Phase transitions in separated braneantibrane at finite temperature

Transcription

1 Phase transitions in separated braneantibrane at finite temperature Vincenzo Calo PhD Student, Queen Mary College London V.C., S. Thomas, arxiv: [hep-th] JHEP-06(2008)063 AYIA NAPA 1

2 Summary General considerations on finite temperature analysis. DantiD system and the tachyon potential. Boundary String Field Theory (BSFT). Tachyon potential at finite temperature and second order phase transitions. Conclusion and possible applications to cosmology. 2

3 1. Finite temperature analysis! Temperature effects that may spoil the metastable vacuum of a theory. (Type IIB flux compactification: KKLT, LVM in progress) " Temperature effects that may render stable a system which is not at zero temperature. (D-antiD branes or non-bps D-brane) 3

4 1.1 Finite temperature effect in flux compactification CS moduli fixed by perturbative effects Kaehler moduli fixed by non-perturbative effects SUSY broken by uplifting sector KKLT, Large volume compactification, etc. etc. 1.4 V tunneling potential at zero temperature ds vacuum is long-lived! Σ 4

5 1.2 Effective potential at finite temperature Can this minimum be spoiled by temperature effects? The moduli potential receive important thermal corrections! V eff = V 0 + V 1 + V T Tree level moduli potential One loop (Coleman - Weinberg) One-loop at finite temperature 5

6 [Buchmuller et al., hepth/ ] V 0 ( ) V T = T 4 1 a 0 + a 2 Φ V Vth (a) V. Veff (b) V th. Constraints for the reheating temperature after inflation! T T crit T T crit (c) V eff (Φ, T ). T T crit V 0 + V T 6

7 a) At high temperature the free energy drives the scalar expectation values to zero. b) The free energy in the squark direction develops a minimum away from zero, while in the meson direction it develops a potential barrier. V eff = V eff = 0 [Fischler et al, hepth/ ] [Abel et al., hepth/ , Why the early universe preferred the non susy vacuum] 7 7

8 1.3 References Abel et al., Why the early universe preferred the non susy vacuum, hepth/ Fischler et al., Metastable susy breaking vacuum in the early universe, hepth/ Anguelova et al., Metastable susy breaking and supergravity at finite temperature, hepth/ Buchmuller et al., Dilaton Destabilization at high temperatures, hepth/ Ratz et al., Maximal temperature in flux compactification, hepth/ Copeland et al., Moduli evolution in the presence of thermal corrections, arxiv: Anguelova, V.C., O KKLT at finite temperature, arxiv: Papineau, Finite temperature behaviour of the ISS-uplifted KKLT model, arxiv: V.C. et al., Large Volume compactification at finite temperature, in preparation 8

9 1I. D-antiD-brane action! Low energy effective action approach [Sen, Garousi] " Boundary String field theory in a nutshell [Finn, Larsen, Takayanagi, Terashima, Uesugi] 9

10 Strings excitations between parallel D-Branes λ λ s 10 10

11 From non-bps D-branes to D-antiD branes D1 D3 D5 D7 D9 Type IIA One non BPS- Dbrane D0 D2 D4 D6 D8 Type IIB 2 non BPS- Dbrane? The exact form of the action for multiple non-bps branes is not known. There are however two proposals. 11

12 S 1 = Tr S 2 = STr Effective Action of two non-bps D9-branes [Sen, Dirac-Born-Infeld action on the tachyon kink and vortex, hep-th/ ] d 10 σv (T )e Φ det(g µν + B µν + 2πα F µν + πα [D µ T D ν T + D ν T D µ T ])( F µν = µ A ν ν A µ i[a µ, A ν ] D µ T = µ T i[a µ, T ]. This action reproduces the one proposed by Sen and it has a vortex solution corresponding to a D(p-1)-brane which is left over after tachyon condensation. [Garousi, On the effective action of the D-anti-D-system, ] d 10 σv (T )e Φ det(g µν + B µν + 2πα F µν + 2πα D µ T D ν T ) Various couplings in this action are consistent with the disk level S- matrix elements in string theory. 12

13 The proposal for the effective action of D9-antiD9 system is to project the effective action of two non-bps D9-brane with A µ = ( 1) F l Since there is no off-diagonal terms for the gauge fields, the theory has gauge symmetry U(1) x U(1). ) ) ( A (1) µ 0 0 A (2) µ, T = ( 0 τ τ 0 For the action S1 before we obtain S 1 = d 10 σv ( τ )e Φ ( det A (1) + det A (2) ) A (n) µν = g µν + B µν + 2πα F (n) µν + πα (D µ τ(d ν τ) + D ν τ(d µ τ) ). [Sen] 13

14 ( ) For the other action, the one with the symmetrized trace, ( ) L 2 = T 9 (πα )Tr ( m 2 T 2 + D µ T D µ T πα F µν F νµ) + T 9 (πα ) 3 (1 ( 2 Tr 3 Dα T D α T F µν F νµ Dα T F µν D α T F µν + 2m2 3 T 2 F µν F νµ + m2 3 T F µνt F µν 4 3 F µα F αβ D β T D µ T 4 3 F αβf µα D β T D µ T 4 6 F µα D β T F αβ D µ T 4 6 F αβd β T F µα D µ T ) Different couplings No soliton solutions found yet 14

15 Space of all twodimensional worldsheet field theories BSFT: Main IDEA [Witten hepth/ ] [Shatashvili, hepth/ ] arbitrary boundary interactions Class. conf. space CFT in the interior Free action defining an open plus closed conformal background General worldsheet theory with boundary interactions S = S 0 + 2π 0 dτ 2π V General boundary perturbation V = i λ i V i 15

16 S = S 0 + 2π 0 dτ 2π V Z e S The proposal for the classical spacetime action is Bosonic string S = Superstrings ( β i ) λ i + 1 S = Z Z(λ) 16

17 The worldsheet action is where the bulk action is S bulk = 1 4π Boundary action S bndy = [Kraus, Larsen, Boundary string field theory of the DD-bar system, hep-th/ ] S = S bulk + S boundary d 2 z ( α [ ] dτ 2 Xµ Xµ + ψ µ ψµ + ψ µ ψ µ [ α 4 T I T I ηi η I + α 2 D µt I ψ µ η I + i 2 (Ẋµ A µ α F µν ψ µ ψ ν ) + i 4 (Ẋµ A IJ µ + 1 ] 2 α Fµν IJ ψ µ ψ ν )η I η J, A ± µ = 1 2 (A ] µ ± ia 12 ) µ ) T = 1 2 (T 1 + it 2 ). D µ T I = µ T I ia IJ µ T J. 17

18 For example, suppose that on the disk: T=const and A=0. BSFT gives the following potential: V (T, T ) = 2T D9 e 2πα T T open string vacuum closed string vacuum 18

19 D9 anti D9 separated Suppose that T is linear. Then S = 2T D9 d 10 T T x e compactify one direction [1 + 2α D µ T Dµ T + (π α ) 2 D9 A µ = A D9 µ A µ turn on a constant Wilson line 4 F 2 µν ] Anti-D9 D9 T-Duality AntiD8 D8 19

20 V 0 (T ) = 2T 9 e T 2 [ 1 + 2α A 2 T 2] m 2 T = A 2 1 2α A cr = 1 2α Metastable minimum [Hashimoto, Dynamical decay of D-anti-D branes, hep-th/ ] We would like to study this system at finite temperature for different values of A. 20

21 III. Tachyon potential at finite temperature 21

22 Strings at finite temperature To consider finite temperature effects, we would like to study the behaviour of an effective potential where V 1 is the one-loop finite temperature piece. If we work in the canonical ensemble the one-loop part of the effective potential is given by the free energy F of open strings

23 Subtleties: BSFT at one loop! The free energy is computed from the one loop amplitude in which time has been compactified on a circle of radius inverse to the temperature. Add an additional boundary to the disk (weighted by e^{!sbndy} ) and then integrating over moduli. Since in BSFT conformal invariance is broken on the boundaries, we are left with an ambiguity in the choice of the Weyl factors of the two boundaries of this worldsheet. [Andreev and Ott, 01] [Craps, Kraus, Larsen 01] 23 23

24 [Andreev and Ott, 01] The proposed one-loop amplitude on the cylinder in the Minkowski spacetime in type II string theory is Z 1 = 16π4 iv p (2πα ) p ( ϑ3 (0 iτ) ϑ 1 (0 iτ) dτ p+1 τ (4πτ) 2 e 4π T 2 τ ( ) 4 ϑ2 (0 iτ) ϑ, 1 (0 iτ) ) 4 If we compactify the Euclidean time on a circumference with radius β F (T, β) = 16π4 V p (2πα ) p dτ p+1 τ (4πτ) 2 e 4π T 2 τ ( ) 4 ( ϑ3 (0 iτ) iβ (ϑ 2 ) ) ϑ (0 iτ) 8π 2 α τ ( ) 4 ( ϑ2 (0 iτ) iβ (ϑ 2 ϑ (0 iτ) 8π 2 α τ 24 ) 1) 24

25 The free energy of a system of Dp-branes can be computed given F (β) = + V p (2πα ) p+1 2 V p (2πα ) p dτ p+1 τ (4πτ) 2 dτ p+1 τ (4πτ) 2 The mass spectrum is given by p d MNS 2 = M 2 R = R I I=1 p d ( mi I=1 ( mi R I ) 2 + ) 2 + D i=p d+1 M NS 2 M R 2 r=1 r=1 ( ) ( D i=p d+1 ( ) ni R 2 i + 1 α ( ni R i α α ) α exp ( ( 1) r exp 2πα M NS 2 τ r2 β 2 ( ( N B + N NS + T 2 2 ( N B + N R + T 2 8πα τ ) 2πα M R 2 τ r2 β M 2 A + M 2 A ) 8πα τ ) ) ( where M 2 A = α A 2 ( T 4 4T ) 25 25

26 1. Expand at large temperature 2. Expand along the open string vacuum, namely around T=0. F (T, x) CV p β H [ ( ( ) ) ] [ [ 1 π (x 2 1) π ( 2α A 2 ( T 4 4T ) + T 2) Γ ( 0, π ( x 2 1 ) Λ ) ] (4 V 0 (T ) = 2T 9 e T 2 [ 1 + 2α A 2 T 2] x = β β H Phase transition Second order phase transitions 26

27 Zero separation V eff H zero temperature V eff critical temperature T β cr β H [1 + Exp V eff 2 ( 16 )] γ π g s T T T > Tcr Β Β cr 1.23 Β H Β

28 V eff Large separation V eff β cr β H [1 + Exp O ( π g s ( 2A 2 α 1 ) (8A 2 α 1) γ )] T V eff V eff T T T

29 Conclusions We have investigated the behaviour of the finite temperature effective potential on the brane-antibrane pair in the case of zero and non-zero separation of the pair. In case of zero separation, the tachyonic open string vacuum at the origin of the tachyon field is stabilized at sufficiently high temperatures. In the case of large separation, there is a new deep minimum which is formed only at finite temperature. Moreover, the metastable open string vacuum at the origin is destabilized at sufficiently high temperature. 29

30 Cosmology prospective Inflation from D anti-d brane annihilation We can consider the cosmological production of D9 anti-d9 branes in the early universe at high temperature and study the formation of lower dimensional D-branes. Rolling tachyon at finite temperature. 30