Evaluating the 6-point remainder function
near the collinear limit
Georgios Papathanasiou
Laboratoire d’Annecy-le-Vieux
de Physique Théorique & CERN
Moriond QCD and High Energy Interactions
March 27, 2014
1310.5735 [hep-th]
work in progress
Outline
Motivation: Why N = 4 SYM?
Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
Main Part: Implications for the 6-pt Amplitude
Conclusions & Outlook
GP — 6-point remainder function near the collinear limit
2/28
N = 4 Super Yang Mills Theory & Why Should We Care
Unique possibility for the nonperturbative investigation of gauge theories
GP — 6-point remainder function near the collinear limit
Motivation: Why N = 4 SYM?
3/28
N = 4 Super Yang Mills Theory & Why Should We Care
Unique possibility for the nonperturbative investigation of gauge theories
I
N = 4 SU (N ) SYM ⇔ Type IIB string theory on AdS5 × S 5 .
GP — 6-point remainder function near the collinear limit
Motivation: Why N = 4 SYM?
3/28
N = 4 Super Yang Mills Theory & Why Should We Care
Unique possibility for the nonperturbative investigation of gauge theories
I
N = 4 SU (N ) SYM ⇔ Type IIB string theory on AdS5 × S 5 .
strongly coupled ⇔ weakly coupled [See talks by Djuric,Raben,Teramond]
GP — 6-point remainder function near the collinear limit
Motivation: Why N = 4 SYM?
3/28
N = 4 Super Yang Mills Theory & Why Should We Care
Unique possibility for the nonperturbative investigation of gauge theories
I
N = 4 SU (N ) SYM ⇔ Type IIB string theory on AdS5 × S 5 .
strongly coupled ⇔ weakly coupled [See talks by Djuric,Raben,Teramond]
I
In the ’t Hooft limit, N → ∞ with λ = gY2 M N fixed:
Integrable structures ⇒ All loop, interpolating observables!
GP — 6-point remainder function near the collinear limit
Motivation: Why N = 4 SYM?
3/28
N = 4 Super Yang Mills Theory & Why Should We Care
Unique possibility for the nonperturbative investigation of gauge theories
I
N = 4 SU (N ) SYM ⇔ Type IIB string theory on AdS5 × S 5 .
strongly coupled ⇔ weakly coupled [See talks by Djuric,Raben,Teramond]
I
In the ’t Hooft limit, N → ∞ with λ = gY2 M N fixed:
Integrable structures ⇒ All loop, interpolating observables!
Ideal theoretical laboratory for developing new computational tools,
GP — 6-point remainder function near the collinear limit
Motivation: Why N = 4 SYM?
3/28
N = 4 Super Yang Mills Theory & Why Should We Care
Unique possibility for the nonperturbative investigation of gauge theories
I
N = 4 SU (N ) SYM ⇔ Type IIB string theory on AdS5 × S 5 .
strongly coupled ⇔ weakly coupled [See talks by Djuric,Raben,Teramond]
I
In the ’t Hooft limit, N → ∞ with λ = gY2 M N fixed:
Integrable structures ⇒ All loop, interpolating observables!
Ideal theoretical laboratory for developing new computational tools,
I
Generalized Unitarity
[Bern,Dixon,Dunbar,Kosower. . . ]
GP — 6-point remainder function near the collinear limit
Motivation: Why N = 4 SYM?
3/28
N = 4 Super Yang Mills Theory & Why Should We Care
Unique possibility for the nonperturbative investigation of gauge theories
I
N = 4 SU (N ) SYM ⇔ Type IIB string theory on AdS5 × S 5 .
strongly coupled ⇔ weakly coupled [See talks by Djuric,Raben,Teramond]
I
In the ’t Hooft limit, N → ∞ with λ = gY2 M N fixed:
Integrable structures ⇒ All loop, interpolating observables!
Ideal theoretical laboratory for developing new computational tools,
I
Generalized Unitarity
I
Method of Symbols
[Bern,Dixon,Dunbar,Kosower. . . ]
[Goncharov,Spradlin,Vergu,Volovich]
GP — 6-point remainder function near the collinear limit
Motivation: Why N = 4 SYM?
3/28
N = 4 Super Yang Mills Theory & Why Should We Care
Unique possibility for the nonperturbative investigation of gauge theories
I
N = 4 SU (N ) SYM ⇔ Type IIB string theory on AdS5 × S 5 .
strongly coupled ⇔ weakly coupled [See talks by Djuric,Raben,Teramond]
I
In the ’t Hooft limit, N → ∞ with λ = gY2 M N fixed:
Integrable structures ⇒ All loop, interpolating observables!
Ideal theoretical laboratory for developing new computational tools,
I
Generalized Unitarity
I
Method of Symbols
[Bern,Dixon,Dunbar,Kosower. . . ]
[Goncharov,Spradlin,Vergu,Volovich]
Then apply to QCD, e.g. NNLO gg → tt̄ for LHC!
[Bonciani,Ferroglia,Gehrmann,Manteuffel,Studerus]
GP — 6-point remainder function near the collinear limit
Motivation: Why N = 4 SYM?
3/28
The Planar MHV 6-point Amplitude
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
4/28
The Planar MHV 6-point Amplitude
I
Planar: Color information reduces to an overall trace, can strip it off.
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
4/28
The Planar MHV 6-point Amplitude
I
I
Planar: Color information reduces to an overall trace, can strip it off.
Maximally Helicity Violating, An (+ · · · + −−): Simplest amplitude.
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
4/28
The Planar MHV 6-point Amplitude
I
I
Planar: Color information reduces to an overall trace, can strip it off.
Maximally Helicity Violating, An (+ · · · + −−): Simplest amplitude.
Remarkably, also dual to null polygonal Wilson loops.
[Alday,Maldacena][Drummond,Korchemsky,Sokatchev][Brandhuber,Heslop,Travaglini]
kn
x1
xn
ki ≡ xi+1 − xi ≡ xi+1,i ,
k1
ki2 = x2i+1,i = 0
x2
An
k2
log Wn = log
HV
AM
n
+ O()
HV
AM
n,tree
x3
k3
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
4/28
The Planar MHV 6-point Amplitude
I
I
Planar: Color information reduces to an overall trace, can strip it off.
Maximally Helicity Violating, An (+ · · · + −−): Simplest amplitude.
Remarkably, also dual to null polygonal Wilson loops.
[Alday,Maldacena][Drummond,Korchemsky,Sokatchev][Brandhuber,Heslop,Travaglini]
kn
x1
xn
ki ≡ xi+1 − xi ≡ xi+1,i ,
k1
ki2 = x2i+1,i = 0
x2
An
k2
log Wn = log
HV
AM
n
+ O()
HV
AM
n,tree
x3
k3
I
Wn = WnBDS
For n = 4, 5, dimensionally regularized An /Wn known to all loops!
[Anastasiou,Bern,Dixon,Kosower][Bern,Dixon,Smirnov]
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
4/28
The Planar MHV 6-point Amplitude
I
I
Planar: Color information reduces to an overall trace, can strip it off.
Maximally Helicity Violating, An (+ · · · + −−): Simplest amplitude.
Remarkably, also dual to null polygonal Wilson loops.
[Alday,Maldacena][Drummond,Korchemsky,Sokatchev][Brandhuber,Heslop,Travaglini]
kn
x1
xn
ki ≡ xi+1 − xi ≡ xi+1,i ,
k1
ki2 = x2i+1,i = 0
x2
An
log Wn = log
k2
HV
AM
n
+ O()
HV
AM
n,tree
x3
k3
I
Wn = WnBDS eRn (u1 ,...,um )
For n = 4, 5, dimensionally regularized An /Wn known to all loops!
[Anastasiou,Bern,Dixon,Kosower][Bern,Dixon,Smirnov]
I
For n ≥ 6, needs to be corrected by ‘remainder function’ Rn of
x2 x2
conformal cross ratios, e.g u = x246 x213 . Focus on R6 (u1 , u2 , u3 ).
36 14
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
4/28
Nonperturbative Definition via the Collinear Limit
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
5/28
Nonperturbative Definition via the Collinear Limit
I
Form square (OPSF). Use conformal transformations to place O at
origin, and P,S,F at ∞ in (x0 , x1 ) plane. It will then be invariant
under dilatations D, boosts M01 , and rotations on (x2 , x3 ) plane M23 .
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
5/28
Nonperturbative Definition via the Collinear Limit
I
I
Form square (OPSF). Use conformal transformations to place O at
origin, and P,S,F at ∞ in (x0 , x1 ) plane. It will then be invariant
under dilatations D, boosts M01 , and rotations on (x2 , x3 ) plane M23 .
Collinear limit: Act with e−τ (D−M01 ) on A and B, and take τ → ∞.
Parametrize u1 , u2 , u3 by group coordinates τ, σ, φ.
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
5/28
Nonperturbative Definition via the Collinear Limit
I
I
I
Form square (OPSF). Use conformal transformations to place O at
origin, and P,S,F at ∞ in (x0 , x1 ) plane. It will then be invariant
under dilatations D, boosts M01 , and rotations on (x2 , x3 ) plane M23 .
Collinear limit: Act with e−τ (D−M01 ) on A and B, and take τ → ∞.
Parametrize u1 , u2 , u3 by group coordinates τ, σ, φ.
Can think of (P O),(SF ) as a color-electric flux tube sourced by q q̄,
and decompose the Wilson loop with respect its excitations.
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
5/28
Nonperturbative Definition via the Collinear Limit
I
I
I
Form square (OPSF). Use conformal transformations to place O at
origin, and P,S,F at ∞ in (x0 , x1 ) plane. It will then be invariant
under dilatations D, boosts M01 , and rotations on (x2 , x3 ) plane M23 .
Collinear limit: Act with e−τ (D−M01 ) on A and B, and take τ → ∞.
Parametrize u1 , u2 , u3 by group coordinates τ, σ, φ.
Can think of (P O),(SF ) as a color-electric flux tube sourced by q q̄,
and decompose the Wilson loop with respect its excitations.
Schematically,
X
W =
e−τ Ei +ipi +imi φ P(0|ψ1 )P(ψ1 |0)
ψi
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
5/28
Nonperturbative Definition via the Collinear Limit
I
I
I
Form square (OPSF). Use conformal transformations to place O at
origin, and P,S,F at ∞ in (x0 , x1 ) plane. It will then be invariant
under dilatations D, boosts M01 , and rotations on (x2 , x3 ) plane M23 .
Collinear limit: Act with e−τ (D−M01 ) on A and B, and take τ → ∞.
Parametrize u1 , u2 , u3 by group coordinates τ, σ, φ.
Can think of (P O),(SF ) as a color-electric flux tube sourced by q q̄,
and decompose the Wilson loop with respect its excitations.
Schematically,
X
W =
e−τ Ei +ipi +imi φ P(0|ψ1 )P(ψ1 |0)
ψi
I
Propagation of square eigenstates
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
5/28
Nonperturbative Definition via the Collinear Limit
I
I
I
Form square (OPSF). Use conformal transformations to place O at
origin, and P,S,F at ∞ in (x0 , x1 ) plane. It will then be invariant
under dilatations D, boosts M01 , and rotations on (x2 , x3 ) plane M23 .
Collinear limit: Act with e−τ (D−M01 ) on A and B, and take τ → ∞.
Parametrize u1 , u2 , u3 by group coordinates τ, σ, φ.
Can think of (P O),(SF ) as a color-electric flux tube sourced by q q̄,
and decompose the Wilson loop with respect its excitations.
Schematically,
X
W =
e−τ Ei +ipi +imi φ P(0|ψ1 )P(ψ1 |0)
ψi
I
Propagation of square eigenstates
I
Transition between squares
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
5/28
Nonperturbative Definition via the Collinear Limit
I
I
I
Form square (OPSF). Use conformal transformations to place O at
origin, and P,S,F at ∞ in (x0 , x1 ) plane. It will then be invariant
under dilatations D, boosts M01 , and rotations on (x2 , x3 ) plane M23 .
Collinear limit: Act with e−τ (D−M01 ) on A and B, and take τ → ∞.
Parametrize u1 , u2 , u3 by group coordinates τ, σ, φ.
Can think of (P O),(SF ) as a color-electric flux tube sourced by q q̄,
and decompose the Wilson loop with respect its excitations.
Schematically,
X
W =
e−τ Ei +ipi +imi φ P(0|ψ1 )P(ψ1 |0)
ψi
I
Propagation of square eigenstates
I
Transition between squares
⇒ WL ‘Operator Product Expansion’ (OPE)
[Alday,Gaiotto,Maldacena,Sever,Vieira]
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
5/28
Wilson Loop OPE & Integrability
In N = 4 SYM, flux tube excitations in 1-1 correspondence with
excitations of an integrable spin chain with hamiltonian D − M01 .
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
6/28
Wilson Loop OPE & Integrability
In N = 4 SYM, flux tube excitations in 1-1 correspondence with
excitations of an integrable spin chain with hamiltonian D − M01 .
The lightest excitations are made of 6 scalars φ, 4+4 fermions ψ, ψ̄ and
1+1 gluons F, F̄ of the theory, with classical ∆ − S = 1, over the
S
vacuum = tr ZD+
Z , Z = φ1 + iφ2 , D+ = D0 + D1
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
6/28
Wilson Loop OPE & Integrability
In N = 4 SYM, flux tube excitations in 1-1 correspondence with
excitations of an integrable spin chain with hamiltonian D − M01 .
The lightest excitations are made of 6 scalars φ, 4+4 fermions ψ, ψ̄ and
1+1 gluons F, F̄ of the theory, with classical ∆ − S = 1, over the
S
vacuum = tr ZD+
Z , Z = φ1 + iφ2 , D+ = D0 + D1
Integrability enables the calculation of the excitation energies E(p),
E(p) = (∆ − S)1 − (∆ − S)vac = 1 +
∞
X
λl E (l) (p)
l=1
to all loops, and implies that for M excitations EM = M + O(λ).[Basso]
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
6/28
Wilson Loop OPE & Integrability
In N = 4 SYM, flux tube excitations in 1-1 correspondence with
excitations of an integrable spin chain with hamiltonian D − M01 .
The lightest excitations are made of 6 scalars φ, 4+4 fermions ψ, ψ̄ and
1+1 gluons F, F̄ of the theory, with classical ∆ − S = 1, over the
S
vacuum = tr ZD+
Z , Z = φ1 + iφ2 , D+ = D0 + D1
Integrability enables the calculation of the excitation energies E(p),
E(p) = (∆ − S)1 − (∆ − S)vac = 1 +
∞
X
λl E (l) (p)
l=1
to all loops, and implies that for M excitations EM = M + O(λ).[Basso]
Thus, weak coupling WL OPE=expansion in terms ∝ e−τ M , M = 1, 2 . . .
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
6/28
Wilson Loop OPE & Integrability
In N = 4 SYM, flux tube excitations in 1-1 correspondence with
excitations of an integrable spin chain with hamiltonian D − M01 .
The lightest excitations are made of 6 scalars φ, 4+4 fermions ψ, ψ̄ and
1+1 gluons F, F̄ of the theory, with classical ∆ − S = 1, over the
S
vacuum = tr ZD+
Z , Z = φ1 + iφ2 , D+ = D0 + D1
Integrability enables the calculation of the excitation energies E(p),
E(p) = (∆ − S)1 − (∆ − S)vac = 1 +
∞
X
λl E (l) (p)
l=1
to all loops, and implies that for M excitations EM = M + O(λ).[Basso]
Thus, weak coupling WL OPE=expansion in terms ∝ e−τ M , M = 1, 2 . . .
For the moment, focus on ‘leading OPE contribution’ O(e−τ ).
GP — 6-point remainder function near the collinear limit Ampitudes, Wilson Loops & the Proposal of Basso,Sever,Vieira
6/28
The Proposal of Basso,Sever,Vieira
To complete WL OPE description, also emission/absorption form factors
P(0|ψ1 ), P(ψ1 |0) needed.
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
7/28
The Proposal of Basso,Sever,Vieira
To complete WL OPE description, also emission/absorption form factors
P(0|ψ1 ), P(ψ1 |0) needed.
I
Obtained at all loops again exploiting the power of integrability.
Related to S-matrix of excitations on top of the GKP string.
[Basso, Rej][Fioravanti,Piscaglia,Rossi]
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
7/28
The Proposal of Basso,Sever,Vieira
To complete WL OPE description, also emission/absorption form factors
P(0|ψ1 ), P(ψ1 |0) needed.
I
Obtained at all loops again exploiting the power of integrability.
Related to S-matrix of excitations on top of the GKP string.
[Basso, Rej][Fioravanti,Piscaglia,Rossi]
I
This approach is generalizable to any n-point amplitude, and also to
multiparticle contributions e−2τ etc.
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
7/28
The Proposal of Basso,Sever,Vieira
To complete WL OPE description, also emission/absorption form factors
P(0|ψ1 ), P(ψ1 |0) needed.
I
Obtained at all loops again exploiting the power of integrability.
Related to S-matrix of excitations on top of the GKP string.
[Basso, Rej][Fioravanti,Piscaglia,Rossi]
I
This approach is generalizable to any n-point amplitude, and also to
multiparticle contributions e−2τ etc.
Leading (single-particle) contribution for 6-pt amplitude reads:[BSV1;BSV2]
Z +∞
du
−τ
R =2 cos φ e
µ(u)e−γ(u)τ +ip(u)σ
−∞ 2π
−σ
Γ
+ cusp
e log(1 + e2σ ) + eσ log(1 + e−2σ ) + O(e−2τ ) ,
4
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
7/28
The Proposal of Basso,Sever,Vieira
To complete WL OPE description, also emission/absorption form factors
P(0|ψ1 ), P(ψ1 |0) needed.
I
Obtained at all loops again exploiting the power of integrability.
Related to S-matrix of excitations on top of the GKP string.
[Basso, Rej][Fioravanti,Piscaglia,Rossi]
I
This approach is generalizable to any n-point amplitude, and also to
multiparticle contributions e−2τ etc.
Leading (single-particle) contribution for 6-pt amplitude reads:[BSV1;BSV2]
Z +∞
du
−τ
R =2 cos φ e
µ(u)e−γ(u)τ +ip(u)σ
−∞ 2π
−σ
Γ
+ cusp
e log(1 + e2σ ) + eσ log(1 + e−2σ ) + O(e−2τ ) ,
4
I
Γcusp = 4g 2 −
4π 2 4
3 g
+
44π 4 6
45 g
GP — 6-point remainder function near the collinear limit
+ O(g 8 ) ,
g2 ≡
λ
(4π)2
.
Main Part: Implications for the 6-pt Amplitude
7/28
The Proposal of Basso,Sever,Vieira
To complete WL OPE description, also emission/absorption form factors
P(0|ψ1 ), P(ψ1 |0) needed.
I
Obtained at all loops again exploiting the power of integrability.
Related to S-matrix of excitations on top of the GKP string.
[Basso, Rej][Fioravanti,Piscaglia,Rossi]
I
This approach is generalizable to any n-point amplitude, and also to
multiparticle contributions e−2τ etc.
Leading (single-particle) contribution for 6-pt amplitude reads:[BSV1;BSV2]
Z +∞
du
−τ
R =2 cos φ e
µ(u)e−γ(u)τ +ip(u)σ
−∞ 2π
−σ
Γ
+ cusp
e log(1 + e2σ ) + eσ log(1 + e−2σ ) + O(e−2τ ) ,
4
4π 2 4
3 g
44π 4 6
45 g
I
Γcusp = 4g 2 −
I
Single Fourier integral left to do, but. . .
+
GP — 6-point remainder function near the collinear limit
+ O(g 8 ) ,
g2 ≡
λ
(4π)2
.
Main Part: Implications for the 6-pt Amplitude
7/28
The Proposal of Basso,Sever,Vieira (cont’d)
Z ∞ ∅
∅
(2gt) dt γ+ (2gt) γ−
−t/2
γ(u) ≡ E(u) − 1 =
−
cos
(ut)e
−
1
,
t 1 − e−t
et − 1
0
Z ∞ ∅
∅
(2gt)
dt γ− (2gt) γ+
p(u) = 2u −
+
sin (ut)e−t/2 ,
−t
t−1
t
1
−
e
e
0
u2 + 41
πg 2
p
µ(u) = −
×
cosh (πu) (x+ x− − g 2 ) (x+ x+ − g 2 )(x− x− − g 2 )
Z∞
2e−t/2 cos(ut) − J0 (2gt) − 1
dt
(J0 (2gt) − 1)
+ f3 (u, u) − f4 (u, u)
exp
t
et − 1
0
±
x = x(u ±
i
2) ,
x(u) =
u+
p
u2 − (2g)2
,
2
∅
where γ±
(t) have known regular Taylor expansions around t = 0,
Ji the i-th Bessel function, and f3 , f4 are similar in structure to
remaining exponential part of µ(u).
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
8/28
Our Contribution
1. Practical question: How do we efficiently evaluate the integral at
higher and higher loops?
2. Conceptual question: Is there basis of functions large enough to
describe the answer at any loop order?
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
9/28
Our Contribution
1. Practical question: How do we efficiently evaluate the integral at
higher and higher loops?
2. Conceptual question: Is there basis of functions large enough to
describe the answer at any loop order?
We proved that
[GP’13]
∞
l−1
X
X
du
µ(u) e−τ γ(u)+ip(u)σ ≡
g 2l
τ n h(l)
n (σ) ,
2π
n=0
l=1
X
(l)
±
±σ s
hn (σ) =
cs,m1 ,...,mr e σ Hm1 ,...,mr (−e−2σ ) , mi ≥ 1 ,
Z
I≡
s,r,mi
where c± are numeric coefficients, and Hm1 ...mn Harmonic Polylogarithms
X
xn1
(HPL):
Hm1 ,...,mr (x) =
mr .
1
nm
1 . . . nr
n1 >n2 >...>nr ≥1
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
9/28
Our Contribution
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
10/28
Our Contribution
Result we obtained consistent with the conjecture, based on a particular
“d log” form for the all-loop integrand, that multiple polylogs is the right
basis for describing the 6-point remainder function (HPLs are subset).
[Arkani-Hamed,Bourjaily, Cachazo,Goncharov,Postnikov,Trnka]
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
10/28
Our Contribution
Result we obtained consistent with the conjecture, based on a particular
“d log” form for the all-loop integrand, that multiple polylogs is the right
basis for describing the 6-point remainder function (HPLs are subset).
[Arkani-Hamed,Bourjaily, Cachazo,Goncharov,Postnikov,Trnka]
Furthermore, our proof constitutes an algorithm for the direct computation
(l)
of the integrals, in principle at any loop order. Implemented to find hn for
any n up to l = 6 loops, and for n = l − 1, l − 2 up to l = 12, thereby
providing new high-loop predictions for the MHV hexagon.
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
10/28
Our Contribution
Result we obtained consistent with the conjecture, based on a particular
“d log” form for the all-loop integrand, that multiple polylogs is the right
basis for describing the 6-point remainder function (HPLs are subset).
[Arkani-Hamed,Bourjaily, Cachazo,Goncharov,Postnikov,Trnka]
Furthermore, our proof constitutes an algorithm for the direct computation
(l)
of the integrals, in principle at any loop order. Implemented to find hn for
any n up to l = 6 loops, and for n = l − 1, l − 2 up to l = 12, thereby
providing new high-loop predictions for the MHV hexagon.
Example
16σ 3
4π 2 σ
4π 2 80
16σ 2
4π 2 32
(4)
h3 =eσ H1 8ζ3 + 9 −16σ 2 + 9 +48σ− 3 − 3 +H2 − 3 +8σ− 9 + 3
32 16σ
4π 2
+H3 3 − 3 + −16σ 2 +64σ− 3 −48 H1,1 +16σH1,2 +(16σ+8)H2,1
40H1,3
+(64σ−64)H1,1,1 + 3 +8H2,2 +16H3,1 −64H1,1,1,1 +(σ→−σ) ,
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
10/28
Plot
MHV hexagon leading OPE term, Τ=0
10 5
Loop order
l=6
1000
l=5
l
I-1M h0
HlL
10 4
l=4
100
l=3
10
l=2
1
0
1
2
3
4
Σ
(l)
h0
Log-linear plot of the
component as a function of σ at different loop
orders l. Its sign is given by (−1)l , as it varies continuously without
vanishing. Increasing l by one increases the magnitude by roughly a factor
of 10, maintaining similar shape.
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
11/28
Plot of h(6) ≡
P5
n=0 τ
n h(6) (σ)
n
Colors of the visible spectrum denote different values of h(6) , increasing
from blue to red. The function is always positive, and monotonically
increasing and decreasing in τ and σ respectively.
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
12/28
The 2-particle OPE Contribution
Very recently, Basso,Sever,Vieira 3 proposed similar integral formulas for
the twist-2, O(e−2τ ) terms of the 6-point remainder function.
(
Rtwist-2 = e−2τ 2 cos(2φ)(WDF + WF F ) + Wφφ + Wψψ̄ + WF F̄
+
Γcusp h
cos(2φ) e2σ log(1 + e−2σ ) + e−2σ log(1 + e2σ ) − 1 ,
2
)
i
+ 2σ − log(1 + e−2σ ) − log(1 + e2σ )
where apart from two scalar, fermion and gluon excitations, we can also
have a 2-gluon bound state DF .
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
13/28
New Results on 2-particule Contributions
We proved that the 2-gluon bound state contribution is similarly
WDF ≡
∞
X
l=1
h̃(l)
n (σ) =
X
g
2l
l−1
X
τ n h̃(l)
n (σ) ,
n=0
kσ s
−2σ
c±
),
s,m1 ,...,mr e σ Hm1 ,...,mr (−e
k = ±2, 0 ,
s,r,mi
GP — 6-point remainder function near the collinear limit
Main Part: Implications for the 6-pt Amplitude
14/28
New Results on 2-particule Contributions
We proved that the 2-gluon bound state contribution is similarly
WDF ≡
∞
X
g
2l
h̃(l)
n (σ) =
τ n h̃(l)
n (σ) ,
n=0
l=1
X
l−1
X
kσ s
−2σ
c±
),
s,m1 ,...,mr e σ Hm1 ,...,mr (−e
k = ±2, 0 ,
s,r,mi
For example,
(1)
h̃0 = − e2σ H21 + e−2σ σ − H21 − 21
h
i
(2)
h̃0 = e2σ 12 − σ H1 + (1 − 2σ)H11 + 2H111 + H3
h
+ e−2σ 21 − σ H1 + (1 − 2σ)H11 + 2H111 + H3 − 1 +
+ 2H11 + (1 − 2σ)H1 − σ +
GP — 6-point remainder function near the collinear limit
π2
3
i
σ
3
2
Main Part: Implications for the 6-pt Amplitude
14/28
Conclusions & Outlook
In this presentation, we talked about
I
The promise N = 4 super Yang-Mills theory offers, as an interacting
4D gauge theory which may be exactly solvable in the planar limit
I
The recent, integrability-based proposal of Basso, Sever and Vieira,
for a nonperturbative description of its amplitudes/Wilson loops
In particular, we proved that the integrals the latter proposal
predicts for part of the 6-pt amplitude evaluate to Harmonic
Polylogarithms at any loop order, and obtained explicit expressions for them up to 12 loops.
Next Stage
I
More terms in the OPE of the 6-pt amplitude. Resummation?
I
Higher point amplitudes/Wilson loops
I
Similar treatment around multi-Regge kinematics
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
15/28
Extension: NMHV Amplitude A(+ + + − −−)
There exists an analogous integral formula for the OPE of certain
components of the NMHV hexagon. [Sever,Vieira,Wang][Basso,Sever,Vieira]
Follows attempts for generalizing the Wilson loop/scattering amplitude
duality beyond MHV. [Mason, Skinner][Caron-Huot]
Starting point, the packaging of all on-shell fields to a superfield
Φ = G+ +η A ΓA + 2!1 η A η B SAB + 3!1 η A η B η C ABCD Γ̄D + 3!1 η A η B η C η D ABCD G−
which allows to combine all n-point amplitudes in a superamplitude An .
Then, it has been argued that MHV and NMHV amplitudes have same IR
divergence structure, allowing one to write [Drummond,Henn,Korchemsky,Sokatchev]
ANMHV
= AMHV
Rn ,
n
n
where Rn the NMHV ratio function.
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
17/28
Extension: NMHV Amplitude
(6134)
R6
∞
l
e−τ X 2l X n (l)
g
τ Fn (σ) + O(e−2τ ) ,
=
2 cosh σ
l=0
n=0
where for any l,
Fn(l) (σ) =
X
cs,m1 ,...,mr σ s Hm1 ,...,mr (−e−2σ ) ,
s,r,mi
and the c’s are numerical coefficients. We similarly employed our
(l)
algorithm implementation in order to obtain explicit expressions for all fn
up to l = 6, and the n = l, l − 1 terms up to l = 12.
16
16
32σ
32
(4)
F4 =− 9 H1 (18ζ3 +4σ 3 +π 2 σ )+ 9 H2 (12σ 2 +π 2 )+ 3 H3 − 3 H4 −64σ(H1,2 +H2,1 )
16
160
+ 3 (12σ 2 +π 2 )H1,1 −256σH1,1,1 −32H2,2 − 3 (H1,3 +H3,1 )+256H1,1,1,1
4
2π 2 σ 2 7π 4
+ 9 (24ζ3 σ+σ 4 )+ 9 + 540 ,
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
18/28
Plots of the leading OPE contribution F (l) ≡
Pl
n=0 τ
n F (l) (σ)
n
As the functions change sign, we have also included the F (l) = 0 plane for
comparison.
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
19/28
Outline of Proof I
I
I
Expand integrand at g 1 with the help of
Z ∞ m −zt
t e
e−t
dt
−
δm,0 = (−1)m+1 ψ (m) (z) , m ≥ 0 ,
1 − e−t
t
0
R
P
Close contour with ∞ semicircle and reduce → (residues).
I
Identify u-dependent building blocks of integrand:
e2iuσ , sech(πu), ψ (m) ( 12 ± iu), ψ (m) ( 32 ± iu), (u ± 2i )−1 ⇒
Only possible locations of poles are are for u = (k + 12 )i, k ∈ Z
I
Employ identities such as
ψ (n) (z) = (−1)n ψ (n) (1 − z) − π
∂n
cot(πz) ,
∂z n
to obtain functions with known (generalized) Taylor expansions
around the poles for general k.
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
20/28
Outline of Proof II
I
Taking residues at u = (k + 12 )i for all k yields
X
±σ s
c±
σ
s,m1 ,...,mr e
Sm (k) =
k
X
1
.
nm
n=1
i=2
k
I
r
(−e−2σ )k Y
Smi (k) ,
k m1
Finally, nest all independent harmonic sums Sm (k), for example
∞
X
n1 =1
1
1
nm
1
∞
X
n2 =1
1
2
nm
2
=
X
n1 >n2
∞
X
X
1
1
1
+
+
m1 m2
m2 m1
m
+m2 .
1
n1 n2
n n1
n
1
n2 >n1 2
n =1
1
(More formally, one employs the quasi-shuffle algebra of Z-sums)
[Moch,Uwer,Weinzierl]
I
In this manner, all sums expressible in terms of (x = −e−2σ )
Hm1 ,...,mr (x) =
X
n1 >n2 >...>nr ≥1
GP — 6-point remainder function near the collinear limit
xn1
mr !
1
nm
1 . . . nr
Conclusions & Outlook
21/28
Scattering Amplitudes: dσ ∝ |A|2
For N = 4, all fields massless and in adjoint of gauge group SU (N ).
~ · p̂ to classify on-shell particle content,
Can thus use helicity h = S
h : −1
−1/2
Q1
G− −−→
0
Q2
1/2
Q3
Γ̄A −−→ ΦAB −−→
1
Q4
ΓA −−→ G+
For the gluons G± , the gluinos Γ, Γ̄, and the scalars Φ. For n gluons,
AnL−loop ({ki , hi , ai })
X
h1
hn
=
Tr(T aσ(1) · · · T aσ(n) ) A(L)
n (σ(1 ), . . . , σ(n ))
σ∈Sn /Zn
+multitrace terms, subleading by powers of 1/N 2 .
(L)
An : color-stripped amplitude, all color factors removed.
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
22/28
Hexagon Coordinates
1
x246 x213
e2σ+τ sechτ
=
,
2 1 + e2σ + 2 eσ−τ cos φ + e−2τ
x236 x214
1
x2 x224
= e−τ sechτ ,
u2 = 15
2
x214 x225
x2 x235
1
u3 = 26
=
.
1 + e2σ + 2 eσ−τ cos φ + e−2τ
x225 x236
u1 =
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
23/28
∅
The γ±
functions
∅
γ−
(t) = 2
∅
γ+
(t) = 2
∞
X
∅
(2n − 1)γ2n−1
J2n−1 (t) ,
n=1
∞
X
∅
(2n)γ2n
J2n−1 (t) ,
n=1
γn∅
where the coefficients
depend on g and obey
Z ∞
∅
γ ∅ (2gt) − (−1)n γ−
(2gt)
dt
γn∅ +
Jn (2gt) +
= 2g δn,1 .
t
t
e −1
0
Can be solved perturbatively in g 1, by Taylor-expanding the i-th Bessel
function of the first kind Ji (z),
∞
z i+2n
X
(−1)n
Ji (z) =
,
k!Γ(i + n + 1) 2
n=0
∅
In any case, from the latter formula it is evident that for small t, γ±
have
a regular Taylor expansion.
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
24/28
The fi functions
f1 (u, v) = 2 κ̃(u) · Q · M · κ(v) ,
f2 (u, v) = 2 κ̃(v) · Q · M · κ(v) ,
f3 (u, v) = 2 κ̃(u) · Q · M · κ̃(v) ,
f4 (u, v) = 2 κ(v) · Q · M · κ(v) ,
where the κ, κ̃ are vectors with elements
(−1)η×j
Z∞
)
dt Jj (2gt)(J0 (2gt) − cos(ut) et/2
κj (u) ≡ −
t
t
e −1
0
Z∞
κ̃j (u) ≡ −
t/2 (−1)η×(j+1)
dt
j+1 Jj (2gt) sin(ut) e
(−1)
t
et − 1
0
with η = 0 for the NMHV case and η = 1 for MHV case, and
∞
X
Qij = δij (−1)i+1 i , M ≡ (1 + K)−1 =
(−K)n ,
n=0
Kij = 2j(−1)j(i+1)
Z∞
dt Ji (2gt)Jj (2gt)
.
t
et − 1
0
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
25/28
Polygamma Functions & Harmonic Numbers
dm
dm+1
ψ(z)
=
log Γ(z)
dz m
dz m+1
∂n
ψ (n) (z) = (−1)n ψ (n) (1 − z) − π n cot(πz) ,
∂z
(n)
(n)
n
ψ (z + 1) = ψ (z) + (−1) n!z −n−1 ,
ψ (m) (z) ≡
For integer arguments,
ψ(k + 1) = −γE + S1 (k)
ψ
(m−1)
(k + 1) = (−1)m (m − 1)!(ζm − Sm (k)) ,
where Sm (k) the generalized harmonic numbers,
k
X
1
Sm (k) =
,
nm
n=1
γE = −ψ(1) ' 0.577 the Euler-Mascheroni constant, and ζm the Riemann
zeta function.
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
26/28
Z-sums
Defined by
[Moch,Uwer,Weinzierl]
i
Z(n; m1 , . . . , mj ; x1 , . . . , xj ) =
X
n≥i1 >i2 >...>ij >0
xjj
xi11
. . . mj ,
1
im
ij
1
or recursively by (Z(n) equal to the unit step function)
Z(n; m1 , . . . , mj ; x1 , . . . , xj ) =
n
X
xi11
Z(i1 − 1; m2 , . . . , mj ; x2 , . . . , xj ) ,
1
im
1
i1 =1
They form a quasi-shuffle algebra, i.e. each product can be expressed as
linear combination of single Z-sums. For example,
Z(l)Z(m1 ,...,mj )
=Z(l,m1 ,...,mj )+Z(m1 ,l,...,mj )+...+Z(m1 ,...,mj ,l)
+Z(m1 +l,...,mj )+...+Z(m1 ,...,mj +l) .
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
27/28
Aside: What functions appear in gauge-theoretic multiloop calculations?
I
Feynman integrals have discontinuities from propagators going
on-shell.
I
Already at 1-loop, apart from logarithms, also the dilogarithm Li2 (x)
appears.
I
Most broad generalization applicable to our case: multiple polylogs,
Lim1 ,...,mr (x1 , . . . , xr ) =
X
n1 >n2 >...>nr ≥1
I
I
I
xn1 1 . . . xnr r
mr .
1
nm
1 . . . nr
HPLs amount to Hm1 ,...,mr (x) = Lim1 ,...,mr (x1 , 1, . . . , 1)
P
Important property: Transcendentality w = ri mi , also inherited by
special numbers Lim1 ,...,mr (1, . . . , 1) = ζm1 ,...,mr , ζ(2) = π 2 /6 etc.
Very fruitful interplay coming from the study of these functions in the
context of number theory and algebraic geometry.
[Bloch,Brown,Goncharov,Zagier. . . ]
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
28/28
Harmonic Polylogarithms
For x ∈ (0, 1) and ai = {−1, 0, 1}, harmonic polylogarithms are defined as
H(x) = 1 ,
(
H(a1 , . . . , an ; x) =
1
logn x
Rn!x
0 dyfa1 (t)H(a2 , . . . , an ; t)
if a1 = . . . an = 0 ,
otherwise,
where the auxiliary functions fa are given by
f−1 (x) =
1
,
1+x
f0 (x) =
1
,
x
f1 (x) =
1
.
1−x
m−1
z }| {
Compact notation: 0, 0, . . . 0, ±1 → ±m . E.g. H(1, 0, −1; x) = H1,−2 (x) .
In terms of Z-sums, Hm1 ,...,mj (x) = Z(∞; m1 , . . . , mj ; x, 1, . . . , 1) .
Weight (or transcendentality): number of ai indices
Depth: number of nonzero ai indices
GP — 6-point remainder function near the collinear limit
Conclusions & Outlook
29/28