leonardo vernazza - physics.iith.ac.in

TWO-PARTON SCATTERING IN THE HIGH-ENERGY LIMIT

Leonardo Vernazza

Higgs Centre for Theoretical Physics,School of Physics and Astronomy

Hyderabad, HEP Remote Video Seminar Series, 25/07/2017

OUTLINE

• Aspects of 2 2 scattering amplitudes in the high-energy limit

• High-energy evolution and the Balitsky-JIMWLK equation

• The three-Reggeon cut

• The two-Reggeon cut

In collaboration with Simon Caron-Huot, Einan Gardi and Joscha Reichel,

Based on arXiv:1701.05241 and work in progress

ASPECTS OF 2 2 SCATTERING AMPLITUDES IN THE HIGH-ENERGY LIMIT

2 2 SCATTERING AMPLITUDES IN THE HIGH-ENERGY LIMIT

• Calculation of scattering amplitudes at high order in perturbation theory is one of the main ingredients for the program of precision physics at the LHC

• Amplitudes are complicated functions of the kinematical invariants, their calculation is non-trivial, and it is subject of intense study.

• Express Feynman integrals in terms of known functions (harmonic polylogarithms, elliptic integrals, etc)

• Amplitudes contains infrared divergences, which must cancel when summing virtual and real corrections.

M = 4↵s

M(0) +

↵s

4M(1) +

↵s

4

2M(2) + ...

. . .

. . .

. . .


• Consider 2 → 2 scattering amplitudes in the high-energy limit:

• The amplitude becomes a function of the ratio |s/t|; here we consider the leading power term in this expansion

s = (p1 + p2)2 t = (p1 p4)

2 > 0

p1

p2 p3

p4

t channels channel

• Information and constraints can be obtained by considering kinematical limits:

• the number of invariants is reduced;

• identify factorisation properties and iterative structures of the amplitude;

• relevant for phenomenology: because of soft and collinear enhancement, differential distributions in specific kinematic limit develops large logarithms, which may spoil the convergence of the perturbative expansion.

M(s, t, µ) = MLP

s

t,t

µ2

1 +O

t

s

.

2 2 SCATTERING AMPLITUDES IN THE HIGH-ENERGY LIMIT• Gluon-gluon scattering amplitude at tree level:

• In the high-energy limit only the second diagram contributes at leading power.

• The amplitude at higher orders contains logarithms of the ratio |s/t|. They can be characterised in terms of Regge poles and cuts: at LL

• The function 𝛼g(t) is known as the Regge trajectory

Mij!ij |LL =

s

t

↵s CA ↵(1)

g (t)

4↵s M(0)ij!ij ,

∼ 1/s ∼ 1/t ∼ 1/u ∼ 1

Regge, Gribov

M(0)ij!ij =

2s

t(T b

i )a1a4(Tbj )a2a3 1423 .

↵(1)g (t) =

r2

t

µ2

µ2!t=

r2

, r = eE(1 )2(1 + )

(1 2) 1 1

22

2 7

33

3 + . . .

• The amplitudes which develop definite factorisation properties in the high-energy limit are the so called even and odd amplitudes, i.e. the projection onto eigenstates of signature, (crossing symmetry s ↔ u)

• M(+) and M(−) are respectively imaginary and real, when expressed in terms of the natural signature-even combination of logs

L log

s

t

i

2

=

1

2

log

s i0

t+ log

u i0

t

.

M(±)(s, t) = 12

M(s, t)±M(s t, t)

.


s

• Determining the amplitude beyond LL requires to understand the structure of Regge cuts.

• Beyond tree level the amplitude has a non-trivial color structure

• Decompose the amplitude in a color orthonormal basis in the t-channel

• Invoking Bose symmetry we deduce

M(s, t) =X

i

c[i] M[i](s, t).

odd: M[8a],M[10+10], even: M[1],M[8s],M[27],M[0](gg scattering) .

8 8 = 1 8s 8a 10 10 27 0


FACTORISATION STRUCTURE• Write the amplitude as the sum of odd and even component

• The amplitude in the high-energy limit has the following factorisation structure

• Focus on the Regge-cut contributions: define a “reduced” amplitude by removing the Reggeized gluon and collinear divergences

. . .

. . .. . . . . .

+

Mij!ij (ZiZj)1 eT2

t ↵g(t)L Mij!ij ,

M(s, t) = M()(s, t) +M(+)(s, t), M(±)(s, t) = 4↵s

X

l,m

↵s

lLm M(±,l,m) .

Odd (M[8a],M[10+10]) Even (M[1],M[8s],M[27],M[0])

LL

NLL

NNLL

THE BALITSKY-JIMWLK EQUATION AND THE THREE REGGEON CUT

. . . . . .

+

• Starting at NNLL, one has mixing between one- and three-Reggeons exchange:

• The mixing between one- and three-Reggeons exchange has significant consequences:

• It is at the origin of the breaking of the simple power law one has up to NLL accuracy. Such breaking appears for the first time at two loops.

• Starting at three loops, there will be a single-logarithmic contribution originating from the three-Reggeon exchange, and from the interference of the one- and three-Reggeon exchange: the interpretation of the Regge trajectory at three loops needs to be clarified.

• Schematically, the whole amplitude at NNLL is composed of

THE ODD AMPLITUDE AT NNLL

. . . . . .

Di

Dj

αg +

Mij!ij |NNLL

= M()

ij!ij |1-Reggeon + 3-Reggeon

+ M(+)

ij!ij |2-Reggeon

.

Del Duca, Glover, 2001; Del Duca, Falcioni, Magnea, LV, 2013

• The high-energy limit correspond to a configuration of forward scattering:

• The high-energy logarithm is the rapidity difference between the target and the projectile:

• This kinematical configuration is described in terms of Wilson lines stretching from −∞ to +∞. The Wilson lines follow the paths of color charges inside the projectile, are null and labelled by transverse coordinates z:

• The idea is to approximate, to leading power, the fast projectile and target by Wilson lines and then compute the scattering amplitude between Wilson lines.

BFKL THEORY ABRIDGED

U(z?) = P exp

igs

Z +1

1A

a+(x

+, x

=0, z?)dx

+T

a

.

. . .

= L log

s

t

i

2

.

Korchemskaya, Korchemsky, 1994, 1996

Babansky, Balitsky, 2002, Caron-Huot, 2013

t = (p1 p4)2= (p2 p3)

2= s

2

(1 cos ), s t ) ! 0

• The Wilson line stretches from −∞ to +∞ and thus develops rapidity divergencies. The regularised Wilson lines obeys the Balitsky-JIMWLK evolution equation: with and TL/R’s are generators for left and right color rotations:

• In our analysis we need only the leading-order conformal invariant kernel Kij

• The number of Wilson lines is not fixed: a projectile necessarily contains multiple color charges at different transverse positions.

THE BALITSKY-JIMWLK EQUATION

d

d

hU(z1) . . . U(zn)

=

nX

i,j=1

Hij ·hU(z1) . . . U(zn)

,

T ai,L = [T aU(zi)]

U(zi), T a

i,R(z) = [U(zi)Ta]

U(zi).

Hij =↵s

22

Z[dzi][dzj ][dz0]Kij;0

hT ai,LT

aj,L + T a

i,RTaj,R Uab

ad(z0)T ai,LT

bj,R + T a

j,LTbi,R

i+O(↵2

s).

Caron-Huot, 2013

z1

z2

z0

z1

z2

x− x+

Balitsky Chirilli, 2013; Kovner, Lublinsky, Mulian, 2013, 2014, 2016

Kij;0 = S(µ2)(1 )2

2

z0i · z0j(z20iz

20j)

1.

• However, in perturbation theory the unitary matrices U(z) will be close to identity and so can be usefully parametrised by a field W

• The color-adjoint field W sources a BFKL Reggeised gluon. A generic projectile, created with four-momentum p1 and absorbed with p4, can thus be expanded at weak coupling asand we introduce the impact factors Di,j, which encode the dependence on the transverse coordinates of the W fields.

• We need to derive the evolution equation for the field W. This is equivalent to switch from the Balitsky-JIMWLK to the BFLK regime.


U(z) = eigs TaWa(z) .

. . .

Caron-Huot, 2013

| i gs D1(t) |W i+ g2s D2(t) |WW i+ g3s D3(t) |WWW i+ . . . = | 1i+ | 2i+ | 3i+ . . . .

• Expand U in powers of W

• The expansion of the color generators follows by using the Backer-Campbell-Hausdorff formula. Then, it is possible to expand the leading Hamiltonian Hij in powers of gs We get

• The first non-linear correction is new

U = eigs WaTa

= 1 + igs Wa T a g2s

2W aW b T aT b i

g3s6W aW bW c T aT bT c

+g4s24

W aW bW cW d T aT bT cT d +O(g5s W5).

H = Hk!k +Hk!k+2 + . . .

Hk!k

=↵s

CA

22

Z[dz

i

][dz0]Kii;0 (Wi

W0)a

W a

i

↵s

22

Z[dz

i

][dzj

][dz0]Kij;0(Wi

W0)x(W

j

W0)y(F xF y)ab

2

W a

i

W b

j

.

Hk!k+2 =

↵2s

3

Z[dz

i

][dz0]Kii;0 (Wi

W0)xW y

0 (Wi

W0)z Tr

F xF yF zF a

W a

i

+↵2s

6

Z[dz

i

][dzj

][dz0]Kij;0 (FxF yF zF t)ab

h(W

i

W0)xW y

0 Wz

0 (Wj

W0)t

W x

i

(Wi

W0)yW z

0 (Wj

W0)t (W

i

W0)xW y

0 (Wj

W0)zW t

j

i 2

W a

i

W b

j

.


Caron-Huot, Gardi, LV, 2017

• The inner product is the scattering amplitude of Wilson lines renormalized to equal rapidity.

• Multi-Reggeon correlators are obtained by Wick contractions

• There are also off-diagonal elements, which can be defined to have zero overlap (at equal rapidity)

• Choosing the 1-W and 3-W states to be orthogonal, combined with symmetry of the Hamiltonian, (boost invariance)

• implies that in this scheme Hk → k+2 = Hk+2 → k. This relation is known as projectile-target duality.


hW1W2|W10W20i = G110G220 +G120G210 +O(g2s),

hW1W2W3|W10W20W30i = G110G220G330 + (5 permutations) +O(g2s),

. . .

hW1W2W3|W4i = hW4|W1W2W3i = 0.

d

dhO1|O2i = 0 , hHO1|O2i = hO1|HO2i hO1|H|O2i,

G110 hW1|W10i = ia1a

01

p21(22)(p1 p01) +O(g2s).

Caron-Huot, 2013

• An m→m+k transition from the leading-order Balitsky-JIMWLK equation is proportional to gs2l+k. Thus for k ≥ 0, all the interactions can be extracted from the leading-order equation.

• Interactions with k < 0 are suppressed by at least gs2l+|k|, which means that they can first appear in the (|k|+1)-loop Balitsky-JIMWLK Hamiltonian.

• Thus to obtain the m→m−2 transition by direct calculation of the Hamiltonian would require three- loop non-planar computation.

• For our purposes this is unnecessary, since the symmetry of H predicts the result.


H

0

BBBBBB@

W(W )2

(W )3

(W )4

(W )5

. . .

1

CCCCCCA=

0

BBBBBB@

H1!1 0 H3!1 0 H5!1 . . .0 H2!2 0 H4!2 0 . . .

H1!3 0 H3!3 0 H5!3 . . .0 H2!4 0 H4!4 0 . . .

H1!5 0 H3!5 0 H5!5 . . .. . . . . . . . . . . . . . . . . .

1

CCCCCCA

0

BBBBBB@

W(W )2

(W )3

(W )4

(W )5

. . .

1

CCCCCCA

0

BBBBBB@

g2s 0 g4s 0 g6s . . .0 g2s 0 g4s 0 . . .g4s 0 g2s 0 g4s . . .0 g4s 0 g2s 0 . . .g6s 0 g4s 0 g2s . . .. . . . . . . . . . . . . . . . . .

1

CCCCCCA

0

BBBBBB@

W(W )2

(W )3

(W )4

(W )5

. . .

1

CCCCCCA.

From LO B-JIMWLK

LO BFKL kernel

Terms in NNLO

B-JIMWLK - predicted by symmetry H = HT

Up to two loops the amplitude reads

with

At three loops we find the following amplitude:

where the loop functions RA,B,C are

RESULT: THE ODD AMPLITUDE AT NNLL TO THREE LOOPS

M(,1)ij!ij =

D(1)

i +D(1)j

M(0)

ij!ij ,

M(,2)ij!ij =

D(2)

i +D(2)j +D(1)

i D(1)j + 2R(2)

(T2

su)2 1

12 (CA)2

M(0)ij!ij ,

R(2) 1

24(r)

2 I[1] = (r)2

62B1,1+()

B1,1()= (r)

2

1

82+

3

43 +

9

824 + . . .

,

Three-Reggeon cut

M(,3,1)ij!ij = 2

R(3)

A T2su[T

2t ,T

2su] + R(3)

B [T2t ,T

2su]T

2su +R(3)

C (CA)3M(0)

ij!ij ,

R(3)A =

1

16(r)

3(IaIc) = (r)3

1

483+

37

243 + . . .

,

R(3)B =

1

16(r)

3(IcIb) = (r)3

1

243+

1

123 + . . .

,

R(3)C =

1

288(r)

3(2IcIaIb) = (r)3

1

8643 35

4323 + . . .

.


COMPARISON BETWEEN REGGE AND INFRARED FACTORIZATION

• The calculation of the amplitude was based solely on evolution equations of the Regge limit.

• Highly nontrivial consistency test: the prediction must be consistent with the known exponentiation pattern and the anomalous dimensions governing infrared divergences.

• Conversely, the prediction for the reduced amplitude gives a constraint on the soft anomalous dimension.

• The infrared divergences of amplitudes are controlled by a renormalization group equation: where Z is given as a path-ordered exponential of the soft-anomalous dimension:

• The soft anomalous dimension for scattering of massless partons (pi2 = 0) is an operators in

color space given, to three loops, by

Mn

pi, µ,↵s(µ2)= Zn

pi, µ,↵s(µ2)Hn

pi, µ,↵s(µ2),

BFKL VS INFRARED FACTORISATION

n

pi,,↵s(

2)= dip.

n

pi,,↵s(

2)+ n (ijkl) .

Becher, Neubert, 2009; Gardi, Magnea, 2009

Becher, Neubert, 2009; Dixon, Gardi, Magnea, 2009; Del Duca, Duhr, Gardi, Magnea, White, 2011; Neubert, LV, 2012, Almelid, Duhr, Gardi, McLeod, White, 2017

Zn

pi, µ,↵s(µ

2)

= P exp

(1

2

Z µ2

0

d2

2n

pi,,↵s(

2)

)

.

• 𝜞dipn involves only pairwise interactions amongst the hard partons: “dipole formula”

• The term 𝜟n(𝜌ijkl) involves interactions of up to four partons: “quadrupole correction”

• The three loop correction has been calculated recently, and readswhere F is a function of cross ratios: 𝜌ijkl = (-sij)(-skl)/(-sik)(-sjl). Explicitly, one has where the ℒ are Brown’s single-valued harmonic polylogarithms, and the constant term reads

n(ijkl) =1X

i=3

↵s

i(i)

n (ijkl).

(3)n (ijkl) =

1

4fabef cde

X

1i<j<k<ln

hTa

iTbjT

ckT

dl F(ikjl, iljk)

+ TaiT

bkT

cjT

dl F(ijkl, ilkj) + Ta

iTblT

cjT

dk F(ijlk, iklj)

i

C

4fabef cde

nX

i=1

X

1j<kn,j,k 6=i

Tai ,T

di Tb

jTck,

F(ikjl, ilkj) = F (1 zijkl) F (zijkl), with F (z) = L10101(z) + 22L001(z) + L100(z)

,


dip.n

pi,,↵s(

2)

= K(↵s)

2

X

i<j

log

sij2

Ti ·Tj +

X

i

i(↵s) .

C = 5 + 223.

Almelid, Duhr, Gardi, 2015, 2016

• In the high-energy limit the dipole formula reduces to

• The quadrupole correction has only one imaginary term at NNLL

• Because of the form of 𝜞dip and 𝜟(𝜌ijkl) in the High-energy limit, the Z factor factorises

• The relevant bit for us is

• The factors K and Q𝜟 involve integrals over the scale

dip.pi,,↵s(

2

)

Regge! K(↵s)

2

LT2

t + iT2

su +

Ctot

2

log

t

2

+

4X

i=1

i(↵s) +Ot

s

,

Zpi, µ,↵s(µ

2)= Z

st, µ,↵s(µ

2)Zi

t, µ,↵s(µ

2)Zj

t, µ,↵s(µ

2),

Zs

t, µ,↵s(µ

2)

= exp

n

K

↵s(µ2)

LT2t + iT2

su

+Q(3)

o

BFKL VS INFRARED FACTORISATIONDel Duca, Duhr, Gardi,

Magnea, White, 2011

Caron-Huot,

Gardi, LV, 2017

K = 1

4

Z µ2

0

d2

2K

↵s(

2)=

1

2

↵s(µ2)

+ . . . ,

Q(3) = (3)

2

Z µ2

0

d2

2

↵s(2)

3

=(3)

6

↵s(µ2)

3

.

(3) = i [T2t , [T

2t ,T

2su]]

34L + O(L0).

• The finite reminder of the amplitude, i.e. the hard function reads

• This equation allows us to pass from directly from the reduced amplitude predicted using BFKL theory, to the hard function.

• In particular, the statement that the left-hand-side H is finite, which is equivalent to the exponentiation of infrared divergences, is a highly nontrivial constraint on our result.

• By using Baker-Campbell-Hausdorff formula we get the hard function at each order in perturbation theory. For instance

Hij!ij

pi, µ,↵s(µ2)

= exp

1n

K

↵s(µ2)

LT2t + iT2

su

+Q(3)

o

· exp

↵g(t)LT2t

ˆMij!ij

pi, µ,↵s(µ2)

.


Re[H(2,0)] =

D(2)

i +D(2)j +D(1)

i D(1)j 2R(2) 1

12 (CA)2

+ 2R(2) + 1

2 (K(1))2 + K(1)↵(1)

g

(T2

su)2

M(0).

Del Duca, Falcioni, Magnea, LV, 2013

• Some coefficients, like the impact factors, are not predicted explicitly from Regge theory.

• The BFLK approach developed here allows us to extract these quantities consistently, and use them to predict higher orders.

• The impact factors at two loops are extracted by taking the projection of the amplitude onto the antisymmetric octet component:

• The effect of the three-Reggeon cut is evident from the color-dependent term. Consistency requires the three equations above to be satisfied simultaneously.


2D(2)g =

H(2,0)[8a]gg!gg

H(0)[8a]gg!gg

(D(1)g )2 + 2R(2)N

2c

12 2R(2)N

2c + 24

4,

D(2)q +D(2)

g =H(2,0)[8a]

qg!qg

H(0)[8a]qg!qg

D(1)q D(1)

g + 2R(2)N2c

12 2R(2)N

2c + 4

4,

2D(2)q =

Re[H(2,0)[8a]qq!qq ]

H(0)[8a]qq!qq

(D(1)q )2 + 2R(2)N

2c

12 2R(2)N

4c 4N2

c + 12

4N2c

.

Dq

αg αg αg

Dq

Dg

Dg Dg

Dq


• At three loops, at NNLL, the calculation of the odd sector within Regge theory giveswhich is consistent with infrared factorisation. This is a rather non-trivial check, given that the two calculations are done in two completely different ways.

Re[H(3,1)] =h↵(3)g + ↵(2)

g

D(1)

i +D(1)j

+ ↵(1)

g

D(2)

i +D(2)j +D(1)

i D(1)j

iT2

t M(0)

+ 2hR(3)

C 112 ↵

(1)g R(2)

i(T2

t )3 M(0) + 2 ↵(1)

g R(2) T2t (T

2su)

2 M(0)

+ 2hR(3)

A + 16 K

(1)2(K(1))2 + 3↵(1)

g K(1) + 3d2i

T2su[T

2t ,T

2su]M(0)

+ 2hR(3)

B 13 K

(1)(K(1))2 + 3↵(1)

g K(1) + 3(↵(1)g )2

i[T2

t ,T2su]T

2su M(0).



must be finite

Re[H(3,1)] =

"↵(3)g + ↵(2)

g

D(1)

i +D(1)j

+ ↵(1)

g

D(2)

i +D(2)j +D(1)

i D(1)j

+ C2A2

864

1

3 152

4 1753

2

#CA M(0)

+ 2 5312

T2su[T

2t ,T

2su]M(0) + 2 3

12[T2

t ,T2su]T

2su M(0) +O().

• We get some parts of the finite amplitude. In the orthonormal basis in the t-channel we have

• The antisymmetric octet amplitude cannot be predicted entirely, given the unknown Regge trajectory at three loops. The component can be predicted exactly, and it agrees with a recent calculation of the gluon-gluon amplitude in N=4 SYM.

• Starting from three loops the “gluon Regge trajectory” is scheme-dependent. We define it to be the 1→1 matrix element of the Hamiltonian, αg(t) = −H1→1/CA, in the scheme where states corresponding to a different number of Reggeon are orthogonal



Re[H(3,1),[8a]] =

CA

h↵(3)g + ↵(2)

g

D(1)

i +D(1)j

+ ↵(1)

g

D(2)

i +D(2)j +D(1)

i D(1)j

i

+ C3A2

864

1

3 152

4 1753

2

CA

2 233

+O()

M(0),[8a],

Re[H(3,1),[10+10]] =p2CA

qC2

A 4

112324

+O()

M(0),[8a].

10 + 10

Henn, Mistlberger, 2016

log

M[8a]gg!gg

M(0)[8a]gg!gg

= L

H1!1(t) +

↵s

32

hNc

2R

(3)A + 2R

(3)B

+N3

cR(3)C

i+O(L0,↵4

s),

• Thanks to a recent calculation of the gluon-gluon amplitude in N=4 SYM, in this theory one hasDefine the Regge trajectory as Then, matching these two results we get

• The amplitude is really a sum of multiple powers. Simply exponentiating the log of the full amplitude at three loops predicts an incorrect four-loop amplitude. The correct, procedure is to exponentiate the BFKL Hamiltonian. With the “trajectory” fixed as above, this procedure does not require any new parameter for the odd amplitude at NNLL to all loop orders.

THE REGGE TRAJECTORY AT THREE LOOPS IN N=4 SYM

log

M[8a],N=4gg!gg

M(0)[8a]gg!gg

L

= Nc

↵s

k1 +

↵s

2k2 +

↵s

3k3 + · · ·

,

Henn, Mistlberger, 2016

↵(1)g |N=4 = k1 =

1

2

24

27

63 3

47

324 + 4

7

1223 31

105

+O(5),

↵(2)g |N=4 = k2 = Nc

2

8

1

3

8

3

164 + 2

71

2423 +

41

85

+O(3)

,

↵(3)g |N=4 = k3 2

hNc

2R(3)

A + 2R(3)B

+N3

cR(3)C

= N2c

2

144

1

3+

494192

1

+

107

14423 +

54

+O()

+N0

c

0 +O()

.


HN=41!1 = Nc

↵s

↵(1)g |N=4 +

↵s

2↵(2)g |N=4 +

↵s

3↵(3)g |N=4 + · · ·

,

Del Duca, Falcioni, Magnea, LV, 2014

THE BALITSKY-JIMWLK EQUATION AND THE TWO REGGEON CUT

. . .


• The even amplitude at NLL is given by

. . .

. . .. . . . . .

+

Odd (M[8a],M[10+10]) Even (M[1],M[8s],M[27],M[0])

LL

NLL

NNLL

i

2sM(+)

NLL = h (+)j,2 |eHL| (+)

i,2 i(LO),i

2sM(+,`)

NLL =1

(` 1)!h (+)

2 |H2!2

`1

| (+)2 i(LO).

• The even amplitude readswith

• The “target averaged wave function” readswith and the initial condition is fixed to

• The function f is the BFKL kernel


M(+,`)NLL = i

(B0)`

(` 1)!

Z[Dk]

p2

k2(k p)2(`1)(p, k)T2

su M(0),

(`1)(p, k) = (2CA T2t )

(`1)(p, k) + (CA T2t )

(`1)(p, k),

(`1)(p, k) =

Z[Dk0] f(p, k, k0)

h(`2)(p, k0) (`2)(p, k)

i, (`1)(p, k) =

1 J(p, k)

2(`2)(p, k),

B0 = r = eE2(1 )(1 + )

(1 2).

f(p, k0, k) =k02

k2(k k0)2+

(p k0)2

(p k)2(k k0)2 p2

k2(p k)2, J(p, k) = 2

Z[Dk0] f(p, k, k0).

(0)(p, k) = 1.

. . .⎫⎪⎪⎪⎬

⎪⎪⎪⎭Ω(ℓ−2)(p, k)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

Ω(ℓ−1)(p, k′)

⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭

M(+,ℓ)(p)

⎫⎬

⎭ H2→2(p, k′, k)

• Up to four loops one gets


M(+,1)NLL = i

B0

2T2

su M(0),

M(+,2)NLL = i

(B0)2

2

1

(2)2+

932

+2744

2 +6352

3 +O(4)

(CA T2

t )T2su M(0),

M(+,3)NLL = i

(B0)3

3!

1

(2)3 113

4 334

8 3575

42 +O(3)

(CA T2

t )2 T2

su M(0),

M(+,4)NLL = i

(B0)4

4!

(CA T2

t )3

1

(2)4+

17552

+O(2)

+ CA(CA T2t )

2

3

8 3

164 1675

8+O(2)

T2

su M(0).

• At four loop a new color structure appear, with a single pole not predicted by the dipole formula of infrared divergences!

• The fact that it arises only at four loops is a consequence of the “top-bottom” symmetry of the ladder. The new color structure appears in the target-averaged wave function already at three loops, but it cancels out due to this symmetry.

(CA −T2t )

(CA −T2t ), CA

(CA −T2t )

Caron-Huot, 2013

• It would be possible to calculate few order higher in perturbation theory; the problem becomes rapidly quite involved.

• However, this is not necessary, if we are interested to know only the infrared singularities. Reconsider the wave function: with where

• The wave function is actually finite. All divergences must arise from the last integration!

• Divergences arises only from the limit k → p or k → 0 limit. Consider one of the two regions, and multiply the result by two.

TWO REGGEON CUT: SOFT APPROXIMATION

(`1)(p, k) = (2CA T2t )

(`1)(p, k) + (CA T2t )

(`1)(p, k),

(`1)(p, k) =

Z[Dk0] f(p, k, k0)

h(`2)(p, k0) (`2)(p, k)

i, (`1)(p, k) =

1 J(p, k)

2(`2)(p, k),

f(p, k0, k) =k02

k2(k k0)2+

(p k0)2

(p k)2(k k0)2 p2

k2(p k)2,

J(p, k) =

p2

k2

+

p2

(p k)2

1.

M(+,`)NLL = i

(B0)`

(` 1)!

Z[Dk]

p2

k2(k p)2(`1)(p, k)T2

su M(0),

finite!

• In the soft limit the integrations becomes trivial (“bubble” integrals). We obtain an all-order solution for the target-averaged wave function where

• It is immediate to get the reduced amplitude

• The result is valid up to the single poles, which allows one to achieve a tremendous simplificationwhere


(`1)s (p, k) =

(CA T2t )

`1

(2)`1

`1X

n=0

(1)n` 1

n

p2

k2

n n1Y

m=0

1 + Bm()

2CA T2t

CA T2t

,

Bn() =Bn()

B0() 1, and Bn() = eE

(1 )

(1 + n)

(1 + + n)(1 n)

(1 2 n).

M(+,`)NLL |s = i

1

(2)`B`

0()

`!(1 + B1) (CA T2

t )`1

X

n=1

(1)n+1

`

n

n2Y

m=0

1 + Bm()

2CA T2t

CA T2t

T2

su M(0) +O(0).

Caron-Huot, Gardi, Reichel, LV, preliminar

M(+,`)NLL |s = i

1

(2)`B`

0()

`!

1R()

CA

CA T2t

1

(CA T2t )

`1 T2su M(0) +O(0),

R() B0()

B1() 1 =

3(1 )(1 + )

(1 2) 1 = 23

3 344 65

5 223 + 106

6 +O(7).

• Expand for a few orders in the strong coupling constant: A new color structure appears every three loops!

• Resumming the amplitude to all loops we get



M(+,`=1,2,3)NLL |s = i

B`0()

`! (2)`(CA T2

t )`1 T2

su M(0) +O(0),

M(+,`=4,5,6)NLL |s = i

B`0()

`! (2)`

(CA T2

t )`1 +R()CA(CA T2

t )`2

T2

su M(0) +O(0),

M(+,`=7,8,9)NLL |s = i

B`0()

`! (2)`

(CA T2

t )`1 +R()CA(CA T2

t )`2

+ R2()C2A(CA T2

t )`3

T2

su M(0) +O(0),

M(+,`=10,11,12)NLL |s = i

B`0()

`! (2)`

(CA T2

t )`1 +R()CA(CA T2

t )`2

+ R2()C2A(CA T2

t )`3 +R3()C3

A(CA T2t )

`4

T2

su M(0) +O(0).

ˆM(+)NLL|s = 4↵s

i

L(CA T2t )

1R()

CA

CA T2t

1 exp

B0()

2

↵s

L(CA T2

t )

1

T2

su M(0)+O(0).

COMPARISON BETWEEN REGGE AND INFRARED FACTORIZATION

• Consider the soft anomalous dimension

• with

• Parameterise the soft anomalous dimension at NLL according to

• Within the dipole formula one has

• Recall now the infrared factorisation formula

• with

TWO REGGEON CUT: BFKL VS INFRARED FACTORISATION

pi,,↵s(

2)=

st,,↵s(

2)+

4X

i=1

i

t,,↵s(

2)+O

t

s

,

M pi, µ,↵s(µ2)= Z

pi, µ,↵s(µ2)H pi, µ,↵s(µ

2),

Zpi, µ,↵s(µ

2)

= P exp

(1

2

Z µ2

0

d2

2pi,,↵s(

2)

)

.

NLL

↵s(

2)=

1X

`=1

(`)NLL

↵s(2)

`

=1X

`=1

(`)NLL

↵s(p2)

` p2

2

`

.

LL

↵s(

2)=

K↵s(2)

2LT2

t , (1)NLL = iT2

su,

↵s(

2)= LL

↵s(

2)+ NLL

↵s(

2)+ NNLL

↵s(

2)+ . . .

• We get the infrared-factorised representation of the reduced amplitude:

• and comparing with the result from the Regge theory allows us to obtain

• Explicitly, for the first few orders we have:

• The result can be used as constraint in a bootstrap approach to the soft anomalous dimension.

TWO REGGEON CUT: BFKL VS INFRARED FACTORISATION


ˆM(+)NLL = 4↵s exp

(B0 1)

2

↵s

L(CA T2

t )

exp

1

2

↵s

LT2

t

P exp

(1

2

Z p2

0

d2

2

LL

↵s(

2)

+ NLL

↵s(

2)

)

M(0)+O(0),

(`)NLL =

i

(` 1)!

"↵s

1R

↵s

2L(CA T2

t ) CA

CA T2t

1#

↵`s

T2su.

(1)NLL = iT2

su, (2)NLL = 0, (3)

NLL = 0,

(4)NLL = iL3 3

24CA(CA T2

t )2 T2

su,

(5)NLL = iL4 4

128CA(CA T2

t )3 T2

su,

(6)NLL = iL5 5

640CA(CA T2

t )4 T2

su,

(7)NLL = i

L6

720

2316

C2A(CA T2

t )4 +

1

32

23 56

CA(CA T2

t )5

T2

su,

(8)NLL = i

L7

5040

33432

C2A(CA T2

t )5 +

3

64(34 37) CA(CA T2

t )6

T2

su.

Almelid, Duhr, Gardi, McLeod, White, 2017

CONCLUSION

• Using the non-linear Balitsky-JIMWLK rapidity evolution equation we have computed the three-Reggeon cut to three loops, at NNLL in the signature-odd sector, and the IR singular part of the two-Reggeon cut to all orders, at NLL in the signature-even sector, for 2 → 2 scattering amplitudes.

• Concerning the three-Reggeon cut, we have shown how to take systematically into account the effect of mixing between states with k and k+2 Reggeized gluons, due non-diagonal terms in the Balitsky-JIMWLK Hamiltonian, which contribute first at NNLL.

• Our results are consistent with a recent determination of the infrared structure of scattering amplitudes at three loops, as well as a computation of 2 → 2 gluon scattering in N = 4 super Yang-Mills theory. Combining the latter with our Regge-cut calculation we extract the three-loop Regge trajectory in this theory.

• The calculation of the infrared singular part of the two-Reggeon cut allows us to extract the soft anomalous dimension to all orders in perturbation theory, in this kinematical limit.

• The information obtained concerning infrared singularities has been/will be used to constrain the structure of the soft anomalous dimension in general kinematics. (See Almelid, Duhr, Gardi, McLeod, White, 2017).

leonardo vernazza - physics.iith.ac.in

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