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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

1

Motivation

2

Scattering of open strings

3

ψ-S-correlators

4

Conclusion

Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 2 / 14

Historical motivation

Excited mesons exhibit Regge behavior, J ≈ α0 + α′m2. 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 Ms ∼ 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 + iσ 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.

V1 V3 V4 V2 1 4 2 3

The Vi’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 + iσ 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.

V1 V3 V4 V2 1 4 2 3

The Vi’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 + iσ 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.

V1 V3 V4 V2 1 4 2 3

The Vi’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 + iσ 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.

V1 V3 V4 V2 1 4 2 3

The Vi’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 + iσ 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.

1 4 2 3 V1 V3 V4 V2

The Vi’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 + iσ 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.

V1 V3 V4 V2 1 4 2 3

The Vi’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 + iσ 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.

V1 V3 V4 V2 1 4 2 3

The Vi’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 + iσ 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.

V1 V3 V4 V2 1 4 2 3

The Vi’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

V1(w1) V3(w3) V4(w4) V2(w2) H V1(z1) V2(z2) V3(z3) V4(z4)

making use of the M¨

• bius transformation w → z(w) = i 1 + w

1 − w .

To calculate the tree-level string scattering amplitude we have to evaluate the correlation function V1(z1) V2(z2) V3(z3) V4(z4) .

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 4 2 3 1 3 2 4 1 2 4 3 , . . .

and integrate over the vertex operator positions zi A(1, . . . , N) ∝

• N
• i=1

dzi

• σ∈SN−1

V1(z1) Vσ(2)(zσ(2)) . . . Vσ(N)(zσ(N)) .

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 VAa(z, ξ, k) = gA λa e−φ(z) ξµ ψµ(z) ei kνX ν(z) , Vψα

β (z, u, k) = gψ λα

β e−φ(z)/2 uλ Sλ(z) ei kνX ν(z) Ξ(z) ,

where ψµ Sα is an external (4d) vector spin field , eikX νis the momentum part, and Ξ is 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 V1(z1) . . . VN(zN) ∝ OX · OΞ

• well-known

·Oψ,S . For the scattering of two gauge bosons and two fermions one needs ψµ(z1) ψν(z2) ψλ(z3) Sα(z4) S ˙

β(z5) =

1 √ 2 (z14 z15 z24 z25 z34 z35)1/2

• (σµ¯

σνσλ)α ˙

β

z45 2 + ηµν σλ

α ˙ β

z14 z25 z12 − ηµλ σν

α ˙ β

z14 z35 z13 + ηνλ σµ

α ˙ β

z24 z35 z23

• ,

where zij ≡ zi − zj. 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

A systematic approach

Substitute all ψ’s: ψµ(z) = − 1

√ 2 ¯

σµ ˙

αα S ˙ α(z) Sα(z) .

Then the previous correlator becomes − 1 2 √ 2 σµ ˙

γγ σµ ˙ δδ σµ ϕ ˙ ϕ

× Sγ(z1) Sδ(z2) Sϕ(z3) Sα(z4) S ˙

γ(z1) S ˙ δ(z2) S ˙ ϕ(z3) S ˙ β(z5) .

An argument from CFT assures that in 4d this factorizes − 1 2 √ 2 σµ ˙

γγ σµ ˙ δδ σµ ϕ ˙ ϕ

× Sγ(z1) Sδ(z2) Sϕ(z3) Sα(z4) · S ˙

γ(z1) S ˙ δ(z2) S ˙ ϕ(z3) S ˙ β(z5) .

This factorizations generalizes to the case with arbitrary many Sα and S ˙

α.

Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 11 / 14

A systematic approach

Substitute all ψ’s: ψµ(z) = − 1

√ 2 ¯

σµ ˙

αα S ˙ α(z) Sα(z) .

Then the previous correlator becomes − 1 2 √ 2 σµ ˙

γγ σµ ˙ δδ σµ ϕ ˙ ϕ

× Sγ(z1) Sδ(z2) Sϕ(z3) Sα(z4) S ˙

γ(z1) S ˙ δ(z2) S ˙ ϕ(z3) S ˙ β(z5) .

An argument from CFT assures that in 4d this factorizes − 1 2 √ 2 σµ ˙

γγ σµ ˙ δδ σµ ϕ ˙ ϕ

× Sγ(z1) Sδ(z2) Sϕ(z3) Sα(z4) · S ˙

γ(z1) S ˙ δ(z2) S ˙ ϕ(z3) S ˙ β(z5) .

This factorizations generalizes to the case with arbitrary many Sα and S ˙

α.

Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 11 / 14

A systematic approach

Substitute all ψ’s: ψµ(z) = − 1

√ 2 ¯

σµ ˙

αα S ˙ α(z) Sα(z) .

Then the previous correlator becomes − 1 2 √ 2 σµ ˙

γγ σµ ˙ δδ σµ ϕ ˙ ϕ

× Sγ(z1) Sδ(z2) Sϕ(z3) Sα(z4) S ˙

γ(z1) S ˙ δ(z2) S ˙ ϕ(z3) S ˙ β(z5) .

An argument from CFT assures that in 4d this factorizes − 1 2 √ 2 σµ ˙

γγ σµ ˙ δδ σµ ϕ ˙ ϕ

× Sγ(z1) Sδ(z2) Sϕ(z3) Sα(z4) · S ˙

γ(z1) S ˙ δ(z2) S ˙ ϕ(z3) S ˙ β(z5) .

This factorizations generalizes to the case with arbitrary many Sα and S ˙

α.

Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 11 / 14

The master formula

We derived the general expression for the correlator with arbitrary many left-handed spin fields Sα1(z1) Sβ1(w1) ... Sαn(zn) Sβn(wn) = (−1)n  

n

• i≤j

(zi − wj)

n

• i>j

(wj − zi)  

1/2

× n

• k<l

zkl wkl −1/2 ×

• ρ∈Sn

sgn(ρ)

n

• m=1

ǫαmβρ(m) zm − wρ(m) . Using this approach every ψ-S-correlator for disk-diagrams can be calculated.

Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 12 / 14

Conclusion

String scattering amplitudes are useful for string theory and for string phenomenology. The tree level diagrams of open string scattering can be mapped onto D and

• H. The maps are biholomorphic.

String scattering amplitudes are calculated by evaluating correlation functions

• f vertex operators.

We have found a general formula to calculate arbitrary ψ-S-correlators for disk amplitudes in 4d. I hope to present some stringy physics results next year.

Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 13 / 14

Thanks for your attention.

Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 14 / 14

Backup: Operator Product Expansions

ψµ(z) ψν(w) = + ηµν z − w + . . . , Sα(z) Sβ(w) = − (z − w)−1/2 ǫαβ + . . . , S ˙

α(z) S ˙ β(w) = + (z − w)−1/2 ǫ ˙ α ˙ β + . . . , ,

Sα(z) S ˙

β(w) = + 1

√ 2(z − w)0 σµ

α ˙ β ψµ(w) + . . . ,

S ˙

α(z) Sβ(w) = + 1

√ 2(z − w)0 ¯ σµ α ˙

β ψµ(w) + . . . ,

ψµ(z) Sα(w) = + 1 √ 2(z − w)−1/2 σµ

α ˙ βS ˙ β(w) + · · · ,

ψµ(z) S ˙

α(w) = + 1

√ 2(z − w)−1/2 ¯ σµ ˙

αβSβ(w) + · · · .

Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 15 / 14

Backup: The MSSM from intersecting D6-branes

l W ± q (a) baryonic U(2) (d) leptonic U(1)R (c) right (b) left e u, d U(3) U(1)L g

from [L¨ ust, Stieberger, Taylor].

gauge bosons:

• pen strings with endpoints attached to same stack of branes,

chiral matter:

• pen strings with endpoints attached to different stacks of branes.

Daniel H¨ artl (MPP) Scattering Amplitudes in String Theory Stringberg Workshop 2009 16 / 14