I. Introduction II. Collision of Domain Walls in 5D Minkowski Space - PowerPoint PPT Presentation

I. Introduction II. Collision of Domain Walls in 5D Minkowski Space III. Reheating by Collision of Branes IV. Fermion Localization at Collision V. Collision of Domain Walls in Asymptotically AdS Space VI. Summary Kei-ichi Maeda Waseda Univ. with G. Gibbons, H. Kudoh, and Y. Takamizu Phys. Rev. D70 (2004) 123514 , D73 (2006) 103508 hep-th/0610286, in preparation

I. Introduction A brane: an interesting object in string theory D3 brane : could be our universe Some interesting cosmological senarios Ekpyrotic (or cyclic) universe Brane inflation (Dvali-Tye , Rolling Tachyon , KKLMMT, ・・・) collision of branes A brane is usually treated as an infinitesimally thin object To discuss “ matter ” on branes (e.g. reheating, localization), we consider a finite thickness of brane. a brane = a domain wall

II. collision of domain walls in 5D Minkowski space 5D scalar field Φ Y. Takamizu & KM: Phys.Rev. D70 (2004) 123514 Ä 2 Å 2 V potential 2 à = @V V = ï 2 Ä ë à @ à 4 ê y ë −η η Φ domain wall domain wall solution à = ë tanh D Φ η 1 0.5 Φ=−η Φ=η y -6 -4 -2 2 4 6 -0.5 −η -1

unit : ë = 1 Collision of two domain walls Two boosted domain walls with velocities v and − v initial condition à( y; 0) = à v ( y + y 0 ; 0) Ä à Ä v ( y Ä y 0 ; 0) Ä 1 Φ 1 -1 0.5 -0.5 6 4 2 -2 -4 -6 6 4 2 -2 -4 -6 y y -0.5 0.5 − y 0 -1 1 y 0 − v v

ê à 2 + à 0 2 ë Ä 2 Å 2 à = 1 + ï à 2 Ä ë _ ö à( t; y ) 2 4 collide once v=0.4 collide twice v=0.2 The number of bounces highly depends on the initial velocity. N-bounce sols. : a fractal structure Anninos, Oliveira and Matzner, PRD 44 (1991) 1147

III. Reheating by Collision of Branes a scalar field σ confined on a brane, which is coupled to Φ L int = 1 2 g 2 à 2 2 W ( ú ) õ ) : evaluated on a domain wall (a brane) à W ( ú τ : a proper time on a domain wall |out > |in > |in > |out > quantization of a scalar field σ with a “time-dependent mass” particle creation

g ñ ù ï D N b n ö 3.69 Ç 10 Ä 7 2.05 Ç 10 Ä 7 1.0 1.414 0.4 1 spectrum: gaussian 1.16 Ç 10 Ä 7 2.05 Ç 10 Ä 7 10 0.447 0 : 01 7.19 Ç 10 Ä 7 3.90 Ç 10 Ä 7 1.0 1.414 0.2 2 2.26 Ç 10 Ä 7 3.91 Ç 10 Ä 7 10 0.447 3.57 Ç 10 Ä 3 2.01 Ç 10 Ä 3 1.0 1.414 0.4 1 1.16 Ç 10 Ä 3 2.05 Ç 10 Ä 3 10 0.447 0.1 6.65 Ç 10 Ä 3 3.81 Ç 10 Ä 3 1.0 1.414 0.2 2 2.24 Ç 10 Ä 3 3.88 Ç 10 Ä 3 10 0.447 quantum creation of particles ö = 20 g 4 N b ; n = 25 Dg 4 N b ë Ä 1 í ì ê g ñ T R m ë ò 10 15 [GeV] N Ä 1 = 4 :mass scale of domain wall b 10 Ä 5 10 10 GeV enough reheating !

III. Fermion Localization at Collision G. Gibbons, KM &Y. Takamizu : hep-th/0610286 5D four-component fermion Ψ coupled to the scalar field Φ M = e M ^ A M D M â + g àâ = 0 Ä A Ä Ä ^ B = Ä A ^ ^ [ ^ ^ A Ä B ] D M = @ M + 1 Ä A ^ ^ B 4 ! ^ BM Ä A ^ ^ ^ A ^ ^ A ; Ä B g = 2 ë B f Ä ê 5 ë † ! † ! Two chiral states â Ä = 1 ^ ê 5 ë 1 Ä Ä â † † 2 + Ä â Ä = â + = â + = 1 Ä † ^ † Ä 1 + Ä â + 2 In Minkowski background ñ @ ñ â + + g àâ Ä = 0 Ä @ 5 â Ä + Ä ñ @ ñ â Ä + g àâ + = 0 @ 5 â + + Ä

Jackiw-Rebbi (76), Rubakov-Shaposhnikov(83) Localization on a brane Randjbar-Daemi-Shaposhnikov(00) Static domain wall Bajc-Gabadadze(00), Kehagias-Tamvakis (01) z è = Ä 1 : antikink è = 1 : kink à = è ë tanh D Assume massless fermion on the brane ñ @ ñ † â + ( x; z ) = † + ( x ) f + ( z ) Ä + = 0 ñ @ ñ † â Ä ( x; z ) = † Ä ( x ) f Ä ( z ) Ä Ä = 0 1 1 f + f Ä Ä @ 5 f Ä + g à f Ä = 0 0.5 0.5 -10 -5 5 10 -10 -5 5 10 @ 5 f + + g à f + = 0 -0.5 -0.5 -1 -1 1 è = 1 f + / positive-chirality: localized (cosh( z=D )) gD 1 è = Ä 1 negative-chirality: localized f Ä / (cosh( z=D )) gD

normalized wave function î Ä ï 1 = 2 h ê z ëi Ä gD ( gD + 1 2 ) f Ü ( z ) = 2 p ô cosh D Ä ( gD ) D wave function of localized fermions on a kink and on an antikink 0 1 0 1 @ A @ A (4) (4) † + ( x ) f + ( z ) † Ä ( x ) f Ä ( z ) â (K) ( x; z ) = â (A) ( x; z ) = (4) (4) † + ( x ) f + ( z ) Ä † Ä ( x ) f Ä ( z ) annihilation operators a K = h â (K) ; â i and a A = h â (A) ; â i

Time-dependent Background 1 ) 3 = 2 e i~ Ä ( t; z : ~ k~ x † † Ä = k ) (2 ô Ansatz 1: 3-space is flat 1 ) 3 = 2 e i~ + ( t; z : ~ k~ x † † + = k ) (2 ô + Ä ( i@ 0 + ~ ( @ 5 + g à) † k~ õ ) † Ä = 0 Ä + ( i@ 0 Ä ~ ( @ 5 Ä g à) † k~ õ ) † + = 0 † ! ~ † ! Ansatz 2: Low energy state k : u-d mixing k ô 0 u: up † † + u Ä u † + = † Ä = d: down † † Ä d + d i@ 0 † Ä u = ( @ 5 + g à) † time-dependence + u : chirality mixing i@ 0 † + u = ( Ä @ 5 + g à) † Ä u u ! d and

0 1 0 1 B C B C B C B C 1 1 B C B C B C B C 0 0 @ A @ A up state â = † + ( z; t ) + † Ä ( z; t ) 1 Ä 1 0 0 wave function on a moving kink r ç + 1 (K) ( ç † (K) ~ + ( z; t ; ù ) = † ( z Ä ù t )) r 2 ) = i ç ù ç + 1 (K) ( ç (K) ~ † Ä ( z; t ; ù † ( z Ä ù t )) ç + 1 2 wave function on a moving antikink r ç + 1 (A) ( ç r † (A) ~ Ä ( z; t ; ù ) = † ( z Ä ù t )) 2 ) = Ä i ç ù ç + 1 (A) ( ç † (A) ~ + ( z; t ; ù † ( z Ä ù t )) ç + 1 2

Fermion Localization on Colliding Branes à( t; z ) : colliding two domain walls (Sec. II) Fermion wave functions 1 -1 0.5 -0.5 6 4 2 v -2 -4 -6 Before collision − v 6 4 2 -2 -4 -6 0.5 -0.5 1 -1 â = â (K) ^ ) a K + â (A) ) a A + â (B) in ( x; z ; ù in ( x; z ; Ä ù in ( x; z ) a B 1 -1 0.5 -0.5 6 4 2 -2 -4 -6 After collision 6 4 2 -2 -4 -6 − v v 0.5 -0.5 -1 1 â = â (K) ) b K + â (A) ) b A + â (B) ^ out ( x; z ; Ä ù out ( x; z ; ù out ( x; z ) b B Mode mixing by domain wall collision â (K) K â (K) K â (A) K â (B) in ( x; z ; ù ) ò ã out ( x; z ; Ä ù ) + å out ( x; z ; ù ) + ç out ( x; z ) â (A) A â (A) A â (K) A â (B) in ( x; z ; Ä ù ) ò ã out ( x; z ; ù ) + å out ( x; z ; Ä ù ) + ç out ( x; z ) Bogoliubov transformation b K = ã K a K + å A a A b A = ã A a A + å K a K

Two cases : (1) same amount of fermion on each brane Initial state 1 1 ~ ~ â + â Ä 0.5 0.5 j KA i = a y K a y A j 0 i -10 -5 5 10 -10 -5 5 10 -0.5 -0.5 -1 -1 after collision K j 2 + j å h N K i ë h KA j b y A j 2 K b K j KA i = j ã A j 2 + j å h N A i ë h KA j b y K j 2 A b A j KA i = j ã (2) one brane is empty 1 1 ~ â + Initial state 0.5 0.5 -10 -5 5 10 6 4 2 -2 -4 -6 j K0 i = a y -0.5 K j 0 i -0.5 -1 -1 after collision h N K i ë h K0 j b y K j 2 K b K j K0 i = j ã h N A i ë h K0 j b y K j 2 A b A j K0 i = j å

Bogoliubov coefficients by solving the Dirac eqs. (1) v=0.8 1.4 1.4 1.4 1.2 1.2 1.2 1 1 1 0.8 0.8 0.8 n n n 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -40 -30 -20 -10 0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 z z z 1.4 1.4 1.4 n + n + 1.2 1.2 1.2 n + n - n - 1 1 1 n - 0.8 0.8 0.8 0.6 0.6 0.6 0.4 0.4 0.4 0.2 0.2 0.2 0 0 0 -40 -30 -20 -10 0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 -40 -30 -20 -10 0 10 20 30 40 z z z Fermions transfer to the vacuum brane (2)v=0.4

Bogoliubov coefficients g = 2 g = 2 : 5 ù K j 2 K j 2 K j 2 K j 2 K j 2 K j 2 j ã j å j ç j ã j å j ç 0.3 0.94 0.056 0.004 0.47 0.53 0.00 0.4 0.87 0.12 0.01 0.57 0.40 0.03 0.6 0.69 0.30 0.01 0.78 0.17 0.05 0.8 0.42 0.55 0.03 0.88 0.02 0.10 K j 2 + j å K j 2 ô 1 j ã The number of fermions are conserved as a whole K j 2 ú 1 A few percent of fermions escape to bulk space j ç Because of the left-right symmetry, K j 2 = j ã K j 2 = j å A j 2 A j 2 j å j ã

(1) collision of two fermion branes 1.2 1 K j 2 + j å A j 2 ô 1 h N K i = j ã 0.8 0.6 n A j 2 + j å K j 2 ô 1 0.4 h N A i = j ã 0.2 0 -40 -30 -20 -10 0 10 20 30 40 z (2) collision of fermion-vacuum branes 1.4 1.2 K j 2 h N K i = j ã 1 0.8 n K j 2 0.6 h N A i = j å 0.4 0.2 0 -40 -30 -20 -10 0 10 20 30 40 z g-dependence 1 0.9 0.8 0.7 | α K | 2 , | β K | 2 0.6 0.5 0.4 0.3 0.2 0.1 0 1 1.5 2 2.5 3 3.5 4 g

h i p p j ã K j 2 ; j å K j 2 = 1 1 Ü sin (3 2 " g = ï + C ã ;å ( ù )) 2 " = Ü 1 The amount of fermions on each wall g ù a n d p depends sensitively on ï Some remarks: (1) g < 2 =D : the localization of fermions on a domain wall is not j ã K j 2 + j å A j 2 = 1 : 28 suécient. g = 1 ; ù = 0 : 8 (2) If we change the incident velocity very little, the number of bounces changes. This causes a drastic change of å nal distribution of fermions on each wall.

IV. collision of domain walls in AdS space Y. Takamizu & KM: Phys.Rev. D73 (2006) 103508 ê @W ë 2 BPS domain wall solution Ä 8 5 W 2 2 í ì V (à) = 3 î @ à W ë 1 à Ä 1 3à 3 Ä 2 superpotential D 3 ê y ë à K ( y ) = tanh D ds 2 = e 2 A K ( y ) ( Ä dt 2 + d x 2 ) + dy 2 n h ê y ëi o + tanh 2 ( y=D ) A K ( y ) = Ä 4 Ä y 2 9 î ln cosh : 5 D 4 D AdS M Eto-Sakai, PRD68(2003)125001 Arai et al., PLB556 (2003) 192-202

Initial setting Two domain walls in asymptotocally AdS background Φ 1 -1 0.5 -0.5 scalar field 6 4 2 -2 -4 -6 6 4 2 -2 -4 -6 y 0.5 -0.5 − y 0 1 -1 y 0 A metric y − y 0 y 0 Lorentz boost − v v