Webs and polylogarithms Mark Harley The University of Edinburgh - - PowerPoint PPT Presentation

Webs and polylogarithms

Mark Harley The University of Edinburgh Giulio Falcioni, Einan Gardi, MH, Lorenzo Magnea, Chris White [arXiv:1407.3477] Numbers and Physics, ICMAT, 18 September 2014

Outline

• Infrared Singularities
• Webs and subtracted webs
• Computing webs
• Interesting properties of subtracted webs
• Open questions

Singularities of Gauge Theories

Loop level scattering amplitudes suffer from divergences

Ultraviolet Infrared Regularise with dimensional regularisation: d = 4 − 2✏

Z ddk 1 k2(k2 + 2p1 · k)(k2 − 2p2 · k)

is log divergent as kµ → 0 to regulate IR

✏ < 0

Infrared Singularities — Cancellation

The virtual infrared singularities cancel with those from real emissions (unlike UV) + +

− → log ✓ q2 µ2 ◆

Large logarithm

Phenomenology: necessary to compute for practical purposes Theory: Interesting insights into perturbative series and computations

Eikonal Approximation

In order to study the infrared singularities of a scattering amplitude we approximate

− →

Remove scale + spin ⇒ External fields replaced by Wilson lines Φβ(0, ∞) = P exp ✓ igs Z ∞ dλβ · A(λβµ) ◆ , β2 6= 0

Factorisation

This approximation gives the IR singularities of an amplitude through factorisation

M = S ⊗ H

S = hΦβ1Φβ2 . . . Φβni0

The hard function, , is a matching coefficient containing information from non-soft underlying amplitude

H

The universal soft function is a product of Wilson lines

Exponentiation — QED

The soft function exponentiates, drastically simplifying computation, e.g. QED form factor SQED = exp ✓ + + + . . . ◆ = 1 + + + + . . . Exponent is formed from only “connected” diagrams

Exponentiation — QCD

QCD is non-abelian (specifically SU(3) gauge theory) and therefore exponentiation is far less simple SQCD = P exp ✓ + + + + . . . ◆ Especially when considering more lines (matrix valued)

Webs

Webs are specific collections of diagrams which contribute to the exponent E.g. 1-2-1:

−

◆

1-1-1-3:

w(2,−1)

1-2-1

= 1 2f abcT a

1 T b 2T c 3

✓

Webs appear in exponent with “connected” colour factors

w(3,−1)

1-1-1-3 = −1

6f adef bceT a

1 T b 2T c 3T d 4 (2A − B − C + 2D − E − F)

−1 6f abef cdeT a

1 T b 2T c 3T d 4 (A + B − 2C + D − 2E + F)

Web combinatorics

The exponent takes the form of the diagrammatic colour and kinematic factors mixed by the “Mixing Matrix” W ≡ X

D,D0

F(D) RD,D0 C(D0) Recent studies have found interesting links between the combinatorics of these matrices and partially ordered sets (posets) Dukes, Gardi, McAslan, Scott, White [arXiv:1310.3127] Dukes, Gardi, Steingrimsson, White [arXiv:1301.6576]

Renormalising the soft function

The Eikonal approximation results in further UV divergences due to introduction of cusp ∝ Z ddk k2 β1 · k β2 · k Need to renormalise the soft function

Sren.(↵ij, ↵s(µ2), ✏IR, µ) = SUV+IR Z(↵ij, ↵s(µ2), ✏UV, µ) = Z(↵ij, ↵s(µ2), ✏UV, µ)

Can determine the UV poles by introducing IR regulator

Sren.(↵ij, ↵s(µ2), µ, m) = S(↵ij, ↵s(µ2), ✏, m) Z(↵ij, ↵s(µ2), ✏, µ)

Renormalisation and the exponent

In QCD the soft function and renormalisation factor are matrices S = exp(w(✏)) Z = exp(⇣(✏, µ)) ,

w = X

n,k

w(n,k)↵n

s ✏k

Applying BCH formula and some physical constraints,

Γ(1) = −2w(1,−1) Γ(2) = −4w(2,−1) − 2 h w(1,−1), w(1,0)i Γ(3) = . . .

,

Γ = X

n

Γ(n)αn

s

dZ d ln µ − ZΓ

Gardi, Smillie, White [arXiv:1108.1357] Mitov, Sterman, Sung (2009-2010)

Subtracted webs

A subtracted web is a web combined with a relevant set of commutators ,

• −

1 2

✓ ◆

(2, −1) (1, 0) (1, −1)

+ 

e w(2)

1-2-1 = −4

• Renormalisation factor, , can not have dependance on IR

regulator therefore neither do subtracted webs

• Owing to this physical symmetries hidden by regularisation

are restored

• Free of subdivergences (only physically relevant single

pole) Z

Multiple Gluon Exchange Webs (MGEWs)

We wish to specialise to a subclass of simple webs involving

• nly multiple gluon exchanges

Easily manifest themselves as iterated multiple- polylogarithmic integrals Lend themselves naturally to development of automated techniques

Kinematics: Two line, colour-singlet case

IR regulated one loop

β1 β2

Exponential regulator

}

γ12 = 2β1 · β2 p β2

1β2 2

w(1)

1-1 = T a

1 T a 2 ↵sµ2✏N1 · 2

Z ∞ ds Z ∞ dt (−(s1 − t2)2)✏−1e−m√

2

1s−m√

2

2t

= T a

1 T a 2  12

Z ∞ d⌧ Z ∞ d (2 + ⌧ 2 − 12⌧)✏−1 e−⌧− = T a

1 T a 2  12 Γ(2✏)

Z 1 dx P(x, 12)

P(x, γ12) = (x2 + (1 − x)2 − γ12x(1 − x))✏−1

λ = τ + σ

x = σ σ + τ

,

Recent formulation in terms of iterated integrals

Kidonakis (2009); Henn, Huber (2012)

Three loop results recently obtained

Grozin, Henn, Korchemsky, Marquard [arXiv:1409.0023] Korchemsky, Radyushkin (1987)

σ τ

Kinematics continued

Let’s choose a more convenient kinematic variable

γij = 2βi · βj p β2

1β2 2

= −αij − 1 αij

threshold lightlike straight-line

w(1,−1) = γ12 4π Z 1 dx P0(x, γ12) = − 1 4π ✓ α12 + 1 α12 ◆ Z 1 dx 1 x2 + (1 − x)2 + x(1 − x)(α12 + 1/α12) = 1 4π 1 + α2

12

1 − α2

12

Z 1 dx ✓ 1 x −

1 1−α12

− 1 x +

α12 1−α12

◆ = 1 4π 2 r(α12) ln(α12)

symmetry is realised through interplay between rational and logarithm

α → 1/α

This structure generalises to any MGEW: products of multiplying multiple-polylogs

r(α) = 1 + α2 1 − α2

r(αij)

General form of MGEW and methodology

Now for MGEW diagrams

Gardi [arXiv:1310.5268] Has a Laurent expansion in , , where is a purely transcendental function of weight multiplying, in some cases, Heaviside functions of {x_i}

✏

(n)

D ({xi}; ✏) =

Z 1  n−1 Y

k=1

dyk(1 − yk)−1+2✏y−1+2k✏

k

• ΘD[{xk, yk}]

F(n) = n Γ(2n✏) Z 1 

n

Y

k=1

dxk k P(x, k)

• (n)

D ({xi}; ✏)

(n)

D (xi; ✏) =

X

k

(n,k)

D

({xi}) ✏k φ(n,k)

D

({xi})

t1 t2

t3

t4

θ(t1 − t2)θ(t2 − t3)θ(t3 − t4)

n − 1 + k

Computing webs

We combine integrands to directly obtain subtracted web

,

• −

1 2

✓ ◆

(2, −1) (1, 0) (1, −1)

+ 

e w(2)

1-2-1 = −4

ln q(x, α) = ln ✓ x − 1 1 − α ◆ + ln ✓ x + α 1 − α ◆ ln e q(x, α) = ln ✓ x − 1 1 − α ◆ − ln ✓ x + α 1 − α ◆

e w(n) = c(n)

i

✓

n

Y

k=1

r(αk) ◆ Z 1 

n

Y

k=1

dxk ✓ 1 xk −

1 1−αk

− 1 xk +

αk 1−αk

◆ G({xi})Θ[{xi}]

General subtracted MGEW: Conjecture: Integrand factorises such that result can be written as sums of products of polylogarithms, each dependent upon a single cusp angle

Mk,l,m(α) = 1 r(α) Z 1 dx γ P0(x, γ) lnk ✓q(x, α) x2 ◆ lnl ✓ x 1 − x ◆ lnm e q(x, α)

MGEW Basis

e w(3)

(1,2,3) = . . . + c(3) 4

4 3r(α13)r2(α23)  M0,1,1(α23)M1,0,0(α13) + 1 8 ✓ M 2

1,0,0(α23) − M0,0,0(α23)M2,0,0(α23)

− 1 12M 4

0,0,0(α23) + 2M0,0,0(α23)M0,2,0(α23)

◆ M0,0,0(α13)

• + . . .

M0,0,0(α) = 2 ln(α) M1,0,0(α) = 2 Li2(α2) + 4 log(α) log

• 1 − α2

− 2 log2(α) − 2 ζ(2) M0,1,1(α) = 2 Li3(α2) − 2 log(α)  Li2(α2) + log2(α) 3 + ζ(2)

• − 2 ζ(3)

M0,2,0(α) = 2 3 log3(α) + 4 ζ(2) log(α)

M2,0,0(α) = − 4  Li3(α2) + 2Li3

• 1 − α2

− 8 log

• 1 − α2

log2(α) + 8 3 log3(α) + 8 ζ(2) log(α) + 4 ζ(3)

Basis conjecture

Holds for every MGEW we have studied . . . ?

Term from 1-2-3 (unsubtracted)

Z 1 dx1 Z 1 dx2 γ1P0(x1, γ1) γ2P0(x2, γ2) Li2(−1 − x1 x2 ) = . . . +

Computing Webs — Method 2

Integrating webs (brute force):

• After shifting to variables, propagators factorise and integrals are in plain

“dlog” form after expansion in

αij

✏

MPLs from and logs from expansion of T({xi}) (n)

D ({xi}, ✏)

P(xk, γk)

• Integrate to get higher weight MPLs (depending on multiple angles)
• Combine with combinatoric factors and commutators to get sub. web
• Look for relations between MPLs which don’t factorise

F(n,−1)

D

= N ✓

n

Y

k=1

r(αk) ◆ Z 1 

n

Y

k=1

dxk ✓ 1 xk −

1 1−αk

− 1 xk +

αk 1−αk

◆ T({xi})ΘD[{xi}]

Term from 1-2-3 (unsubtracted)

Z 1 dx1 Z 1 dx2 γ1P0(x1, γ1) γ2P0(x2, γ2) Li2(−1 − x1 x2 ) = . . . +

MGEW Basis symbols

η = α 1 − α2

Mk,l,m(α) = 1 r(α) Z 1 dx γ P0(x, γ) lnk ✓q(x, α) x2 ◆ lnl ✓ x 1 − x ◆ lnm e q(x, α)

Physical constraints: The symbol

• Subtracted web: crossing symmetry is restored.

Leaves result unchanged up to discontinuity

α → −α

• Expect a divergence at and a zero at

α → 0, −1

1

threshold lightlike straight-line threshold straight-line

Coupled with knowledge of integrands constrains symbol entries {α2, 1 − α2}

r(α) = 1 + α2 1 − α2

Symbol conjecture:

Outlook and open problems

• Still want to understand better the symbol alphabet

constraints

• Why ?
• Does the basis of functions describe all MGEWS?
• Can we prove factorisation to all orders?
• What can we say about non-MGEWs?

η = α 1 − α2