The Pink Lecture Series: Scattering Amplitudes in String Theory Daniel H¨ artl Max-Planck-Institut f¨ ur Physik, M¨ unchen IMPRS Young Scientist Workshop at Ringberg Castle July 29, 2009 Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 1 / 14
Outline Motivation 1 Scattering of open strings 2 ψ - S -correlators 3 Conclusion 4 Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 2 / 14
Historical motivation Excited mesons exhibit Regge behavior, J ≈ α 0 + α ′ m 2 . In 1968 Gabriele Veneziano worked out the Veneziano amplitude, where the resonances showed this behaviour at high energies. Later this amplitude was understood to be the four tachyon amplitude in string theory. Scattering amplitudes helped to identify the massless spin 2 state in the closed string spectrum as the graviton. String theory is a candidate for quantum gravity. Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 3 / 14
Motivation For string phenomenology the 10d action must be reduced to 4d. These effective actions can be calculated with the help of scattering amplitudes. Substantial progress (recursion relations, symmetry considerations) has been made in calculating scattering amplitudes in SYM and SUGRA. ⇒ Similar results in string theory? For small string coupling and low string scale M s ∼ O (TeV) string theory gives corrections to SM processes [L¨ ust, Stieberger, Taylor] . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 4 / 14
Open string interactions Strings can interact only in a limited number of ways, i.e. by joining and splitting. Open string diagrams then look like . String diagrams are 2d Minkowskian surfaces in 10d space-time. After performing a Wick rotation they become 2d Euclidean surfaces with coordinates τ and σ . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 5 / 14
Disk diagrams With the complex coordinate z = τ + i σ open string diagrams are obviously subsets of the complex plane. ⇒ Use powerful tools from complex analysis! Using the Riemann mapping theorem tree-level diagrams can be mapped onto the unit disk D . V 1 V 2 1 2 4 3 V 4 V 3 The V i ’s are vertex operators creating and annihilating string states. Riemann mapping theorem If U ⊂ C is open and simply connected, there exists a unique biholomorphic mapping f from U onto the open unit disk D . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 6 / 14
Disk diagrams With the complex coordinate z = τ + i σ open string diagrams are obviously subsets of the complex plane. ⇒ Use powerful tools from complex analysis! Using the Riemann mapping theorem tree-level diagrams can be mapped onto the unit disk D . V 1 V 2 1 2 4 3 V 4 V 3 The V i ’s are vertex operators creating and annihilating string states. Riemann mapping theorem If U ⊂ C is open and simply connected, there exists a unique biholomorphic mapping f from U onto the open unit disk D . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 6 / 14
Disk diagrams With the complex coordinate z = τ + i σ open string diagrams are obviously subsets of the complex plane. ⇒ Use powerful tools from complex analysis! Using the Riemann mapping theorem tree-level diagrams can be mapped onto the unit disk D . V 1 V 2 1 2 4 3 V 4 V 3 The V i ’s are vertex operators creating and annihilating string states. Riemann mapping theorem If U ⊂ C is open and simply connected, there exists a unique biholomorphic mapping f from U onto the open unit disk D . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 6 / 14
Disk diagrams With the complex coordinate z = τ + i σ open string diagrams are obviously subsets of the complex plane. ⇒ Use powerful tools from complex analysis! Using the Riemann mapping theorem tree-level diagrams can be mapped onto the unit disk D . V 1 V 2 1 2 4 3 V 4 V 3 The V i ’s are vertex operators creating and annihilating string states. Riemann mapping theorem If U ⊂ C is open and simply connected, there exists a unique biholomorphic mapping f from U onto the open unit disk D . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 6 / 14
Disk diagrams With the complex coordinate z = τ + i σ open string diagrams are obviously subsets of the complex plane. ⇒ Use powerful tools from complex analysis! Using the Riemann mapping theorem tree-level diagrams can be mapped onto the unit disk D . V 1 V 2 1 2 4 3 V 4 V 3 The V i ’s are vertex operators creating and annihilating string states. Riemann mapping theorem If U ⊂ C is open and simply connected, there exists a unique biholomorphic mapping f from U onto the open unit disk D . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 6 / 14
Disk diagrams With the complex coordinate z = τ + i σ open string diagrams are obviously subsets of the complex plane. ⇒ Use powerful tools from complex analysis! Using the Riemann mapping theorem tree-level diagrams can be mapped onto the unit disk D . V 1 V 2 1 2 4 3 V 4 V 3 The V i ’s are vertex operators creating and annihilating string states. Riemann mapping theorem If U ⊂ C is open and simply connected, there exists a unique biholomorphic mapping f from U onto the open unit disk D . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 6 / 14
Disk diagrams With the complex coordinate z = τ + i σ open string diagrams are obviously subsets of the complex plane. ⇒ Use powerful tools from complex analysis! Using the Riemann mapping theorem tree-level diagrams can be mapped onto the unit disk D . V 1 V 2 1 2 4 3 V 4 V 3 The V i ’s are vertex operators creating and annihilating string states. Riemann mapping theorem If U ⊂ C is open and simply connected, there exists a unique biholomorphic mapping f from U onto the open unit disk D . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 6 / 14
Disk diagrams With the complex coordinate z = τ + i σ open string diagrams are obviously subsets of the complex plane. ⇒ Use powerful tools from complex analysis! Using the Riemann mapping theorem tree-level diagrams can be mapped onto the unit disk D . V 1 V 2 1 2 4 3 V 4 V 3 The V i ’s are vertex operators creating and annihilating string states. Riemann mapping theorem If U ⊂ C is open and simply connected, there exists a unique biholomorphic mapping f from U onto the open unit disk D . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 6 / 14
Mapping into the half plane Furthermore the unit disk D can be mapped to the upper complex half-plane H V 1 ( w 1 ) V 2 ( w 2 ) H V 4 ( w 4 ) V 3 ( w 3 ) V 1 ( z 1 ) V 2 ( z 2 ) V 3 ( z 3 ) V 4 ( z 4 ) obius transformation w �→ z ( w ) = i 1 + w making use of the M¨ 1 − w . To calculate the tree-level string scattering amplitude we have to evaluate the correlation function � V 1 ( z 1 ) V 2 ( z 2 ) V 3 ( z 3 ) V 4 ( z 4 ) � . Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 7 / 14
The full amplitude In QFT: sum over all Feynman diagrams contributing to a certain process e − e − e − e − e + e + e + . e + In string theory: sum over all cyclic non-equivalent configurations 1 2 1 4 1 2 , . . . 4 3 3 4 2 3 and integrate over the vertex operator positions z i � N � � A (1 , . . . , N ) ∝ d z i � V 1 ( z 1 ) V σ (2) ( z σ (2) ) . . . V σ ( N ) ( z σ ( N ) ) � . i =1 σ ∈ S N − 1 Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 8 / 14
Vertex operators Massless strings with both ends attached to a stack of N D-branes are U( N ) gauge bosons. Massless strings with the ends attached to different stacks of D-branes represent chiral matter. The vertex operators for these states are V A a ( z , ξ, k ) = g A λ a e − φ ( z ) ξ µ ψ µ ( z ) e i k ν X ν ( z ) , β e − φ ( z ) / 2 u λ S λ ( z ) e i k ν X ν ( z ) Ξ( z ) , β ( z , u , k ) = g ψ λ α V ψ α ψ µ vector spin field , e ikX ν is the momentum part, and Ξ is where is an external (4d) S α an internal (6d) field. Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 9 / 14
ψ - S -correlators The correlation function of vertex operators factorizes � V 1 ( z 1 ) . . . V N ( z N ) � ∝ �O X � · �O Ξ � ·�O ψ, S � . � �� � well-known For the scattering of two gauge bosons and two fermions one needs 1 � ψ µ ( z 1 ) ψ ν ( z 2 ) ψ λ ( z 3 ) S α ( z 4 ) S ˙ β ( z 5 ) � = √ 2 ( z 14 z 15 z 24 z 25 z 34 z 35 ) 1 / 2 � � z 45 z 14 z 25 z 14 z 35 z 24 z 35 2 + η µν σ λ − η µλ σ ν + η νλ σ µ ( σ µ ¯ σ ν σ λ ) α ˙ , β α ˙ α ˙ α ˙ β z 12 β z 13 β z 23 where z ij ≡ z i − z j . Tedious calculations which get more difficult when more spin fields are involved! Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 10 / 14
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