TH

E

U N I V E R S

I TY

OF

ED I N B U

RG

H

Radcor-Loopfest 2015

Long-distance singularities in multi-leg scattering amplitudes

Einan Gardi

Higgs Centre for theoretical physics, Edinburgh !

19 June, 2015

new 3-loop result for the soft anomalous dimension - work with Øyvind Almelid and Claude Duhr

long-distance singularities in multi-leg scattering amplitudes

Plan of the talk!

Soft singularities: fixed-angle factorization, Wilson lines, rescaling symmetry.!

The soft anomalous dimension for massless partons: the dipole formula.!

Quadrupole interaction at 3-loop. !

Special kinematics: Regge limit, collinear limit.

The soft (Eikonal) approximation and rescaling symmetry

Tp

k < < pp + k

Eikonal Feynman rules:!

Assuming such that all components of are small:

Rescaling invariance: only the direction and the colour charge of the emitter matter. equivalent to emission from a Wilson line:

k ⌧ pk

u(p)⇣�igsT

(a)�µ⌘ i(p/+ k/+m)

(p+ k)2 �m2 + i⇥�⇥ u(p)gsT

(a) pµ

p · k + i⇥

gsT(a) pµ

p · k + i"= gsT

(a) �µ

� · k + i"

��i(⇥, 0) � P exp

⇢igs

Z 1

0d⇥� ·A(⇥�)

�

This symmetry is realised differently for lightlike and massive Wilson lines.

IR Singularities from Wilson lines

5"hard"gluon"amplitude"" 5"Wilson"line"amplitude""

p1

p2 p3

p4

p5 �1

�2�3

�4

�5

Factorization at fixed angles: all kinematic invariants are simultaneously taken large pi · pj = Q2�i · �j � �2

Soft singularities factorise to all orders & computed from a product of Wilson lines:

MJ(pi, ✏IR) =X

K

SJK (�ij , ✏IR)HK(pi)

�ij =2�i · �jq�2i �

2j

S = h��1 ⌦ ��2 ⌦ . . .��niis a product of Wilson lines:S

Due to rescaling symmetry it only depends on angles:

IR Singularities for AmplitudEs with massless legs

M✓piµ,�s, ⇥

◆= exp

(� 1

2

Z µ2

0

d⇤2

⇤2��⇤2/sij ,�s(⇤

2, ⇥)�)H

✓piµ,�s

◆

The Dipole Formula: simple ansatz for the singularity structure of multi-leg massless amplitudes

Becher & Neubert, EG & Magnea (2009)

Complete two-loop calculation by Dixon, Mert-Aybat and Sterman in 2006 !(confirming Catani’s predictions from 1998).! !Generalization to all orders motivated by constraints based on soft/jet factorisationand rescaling symmetry. !!!

Dip.

Jet 1

Jet 2Jet 3

Jet 4

Jet 5

Soft

�(�,↵s) =1

4b�K (↵s)

X

(i,j)

ln

✓�2

�sij

◆Ti · Tj +

nX

i=1

�Ji (↵s)

Exponentiation through solution of renormaliaztion-group equations:

Factorization of AmplitudEs with massless legs

MN (pi/µ, ✏) =X

L

SNL (�i · �j , ✏) HL

✓2pi · pjµ2

,(2pi · ni)2

n2iµ

2

◆

⇥nY

i=1

J i

✓(2pi · ni)2

n2iµ

2, ✏

◆�J i

✓2(�i · ni)2

n2i

, ✏

◆

Fixed angle scattering with lightlike partons!

IR singularities can be factorised - all originate in soft and collinear regions

!

Double counting of soft-collinear region is removed by dividing by eikonal jets. !

Kinematic dependence of the soft function is now on , violating rescaling symmetry. This collinear anomaly is restored by the eikonal jets. This implies an all-order constraint on the soft function, leading to the Dipole Formula.

pi · pj = Q2�i · �j � �2 p2i = 0

�i · �j

Becher & Neubert, EG & Magnea (2009)

Jets (colour singlet)

Lightlike Wilson lines

Corrections to the dipole formula

First possible corrections to the Dipole Formula: Functions of conformally-invariant cross ratios at 3-loops, 4 legs:

� = �Dip. +�(⇢ijkl)

is highly constrained by: Non-Abelian exponentiation! Bose symmetry Transcendental weight Collinear limits Regge limit

�(⇢ijkl)

!

EG & Magnea, Becher & Neubert (2009) Dixon, EG & Magnea (2010) Del Duca, Duhr, EG, Magnea & White (2011) Ahrens & Neubert &Vernazza (2012) Caron-Huot (2013)

⇢ijkl =(pi · pj)(pk · pl)(pi · pk)(pj · pl)

ifabeif cdeT a1 T

b2T

c3T

d4

The structure of the Soft Anomalous dimension: Massless vs. Massive partons

known @ 3-loop* known @ 2-loopsprogress @ 3-loop

starts @ 3-loop - in progress

known @ 3-loop forbidden to all loops 3-loop done! **

Dip. Tri.

Quad.

1

23

1

12

2 3

4

massless

massive

T1 · T2 ifabcT a1 T

b2T

c3 ifabeif cdeT a

1 Tb2T

c3T

d4

* Grozin, Henn, Korchemsky & Marquard, Phys. Rev. Lett. 114, 062006 (2015)

**Almelid, Duhr, EG - to appear

/ planar

cusp

Computing IR Singularities at 3-loops

Single connected subgraph

Two connected subgraphs

Three connected subgraphs (multiple gluon exchanges)

Classes of three-loop webs connecting four Wilson lines

4

1 2

3 4

1 2

3

4

1 2

34

1 2

3

4

1 2

3

Each web depends on all six angles - can form conformally-invariant cross ratios (cicrs)

Depends on only.Cannot form cicrs - yields products of logs for near lightlike kinematics

Depends on 3 angles only!Cannot form cicrs - yields products of logs for near lightlike kinematics

�14, �23, �24, �34

dual momentum Box integral

x

µi = �

µi si

W4g =

Box(p1, p2, p3, p4)

Parametrise the positions along the Wilson lines by

0

BB@

s1s2s3s4

1

CCA = �

0

BB@

cac(1� a)(1� c)b

(1� c)(1� b)

1

CCA

Integration over yields an overall UV pole.!Remaining integrations can be done in 4 dimensions.

� 1/✏

Ø. Almelid, C. Duhr, EG

W4g =g

6s N 4

C4g

Z “1”

0ds1ds2ds3ds4 Box(x1 � x4, x2 � x1, x3 � x2, x4 � x3)

C4g =T a1 T

b2T

c3T

d4

hfabef cde(�13�24 � �14�23) + fadef bce(�12�34 � �13�24) + facef bde(�12�34 � �14�23)

i

pi = xi � xi�1Define auxiliary momenta

z

The z integral is a 4-mass

Connected three-loop webs with two 3-gluon vertices

A similar mapping - but with a diagonal box

W(3g)2 =

⇢1234 = zz

⇢1432 = (1� z)(1� z)

may have non-trivial kinematic dependence in the limit!!! ! ! ! !!!

⇢ijkl =�ij �kl�ik �jl

=(�i · �j) (�k · �l)

(�i · �k)(�j · �l)

�2i ! 0

Ø. Almelid, C. Duhr, EG

W4g and W(3g)2

We extract the asymptotic near-lightlike behaviour using the Mellin-Barnes technique. The remaining MB integral is three-fold, and can be converted into !an iterated parameter integral and be expressed in terms of polylogarithms.

Connected webs: results and Bose symmetry

4

1 2

3

⇢ijkl =�ij �kl�ik �jl

=(�i · �j) (�k · �l)

(�i · �k)(�j · �l)

⇢1234 = zz

⇢1432 = (1� z)(1� z)

The permutation symmetry of the colour factors is mapped onto the kinematics

is symmetricunder these transformationsg1(z, z, {�ij})

w(3,�1)4g =

⇣↵s

4⇡

⌘3Ta

1Tb2T

c3T

d4

hfabef cde (zz � z � z)

+ fadef bce (1� zz) + facef bde (1� z � z)i 1

z � zg1(z, z, {�ij})

2

summing the Connected webs results

4

1 2

3

4

1 2

3 4

1 2

3 4

1 2

3

w(3,�1)(12)(34) =

⇣↵s

4⇡

⌘3Ta

1Tb2T

c3T

d4 fabef cde

hg0(z, z, {�ij})�

zz � z � z

z � zg1(z, z, {�ij})

i

w(3,�1)4g =

⇣↵s

4⇡

⌘3Ta

1Tb2T

c3T

d4

hfabef cde (zz � z � z)

+ fadef bce (1� zz) + facef bde (1� z � z)i 1

z � zg1(z, z, {�ij})

cancels in the sum!Pure function of uniform weight 5 (N=4 SYM property)Symbol alphabet relating to collinear/Regge limits{z, z, 1� z, 1� z}

From the connected webs to the full quadrupole term in the soft anon. dim.

w(3,�1)

con. =⇣↵s

4⇡

⌘3

Ta1

Tb2

Tc3

Td4

hfadef bce F

1

con.(z, z, {�ij}) + fabef cde F2

con.(z, z, {�ij})i

After applying Jacobi Identity one finds

Fncon.

(z, z, {�ij}) = Fncon.

(z, z) +Qcon.n ({log(�ij)})

and the functions separate into a polylogarithmic function of depending only on conformally invariant cross ratios via , and a function involving purely logarithmic dependence on individual cusp angles:

Rescaling symmetry implies that the quadrupole contribution to the light-like soft anomalous dimension would depend exclusively on !Indeed, we so far put aside all non-connected webs… These, in the light-like asymptotics, only involve logarithms, similarly to .These must cancel any dependence on which isn’t rescaling invariant.!!One can infer the final, rescaling-invariant answer from connected webs alone!!!

Qcon.n ({log(�ij)})

{z, z}

ln(�ij)

{z, z}

Regge limit constraints

Korchemskaya and Korchemsky (1996) Del Duca, Duhr, EG, Magnea & White (2011)

1

t�! 1

t

✓s

�t

◆↵(t)

The high-energy limit is dominated by t-channel exchange. Large Logs of (-t/s) are summed through Reggeization:

Long-distance singularities of the Regge trajectory are controlled by

Reggeization is proven through NLL, and the structure of the singularities (in the real part of the amplitude) is fully explained by the dipole formula. Therefore the quadrupole term can start contributing at i↵3

s log2(�t/s) or ↵3

s log(�t/s)

↵(t) =1

4(T2 +T3)

2

Z �t

0

d�2

�2b�K(↵s(�

2, ✏))

The quadrupole function

are the single-valued harmonic polylogarithms introduced by  Francis Brown in 2009. They are defined in the region where !

Ø. Almelid, C. Duhr, EG

�(z, z) = 16⇣↵s

4⇡

⌘3"fabefcde (F (1� 1/z)� F (1/z))

� facefbde (F (z)� F (1� z))

+ fadefbce (F (1/(1� z))� F (1� 1/(1� z)))

#

F (z) = L10101(z) + 2⇣2⇣L100(z) + L001(z)

⌘+ 6⇣4 L1(z)

L10...(z)

z = z⇤

the collinear limit

ML (p1, p2, {pj};µ)1k2�!Sp (p1, p2;µ) ML�1 (P, {pj};µ)

In particular, IR singularities of the splitting amplitude are those present in L parton scattering (with 1||2) and not in L-1 parton scattering:!!

�Sp = �L � �L�1

Surprise in the collinear limit

At 3-loops (beyond the planar limit — c.f. Kosower 1999) the splitting amplitude resolves the colour and directions of the rest of the process!!!

� (z, z)1k2�! �

⇣↵s

⇡

⌘3 X

j,k 6={1,2}

✓�3⇣4 ln

✓P

2

2x(1� x)

pj · pk(P · pj)(P · pk)

◆+ 2⇣2⇣3 + ⇣5

◆[[T1 · T2, T1 · Tj ] , T2 · Tk]

Becher & Neubert (2009) EG & Magnea & Dixon (2010)

Previous expectation: the splitting amplitude should not depend on the rest of the process, thus should vanish in any collinear limit. !

We find, however, forp1 ! xP

p2 ! (1� x)P

�(z, z)

conclusions

IR singularities of massless scattering amplitudes are now known to 3-loops.!

The first correction to the dipole formula takes the form of a quadrupole interaction simultaneously correlating colour and kinematics of 4 patrons. !

The quadrupole term is expressed in terms of single-valued harmonic polylogarithms of weight 5, depending on . These variables are simple algebraic functions of conformally-invariant cross ratios, and they manifest the symmetries and analytic structure of the quadruple interaction.!

Contrary to previous expectations, 3-loop splitting amplitudes acquire sensitivity to the colour flow and directions of the remaining partons.

{z, z}

Many thanks to my students and collaborators!

Analytic structure explored by taking the Regge limit

Euclidean Scattering momentum conserving Regge limit

analyticcontinuation

�s12 = �s34 = sei⇡

�s23 = �s14 = �t > 0

�s13 = �s14 = �u = s+ t > 0

zz =(�s12)(�s34)

(�s13)(�s24)=

s2ei2⇡

(s+ t)2

(1� z)(1� z) =(�s14)(�s23)

(�s13)(�s24)=

(�t)2

(s+ t)2

z = z = s/(s+ t) z = z ! 1

-2 -1 1 2Re(z)

-2

-1

1

2Im(z)

Regge limits in different channels correspond to discontinuities around z = 0, 1,1

ln(zz) �! ln(zz) + 2⇡i

multiple gluon exchange webs

Conjecture: (1) is made exclusively of powers of logs and Heaviside functions.!!(2) is a sum of products of polylogs each depending on a single ! each having a Symbol with alphabet !

�(n) 3 w

(n,�1) =⇣↵s

4⇡

⌘nCi1,i2,...in+1

Zdx1dx2 . . . dxn ⇥

nY

k=1

p0(xk,↵k)⇥

Gn�1

⇣x1, x2, . . . , xn; q(x1,↵1), q(x2,↵2), . . . q(xn,↵n)

⌘

Gn�1

The combinations of diagrams appearing in the exponent (subtracted webs) are much simpler than individual diagrams:

w(n,�1) ↵�↵, 1� ↵2

EG (arXiv:1310.5268) JHEP 1404 (2014) 044

A restricted basis of harmonic polylogarithmic functions was constructed, which is conjectured to be sufficient for all multiple gluon exchange webs!

Falcioni, EG, Harley, Magnea, White (arXiv:1407.3477) JHEP 1410 (2014) 10

p0(x,↵ij) =2�i · �j

(x�i � (1� x)�j)2= r(↵ij)

"1

x� 11�↵ij

� 1

x+ ↵ij

1�↵ij

#

Computing IR Singularities Using non-lightlike Wilson lines

To determine the renormalization Z of a product of Wilson lines we put an exponential IR cutoff along the Wilson lines and compute the UV singularity in dimensional regularisation,!

Define the soft anomalous dimension!

The anomalous dimension is determined by the coefficient of the single UV pole of Z - it is independent of the IR regularisation!

dZ

d lnµ= �Z�

Product of non-lightlike Wilson lines is multiplicatively renormalizable

IR - UV relation:

Arefeva, Dotsenko-Vergeles, Brandt-Neri-Sato (81)

Korchemsky-Radyushkin (87)

S = h��1 ⌦ ��2 ⌦ . . .��ni

S (�ij , ✏IR) = SUV+IR| {z }=1

Z (�ij , ✏UV)

1/✏

D = 4� 2✏, ✏ > 0

Diagrammatic SOft gluon exponentiation

Abelian  case  (1961)                                                                      Only connected diagrams contribute to the exponent !

Non-­‐abelian,  colour  singlet  (two-­‐line)  case  (1983) Only irreducible diagrams contribute to the exponent  

Non-­‐abelian,  mul<-­‐line  case  (2010)  Also reducible diagrams contribute - webs are formed by sets of diagrams.

Yennie-Frautchi-Suura

EG, Leanen, Stavenga & White Mitov, Sterman & Sung

GatheralFrenkel & Taylor

Theorem: all colour structures in the exponent correspond to connected graphs EG, Smillie & White (2013)

Webs are the diagrams contributing to , the exponent of the correctorw

S = expw = h��1 ⌦ ��2 ⌦ . . .��ni

Three-loop web: example

The entire web contributes:

=

0

BB@

F(3a)F(3b)F(3c)F(3d)

1

CCA

T

1

6

0

BB@

1 �1 �1 1�2 2 2 �2�2 2 2 �21 �1 �1 1

1

CCA

0

BB@

C(3a)C(3b)C(3c)C(3d)

1

CCA

RWeb mixing matrixKinematics Colour

(a) (b) (c) (d)

Three-loop web: example

=

0

BB@

F(3a)F(3b)F(3c)F(3d)

1

CCA

T

1

6

0

BB@

1 �1 �1 1�2 2 2 �2�2 2 2 �21 �1 �1 1

1

CCA

0

BB@

C(3a)C(3b)C(3c)C(3d)

1

CCA

(a) (b) (c) (d)

=1

6

✓F(3a)� 2F(3b)� 2F(3c) + F(3d)

◆

| {z }⇥✓C(3a)� C(3b)� C(3c) + C(3d)

◆

| {z }

subdivergences cancel

three-loop web result

(a) (b) (c) (d)

EG (arXiv:1310.5268) JHEP 1404 (2014) 044 S [U1(↵)] = �4↵⌦ ↵

1� ↵2

S [U2(↵)] = 4↵⌦ ↵

1� ↵2⌦ ↵

1� ↵2

S [⌃2(↵)] = 2↵⌦ ↵⌦ ↵

w(3)1221 =� fabef cdeT a

1 Tb2T

c3T

d4

⇣↵s

4⇡

⌘3r(↵12)r(↵23)r(↵34)

"� 8U2(↵12) ln↵23 ln↵34 � 8U2(↵34) ln↵12 ln↵23 + 16

⇣U2(↵23)� 2⌃2(↵23)

⌘ln↵12 ln↵34

� 2 ln↵12 U1(↵23)U1(↵34)� 2 ln↵34 U1(↵12)U1(↵23) + 4 ln↵23 U1(↵12)U1(↵34)

#

↵ = 0

↵ = �1

↵ = 1

special points: lightlike limit production at threshold

straight line

Very simple structure: sum of products of polylogs of individual angles!↵ij +

1

↵ij=

�2�i · �jq�2i �

2j

= ��ij