BCS Instability and Finite Temperature Corrections to Tachyon Mass in Intersecting D1-Branes

A holographic description of BCS superconductivity is given in arxiv:1104.2843. This model was constructed by insertion of a pair of D8-branes on a D4-background. The spectrum of intersecting D8-branes has tachyonic modes indicating an instability which is identified with the BCS instability in superconductors. Our aim is to study the stability of the intersecting branes under finite temperature effects. Many of the technical aspects of this problem are captured by a simpler problem of two intersecting D1-branes on flat background. In the simplified set-up we compute the one-loop finite temperature corrections to the tree-level tachyon mass using the frame-work of SU(2) Yang-Mills theory in (1 + 1)-dimensions. We show that the one-loop two-point functions are ultraviolet finite due to cancellation of ultraviolet divergence between the amplitudes containing bosons and fermions in the loop. The amplitudes are found to be infrared divergent due to the presence of massless fields in the loops. We compute the finite temperature mass correction to all the massless fields and use these temperature dependent masses to compute the tachyonic mass correction. We show numerically the existence of a transition temperature at which the effective mass of the tree-level tachyons becomes zero, thereby stabilizing the brane configuration.


MOTIVATION
• In conventional QCD The Nambu, Jona-Lasinio-model of chiral symmetry breaking elucidates certain apparent similarities between chiral symmetry breaking and the BCS instability in superconductors.
• Inspired by this similarity, a holographic model of BCS superconductivity has been proposed within the broken chiral symmetric scenario in the Sakai Sugimoto model.(N. Sarkar, S. Sarkar, B. Sathiapalan, K. Rama) • proposal: BCS instability (Cooper pairing between Baryons) in the boundary(D4 wrapped on S 1 ) corresponds to tachyonic instability in the bulk (D8).

MOTIVATION: INTERSECTING D8-BRANES
• The formation of Cooper pairs in the boundary: introduce a finite Baryon number density on the boundary theory i.e. a Chemical Potential for Baryon number.
• How?: A point source of Baryon number in the bulk which creates a cusp singularity in the bulk. For two D8-branes, SU (2) is broken and the branes intersect at one angle between them.(Bergman, Lifschytz, Lippert) • In the SS-model a configuration of two intersecting D8-branes were found to have a tachyonic instability in the bulk spectrum which is proposed to correspond to Cooper pairing instability in the boundary theory. (B. Sathiapalan, et.al.) • The tachyon mode is identified as the lowest mode in the open string excitation between the intersecting branes. • Another way of stabilizing: Finite temperature field theory.
• Computation: Finite temperature one-loop mass-squared corrections to the tree-level tachyon.
• Finite temperature effects : Existence of T c at which the effective mass-squared of the tachyon vanish. Our main goal is to calculate the T c .
• However this problem is difficult to handle in the case of D8-branes on a curved D4-background. But many of the technical features are captured by a much simpler set-up consisting of two intersecting D1-branes on a flat background.
• We choose to study the finite temperature effects in this simpler set-up. We are able to do so because the tachyon dynamics is a local phenomenon and not influenced significantly by curvature effects.

Validity
• The low energy theory on the brane can be described by the DBI action for the massless fields on the brane. This is valid as long as only energies << 1 α are being probed. • We can study this as a quantum theory with a cutoff Λ < 1 √ α and proceed to study the corrections due to the massless mode quantum and thermal fluctuations.
• Thermal corrections should be unambiguously finite.
• The background fields : The Lagrangian for the background fields decouple into two pieces, one for each of these doublets.
• In each doublet the fields satisfy a set of coupled differential equations.
• There are two sectors of solutions: , m 2 n = 0. Two different sets of normalized eigenfunctions for each of these doublet fields.
Normalized Eigenfunctions: INTERSECTING Dp-BRANES:SPECTRUM OF BOSONS • We turn on all the other fields as fluctuations. For the other bosonic fields: where I = 1 • The third gauge components of all fields are massless The fermions in the picture play a crucial role in ensuring the UV finiteness of one-loop computations.
• We shall restrict our discussion to only D1-branes now. We have a complete calculation for this case. For D2 and D3branes the work is still in progress.
• The fermions: sixteen left and sixteen right moving Majorana-Weyl fermions, grouped into two different sets of eight pairs distinguished by their e.o.m.
and their complex conjugates.
• L 3 i and R 3 i are massless fermions(plane waves).

TACHYON INSTABILITY
• The bosonic doublets ζ k are eigenvectors corresponding to the mass squared eigenvalue: where k = 0 corresponds to tachyonic modes. Step 1: Calculate the finite T 1-loop mass-corrections for the massless fields namely, Φ 3 1 , Φ 3 I , (I = 1) and A 3 x . • m 2 n = 0: Infinitely degenerate massless modes corresponding to the zero eigenvalue sector: diagonalized mass matrices as a function of temperature (numerically). • Step 2: These temperature dependent masses modify the propagators in the tachyonic amplitudes.
• The tachyon two-point functions are computed self-consistently (numerical computation).

FINITE TEMPERATURE CORRECTIONS
• UV problem: for all fields.
• Finite T 1-loop bosonic and fermionic amplitudes: Each term is UV divergent.
• Sum over discrete momemtum n (fields coupled to the background are massive) • integral over continuous momemtum (massless modes).
• Compute the integrals involved in the vertices and expand the sums over n about n = ∞: leading order 1 √ n .
• Cancellation between Bosonic and fermionic terms yeilds finite answer.
SUDIPTO PAUL CHOWDHURY BCS Instability and Finite Temperature Corrections to Tachyon

FINITE TEMPERATURE CORRECTIONS
• No divergence from temp-dependent part.
• One-loop corrections to the tachyon mass term: set all external momenta in the Feynman diagrams = 0 and integrate/sum over the loop momenta. One-loop diagrams: • The parameter q provides a scale for supersymmetry breaking. The effective mass of the tree-level tachyon • m 2 0 : Quantum corrections (T = 0). Only true for 1 + 1-dimensions.

FINITE TEMPERATURE CORRECTIONS (MASSLESS FIELDS)
Sample plot for massless field :

CONCLUSION
• The finite temperature effects remove the tachyon instability in intersecting D1-branes and stabilize the configuration.
• The effective mass-squared of the tree-level tachyon grows linearly with temperature as expected in (1 + 1)-dimensions.
• The zero temperature quantum corrections are independent of the parameter q (1 + 1-dim.).
• At finite temperature the superconducting instability transits into a stable normal phase.
• This phenomenon bears the hallmark of a phase transition.

FUTURE DIRECTIONS
• To do the full stability analysis we must compute the full finite temperature effective action for the tachyon, which calls for computing higher point functions.
• Our results can be generalized to higher dimensional branes(D2 and D3) without much difficulty. It will be interesting to study the issue of phase transition in higher dimensions. (ongoing) • By scaling arguments(scaling the integrals by powers of β) we see that the finite temperature bhaviour in p + 1-dims (p > 1)is T p−1 .
• Question of adding α -corrections in the loop may be interesting.
• Open string world-sheet perspective : calculating the annular amplitudes at finite T.

THANK YOU!
SUDIPTO PAUL CHOWDHURY BCS Instability and Finite Temperature Corrections to Tachyon

FINITE TEMPERATURE CORRECTIONS
• The one-loop corrections from the bosonic diagrams with 4-pont vertex • where F 's denote the four point vertices in this expression.

FINITE TEMPERATURE CORRECTIONS
• The one-loop corrections from the bosonic diagrams with 3-pont vertex • After performing the Matsubara sums, the mass correction for the four-point vertices become SUDIPTO PAUL CHOWDHURY BCS Instability and Finite Temperature Corrections to Tachyon

FINITE TEMPERATURE CORRECTIONS
The mass correction for the three-point vertices after the Matsubara sum assumes the form SUDIPTO PAUL CHOWDHURY BCS Instability and Finite Temperature Corrections to Tachyon

FINITE TEMPERATURE CORRECTIONS
• The fermionic corrections are accompanied with diagrams with only 3-point vertices.