String-brane interactions from large to small distances Giuseppe - PowerPoint PPT Presentation

String-brane interactions from large to small distances Giuseppe D’Appollonio Universit` a di Cagliari and INFN DAMTP Cambridge Galileo Galilei Institute, Firenze, 10 April 2019 String-brane interactions from large to small distances Giuseppe D’Appollonio 1 / 38

Outline Strings, high energy and curved spacetimes The eikonal operator: classical gravity effects and stringy corrections The eikonal operator from the worldsheet σ -model String absorption: closed-open transition and the infall of a probe into a singularity String-brane interactions from large to small distances Giuseppe D’Appollonio 2 / 38

String backgrounds String theory from a worldsheet perspective is a perturbative series sum over surfaces or genus espansion around a given classical background a (super)conformal σ -model with c = 26 (15) A good understanding of string compactifications, i.e. vacua of the form R 1 ,d − 1 × K Main examples: toroidal compactifications, orbifolds, compact Wess-Zumino-Witten models and their GKO cosets, Gepner models. Many lessons: importance of winding states and twisted sectors, T-duality, non-geometric backgrounds... String-brane interactions from large to small distances Giuseppe D’Appollonio 3 / 38

String backgrounds Our understanding of curved spacetimes in string theory is more limited. Several classes of known examples: Plane waves (Horowitz and Steif, 1990) Lorentzian orbifolds (Horowitz and Steif, 1991; Liu, Moore and Seiberg, 2002) Non-compact WZW models and their cosets (Witten, 1991; Nappi and Witten, 1992 and 1993) Change of strategy: analyze the high-energy dynamics of a string in the background of a collection of D-branes. String-brane interactions from large to small distances Giuseppe D’Appollonio 4 / 38

String-brane system at high energy The string-brane system provides an excellent framework for the study of string dynamics in curved spacetimes Microscopic description at weak coupling in terms of open strings Geometric description at strong coupling, extremal p -branes Unitary S-matrix The high-energy limit makes the dynamics both interesting and (to a certain extent...) tractable The large energy causes an enhancement of classical and quantum gravity effects The large energy reveals the full string dynamics: interactions of states of arbitrary mass and spin. Inelastic processes grows in variety and importance. String-brane interactions from large to small distances Giuseppe D’Appollonio 5 / 38

Extremal p-branes Low energy effective action in the string frame � � � e − 2 ϕ � R + 4( ∇ ϕ ) 2 � d 10 x √− g 1 2 (8 − p )! F 2 S = − . p +2 (2 π ) 7 l 8 s BPS solutions carrying R-R charges Horowitz and Strominger, 1991 � 1 ds 2 = η µν dx µ dx ν + H ( r ) δ ij dx i dx j , e ϕ = gH 3 − p � , 4 H ( r ) � R � 7 − p � R 7 − p ∗ F p +2 = N , H ( r ) = 1 + , = d p gN , l 7 − p r S 8 − p s � 7 − p � R 7 − p 5 − p 2 Γ λ = gN , d p = (4 π ) , τ p = (2 π ) 7 d p g 2 α ′ 4 . 2 String-brane interactions from large to small distances Giuseppe D’Appollonio 6 / 38

String-brane collisions String-brane interactions from large to small distances Giuseppe D’Appollonio 7 / 38

String-brane collisions Relevant scales � R � 7 − p ∼ α ′ ER 7 − p , b 8 − p α ′ s ≫ 1 , ∼ g N , b c ∼ R T l s Various possible processes as the impact parameter is varied b ≫ b T ≫ R elastic scattering b T ≥ b ≫ R string tidal excitations  R ≪ l s creation of open strings    b < b c    R ≫ l s infall into the singularity Fixed effective background: extremal p -brane metric String-brane interactions from large to small distances Giuseppe D’Appollonio 8 / 38

String-brane collisions String-brane interactions from large to small distances Giuseppe D’Appollonio 9 / 38

The eikonal operator Regge limit of the disk amplitude (tree level) � � − α ′ e − iπ α ′ t 4 ( α ′ s ) 1+ α ′ t 4 , A 1 ( s, t ) ∼ Γ 4 t s = E 2 , t = − ( p 1 + p 2 ) 2 Grows too fast with energy. Include higher-orders. String-brane interactions from large to small distances Giuseppe D’Appollonio 10 / 38

The eikonal operator The result is the eikonal operator δ ( s,b ) , S ( s, b ) = e 2 i ˆ � 2 π dσ : A 1 ( s, b + X ( σ )) : 2ˆ δ ( s, b ) = 2 π 2 E 0 Amati, Ciafaloni e Veneziano (1987) GD, Di Vecchia, Russo e Veneziano (2010). In impact parameter space the tree-level amplitude is � � � � R p iπ √ s � 7 − p 6 − p Γ � R 7 − p b 2 A 1 ( s, b ) ∼ s √ π πα ′ s 2 − p � � � l 2 b 6 − p + e s ( s ) ln α ′ s l s ( s ) 7 − p 7 − p Γ Γ 2 2 √ ln α ′ s l s ( s ) is the effective string length l s ( s ) = l s String-brane interactions from large to small distances Giuseppe D’Appollonio 11 / 38

The eikonal operator Two main effects deflection of the trajectory excitation of the internal degrees of freedom of the string: tidal forces ∂δ ( s,b ) θ = − 2 Scattering angle E ∂b The leading and next-to-leading terms � � � 8 − p � � 15 − 2 p � � R p � 7 − p � R p � 2(7 − p ) Θ p = √ π Γ Γ + 1 2 2 � 7 − p � + . . . b 2 Γ (6 − p ) b Γ 2 are in perfect agreement with the classical deflection of a massless point-like probe in the p-brane background. String-brane interactions from large to small distances Giuseppe D’Appollonio 12 / 38

Tidal excitation at leading order � ln( α ′ s ) When b ≫ R ≫ l s � � ∂ 2 A 1 ( s, b ) X ) ∼ 1 A 1 ( s, b ) + 1 X i ˆ X j + ... 2 ˆ δ ( s, b + ˆ ˆ 2 E 2 ∂b i ∂b j � 2 π where ¯ 1 Q ≡ dσ : Q ( σ ) : and the string coordinates are 2 π 0 � � A i � � ¯ A i α ′ X i = i n e inσ + n n e − inσ n [ A i n , A j m ] = nδ ij δ n + m, 0 , 2 n � =0 Matrix of the second derivatives of the eikonal phase � � ∂ 2 A 1 ( s, b ) 1 δ ij − b i b j + Q � ( s, b ) b i b j 4 √ s = Q ⊥ ( s, b ) b 2 b 2 ∂b i ∂b j where d 2 A 1 ( s, b ) 1 1 d A 1 ( s, b ) 1 Q ⊥ ( s, b ) = 4 √ s , Q � ( s, b ) = 4 √ s db 2 b db String-brane interactions from large to small distances Giuseppe D’Appollonio 13 / 38

Tidal excitation at leading order At leading order in R b � 8 − p � √ π √ s Γ � R 7 − p 2 � 7 − p Q ⊥ ( s, b ) = − b 8 − p , Q � ( s, b ) = − (7 − p ) Q ⊥ ( s, b ) 2 Γ 2 String corrections contribute a non-vanishing imaginary part to the eikonal phase � � 2 � � � 2 √ s Im A 1 ( s,b ) (2 πα ′ | Q ⊥ | ) 8 − p 1 � � 0 | e 2 i ˆ � ∼ e − 7 − p e − πα ′ (7 − p ) | Q ⊥ | δ ( s, b ) | 0 � The absorption of the elastic channel due to string excitations becomes non negligible for b ≤ b T � 8 − p � 2 α ′ √ πs (7 − p )Γ = π b 8 − p 2 � R 7 − p � 7 − p p T Γ 2 String-brane interactions from large to small distances Giuseppe D’Appollonio 14 / 38

Tidal excitation and the pp-wave limit Can we reproduce these results starting with the sigma-model for the extremal p-brane metric? � � p � � � ds 2 = α ( r ) − dt 2 + dr 2 + r 2 ( dθ 2 + sin 2 θd Ω 2 ( dx a ) 2 + β ( r ) 7 − p ) a =1 � where β ( r ) = 1 /α ( r ) = H ( r ). We focus on a small neighborhood around a null geodesic expanding the sigma model action in Fermi coordinates. The leading term in energy corresponds to the Penrose limit of the brane background p 7 − p � � y 0 ) du 2 , ds 2 x 2 y 2 y 2 x a , ˆ y i , ˆ = 2 dud ˆ v + d ˆ a + d ˆ i + d ˆ 0 + G ( u, ˆ a =1 i =1 � √ α u ( √ βr sin ¯ p 7 − p � � βr 2 − b 2 α ∂ 2 a + ∂ 2 i + ∂ 2 θ ) u x 2 y 2 u y 2 G = √ α ˆ √ βr sin ¯ ˆ � ˆ 0 . βr 2 − b 2 α θ a =1 i =1 Blau, Figueroa-O’Farrill and Papadopoulos, 2002 String-brane interactions from large to small distances Giuseppe D’Appollonio 15 / 38

The bosonic part of the string sigma model becomes � � 2 π 1 dσ η αβ ∂ α U∂ β U G ( U, X a , Y i , Y 0 ) S ∼ S 0 − dτ 4 πα ′ 0 where S 0 is the string action in Minkowski space. We work in the light-cone gauge U ( σ, τ ) = α ′ Eτ and evaluate the transition amplitudes in the impulsive approximation. The integrals over u then decouple from the string coordinates and simply provide c -number coefficients to the quadratic action of the fluctuations � � � 2 π p 7 − p � � S − S 0 ∼ E dσ X 2 Y 2 i ( σ, 0) + c 0 Y 2 c x a ( σ, 0) + c y 0 ( σ, 0) 2 2 π 0 a =1 i =1 u ( √ βr sin ¯ � + ∞ � ∞ ∂ 2 θ ) E c y = du G y ( u ) = 2 √ βr sin ¯ = ⇒ 2 c y = Q ⊥ ( s, b ) θ −∞ 0 � � + ∞ � ∞ βr 2 − b 2 α ∂ 2 E u � c 0 = du G 0 ( u ) = 2 = ⇒ 2 c 0 = Q � ( s, b ) βr 2 − b 2 α −∞ 0 String-brane interactions from large to small distances Giuseppe D’Appollonio 16 / 38