Advances in QCD Theory Christopher Lee Los Alamos National - - PowerPoint PPT Presentation

Advances in QCD Theory

Christopher Lee 
 Los Alamos National Laboratory
 Theoretical Division August 3, 2017

Advances in QCD Theory

Christopher Lee 
 Los Alamos National Laboratory
 Theoretical Division August 3, 2017

A handful of

Fermi and Los Alamos

Valles Caldera, near Los Alamos, May 8, 1945

Fermilab and Los Alamos

• Drell-Yan: E772/E789/E866 (NuSea)/E906 (SeaQuest), E1039

Geoff Mills, 1955-2017

My charge

“QCD theory developments 


• ver the last ~2 years”

Two years ago

Christopher Lee,
 May 27, 1922 
 — June 7, 2015

750 GeV, 3.6σ

December 2015 
 — August 2016

Historic Day: July 4

Historic Day: July 4

2012

mH = 125 GeV/c2

San Diego
 “Big Bay Boom”

Historic Day: July 4

2012

mH = 125 GeV/c2

San Diego
 “Big Bay Boom”

2017 mAML = 1.813 × 1027 GeV/c2

Andreas Maria Lee,
 Los Alamos, NM

New observables
 Jet algorithms, substrucure, grooming

QCD

EFTs


SCET, NRQCD, 
 HQET, χPT

Parton showers, Monte Carlo PDFs


quasi-PDFs, nPDFs

Lattice QCD Unitarity, Amplitudes AdS/CFT


AdS/QCD

Phases of QCD


Confinement, QGP

pQCD


NkLO, factorization, resummation

…

• Soft Collinear Effective Theory
• N-Jettiness, SCETI, N3LL resummation
• SCETII and N3LL resummed TMD distributions
• Subtractions and NNLO cross sections
• Non-Global Logarithms, fixed order and resummed
• Jet substructure and SCET+
• Outlook

Outline

Formation of Jets in QCD

p

K

π π

ρ

αs . 1

e or p e or p

soft and collinear enhancements Perturbative soft and collinear splittings happen at intermediate time

αs . 1

Hadronization at late time at low energy scale ΛQCD

αs 1

probability

• f splitting ∼

1 Eg(1 − cos θ)

Production of a new jet suppressed by

αs ⌧ 1 θ Eg

• Need to resum large perturbative logs
• Separate pert. and non-pert. physics
• Both are problems of scale

separation: a job for EFT

History of Jets in QCD

• Existence of gluons:
• Measurements of strong coupling:
• Boosted heavy particles in SM and BSM

b b H

Higgs Jets Top Jets

t b W

10 PDG, RPP (2015-16)

event shapes

Separation of scales

• Large logs in QCD arise from large ratios of physical scales defining the

measurement or degree of exclusivity of a jet cross section.

• For jet cross sections, these are precisely ratios of hard to soft scales

and ratios of collinear momentum components.

• e.g. measurement of jet mass

pc ∼ ⇣ Q, m2

J

Q , mJ ⌘

pS ∼ ⇣m2

J

Q , m2

J

Q , m2

J

Q ⌘

p2

J = (pc + ps)2 = m2 J

µH = Q µJ = mJ

µS = m2

J

Q

Hierarchy of scales Hard Jet Soft Factorize cross section into pieces depending

• n only one of these

scales at a time.

p = (¯ n · p, n · p, p⊥)

• Modern tools for high precision resummation, factorization of

perturbative and nonperturbative effects

Bauer, Fleming, Luke, Pirjol, Stewart (1999-2001)

⊗

C(Q, µ)×

hard matching coefficient decoupled collinear jet/beam functions decoupled soft function collinear to n2 collinear to n1

SCET: QCD:

Power expansion

hard scale

µH = Q jet/beam scale

µJ,B = Q√τ

soft scale µS = Qτ

N3LL Next-to-
 Leading Log 
 (NLL) Leading
 Log (LL) NNLL

ln σ(τ) ∼ αs(ln2 τ + ln τ) + α2

s(ln3 τ + ln2 τ + ln τ)

+ α3

s(ln4 τ + ln3 τ + ln2 τ + ln τ)

+ . . . . . . . . . . . .

• RG Evolution
• Resummation of large logs

Soft Collinear Effective Theory

Soft Collinear Effective Theory

• SCETI
• SCETII

Theory for jets constrained by mass Theory for jets constrained by transverse momentum


• r for exclusive collinear hadrons

soft collinear

hard Q

anti-collinear

p2 ∼ Q2λ4 p2 ∼ Q2λ2

Qλ2

p2 ∼ Q2

a = 0

Remove hard modes from theory

E + pz

E − pz

Q Qλ2

soft collinear

hard

Q

anti-collinear

p2 ∼ Q2

a = 0

E + pz

E − pz

Qη Qη2 Q Qη Qη2

p2 ∼ Q2η2

Remove hard modes

• Hard, collinear, soft all separated by virtuality
• Collinear/soft decoupling and factorization
• Dim. Reg. regulates all divergences
• Hard separated from coll. and soft by virtuality,

collinear & soft separated by rapidity

• Inherits SCETI collinear-soft decoupling
• Dim. Reg. regulates virtuality divergences but not

rapidity divergences need additional regulator

Bauer, Fleming, Luke (2000)
 Bauer, Fleming, Pirjol, Stewart (2001) Bauer, Fleming, Pirjol, Rothstein, Stewart (2002) Chiu, Jain, Neill, Rothstein (2011, 2012)

Challenges to Precision Jet Cross Sections

• Jet cross sections typically depend on
• choice of jet algorithm
• jet sizes
• jet vetoes (for exclusive jet cross sections)
• These parameters generate a number of logarithms (non-

global logs, logs of radii R, etc.) in perturbation theory which are challenging to resum

• N-Jettiness: a global observable picking out N-jet final

states by measurement of a single parameter, logs of which can be resummed in perturbation theory by standard RGE

N-jettiness

• A global event shape measuring degree to which final state is N-jet-like. 


(small N-jettiness vetoes events with more than N jets.)

p p

τN = 2 Q2 X

k

min{qA · pk, qB · pk, q1 · pk, . . . , qN · pk}

Stewart, Tackmann, Waalewijn (2010)

groups particles into regions, according to which vector qi is closest. # beams # jets qA qB q1 qN Factorization and Resummation-friendly

N3LL resummation with SCET

τ

0.24 0.16 0.20 0.30 0.28 0.26 0.22 0.18 0.0 1.0 1.4 0.4 0.2 0.6 0.8 1.2

σ dσ dτ

N LL

3

’

N LL

3

NNLL NNLL NLL

’ ’

Sum Logs, with S + gap

mod

1

Q=mZ

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.0 0.5 1.0 1.5

τ1 τ1 dσ /daτ1

Q=100 GeV x=0.1

NLL NNLL N3LL

Abbate, Fickinger, Hoang, Mateu, Stewart (2010)

e+e- Thrust
 (2-Jettiness) e+e- Hemisphere Jet Mass

Chien, Schwartz (2010)

O(αs) O(α2

s)

O(α3

s)

τ

0.24 0.16 0.20 0.30 0.28 0.26 0.22 0.18 0.0 1.0 1.4 0.4 0.2 0.6 0.8 1.2

σ dσ dτ

Fixed Order

1

Q=mZ

Compare fixed order: e+e- C Parameter

Hoang, Kolodrubetz, Mateu, Stewart (2014)

DIS ep 1-Jettiness

Kang, CL, Stewart (preliminary, 2017)

High precision strong coupling

PDG, RPP (2015) from SCET predictions for e+e- 
 event shapes

present extractions

Hoang, Kolodrubetz, Mateu, Stewart (2015)

NNLL resummation for generic observables

Banfi, LoopFest 2017

• We do not always have a factorization theorem available to make SCET

and its RG evolution to achieve resummation

• Monte Carlo implementation ARES (successor to NLL CAESAR) of

emission amplitudes needed for NNLL

Banfi, McAslan, Monni, Zanderighi (2016)

Cambridge /Durham Jet Rates

d dq2

T dy = 0H(Q2, µ)

Z d2~ qT sd2~ qT 1d2~ qT 2( ~ qT

2 − | ~

qT s + ~ qT 1 + ~ qT 2|2) × S(~ qT s, µ, ⌫)f ⊥

1 (x1 = Q

√sey, Q, ~ qT 1, µ, ⌫)f ⊥

2 (x2 = Q

√se−y, Q, ~ qT 2, µ, ⌫)

soft collinear

hard

Q

anti-collinear

p2 ∼ Q2

a = 0

E + pz

E − pz

Qη Qη2 Q Qη Qη2

p2 ∼ Q2η2 Chiu, Jain, Neill, Rothstein (2011, 2012)

High precision pT resummation at LHC

• SCETII
• Rapidity Renormalization Group

dσ dq2

T dy = σ0π(2π)2H(Q2, µ)

Z db bJ0(bqT )e S(b, µ, ν) e f ⊥

1 (b, Q, x1, µ, ν) e

f ⊥

2 (b, Q, x2, µ, ν)

e f(~ b) ≡ Z d2qT (2⇡)2 ei~

b· ~ qT f( ~

qT )

e f(~ b) ≡ 1 2⇡ e f(b) , b ≡ |~ b|

New rapidity regulator and 3-loop anomalous dimension

• Computation of beam or soft functions requires regulation of rapidity divergences:
• Regulator: shift separation of soft Wilson lines defining soft function in Euclidean time

Li, Neill, Zhu (2016)

N3LL resummed pT spectrum

Li, Neill, Schulze, Stewart, Zhu 
 (SCET2016, Argonne Advances in QCD 2016)

• 3-loop soft function diagrams:
• 3-loop rapidity anomalous dimension:
• N3LL resummed results:

resummed envelope

NNLO Subtractions

Moult (LoopFest 2017)

N-Jettiness Subtractions

• Exploit factorization and 2-loop computations of ingredients for small τN

Boughezal, Liu, Focke, Petriello (2015)
 Gaunt, Stahlhofen, Tackmann, Walsh (2015)

• High precision, numerical stability requires power corrections:

Subleading Power Corrections

• SCET well formulated to compute power corrections:

Moult, Rothen, Stewart, Tackmann, Zhu (2016)

• Also computable in fixed-order QCD, dramatic improvement in independence:

τcut

Boughezal, Liu, Petriello (2016)

NNLO Results V+jet

Boughezal, Liu, Petriello (2016)

• N-jettiness subtraction method
• vs. antenna subtraction:
• vs. data:

NNLO Revolution

• X. Liu DPF 2017

Non-global logs

• Global observable:


(thrust, N-jettiness)

• Non-global observable:


(double hemisphere mass, 
 jet vetoes)

• D. Neill SCET 2017

Non-global logs

• Start to spoil “global” resummation at 2 loops:

σ(mH/mL) = σgl(mH/mL)  1 + α2

s

(2π)2 CF CA π2 3 ln2 mH mL + · · ·

• Dasgupta, Salam (2002)

Dasgupta, Salam (2001)

Sng = exp  −CF CA π2 3 ✓1 + (at)2 1 + (bt)c ◆ t2

• Conjecture / fit to Monte Carlo resummation (large Nc):

t = 1 4πβ0 ln 1 1 − 2β0αsL

L = ln mH mL

a = 0.85CA , b = 0.86CA , c = 1.33

Fixed-order computations

• Soft functions for non-global observables in SCET, two-loop computations, 


and subleading (single) NGLs

Kelley, Schabinger, Schwartz, Zhu (2011); Hornig, CL, Stewart, Walsh, Zuberi (2011)

• 5 loops:

Schwartz, Zhu (2014)

• 12 loops!

Caron-Huot

Factorization and Resummation of NGLs

RG evolve, integrate over to obtain original non-global distribution, now resummed!

Larkoski, Moult, Neill (2015)

Resummed computations

Larkoski, Moult, Neill (2015)

Singularities and Buffers

• Buffer region and singularities in L

reproduced by resummed calculation with jets, but not fixed-order calculation with partons

• Take fixed order series and apply

conformal mapping obeying proper singularities in L and buffer region:

Larkoski, Moult, Neill (2016)

Conformal improvement 


• f fixed-order NGLs

Caron-Huot Larkoski, Moult, Neill (2016)

Jet Substructure

1-prong 2-prong 3-prong:

Energy Correlators

Moult, Necib, Thaler (2016)

3-prong (top): good discriminants: 2-prong: 1-prong (q vs g): definite power counting, amenable to factorization and precision calculation

• cf. Basham, Brown,

Ellis, Love (1978); 
 Larkoski, Salam, Thaler (2013)

Grooming and Soft Drop

contamination:

Larkoski, Marzani, Soyez, Thaler (2014)

grooming:

Soft Drop

• Simplifies theoretical calculations:

β = 0

Groomed substructure

• Top tagging:
• q vs g:

Moult, Necib, Thaler (2016)

Groomed substructure and SCET+

• Soft drop groomed energy correlators:

Frye, Larkoski, Schwartz, Yan (2016) SCET+: Bauer, Tackmann, Walsh, Zuberi (2011) free of NGLs; correlated hierarchical emissions groomed away

NNLL substructure calculations

Frye, Larkoski, Schwartz, Yan (2016)

Many other EFT directions

• Connection of NGLs and small-x evolution (BFKL)
• SCET with Glauber modes for factorization violating

effects, small-x resummation, forward scattering

• I. Rothstein and I. Stewart (2016)
• SCETG for jets in heavy-ion collisions
• A. Idilbi and A. Majumder (2008)
• G. Ovanesyan and I.

Vitev (2011)

• SCETEW for resummation of electroweak logs in colliders,

dark matter production and annihilation

Chiu, Golf, Kelley, Manohar (2007)
 Ovanesyan, Slatyer, Stewart (2014) Baumgart, Rothstein, Vaidya (2014)
 etc.

• SCET + NRQCD for improved description of quarkonia 


in jets, discriminate production mechanisms

Baumgart, Leibovich, Mehen, Rothstein (2014)
 Bain, Dai, Leibovich, Makris, Mehen (2016-17)

In the last several years, we have gained a collection of EFT and other powerful tools for high precision calculations of observables in multiscale jet-like processes, making possible fixed-order and resummed calculations to

• rders previously unachievable and the solution
• f problems previously intractable in QCD.

The future holds great promise

Extra slides

Jet Algorithms and Radii

• Example: e+e- to two jet cross section:

R

jet radius jet veto

E0

• One-loop cross section in QCD:
• in a cone algorithm:


• in a kT
• type recombination (or Sterman-Weinberg) algorithm:


σ2-jet σ0 = 1 + αsCF π ✓ −4 ln 2E0 Q ln R − 3 ln R − 1 2 + 3 ln 2 ◆ σ2-jet σ0 = 1 + αsCF π ✓ −4 ln 2E0 Q ln R − 3 ln R − π2 3 + 5 2 ◆

• Natural to use SCET to factorize and resum, but structure of logs is

surprisingly subtle.

Soft and Soft-Collinear phase space

• collinear and soft phase space for cone and kT algorithms:

Chien, Hornig, CL (2015)

p+

g

p−

g

Q

Q

p+

g = Rp− g

p+

g = p− g /R

cn c¯

n

s

p+

g

p−

g

Q 2Λ 2Λ Q s

c¯

n

cn

pg = (p−

g , p+ g , p⊥ g )

2E0 2E0

Soft and Soft-Collinear phase space

• Soft phase space splits into two, single-scale-sensitive regions:

Chien, Hornig, CL (2015)

p+

g = Rp− g

p+

g = p− g /R

p−

g

p+

g

2E0

ss

scn sc¯

n

=

−

E < E0

E < E0

Ss(E0, µ) −2Ssc(E0R, µ) Sveto(E0, R, µ)

αsCF 4π ✓ 8 ln2 µ 2E0 − π2 ◆

Ellis, Hornig, CL, Vermilion, Walsh (2010)

αsCF 4π ✓ −8 ln2 µ 2E0R + π2 3 ◆

αsCF 4π ✓ 8 ln R ln µ2 4E2

0R − 2π2

3 ◆

2E0

SCET++

µS = Qτ/R

Hard scale Jet scale Global soft (veto) scale µH = Q

µΛ = 2Λ

µsc = 2ΛR µJ = Q√τ

Csoft scale Soft-collinear
 scale

Q(1, τ, √τ)

∼ (Q, Q, Q)

Qτ ⇣ 1 R2 , 1, 1 R ⌘

(E0, E0, E0)

E0(1, R2, R)

Chien, Hornig, CL (2015)

• Integrated jet thrust in e+e-:

0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.0 0.2 0.4 0.6 0.8 1.0 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 0.0 0.2 0.4 0.6 0.8 1.0

No s-c refactorization

τ

NLL NNLL

R = 0.2, Λ = 10 GeV, Q = 100 GeV

With s-c refactorization

τ

σc(τ, Λ, R) NLL NNLL

R = 0.2, Λ = 10 GeV, Q = 100 GeV

• Improved perturbative convergence thanks to additional logs resummed

after soft-collinear refactorization

Chien, Hornig, CL (2015)

Resummed jet thrust cross section

Resummed jet thrust cross section

• pp jet angularity differential distribution:
• Larger impact on differential shape
• A. Hornig,
• Y. Makris, T. Mehen (2016)

without soft-collinear refactorization with soft-collinear refactorization

NP Corrections

• Reminder: Dokshitzer-Webber model

CL, Sterman (2006, 2007)

• bservable dependent, 


calculable coefficient universal nonperturbative parameter

conjecture from single soft gluon emission: Dokshitzer, Webber (1995, 1997)

ce hei = heiPT + ce Ω1 Q Ω1

• SCET: First rigorous proof (and field theory definition of ) 


from factorization theorem and boost invariance of soft radiation:

soft radiation sees only direction, not energy, of original collinear partons, invariant to boosts along z

Ω1

Ω1 = 1 NC Trh0|Y

† ¯ nY † nET (η)YnY ¯ n|0i

proof to all orders in soft gluon emission:

“energy flow”

• perator

(one for each of ee, ep, pp)

Momentum Flow Operators

ET (η)|Xi = X

i∈X

|pi

T |δ(η ηi)|Xi

pT η

ET(η) = 1 cosh3 η 2π dφ lim

R→∞ R2

∞ dt ˆ niT0i(t, Rˆ n) .

generic form of event shapes: e(X) = 1 Q X

i∈X

fe(ηi)|pi

T |

fτa(η) = e−|η|(1−a)

e.g. angularities

ˆ e |X⟩ ≡ e(X) |X⟩ = 1 Q ∞

−∞

dη fe(η)ET(η; ˆ t) |X⟩

• perator action in terms of

transverse momentum flow operator: construct out of energy-momentum tensor of QCD: measures total transverse momentum 
 flowing through slice of sphere at rapidity
 from collision time t=0 to detector at

|pT |

η

t → ∞ R → ∞

since Lagrangian of SCET factors into collinear and soft sectors, so does the energy-momentum tensor:

Tµν → T n

µν + T ¯ n µν + T s µν

Belitsky, Korchemsky, Sterman (2001) Bauer, Fleming, CL, Sterman (2008)

Proof of universality

∆heis = 1 Q Z ∞

−∞

dη fe(η) 1 NC Trh0|T[Y †

nY¯ n]ET (η)T[Y † ¯ nYn]|0i

Lorentz boosts by rapidity along z:

α

Λ−1

α Λα

Yn = P exp h ig Z ∞ ds n · As(ns) i

Yn

|0i |0i

ET (η) ET (η + α) ∆heis = 1 Q ⇢Z ∞

−∞

dη fe(η) ⇢ 1 NC Trh0|T[Y †

nY¯ n]ET (0)T[Y † ¯ nYn]|0i

• ce

Ω1

CL, Sterman (2006, 2007)

• In general NP part of soft function must be modeled and is observable-dependent:
• The universality of the first moment, however, can be proven exactly:

S(e, µ, Λ) = Z 1 de0SPT(e − e0, µ)FNP(e0, Λ)

cτa = 2 1 − a

for e+e- scaling is obeyed well by LEP data

cτ = 2

cC = 3π

e.g.

e+e- Thrust: Precision extraction of

τ

0.24 0.16 0.20 0.30 0.28 0.26 0.22 0.18 0.0 1.0 1.4 0.4 0.2 0.6 0.8 1.2

σ dσ dτ

N LL

3

’

N LL

3

NNLL NNLL NLL

’ ’

Sum Logs, with S + gap

mod

1

Q=mZ

Abbate, Fickinger, Hoang, Mateu, Stewart (2010)

NNNLL perturbative prediction + nonperturbative soft power correction led to most precise extraction of strong coupling from event shapes

NNNLL resummed perturbative distribution
 Becher, Schwartz (2008)

Abbate, Fickinger, Hoang, Mateu, Stewart (2010)

• 7.5% shift from NP

power corrections

αs

(2-jettiness)

O(αs) O(α2

s)

O(α3

s)

τ

0.24 0.16 0.20 0.30 0.28 0.26 0.22 0.18 0.0 1.0 1.4 0.4 0.2 0.6 0.8 1.2

σ dσ dτ

Fixed Order

1

Q=mZ

Compare fixed order:

e+e- Thrust: Precision extraction of

τ

0.24 0.16 0.20 0.30 0.28 0.26 0.22 0.18 0.0 1.0 1.4 0.4 0.2 0.6 0.8 1.2

σ dσ dτ

N LL

3

’

N LL

3

NNLL NNLL NLL

’ ’

Sum Logs, with S + gap

mod

1

Q=mZ

Abbate, Fickinger, Hoang, Mateu, Stewart (2010)

NNNLL perturbative prediction + nonperturbative soft power correction led to most precise extraction of strong coupling from event shapes

NNNLL resummed perturbative distribution
 Becher, Schwartz (2008)

Abbate, Fickinger, Hoang, Mateu, Stewart (2010)

• 7.5% shift from NP

power corrections

Generically, better perturbative calculations + rigorous treatment of nonperturbative corrections gives smaller
 αs

αs

(2-jettiness)

O(αs) O(α2

s)

O(α3

s)

τ

0.24 0.16 0.20 0.30 0.28 0.26 0.22 0.18 0.0 1.0 1.4 0.4 0.2 0.6 0.8 1.2

σ dσ dτ

Fixed Order

1

Q=mZ

Compare fixed order:

Beam Function and PDFs

B(ωk+, x, k2

⊥, µ) = θ(ω)

ω Z dy− 4π eik+y−/2hPn(P −)|¯ χn ⇣ y− n 2 ⌘ δ(xP − ¯ n · P)δ(k2

⊥ P2 ⊥)χn(0)|Pn(P −)i

f(x, µ) = θ(ω)hPn(P −)|¯ χn(0)δ(xP − ¯ n · P)χn(0)|Pn(P −)i Measure small light-cone momentum
 and transverse momentum


• f initial state radiation

k+ = t/P −

k⊥

transverse momentum dependent beam function: match onto PDF

u u d u u d

Bq(t, x, k2

⊥, µ) =

X

j

Z 1

x

dξ ξ Iij ⇣ t, x ξ , k2

⊥, µ

⌘ fj(ξ, µ) x ξ

Generalized Beam Function to 1-loop

Tells us that PDFs should be evaluated at the beam radiation scale t

Bq(t, x, k2

⊥, µ) =

X

j

Z 1

x

dξ ξ Iij ⇣ t, x ξ , k2

⊥, µ

⌘ fj(x, µ)

B(t, x, µ) = Z d2k⊥B(t, x, k2

⊥, µ)

• rdinary beam function:

now known to 2 loops; anomalous dimension known to 3 loops

Stewart, Tackmann, Waalewijn (2009)

u u d u u d

x ξ

Jain, Procura, Waalewijn (2009) Gaunt, Stahlhofen, Tackmann (2014)

Universal nonperturbative shift in 3 versions of DIS 1-jettiness:
 Surprising relation also to leading NP correction to jet mass in pp to 1 jet

POWER CORRECTIONS IN PP AND DIS

• D. Kang, CL, I. Stewart (2013)

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 10 20 30 40 50 60

ta dsêdta

Q=80 GeV x=0.2 W=0.35 GeV

NLO PT NNLL PT NNLL PT + NP

2Ω1/Q

Using factorization theorems and boost invariance properties of soft Wilson lines, can prove that:

Ωa

1 = Ωb 1 = Ωc 1

Boost

Stewart, Tackmann, Waalewijn (2014)

Experimental Coverage

Preliminary: Kang, CL, Stewart (2016)

HERA

EIC


(high)

EIC


(low)

existing HERA event shape analyses

New analyses of HERA data for 1-jettiness under way!

preliminary theoretical uncertainty (N3LL)

TMD resummation

σ(b, z1, z2; µi, νi; µ, ν) = Utot(µi, νi; µ, ν) H(Q2, µH)e S(b; µL, νL) × e f⊥(b, z1; µL, νH) e f⊥(b, z2; µL, νH)

Utot(µi, νi, µ, ν) = exp ⇢ 4KΓ(µL, µH) − 4ηΓ(µL, µH) ln Q µL − KγH(µL, µH) + h − 4 ηΓ(1/b0, µL) + γR S ⇥ αs(1/b0) ⇤i ln νH νL

• Standard scale choices:

µL = νL = 2 beγE ≡ 1 b0

µH = νH = Q

Landau pole

TMD resummation in momentum space

Our scale choices: automatic damping of b integrand using terms actually in perturbative series

ν∗

L = νL(µLb0)−1+n

n = 1 2  1 − αsβ0 2π log ✓νH νL ◆

νL ∼ µL

µH ∼ νH ∼ Q

chosen in momentum space, after b integration Analytic formula:

Ib = 2C πq2

T ∞

X

n=0

Im ⇢ c2nH2n(α, a0) + iγE β d2n+1H2n+1(β, b0)

• Hn(α, a0) = e

−A(L−iπ/2)2 1+a0A

1 √1 + a0A (−1)nn! (1 + a0A)n

bn/2c

X

m=0

1 m! 1 (n − 2m)! n [A(α2 − a0) − 1](1 + a0A)

• m

(2αz0)n2m

Kang, CL, Vaidya (2017)