A new regulator for rapidity divergence and pT resummation for - PowerPoint PPT Presentation

A new regulator for rapidity divergence and pT resummation for Higgs production at N3LL Hua Xing Zhu with Ye Li, Duff Neill, 1604.00392, 1604.01404; and work in progress LoopFest XV, Buffalo 1

Entering the data rich era for Higgs physics Many interesting talks on Higgs pT distribution in this workshop ❖ top-quark mass effects Neumann ; High energy resummation: Forte ; ❖ Fully-differential distribution: Mistlberger ; Light-quark mass effects: Penin ❖ 2

Sudakov small pT resummation Collins-Soper-Sterman, 1985 At small pT differential distribution contains large logarithms: ❖ Fourier transform ln m M 2 1 α n H s ln m +1 ( M 2 α n H b 2 ) s q 2 q 2 Z h i i ~ d 2 ~ T T q T exp b · ~ q T Z m 2 d ¯ ✓ m 2 µ 2  ◆ � H H ln σ ( b ) ∼ − ln A [ α s (¯ µ )] + B [ α s (¯ µ )] ¯ ¯ µ 2 µ 2 1 /b 2 h i ln 2 ( b 2 m 2 H ) + ln( b 2 m 2 = α s H ) h i ln 3 ( b 2 m 2 H ) + ln 2 ( b 2 m 2 α 2 H ) + ln( b 2 m 2 H ) s h i ln 4 ( b 2 m 2 H ) + ln 3 ( b 2 m 2 H ) + ln 2 ( b 2 m 2 α 3 H ) + ln( b 2 m 2 H ) s + . . . LL NLL NNLL N3LL A 1 A 2 A 3 A 4 See von Manteuffel’s talk B 1 B 2 B 3 This talk! Current states of the art 3

Factorization of pT distribution in SCET No operator definition for A and B. Going to higher order in ❖ logarithmic accuracy highly non-trivial and difficult Soft-Collinear Effective Theory can help! ❖ hard function Beam function soft function d 2 ~ 1 d � Z b ⊥ (2 ⇡ ) 2 e i ~ b ⊥ · ~ Q T [ B ⊗ B ]( ~ b ⊥ , µ, ⌫ ) · S ⊥ ( ~ Q T dY dQ 2 ∼ H ( µ ) b ⊥ , µ, ⌫ ) d 2 ~ � Cross section in SCET factorize into Wilson coefficients from ❖ integrating hard off-shell mode (hard function), matrix element of collinear fields (the beam function), and matrix element of soft Wilson line (soft function). Individual function contain UV and rapidity divergence. After ❖ regularization and renormalization: μ and ν dependence 4

Origin of rapidity divergence ⇤ ¯ S † n ( ~ S † ⇥ ⇥ ⇤ S ⊥ ( b ) = Tr h 0 | T n S n (0) b ) | 0 i T n S ¯ unregulated soft function ¯ Z n n dx a dx b D + ( x 2 ab ) ∼ x a Z ∞ Z ∞ 1 dt 1 dt 2 ∼ ( t 1 t 2 + ~ b 2 ⊥ ) 1 − ✏ x b 0 0 ¯ ¯ n n r = t 1 v = t 1 t 2 change of variable t 2 Z ∞ Z ∞ dr dv unregulated rapidity divergence ∼ ( v 2 + ~ ⊥ ) 1 − ✏ r b 2 0 0 5

There are many different proposals for rapidity regulator in the ❖ market: Ji, Ma, Yuan ’05: Tilting the Wilson line off light-cone ❖ Mantry, Petriello ’10: fully unintegrated colinear matrix element ❖ Becher, Neubert ’11; Becher, Bell, ’12: asymmetric analytic regulator ❖ Echevarria, Idilbi and Scimemi ’11, Delta regulator ❖ Collins ’11: Tilting the Wilson line off light-cone with square-root soft ❖ subtraction Chiu, Jain, Neill, Rothstein ’12: CMU Rapidity Regulator ❖ …… ❖ Our original goal was trying to use one of these regulator to ❖ compute anomalous dimension associated with three-loop rapidity divergence (which can then be related to B 3 ). We end up finding yet a new regulator for rapidity divergence which worths exploring. 6

A new regulator for rapidity divergence x + x − x ⊥ h i h i S ⊥ ( ~ S † n (0 , 0 , ~ unregulated n S n (0 , 0 , ~ S † b ⊥ ) = Tr h 0 | T 0 ⊥ ) b ⊥ ) | 0 i T n S ¯ ¯ soft function regulated soft function h i h i ⊥ ( ~ S † n ( ib 0 ⌧ / 2 , ib 0 ⌧ / 2 , ~ S reg n S n (0 , 0 , ~ S † b ⊥ ) = lim τ → 0 Tr h 0 | T 0 ⊥ ) T n S ¯ b ⊥ ) | 0 i ¯ z b 0 = 2 e − γ E t ν = 1 | ~ b ⊥ | τ O ib 0 ν 7

Properties of the new rapidity regulator Admit operator definition. Can be used to defined transverse- ❖ momentum dependent PDF non-perturbatively mass-like regulator v.s. analytic regulator ❖ 1 1 analytic regulator can not be dropped! η η ln τ 1 τ ln τ new rapidity regulator Manifestly gauge invariant for non-singular gauge at infinite ❖ z t Semi-infinite Wilson lines | ~ b ⊥ | O ib 0 ν Preserve non-Abelian exponentiation theorem for soft Wilson loops ❖ 8

Computing the three-loop soft function with the new regulator Z b 2 0 / ~ b 2 d ln S ⊥ ( ~ µ 2 b ⊥ , µ, ⌫ ) d ¯ ⊥ h i h i ↵ s ( b 0 / | ~ = ↵ s (¯ µ ) + � r b ⊥ | ) µ 2 Γ cusp d ln ⌫ 2 ¯ µ 2 γ 0 B 1 Gehrmann, Lubbert, ❖ r Yang (2012,2014) ❖ Davies, Webber, Stirling (1985) Echevarria, Scimemi, ❖ Grazzini, de Florian (2000) ❖ Vladimirov (2015) Luebbert, Oredsson, γ 1 ❖ B 2 Stahlhofen (2016) r γ 2 Previously unknown B 3 r 9

Double differential soft function To get the three-loop pT soft function, we take a detour ❖ Lifting τ as a dynamical variable, the soft function become double ❖ differential soft function Taking the τ ->0 limit afterwards to recover the pT soft function ❖ Z Z p T e − i ~ Mantry, Petriello, ’09 dE X e − ⌧ E X d 2 ~ b · ~ p T S ( b, ⌧ , µ ) = Lustermans, Waalewijn, Zeune, ’16 double differential soft function h 0 | T [ S † p T � ~ X n S n | X i � ( E X � P 0 X ) � (2) ( ~ P X,T ) h X | S † · n | 0 i n S ¯ ¯ X τ → 0 ~ b → 0 ln n (1 − z ) ln n p 2 T pT soft function Threshold soft function 1 − z p 2 T All integrals in this limit known to three loop! ❖ Anastasiou, et al, ’14 ❖ Li et al, ’14 10

Two-loop double differential soft function Two-loop result can be extracted from 1105.5171 ( Ye Li, Mantry, Petriello ) ❖ Original expression written in terms of classical polylogarithms ❖ (Li2, Li3, Li4, Nielsen’s polylogarithms). Can easily converted to HPL representation S (1) ( b, µ = 1 / τ ) =4 C F H 0 , 1 ( x ) + π 2 C F 3 one-loop squared − 4 3 π 2 H 0 , 1 ( x ) + 268 9 H 0 , 1 ( x ) + 44 h S (2) ( b, µ = 1 / τ ) = C A C F 3 H 0 , 0 , 1 ( x ) − 44 i 3 H 0 , 1 , 1 ( x ) − 8 H 0 , 0 , 0 , 1 ( x ) − 16 H 0 , 0 , 1 , 1 ( x ) − 8 H 0 , 1 , 0 , 1 ( x ) − 16 H 0 , 1 , 1 , 1 ( x ) 4 C F H 0 , 1 ( x ) + π 2 C F − 40 9 H 0 , 1 ( x ) − 8 3 H 0 , 0 , 1 ( x ) + 8 + 1 h i h i 2 + C F n f 3 H 0 , 1 , 1 ( x ) 2 3 + 67 π 2 − π 4 81 − 5 π 2 − 22 ζ (3) + 2428 h 4 ζ (3) − 328 h i i + C A C F + C F n f 9 81 54 3 9 27 x = − b 2 S (3) ( b, τ ) =? threshold constant b 2 0 τ 2 11

We will do the calculation in N=4 Supersymmetric Yang-Mills ❖ theory first. Due to the maximal supersymmetry, the ansatz will be much simpler than QCD: uniform transcendentally N=4 SYM will capture the most complicated part of QCD ❖ QCD can be reconstructed from N=4 SYM by appropriate ❖ combination of color and matter content transcendental weight N=4 SYM QCD pure gluon fermion scalar 6 = ∅ 5 3 loop ∅ 4 ∅ 3 ∅ 2 [N=4 SYM] = 1 gluon + 4 majorana fermion + 3 complex scalar 12

Ansatz for N=4 SYM The ansatz has uniform degree of transcendentality ❖ Ci are rational coefficients need to be fixed ❖ 13

Fixing the coefficients by expanding in small impact parameter The ansatz admits a simple Taylor series expansion around b=0. ❖ x = − b 2 b 2 0 τ 2 14

Fixing the coefficients by expanding in small impact parameter On the other hand, the coefficients can be obtained from direct ❖ calculation Z Z p T e − i ~ d 2 ~ dE X e − ⌧ E X b · ~ p T S ( b, ⌧ , µ ) = h 0 | T [ S † p T � ~ X n S n | X i � ( E X � P 0 X ) � (2) ( ~ P X,T ) h X | S † · n | 0 i n S ¯ ¯ X p T ) + 1 p T ) 2 + 1 p T ) 3 + 1 p T ) 4 + . . . e − i ~ p T = 1 + ( − i ~ 2!( − i ~ 3!( − i ~ 4!( − i ~ b · ~ b · ~ b · ~ b · ~ b · ~ x 2 x x = − b 2 b 2 0 τ 2 15

Obtained a system of linear equation ❖ The system is overdetermined. If there is a solution, it is unique ❖ + …… 16

The N=4 SYM solution one loop two loop ⇣ b ⊥ , ⌧ , µ = ⌧ − 1 ) = c s, N =4 ( ~ S N =4 + N 3 16 ⇣ 2 H 4 + 48 ⇣ 2 H 2 , 2 + 64 ⇣ 2 H 3 , 1 + 3 3 c 96 ⇣ 2 H 2 , 1 , 1 +120 ⇣ 4 H 2 +48 H 6 +24 H 2 , 4 +40 H 3 , 3 +72 H 4 , 2 +128 H 5 , 1 +16 H 2 , 1 , 3 + three loop 56 H 2 , 2 , 2 +80 H 2 , 3 , 1 +80 H 3 , 1 , 2 +144 H 3 , 2 , 1 +224 H 4 , 1 , 1 +64 H 2 , 1 , 1 , 2 +96 H 2 , 1 , 2 , 1 + ⌘ 160 H 2 , 2 , 1 , 1 + 256 H 3 , 1 , 1 , 1 + 192 H 2 , 1 , 1 , 1 , 1 All terms at given loop are integers with uniform sign ❖ Alternating sign between different loop order ❖ These are highly non-trivial check of the correctness of the result! ❖ 17

We are ultimately interested in QCD. Two different approaches ❖ Direct Feynman diagram calculation of the fermionic ❖ contribution (method of differential equation, many integrals, known, 12 new integrals) [N=4 SYM] = 1 gluon + 4 majorana fermion + 3 complex scalar Similar to N=4 SYM case, we made an ansatz and try to fix the ❖ coefficient by expanding in small impact parameter most complicated terms (highest weight terms) given by N=4 ❖ SYM new complication: need new terms in the ansatz ❖ 18

Full three-loop double differential soft function in QCD Cancel in N=1 SYM 19

Taking the τ → 0, rapidity divergence manifest as Log( τ ) ❖ 20

Intriguing relation between rapidity anomalous dimension and threshold anomalous dimension  � 1 Control of threshold logarithms 1 − z + constant term in threshold soft function  1 � Control of pT distribution P 2 T ∗ 21