[PPT] - On-shell recursion for string theory amplitudes on the disc and PowerPoint Presentation

SLIDE 1

On-shell recursion for string theory amplitudes on the disc and sphere

Rutger Boels

Niels Bohr International Academy, Copenhagen

based on: R.B., Daniele Marmiroli and Niels Obers arXiv:1002.xxxx [hep-th]

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 1 / 10

SLIDE 2

Motivation

scattering amplitudes are interesting.

direct link between theory and experiment simplest information to calculate from string theory or QFT

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 2 / 10

SLIDE 3

Motivation

scattering amplitudes are interesting.

direct link between theory and experiment simplest information to calculate from string theory or QFT contain much physical information is there anything interesting left to calculate?

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 2 / 10

SLIDE 4

Motivation

scattering amplitudes are interesting.

direct link between theory and experiment simplest information to calculate from string theory or QFT contain much physical information is there anything interesting left to calculate? → YES! increasing complexity with #particles, #loops in QFT still calculations needed even for LHC

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 2 / 10

SLIDE 5

Motivation

scattering amplitudes are interesting.

direct link between theory and experiment simplest information to calculate from string theory or QFT contain much physical information is there anything interesting left to calculate? → YES! increasing complexity with #particles, #loops in QFT still calculations needed even for LHC last few years quantum leaps in calculational technology in QFT

◮ surprisingly simple results (especially with susy) ◮ see also talks by [Henn] and [Broedel] Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 2 / 10

SLIDE 6

Motivation

scattering amplitudes are interesting.

direct link between theory and experiment simplest information to calculate from string theory or QFT contain much physical information is there anything interesting left to calculate? → YES! increasing complexity with #particles, #loops in QFT still calculations needed even for LHC last few years quantum leaps in calculational technology in QFT

◮ surprisingly simple results (especially with susy) ◮ see also talks by [Henn] and [Broedel]

what about ‘strings’? Just in a flat background?

◮ see also talk by [Mafra] Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 2 / 10

SLIDE 7

Fields vs strings state-of-the-art

field theory strings in flat background all Yang-Mills, gravity tree amplitudes in D = 4 all 1-loop (massless) N = 1 amplitudes all order conjectures in N = 4 6 point amplitude at tree level

[Stieberger-Oprisa, 02]

all multiplicity α′2, α′3 corrections to super Yang-Mills four point from effective action

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 3 / 10

SLIDE 8

Fields vs strings state-of-the-art unacceptable, strings have much more symmetry

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 3 / 10

SLIDE 9

Fields vs strings state-of-the-art

field theory strings in flat background all Yang-Mills, gravity tree amplitudes in D = 4 all 1-loop (massless) N = 1 amplitudes all order conjectures in N = 4 6 point amplitude at tree level

[Stieberger-Oprisa, 02]

all multiplicity α′2, α′3 corrections to super Yang-Mills four point from effective action

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 3 / 10

SLIDE 10

Fields vs strings state-of-the-art

field theory strings in flat background all Yang-Mills, gravity tree amplitudes in D = 4 all 1-loop (massless) N = 1 amplitudes all order conjectures in N = 4 analytic progress based on ‘analytic S-matrix’ 6 point amplitude at tree level

[Stieberger-Oprisa, 02]

all multiplicity α′2, α′3 corrections to super Yang-Mills four point from effective action whole theory based on ‘analytic S-matrix’

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 3 / 10

SLIDE 11

Main idea of our work

analytic S-matrix program (sixties)

construct scattering amplitudes from their physical singularities superseded by Lagrangian based approaches in seventies until recently, only success: CFT

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 4 / 10

SLIDE 12

Main idea of our work

analytic S-matrix program (sixties)

construct scattering amplitudes from their physical singularities superseded by Lagrangian based approaches in seventies until recently, only success: CFT and string theory "Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories" [Veneziano, 68]

✎ ✍ ☞ ✌ ✎ ✍ ☞ ✌

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 4 / 10

SLIDE 13

Main idea of our work

analytic S-matrix program (sixties)

construct scattering amplitudes from their physical singularities superseded by Lagrangian based approaches in seventies until recently, only success: CFT and string theory "Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories" [Veneziano, 68]

✎ ✍ ☞ ✌ ✎ ✍ ☞ ✌

revival inspired by [Witten, 03]

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 4 / 10

SLIDE 14

Main idea of our work

analytic S-matrix program (sixties)

construct scattering amplitudes from their physical singularities superseded by Lagrangian based approaches in seventies until recently, only success: CFT and string theory "Construction of a crossing-symmetric, Regge behaved amplitude for linearly rising trajectories" [Veneziano, 68]

✎ ✍ ☞ ✌ ✎ ✍ ☞ ✌

revival inspired by [Witten, 03] exporting new QFT techniques to string theory natural (cf. [Stieberger-Taylor, 06-]) this talk: on-shell recursion [Britto-Cachazo-Feng-(Witten), 04,05]

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 4 / 10

SLIDE 15

On-shell recursion relations

new input since 60s

if there is no nice complex parameter to play with: introduce one

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 5 / 10

SLIDE 16

On-shell recursion relations

amplitudes are above all functions of momenta (and quantum #s)

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 5 / 10

SLIDE 17

On-shell recursion relations

amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while remaining on-shell? BCFW: → pµ

i → ˆ

pµ

i = pµ i + z nµ

pµ

j → ˆ

pµ

j = pµ j − z nµ

(pµ

i nµ) = (pµ j nµ) = (nµnµ) = 0

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 5 / 10

SLIDE 18

On-shell recursion relations

amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while remaining on-shell? BCFW: → pµ

i → ˆ

pµ

i = pµ i + z nµ

pµ

j → ˆ

pµ

j = pµ j − z nµ

(pµ

i nµ) = (pµ j nµ) = (nµnµ) = 0

amplitude A → A(z) A(0) =

z=0

A(z) z = −

Resz=finite + Resz=∞
Rutger Boels (NBIA)
n-shell recursion in string theory

Nordic String Meeting 2010 5 / 10

SLIDE 19

On-shell recursion relations

amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while remaining on-shell? BCFW: → pµ

i → ˆ

pµ

i = pµ i + z nµ

pµ

j → ˆ

pµ

j = pµ j − z nµ

(pµ

i nµ) = (pµ j nµ) = (nµnµ) = 0

amplitude A → A(z) A(0) =

z=0

A(z) z = −

Resz=finite + Resz=∞
finite z residues: lower point amplitudes

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 5 / 10

SLIDE 20

On-shell recursion relations

amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while remaining on-shell? BCFW: → pµ

i → ˆ

pµ

i = pµ i + z nµ

pµ

j → ˆ

pµ

j = pµ j − z nµ

(pµ

i nµ) = (pµ j nµ) = (nµnµ) = 0

amplitude A → A(z) A(0) =

z=0

A(z) z = −

Resz=finite + Resz=∞
?

finite z residues: lower point amplitudes → recursion!

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 5 / 10

SLIDE 21

On-shell recursion relations

amplitudes are above all functions of momenta (and quantum #s) change D-dim momenta while remaining on-shell? BCFW: → pµ

i → ˆ

pµ

i = pµ i + z nµ

pµ

j → ˆ

pµ

j = pµ j − z nµ

(pµ

i nµ) = (pµ j nµ) = (nµnµ) = 0

amplitude A → A(z) A(0) =

z=0

A(z) z = −

Resz=finite + Resz=∞
?

finite z residues: lower point amplitudes → recursion! string amplitudes as infinite sums over three point amplitudes z → ∞ related to UV (∼ Regge) behavior different possible shifts related by crossing symmetry

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 5 / 10

SLIDE 22

4 point example

[RB-Larsen-Obers-Vonk,08]

Veneziano amplitude

A4 = Apart

4 (s, t) (Tr)1 + Apart 4 (t, u) (Tr)2 + Apart 4 (u, s) (Tr)3

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 6 / 10

SLIDE 23

4 point example

[RB-Larsen-Obers-Vonk,08]

Veneziano amplitude

A4 = Apart

4 (s, t) (Tr)1 + Apart 4 (t, u) (Tr)2 + Apart 4 (u, s) (Tr)3

definiteness: shift particles 1 and 2 ˆ s = s ˆ t = t − z′ ˆ u = u + z′ with z′ = 2α′(pµ

3nµ)z (special 4-pt kinematics)

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 6 / 10

SLIDE 24

4 point example

[RB-Larsen-Obers-Vonk,08]

Veneziano amplitude

A4 = Apart

4 (s, t) (Tr)1 + Apart 4 (t, u) (Tr)2 + Apart 4 (u, s) (Tr)3

definiteness: shift particles 1 and 2 ˆ s = s ˆ t = t − z′ ˆ u = u + z′ with z′ = 2α′(pµ

3nµ)z (special 4-pt kinematics)

Apart

4 (s, t) = ∞

n=0

(−1)n Γ(n + 1) Γ(α′s − 1) Γ(α′s − 1 − n)

1

α′t − 1 + n

Apart

4 (t, u) = ∞

n=0

(−1)n Γ(n + 1) Γ(1 − α′s + n) Γ(1 − α′s)

1

α′t − 1 + n + 1 α′u − 1 + n

Rutger Boels (NBIA)
n-shell recursion in string theory

Nordic String Meeting 2010 6 / 10

SLIDE 25

4 point example

[RB-Larsen-Obers-Vonk,08]

Veneziano amplitude

A4 = Apart

4 (s, t) (Tr)1 + Apart 4 (t, u) (Tr)2 + Apart 4 (u, s) (Tr)3

definiteness: shift particles 1 and 2 ˆ s = s ˆ t = t − z′ ˆ u = u + z′ with z′ = 2α′(pµ

3nµ)z (special 4-pt kinematics)

Apart

4 (s, t) = ∞

n=0

(−1)n Γ(n + 1) Γ(α′s − 1) Γ(α′s − 1 − n)

1

α′t − 1 + n

Apart

4 (t, u) = ∞

n=0

(−1)n Γ(n + 1) Γ(1 − α′s + n) Γ(1 − α′s)

1

α′t − 1 + n + 1 α′u − 1 + n

→ all four point amplitudes

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 6 / 10

SLIDE 26

On-shell recursion relations in string theory

all open string amplitudes have a Koba-Nielsen type integrand

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 7 / 10

SLIDE 27

On-shell recursion relations in string theory

all open string amplitudes have a Koba-Nielsen type integrand Koba-Nielsen amplitudes @ large-z color-adjacent shifts lim

z→∞ A(z) ∼ zα′(pi+pi+1)2+1

G0 + O

1 z

non-adjacent shifts

lim

z→∞ A(z) ∼ e±(α′)zzα′(pi+pj)2+1

G′

0 + O

1 z

if α′(pi + pi+1)2 < − 2 then no residue at infinity

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 7 / 10

SLIDE 28

On-shell recursion relations in string theory

all open string amplitudes have a Koba-Nielsen type integrand Koba-Nielsen amplitudes @ large-z color-adjacent shifts lim

z→∞ A(z) ∼ zα′(pi+pi+1)2+1

G0 + O

1 z

non-adjacent shifts

lim

z→∞ A(z) ∼ e±(α′)zzα′(pi+pj)2+1

G′

0 + O

1 z

if α′(pi + pi+1)2 < − 2 then no residue at infinity

direct integral proof for adjacent shifts non-adjacent shifts through monodromy relations [Plahte, 70]

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 7 / 10

SLIDE 29

On-shell recursion relations in string theory

all open string amplitudes have a Koba-Nielsen type integrand Koba-Nielsen amplitudes @ large-z color-adjacent shifts lim

z→∞ A(z) ∼ zα′(pi+pi+1)2+?

G0 + O

1 z

non-adjacent shifts

lim

z→∞ A(z) ∼ e±(α′)zzα′(pi+pj)2+?

G′

0 + O

1 z

if α′(pi + pi+1)2 < ? then no residue at infinity

direct integral proof for adjacent shifts non-adjacent shifts through monodromy relations [Plahte, 70]

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 7 / 10

SLIDE 30

On-shell recursion relations in string theory

all open string amplitudes have a Koba-Nielsen type integrand Koba-Nielsen amplitudes @ large-z color-adjacent shifts lim

z→∞ A(z) ∼ zα′(pi+pi+1)2+?

G0 + O

1 z

non-adjacent shifts

lim

z→∞ A(z) ∼ e±(α′)zzα′(pi+pj)2+?

G′

0 + O

1 z

if α′(pi + pi+1)2 < ? then no residue at infinity

direct integral proof for adjacent shifts non-adjacent shifts through monodromy relations [Plahte, 70] all tree level amplitudes in a flat background obey on-shell recursion in any string theory, depending on kinematic invariant proven for open string / argued for closed string (see later) conjecture: ? universal for shifted particles

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 7 / 10

SLIDE 31

CFT point of view

how generic are recursion relations in string theory? curved backgrounds? loops? → study CFT

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 8 / 10

SLIDE 32

CFT point of view

how generic are recursion relations in string theory? curved backgrounds? loops? → study CFT

adjacent shifts from CFT argument (based on [Brower et.al., 06])

:eip1X(y) ::eip2X(0) := 1 y 2α′p1p2 :ei(p1X(y)+p2X(0))+iznµ(Xµ(y)−Xµ(0)) : Taylor expand exponential in y, assume yz ∼ 1 and integrate

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 8 / 10

SLIDE 33

CFT point of view

how generic are recursion relations in string theory? curved backgrounds? loops? → study CFT

adjacent shifts from CFT argument (based on [Brower et.al., 06])

:eip1X(y) ::eip2X(0) := 1 y 2α′p1p2 :ei(p1X(y)+p2X(0))+iznµ(Xµ(y)−Xµ(0)) : Taylor expand exponential in y, assume yz ∼ 1 and integrate shifting gluon legs in the bosonic string An(z) ∼ ˆ ζµ

1 ˆ

ζ2

ν 1

z α′(p1+p2)2 z

ηµν + α′Aµν
+ Bµν + O

1 z

analysis reduces to [Arkani-Hamed, Kaplan, 08]

same result through integral derivation

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 8 / 10

SLIDE 34

CFT point of view

how generic are recursion relations in string theory? curved backgrounds? loops? → study CFT

adjacent shifts from CFT argument (based on [Brower et.al., 06])

:eip1X(y) ::eip2X(0) := 1 y 2α′p1p2 :ei(p1X(y)+p2X(0))+iznµ(Xµ(y)−Xµ(0)) : Taylor expand exponential in y, assume yz ∼ 1 and integrate shifting gluon legs in the superstring An(z) ∼ ˆ ζµ

1 ˆ

ζ2

ν 1

z α′(p1+p2)2 z

ηµν(1 + 2α′p1 · p2)
+ Bµν + O

1 z

analysis reduces to [Arkani-Hamed, Kaplan, 08]

same result through integral derivation (much more involved)

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 8 / 10

SLIDE 35

more CFT

large z in CFT only depends on adjacent shifted particles

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 9 / 10

SLIDE 36

more CFT

large z in CFT only depends on adjacent shifted particles applies to

◮ B-field backgrounds ◮ constant Abelian backgrounds [RB, to appear] (new in field theory) ◮ closed strings as ‘square’ of open string (also through KLT) Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 9 / 10

SLIDE 37

more CFT

large z in CFT only depends on adjacent shifted particles applies to

◮ B-field backgrounds ◮ constant Abelian backgrounds [RB, to appear] (new in field theory) ◮ closed strings as ‘square’ of open string (also through KLT)

CFT derivation monodromy relations [Plahte, 70] for non-adjacent shifts :V1(z1): :V2(z2): ≡ :V2(z2): :V1(z1): R12 in flat backgrounds R12 ∼ e±2πiα′(p1p2)Sign(

z1−z2)

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 9 / 10

SLIDE 38

more CFT

large z in CFT only depends on adjacent shifted particles applies to

◮ B-field backgrounds ◮ constant Abelian backgrounds [RB, to appear] (new in field theory) ◮ closed strings as ‘square’ of open string (also through KLT)

CFT derivation monodromy relations [Plahte, 70] for non-adjacent shifts :V1(z1): :V2(z2): ≡ :V2(z2): :V1(z1): R12 in flat backgrounds R12 ∼ e±2πiα′(p1p2)Sign(

z1−z2)

z1

′ 0| : V(z1) : . . . : V(zn) : |0 = 0 → A(1, 2, . . . , n)+R12A(2, 1, 3, . . . , n)+R12R23A(2, 3, 1, . . . , n)+. . . = 0

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 9 / 10

SLIDE 39

more CFT

large z in CFT only depends on adjacent shifted particles applies to

◮ B-field backgrounds ◮ constant Abelian backgrounds [RB, to appear] (new in field theory) ◮ closed strings as ‘square’ of open string (also through KLT)

CFT derivation monodromy relations [Plahte, 70] for non-adjacent shifts :V1(z1): :V2(z2): ≡ :V2(z2): :V1(z1): R12 in flat backgrounds R12 ∼ e±2πiα′(p1p2)Sign(

z1−z2)

z1

′ 0| : V(z1) : . . . : V(zn) : |0 = 0 → A(1, 2, . . . , n)+R12A(2, 1, 3, . . . , n)+R12R23A(2, 3, 1, . . . , n)+. . . = 0 general CFT: R12 obeys the Yang-Baxter equation

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 9 / 10

SLIDE 40

Conclusions and outlook

Have done

n-shell recursion in tree level strings in flat backgrounds

◮ proven for open strings ◮ argued for closed strings

very natural, CFT interpretation

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 10 / 10

SLIDE 41

Conclusions and outlook

Have done

n-shell recursion in tree level strings in flat backgrounds

◮ proven for open strings ◮ argued for closed strings

very natural, CFT interpretation To do study curved backgrounds, loops calculate amplitudes, extract information

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 10 / 10

SLIDE 42

Conclusions and outlook

Have done

n-shell recursion in tree level strings in flat backgrounds

◮ proven for open strings ◮ argued for closed strings

very natural, CFT interpretation To do study curved backgrounds, loops calculate amplitudes, extract information Dream of

n-shell recursion ↔ crossing symmetry in CFT

Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 10 / 10

SLIDE 43

Conclusions and outlook

Have done

n-shell recursion in tree level strings in flat backgrounds

◮ proven for open strings ◮ argued for closed strings

very natural, CFT interpretation To do study curved backgrounds, loops calculate amplitudes, extract information Dream of

n-shell recursion ↔ crossing symmetry in CFT

◮ is there a string amplitude bootstrap? Rutger Boels (NBIA)

n-shell recursion in string theory

Nordic String Meeting 2010 10 / 10