Disk Scattering of Open and Closed Strings Stephan Stieberger, MPP M - PowerPoint PPT Presentation

Disk Scattering of Open and Closed Strings Stephan Stieberger, MPP M¨ unchen New Perspectives in String Theory The Galileo Galilei Institute for Theoretical Physics, May 20, 2009

Outline I. Higher point open superstring amplitudes (tree) St. St., T.R. Taylor 2006–2008. • Universal properties and relations II. Open & closed vs. pure open string disk amplitudes St. St., to appear very soon. • Sort of generalized KLT on the disk

I. Recent results for N –point open superstring amplitudes N –point open string disk amplitudes in background with CFT description St. St., T.R. Taylor 2006–2008 Motivation: Recent results in YM in spinor basis: compact expressions, recursion relations, . . . • Computed N –point open superstring disk amplitude involving members of vector multiplets to all orders in α ′ , • Compact representation to all orders in α ′ , • Derived SUSY Ward identities to all orders in α ′ Universal Properties • completely model independent • universal to all string compactifications • any numbers of supersymmetries

Examples with members of vector multiplets • 5–gluon MHV amplitude in superstring theory √ 2 , g + 3 , g + 4 , g + Tr( T 1 . . . T 5 ) ( 2 g Y M ) 3 α ′ A ( g − 1 , g − 5 ) = � 1 2 � 2 × � 3 4 � 2 � 4 5 � ( � 4 1 � [1 5] K 1 + � 4 2 � [2 5] K 2 ) • Supersymmetric Ward identities in string theory � 12 � 2 A ( g − 1 , g − 2 , g + 3 , g + 4 , . . . , g + � 34 � 2 A ( φ − 1 , φ − 2 , φ + 3 , φ + 4 , g + 5 , . . . , g + N ) = N ) • N –gluon MHV amplitude in superstring theory 2 , g + 3 , g + 4 , . . . , g + 1 − α ′ 2 ζ (2) A ( g − 1 , g − N ; α ′ ) � 2 F ( N ) � = 2 , g + 3 , g + 4 , . . . , g + N ) + O ( α ′ 3 ) A ( g − 1 , g − ×

Recent results for N –point open superstring amplitudes Note: SUSY transformations within one multiplet (VM) using dz N conserved SUSY charges Q I α , I = 1 , . . . , N , with Q I � 2 πi V I • α = α ( z ) • (Space–time) SUSY transformation of open string vertex operator O dw on world–sheet disk [ Q I ( η I ) , O ( z ) ] := � 2 πi η α I V α ( w ) O ( z ) C z generates SUSY Ward identitites (valid to all orders in α ′ ) c.f. also talk at Strings 2008.

Generalizations and Task • Include chiral multiplets (N=1) • Use of world–sheet supercurrent T F • Include closed strings to probe brane/bulk couplings − → Derive relations between different types of amplitudes − → Amplitudes of open and closed string moduli

First look: N –point parton amplitudes in D = 4 � g = gluon V M Consider superstring disk amplitudes involving both χ = gaugino z 2 z 2 � ψ = fermion E.g.: a a CM a a φ = scalar A a 2 A a 2 z 3 z 3 z 1 z 1 in D =4 A a 3 A a 3 A a 1 A a 1 a a a a ψ β 5 ψ α 4 A a 5 A a 4 α 5 β 4 z 4 z 4 z 5 z 5 b a A ρ ( g − 1 , g + 2 , g + 3 , q − q + [ V (5) ( s j ) − 2 i P (5) ( s j ) ǫ (1 , 2 , 3 , 4) ] 4 , ¯ 5 ) = ρ ( g − 1 , g + 2 , g + 3 , q − q + A FT × 4 , ¯ 5 ) A ρ ( g − 1 , g − 2 , g + 3 , g + 4 , g + [ V (5) ( s j ) − 2 i P (5) ( s j ) ǫ (1 , 2 , 3 , 4) ] 5 ) = ρ ( g − 1 , g − 2 , g + 3 , g + 4 , g + A FT × 5 ) Striking relation to all orders in α ′ !

N –point parton amplitudes in D = 4 � 12 � 4 ρ ( g − 1 , g − 2 , g + 3 , g + 4 , g + A FT i g 3 5 ) = Y M � 12 �� 23 �� 34 �� 45 �� 51 � with: � 14 � 4 � 15 � ρ ( g − 1 , g + 2 , g + 3 , q − q + A FT 4 g 3 4 , ¯ 5 ) = Y M � 12 �� 23 �� 34 �� 45 �� 51 � s 2 s 5 f 1 + 1 V (5) ( s i ) = 2 ( s 2 s 3 + s 4 s 5 − s 1 s 2 − s 3 s 4 − s 1 s 5 ) f 2 and: ǫ ( i, j, m, n ) = α ′ 2 ǫ αβµν k α i k β j k µ P (5) ( s i ) m k ν = f 2 , n � 12 � 4 A ρ ( g − 1 , g − 2 , g + 3 , g + 4 g 2 Y M V (4) ( s j ) C.f.: 4 ) = � 12 �� 23 �� 34 �� 41 � � 13 � 4 � 14 � A ρ ( g − 1 , g + 2 , q − q + 2 g 2 Y M V (4) ( s j ) 3 , ¯ 4 ) = � 12 �� 23 �� 34 �� 41 �

N –point parton amplitudes in D = 4 A ( g a 1 . . . g a N )        A ( χ a 1 χ a 2 g a 1 . . . g a N − 2 )      Relations can be generalized to: A ( ψ a 1 ψ a 2 g a 1 . . . g a N − 2 )         A ( φ a 1 φ a 2 g a 1 . . . g a N − 2 )    No intermediate exchange of KKs nor windings ! g N g N g N − 1 q 1 q 1 g N − 1 g KK g 5 g 4 q 4 q 2 q 2 g 3 q 3

Amplitudes important for low string scale physics Most relevant for signals from low string scale effects in QCD jets • No intermediate exchange of KKs, windings nor emmission of graviton • Useful for model–independent low–energy predictions • Universal deviation from SM in jet distribution L¨ ust, St.St., Taylor, arXiv:0807.3333 ; Anchordoqui, Goldberg, Nawata, L¨ ust, St.St., Taylor, arXiv:0808.0497, arXiv:0904.3547 ; L¨ ust, Schlotterer, St.St., Taylor, to appear

Appendix: Chiral matter vertex operator Vertex operator of chiral fermion ( a, b ) a ( a,b ) b e − 1 Ξ a ∩ b ( z ) e ik ρ X ρ ( z ) V ( − 1 / 2) β ] β 1 ( z, u, k ) = g ψ [ T α 2 φ ( z ) u λ S λ ( z ) ψ α α 1 β [ g ψ =(2 α ′ ) 1 / 2 α ′ 1 / 4 e φ 10 / 2 ] Boundary changing operator Ξ a ∩ b ( z ), with h = 3 8 and: 1 � Ξ a ∩ b ( z 1 ) Ξ a ∩ b ( z 2 ) � = ( z 1 − z 2 ) 3 / 4

II. Disk scattering of open and closed strings   N o N c N o N c � � V − 1 d 2 z i  � � � � � � A = : V o ( x j ) : : V c ( z i , z i ) : � dx j CKG  I π H + j =1 i =1 j =1 i =1 π ∈ S No / Z 2 z N c H + Λ i Π i z 1 z N c − 1 z 2 disk z 3 Π j Λ j x N o − 1 x N o x 1 x 2 x 3 D-brane stack V o ( x i ) = open string vertex operators inserted at x i on the boundary of the disk V c ( z i , z i ) = closed string vertex operators inserted at z i inside the disk

Example: Two open and two closed strings on the disk With PSL (2 , R ) transformation three arbitray points w 1 , w 2 ∈ R and w 3 ∈ C may be mapped to the points x 1 , x 2 and z 1 : x 1 = −∞ , z 1 = − ix , x 2 = 1 , z 1 = ix , z 2 = z , z 2 = z Choice: H + z 2 = z with z ∈ H + and x ∈ R + z 1 = ix x 1 = −∞ x 2 = 1 � ∞ A (1 , 2 , 3 , 4) −∞ dx � c ( −∞ ) c (1) c ( ix ) � = � C d 2 z � : V o ( −∞ ) : : V o (1) : : V c ( − ix, ix ) : : V c ( z, z ) : � ×

Two open & two closed strings versus six open strings on the disk • generic structure of world–sheet disk amplitude of two open & two closed strings : ∞ � � α 1 ,λ 1 ,γ 1 ,β 1 � � W ( κ,α 0 ) dx x α 0 (1 + ix ) α 1 (1 − ix ) α 2 d 2 z (1 − z ) λ 1 (1 − z ) λ 2 = α 2 ,λ 2 ,γ 2 ,β 2 −∞ C ( z − z ) κ ( z − ix ) γ 1 ( z − ix ) γ 2 ( z + ix ) β 1 ( z + ix ) β 2 × Oprisa • generic structure of world–sheet disk amplitude of six open strings : St.St., 2005 1 1 1 � � n 1 ,n 2 ,n 3 � � � dz x p 23 + n 1 y p 23 + k 24 + p 34 + n 2 z p 16 + n 3 F = dx dy n 4 ,n 5 ,n 6 ,n 7 ,n 8 ,n 9 0 0 0 (1 − x ) p 34 + n 4 (1 − y ) p 45 + n 5 (1 − z ) p 56 + n 6 (1 − xy ) p 35 + n 7 × (1 − yz ) p 46 + n 8 (1 − xyz ) p 36 + n 9 × n i ∈ Z ,

Two open & two closed strings versus six open strings on the disk After splitting the complex integral into holomorphic and anti–holomorphic pieces: Analytic continuation, introduce ξ = z 1 + iz 2 , η = z 1 − iz 2 , ρ = ix , ρ, ξ, η ∈ R . ∞ � � α 1 ,λ 1 ,γ 1 ,β 1 dρ | ρ | α 0 | 1 + ρ | α 1 | 1 − ρ | α 2 W ( κ,α 0 ) 1 � = 2 α 2 ,λ 2 ,γ 2 ,β 2 −∞ ∞ ∞ dη | 1 − ξ | λ 1 | ξ − ρ | γ 1 | ξ + ρ | β 1 � � × dξ −∞ −∞ | 1 − η | λ 2 | η − ρ | γ 2 | η + ρ | β 2 | ξ − η | κ Π( ρ, ξ, η ) × Six open strings, with: Answer: 1 k k 1 3 2 z 1 = −∞ , z 3 = − ρ, z 2 = 1 , z 4 = ρ, z 5 = ξ, z 6 = η k 1 k 2 5 1 2 k2 2 k p 1 = k 1 , p 2 = k 2 , 6 p 3 = p 4 = 1 p 5 = p 6 = 1 2 k 3 , 2 k 4 k 1 k1 4 2

Two open & two closed strings versus six open strings on the disk

Two open & two closed strings versus six open strings on the disk After inspecting phase Π( ρ, ξ, η ) : � � α 1 ,λ 1 ,γ 1 ,β 1 W ( κ,α 0 ) = σ γ sin( πβ 2 ) [ A (163542) + A (163524) + A (164532)] α 2 ,λ 2 ,γ 2 ,β 2 + sin( πλ 2 ) [ A (134526) + A (143526)] + σ λ σ γ sin( πγ 2 ) A (132546) + R  A (163542) : z 1 < z 6 < z 3 < z 5 < z 4 < z 2    A (163524) : z 1 < z 6 < z 3 < z 5 < z 2 < z 4      A (134526) :  z 1 < z 3 < z 4 < z 5 < z 2 < z 6  with the six open string orderings A (132546) : z 1 < z 3 < z 2 < z 5 < z 4 < z 6     A (164532) : z 1 < z 6 < z 4 < z 5 < z 3 < z 2      A (143526) : z 1 < z 4 < z 3 < z 5 < z 2 < z 6 