Stephan Stieberger, MPP München

New relations between gauge and gravity amplitudes

in field and string theory

Eurostrings March 23-27, 2015 DAMTP Cambridge

I. Amplitude relations

•relations between different amplitudes within one theory

•relations between amplitudes from different theories

supersymmetric Ward identities in gauge and gravity theory

relations between gauge and gravity amplitudes: (perturbative) Kawai-Lewellen-Tye (KLT) relations

•relations among same amplitudes within one theory

gauge theory: cyclicity, reflection, parity,

Kleiss-Kuijf (KK), Bern-Carrasco-Johansson (BCJ) relations

Tree-level N-point QCD amplitude:

AN = gN�2YM

X

⇧2SN�1

Tr(T a1T a⇧(2) . . . T a⇧(N)) AYM (1,⇧(2), . . . ,⇧(N))

S = KLT kernel

Bern, Dixon, Perelstein, Rozowsky (1998)

S[⇢|�] := S[ ⇢(2, . . . , N � 2) | �(2, . . . , N � 2) ]

=N�2Y

j=2

⇣s1,j⇢ +

j�1X

k=2

✓(j⇢, k⇢) sj⇢,k⇢

⌘

Supergravity graviton N-point tree-level amplitude:

sij = ↵0(ki + kj)2

MFT (1, . . . , N) = (�1)N�3 N�2X

�2SN�3

AYM (1,�(2, 3, . . . , N � 2), N � 1, N)

⇥X

⇢2SN�3

S[⇢|�] AYM (1, ⇢(2, 3, . . . , N � 2), N,N � 1)

KLT MFT (1, . . . , 4) = s12 AYM (1, 2, 3, 4) ˜AYM (1, 2, 4, 3)

graviton amplitude = (gauge amplitude) ⇥ (gauge amplitude)

Structure of string amplitudes has deep impact on the form and organization of quantum field theory amplitudes

== × from monodromiesof world-sheet

many relations in field-theory emerge from properties of string world-sheet:

monodromy on world-sheet yield KLT, BCJ, … relations

Properties of scattering amplitudes in both gauge and gravity theories suggest a deeper understanding from string theory

• Closed string amplitudes as single-valued open string amplitudes, Nucl. Phys. B881 (2014) 269-287, [arXiv:1401.1218]

• Graviton as a Pair of Collinear Gauge Bosons, Phys. Lett. B739 (2014) 457-461, [arXiv:1409.4771]

• Graviton Amplitudes from Collinear Limits of Gauge Amplitudes, [arXiv:1502.00655]

based on: St.St., T.R. Taylor:

•relations among amplitudes from different string vacua

amplitudes are key players in establishing string dualities

II. Heterotic gauge amplitudes as single-valued type I gauge amplitudes

Tree-level N-point type I open superstring gauge amplitude:

Tree-level N-point heterotic closed string gauge amplitude:

Result: AHET(⇧) = sv�AI(⇧)

�

sv= single-valued projection

AHETN = (gHET

YM )N�2X

⇧2SN/ZN

Tr(T a⇧(1)T a⇧(2) . . . T a⇧(N)) AHET(⇧(1), . . . ,⇧(N)) +O(1/N2c )

AIN = (gIYM )N�2

X

⇧2SN/ZN

Tr(T a⇧(1)T a⇧(2) . . . T a⇧(N)) AI(⇧(1), . . . ,⇧(N))

E.g.:

What are these amplitudes describing ?

↵0 � expansion : e��I ⇣n1,...,nr ↵0lTr(F 2+l

) ,rX

i=1

ni = l, l 6= 0, 1

e��I ⇣2 ↵02 Tr(F 4) Born-Infeld type couplings

sv-projection

E.g.:

no tree-level term (consistent with heterotic-type I duality)

TrF 4

⇣n1,...,nr := ⇣(n1, . . . , nr) =X

0<k1<...<kr

rY

l=1

k�nll , nl 2 N+ , nr � 2 ,

e�2�H ⇣SV (2) ↵02 Tr(F 4) = 0

↵0 � expansion : e�2�H ⇣SV(n1, . . . , nr) ↵

0lTr(F 2+l

)

⇣SV (2) = 0

e.g. N=4:

No KLT relations necessary !

KLT:Z

Cd

2z

|z|2s |1� z|2u

z (1� z) z

= sin(⇡u)✓Z 1

0x

s�1 (1� x)u�1

◆ ✓Z 1

1x

t�1 (1� x)u

◆

Z

Cd

2z

|z|2s |1� z|2u

z (1� z) z

= sv✓Z 1

0dx x

s�1 (1� x)u

◆

1s

�(s) �(u) �(t)�(�s) �(�u) �(�t)

= sv✓

�(s) �(1 + u)�(1 + s + u)

◆

s = ↵0(k1 + k2)2

t = ↵0(k1 + k3)2

u = ↵0(k1 + k4)2

iterated real integral on RP1\{0, 1,1}complex integral on P1\{0, 1,1}

= sv

This is generalized to any closed string amplitude: closed string amplitudes as

single-valued open string amplitudes

Complex vs. iterated integrals:

D(⇡) =�zj 2 R | 0 < z⇡(2) < . . . < z⇡(N�2) < 1

Z

CN�3

0

@N�2Y

j=2

d2zj

1

A

N�1Qi<j

|zij |↵0sij

z1,⇢(2) z⇢(2),⇢(3) . . . z⇢(N�3),⇢(N�2)

1

z1,⇡(2) z⇡(2),⇡(3) . . . z⇡(N�2),N�1

= sv

Z

D(⇡)

0

@N�2Y

j=2

dzj

1

A

N�1Qi<j

|zij |↵0sij

z1,⇢(2) z⇢(2),⇢(3) . . . z⇢(N�3),⇢(N�2)

zij := zi � zj

⇢,⇡ 2 SN�3

Multiple zeta-values in superstring theoryDisk integrals: iterated real integral on RP1\{0, 1,1}

Terasoma & Brown: the coefficients of the Taylor expansion of the Selberg integrals w.r.t. the variables

can be expressed as linear combinations of MZVs oversij

Q

⇣n1,...,nr := ⇣(n1, . . . , nr) =X

0<k1<...<kr

rY

l=1

k�nll , nl 2 N+ , nr � 2 ,

Commutative graded - algebra:Q Z =M

k�0

Zk , dimQ(ZN ) = dN

with: (Zagier)dN = dN�2 + dN�3, d0 = 1, d1 = 0, d2 = 1, . . .

Expand w.r.t. : ↵0

V �1CKG

Z

zi<zi+1

0

@5Y

j=1

dzj

1

AY

1i<j5

|zij |sijz12z23z35z54z41

= ↵0�2✓

1

s12s45+

1

s23s45

◆+ ⇣(2)

✓1� s34

s12� s12

s45� s23

s45� s51

s23

◆+O(↵0)

• MZVs occur as the values at unity of MPs

multiple polylogarithms:

(Commutative) graded Q–algebra:

Z =!

k≥0

Zk , dimQ(ZN) = dN ,

with: dN = dN−2 + dN−3, d0 = 1, d1 = 0, d2 = 1, . . . (Zagier)

w 2 3 4 5 6 7 8 9 10 11 12

Zw ζ2 ζ3 ζ22 ζ5 ζ23 ζ7 ζ3,5 ζ9 ζ3,7 ζ3,3,5 ζ2 ζ33 ζ1,1,4,6 ζ2 ζ3,7

ζ2 ζ3 ζ32 ζ2 ζ5 ζ3 ζ5 ζ33 ζ3 ζ7 ζ3,5 ζ3 ζ2 ζ9 ζ3,9 ζ22 ζ3,5

ζ22 ζ3 ζ2 ζ23 ζ2 ζ7 ζ25 ζ11 ζ22 ζ7 ζ3 ζ9 ζ2 ζ25

ζ42 ζ22 ζ5 ζ2 ζ3,5 ζ23 ζ5 ζ32 ζ5 ζ5 ζ7 ζ2 ζ3 ζ7

ζ32 ζ3 ζ2 ζ3 ζ5 ζ42 ζ3 ζ43 ζ22 ζ3 ζ5

ζ22 ζ23 ζ32 ζ23

ζ52 ζ62

dw 1 1 1 2 2 3 4 5 7 9 12

E .g . weight 12 : ζ5,7 = 149

ζ3,9 + 283

ζ5 ζ7 −7762241576575

ζ62

Blumlein, Broadhurst, Vermaseren

Single-valued MZVs

• special class of MZVs, which occurs as the values at unity of SVMPs

SVMPs: multiple polylogarithms can be combined with their complex conjugates to remove monodromy at

rendering the function single-valued on . z = 0, 1,1P1\{0, 1,1}

L2(z) = D(z) = Im {Li2(z) + ln |z| ln(1� z)}

Ln(z) = Ren

(nX

k=1

(� ln(|z|)n�k

(n� k)!Lik(z) +

lnn |z|(2n)!

)

with: Ren =

(Im, n even

Re, n odd

(Bloch-Wigner dilogarithm)

Ln(1) = Ren {Lin(1)} =

(0, n even

⇣n, n odd

(Zagier)

⇣sv(n1, . . . , nr) 2 R

polylogarithms : ln(z), Li1(z) = �ln(1� z), Lia(z), Lia1,...,ar (1, . . . , 1, z)

• coefficients of the Deligne associator W:

d

dzLe0,e1(z) = Le0,e1(z)

✓e0

z+

e1

1� z

◆(reduced) KZ equation:

with generators e0 and e1

of the free Lie algebra g

its unique solution can be given as generating series of multiple polylogarithms:

Le0,e1(z) =X

w2{e0,e1}⇥Lw(z) w

with the symbol w 2 {e0, e1}⇥

denoting a non-commutative word

w1w2 . . . in the letters wi 2 {e0, e1}

F. Brown (2004) defines generating series of SVMPs:

Drinfeld associator Z:

Z(e0, e1) := Le0,e1(1) =X

w2{e0,e1}⇥⇣(w) w = 1 + ⇣2 [e0, e1] + ⇣3 ( [e0, [e0, e1]]� [e1, [e0, e1] ) + . . .

Le0,e1(z) = L�e0,�e01(z)�1 Le0,e1(z)

e01 determined recursively by fixed–point equation:

Z(�e0,�e01) e01 Z(�e0,�e01)�1

= Z(e0, e1) e1 Z(e0, e1)�1

Deligne associator W:

W (e0, e1) = 1 + 2 ⇣3 ([e0, [e0, e1]]� [e1, [e0, e1]) + . . . F. Brown (2013)

W (e0, e1) := L(1) = Z(�e0,�e01)�1 Z(e0, e1) =

X

w2{e0,e1}⇥⇣sv(w) w

⇣(e1en1�10 . . . e1e

nr�10 ) = ⇣n1,...,nr

⇣(w1)⇣(w2) = ⇣(w1 w2) and ⇣(e0) = 0 = ⇣(e1)

L1 = 1,

Len0

=1n!

lnn z,

Len1

=1n!

lnn(1� z)

F. Brown (2013):

There is a natural homomorphism:

⇣sv(2) = 0

⇣sv(2n+ 1) = 2 ⇣2n+1

⇣sv(3, 5) = �10 ⇣3 ⇣5

sv : ⇣n1,...,nr �! ⇣sv(n1, . . . , nr)

⇣sv(3, 5, 3) = 2 ⇣3,5,3 � 2 ⇣3 ⇣3,5 � 10 ⇣23 ⇣5

⇣sv(3, 3, 5) = 2 ⇣3,3,5 � 5 ⇣23 ⇣5 + 90 ⇣2 ⇣9 +12

5⇣22 ⇣7 �

8

7⇣32 ⇣25

↵0

unexpected relation between

open and closed string amplitudes

(beyond KLT)

new string duality(to all orders in i.e. beyond BPS)

AHET(⇧) = sv�AI(⇧)

�Result:

↵0

• By applying naively KLT relations we would not have arrived at these relations

• Much deeper connection between open and closed string amplitudes than what is implied by KLT relations

• Full - dependence of closed string amplitude is entirely encapsulated by open string amplitude

• Any closed string amplitude can be written as single-valued image of open string amplitude

• Various connections between different amplitudes of different vacua can be established

New kind of duality relating amplitudes involving full tower of massive string excitations

(not just BPS states as in most examples of string dualities)

III. Mixed amplitudes in field- and string theory

Mixed amplitudes involving open and closed strings:

x1 x2 x3

H+

xNo−1 xNo

z2

zNc−1

zNc

z3

z1

“Doubling trick”: • convert disk correlators to the standard

holomorphic ones by extending the fields to the entire complex plane.

• integration over positions of world-sheet symmetric closed string states (such as graviton or dilaton) can be extended from the half-plane to the full complex plane

St.St. arXiv:0907.2211

open closed strings: point pure open string amplitude

monodromy problem on the complex plane

si,j ⌘ sij = 2↵0kikj(dN2e � 2) (bN

2c � 1) terms

relations between amplitudes involving open & closed strings and

pure open string amplitudes

A(1, 2, . . . , N�2; q)

=

dN2 e�1X

l=2

lX

i=2

sin

0

@⇡lX

j=i

sj,N�1

1

A A(1, . . . , i� 1, N, i, . . . , l, N � 1, l + 1, . . . , N � 2)

+N�3X

l=dN2 e

N�2X

i=l+1

sin

0

@⇡iX

j=l+1

sj,N�1

1

A A(1, . . . , l, N � 1, l + 1, . . . , i, N, i+ 1, . . . , N � 2)

Nc = 1

A(1, 2, 3; q) = sin(⇡s24) A(1, 5, 2, 4, 3) ,

A(1, 2, 3, 4; q) = sin(⇡s25) A(1, 6, 2, 5, 3, 4) + sin(⇡s45) A(1, 2, 3, 5, 4, 6) ,

A(1, 2, 3, 4, 5; q) = sin(⇡s26) A(1, 7, 2, 6, 3, 4, 5) + sin(⇡s36) A(1, 2, 7, 3, 6, 4, 5)

+ sin[⇡(s36 + s26)] A(1, 7, 2, 3, 6, 4, 5) + sin(⇡s56) A(1, 2, 3, 4, 6, 5, 7)

(in collinear limit)

Examples:

take field-theory limit:

with SYM amplitude:

“graviton appears as a pair of collinear gauge bosons”

yields Einstein-Yang-Mills for any kinematical configuration

AEYM (1+, 2+, 3�; q��) = ⇡ s24 AYM(1+, 5�, 2+, 4�, 3�)

MHV case: Bern, De Freitas, Wong, arXiv:hep-th/9912033 from squares of open string amplitudes (heterotic string)

IV. Graviton amplitudes from gauge amplitudes

express N-graviton amplitude in Einstein’s gravity as collinear limits of

certain linear combinations of pure SYM amplitudes in which each graviton is represented by two gauge bosons

no string theory ! but motivated from string theory

(2N-2 gluons become collinear without producing poles)

Proof: contributions from factorization on triple pole

p

::

q

::

N−1

1

(N−2)π

N−1

(N−2)ρ

1

l = p−q2

yields:

AE [k1,�1; . . . ; kN�1,�N�1; kN ,�N = +2]

=X

⇡,⇢2SN�3

S[⇡|⇢] AYM [pN , µN = +1;N�1, 1,⇡(2, 3, . . . , N�2)]

⇥AYM [1, ⇢(2, . . . , N�2), N�1; qN , ⌫N = +1]

9>>>>=

>>>>;

AYM [p+,�l

�,�p

�N ] =

x

3

2hpqi,

AYM [q�, l+,�q

�N ] =

x

2hpqi

AYM [p,N�1, 1,⇡(2, 3, . . . , N�2), 1, ⇢(2, . . . , N�2), N�1, q] !✓

4

spq

◆3

⇥

⇥ AYM [p+,�l�,�p�N ]⇥AYM [pN , µN = +1;N�1, 1,⇡(2, 3, . . . , N�2)]

⇥ AYM [1, ⇢(2, . . . , N�2), N�1; qN , ⌫N = +1]⇥AYM [q�, l+,�q�N ]

s3pq ⇠ (p� q)6

p

::

q

::

N−1

1

(N−2)π

N−1

(N−2)ρ

1

−l l

pN qN

Concluding remarks

• new kind of duality working beyond usual BPS protected operators

• graviton scattering unified into gauge amplitudes

• growing set of interconnections between open & closed amplitudes with

gauge theory and supergravity amplitudes