JET BROADENING IN EFFECTIVE FIELD THEORY WHEN … · particle physics seminar vienna april 2013....

JET BROADENING IN EFFECTIVE FIELD THEORY:WHEN DIMENSIONAL REGULARISATION FAILS

[ G U I D O B E L L ]

based on: T. Becher, GB, M. Neubert, Phys. Lett. B 704 (2011) 276

T. Becher, GB, Phys. Lett. B 713 (2012) 41

T. Becher, GB, JHEP 1211 (2012) 126

PARTICLE PHYSICS SEMINAR VIENNA APRIL 2013

OUTLINE

EVENT-SHAPE VARIABLES

FACTORISATION

REVIEW OF THRUST ANALYSIS

FACTORISATION BREAKDOWN FOR BROADENING

ANALYTIC REGULARISATION IN SCET

RESUMMATION

COLLINEAR ANOMALY

NNLL RESUMMATION

J E T B R O A D E N I N G I N E F F E C T I V E F I E L D T H E O R Y G U I D O B E L L

PA R T I C L E P H Y S I C S S E M I N A R – V I E N N A A P R I L 2 0 1 3

OUTLINE

FACTORISATION

RESUMMATION

COLLINEAR ANOMALY

NNLL RESUMMATION

Canonical event shape

Thrust:

I T =1Q

|~pi · ~nT |)

two-jet like: T ' 1 spherical: T ' 1/2

Thrust distribution precisely measured at LEP (τ = 1− T )MEASUREMENTS OF THRUST

Will later use ALEPH and OPAL LEP I & II results

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.450.0

dΣ dΤ

0.0 0.1 0.2 0.3 0.40

dΣ dΤ

0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.450.0

dΣ dΤ

Q = 91.2

ALEPHin the two-jet region τ ' 0

dσdτ '

αsCF2π

[− 4 ln τ+3

τ+ . . .

]⇒ Sudakov logs require resummation

Motivation

Why e+e− event shapes in 2013?

I clean environment to test understanding of QCD

I perturbation theory + resummation + non-perturbative effects

I same methods are applied at the LHC: soft gluon resummation (⇒ thrust)

pT resummation (⇒ broadening)

I precision determination of αs

I traditionally based on a fit to six event shapes (T , ρH ,BT ,BW ,C, y3)

I NLO + NLL: αs(MZ ) = 0.1202± 0.0003 (stat)± 0.0009 (exp)± 0.0013 (had)± 0.0047 (theo)

[LEP QCD working group 04]

I NNLO + NLL: αs(MZ ) = 0.1224± 0.0009 (stat)± 0.0009 (exp)± 0.0012 (had)± 0.0035 (theo)

[Dissertori et al 09]

I ⇒ further improvements require to go beyond NLL resummation!

Motivation

Why e+e− event shapes in 2013?

I clean environment to test understanding of QCD

I perturbation theory + resummation + non-perturbative effects

I same methods are applied at the LHC: soft gluon resummation (⇒ thrust)

pT resummation (⇒ broadening)

I precision determination of αs

I traditionally based on a fit to six event shapes (T , ρH ,BT ,BW ,C, y3)

I NLO + NLL: αs(MZ ) = 0.1202± 0.0003 (stat)± 0.0009 (exp)± 0.0013 (had)± 0.0047 (theo)

[LEP QCD working group 04]

I NNLO + NLL: αs(MZ ) = 0.1224± 0.0009 (stat)± 0.0009 (exp)± 0.0012 (had)± 0.0035 (theo)

[Dissertori et al 09]

I ⇒ further improvements require to go beyond NLL resummation!

Beyond NLL?

Traditional resummations are based on the coherent branching algorithm[Catani, Trentadue, Turnock, Webber 93]

I sums probabilities for independent gluon emissions

I apparently hard to extend beyond NLL

In SCET resummations are formulated in an operator language on the amplitude level

I extension to higher orders requires standard EFT techniques

I thrust analysis extended by two orders to N3LL accuracy [Becher, Schwartz 08]

I field theoretical treatment of power corrections [Abbate, Fickinger, Hoang, Mateu, Stewart 10]

I two-dimensional fit to world thrust data

I NNLO + N3LL: αs(MZ ) = 0.1135± 0.0002 (exp)± 0.0005 (had)± 0.0009 (pert)

world average: αs(MZ ) = 0.1184± 0.0007 almost 4σ below world average?

0.1150.1140.1130.112 0.116 0.117 0.118 0.119 0.120 0.121 0.122

Global Thrust Analysis

(AFHMS 10)'

Nonsinglet DIS (Blumlein 07)'

Bethke avg.( 06)'

Bethke avg.( 09)'

PDG avg.( 05)'

Electroweak(Gfitter 08)'

Lattice (HPQCD 08)'

Tau decay (Beneke et al 08)' Tau decay (Davier et al 08)'

Tau decay (Bethke avg. 09)'

Beyond NLL?

Traditional resummations are based on the coherent branching algorithm[Catani, Trentadue, Turnock, Webber 93]

I sums probabilities for independent gluon emissions

I apparently hard to extend beyond NLL

In SCET resummations are formulated in an operator language on the amplitude level

I extension to higher orders requires standard EFT techniques

I thrust analysis extended by

[Becher, Schwartz 08]

I field theoretical treatment of power corrections [Abbate, Fickinger, Hoang, Mateu, Stewart 10]

I two-dimensional fit to world thrust data

I NNLO + N3LL: αs(MZ ) = 0.1135± 0.0002 (exp)± 0.0005 (had)± 0.0009 (pert)

I world average: αs(MZ ) = 0.1184± 0.0007 almost 4σ below world average?

0.1150.1140.1130.112 0.116 0.117 0.118 0.119 0.120 0.121 0.122

Global Thrust Analysis

(AFHMS 10)'

Nonsinglet DIS (Blumlein 07)'

Bethke avg.( 06)'

Bethke avg.( 09)'

PDG avg.( 05)'

Electroweak(Gfitter 08)'

Lattice (HPQCD 08)'

Tau decay (Beneke et al 08)' Tau decay (Davier et al 08)'

Tau decay (Bethke avg. 09)'

Precision thrust analysis

from scan of theoryparameters

485= 0.91

αs(mZ)

Αs�mZ from global thrust fits

! perturbative error

O�Αs

0.1300 0.0047

"N LL3

summation

0.1194 0.0028

" multijet boundary

0.1245 0.0034

" Power Corrections

0.1152 0.0021" R#scheme

0.1140 0.0009 " b#mass & QED

0.1135 0.0009

distribution: αs(MZ ) = 0.1135± 0.0002 (exp)± 0.0005 (had)± 0.0009 (pert) [Abbate et al 10]

moment: αs(MZ ) = 0.1140± 0.0004 (exp)± 0.0013 (had)± 0.0007 (pert) [Abbate et al 12]

NNLO + NNLL: αs(MZ ) = 0.1131 +0.0028−0.0022 [Monni, Gehrmann, Luisoni 12]

Event shape studies in SCET

Heavy jet mass: [Chien, Schwartz 10, Abbate et al in progress]

I ρH =1

L ,M2R)

hemisphere jet masses M2L/R =

( ∑i∈L/R

I similar to thrust ⇒ again N3LL resummation

I non-perturbative effects more involved (hadron masses, . . . ) [Mateu, Stewart, Thaler 12]

Total and wide jet broadening: [Chiu, Jain, Neill, Rothstein 11;Becher, GB, Neubert 11]

IbT = bL + bR

bW = max (bL, bR)hemisphere jet broadenings bL/R = 1

∑i∈L/R

|~pi × ~nT |

I orthogonal to thrust (measure transverse momentum distribution)

I different type of factorisation formula ⇒ aim at NNLL resummation

Event shape studies in SCET

Heavy jet mass: [Chien, Schwartz 10, Abbate et al in progress]

I ρH =1

L ,M2R)

hemisphere jet masses M2L/R =

( ∑i∈L/R

I similar to thrust ⇒ again N3LL resummation

I non-perturbative effects more involved (hadron masses, . . . ) [Mateu, Stewart, Thaler 12]

Total and wide jet broadening: [Chiu, Jain, Neill, Rothstein 11;Becher, GB, Neubert 11]

IbT = bL + bR

bW = max (bL, bR)hemisphere jet broadenings bL/R = 1

∑i∈L/R

|~pi × ~nT |

I orthogonal to thrust (measure transverse momentum distribution)

I different type of factorisation formula ⇒ aim at NNLL resummation

OUTLINE

FACTORISATION

RESUMMATION

COLLINEAR ANOMALY

NNLL RESUMMATION

Thrust in SCET

In the two-jet limit τ → 0 the thrust distribution factorises as [Fleming, Hoang, Mantry,Stewart 07; Schwartz 07]

dσdτ

= H(Q2, µ)

∫dp2

R J(p2L , µ) J(p2

R , µ) S(τQ −

p2L + p2

multi-scale problem: Q2 � p2L ∼ p2

R ∼ τQ2 � τ2Q2

hard collinear soft

H(Q2) J(p2R)J(p2L)

S(µ2S)

Hard function:

I on-shell vector form factor of a massless quark

H(Q2) =

I known to three-loop accuracy [Baikov, Chetyrkin, Smirnov, Smirnov, Steinhauser 09;Gehrmann, Glover, Huber, Ikizlerli, Studerus 10]

I also enters Drell-Yan and DIS in the endpoint region

Jet function:

I imaginary part of quark propagator in light-cone gauge

I J(p2) ∼ Im[F.T.

⟨0∣∣∣ n/n/

4 W†(0)ψ(0) ψ(x)W (x)n/n/4

∣∣∣0⟩] W (x) = P exp

−∞ds n · A(x + sn)

I known to two-loop accuracy (anomalous dimension to three-loop) [Becher, Neubert 06]

I also enters inclusive B decays and DIS in the endpoint region

Soft function:

I matrix element of Wilson lines along the directions of energetic quarks

I S(ω) =∑

∣∣∣⟨X∣∣∣S†n (0) Sn(0)

∣∣∣0⟩∣∣∣2 δ(ω − n · pXn − n · pXn) Sn(x) = P exp

−∞ds n · As(x + sn)

I known to two-loop accuracy (anomalous dimension to three-loop)[Kelley, Schwartz, Schabinger, Zhu 11; Monni, Gehrmann,

Luisoni 11; Hornig, Lee, Stewart, Walsh Zuberi 11]

Thrust in SCET

dσdτ

= H(Q2, µ)

∫dp2

R J(p2L , µ) J(p2

R , µ) S(τQ −

p2L + p2

R ∼ τQ2 � τ2Q2

hard collinear soft

H(Q2) J(p2R)J(p2L)

S(µ2S)

Hard function:

H(Q2) =

Jet function:

I J(p2) ∼ Im[F.T.

⟨0∣∣∣ n/n/

4 W†(0)ψ(0) ψ(x)W (x)n/n/4

∣∣∣0⟩] W (x) = P exp

−∞ds n · A(x + sn)

Soft function:

I S(ω) =∑

∣∣∣⟨X∣∣∣S†n (0) Sn(0)

Thrust in SCET

dσdτ

= H(Q2, µ)

∫dp2

R J(p2L , µ) J(p2

R , µ) S(τQ −

p2L + p2

R ∼ τQ2 � τ2Q2

hard collinear soft

H(Q2) J(p2R)J(p2L)

S(µ2S)

Hard function:

H(Q2) =

Jet function:

I J(p2) ∼ Im[F.T.

⟨0∣∣∣ n/n/

4 W†(0)ψ(0) ψ(x)W (x)n/n/4

∣∣∣0⟩] W (x) = P exp

−∞ds n · A(x + sn)

Soft function:

I S(ω) =∑

∣∣∣⟨X∣∣∣S†n (0) Sn(0)

Thrust in SCET

dσdτ

= H(Q2, µ)

∫dp2

R J(p2L , µ) J(p2

R , µ) S(τQ −

p2L + p2

R ∼ τQ2 � τ2Q2

hard collinear soft

H(Q2) J(p2R)J(p2L)

S(µ2S)

Hard function:

H(Q2) =

Jet function:

I J(p2) ∼ Im[F.T.

⟨0∣∣∣ n/n/

4 W†(0)ψ(0) ψ(x)W (x)n/n/4

∣∣∣0⟩] W (x) = P exp

−∞ds n · A(x + sn)

Soft function:

I S(ω) =∑

∣∣∣⟨X∣∣∣S†n (0) Sn(0)

How does resummation work (roughly)?

Let us have a closer look at the one-loop expressions

H(Q2, µ) = 1 +

[− 2 ln2 Q2

µ2+ 6 ln

µ2− 16 +

J(p2, µ) = δ(p2) +

[(4 ln(p2/µ2)− 3

)[µ2 ]

∗+ (7− π2) δ(p2)

S(ω, µ) = δ(ω) +αsCF

[(−16 ln(ω/µ)

∗+π2

3δ(ω)

General structure:

I logarithms ⇔ divergences

I anomalous dimensions of EFT operators ⇒ resum logs via RG techniques

d lnµH(Q2

, µ) =

[2Γcusp(αs) ln

µ2+ 4γq (αs)

I finite terms ⇒ accounted for in matching calculations

Notice: there is no large log when each function is evaluated at its natural scale!

H(Q2, µ) = 1 +

[− 2 ln2 Q2

µ2+ 6 ln

µ2− 16 +

J(p2, µ) = δ(p2) +

[(4 ln(p2/µ2)− 3

)[µ2 ]

∗+ (7− π2) δ(p2)

[(−16 ln(ω/µ)

∗+π2

3δ(ω)

General structure:

d lnµH(Q2

, µ) =

[2Γcusp(αs) ln

µ2+ 4γq (αs)

H(Q2, µ) = 1 +

[− 2 ln2 Q2

µ2+ 6 ln

µ2− 16 +

J(p2, µ) = δ(p2) +

[(4 ln(p2/µ2)− 3

)[µ2 ]

∗+ (7− π2) δ(p2)

[(−16 ln(ω/µ)

∗+π2

3δ(ω)

General structure:

d lnµH(Q2

, µ) =

[2Γcusp(αs) ln

µ2+ 4γq (αs)

H(Q2, µ) = 1 +

[− 2 ln2 Q2

µ2+ 6 ln

µ2− 16 +

J(p2, µ) = δ(p2) +

[(4 ln(p2/µ2)− 3

)[µ2 ]

∗+ (7− π2) δ(p2)

[(−16 ln(ω/µ)

∗+π2

3δ(ω)

General structure:

d lnµH(Q2

, µ) =

[2Γcusp(αs) ln

µ2+ 4γq (αs)

Angularities

Interesting class of event shape variables [Berger, Kucs, Sterman 03]

I τa =1Q

Ei (sin θi )a (1− | cos θi |)1−a

I interpolates between thrust (a = 0) and broadening (a = 1)

I infrared safe for a < 2, but standard factorisation only for a < 1

SCET analysis [Hornig, Lee, Ovanesyan 09]

I relevant scales: µ2H ∼ Q2 � µ2

J ∼ Q2τ2

2−aa � µ2

S ∼ Q2τ2a

I thrust: µ2H ∼ Q2 � µ2

J ∼ Q2τ � µ2S ∼ Q2τ2 (SCETI)

I broadening: µ2H ∼ Q2 � µ2

J ∼ Q2B2 ∼ µ2S ∼ Q2B2 (SCETII)

⇒ factorisation formula for broadening will be different (and more complicated)

Jet broadening

In the two-jet limit bL ∼ bR → 0 expect that the broadening distribution factorises as

dbL dbR= H(Q2, µ)

∫dbs

∫dd−2p⊥L

∫dd−2p⊥R

JL(bL − bsL, p⊥L , µ) JR(bR − bs

R , p⊥R , µ) S(bs

L, bsR ,−p⊥L ,−p⊥R , µ)

two-scale problem: Q2 � b2L ∼ b2

I relevant modes have p⊥coll ∼ p⊥soft ∼ bL,R

I jet recoils against soft radiation

Hard function:

I precisely the same object as for thrust

I recall the RG equation

d lnµH(Q2, µ) =

[2Γcusp(αs) ln

µ2+ 4γq(αs)

]H(Q2, µ)

I ⇒ there is a hidden Q-dependence in the second line!

I thrust µ2JµS

= τQ2

τQ = Q ⇔ broadening µ2JµS

Some manipulations:

I Laplace transform bL,R → τL,R

I Fourier transform p⊥L,R → x⊥L,R

I define dimensionless variable zL,R =2|x⊥L,R |τL,R

⇒ the naive factorisation theorem takes the form

dτL dτR= H(Q2, µ)

∫ ∞0

dzR J L(τL, zL, µ) J R(τR , zR , µ) S(τL, τR , zL, zR , µ)

Jet broadening

dbL dbR= H(Q2, µ)

∫dbs

∫dd−2p⊥L

∫dd−2p⊥R

L, bsR ,−p⊥L ,−p⊥R , µ)

Hard function:

d lnµH(Q2, µ) =

[2Γcusp(αs) ln

µ2+ 4γq(αs)

]H(Q2, µ)

I thrust µ2JµS

= τQ2

Some manipulations:

∫ ∞0

Jet broadening

dbL dbR= H(Q2, µ)

∫dbs

∫dd−2p⊥L

∫dd−2p⊥R

L, bsR ,−p⊥L ,−p⊥R , µ)

Hard function:

d lnµH(Q2, µ) =

[2Γcusp(αs) ln

µ2+ 4γq(αs)

]H(Q2, µ)

I thrust µ2JµS

= τQ2

Some manipulations:

∫ ∞0

Jet function

The quark jet function for broadening reads

I J (b, p⊥) ∼∑

δ(n · pX − Q) δd−2(p⊥X − p⊥) δ(

b − 12

∑i∈X

|p⊥i |) ∣∣∣⟨X

∣∣∣ψ(0)W (0)n/n/4

∣∣∣0⟩∣∣∣2I delta-functions ensure that jet has given energy, p⊥ and b

I tree level: J (b, p⊥) = δ(

b − 12 |p⊥|)⇒ J (τ, z) = z(

1+z2)3/2 +O(ε)

At one-loop the calculation involves

Ix x x x x x x x

I Wilson-line diagrams are not well-defined in dimensional regularisation!

I∫ Q

dk−k−

diverges in the soft limit (DR regularises dd−2k⊥)

I this does not happen for thrust or any SCETI problem

Jet function

The quark jet function for broadening reads

I J (b, p⊥) ∼∑

δ(n · pX − Q) δd−2(p⊥X − p⊥) δ(

b − 12

∑i∈X

|p⊥i |) ∣∣∣⟨X

∣∣∣ψ(0)W (0)n/n/4

∣∣∣0⟩∣∣∣2I delta-functions ensure that jet has given energy, p⊥ and b

I tree level: J (b, p⊥) = δ(

b − 12 |p⊥|)⇒ J (τ, z) = z(

1+z2)3/2 +O(ε)

Ix x x x x x x x

I Wilson-line diagrams are not well-defined in dimensional regularisation!

I∫ Q

dk−k−

diverges in the soft limit (DR regularises dd−2k⊥)

I this does not happen for thrust or any SCETI problem

Momentum modes

Thrust (SCETI) Broadening (SCETII)

collinear

soft anti-collinear

collinear

anti-collinear

Qbb2/Q

I thrust: p2s � p2

I broadening: p2s ∼ p2

I ⇒ cannot distinguish soft mode from collinear mode when radiated into jet direction

I ⇒ need additional regulator that distinguishes modes by their rapidities

Regularisation in SCETII

The regularisation of individual diagrams is largely arbitrary, one could use e.g.

p2 + iε→

1p2 −∆ + iε

,(ν2)α(

p2 + iε)1+α

, . . .

I trivial for QCD, but regularises ill-defined EFT diagrams

I spoils gauge-invariance and eikonal structure of Wilson line emissions

In a massless theory it is sufficient to regularise phase space integrals [Becher, GB 11]

dd k δ(k2) θ(k0) ⇒∫

dd k(ν+

)αδ(k2) θ(k0)

I does not modify SCET at all ⇒ keeps gauge-invariance and eikonal structure

I analytic, minimal and adopted to the problem (LC propagators)

Regularisation in SCETII

The regularisation of individual diagrams is largely arbitrary, one could use e.g.

p2 + iε→

1p2 −∆ + iε

,(ν2)α(

p2 + iε)1+α

, . . .

I trivial for QCD, but regularises ill-defined EFT diagrams

I spoils gauge-invariance and eikonal structure of Wilson line emissions

In a massless theory it is sufficient to regularise phase space integrals [Becher, GB 11]

dd k δ(k2) θ(k0) ⇒∫

dd k(ν+

)αδ(k2) θ(k0)

I does not modify SCET at all ⇒ keeps gauge-invariance and eikonal structure

I analytic, minimal and adopted to the problem (LC propagators)

Why does it work?

Our new prescription amounts to

dd k δ(k2) θ(k0) ⇒∫

dd k(ν+

)αδ(k2) θ(k0)

I virtual corrections do not need regularisation

I matrix elements of Wilson lines in QCD ⇒ the same for thrust and broadening

I technical reason:∫

dd−2k⊥ f (k⊥, k+) ∼ k−ε+

I required for observables sensitive to transverse momenta

I f (k⊥, k+) ∼ δd−2(k⊥ − p⊥) ⇒ factor k−ε+ absent ⇒ reinstalled as k−α+

I can show that the prescription regularises all LC singularities in SCET [Becher, GB 11]

I not sufficient for cases where virtual corrections are ill-defined

I examples: electroweak Sudakov corrections, Regge limits

Why does it work?

Our new prescription amounts to

dd k δ(k2) θ(k0) ⇒∫

dd k(ν+

)αδ(k2) θ(k0)

I virtual corrections do not need regularisation

I matrix elements of Wilson lines in QCD ⇒ the same for thrust and broadening

I technical reason:∫

dd−2k⊥ f (k⊥, k+) ∼ k−ε+

I required for observables sensitive to transverse momenta

I f (k⊥, k+) ∼ δd−2(k⊥ − p⊥) ⇒ factor k−ε+ absent ⇒ reinstalled as k−α+

I can show that the prescription regularises all LC singularities in SCET [Becher, GB 11]

I not sufficient for cases where virtual corrections are ill-defined

I examples: electroweak Sudakov corrections, Regge limits

Jet function revisited

With the additional regulator in place, the jet functions can be evaluated

I JL(b, p⊥ = 0) = δ(b) +CFαs

eεγE

Γ(1− ε)

1− ε +4 Γ(2 + α) Γ(α)

Γ(2 + 2α)

I JR (b, p⊥ = 0) = δ(b) +CFαs

eεγE

Γ(1− ε)

1− ε +4 Γ(−α)

Γ(2− α)

I ordered limit α→ 0, ε→ 0 generates a pole in the analytic regulator

I note the characteristic scaling(ν+k+

)αin each region

For p⊥ 6= 0 the computation is considerably more involved (→ later)

I J L(τ, z) = J (0)L (τ, z)

ε+ ln

2) + 2 ln

√1 + z2 + 1

)(Qν+τ

+ . . .

I divergent term has non-trivial z-dependence

Soft function

The soft function for broadening readsI

S(bL, bR , p⊥L , p⊥R ) ∼∑

δd−2(p⊥XL

− p⊥L ) δd−2(p⊥XR− p⊥R )

bL − 12

∑i∈XL

|p⊥L,i |)δ(

bR − 12

∑j∈XR

|p⊥R,j |) ∣∣∣⟨XLXR

∣∣∣S†n (0) Sn(0)∣∣∣0⟩∣∣∣2

I split final state into left and right-moving particles

I tree level: S(bL, bR , p⊥L , p⊥R ) = δ(bL) δ(bR ) δd−2(p⊥L ) δd−2(p⊥R ) ⇒ S(τL, τR , zL, zR ) = 1

⇒ S(τL, τR , zL, zR ) = 1 +CFαs

ε+ ln

+ 2 ln

√1 + z2

)(ν+τL

)α− (L↔ R) + . . .

Anomalous Q dependence

Let us now put the jet and soft functions together

J L(τL, zL) J R (τR , zR ) S(τL, τR , zL, zR ) = J (0)L (τL, zL) J (0)

R (τR , zR )

α− ln

(Qν+τ

α+ ln

(ν+τL

ε+ ln

+ 2 ln

√1 + z2

α+ ln

α− ln

(ν+τR

ε+ ln

+ 2 ln

√1 + z2

)+ . . .

I well-defined without additional regulators

I similarly the artificial scale ν+ drops out

I the hidden Q dependence shows up!

⇒ the naive factorisation formula does not achieve a proper scale separation

How can we resum a logarithm that appears in a matching calculation?

R (τR , zR )

[(− ln

(Qν+τ

+ ln(ν+τL

ε+ ln

+ 2 ln

√1 + z2

)− ln

(ν+τR

ε+ ln

+ 2 ln

√1 + z2

)+ . . .

R (τR , zR )

[(− ln

) )( 1

ε+ ln

+ 2 ln

√1 + z2

(− ln

) )( 1

ε+ ln

+ 2 ln

√1 + z2

)+ . . .

OUTLINE

FACTORISATION

RESUMMATION

COLLINEAR ANOMALY

NNLL RESUMMATION

Collinear anomaly

Can show that the Q dependence exponentiates using and extending arguments from

I electroweak Sudakov resummation [Chiu, Golf, Kelley, Manohar 07]

I pT resummation in Drell-Yan production [Becher, Neubert 10]

Start from the logarithm of the product of jet and soft functions

I ln P = lnJ L

(ln(Qν+τ

; τL, zL

)+ lnJ R

(ln( ν+

; τR , zR

)+ lnS

(ln(ν+τL

); τL, τR , zL, zR

)/ | \

collinear: k+ ∼ b2Q anticollinear: k+ ∼ Q soft: k+ ∼ b

I use that product does not depend on ν+ and that it is LR symmetric

I ⇒ ln P =k2(µ)

4 ln2(Q2 τLτR)− FB(τL, zL, µ) ln

(Q2τ2

L)− FB(τR , zR , µ) ln

(Q2τ2

+ ln W (τL, τR , zL, zR , µ)

I RG invariance implies k2(µ) = 0 to all orders

I ⇒ P(Q2, τL, τR , zL, zR , µ) =

2L)−FB (τL,zL,µ) (Q2

τ2R)−FB (τR ,zR ,µ) W (τL, τR , zL, zR , µ)

Collinear anomaly

Can show that the Q dependence exponentiates using and extending arguments from

I electroweak Sudakov resummation [Chiu, Golf, Kelley, Manohar 07]

I pT resummation in Drell-Yan production [Becher, Neubert 10]

Start from the logarithm of the product of jet and soft functions

I ln P = lnJ L

(ln(Qν+τ

; τL, zL

)+ lnJ R

(ln( ν+

; τR , zR

)+ lnS

(ln(ν+τL

); τL, τR , zL, zR

)/ | \

collinear: k+ ∼ b2Q anticollinear: k+ ∼ Q soft: k+ ∼ b

I use that product does not depend on ν+ and that it is LR symmetric

I ⇒ ln P =k2(µ)

4 ln2(Q2 τLτR)− FB(τL, zL, µ) ln

(Q2τ2

L)− FB(τR , zR , µ) ln

(Q2τ2

+ ln W (τL, τR , zL, zR , µ)

I RG invariance implies k2(µ) = 0 to all orders

I ⇒ P(Q2, τL, τR , zL, zR , µ) =

Final factorisation formula

The corrected all-order generalisation of the naive factorisation formula becomes[Becher, GB, Neubert 11]

dτL dτR= H(Q2

∫ ∞0

dzR(Q2τ

To NLL the Mellin inversion can be performed analytically

dbT= H(Q2

, µ)e−2γEη

Γ(2η)

)2ηI2(η)

dbW= H(Q2

, µ)2η e−2γEη

Γ2(1 + η)

)2ηI2(η)

η =CFαs(µ)

µ2= O(1)

The non-trivial z-dependence of the anomaly coefficient is encoded in

I(η) =

∫ ∞0

1 + z2)3/2

(√1 + z2 + 1

)−η=

1 + η2F1(η, 1 + η, 2 + η,−1)

Final factorisation formula

The corrected all-order generalisation of the naive factorisation formula becomes[Becher, GB, Neubert 11]

dτL dτR= H(Q2

∫ ∞0

dzR(Q2τ

To NLL the Mellin inversion can be performed analytically

dbT= H(Q2

, µ)e−2γEη

Γ(2η)

)2ηI2(η)

dbW= H(Q2

, µ)2η e−2γEη

Γ2(1 + η)

)2ηI2(η)

η =CFαs(µ)

µ2= O(1)

The non-trivial z-dependence of the anomaly coefficient is encoded in

I(η) =

∫ ∞0

1 + z2)3/2

(√1 + z2 + 1

)−η=

1 + η2F1(η, 1 + η, 2 + η,−1)

Comparison with literature

Traditional resummation

I pioneering work missed quark recoil effects ⇒ valid to LL [Catani, Turnock, Webber 92]

I first NLL resummation by Dokshitzer et al [Dokshitzer, Lucenti, Marchesini, Salam 98]

I we find complete analytical agreement with this work

Resummation using SCET [Chiu, Jain, Neill, Rothstein 11,12]

I start from same naive factorisation formula

I modify Wilson-line propagators to regularise rapidity divergences

I treat additional divergences in a ”rapidity renormalization group”

I 2011 paper missed quark recoil effects ⇒ valid only to LL

I 2012 paper in agreement with Dokshitzer et al

Beyond NLL

The extension to NNLL requires three ingredients

I one-loop soft function

I one-loop jet function

I two-loop anomaly coefficient

The calculation of the one-loop soft function is straight-forward [Becher, GB, Neubert 11]

⇒ S(τL, τR , zL, zR ) = 1 +αsCF

2L)ε (

ν+τL)α [ 4

ε+ 2 ln

( 1 +√

1 + z2L

+ 8 Li2

√1 + z2

L − 1√1 + z2

)+ 4 ln2

( 1 +√

1 + z2L

]− (L↔ R)

One-loop jet function

The calculation of the one-loop jet function is surprisingly complicated

∼∫

dd q δ(q2) θ(q0)

∫dd k

)αδ(k2) θ(k0)

nq (nk + nq)

nk (q + k)2

× δ(

Q − nq − nk)δd−2

(p⊥ − q⊥ − k⊥

b − 12 |q⊥| −

12 |k⊥|

∼∫ 1

0dη η (1− η)−1+α

∫ 1+y

1−ydξ

ξ(2− ξ)1−2α(ξ(2− ξ)− 1 + y2)− 1

2−ε

(ξ − 2yη)2 + 4η(1− y)(1 + y − ξ)

I non-trivial angle complicates calculation

I expansion in α and ε is subtle

I ⇒ have to keep (2b − p)−1−ε, (2b − p)−1−2ε, . . . to all orders

I computed the integrals in closed form without expanding in ε

I ⇒ hypergeometric functions of half-integer parameters

I performed Laplace + Fourier transformations analytically

Two-loop anomaly coefficient

Most easily extracted from the two-loop soft function

I again two particles in final state

I but requires to go one order higher in ε-expansion

I encounter Nielsen polylogs and elliptic integrals

dB2 (z) = CA

1 + z2

9h1(z) +

67 + 2z2

9h2(z)− 8 h3(z) + 32 S1,2

z−z+

)− 8 Li3

z−z+

)+ 8 S1,2(−w)− 24 Li3(−w)− 24 S1,2(1− w) + 8 Li3(1− w) + 24 S1,2

( 1− w

)+ a few more lines

+ TF nf

{ 2(1 + z2)

9h1(z)−

2(13 + 2z2)

9h2(z)−

3ln2 z+ −

9ln z+ +

9z2 −

+4w(5− z2)

9ln( 1 + w

2w(11 + 2z2)

I with w =√

1 + z2 and z± = (w ± 1)/4

A glimpse at the data [Becher, GB 12]

Total broadening Wide broadening

0.0 0.1 0.2 0.3 0.40

0.00 0.05 0.10 0.15 0.20 0.25 0.300

I theory uncertainty significantly reduced in fit region for αs extraction

I without matching to fixed-order calculation

I without estimate of non-perturbative corrections

Compare with fixed order

Confront with output of fixed-order MC generators (EVENT2, EERAD3)

-10 -8 -6 -4 -2 0-40

-7 -6 -5 -4 -3 -2 -1 0-8-6-4-2

Nc n f

-7 -6 -5 -4 -3 -2 -1 0

n f �Nc

-7 -6 -5 -4 -3 -2 -1 0

1�Nc2

-7 -6 -5 -4 -3 -2 -1 0-0.1

-7 -6 -5 -4 -3 -2 -1 0-0.3

10-4DCHBTL�CF

lnHBTL

NLO NNLO−NNLL

NLO−NNLL

⇒ we obtain the right logarithmic terms for small values of L = ln BT

Conclusions

Resummation beyond standard RG techniques via collinear anomaly

I we proposed an analytic phase space regularisation for SCETII problems

dd k δ(k2) θ(k0) ⇒∫

dd k(ν+

)αδ(k2) θ(k0)

I respects symmetries of EFT, well-suited for efficient calculations

We determined all ingredients to perform NNLL resummation for jet broadening

I allows for precision determinations of αs from bT and bW distributions

The formalism is relevant for many interesting LHC observables

I Higgs production, t t , jet vetoes, jet substructure, . . .

Transverse momentum-dependent PDFs

Central ingredient for pT resummation at hardon colliders

I Bq/N (z, xT ;µ) =1

∫dt e−izt n·p ∑∫

n/αβ2〈N(p)| χα(t n+x⊥) |X〉〈X |χβ (0) |N(p)〉

⇒ ill-defined in DimReg because of unregularized rapidity divergences

Many attempts to find an optimal definition, e.g. [Collins 11]

I Bq/N (z, xT ; ζA;µ) = limy1,2→±∞

Bunsubq/N (z, xT ; yp − y2)

√√√√ S(xT , y1, yn)

S(xT , y1, y2) S(0)(xT , yn, y2)

”This definition seems unexpectedly complicated”

We propose a minimal modification of the naive definition [Becher, GB 11]

n/αβ2〈N(p)| χα(t n + x⊥) |X〉〈X |χβ (0) |N(p)〉

⇒ the only definition that has shown to work at two-loop order [Gehrmann, Lubbert, Yang 12]

√√√√ S(xT , y1, yn)

S(xT , y1, y2) S(0)(xT , yn, y2)

√√√√ S(xT , y1, yn)

S(xT , y1, y2) S(0)(xT , yn, y2)

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