Weinbergweinberg , who found that in $d=4$ the theory undergoes a first-order phase transition. This conclusion was verified independently by Halperin, Lubensky and Ma(HLM)HaLuMa , who also carried out an $\epsilon$ expansion in $d=4-\epsilon$ dimensions to first order in $\epsilon$. They also showed $d=3$ by integrating out the gauge degrees of freedom that the transition becomes weakly first order. Generalizing to $M$ complex boson fields they found for $M>M_{\text{crit}}=182.95$ two more fixed points appear, as shown in Fig. 1. It is seen that for $M>M_{\text{crit}}$ there is a stable fixed point in the zero mass plane indicating a second order phase transition. Halperin, Lubensky, and Ma also calculated the critical exponents for the transition in the $\epsilon$ expansion and in fixed dimension $d=3$ in the large-$M$ approximation.

The case of an $SU(2)$ gauge field coupled to $M$ fundamental bosons has been studied more recently by Arnold and Yaffe.arnold . They found a picture very similar Fig. 1 in the $\epsilon$ expansion. To $O(\epsilon)$ they found that for $M>M_{\text{crit}}=359$ there are two charged fixed points. One of them is attractive in the $b-g^{2}$ plane, again indicating a second-order phase transition. The $SU(2)\times U(1)$ case is known as the electroweak phase transition.

It is known from several numerical studieslaine ; laine2 ; laine3 in lattice gauge theory that in the case of $M=1$ there exists a critical ratio of the couplings such that for $b/g^{2}>C$ there is no phase transition at all and for $b/g^{2}<C$ the transition is first order. The second order phase transition exists only if $b/g^{2}=C$. The reason is that for $b/g^{2}>C$ no symmetry is broken in the $SU(2)$ transition.

But this picture changes in a very significant way when more than one species/flavours of boson are introduced (these transform as higher representations under the gauge group). In that case as Fradkin and Shenkerfradkin show in lattice gauge theory, a phase transition does occur for all the values of ratio of couplings. In a gauge theory with a non-trivial center, the center survives for higher representations in unitary gauge if the boson is in the adjoint representation. Introducing $M$ species of bosons leads to a global $U(M)$ symmetryarnold2 . In the unitary gauge the $SU(N)$ gauge symmetry breaks down but this $U(M)$ symmetry survives. The phase transition corresponds to spontaneous breaking of this $U(M)$ symmetry.

In this paper we study $SU(N)\times U(1)$ theory with $M$ flavours of bosons. Such a theory arises in a completely different context, the study of $SU(M)$ antiferromagnets on a square latticesachdev .

The Hamiltonian of this model is,

[TABLE]

Where ${\hat{S}^{\beta}}_{\alpha}(i)$ are the generators of $SU(M)$ and $\langle i,j\rangle$ represents nearest neighbour sum on this bipartite square lattice. The representation of the spins sitting in two sublattices ($A$ and $B$) can be described using the two integers describing the Young tableau, $n_{c}$ and $M$. The representation of the spins are described in Fig. 2. For the $A$ sublattice the number of boxes in the column of the young tableau is $N$ where for the $B$ sub-lattice the boxes in the column is $M-N$. The number of boxes in every row is fixed to be $n_{c}$.

Now, we introduce boson (Schwinger bosonauerbach ; auerbach2 ) operators $b^{\alpha a}(i)$ for sublattice $A$ and $\bar{b}_{\alpha a}(j)$ on each sublattice $B$ with the constraint,

[TABLE]

And the spin operators will be,

[TABLE]

In the functional integral representation of the partition function of the $\mathcal{H}$ can be written as,

[TABLE]

Where,

[TABLE]

Where $\lambda(i)$ is the lagrange multiplier which fixes the number of bosons per site to be $n_{c}$. This Lagrangian is local $U(N)$ invariant. The field ${Q^{a}_{b}}_{i,i+\hat{n}}$ is the Hubbard-Stratonovich field. Now we can do a mean field approximation of this theory and in that mean field approximation. In which $\lambda$ and $Q$ becomes constant. The fluctuation of this these fields around that mean field value will be,

[TABLE]

Where $B_{\mu}$ is the $U(N)$ gauge field and $q$ is amplitude fluctuation. One can do a long wavelength approximation to this to get action,

[TABLE]

The gauge field can be broken into a $U(1)$ and a $SU(N)$ part(trace and trace less part),

[TABLE]

A gradient expansion of this will give us the $SU(N)\times U(1)$ theory with $M$ flavour of bosons.

The phases $SU(M)$ antiferromagnet are known for $N=1$. We want to check how the order of the phase transition depends on the number of flavor for the $SU(N)\times U(1)$. We want to check this in two ways. First we can try to integrate out the gauge field which we will do for $M=1$ and $N=2$ to show that for a single flavor in fundamental representation there is no second order transition at least for $N=2$. Next, as we want to study the the theory that arises from the $SU(M)$ anti-ferromagnets. We will study the RG flow of this theory for arbitrary $M$ and $N$ and the fixed point structure of the theory.

The phase transition in $SU(M)$ magnets(Heisenberg model) has been studied numerically before. Kawashima and Tanabenaoki found evidence of emergent $U(1)$ symmetry of the ground state space of the $SU(M)$ Heisenberg model with the fundamental representation. Beach et al.beach developed a quantum Monte Carlo algorithm to simulate this model for continuous $M$ in total singlet basis and found a phase transition between Neél and VBC columnar phase occurring at $M_{c}=4.57(5)$. They also identified the phase transition to be second order with critical exponents, $z=1$ and $\beta/\nu=0.81(3)$.

II Effect of gauge fluctuations

First we will try to integrate out the gauge field to see what happens for $M=1$(in the Unitary gauge) to the action defined as,

[TABLE]

Where,

[TABLE]

and as usual,

[TABLE]

This in pure $U(1)$ case leads to a weak-first order phase transition as the gauge field around mean field approximation of the order parameter picks up a mass(in other words this will give us Meissner effect with a penetration depth defined by the mass).HaLuMa

The $\psi$ field has $N$ components. Now, the minimum of this action is when all the fluctuations of fields are zero and $\absolutevalue{\psi}=const$. This value of the constant is well known, i.e.

[TABLE]

Now for $N=2$ one can choose a gauge to make, $\psi_{1}=0$ and $\psi_{2}=\sqrt{-a/b}$.

Now the question comes of the Ginzburg-Criteria which we can find after a few calculations is,

[TABLE]

In the case of superconductor theory we actually know the microscopic theory(BCS theory) and from there one can exactly find these coefficients $a,b$ in terms of microscopic parameters.gor This ensures that the Ginzburg Criteria is met and we can actually use constant Mean Field solution. In our case we don’t know the microscopic theory but for now we will assume that the $\psi$(order parameter) fluctuation is very small and we can use the mean field value of the field.

Next we need to consider the case where we choose a specific gauge and want to calculate the effect of the gauge field fluctuations. Again we will do it for $N=2$. We choose the gauge such that $\psi_{1}=0$ and $\psi_{2}=v$. For $N=2$ the generators are,

[TABLE]

We find that the mass matrix of the fields is not diagonalized. After the mass matrix diagonalization we find that there will be $3$ gauge fields with mass and one massless gauge field(all U(1)) but interacting with each other. The massive fields are $W^{1}_{\mu},W^{2}_{\mu}$ with mass square, $m_{1}^{2}=m_{2}^{2}=(1/2)g^{2}v^{2}$ and $Z_{\mu}$ with mass square, $m_{Z}^{2}=v^{2}(g^{2}+4y^{2})/2$. We will also have a massless field $A_{\mu}$. The definition of $A_{\mu}$ and $z_{\mu}$ is,

[TABLE]

where, $\sin\theta_{W}=g/\sqrt{g^{2}+4y^{2}}$ and $\cos\theta_{W}=2y/\sqrt{g^{2}+4y^{2}}$.

As mentioned before we want to calculate,

[TABLE]

For our gauge we find that,

[TABLE]

These averages can be calculated to be, for small $v$

[TABLE]

Putting all this to 19 and integrating over $v$ we get,

[TABLE]

This introduces a first order phase transition exactly like in $U(1)$-case.HaLuMa . From this one can calculate the size of the phase transition etc.

III Beta functions and fixed points

The more general way to find $\beta$-function is to carry out RG calculations in $d=4-\epsilon$ and for general $N$ using dimensional regularization. we define here for simplicity of the calculations $\alpha_{1}=y^{2}$ and $\alpha_{2}=g^{2}$.pes ; sred ; igor

Thus the beta functions are,

[TABLE]

[TABLE]

One can easily see from the structure of the $\beta$-function that for $N=1$, $\beta_{\alpha_{1}},\beta_{a},\beta_{b}$ completely decouples from the $\alpha_{2}$ and as one can check that it has the correct structure for $U(1)$ gauge theory with multiple scalar.HaLuMa ; igor Next one can look into the fixed point structure of this theory. There are $8$ possible fixed points of these $\beta$-functions. Two of them are the old Gaussian and the Wilson-Fisher fixed point and fixed points where there is no $SU(N)$ or $U(1)$ charge.HaLuMa As before the $U(1)$-charged fixed points do not exist for $NM<182.952$. There are four more fixed points that arise in the theory and one of them is critical as that one is completely stable in all direction except for the temperature(mass) direction. This point is doubly charged. But this fixed point does not exist for $M<M_{\text{crit}}$. This $M_{\text{crit}}$ is different for different values of $N$. For example for $N=2$, $M_{\text{crit}}=1277.47$. There are two singly charged ($SU(N)$ charge) fixed points also. This $SU(N)$ charged fixed points also have some critical value of $M$ as a function of $N$. As previously calculated for $N=2$ this critical value is $359$.arnold

IV Critical exponents

The critical exponents of this phase transition can be easily calculated in the regular way and we can see that $\nu\rightarrow 1$ and $\eta\rightarrow 0$ as $M\rightarrow\infty$ for $\epsilon=1$. In terms of fixed point value of the parameters($a^{*}=0,b^{*},\alpha_{1}^{*},\alpha_{2}^{*}$)kiom

[TABLE]

As we have seen these beta functions has a very interesting structure of fixed points (we have $M>N$). There are $8$ fixed points but not all of them exists at every value of $M$ and $N$. The $M$ and $N$ comes from the microscopic theory.sachdev . For $N=1$ the theory contains only the abelian gauge field. The question one needs to ask is for what values of $N$ and $M$ there exist a doubly charged critical point. We can easily find out the relation between $N$ and $M_{\text{cric}}$. That relation is quadratic,

[TABLE]

The Region on $N-M$ plane for which the theory has a critical point is in shaded region of Fig. 5

This critical exponents can also be calculated also in fixed dimension($d=3$) in the large $M$ limit. Where the coupling constants are $b\sim O(1/M),y\sim O(1/\sqrt{M}),g\sim O(1/\sqrt{M})$. This method is similar to what is described by Mama . From this calculation we get for $M$- complex fields in fundamental representation of $SU(N)$,

[TABLE]

This result matches with the already known results for $N=1$.HaLuMa ; sakhi

V Discussion and Conclusion

From this analysis we found that for $SU(M)$ anti-ferromagnets there is a temperature driven phase transition for a very large $M$ compared to $N$(representation of the spin). This critical value $M_{\text{crit}}$ can be calculated for as a function of $N$.

The critical exponents of this second order phase transition are calculated in both $\epsilon$ expansion and in large-$M$ expansion. The next question one should ask is what are the phases that lie on the either side of the phase transition.

It has already been discovered numerically that for $M=1$ there is no electro-weak phase transition at all for large value of $b/g^{2}$.laine ; laine2 ; laine3 For large $M$ there is a phase transition. This phase transition corresponds to the breaking of the left over symmetry($U(M)$ flavour symmetry).fradkin ; arnold2 The question still remains that what will be order parameter in that limit. It is known that those phases are connected to conventional Higgs and confinement phase.fradkin One needs to study the lattice model rather than the coarse-grained theory to identify the phases.

All this analysis has been done when there is no topological term. The critical exponents can also be calculated if there is a topological term. The $U(1)$ case has been calculated recentlysakhi but $SU(N)\times U(1)$ case is not known. That can provide a better understanding of the topological phases in the actual lattice model for $M>1$ which is not known though the $M=1$ case has been studied.sachdev

I plan to study in future the effect of the topological term in the Lagrangian and also the phases in the case of $M>1$ case on the lattice. There is also the case of non-bipartite lattice one may consider to study.

VI Acknowledgement

I thank Dr. Ganpathy Murthy for his help and valuable discussion to understand a number of shuttle issues about this problem. I am very thankful to Dr. Ribhu Kaul for introducing me to this area of Physics. I am very grateful to Dr. Michael Eides for his very helpful comments and discussions. I regard Dr. Peter Arnold for very helpful discussions.

I thank NSF(DMR-1306897) and University of Kentucky for supporting this project.

Appendix A Calculation of beta function in dimensional regularization

We have to calculate the correction to the boson field propagator.

The above diagrams we need to calculate to find out the correction to the boson propagator. Where the propagator definitions are,

One can calculate these diagrams easily to get in order $1/\epsilon$,

[TABLE]

[TABLE]

[TABLE]

The $diagram~{}a4,diagram~{}a5$ can be calculated very simply and as the gauge theory is massless thus diagrams contribute zero. The results of these diagrams are,

[TABLE]

From this we can easily calculate the $Z$ values in the normalization assred ,

[TABLE]

The diagrams that will contribute to the $U(1)$ gauge propagator,

We can calculate the diagram to be,

[TABLE]

[TABLE]

As we can see these diagram add to give a transverse field as expected and also,

[TABLE]

Similarly one need to calculate the correction to the non-abelian gauge propagator.

We similarly can calculate in the Feynman gauge pes to calculate these diagrams,

[TABLE]

[TABLE]

The calculation for the $diagram~{}c3,diagram~{}c4,diagram~{}c5$ is straight forward in transverse gauge,pes

[TABLE]

This gives,

[TABLE]

Next is the correction to the four boson vertex,

This diagrams can be calculated as,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

Next we need to look for 3-point $U(1)$-boson-boson vertex,

The value of the diagrams will be,

[TABLE]

[TABLE]

[TABLE]

Thus, we can write down $Z_{1}$ as,

[TABLE]

Next we need to look for 3-point $SU(N)$-boson-boson vertex,

The evaluation of this diagrams will be,

[TABLE]

[TABLE]

$diagram~{}f3$ is zero similarly to the $U(1)$ case.

[TABLE]

Similarly $diagram~{}f5$ is zero for transverse gauge.

[TABLE]

Now, as before we can calculate $Z_{2}$ here as,

[TABLE]

Using,

[TABLE]

One can calculate the beta functions.

Appendix B Critical exponents in large $M$ limit

Next we need to calculate the critical exponents in the philosophy of ma . I will also discuss the calculation done by ma but only in $d=3$.

We will calculate $\eta$ and $\gamma$ directly then from that using scaling relation one can calculate $\nu=\gamma/(2-\eta)$. For this lets start with when we have no gauge field. Then the action looks like,

[TABLE]

Now, we can introduce a hubbard-stratonovich field $\chi$ to reduce the four point vertex into a a three point vertex, i.e.

[TABLE]

Thus the propagator of the $\chi$ field is $b$. Now we want to calculate the correction to the $\psi$ propagator to $O(1/M)$.

We can add as many boson loop as we need in the hubbard propagator. Thus, adding all those,

[TABLE]

Where, $\pi$ is called the fundamental bubble,

[TABLE]

Thus the correction to the propagator is,

[TABLE]

B.1 Critical exponent $\eta$

Now, the critical exponent $\eta$ is defined as(in mass zero limit, the case here),

[TABLE]

Thus, we need to calculate

[TABLE]

From this we need to find the coefficient of $-k^{2}\log(k)$, which can be done easily as only small $l$ region that contributes and $\pi(l)$ diverges for small $l$ and we can neglect $1$ w.r.t. $\pi(l)$ and found to be

[TABLE]

Next if we introduce a $U(1)$ gauge field then we can see the following diagrams contribute to $O(1/M)$,

Lets concentrate on the mixed diagram,

This mixed term goes to zero very simply. Thus we just need to calculate the correction to the $U(1)$ propagator,

[TABLE]

Thus the corrected $U(1)$ Propagator will be,

[TABLE]

And we will represent the as double line diagram,

Now, we want to collect the correction to the $\psi$ propagator upto $O(1/M)$,

[TABLE]

Again we find the $-k^{2}\log k$ coefficient,

[TABLE]

Next we need to introduce $SU(N)$ gauge field. Again the mixed diagrams will cancel. Now the correction to the $SU(N)$ propagator will be,

Now it can be easily checked that for self interaction of the gauge field is suppressed by $O(1/M)$ thus we can drop them from the calculation,

[TABLE]

We can now define the exact propagator as double line,

Now, we want to collect the correction to the $\psi$ propagator upto $O(1/M)$,

This calculation is exactly like $U(1)$ case which gives,

[TABLE]

Which gives,

[TABLE]

B.2 calculating $\gamma$

Next as in ma we will switch on the mass. Then the fundamental bubble becomes,

[TABLE]

Now, to calculate $\gamma$ we have above the critical temperature(massive),

[TABLE]

Thus the leading diagram contributing to this is,

And this contribution is,

[TABLE]

Thus from leading calculation we get,

[TABLE]

Thus next order the diagram g19 does not contribute and can be shown easily.

The diagram g20 will contribute to the calculation of $\gamma$

And that gives,

[TABLE]

[TABLE]

All we need from this is to find the $\log[a]$ term of that integral. Defining the intregral to be $L$ we get,

[TABLE]

Now expanding and integrating back we get,

[TABLE]

[TABLE]

Now we want to introduce $U(1)$ gauge field then the fundamental bubble is,

Now,

[TABLE]

Thus,

[TABLE]

Now, similar to the case of the Hubbard-Stratonovich field the digrams that will contribute are diagram g23 and g24.

Now from this one can calculate,

[TABLE]

Where $\Sigma_{c}(a,l)$ is defined as sum of the diagram g25 and g26.

This gives,

[TABLE]

This gives,

[TABLE]

[TABLE]

In case of $SU(N)$ gauge field the fundamental bubble comes from diagram g27,g28

Then calculation similar to $U(1)$ case gives.

[TABLE]

Then diagram g31,g32 gives the contribution to $\gamma$,

Now,

[TABLE]

Where $\Sigma_{c}(a,l)$ is diagram g29+g30.

Thus similar to $U(1)$ case,

[TABLE]

[TABLE]

B.3 Calculation of $\nu$

From scaling law we have,

[TABLE]

Thus here we get,

[TABLE]