The Regge limit in QCD and in N=4 super-Yang-Millsdelducav/ETHtalk-may17.pdf · The Regge limit in...

The Regge limitin QCD

and in N=4 super-Yang-Mills

Vittorio Del Duca

ETH Zürich & INFN LNF

ETH 30 May 2017

Production of two high-energy jets as recorded by CMS at the LHC

© 2017 CERN

At the LHC, particles are produced throughthe head-on collisions of protons, and in particular through the collisions of quarks and gluons within the protons

The probability of a collision event is computed through the cross section,which is given as an integral of (squared) scattering amplitudes over the phase space of the produced particles

The scattering amplitudes arefundamental objects of particle physics

The scattering amplitudes are given asa power series (loop expansion) in the strong and/or electroweak couplings

Scattering amplitudes

The scattering amplitudes are given as a loop-momentum expansion in the strong and/or electroweak couplings

The more terms we know in the loop expansion,the more precisely we can compute the cross section

As a matter of fact, we know any amplitude of interest at one loop, several (2 → 2) amplitudes at two loops, just a couple (2 → 1) at three loops, and nothing beyond that

What outcome do we expect fromthe loop expansion of an amplitude?

From renormalisability and the infrared structure of the amplitude, we expect that the divergent parts are logarithmic functions of the external momenta

But, except for unitarity, we have little guidance for the finite parts.Heuristically, we know that:— at one loop, logarithmic and dilogarithmic functions of the external momenta occur— beyond one loop, higher polylogarithmicfunctions appear and elliptic functions may appear

In the last few years, a lot of progress has been made in understanding the analytic structure of multi-loop amplitudes, in particular on how the polylogarithmicfunctions appear at any loop level

In particular, a lot of progress has been made:— in N=4 Super Yang-Mills (SYM)— in the Regge limit of QCD— in the Regge limit of N=4 SYM

One of the most remarkable discoveries in elementary particle physics has been that of the existence of the complex plane …

incipitThe analytic S-matrix

Eden Landshoff Olive Polkinghorne 1966

Regge limitin the

leading logarithmic approximation

In perturbative QCD, in the Regge limit s » t, any scattering process is dominated by gluon exchange in the t channel

Regge limit of QCD

For a 4-gluon tree amplitude, we obtain

Mgg!ggaa0bb0 (s, t) = 2 g2s

(T c)aa0C⌫a⌫a0 (pa, pa0)

�s

t

(Tc)bb0C⌫b⌫b0 (pb, pb0)

�

C⌫a⌫a0 (pa, pa0) are called impact factors

leading logarithms of s/t are obtained by the substitution1

t! 1

t

✓s

�t

◆↵(t)

α(t) is the Regge gluon trajectory, with infrared coefficients

↵(t) =↵s(�t, ✏)

4⇡↵(1) +

✓↵s(�t, ✏)

4⇡

◆2

↵(2) + O �↵3s

�

↵(1) = CAb�(1)K

✏= CA

2

✏↵(2) = CA

�b0✏2

+ b�(2)K

2

✏+ CA

✓404

27� 2⇣3

◆+ nf

✓�56

27

◆�


s

t


� "✓s

�t

◆↵(t)

+

✓�s

�t

◆↵(t)#


�

in the Regge limit, the amplitude is invariant under s ↔ u exchange.

To NLL accuracy, the amplitude is given by Fadin Lipatov 1993

↵s(�t, ✏) =

✓µ2

�t

◆✏

↵s(µ2)


s

t


� "✓s

�t

◆↵(t)

+

✓�s

�t

◆↵(t)#


�Regge limit of the gluon-gluon amplitude

strip colour off & expand at one loop

the Regge gluon trajectory is universal;the one-loop gluon impact factor is a polynomial in t, ε, starting at 1/ε2

perform the Regge limit of the quark-quark amplitude → get one-loop quark impact factor

if factorisation holds, one can obtain the one-loop quark-gluon amplitude by assembling the Regge trajectory and the gluon and quark impact factorsthe result should match the quark-gluon amplitude in the high-energy limit: it does

m(1)gg!gg = 2 g2s

s

t

✓↵(1)(t) ln

✓s

�t

◆+ 2C(1)

gg (t)

◆

Amplitudes in the Regge limit

Balitski Fadin Kuraev Lipatov

BFKL is a resummation of multiple gluon radiationout of the gluon exchanged in the t channel

the resummation yields an integral (BFKL) equation for the evolution of the gluon propagator in 2-dim transverse momentum space

the Leading Logarithmic (BFKL 1976-77) and Next-to-Leading Logarithmic (Fadin-Lipatov 1998) contributions in log(s/|t|) of the radiative corrections to the gluon propagator in the t channel are resummed to all orders in αs

the BFKL equation is obtained in the limit of strong rapidity orderingof the emitted gluons, with no ordering in transverse momentum - multi-Regge kinematics (MRK)

the solution is a Green’s function of the momenta flowing in and out of the gluon ladder exchanged in the t channel

Mueller-Navelet jets

pa = xaPA pb = xbPB

Dijet production cross section with two tagging jets in the forward and backward directions

incoming parton momenta

S: hadron centre-of-mass energy

s = xaxbS: parton centre-of-mass energy

ETj: jet transverse energies

�y = |yj1 � yj2 | ' log

s

ETj1ETj2

is the rapidity interval between the tagging jets

gluon radiation is considered in MRK and resummed through the LL BFKL equation

Mueller-Navelet evaluated the inclusive dijet cross section up to 5 loops

Mueller Navelet 1987

d�gg

dp21?dp22?d�jj

=⇡

2

CA↵s

p21?

�f(~q1?, ~q2?,�y)

CA↵s

p22?

�

the cross section for dijet production at large rapidityintervals

can be described through the BFKL Green’s function

s = xaxbS , t = �q

p

21?p

22?

�y = y1 � y2 = ln

✓s

�t

◆� 1

with

f(~q1?, ~q2?,�y) =1

(2⇡)2p

q21? q22?

+1X

n=�1ein�

Z +1

�1d⌫

✓q21?q22?

◆i⌫

e⌘ �⌫,n

⌘ ⌘ CA ↵s

⇡�y

�⌫,n = �2�E �

✓1

2+

|n|2

+ i⌫

◆�

✓1

2+

|n|2

� i⌫

◆

with

and the LL BFKL eigenvalue

and ɸ the angle between q12 and q2

2

Mueller-Navelet dijet cross section

Mueller-Navelet dijet cross section

azimuthal angle distribution (ɸjj = ɸ-π)

d�gg

d�jj=

⇡(CA↵s)2

2E2?

"�(�jj � ⇡) +

1X

k=1

1X

n=�1

ein�

2⇡fn,k

!⌘k#

fn,k =1

2⇡

1

k!

Z 1

�1d⌫

�k⌫,n

⌫2 + 14

with

the dijet cross section is �gg =⇡(CA↵s)2

2E2?

1X

k=0

f0,k ⌘k

with

f0,0 = 1 ,

f0,1 = 0 ,

f0,2 = 2⇣2 ,

f0,3 = �3⇣3 ,

f0,4 =53

6⇣4 ,

f0,5 = � 1

12(115⇣5 + 48⇣2⇣3)

Mueller Navelet 1987

N=4 Super Yang Millsmaximal supersymmetric theory (without gravity)conformally invariant, β fn. = 0

spin 1 gluon4 spin 1/2 gluinos6 spin 0 real scalars

‘t Hooft limit: Nc →∞ with λ = g2Nc fixed

only planar diagrams

AdS/CFT duality Maldacena 97

large-λ limit of 4dim CFT ↔ weakly-coupled string theory

(aka weak-strong duality)

use N=4 SYM as a computational lab:

to learn techniques and tools to be used in Standard Modelcalculations

to learn about the bases of special functions which may occurin the scattering processes

amplitudes in N=4 SYM are much simplerthan in Standard Model processes

N=4 Super Yang Mills

N=4 Super Yang Mills

In the last years, a huge progress has been made in understanding the analytic structure of the S-matrix of N=4 SYM

Besides the ordinary conformal symmetry,in the planar limit the S-matrix exhibits a dual conformal symmetry

Accordingly, the analytic structure of the scattering amplitudes ishighly constraint

Drummond Henn Smirnov Sokatchev 2006

4- and 5-point amplitudes are fixed to all loops by the symmetriesin terms of the one-loop amplitudes and the cusp anomalous dimension

Anastasiou Bern Dixon Kosower 2003, Bern Dixon Smirnov 2005Drummond Henn Korchemsky Sokatchev 2007

Beyond 5 points, the finite part of the amplitudes is given in terms of a remainder function R. The symmetries only fix the variables of R (some conformally invariant cross ratios) but not the analytic dependence of R on them

for n = 6, the conformally invariant cross ratios are

thus x2k,k+r = (pk + . . . + pk+r�1)2

u1 =x2

13x246

x214x

236

u2 =x2

24x215

x225x

214

u3 =x2

35x226

x236x

225

1

2

6

3

4

5pi = xi � xi+1xi are variables in a dual space s.t.

for n points, dual conformal invariance implies dependence on 3n-15 independent cross ratios

u1i =x

2i+1,i+5x

2i+2,i+4

x

2i+1,i+4x

2i+2,i+5

, u2i =x

2N,i+3x

21,i+2

x

2N,i+2x

21,i+3

, u3i =x

21 i+4x

22,i+3

x

21,i+3x

22,i+4

Scattering amplitudes

Scattering amplitudes of gluons are functions of:— the external momenta— colour— helicities: — maximally helicity violating (- - ++…+) — next-to-maximally helicity violating (- - - ++…+) — and so on

Scattering amplitudes in N=4 SYM

The progress in understanding the analytic structure of the S-matrix inplanar N=4 SYM is also due to an improved understanding of themathematical structures underlying the scattering amplitudes

n-point amplitudes are expected to be written in terms of iterated integrals (on the space of configurations of points in 3-dim projective space Confn(CP3) )

The simplest case of iterated integrals are the iterated integralsover rational functions, i.e. the multiple polylogarithms

G(a, ⌥w; z) =⇤ z

0

dt

t� aG(⌥w; t) , G(a; z) = ln

�1� z

a

⇥ Goncharov 2001

Golden Paulos Spradlin Volovich 2014

It is thought that maximally helicity violating (MHV) and next-to-MHV(NMHV) amplitudes can be expressed in terms of multiple polylogarithms of uniform transcendental weight

Arkani-Hamed et al. Scattering Amplitudes and the Positive Grassmannian 2012

G(0, 1; z) = �Li2(z)


MHV and NMHV amplitudes feature maximal transcendentality, i.e. L-loop amplitudes are expressed in terms of multiple polylogarithms of weight 2L only

MHV amplitudes are pure, i.e. the coefficients of the multiple polylogarithms are (rational) numbers

6-pt (N)MHV amplitudes are known analytically up to 5(4) loops

Duhr Smirnov VDD 2009Goncharov Spradlin Vergu Volovich 2010Dixon Drummond Henn 2011Dixon Drummond von Hippel Pennington 2013Dixon Drummond Duhr Pennington 2014Caron-Huot Dixon von Hippel McLeod 2016

Dixon Drummond Henn 2011Dixon von Hippel 2014Dixon von Hippel McLeod 2015

7-pt MHV amplitudes are known analytically at two loopsGolden Spradlin 2014

No analytic result is known beyond 7 points.The cluster algebra which defines the multiple polylogarithmsis infinite starting from 8 points


Golden Goncharov Spradlin Vergu Volovich 2013

Multi-Regge limit of N=4 SYM

In the Euclidean region (where all Mandelstam invariants are negative),amplitudes in MRK factorise completely in terms of building blockswhich are expressed in terms of Regge poles and can be determinedto all orders through the 4-pt and 5-pt amplitudes. Thus the remainder functions R vanish at all points Duhr Glover VDD 2008

After analytic continuation to some regions of the Minkowski space,the amplitudes develops cuts which are described by a dispersion relationfor octet exchange, which is similar to the singlet BFKL equation in QCD

Bartels Lipatov Sabio-Vera 2008

Accordingly, 6-pt amplitudes have been thoroughly examined,both at weak and at strong coupling

In particular, 6-pt amplitudes at weak coupling can be expressed in terms of single-valued harmonic polylogarithms

Dixon Duhr Pennington 2012

Regge factorisation of the n-pt amplitude

mn(1, 2, . . . , n) = s [g C(p2, p3)]1

tn�3

��sn�3

⇥

⇥�(tn�3)

[g V (qn�3, qn�4, �n�4)]

· · ·⇤ 1t2

��s2

⇥

⇥�(t2)

[g V (q2, q1, �1)]1t1

��s1

⇥

⇥�(t1)

[g C(p1, pn)]

n-pt amplitude in the multi-Regge limit

s⇥ s1, s2, . . . , sn�3 ⇥ �t1,�t2 . . . ,�tn�3

the l-loop n-pt amplitude can be assembledusing the l-loop trajectories, vertices andcoefficient functions, determined through thel-loop 4-pt and 5-pt amplitudes

y3 ⇥ y4 ⇥ · · ·⇥ yn; |p3�| ⇤ |p4�|... ⇤ |pn�|

in Euclidean space, no violation of the BDS ansatz canbe found in the multi-Regge limit

Duhr Glover VDD 2008

in MRK, there is no ordering in transverse momentum,i.e. only the n-2 transverse momenta are non-trivial

dual conformal invariance in transverse momentum spaceimplies dependence on n-5 cross ratios of the transverse momenta

zi =(x1 � xi+3) (xi+2 � xi+1)

(x1 � xi+1) (xi+2 � xi+3)= � qi+1 ki

qi�1 ki+1

i = 1, . . . , n� 5

ℳ0,p = space of configurations of p points on the Riemann sphere

Moduli space of Riemann spheres

ℳ0,n-2 is the space of the MRK, with dim(ℳ0,n-2) = n-5

Because we can fix 3 points at 0, 1, ∞, its dimension is dim(ℳ0,p)= p-3

Its coordinates can be chosen to be the zi’s,i.e. the cross ratios of the transverse momenta

Drummond Druc Duhr Dulat Marzucca Papathanasiou Verbeek VDD 2016

Iterated integrals on ℳ0,n-2

on ℳ0,n-2, the singularities are associated to degenerate configurationswhen two points merge xi → xi+1

i.e. when momentum pi becomes soft pi → 0

iterated integrals on ℳ0,p can be written as multiple polylogarithms Brown 2006

amplitudes in MRK can be written in terms of multiple polylogarithms

massless amplitudes may have branch points when Mandelstam invariantsvanish sij → 0 or become infinite sij → ∞, but branch cuts are constrained by unitarity

analytic structure of amplitudes is constrained by unitarityand the optical theorem Disc(M) = iMM†

Goncharov 2002

algebra is a vector space with a product μ: A ⊗ A → A μ(a⊗b) = a⋄bthat is associative A ⊗ A ⊗ A → A ⊗ A → A (a⋄b)⋄c = a⋄(b⋄c)

μ puts together; Δ decomposes

coalgebra is a vector space with a coproduct Δ: B → B ⊗ Bthat is coassociative B → B ⊗ B → B ⊗ B ⊗ B �(a) =

X

i

a(1)i ⌦ a(2)i

Duhr 2012

a Hopf algebra is an algebra and a coalgebra, such that product and coproduct are compatible Δ(a⋄b) = Δ(a)⋄Δ(b)

multiple polylogarithms form a Hopf algebra with a coproduct

�(Lw) =wX

k=0

�k,w�k(Lw) =wX

k=0

Lk ⌦ Lw�k

in particular, on classical polylogarithms

�(ln z) = 1⌦ ln z + ln z ⌦ 1

��Li2(z)

�= 1⌦ Li2(z) + Li2(z)⌦ 1� ln(1� z)⌦ ln z

Hopf algebra and the coproduct

the two entries of the coproduct are the discontinuity and the derivative

�Disc = (Disc⌦ id)� �@ = (id⌦ @)�

then the coproduct of an amplitude is related to unitarity

thus, for massless amplitudes

�(M) = ln(sij)⌦ . . .

in particular, for amplitudes in MRK

�(M) = ln |xi � xj |2 ⌦ . . .

except for the soft limit pi → 0, in MRK the transverse momenta never vanish

|xi � xj |2 6= 0 single-valued functions

therefore, n-point amplitudes in MRK of planar N=4 SYM can be writtenin terms of single-valued iterated integrals on ℳ0,n-2


for n=6, iterated integrals on ℳ0,4 are harmonic polylogarithmsthus, 6-point amplitudes in MRK of can be written in terms ofsingle-valued harmonic polylogarithms (SVHPL) Dixon Duhr Pennington 2012

Harmonic polylogarithms

classical polylogarithms Lim(z) =

Z z

0dz0

Lim�1(z0)

z0

with Remiddi Vermaseren 1999

harmonic polylogarithms (HPLs)

H(a, ~w; z) =

Z z

0dt f(a; t)H(~w; t) f(�1; t) =

1

1 + t, f(0; t) =

1

t, f(1; t) =

1

1� t

{a, ~w} 2 {�1, 0, 1}

HPLs obey the differential equations

d

dzH0!(z) =

H!(z)

z,

d

dzH1!(z) =

H!(z)

1� z

subject to the constraints

H(z) = 1, H~0n(z) =

1

n!lnn z, lim

z!0H! 6=~0n

(z) = 0

HPLs form a shuffle algebra

H!1(z)H!2(z) =X

!

H!(z) with ω the shuffle of ω1 and ω2

HPLs are multi-valued functions on the complex plane

Single-valued polylogarithms

Single-valued functions are real analytic functions on the complex plane

Because the discontinuities of the classical polylogarithms are known

�Lin(z) = 2⇡ilog

n�1 z

(n� 1)!

one can build combinations of classical polylogarithmssuch that all branch cuts cancel on the punctured plane C/{0,1}(Riemann sphere with punctures)

An example is the Bloch-Wigner dilogarithm

D2(z) = Im[Li2(z)] + arg(1� z) log |z|

Single-valued harmonic polylogarithms

define a function ℒ that is real-analytic and single-valued onand that has the same properties as the HPLs

C/{0, 1}

it obeys the differential equations

@

@zL0!(z) =

L!(z)

z

@

@zL1!(z) =

L!(z)

1� z

subject to the constraints

Le(z) = 1 , L~0n(z) =

1

n!lnn |z|2 lim

z!0L! 6=~0n

(z) = 0

the SVHPLs ℒω(z) also form a shuffle algebra

L!1(z)L!2(z) =X

!

L!(z) with ω the shuffle of ω1 and ω2

Brown 2004SVHPLs can be explicitly expressed as combinations of HPLssuch that all the branch cuts cancel

examples

L0(z) = H0(z) +H0(z) = ln |z|2

L1(z) = H1(z) +H1(z) = � ln |1 + z|2

L0,1(z) =1

4

⇥�2H1,0 + 2H1,0 + 2H0H1 � 2H0H1 + 2H0.1 � 2H0,1

⇤

= Li2(z)� Li2(z) +1

2ln |z|2 (ln(1� z)� ln(1� z))

Single-valued multiple polylogarithms

Single-valued multiple polylogarithms (SVMPL) can be constructed througha map that to each multiple polylogarithm associates its single-valued version

Brown 2004, 2013, 2015examples of SVMPLs

Ga,b(z) = Ga,b(z) +Gb,a(z) +Gb(a)Ga(z) +Gb(a)Ga(z)

�Ga(b)Gb(z) +Ga(z)Gb(z)�Ga(b)Gb(z)

Ga(z) = Ga(z) +Ga(z) = ln��1�

z

a

��2

MRK in N=4 SYM

Beyond 6 points, only 2-loop MHV amplitudes were known in MRK at LLA

In MRK at LLA, the 2-loop n-pt remainder function Rn(2)

can be written as a sum of 2-loop 6-pt remainder functions R6(2)

Prygarin Spradlin Vergu Volovich 2011Bartels Kormilitzin Lipatov Prygarin 2011Bargheer Papathanasiou Schomerus 2015

In MRK at LLA, we can compute all MHV amplitudes at L loopsin terms of amplitudes with up to (L+4) points


We showed explicitly that the MHV amplitudes at:• 2 loops are determined by the 2-loop 6-pt amplitudes• 3 loops are determined by the 6- and 7-pt amplitudes through 3 loops• 4 loops are determined by the 6-, 7-, and 8-pt amplitudes through 4 loops• 5 loops are determined by the 6-, 7-, 8- and 9-pt amplitudes through 5 loops

We computed also all non-MHV amplitudes up 8 points and 4 loops

In MRK, 6-pt MHV and NMHV amplitudes are known at any number of loopsLipatov Prygarin 2010-2011Dixon Duhr Pennington 2012Pennington 2012Lipatov Prygarin Schnitzer 2012

MRK factorisation in N=4 SYM

For the helicities h1, …, hN-4 define the ratio

factorisation in MRK at LLA

with 𝛕k = function of cross ratios, and with coefficients

g(i1,...,iN�5)h1,...,hN�4

(z1, . . . , zN�5) =(�1)N+1

2

"N�5Y

k=1

+1X

nk=�1

✓zkzk

◆nk/2 Z +1

�1

d⌫k2⇡

|zk|2i⌫kEik⌫knk

#

⇥ �h1(⌫1, n1)

2

4N�5Y

j=2

Chj (⌫j�1, nj�1, ⌫j , nj)

3

5 ��hN�4(⌫N�5, nN�5)

where:the 𝛘’s are the 2 impact factors,the C’s are the N-6 central-emission verticesthe E’s are the N-5 BFKL-like eigenvalues for octet exchange

…

1

23

4

5

NN-1

N-2

i1

i2

iN-5

iN-4

-

-

+

+

h1

h2

hN-5

hN-4

�h1

�hN�4

ChN�5

Ch2

Rh1,...,hN�4 =

AN (�,+, h1, . . . , hN�4,+,�)

ABDSN (�,+, . . . ,+,�)

�

|MRK, LLA

Rh1,...,hN�4 (⌧1, z1, . . . , ⌧N�5, zN�5)

⇡2⇡i1X

i=2

X

i1+...+iN�5=i�1

ai

N�5Y

k=1

1

ik!lnik ⌧k

!g(i1,...,iN�5)h1,...,hN�4

(z1, . . . , zN�5)

Convolutions

we use the Fouries-Mellin (FM) transform

F [F (⌫, n)] =1X

n=�1

⇣zz

⌘n/2Z +1

�1

d⌫

2⇡|z|2i⌫ F (⌫, n)

which maps products into convolutions

F [F ·G] = F [F ] ⇤ F [G] = (f ⇤ g)(z) = 1

⇡

Zd2w

|w|2 f(w) g⇣ z

w

⌘

we compute the integral through the residue formulaZ

d2z

⇡f(z) = Resz=1F (z)�

X

i

Resz=aiF (z) Schnetz 2013

where F is the antiholomorphic primitive of f @zF = f

through the FM transform of the BFKL eigenvalue

E(z) = F [E⌫n]

we can write the recursion

g(i1,...,ik+1,...,iN�5)+...+ (z1, . . . , zN�5) = E(zk) ⇤ g(i1,...,iN�5)

+...+ (z1, . . . , zN�5)

g(0,...,0,ia1 ,0,...,0,ia2 ,0,...,0,iak

,0,...,0)+...+ (⇢1, . . . , ⇢N�5) = g

(ia1 ,ia2 ,...,iak)

+...+ (⇢ia1, ⇢ia2

, . . . , ⇢iak)

which implies that we can drop all the propagators without a log

i1

01

23

4

5

-

-

+

+

h1

h2

�h1

Ch2

67

h3

�h3

example for N=7, with h1 = h2

which connects amplitudes with a different number of legs

1

23

4

5

-

-

+

+

h1�h1

6 h3

�h3

i1

Convolutions and factorization

in fact, if all indices are zero except for one

g(0,...,0,ia,0,...,0)+...+ (⇢1, . . . , ⇢N�5) = g(ia)++ (⇢a)

which implies that

which shows, as previously stated, that in MRK at LLA, the 2-loop n-pt remainder function Rn(2)

can be written as a sum of 2-loop 6-pt amplitudes, in terms of SVHPLs

R(2)+...+ =

X

1iN�5

ln ⌧i g(1)++(⇢i)

with

g(1)++(⇢1) = �1

4G0,1 (⇢1)�

1

4G1,0 (⇢1) +

1

2G1,1 (⇢1)

At 3 loops, the n-pt remainder function Rn(3) can be written

as a sum of 3-loop 6-pt and 7-pt amplitudes

R(3)+...+ =

1

2

X

1iN�5

ln2 ⌧i g(2)++(⇢i) +

X

1i<jN�5

ln ⌧i ln ⌧j g(1,1)+++(⇢i, ⇢j)

with

g(2)++(⇢1) =� 1

8G0,0,1 (⇢1)�

1

4G0,1,0 (⇢1) +

1

2G0,1,1 (⇢1)�

1

8G1,0,0 (⇢1)

+1

2G1,0,1 (⇢1) +

1

2G1,1,0 (⇢1)� G1,1,1 (⇢1)

g(1,1)+++(⇢1, ⇢2) = �1

8G0,1,⇢2 (⇢1)�

1

8G0,⇢2,1 (⇢1) +

1

8G1,1,⇢2 (⇢1)�

1

8G1,⇢2,0 (⇢1)

� 1

8G⇢2,1,0 (⇢1) +

1

8G⇢2,1,1 (⇢1) +

1

4G1,⇢2,1 (⇢1)�

1

4G1 (⇢2)G1,⇢2 (⇢1)

+1

8G1 (⇢1)G0,0 (⇢2)�

1

8G0 (⇢2)G0,1 (⇢1) +

1

8G1 (⇢2)G0,1 (⇢1)�

1

8G⇢2 (⇢1)G0,1 (⇢2)

+1

8G1 (⇢2)G0,⇢2 (⇢1)�

1

8G0 (⇢2)G1,0 (⇢1) +

1

8G1 (⇢2)G1,0 (⇢1) +

1

8G0 (⇢2)G1,1 (⇢1)

� 1

8G1 (⇢2)G1,1 (⇢1)�

1

8G1 (⇢1)G1,1 (⇢2) +

1

8G⇢2 (⇢1)G1,1 (⇢2) +

1

8G0 (⇢2)G1,⇢2 (⇢1)

+1

8G1 (⇢2)G⇢2,0 (⇢1)�

1

8G1 (⇢2)G⇢2,1 (⇢1)

Note that Rn(3) cannot be written only in terms of SVHPLs, but SVMPLs are necessary

At 4 loops, the n-pt remainder function Rn(4) can be written

as a sum of 4-loop 6-pt, 7-pt and 8-pt amplitudes,and in general, we can compute all MHV amplitudes at L loops in terms of amplitudes with up to (L+4) points


Note also that because the convolutions with pure functions are pure,so are the MHV amplitudes

MRK factorisation works also for non-MHV amplitudes,however at each loop the number of building blocks is infinite,and the helicity flips make the non-MHV amplitudes not pure

We displayed that explicitly up to 5 loops, showing that the n-pt remainder function Rn(5)

can be written as a sum of 5-loop 6-, 7-, 8- and 9-pt amplitudes

We constructed explicitly all non-MHV amplitudes through 4 loop and 8 points

Mueller-Navelet jets and SVHPLs

Mueller & Navelet evaluated analytically the inclusive dijet cross section up to 5 loops. We evaluated it analytically up to 13 loops

The singlet LL BFKL ladder in QCD, and thus the dijet cross sectionin the high-energy limit, can also be expressed in terms of SVHPLs, i.e. in terms of single-valued iterated integrals on ℳ0,4

Dixon Duhr Pennington VDD 2013

Also, we could evaluate analytically the dijet cross section differentialin the jet transverse energies or the azimuthal angle between the jets(up to 6 loops)

use complex transverse momentum

and a complex variable

the Green's function can be expanded into a power series in

where the coefficient functions fk are given by the Fourier-Mellin transform

the fk have a unique, well-defined value for every ratio of the magnitudes of the two jet transverse momenta and angle between them.So, they are real-analytic functions of w

BFKL Green’s function and single-valued functions

qk

⌘ qxk

+ iqyk

z ⌘ q1q2

⌘µ = ↵µ y

fLL(q1, q2, ⌘µ) =1

2�(2)(q1 � q2) +

1

2⇡pq21 q

22

1X

k=1

⌘kµk!

fLLk (z)

fLLk (z) = F

⇥�k⌫n

⇤=

+1X

n=�1

⇣zz

⌘n/2Z +1

�1

d⌫

2⇡|z|2i⌫ �k

⌫n

generating functional of SVHPLs

to all orders in η the BFKL Green's function can be written in terms of a generating functional of SVHPLs Dixon Duhr Pennington VDD 2013

writing the coefficient function fk as

we obtain that the first few functions Fk are

note that fk has weight k-1

fLLk (z) =

|z|2⇡ |1� z|2 Fk(z)

F1(z) = 1 ,

F2(z) = 2G1(z)� G0(z) ,

F3(z) = 6G1,1(z)� 3G0,1(z)� 3G1,0(z) + G0,0,0(z) ,

F4(z) = 24G1,1,1(z) + 4G0,0,1(z) + 6G0,1,0(z)� 12G0,1,1(z) + 4G1,0,0(z)

� 12G1,0,1(z)� 12G1,1,0(z)� G0,0,0(z) + 8 ⇣3

this allows us to write the azimuthal angle distribution as

d�gg

d�jj=

⇡(CA↵s)2

2E2?

"�(�jj � ⇡) +

1X

k=1

ak(�jj)

⇡⌘k

#

where the contribution of the kth loop is

Azimuthal angle distribution

with

A1(�jj) = �1

2H0 ,

A2(�jj) = H1,0 ,

A3(�jj) =2

3H0,0,0 � 2H1,1,0 +

5

3⇣2H0 � i⇡ ⇣2 ,

A4(�jj) = �4

3H0,0,1,0 �H0,1,0,0 �

4

3H1,0,0,0 + 4H1,1,1,0 � ⇣2

✓2H0,1 +

10

3H1,0

◆+

4

3⇣3 H0 + i⇡

⇣2⇣2H1 � 2⇣3

⌘,

A5(�jj) = �46

15H0,0,0,0,0 +

8

3H0,0,1,1,0 + 2H0,1,0,1,0 + 2H0,1,1,0,0 +

8

3H1,0,0,1,0 + 2H1,0,1,0,0

+8

3H1,1,0,0,0 � 8H1,1,1,1,0 � ⇣2

✓33

5H0,0,0 � 4H0,1,1 � 4H1,0,1 �

20

3H1,1,0

◆

� ⇣3

✓2H0,1 +

8

3H1,0

◆+

217

15⇣4H0 + i⇡

⇣2

✓10

3H0,0 � 4H1,1

◆+ 4⇣3H1 �

10

3⇣4

�

where Hi,j,... ⌘ Hi,j,...(e�2i�jj ) Dixon Duhr Pennington VDD 2013

Transverse momentum distribution

d�gg

dp21?dp22?

=⇡(CA↵s)2

2p21?p22?

"�(p21? � p22?) +

1

2⇡p

p21? p22?b(⇢; ⌘)

#

where ⇢ = |w|

with

B1(⇢) = 1 ,

B2(⇢) = �1

2H0 � 2H1 ,

B3(⇢) =1

6H0,0 + 2H0,1 +H1,0 + 4H1,1 ,

B4(⇢) = � 1

24H0,0,0 �

4

3H0,0,1 �H0,1,0 � 4H0,1,1 �

1

3H1,0,0 � 4H1,0,1 � 2H1,1,0 � 8H1,1,1 +

1

3⇣3 ,

B5(⇢) =1

120H0,0,0,0 +

2

3H0,0,0,1 +

2

3H0,0,1,0 +

8

3H0,0,1,1 +

1

3H0,1,0,0 + 4H0,1,0,1

+ 2H0,1,1,0 + 8H0,1,1,1 +1

12H1,0,0,0 +

8

3H1,0,0,1 + 2H1,0,1,0 + 8H1,0,1,1

+2

3H1,1,0,0 + 8H1,1,0,1 + 4H1,1,1,0 + 16H1,1,1,1 + ⇣3

✓� 1

12H0 �

2

3H1

◆,

where Hi,j,... ⌘ Hi,j,...(⇢2) Dixon Duhr Pennington VDD 2013

Mueller-Navelet dijet cross section reloaded

the MN dijet cross section is

the first 5 loops were computed by Mueller-Navelet.We computed it through the 13 loops

Dixon Duhr Pennington VDD 2013

Regge limitin the

next-to-leading logarithmicapproximation

BFKL theorythe BFKL equation describes the evolution of the gluon propagatorin 2-dim transverse momentum space

! f!(q1, q2) =1

2�(2)(q1 � q2) +

Zd2kK(q1, k) f!(k, q2)

the solution is given in terms of eigenfunctions Φνn and an eigenvalue ωνn

f!(q1, q2) =+1X

n=�1

Z +1

�1d⌫

1

! � !⌫n�⌫n(q1)�

⇤⌫n(q2)

as a function of rapidity, the solution is

f(q1, q2, y) =+1X

n=�1

Z +1

�1d⌫ �⌫n(q1)�

⇤⌫n(q2) e

y !⌫n

we expand kernel K, eigenfunctions Φνn and eigenvalue ωνn in powers of ↵µ =NC

⇡↵S(µ

2)

K(q1, q2) = ↵µ

1X

l=0

↵lµ K

(l)(q1, q2) !⌫n = ↵µ

1X

l=0

↵lµ !

(l)⌫n �⌫n(q) =

1X

l=0

↵lµ �

(l)⌫n(q)

At LLA

!(0)⌫n = �2�E �

✓|n|+ 1

2+ i⌫

◆�

✓|n|+ 1

2� i⌫

◆�(0)

⌫n (q) =1

2⇡(q2)�1/2+i⌫ ein✓

note that in N=4 SYM the eigenfunctions and the eigenvalue are the same

At NLLA in QCD and in N=4 SYM, the eigenvalue is Fadin Lipatov 1998Kotikov Lipatov 2000, 2002Chirilli Kovchegov 2013Duhr Marzucca Verbeek VDD 2017

with one-loop beta function and two-loop cusp anomalous dimension

and with

�(1)⌫n = @2⌫�⌫n

�(2)⌫n = �2�(n, �)� 2�(n, 1� �)

�(3)⌫n =��( 12 + i⌫)�( 12 � i⌫)

2i⌫

✓1

2+ i⌫

◆�

✓1

2� i⌫

◆�

⇥�n0

✓3 +

✓1 +

Nf

N3c

◆2 + 3�(1� �)

(3� 2�)(1 + 2�)

◆� �|n|2

✓✓1 +

Nf

N3c

◆�(1� �)

2(3� 2�)(1 + 2�)

◆�

�0 =11

3� 2Nf

3Nc�(2)K =

1

4

✓64

9� 10Nf

9Nc

◆�⇣2

2

Φ(n,γ) is a sum over linear combinations of ψ functionsand γ is a shorthand γ = 1/2 + iν

+3

2⇣3!(1)

⌫n = +1

4�(3)⌫n + �(2)

K �⌫n � 1

8�0�

2⌫n

1

4�(1)⌫n +

1

4�(2)⌫n

In blue we labeled the terms which occur only in QCD,in red the ones which occur in QCD and in N=4 SYM

�⌫n = !(0)⌫n

BFKL eigenvalue at NLLA

At NLLA, the expansion of the BFKL ladder is

f(q1, q2, y) = fLL(q1, q2, ⌘µ) + ↵µ fNLL(q1, q2, ⌘µ) + . . . , ⌘µ = ↵µ y

fNLL contains the NLO corrections to the eigenvalue and to the eigenfunctions,however if we use the scale of the strong coupling to be the geometric mean of the transverse momenta at the ends of the ladder, then we can use the LOeigenfunctions instead of the NLO ones

At NLLA in QCD, the eigenfunction is

�⌫n(q) = �(0)⌫n (q)

1 + ↵µ

�0

8ln

q2

µ2

✓@⌫P

�⌫n

@⌫�⌫n+ i ln

q2

µ2P

�⌫n

@⌫�⌫n

◆+O(↵2

µ)

�

Chirilli Kovchegov 2013Duhr Marzucca Verbeek VDD 2017

f(q1, q2, y) =+1X

n=�1

Z +1

�1d⌫ �(0)

⌫n (q1)�(0)⇤⌫n (q2) e

y ↵S(s0)[!(0)⌫n+↵S(s0)!

(1)⌫n ] + . . .

Duhr Marzucca Verbeek VDD 2017

µ2 = s0 =qq21q

22with

BFKL eigenfunctions at NLLA

At NLLA, the BFKL ladder is

fNLL(q1, q2, ⌘s0) =1

2⇡p

q21q22

1X

k=1

⌘ks0k!

fNLLk+1 (z) ⌘s0 = ↵S(s0) y

with coefficients given by the Fourier-Mellin transform

fNLLk (z) = F

h!(1)⌫n �k�2

⌫n

i=

+1X

n=�1

⇣zz

⌘n/2Z +1

�1

d⌫

2⇡|z|2i⌫ !(1)

⌫n �k�2⌫n �⌫n = !(0)

⌫n

+3

2⇣3!(1)

⌫n = +1

4�(3)⌫n + �(2)

K �⌫n � 1

8�0�

2⌫n

1

4�(1)⌫n +

1

4�(2)⌫n

using the explicit form of the eigenvalue

the coefficients can be written as

fNLLk (z) =

1

4C(1)

k (z) +1

4C(2)

k (z) +1

4C(3)

k (z) + �(2)K fLL

k�1(z)�1

8�0 f

LLk (z) +

3

2⇣3 f

LLk�2(z)

with C(i)k (z) = F

h�(i)⌫n �k�2

⌫n

i

the weight of fNLLk is

weight(fNLLk)= k k kk � 1k � 2 w k0 w k

Fourier-Mellin transform

are SVHPLs of uniform weight k with singularities at z=0 and z=1

C(3)k (z) are MPLs of type G(a1, . . . , an; |z|) with ak 2 {�i, 0, i}

they are SV functions of z because they have no branch cut on the positive real axis,and have weight 0 ≤ w ≤ k

C(1)k (z)

C(2)k (z) one needs Schnetz’ generalised SVMPLs with singularities at

z =↵ z + �

� z + �, ↵,�, �, � 2 C

ForSchnetz 2016

are Schnetz’ generalised SVMPLsthen one can show that C(2)k (z)

with singularities atG(a1, . . . , an; z) ai 2 {�1, 0, 1,�1/z}Duhr Marzucca Verbeek VDD 2017

Interestingly, in transverse momentum space at NLLA, the maximal weight of the BFKL ladder in QCD is not the same as the one of the ladder in N=4 SYM

In moment space, the maximal weight of the BFKL eigenvalue and of the anomalous dimensions of the leading twist operators which control the Bjorken scaling violations in QCD is the same as the correspondingquantities in N=4 SYM Kotikov Lipatov 2000, 2002

Kotikov Lipatov Velizhanin 2003


SV functions

one can consider the BFKL eigenvalue at NLLA in a SU(Nc) gauge theorywith scalar or fermionic matter in arbitrary representations

!(1)⌫n =

1

4�(1)⌫n +

1

4�(2)⌫n +

1

4�(3)⌫n (Nf , Ns) +

3

2⇣3 + �(2)(nf , ns)�⌫n � 1

8�0(nf , ns)�

2⌫n

Kotikov Lipatov 2000

with �0(nf , ns) =11

3� 2nf

3Nc� ns

6Nc�(2)(nf , ns) =

1

4

✓64

9� 10 nf

9Nc� 4 ns

9Nc

◆� ⇣2

2

nf =X

R

nRf TR ns =

X

R

nRs TR Tr(T a

RTbR) = TR �ab TF =

1

2

BFKL ladder in a generic SU(Nc) gauge theory

number of scalars (Weyl fermions) in the representation Rns(nf ) =

�(3)⌫n (Nf , Ns) = �(3,1)⌫n (Nf , Ns) + �(3,2)⌫n (Nf , Ns)

withN

x

=1

2

X

R

n

R

x

T

R

(2CR

�N

c

) , x = f, s

Necessary and sufficient conditions for a SU(Nc) gauge theory to have a BFKL ladderof maximal weight are:

— the one-loop beta function must vanish— the two-loop cusp AD must be proportional to ζ2

— must vanish → �(3,2)⌫n

There is no theory whose BFKL ladder has uniform maximal weight which agrees with the maximal weight terms of QCD


2Nf = N2c + Ns


Matter in the fundamental and in the adjoint

We solve the conditions above for matter in the fundamental F and in the adjoint Arepresentations. We obtain:

2nFf = nF

s 2nAf = 2 + nA

s

which describes the spectrum of a gauge theory with N supersymmetriesand nF = nfF chiral multiplets in F and nA = nfA - N chiral multiplets in A

There are four solutions to those conditions

— the first is N=4 SYM— the second is N=2 superconformal QCD with Nf = 2Nc hypermultiplets— the third is N=1 superconf. QCD

because the one-loop beta function is fixed by matter loops in gluon self-energies,we are only sensitive to the matter content of a theory, and not to its details(like scalar potential or Yukawa couplings)


Conclusions at LLA

We computed the Mueller-Navelet dijet cross section through 13 loops

n-point amplitudes in the LLA of the Multi-Regge limit of N=4 SYM can be fully expressed in terms of single-valued iterated integrals on ℳ0,n-2 i.e. in terms of single-valued multiple polylogarithms

We showed that one can compute all MHV amplitudes at L loopsin terms of amplitudes with up to (L+4) points

We displayed that explicitly up to 5 loops, showing that the n-pt remainder function Rn

(5) can be written as a sumof 5-loop 6-, 7-, 8- and 9-pt amplitudes

We constructed explicitly all non-MHV amplitudes through4 loop and 8 points

The singlet BFKL ladder, and thus the dijet cross section in the LLAof the high-energy limit of QCD, can also be expressed in terms of of single-valued iterated integrals on ℳ0,4, i.e. in terms of SVHPLs

We expect that the n-jet cross section in the LLA of the high-energy limit of QCD can be written in terms of single-valued iterated integrals on ℳ0,n+2

Conclusions at NLLA

The singlet BFKL ladder at NLLA in QCD and in N=4 SYM canalso be expressed in terms of SVMPLs, but generalised tonon-constant values of the roots

In transverse momentum space at NLLA, the maximal weight of theBFKL ladder in QCD is not the same as the one of the ladder in N=4 SYM

There is no SU(Nc) gauge theory whose BFKL ladder has uniformmaximal weight which agrees with the maximal weight terms of QCD

We found four theories with matter in the fundamental and in the adjointrepresentations, whose BFKL ladder has uniform maximal weight

The Regge limit in QCD and in N=4 super-Yang-Millsdelducav/ETHtalk-may17.pdf · The Regge limit in...

Documents

Transcript of The Regge limit in QCD and in N=4 super-Yang-Millsdelducav/ETHtalk-may17.pdf · The Regge limit in...