Stephan Stieberger, MPP München
New relations between gauge and gravity amplitudes
in field and string theory
HKUST Jockey Club Institute for Advanced Study
Hongkong University January 20, 2017
I. Amplitude 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))
(real part) field–theory relations (Kleiss–Kuijf relations):
AY M (1, 2, . . . , N) +AY M (2, 1, 3, . . . , N � 1, N) + . . .+AY M (2, 3, . . . , N � 1, 1, N) = 0
(imaginary part) field–theory relations (BCJ relations):
s12 AY M (2, 1, 3, . . . , N � 1, N) + . . .+ (s12 + s13 + . . .+ s1N�1) AY M (2, 3, . . . , N � 1, 1, N) = 0
AYM (1,�(2, . . . , N � 2), N � 1, N)
(N-3)! dimensional basis of subamplitudes
•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 amplitudes from different string vacua
amplitudes are key players in establishing string dualities
Many relations in field-theory emerge from properties of string world-sheet:
monodromy on world-sheet yield KLT, BCJ, … relations
Structure of string amplitudes has deep impact on the form and organization of
quantum field theory amplitudes
== × from monodromiesof world-sheet
Properties of scattering amplitudes in both gauge and gravity theories suggest a deeper understanding from string theory
KLT
MFT (1, . . . , 4) = s12 AYM (1, 2, 3, 4) ˜AYM (1, 2, 4, 3)
graviton amplitude = (gauge amplitude) ⇥ (gauge amplitude)
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)
• 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, Phys. Lett. B744 (2015) 160-162 [arXiv:1502.00655]
• Subleading Terms in the Collinear Limit of Yang-Mills Amplitudes, Phys. Lett. B750 (2015) 587-590 [arXiv:1508.01116]
• New Relations for Einstein-Yang-Mills Amplitudes, Nucl. Phys. B913 (2016) 151-162, [arXiv:1606.09616]
based on: St.St., T.R. Taylor:
C+
Disk
z z1 2 zNzN−1
1
2
N
conformaltransformation
iterated real integral on RP1\{0, 1,1}
complex integral on P1\{0, 1,1}
Z
Cd2z
|z|2s |1� z|2u
z (1� z) z
Z 1
0dx x
s�1 (1� x)u
e.g. N=4:
s = ↵0(k1 + k2)2
t = ↵0(k1 + k3)2
u = ↵0(k1 + k4)2
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
◆
No KLT relations necessary !
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)
◆
= sv
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
J [⇢|⇡] Z⇡(⇢)
J = sv (Z)
= sv
0
BBBBBBBBB@
Z
0<z2<z3<1
dz2 dz3
4Qi<j
|zij |sij
z12z23
Z
0<z2<z3<1
dz2 dz3
4Qi<j
|zij |sij
z13z32
Z
0<z3<z2<1
dz2 dz3
4Qi<j
|zij |sij
z12z23
Z
0<z3<z2<1
dz2 dz3
4Qi<j
|zij |sij
z13z32
1
CCCCCCCCCA
0
BBBBBBBBBB@
Z
z2,z32C
d2z2 d2z3
4Qi<j
|zij |2sij
z12z23 z12z23z34
Z
z2,z32C
d2z2 d2z3
4Qi<j
|zij |2sij
z13z32 z12z23z34
Z
z2,z32C
d2z2 d2z3
4Qi<j
|zij |2sij
z12z23 z13z32z24
Z
z2,z32C
d2z2 d2z3
4Qi<j
|zij |2sij
z13z32 z13z32z24
1
CCCCCCCCCCA
N=5:Complex vs. iterated integrals
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
Z23(23) ⌘ 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)
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
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(⇧)
�
AIN = (gI
Y M )N�2X
⇧2SN /Z2
Tr(T a⇧(1) . . . T a⇧(N)) AI(⇧(1), . . . ,⇧(N))
AHETN = (gHET
Y M )N�2X
⇧2SN /Z2
Tr(T a⇧(1) . . . T a⇧(N)) AHET(⇧(1), . . . ,⇧(N)) +O(1/N2c )
sv= single-valued projection
↵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
relations between amplitudes involving open & closed strings and
pure open string amplitudes
St.St. arXiv:0907.2211
open closed strings: point pure open string amplitude
monodromy problem on the complex plane
“Doubling trick”: • convert disk correlators to the standard
holomorphic ones by extending the fields to the entire complex plane.
“KLT trick”: • integration over complex positions of closed
string states can be disentangled into real ones by introducing monodromy phases
sij ⌘ si,j = ↵0(ki + kj)2 = 2↵0kikj
⌧i(⇢) =
(sign(⇢�1(i)� ⇢�1(i+ 1)) (si,N�1 + si+1,N�1) , 3 i N � 3 ,
sN�2,N�1 , i = N�2
A(1, 2, . . . , N�2; q1, q2) = (�1)N e�⇡i(s1,N+s2,N�1)N�2X
l=2
(�1)l sin(⇡sl,N�1) e⇡i(�1)l sl,N�1
⇥X
⇢2{OP (↵,�t),l}
e⇡i
bN�32
cPk=1
⌧2k+1(⇢)S(⇢) A(1, ⇢, N � 1, N)
Nc = 1
S(⇢) ⌘ S[⇢(2, . . . , N � 2) ] =
N�2Y
i=2
N�2Y
j=i+1
exp
�⇡i ✓(⇢�1
(i)� ⇢�1(j)) si,j
E.g.:
A(1, 2, 3; q1, q2) = e�⇡is24⇥e�⇡is51 sin(⇡s34) A(1, 2, 3, 4, 5)� sin(⇡s24) A(1, 3, 2, 4, 5)
⇤N=5
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)
E.g.:
take collinear limit:
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�)
q1 = kN�1 =1
2q,
q2 = kN =1
2q
graviton is replaced by two gluons in collinear configurations
generalization to arbitrary collinear configuration:
q1 = kN�1 = x q,
q2 = kN = (1� x) q
A(1, 2, 3; q1, q2) = (1� x)
g
2s24 A(1, 5, 2, 4, 3) ,
A(1, 2, 3, 4; q1, q2) = (1� x)
g
2
n
s25 A(1, 6, 2, 5, 3, 4) + s45 A(1, 2, 3, 5, 4, 6)o
,
A(1, 2, 3, 4, 5; q1, q2) = (1� x)
g
2
n
s26 A(1, 7, 2, 6, 3, 4, 5) + s36 A(1, 2, 7, 3, 6, 4, 5)
+ (s36 + s26) A(1, 7, 2, 3, 6, 4, 5) + s56 A(1, 2, 3, 4, 6, 5, 7)o
graviton is replaced by two gluons in arbitrary collinear configurations
can explicitly be checked !
x is a free real parameter !
E.g.:
highly non-trivial !
based on these results we may take: x = 0
soft-gluon limit of (N-1)-th gluon
q1 = kN�1 ! 0,
q2 = kN ! q
with: limx!0
x (1� x) A(. . . ,m, xq
+, n, . . . ) = g
hmnihmqihqni A(. . . ,m, n, . . . ) ,
limx!0
x (1� x) A(. . . ,m, xq
�, n, . . . ) = g
[mn]
[mq][qn]A(. . . ,m, n, . . . )
IV. New relations between Einstein-Yang-Mills and Yang-Mills amplitudes
n
xl =lX
k=1
pk(N+1)-point on both sides !
AEYM (1, 2, . . . , N ; q±±) =
g
N�1X
l=1
(✏±P ·xl) A(1, 2, . . . , l, q±, l+1, . . . , N)
in any dimension D
V. 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)
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
by combining field and string theory structures obtain information on a possible alternative or dual
description of perturbative string amplitudes: obtain amplitudes from first principles
