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
1
with Ye Li, Duff Neill, 1604.00392, 1604.01404; and work in progress
❖
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
3
LL NLL NNLL N3LL
ln σ(b) ∼ − Z m2
H
1/b2
d ¯ µ2 ¯ µ2 ln ✓m2
H
¯ µ2 ◆ A[αs(¯ µ)] + B[αs(¯ µ)]
h ln2(b2m2
H) + ln(b2m2 H)
i α2
s
h ln3(b2m2
H) + ln2(b2m2 H) + ln(b2m2 H)
i α3
s
h ln4(b2m2
H) + ln3(b2m2 H) + ln2(b2m2 H) + ln(b2m2 H)
i + . . .
A1 A2 B1 A3 B2 A4 B3 Current states of the art Collins-Soper-Sterman, 1985 This talk!
❖
At small pT differential distribution contains large logarithms: αn
s
1 q2
T
lnm M 2
H
q2
T
αn
s lnm+1(M 2 Hb2)
Fourier transform Z d2~ qT exp h i~ b · ~ qT i See von Manteuffel’s talk
❖
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!
❖
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
1
d2 ~ QT dY dQ2 ∼ H(µ) Z d2~ b⊥ (2⇡)2 ei~
b⊥· ~ QT [B ⊗ B](~
b⊥, µ, ⌫) · S⊥(~ b⊥, µ, ⌫)
Beam function hard function soft function
5
¯ nSn(0)
nS¯ n(~
¯ n n ¯ n n
change of variable
ab)
⊥)1−✏
unregulated rapidity divergence
⊥)1 − ✏
unregulated soft function
❖
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 B3). We end up finding yet a new regulator for rapidity divergence which worths exploring.
6
7
S⊥(~ b⊥) = Trh0|T h S†
¯ nSn(0, 0,~
0⊥) i T h S†
nS¯ n(0, 0,~
b⊥) i |0i
unregulated soft function regulated soft function
O |~ b⊥|
ib0 ν
t z
Sreg
⊥ (~
b⊥) = lim
τ→0 Trh0|T
h S†
¯ nSn(0, 0,~
0⊥) i T h S†
nS¯ n(ib0⌧/2, ib0⌧/2,~
b⊥) i |0i
x+ x− x⊥ b0 = 2e−γE ν = 1 τ
❖
Admit operator definition. Can be used to defined transverse- momentum dependent PDF non-perturbatively
❖
mass-like regulator v.s. analytic regulator
❖
Manifestly gauge invariant for non-singular gauge at infinite
❖
Preserve non-Abelian exponentiation theorem for soft Wilson loops
8
ln τ 1 τ ln τ new rapidity regulator analytic regulator 1 η 1 η can not be dropped!
O |~ b⊥|
ib0 ν
t z
Semi-infinite Wilson lines
9
d ln S⊥(~ b⊥, µ, ⌫) d ln ⌫2 = Z b2
0/~
b2
⊥
µ2
d¯ µ2 ¯ µ2 Γcusp h ↵s(¯ µ) i + r h ↵s(b0/|~ b⊥|) i
r
r
r
Previously unknown
❖ Davies, Webber, Stirling (1985) ❖
Grazzini, de Florian (2000)
❖
Gehrmann, Lubbert, Yang (2012,2014)
❖
Echevarria, Scimemi, Vladimirov (2015)
❖
Luebbert, Oredsson, Stahlhofen (2016)
❖
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
10
S(b, ⌧, µ) = Z dEXe−⌧EX Z d2~ pT e−i~
b·~ pT
· X
X
h0|T[S†
¯ nSn|Xi(EX P 0 X)(2)(~
pT ~ PX,T )hX|S†
nS¯ n|0i
double differential soft function Mantry, Petriello, ’09 Lustermans, Waalewijn, Zeune, ’16
pT soft function
Threshold soft function All integrals in this limit known to three loop!
❖ Anastasiou, et al, ’14 ❖ Li et al, ’14
lnn(1 − z) 1 − z lnn p2
T
p2
T
11
S(1)(b, µ = 1/τ) =4CF H0,1(x) + π2CF 3 S(2)(b, µ = 1/τ) =CACF h − 4 3π2H0,1(x) + 268 9 H0,1(x) + 44 3 H0,0,1(x) − 44 3 H0,1,1(x) − 8H0,0,0,1(x) − 16H0,0,1,1(x) − 8H0,1,0,1(x) − 16H0,1,1,1(x) i + CF nf h − 40 9 H0,1(x) − 8 3H0,0,1(x) + 8 3H0,1,1(x) i + 1 2 h 4CF H0,1(x) + π2CF 3 i
2
+ h − 22ζ(3) 9 + 2428 81 + 67π2 54 − π4 3 i CACF + h4ζ(3) 9 − 328 81 − 5π2 27 i CF nf
threshold constant
❖
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
0τ 2
❖
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
12
transcendental weight QCD N=4 SYM 6 5 4 3 2 =
pure gluon fermion scalar
[N=4 SYM] = 1 gluon + 4 majorana fermion + 3 complex scalar
3 loop
❖
The ansatz has uniform degree of transcendentality
❖
Ci are rational coefficients need to be fixed
13
❖
The ansatz admits a simple Taylor series expansion around b=0.
14
0τ 2
❖
On the other hand, the coefficients can be obtained from direct calculation
15
S(b, ⌧, µ) = Z dEXe−⌧EX Z d2~ pT e−i~
b·~ pT
· X
X
h0|T[S†
¯ nSn|Xi(EX P 0 X)(2)(~
pT ~ PX,T )hX|S†
nS¯ n|0i
e−i~
b·~ pT = 1 + (−i~
b · ~ pT ) + 1 2!(−i~ b · ~ pT )2 + 1 3!(−i~ b · ~ pT )3 + 1 4!(−i~ b · ~ pT )4 + . . .
0τ 2
❖
Obtained a system of linear equation
❖
The system is overdetermined. If there is a solution, it is unique
16
+ ……
❖
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
SN =4
3
(~ b⊥, ⌧, µ = ⌧ −1) = cs,N =4
3
+ N 3
c
⇣ 16⇣2H4 + 48⇣2H2,2 + 64⇣2H3,1 + 96⇣2H2,1,1 +120⇣4H2 +48H6 +24H2,4 +40H3,3 +72H4,2 +128H5,1 +16H2,1,3 + 56H2,2,2 +80H2,3,1 +80H3,1,2 +144H3,2,1 +224H4,1,1 +64H2,1,1,2 +96H2,1,2,1 + 160H2,2,1,1 + 256H3,1,1,1 + 192H2,1,1,1,1 ⌘
two loop three loop
❖
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)
❖
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
[N=4 SYM] = 1 gluon + 4 majorana fermion + 3 complex scalar
19
Cancel in N=1 SYM
❖
Taking the τ → 0, rapidity divergence manifest as Log(τ)
20
21
Control of pT distribution
1 P 2
T
Control of threshold logarithms
1 1 − z
constant term in threshold soft function
❖
Resummation performed in b space
❖
Perturbative order of various ingredients:
❖
Two-loop hard function, beam function, soft function
❖
Three-loop normal anomalous dimension
❖
Three-loop splitting function
❖
Three-loop rapidity anomalous dimension (new)
❖
Four-loop cusp anomalous dimension (Pade approximation)
❖
Scale uncertainties estimated by varying hard scale, beam and soft μ scale, soft ν scale.
❖
Simple b* scheme for non-perturbative effects
❖
Light quark mass effects included at fixed order
22
b∗ = b p 1 + b2/b2
max
❖
NLO full: LO H+j production; NNLO full: NLO H+j production
23
σ
s
Accurate through to
24
+ + +
σ (μ)/ σ (μ = )/
μ=() + + +
σ
+
25
+ + +
σ
+
+ + +
σ μ σ μ =
/ μ μ=()/
26
+ + +
σ
+
+ + +
σ (ν)/ σ ν =
ν ν=()/
27
=- =- =-
[-*] [-*] [-*] [-*]
=-
28
+ + +
σ (μ)/ σ (μ = )/
+ + +
σ
+
❖
Introduce a new regulator for rapidity divergence in SCET description of transverse-momentum distribution.
❖
Analytic calculation of the resulting three-loop soft function through three- loops for the first time, extracting the rapidity anomalous dimension (also known as collinear anomaly d2)
❖
Lifting the rapidity regulator as an dynamical variable: double differential soft function
❖
Compute the double differential soft function (the N=4 part) by making an ansatz, and then fixing the coefficient using expansion around b=0. Two different method for the remaining QCD part.
❖
Intriguing relation between rapidity anomalous dimension and soft anomalous dimension.
❖
N3LL pT resummation for Higgs production (except for four-loop cusp)
❖
Significant reduction of uncertainties. About 10% total uncertainties in the resumed region.
29
30