Finite Temperature Corrections to Tachyon Mass in Intersecting D-Branes

We continue with the analysis of finite temperature corrections to the Tachyon mass in intersecting branes which was initiated in arxiv:1403.0389. In this paper we extend the computation to the case of intersecting D3-branes by considering a setup of two intersecting branes in flat-space background. A holographic model dual to BCS superconductor consisting of intersecting D8-branes in D4-brane background was proposed in arxiv:1104.2843. The background considered here is a simplified configuration of this dual model. We compute the one-loop Tachyon amplitude in the Yang-Mills approximation and show that the result is finite. Analyzing the amplitudes further we numerically compute the transition temperature at which the Tachyon becomes massless. The analytic expressions for the one-loop amplitudes obtained here reduce to those for intersecting D1-branes obtained in arxiv:1403.0389 as well as those for intersecting D2-branes.

in the temporal gauge (A a 0 = 0), the wave-functions that diagonalize the quadratic part of the Lagrangian consist of Hermite Polynomials instead of exponential plane-waves which we have in the usual case. The corresponding momentum modes are discrete and are denoted by n. The wave functions have been derived in [1] for the D1 branes. These functions have been adapted for the present case of D3 branes with some modifications. The additional complication in the present case is the presence of extra momentum degrees of freedom coming from the directions transverse to the plane of intersection. There are two such extra momentum directions relative to the D1 brane case. We call these momentum modes k = (k 2 , k 3 ).
The supersymmetric Yang-Mills theory on the D3 brane is finite. Although supersymmetry is completely broken for the intersecting branes, the ultraviolet property of the theory on the intersecting branes is unaffected. The theory is still finite in the ultraviolet. We shall show analytically the ultraviolet finiteness for all the one-loop two-point amplitudes computed in the paper. Specifically we show that for large n, independent of the value of k, the contributions from the bosons and fermions in the loop cancel. Similar cancellation has also been shown for the large k for the two-point amplitude for the "massless" fields.
The one-loop amplitude for the Tachyon involves fields propagating in the loop that are massless at treelevel. In order to deal with the resulting infrared divergences arising out of these massless fields we first find the one-loop correction to the masses for these fields that are massless at tree-level at finite temperature.
These masses are then used to modify the propagators for the "massless" fields. The resulting one-loop Tachyon amplitude is thus infrared finite. The analytic expressions obtained here reduce to those for the intersecting D1 and D2 branes once the extra transverse momenta/momentum is set to zero. The final evaluation of the temperature corrected masses for the "massless" as well as the Tachyon field could not however be performed analytically. We had to resort to numerics due to the complicated nature of the expressions involved. This paper is organized as follows. In the following section (2) we analyze the quadratic part of the Yang-Mills Lagrangian with the background Φ 3 1 = qx. We derive the various wave-functions both for the bosons and fermions and then write out the the Lagrangian in terms of momentum modes. The propagators for the various fields can then be read out. In section (3) we compute the one-loop two-point Tachyon amplitude. In the subsection (3.1.1) we analytically show the finiteness of the Tachyon amplitude for large n (ultraviolet) and for all values of the momenta k. The one loop amplitude for the massless fields namely Φ 3 I (I = 1, .., 6) and A 3 µ (µ = 1, 2, 3) have been worked out in section (4). These have been done in subsections (4.1), (4.2) for fields Φ 3 1 and Φ 3 2 respectively. The contributions from the bosons and the fermions for fields A 3 µ have been worked out separately in subsections (4.3.1), (4.3.2), (4.3.3) and (4.3.4). In section (5) we show the numerical results for the computation of all the two point amplitudes including transition temperature at which the Tachyon becomes massless. We conclude the paper with some outlook in section (6). Various details have been worked out in the appendices. We summarize our notations in appendix A. We list the various parameters, variables used in the paper in Table 1 of this appendix. We review the dimensional reduction of D = 10 super Yang-Mills to D = 4 in appendix B. The propagators and vertices for the Tachyon and various massless fields have been worked out in appendices C and E respectively. In appendix D we give details of the one-loop tachyon amplitude incorporating the corrected propagators for the fields with tree-level mass zero.

.1 Bosons
In the Yang-Mills approximation the intersecting brane configuration with one angle is obtained by putting the background value of Φ 3 1 equal to qx. In this background we first write down the quadratic part of the bosonic action in the temporal gauge A a 0 = 0. Various fields decouple at this quadratic level. We first write down below the various decoupled parts of the resulting from the action (B.183), q 2 x 2 (Φ b J ) 2 (I = 2, 3; a = 1, 2) (J = 1, 2, 3; b = 1, 2, 3) (2.3) Now defining the quadratic bosonic part of the action can be written as, can be obtained from O B by replacing q in O B by −q. We have identified z 1 = x. In the following we shall identify z 0 = −iτ and (z 2 , z 3 ) ≡ y.
The eigenvalues corresponding to ζ n (x) are (2n − 1)q and those corresponding toζ n (x) are all zero. Thus the spectrum in the latter case is completely degenerate. In the non-zero eigenvalue sector we do not have normalizable eigenfunctions corresponding to n = 1. However unlike this sector, in the zero eigenvalue sector we have normalizable eigenfunction for n = 1, which is simply H 1 ( √ qx) but there is no normalizable eigenfunctions for n = 0 in this sector.
L(A 3 µ , Φ I ,Φ J ) given by (2.3) yields decoupled eigenvalue equations for the scalar fields. The scalar fields with gauge components (a, b = 1, 2) can be expanded using the basis of harmonic oscillator wavefunctions, . Thus for example, The scalar fields with the gauge component (a, b = 3) and the gauge fields A 3 i can be expanded using the basis for plane wave as We can thus write down the third term in the action (2.5) in terms of the momentum modes as, Here (I = 2, 3; a = 1, 2), (J = 1, 2, 3; b = 1, 2, ), (i, j = 1, 2, 3) and k 2 = (ω 2 m + k 2 x + k 2 ).

Fermions
The quadratic terms involving fermions in the action (B.184) are given by The terms with gauge indices 1 and 2 decouple into four different sets. Since α 1 is off-diagonal these sets are (λ 1 1 , λ 2 4 ), (λ 1 2 , λ 2 3 ), (λ 1 3 , λ 2 2 ), (λ 1 4 , λ 2 1 ). Defining, The Lagrangian (2.30) can thus be written as where the operators O F andÕ F are given by, where i = 0, 2, 3. The eigenfunctions of the matrix operators have been obtained in [1]. Adopting the same notation as in [1] respectively. Here, We now list some important relations satisfied by the eigenfunctions We can now write down the mode expansions for the fermions where θ i are four component fermions. Further The quadratic action in terms of the momentum modes is then 3 One-loop two-point tachyon amplitude In this section we compute the two-point amplitude for the C(m ′′ , n ′′ , k ′′ ) fields. The required propagators and the vertices are worked out in the appendix (C). Using the four point vertices we first write down the contributions from bosons in the loop involving four-point and three-point vertices separately as shown in Figures 1 and 2. This is given by x , k ′′ ) and (ωm′′ ,k ′′ x ,k ′′ ). In the above diagrams i = 2, 3.
We now write down the contribution from fermions in the loop. The corresponding Feynman diagram is shown in Figure 3.
We will now be interested in evaluating the two point function with the external momenta set to zero.
(ω m ′′ =ω m ′′ = k ′′ =k ′′ = n ′′ =ñ ′′ = 0). Evaluating the trace in (3.65) the amplitude reduces to where, Now define the sum over the fermionic frequencies as, Then after performing the sum over the Matsubara frequencies the amplitude can be written as

Ultraviolet finiteness
In this section we show that the one-loop amplitude is finite in the UV (large n). Finiteness of the amplitudes in the UV is expected as the underlying N = 4 SYM theory is finite. In the present case supersymmetry is only broken by the background. Nevertheless we demonstrate this finiteness explicitly by showing the cancellation of contributions from bosons and fermions in the loop in the UV, which in this case is the large n limit. This exercise is particularly useful in order to ensure that the counting of the modes in the loops has been done correctly. In the following we proceed along a slightly alternate route to demonstrate the UV finiteness than that was followed in [1] for the case of intersecting D1 branes.
We first note that the factors coming from the three-point vertices are of the form The above asymptotic expression shows that the leading large n contribution from the l integrals come from the regionl 2 = 2n. We will thus, in all the contributions obtained above, replacel 2 by 2n in the propagators. This leaves us with l integrals only over the vertices. We can now perform these integrals over l separately noting that To illustrate this further, consider the integral where the function I(l 2 , ω, |k| 2 ) contains propagators and other polynomials inl 2 in the numerator.
According to the above in the large n limit the leading value of the integral (3.72) will be given by 2n I(2n, ω, |k| 2 ). As a first example let us take I(l 2 , ω, |k| 2 ) =l 2 . Performing the integral (3.72) directly and then taking the large n limit we get The large n leading term in the expression (3.73) can also obtained as described above.
As a second example let us consider the integral that involves both the sum over n as well as the integral over l (which appears in many of the expressions for the two-point function). A typical such integral is of the form, We first perform the integral over l in (3.74). The result in the large n limit reads (for simplicity we have set q = 1) where we have assumed 2nq >> ω 2 m + |k| 2 . Further terms with odd powers of ω m have been dropped. This is exactly as we had proceeded in [1]. Following the other route, we have The leading term in the expressions (3.75) and (3.76) match when we assume 2nq >> ω 2 m + |k| 2 . The difference in the sub-leading pieces is expected as we have restricted ourselves to leading contribution as coming froml 2 = 2n in the second approach.
Although this expansions with the assumption 2nq >> ω 2 m + |k| 2 is needed to show the agreement of the resulting leading behaviours in the two approaches it turns out that it will not be necessary in the ultimate demonstration of finiteness of the full two-point amplitude at large n. In other words in the following we work with the full expression (3.76) and the assumption that 2nq >> ω 2 m + |k| 2 is not invoked. With these observations we proceed to write down the asymptotic contributions from each of the terms For the terms containing four-point vertices (Feynman diagrams in Figure 1) we can directly take the n → ∞ limit. Using the following asymptotic forms worked out in [1] we can now write, in the expressions of (IV ), (V ) and (V I) and doing the l integral as elaborated above, we get (IV ) = (III) and We now write down the asymptotic forms of the contributions (V II) − (XI) corresponding to the diagrams in Figure 2. Here we implement the steps described at the beginning of this section.
Adding all the contributions (I) − (XI) we get Now following the above observations it is easy to see that that the large n limit of (3.66) is exactly same as (3.86) coming from fermions in the loop but with opposite sign. Thus the one loop amplitude is finite in the UV (large n).
We should note that the finiteness in the large n limit as shown here holds irrespective of the values of ω 2 m and |k| 2 . That is for every fixed value of ω 2 m and |k| 2 the amplitudes are finite in the large n limit. It would similarly be useful to give an analytic proof of the finiteness of the amplitudes for fixed values of ω 2 m and 2nq but large |k|. Unfortunately we are not able to demonstrate this analytically as the closed form expressions (in n) for the integrals (3.47,3.48,3.49) that involves product of four Hermite polynomials could not be obtained. Such finiteness of the amplitudes for the massless fields however can be shown both for large n and fixed |k| as well as large |k| and fixed n which we shall see in the following sections.

Infrared issues
Some of the contributions to the two point amplitude (I − XI) diverges in the infrared. These infrared divergences occur from two sources: (i) There are poles of the from of 1/ω 2 m . These poles are artifacts of the A a 0 = 0 gauge and one should use prescriptions to remove them. See for example [40]- [43] and references therein. In our case this means that we shall simply drop terms that are proportional to m There are also genuine infrared divergences due to tree-level "massless" modes propagating in the loop.
These modes are the fields with the gauge index a = 3, namely Φ 3 J /Φ 3 J (J = 1, 2, 3) and A 3 i (i = 1, 2, 3). This is because these fields do not couple to the background value of the field Φ 3 1 , and hence the tree-level spectrum for the fields with gauge index equal to 3 is not gapped unlike the fields with gauge indices equal to 1, 2. To ultimately get a finite answer for the one loop amplitude we follow the procedure implemented in [1]. We first find the one-loop correction to the propagators for these fields and then use the corrected propagators to evaluate the one-loop two-point tachyon amplitude. The corrected propagator will involve one-loop corrections to the zero tree-level masses for the fields Φ 3 I , A 3 µ . These corrections will be computed in the next section.
We now give some details on how to incorporate the one-loop masses in the propagators for the tree-level "massless" modes. The quadratic action for the Φ 3 I /Φ 3 I / fields (see eqn 2.28) including the mass terms is written as, Here (J = 1, 2, 3) and k 2 = (ω 2 m + k 2 x + k 2 ). Due to the remaining SO(5) symmetry, except for m 2 are all equal. We thus write down the modified propagators as: For the A 3 i fields writing down the modified propagator requires a bit more work. The one-loop correction breaks the SO(3) in (2.28) to SO (2). As a result the one-loop mass for A 3 1 (m 1 ) field is different from those of A 3 2 and A 3 3 fields which are equal (say m 2 ). Introducing a vector u i ≡ (1, 0, 0), we can write down the quadratic part of the action as Here (i, j = 1, 2, 3) and k 2 = (ω 2 m +k 2 x +k 2 ). For the one-loop amplitude there are momentum dependent corrections as well. Correspondingly the termÃ 3 j also arises in the one-loop effective action (3.89). The correction coefficients a, b and c are different due to the breaking of SO(3) invariance. In the infrared limit however only the mass terms in (3.89) provide the necessary regulation. Thus for simplicity we shall work with only the non-zero masses m 1 and m 2 . Following these observations the corrected propagator takes the following form (see appendix D for further details) where, the functions A, B, C and D are given by With these modifications we can re-write the corresponding modified expressions in (I)-(XI). For the Φ 3 I fields the mass terms in the propagators in the loop are easy to implement. However for the A 3 i some more work is needed. These expressions are listed in appendix D. The numerical results incorporating the above are analyzed in section 5.

Two point amplitude for massless fields
In the following sections we shall compute the one-loop corrections to the masses for the fields,

Two point amplitude for Φ 3 1 field
In this section we compute the two point amplitude for Φ 3 1 field. Using the vertices listed in appendix (E.1), we first write down the contributions to the one loop amplitude from bosons in the loop, This contribution consisting of the four-point vertices as shown in figure (4) is given by, and the contributions from the three-point vertices (see figure (5)) is Where For computing the above amplitudes at zero external momentum we list the various exact functions below In the above expressions i = 2, 3 andk i = k i / √ q. We first analyze the contributions to the expressions (4.92) and (4.93) in the large n limit. This will give us their ultraviolet behavior. We write down the contributions from each of the terms separately, The full contribution from (4.95) and (4.96) simplifies to where in the last line of the above equation we have kept the zero temperature UV divergent piece.
We now compute the the exact expression for the two point amplitude for all n. To do this we write down term by term each of which corresponds to the Feynman diagrams in the figures (4) and (5). We where, where, Figure 6: Feynman diagrams involving three-point vertices We now write down the contribution from fermions in the loop The factor of 4 in front comes from summing over the contributions from all the four sets of fermions (eqn. E.207).
We will now be interested in evaluating the two point function with the external momenta set to zero.
(ω m ′′ =ω m ′′ = k ′′ =k ′′ = 0). At zero external momenta the exact form of this contribution to the two point amplitude is We now perform the sum over the Matsubara frequencies and write down the the contributions as where g(λ ′ n , β) is defined in equation (3.68). We can extract the zero temperature contributions from the above equation. To study the UV behaviour of the zero temperature contributions we take the large n limit.
In this limit the expression reduce to Thus equations (4.97) and (4.110) show that the UV divergences cancel (for large n) between that contributions from the bosons and fermions in the loop.

Large k
In the following we list the large k (for fixed n) behaviour of the integrands in Σ 1 at zero temperature. So the in the limit k >> nq the various expressions reduce to, The above O(1/|k|) contributions from the bosons in the loop in the large k limit adds up to This is exactly the same in magnitude and opposite in sign as the O(1/|k|) term from (4.109). This shows that in this regime of large k and fixed n the integrals converge.

Two point amplitude for Φ 3 I fields
There is an unbroken SO(5) invariance in the theory. We thus only consider the amplitude for Φ 3 2 field. Using the vertices listed in the appendix (E.2) we first compute the amplitude with bosons in the loop. Figure 7: Feynman diagrams contributing to Σ 1

PSfrag
(2)i 3Ṽ (2)j * 3 (IV ) (V ) Figure 8: Feynman diagrams contributing to Σ 2 where The vertices at zero external momentum are given by 3 (n, n ′ , 0, 0) = 0 ,G i 3 (n, n ′ , 0, 0) = ik i δ n,n ′ (4.115) In the above expressions i = 2, 3 andk i = k i / √ q. Using these zero momentum vertices the amplitudes corresponding to the diagrams in figures (7) and (8) are as follows with, where (IV ) = 0 (4.122) In the large n limit the above amplitude reduces to Figure 9: Feynman diagrams involving three-point vertices We now turn to the fermions. The contribution from fermions in the loop is given by factor of 2 in front is due to the contributions from the two sets of fermions (eqn. E.209).
For zero external momenta (ω m ′′ =ω m ′′ = k ′′ =k ′′ = 0), G 2 f (n, n ′ , 0) = G 2 * f (n, n ′ , 0) = 0. So the amplitude simplifies as In the above equation we have summed over the Matsubara frequencies. g(λ ′ n , β) is given by eqn. (3.68). To show that UV divergence cancellation we take the limit β → ∞. The zero temperature contribution is This is opposite in sign to the contribution from the bosons in the loop thus showing the finiteness of the one-loop amplitude in the UV.

Large k
As we have done before we list behaviour of the integrands in Σ 1 at zero temperature in the limit k >> nq.
We also get the same O(1/|k|) contribution from (4.127) having opposite sign.

Two point amplitude for A 3 µ fields
In this section we compute the two point amplitude for A 3 µ fields. We compute the amplitudes for µ = 1 and µ = 1 in the following two subsections.

µ = 1 : Bosons
In this part we compute the contribution with bosons in the loop for the two-point function for the A 3 1 Using the vertices listed in appendix (E.3), we first write down the contributions to the one loop amplitude from bosons in the loop, The contribution consisting of the four-point vertices as shown in Figure 10 is given by, and the contributions from the three-point vertices as shown in Figure 11) is Since we are interested in the amplitudes at zero external momentum we list the various exact functions at this momentum below and (ωm′′ ,k δ n−2,n ′ + 2(n − 1)(n + 1) (2n − 1)(2n + 1) δ n,n ′ ,G 11 3 (0, n, n ′ ) = 0, In the above expressions i = 2, 3 andk i = k i / √ q. Using the above vertices (eqn. This is same as the the contribution obtained for Σ 1 where, (I), (II) and (III) are given by eqns.
We now write down the contribution towards the two point amplitude amplitude from the three point vertices as shown in Figure 11. Using (3.52), this is given by where,  4 (4.139)

µ = 2, 3 : Bosons
We now compute the contribution with bosons in the loop for the two-point function for the A 3 2 . Due to the underlying symmetry the amplitude for µ = 3 is same as the one for µ = 2. Using the vertices listed in appendix (E.3) the contribution consisting of the four-point vertices as shown in Figure 12 is given by, Figure 12: Feynman diagrams contributing to Σ 1 and the contributions from the three-point vertices as shown in Figure 13 is where At zero external momentum the various functions have the following forms In the above expressionsk i = k i / √ q. Using these vertices the various diagrams in Figures 12 and 13 reduce to

= (I) + (II) + (III) (4.146)
Similarly, It is now not difficult to show that in the large n limit the full contribution to the two point amplitude reduces to At zero temperature in the limit k >> nq the various O(1/|k|) contributions reduce to, which adds up to (4.154) The contribution from fermions in the loop is given by
Summing over the Matsubara frequencies we write down various contributions and using the form for g(λ ′ n , β) given by (3.68), we get In the large n limit and at zero temperature (β → ∞) the above contribution has the form This is equal in magnitude to the contribution from bosons in the loop but with opposite sign thus showing that the amplitude is ultraviolet finite. A similar analysis such as the one that was done for the two-point amplitude for the Φ 3 1 field at zero temperature and in the limit k >> nq can again be done here. The O(1/|k|) contributions from the bosons and the fermions in the loop can be shown to cancel as done exactly at the end of section (4.1).

µ = 2, 3 : Fermions
From eqn. (4.155) we can now write down the contribution from the fermions in the loop for µ = 2, 3 Now summing over the Matsubara frequencies we arrive at the following expressions where g(λ ′ n , β) is defined in (3.68). From this the zero temperature contribution in the large n limit is Thus confirming the cancellation with the contribution from bosons in the loop. Similarly we can expand the integrand at zero temperature and in the limit k >> nq, which gives Comparing with (4.154) we see similar convergence of the full amplitude for large k with nq fixed.

Numerical Results
This section is devoted to study the two-point functions for the massless as well as the Tachyon fields numerically. The analytical expressions for the two-point amplitudes derived in the previous sections can be reduced to the case of intersecting D2 branes by setting one of the momenta transverse to the x direction as zero. However here we restrict ourselves only to the case of intersecting D3 branes. We give the numerical results for the tree-level massless modes and the Tachyon in separate subsections. As explained before the corrections to the masses for the tree-level massless modes are needed to cure the the infrared divergences in the Tachyon two-point amplitude. The computations are done for the Yang-Mills coupling, g 2 = 1/100.

Numerical Results: Tree-level massless modes
In this section we discuss the numerical computation of the two-point functions for the modes that are massless at tree-level, namely the Ψ ≡ (Φ 3 1 , Φ 3 I (I = 2, · · · , 6), At finite temperature there are two dimensionful parameters q, T . The one loop mass 2 corrections go as m 2 = qf (T / √ q). For T / √ q >> 1 we expect the m 2 to go as T 2 in 3 + 1 dimensions. To see this explicitly using the standard scaling arguments consider a simple representative of the temperature dependent part of the amplitudes for the tree-level massless fields, Let us now rescale the momenta as k ′ = βk. Further the the term β 2 λ n in the exponent, in the limit β 2 q → 0 can be replaced by a continuous variable, say l ′ 2 . Thus replacing we may write (5.168) as The integrals are independent of β and hence we expect the T 2 behaviour. This assumes that the momentum limits in (5.168) go from zero to infinity. In the numerics however we have to work with finite limits on the momenta. The plots as shown in the figures here are for |k|/ √ q integrated over 0 to 1000 and n is summed over up to 10.

Numerical Results: Tachyon
The tachyon two-point one-loop amplitude has various massless fields propagating in the loop. As discussed earlier, to regulate the infrared divergences due to the massless fields we incorporate the one-loop corrected masses for the tree-level massless fields. The behaviour of these masses as a function of temperature is computed numerically in the previous section. Here however in the study of the behaviour of the tachyon mass as a function of temperature, to simplify the numerical computation, we take fixed non-zero values for There exists a critical temperature T c , where the one-loop effective mass of the tree-level tachyon is zero.
In the plots (23) and (24), the critical value of √ qβ namely √ qβ c = 0.062 or the critical temperature Tc √ q = 15.96. The zero temperature quantum correction for the tachyon mass-squared is found to be m 2 0tach /q = 0.259. This behaviour bears the hallmark of a phase-transition. In the brane picture the mass squared becoming positive indicates the dissolution of the tachyonic mode, by way of which stability is achieved. This is also analogous to the transition from the superconducting BCS phase to the normal phase as studied in [1,2].
Following the scaling arguments similar to that for the massless fields we here too we expect a T 2 behaviour for the tachyon mass for T >> √ q. However note that the plot in fig (24) is almost linear in T √ q which is a deviation from the T 2 behaviour. This is an artifact of the numerical computation. The T 2 behaviour and the scalings of the momenta are valid for infinite sums and integrals. In order to perform the the numerical calculations, due to technical limitations we had to simplify the computations by summing and integrating over relatively small range for the momenta, namely n, l, k ∼ 0 − 10. For sufficiently large range of momentum sums and integrations, the T 2 behaviour can presumably be reproduced in the numerics.

Discussion and Summary
In this paper we have computed the one-loop correction to the tachyon mass for the intersecting D3 branes at finite temperature. The calculation is extension of our previous work [1], The infrared divergence appearing in the one-loop tachyon amplitudes are of two kinds. Ones that appear as artifacts of the A a 0 = 0 gauge can be removed prescriptions as discussed in [40]- [43]. The genuine infrared divergences appearing from massless modes in the loop are cured by using the one-loop corrected propagator that incorporates the one-loop masses for the tree-level massless fields. Finally the amplitudes are all finite both in the infrared as well as in the ultraviolet.
It can be seen that the analytic expressions for the amplitudes obtained in the paper reduce to the ones for the D1 branes. Similarly one can easily find the corresponding expressions for the case of intersecting D2 branes as well by setting one of the momentum modes transverse to the x direction to zero. In this paper we have only presented here the numerical results for the case of D3 branes. It is shown that all the masses including the Tachyon mass grows as T 2 as expected. From the numerics we have found the critical temperature T c when the Tachyon becomes massless. Thus for temperatures above the critical temperature the intersecting configuration is favoured. Although the configuration of intersecting branes here is in flat space the results obtained in this simplified model is consistent with the strong coupling BCS holographic model proposed in [2]. It was shown in [2] that at zero temperature there exists solutions in the Yang-Mills approximation which correspond to breaking of U (1) symmetry thus signaling condensate formation and the brane configuration is smoothed out.
The existence of the critical temperature denotes a phase transition. To analyze the nature of the transition at lower temperatures one needs to analyze the full tachyon potential. This knowledge would in turn shed light on the the BCS transition in the dual theory. In the following we conclude with some observations on the effective potential. Here we restrict ourselves to the case of D3 branes. In the Yang-Mills approximation, from the point of view of (2 + 1)D, the modes C(m, n, k) have masses (2n − 1)q. Thus there is an infinite tower of states including the Tachyon with masses proportional to q. In this scenario it might appear that the notion of Tachyon effective action obtained by integrating out massive modes is ill-defined.
There have however been extensive studies of Tachyon effective action in open string theory where the effective actions were proposed from the consistency with world-sheet conformal field theory (see [44] for review). Similar techniques should be employed for the construction and further study of effective potential in the present case. For approaches along these directions see [34]- [39]. It would further be interesting to generalize such potential to the case of finite temperature and for nontrivial background geometries.
Alternatively from the point of view of (3 + 1)D the discrete n-modes are momenta for the field ζ(z) (see eqn (2.16)) along the x direction. We can thus construct a (3 + 1)D effective action for the field ζ(z).
This in the zero external-momentum limit including n = 0 reduces to the potential in terms of only the C(0, 0, 0) modes. This potential in principle be computed using perturbative techniques utilized in the paper and would involve computation of N -point amplitudes for the C(0, 0, 0) modes. Compared to the effective potential in the (2 + 1)D theory this would thus give the effective potential when all the modes with positive mass-squared are set to zero i.e. the massive modes only appear in loops. This is exactly what is done in this paper. Here for simplicity we have ignored C(0, n, 0)C(0, n ′ , 0) , amplitudes for n = n ′ .
Nevertheless computation of an effective potential only in terms of C(0, 0, 0) for the (3 + 1)D theory should be interesting in its own right.

B Dimensional Reduction
Following [46] and [47] we give below the details of the dimensional reduction of D = 10, N = 1 to D = 4.
The action for D = 10, N = 1 SYM 1 , To proceed with the reduction, the gamma-matrices are first decomposed as, In this representation, The dimensionally reduced action can now be written as, where the Bosonic part if the action is, and the Fermionic part is, The α and β matrices satisfy An explicit representation of these matrices is given by the following We end this section by writing down the explicit forms of the 4D γ-matrices C Propagators and Vertices for computation of two point C(m, n, k) amplitude
Propagator for the A 3 i fields (i = 1, 2, 3): We also have additional vertices as V 1 2 for the A 1 i (i = 1) fields in place of Φ 1 I /Φ I 1 fields.
We also have additional vertices as V 2 2 for the A 2 i (i = 1) fields in place of Φ 2 I /Φ I 2 fields.
We also have additional vertices as V 3 2 for the A 3 i (i = 1) fields in place of

C.2 Fermions
The terms in the action (B.184) that contain C(m, n, k) are where · · · contain other similar terms involving the other three sets (λ 1 2 , λ 2 3 ), (λ 1 3 , λ 2 2 ), (λ 1 4 , λ 2 1 ). Inserting the mode expansions of the fields we can rewrite the action as m,m ′ ,m ′′ n,n ′ The required vertices are now identified as C(m, n, k) and similarly there are three other sets of vertices coming from the other terms in the action.