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 1 Z d d k is log divergent as k µ → 0 k 2 ( k 2 + 2 p 1 · k )( k 2 − 2 p 2 · k ) Regularise with dimensional regularisation: d = 4 − 2 ✏ to regulate IR ✏ < 0

Infrared Singularities — Cancellation The virtual infrared singularities cancel with those from real emissions (unlike UV) + + ✓ q 2 ◆ Large logarithm log − → µ 2 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 − → External fields replaced Remove scale + spin ⇒ by Wilson lines Z ∞ ✓ ◆ β 2 6 = 0 , Φ β (0 , ∞ ) = P exp d λβ · A ( λβ µ ) ig s 0

Factorisation This approximation gives the IR singularities of an amplitude through factorisation M = S ⊗ H The hard function , , is a matching coefficient containing H information from non-soft underlying amplitude S = h Φ β 1 Φ β 2 . . . Φ β n i 0 The universal soft function is a product of Wilson lines

Exponentiation — QED The soft function exponentiates, drastically simplifying computation, e.g. QED form factor ✓ ◆ S QED = 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 ✓ S QCD = 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 w (2 , − 1) 2 f abc T a 1 T b 2 T c 3 − 1-2-1 1-1-1-3: 1-1-1-3 = − 1 w (3 , − 1) 6 f ade f bce T a 1 T b 2 T c 3 T d 4 (2 A − B − C + 2 D − E − F ) − 1 6 f abe f cde T a 1 T b 2 T c 3 T d 4 ( A + B − 2 C + D − 2 E + F ) Webs appear in exponent with “connected” colour factors

Web combinatorics The exponent takes the form of the diagrammatic colour and kinematic factors mixed by the “Mixing Matrix” X F ( D ) R D,D 0 C ( D 0 ) W ≡ D,D 0 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 d d k Z ∝ k 2 β 1 · k β 2 · k Need to renormalise the soft function S ren. ( ↵ ij , ↵ s ( µ 2 ) , ✏ IR , µ ) = S UV + IR Z ( ↵ ij , ↵ s ( µ 2 ) , ✏ UV , µ ) = Z ( ↵ ij , ↵ s ( µ 2 ) , ✏ UV , µ ) Can determine the UV poles by introducing IR regulator S ren. ( ↵ ij , ↵ s ( µ 2 ) , µ, m ) = S ( ↵ ij , ↵ s ( µ 2 ) , ✏ , m ) Z ( ↵ ij , ↵ s ( µ 2 ) , ✏ , µ )

Renormalisation and the exponent In QCD the soft function and renormalisation factor are matrices S = exp( w ( ✏ )) Z = exp( ⇣ ( ✏ , µ )) , X w ( n,k ) ↵ n s ✏ k w = n,k Applying BCH formula and some physical constraints, Γ (1) = − 2 w (1 , − 1) h w (1 , − 1) , w (1 , 0) i Γ (2) = − 4 w (2 , − 1) − 2 Γ (3) = . . . dZ X Γ ( n ) α n d ln µ − Z Γ Γ = , s n Mitov, Sterman, Sung (2009-2010) Gardi, Smillie, White [arXiv:1108.1357]

Subtracted webs A subtracted web is a web combined with a relevant set of commutators (1 , − 1) (2 , − 1) (1 , 0) ◆ � ✓  1 w (2) e + 1-2-1 = − 4 − , 2 • Renormalisation factor, , can not have dependance on IR Z regulator therefore neither do subtracted webs • Owing to this physical symmetries hidden by regularisation are restored • Free of subdivergences (only physically relevant single pole)

Multiple Gluon Exchange Webs (MGEWs) We wish to specialise to a subclass of simple webs involving only 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 β 1 σ IR regulated one loop τ Exponential regulator β 2 } Z ∞ Z ∞ dt ( − ( s � 1 − t � 2 ) 2 ) ✏ − 1 e − m √ 1 s − m √ � 2 � 2 w (1) 2 ↵ s µ 2 ✏ N � 1 · � 2 1-1 = T a 1 T a 2 t ds 0 0 Z ∞ Z ∞ d � ( � 2 + ⌧ 2 − � 12 ⌧� ) ✏ − 1 e − ⌧ − � = T a 1 T a 2  � 12 d ⌧ γ 12 = 2 β 1 · β 2 0 0 Z 1 p β 2 1 β 2 2 = T a 1 T a 2  � 12 Γ (2 ✏ ) dx P ( x, � 12 ) 0 P ( x, γ 12 ) = ( x 2 + (1 − x ) 2 − γ 12 x (1 − x )) ✏ − 1 σ λ = τ + σ x = , σ + τ Korchemsky, Radyushkin (1987) Recent formulation in terms of iterated integrals Kidonakis (2009); Henn, Huber (2012) Three loop results recently obtained Grozin, Henn, Korchemsky, Marquard [arXiv:1409.0023]

Kinematics continued Let’s choose a more convenient kinematic variable γ ij = 2 β i · β j = − α ij − 1 p α ij β 2 1 β 2 2 Z 1 threshold lightlike straight-line w (1 , − 1) = γ 12 dx P 0 ( x, γ 12 ) 4 π 0 ◆ Z 1 ✓ = − 1 1 1 α 12 + dx x 2 + (1 − x ) 2 + x (1 − x )( α 12 + 1 / α 12 ) 4 π α 12 0 Z 1 1 + α 2 ✓ ◆ = 1 1 1 12 dx − 1 − α 2 1 4 π x + α 12 x − 0 12 1 − α 12 1 − α 12 r ( α ) = 1 + α 2 = 1 4 π 2 r ( α 12 ) ln( α 12 ) 1 − α 2 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 ij )

General form of MGEW and methodology Now for MGEW diagrams Z 1 n  � F ( n ) =  n Γ (2 n ✏ ) � ( n ) Y dx k � k P ( x, � k ) D ( { x i } ; ✏ ) 0 k =1 Z 1  n − 1 � � ( n ) Y dy k (1 − y k ) − 1+2 ✏ y − 1+2 k ✏ D ( { x i } ; ✏ ) = Θ D [ { x k , y k } ] k 0 k =1 t 1 t 2 t 3 t 4 θ ( t 1 − t 2 ) θ ( t 2 − t 3 ) θ ( t 3 − t 4 ) � ( n ) � ( n,k ) X ( { x i } ) ✏ k Has a Laurent expansion in , , D ( x i ; ✏ ) = ✏ D k φ ( n,k ) where is a purely transcendental function of weight ( { x i } ) n − 1 + k D multiplying, in some cases, Heaviside functions of {x_i} Gardi [arXiv:1310.5268]

Computing webs We combine integrands to directly obtain subtracted web (1 , − 1) (2 , − 1) (1 , 0) ◆ � ✓  1 w (2) e + 1-2-1 = − 4 − , 2 General subtracted MGEW: ✓ ◆ Z 1  ✓ ◆� n n Y Y 1 1 w ( n ) = c ( n ) r ( α k ) G ( { x i } ) Θ [ { x i } ] e dx k − i 1 x k + α k x k − 0 1 − α k 1 − α k k =1 k =1 Conjecture: Integrand factorises such that result can be written as sums of products of polylogarithms, each dependent upon a single cusp angle ✓ q ( x, α ) ◆ ✓ ◆ Z 1 1 x ln m e dx γ P 0 ( x, γ ) ln k ln l M k,l,m ( α ) = q ( x, α ) r ( α ) x 2 1 − x 0 ✓ ◆ ✓ ◆ ✓ 1 ◆ ✓ ◆ 1 α α ln q ( x, α ) = ln + ln x + ln e q ( x, α ) = ln − ln x + x − x − 1 − α 1 − α 1 − α 1 − α

MGEW Basis  ✓ 4 M 0 , 1 , 1 ( α 23 ) M 1 , 0 , 0 ( α 13 ) + 1 w (3) (1 , 2 , 3) = . . . + c (3) 3 r ( α 13 ) r 2 ( α 23 ) M 2 1 , 0 , 0 ( α 23 ) − M 0 , 0 , 0 ( α 23 ) M 2 , 0 , 0 ( α 23 ) e 4 8 ◆ � − 1 12 M 4 0 , 0 , 0 ( α 23 ) + 2 M 0 , 0 , 0 ( α 23 ) M 0 , 2 , 0 ( α 23 ) M 0 , 0 , 0 ( α 13) + . . . M 0 , 0 , 0 ( α ) = 2 ln( α ) − 2 log 2 ( α ) − 2 ζ (2) M 1 , 0 , 0 ( α ) = 2 Li 2 ( α 2 ) + 4 log( α ) log 1 − α 2 � � Li 2 ( α 2 ) + log 2 ( α )  � M 0 , 1 , 1 ( α ) = 2 Li 3 ( α 2 ) − 2 log( α ) + ζ (2) − 2 ζ (3) 3 M 0 , 2 , 0 ( α ) = 2 3 log 3 ( α ) + 4 ζ (2) log( α )  1 − α 2 � � log 2 ( α ) Li 3 ( α 2 ) + 2Li 3 1 − α 2 � � � M 2 , 0 , 0 ( α ) = − 4 − 8 log + 8 3 log 3 ( 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 Z 1 dx 2 γ 1 P 0 ( x 1 , γ 1 ) γ 2 P 0 ( x 2 , γ 2 ) Li 2 ( − 1 − x 1 ) = . . . + dx 1 x 2 0 0

Computing Webs — Method 2 Integrating webs (brute force): • After shifting to variables, propagators factorise and integrals are in plain α ij “dlog” form after expansion in ✏ ◆ Z 1 n n ✓  ✓ ◆� 1 1 F ( n, − 1) Y Y = N r ( α k ) T ( { x i } ) Θ D [ { x i } ] dx k − D 1 x k + α k x k − 0 1 − α k 1 − α k k =1 k =1 � ( n ) MPLs from and logs from expansion of T ( { x i } ) D ( { x i } , ✏ ) P ( x k , γ k ) • 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

Term from 1-2-3 (unsubtracted) Z 1 Z 1 dx 2 γ 1 P 0 ( x 1 , γ 1 ) γ 2 P 0 ( x 2 , γ 2 ) Li 2 ( − 1 − x 1 ) = . . . + dx 1 x 2 0 0

MGEW Basis symbols α η = 1 − α 2 ✓ q ( x, α ) ◆ ✓ ◆ Z 1 1 x ln m e dx γ P 0 ( x, γ ) ln k ln l M k,l,m ( α ) = q ( x, α ) r ( α ) x 2 1 − x 0