Towards alternative approach for top pair production at NNLO

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Towards alternative approach for top pair production at NNLO

Sebastian Sapeta

IFJ PAN Kraków

In collaboration with Michał Czakon and Ren´e ´Angeles-Martinez

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Top pair production: the status of QCD calculations

I A single complete NNLO result for total and differential cross section obtained with STRIP- PER methodology [Czakon, Fie- dler, Mitov ‘13; Czakon, Heymes, Mi- tov ‘16]

I Flavour off-diagonal channels at NNLO fromq_Tsubtraction[Bon- ciani, Catani, Grazzini, Sargsyan, Tor- re ‘15]

I Approximate NNLO[Broggio, Pa- panastasiou, Signer ‘14] and N³LO

[Kidonakis ‘14]

I Soft and small-mass resummation at NNLL[Pecjak, Scott, Wang, Yang ‘16]

I Small-qT resummation at NNLL

[Li, Li, Shao, Yang, Zhu ‘13; Catani, Grazzini, Torre ‘14]

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The q

T

slicing method

[Catani, Grazzini ‘07, ‘15]

p+p→F(qT) +X

σ_N^FmLO= Z q_T,cut

0

dq_T dσ^F_NmLO

dqT

+ Z ∞

q_T,cut

dq_T dσ^F_NmLO

dqT

= Z q_T,cut

0

dqT

dσ^F_NmLO

dq_T + Z ∞

qT,cut

dqT

dσ^F^+jet

N^m−1LO

dq_T

enough to know in

small-q_T approximation known

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The q

T

slicing method

p+p→F(qT) +X

0

dq_T dσ^F_NmLO

dqT

+ Z ∞

q_T,cut

dq_T dσ^F_NmLO

dqT

= Z q_T,cut

0

dqT

dσ^F_NmLO

dq_T + Z ∞

qT,cut

dqT

dσ^F^+jet

N^m−1LO

dq_T

enough to know in

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The q

T

slicing method

p+p→F(qT) +X

0

dq_T dσ^F_NmLO

dqT

+ Z ∞

q_T,cut

dq_T dσ^F_NmLO

dqT

= Z q_T,cut

0

dqT

dσ^F_NmLO

dq_T + Z ∞

qT,cut

dqT

dσ^F^+jet

N^m−1LO

dq_T

enough to know in

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Soft Collinear Effective Theory (SCET)

SCET'QCD _{IR limit}

QCD fields written as sums of collinear, anti-collinear and soft components:

φ(x) =φc(x) +φ¯c(x) +φs(x)

The new fields decouple in the Lagrangian

LSCET=Lc+L¯c+Ls

The above separation facilitates proofs of factorization theorems.

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Soft Collinear Effective Theory (SCET)

LSCET=Lc+L¯c+Ls

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Soft Collinear Effective Theory (SCET)

L_SCET=L_c+L_¯_c+L_s

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Small-q

T

factorization in SCET

whereF=H,V,VV,t¯t

dσ =B1⊗ B2⊗ H ⊗ S+O q_T²

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Small-q

T

Gluons’ momenta in light-cone coordinates

k_i^µ= k_i⁺,k_i⁻,k^⊥_i

Regions

collinear k_i^µ∼(1, λ², λ)Q² B₁ anti-collinear k_i^µ∼(λ²,1, λ)Q² B2

hard k_i^µ∼(1,1,1)Q² H soft k_i^µ∼(λ, λ, λ)Q² S

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Small-q

T

Gluons’ momenta in light-cone coordinates

k_i^µ= k_i⁺,k_i⁻,k^⊥_i

Regions

collinear k_i^µ∼(1, λ², λ)Q² B₁ anti-collinear k_i^µ∼(λ²,1, λ)Q² B2

hard k_i^µ∼(1,1,1)Q² H soft k^µ∼(λ, λ, λ)Q² S

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Top pair production at small-q

T

through NNLO

dσ^NNLO

dq_Tdy dM dcosθ =X

i,¯i

B_i/N₁⊗ B¯i/N₂⊗Tr[H_i¯i⊗ S_i¯i]

where

qT,y,M : transverse momentum, rapidity, mass of top quark pair θ : scattering angle of the top quark int¯t rest frame

B - known up to NNLO[Gehrmann, L¨ubbert, Yang ’12, ’14]

H - known up to NNLO[Czakon ‘08; Baernreuther, Czakon, Fiedler ‘13]

S - known up to NLO in small-q_T limit[Li, Li, Shao, Yan, Zhu ‘13; Catani, Grazzini, Torre ‘14](up to NNLO in the threshold limit with massless tops[Ferroglia, Pecjak, Yang ‘12])

Calculating the missing NNLO correction to the soft function in the small-qT limit,S, is the aim of our work.

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Top pair production at small-q

T

through NNLO

dσ^NNLO

dq_Tdy dM dcosθ =X

i,¯i

where

S - known up to NLO in small-q_T limit[Li, Li, Shao, Yan, Zhu ‘13;

Catani, Grazzini, Torre ‘14](up to NNLO in the threshold limit with massless tops[Ferroglia, Pecjak, Yang ‘12])

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Top pair production at small-q

T

through NNLO

dσ^NNLO

dq_Tdy dM dcosθ =X

i,¯i

where

S - known up to NLO in small-q_T limit[Li, Li, Shao, Yan, Zhu ‘13;

Catani, Grazzini, Torre ‘14](up to NNLO in the threshold limit with massless tops[Ferroglia, Pecjak, Yang ‘12])

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Rapidity divergences and analytic regulator

Modification of the measure[Becher, Bell ‘12]

Z

d^dkδ⁺(k²)→ Z

d^dk ν+

k₊ ^α

δ⁺(k²)

I The regulator is necessary at intermediate steps of the calculation.

I Rapidity divergences do not appear in QCD, hence, the complete SCET result has to stay finite in the limitα→0.

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Rapidity divergences and analytic regulator

Modification of the measure[Becher, Bell ‘12]

Z

d^dkδ⁺(k²)→ Z

d^dk ν+

k₊ ^α

δ⁺(k²)

I The regulator is necessary at intermediate steps of the calculation.

I Rapidity divergences do not appear in QCD, hence, the complete SCET result has to stay finite in the limitα→0.

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Kinematics and notation

Partonic process

q(p1) + ¯q(p2) → t(p3) + ¯t(p4) +P

ig(ki)

Invariants

ˆs= (p1+p2)² M²= (p3+p4)² β= r

1−4m²_t M² t₁= (p₁−p₃)²−m²_t u₁= (p₁−p₄)−m_t²

Small-qT limit ˆ

s,M²,t₁²,u₁²,m²_t q_T² = (p₃+p₄)²_T Λ²_QCD

Momenta

n= (1,0,0,1), n¯= (1,0,0,−1)

k_i^µ = (n·ki)¯n^µ

2 + (¯n·ki)n^µ 2 +k_i⊥^µ

p^µ₁ =mtn, p₂^µ=mt¯n, p^µ_3,4=mtv_3,4^µ +λ^µ_3,4

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Kinematics and notation

Partonic process

q(p1) + ¯q(p2) → t(p3) + ¯t(p4) +P

ig(ki) Invariants

ˆs= (p1+p2)² M²= (p3+p4)² β= r

1−4m²_t M² t₁= (p₁−p₃)²−m²_t u₁= (p₁−p₄)−m_t²

Small-qT limit ˆ

Momenta

n= (1,0,0,1), n¯= (1,0,0,−1)

2 + (¯n·ki)n^µ 2 +k_i⊥^µ

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Kinematics and notation

Partonic process

q(p1) + ¯q(p2) → t(p3) + ¯t(p4) +P

ig(ki) Invariants

ˆs= (p1+p2)² M²= (p3+p4)² β= r

1−4m²_t M² t₁= (p₁−p₃)²−m²_t u₁= (p₁−p₄)−m_t²

Small-qT limit ˆ

Momenta

n= (1,0,0,1), n¯= (1,0,0,−1)

2 + (¯n·ki)n^µ 2 +k_i⊥^µ

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Kinematics and notation

Partonic process

q(p1) + ¯q(p2) → t(p3) + ¯t(p4) +P

ig(ki) Invariants

ˆs= (p1+p2)² M²= (p3+p4)² β= r

1−4m²_t M² t₁= (p₁−p₃)²−m²_t u₁= (p₁−p₄)−m_t²

Small-qT limit ˆ

Momenta

n= (1,0,0,1), n¯= (1,0,0,−1)

2 + (¯n·ki)n^µ 2 +k_i⊥^µ

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Soft function

I Represents corrections coming from exchanges ofreal, soft gluons, whose transverse momenta sum up to a fixed valueqT.

S(q_T,v₃,v₄)∝X

δ(q_T− |P

ik_i⊥|)Q

iδ⁺(k_i²)

I external momenta: Born kinematics

I eikonal Feynman rules

S_i¯i =P∞

n=0S⁽ⁿ⁾_i_¯_i ^α_4π^sn

S⁽ⁿ⁾_i_¯_i =P

{j}wⁱ_{j}^¯ⁱ I_{j}

colour matrices phase space integrals

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Soft function

δ(q_T− |P

ik_i⊥|)Q

iδ⁺(k_i²)

S_i¯i =P∞

n=0S⁽ⁿ⁾_i_¯_i ^α_4π^sn

S⁽ⁿ⁾_i_¯_i =P

{j}wⁱ_{j}^¯ⁱ I_{j}

colour matrices phase space integrals

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Soft function

δ(q_T− |P

ik_i⊥|)Q

iδ⁺(k_i²)

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Soft function at NLO

I Known in analytic form

[Li, Li, Shao, Yan, Zhu ‘13; Catani, Grazzini, Torre ‘13]

S⁽¹⁾_i_¯_i =4L_⊥

2w¹³_i_¯_i ln −t1

m_tM +2w²³_i_¯_i ln −u1

m_tM +w³³_i_¯_i

−4 w¹³_i_¯_i +w²³_i_¯_i Li2

1− t1u1

m_t²M²

+4w³³_i_¯_i ln t1u1

m²_tM²

−2w³⁴_i_¯_i 1+β_t² βt

hL_⊥lnx_s−Li₂

−xstg²θ 2

+Li₂

−1 xs

tg²θ 2

+4 lnxsln cosθ 2 i

, where L_⊥ =ln x_T²µ² 4e^−2γ^E

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Soft function at NLO

I Known in analytic form

[Li, Li, Shao, Yan, Zhu ‘13; Catani, Grazzini, Torre ‘13]

S⁽¹⁾_i_¯_i =4L_⊥

2w¹³_i_¯_i ln −t1

m_tM +2w²³_i_¯_i ln −u1

m_tM +w³³_i_¯_i

−4 w¹³_i_¯_i +w²³_i_¯_i Li2

1− t1u1

m²_tM²

+4w³³_i_¯_i ln t1u1

m_t²M²

341+β_t²h

2θ 1 ₂θ

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Soft function at NNLO

Three distinct groups of diagrams:

I bubble ^I single-cut

I double-cut

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NNLO integrals

Example:

I3gv,ij=

Z d^dk1d^dk2δ⁺(k₁²)δ⁺(k₂²)δ((k1+k2)²_T−q_T²)

(n·k₁)^α(n·k₂)^α(n_i·k₁) (n_j·(k₁+k₂)) (k₁+k₂)²+· · ·

I a range of overlapping singularities

I complication introduced by δ((k1+k2)²_T−q_T²)which additionally couples gluon’s momenta

I divergent in the limits→0 andα→0

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NNLO integrals

Example:

I3gv,ij=

Z d^dk1d^dk2δ⁺(k₁²)δ⁺(k₂²)δ((k1+k2)²_T−q_T²)

(n·k₁)^α(n·k₂)^α(n_i·k₁) (n_j·(k₁+k₂)) (k₁+k₂)²+· · ·

I a range of overlapping singularities

I complication introduced by δ((k1+k2)²_T−q_T²)which additionally couples gluon’s momenta

I divergent in the limits→0 andα→0

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NNLO integrals

Our strategy:

1. Represent each integral as

IG = Z

IG× WG

boundary integral simpler but encoding all α, →0 singularities

weight

complicated but finite

2. Usesector decompositionto calculate series expansion ofIG

IG = X

m,n=−1

amnα^mⁿ

3. Convolute the above with W_G and perform numerical integration of the coefficients

IG = X

m,n=−1

Z

amn× WG

α^mⁿ= X

m,n=−1

cmnα^mⁿ

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NNLO integrals

Our strategy:

IG = Z

IG× WG

weight

IG = X

m,n=−1

amnα^mⁿ

IG = X

m,n=−1

Z

amn× WG

α^mⁿ= X

m,n=−1

cmnα^mⁿ

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NNLO integrals

Our strategy:

IG = Z

IG× WG

weight

IG = X

m,n=−1

amnα^mⁿ

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Sector decomposition

I A method to disentangle overlapping singularities[Binoth, Heinrich, ‘00;

Borowka, Heinrich, Jahn, Jones, Kerner, Schlenk, Zirke ‘17]

Z 1

0

dx dy W(x,y) (x+y)²⁺ =

Z 1

0

dx dy W(x,y) (x+y)²⁺

h

(1)

z }| { Θ(x−y) +

(2)

z }| { Θ(y−x)i

...

(1) y=x t (2) x=y t ...

= Z 1

0

dx dt W(x,tx) (1+t)²⁺x¹⁺+

Z 1

0

dt dy W(ty,y) (1+t)²⁺y¹⁺

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Sector decomposition

Two types of singularities

I Endpoint,e.g. soft:

k₁⁺,k₁⁻,k₁^⊥

→0

I Manifold,e.g. collinear

k1·k2→0

0 20 40

k+ 20 0 40

l+

-1.0 -0.5 0.0 0.5 1.0

y_T

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Sector decomposition

Two types of singularities

I Endpoint,e.g. soft:

k₁⁺,k₁⁻,k₁^⊥

→0

I Manifold,e.g. collinear

k1·k2→0

0 20 40

k+ 20 0 40

l+

-1.0 -0.5 0.0 0.5 1.0

y_T

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The strategy

Given the integral:

I Analytically integrate 3 out of 2d dimensions

I Map the remaining variables to a unit hypercube (split the original integral into a sum if necessary)

I Apply sector decomposition to disentangle overlapping singularities

I Expand the result in Laurent series inandα

IG = Z

d^dk1d^dk2IG × WG

=X

j

Z

Q2d−3

i=1 dxi(IG × WG)_j

=X

j,k

Z

Q2d−3

i=1 dx_i(I_G × W_G)_jk

=X

j,k

X

m,n=−1

Z

a_mn× W_G

α^mⁿ

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Validation with the bubble

I Laboratory to stress-test sector decomposition-based methodology

I Non-trivial tensor structure →challenging numerators

I Solvable analytically: direct cross check of our sector decomposition-based implementation

I Comparable with nf part of Renormalization Group prediction

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Validation with the bubble

I Laboratory to stress-test sector decomposition-based methodology

I Non-trivial tensor structure →challenging numerators

I Solvable analytically: direct cross check of our sector decomposition-based implementation

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Renormalization

dσ

dΦ =B⁽⁰⁾₁ ⊗ B₂⁽⁰⁾⊗Trh

H⁽⁰⁾⊗ S⁽⁰⁾i

=ZBB⁽⁰⁾₁ ⊗ZBB⁽⁰⁾₂ ⊗Trh

Z^†_HH⁽⁰⁾ZH⊗Z^†_SS⁽⁰⁾ZS

i

=B₁(µ)⊗ B₂(µ)⊗Tr[H(µ)⊗ S(µ)]

finite

separately divergent

separately finite

d dµ

dσ

dΦ =0 → Renormalization Group Equations for B,HandS

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Renormalization

S(µ) =Z^†_s(µ, )S_bare()Z_s(µ, )

I RG equation d

dlnµS_i¯i(µ) =−γ_i^s†_¯_i S_i¯i(µ)−S_i¯i(µ)γ_i^s_¯_i

I Soft anomalous dimension

γ^s =−Z⁻¹_s dZs

dlnµ

Specifically, at the orderα²_s, we get

S⁽²⁾

|{z}

finite part only

=

pole part only

z }| {

Z^†(2)_s S˜⁽⁰⁾_bare+ ˜S⁽⁰⁾_bareZ⁽²⁾_s +Z^†(1)_s S˜⁽⁰⁾_bareZ⁽¹⁾_s

+ Z^†(1)_s S˜⁽¹⁾_bare+ ˜S⁽¹⁾_bareZ⁽¹⁾_s + ˜S⁽²⁾_bare−β0

S˜⁽¹⁾_bare

| {z }

finite + pole part

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Renormalization

S(µ) =Z^†_s(µ, )S_bare()Z_s(µ, )

I RG equation d

dlnµS_i¯i(µ) =−γ_i^s†_¯_i S_i¯i(µ)−S_i¯i(µ)γ_i^s_¯_i

I Soft anomalous dimension

γ^s =−Z⁻¹_s dZs

dlnµ Specifically, at the orderα²_s, we get

S⁽²⁾

|{z}

finite part only

=

pole part only

z }| {

Z^†(2)_s S˜⁽⁰⁾_bare+ ˜S⁽⁰⁾_bareZ⁽²⁾_s +Z^†(1)_s S˜⁽⁰⁾_bareZ⁽¹⁾_s

+ Z^†(1)_s S˜⁽¹⁾_bare+ ˜S⁽¹⁾_bareZ⁽¹⁾_s + ˜S⁽²⁾_bare−β0

S˜⁽¹⁾_bare

| {z }

finite + pole part

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Bubble part of the soft function from differential equations

∝

Z d^dqδ(q_T−1)θ⁺(q²)n^µ_in^ν_j q⁴(n_i·q) (nj·q)

!

µν

where

!

µν

=

Z d^dk Nµνδ⁺(k²)δ⁺((q−k)²) (n·k)^α(n·(q−k))^αk²(q−k)²

=T00g^µ,ν+Tqqq^µq^ν+Tnnn^µn^ν+Tqn (n^µq^ν+q^µn^ν)

I Topology:

Z d^dk

(n·k)â¹^+2α (¯n·k)â² (v₃·k)â³ (v₄·k)â⁴(k²−m²)â⁵((n·k) (¯n·k)−m²−1)â⁶

I IBPs →reduction→DE→solutions →R

dm²→Ijk(β, θ)

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Bubble part of the soft function from differential equations

∝

!

µν

where

!

µν

=

I Topology:

Z d^dk

(n·k)â¹^+2α(¯n·k)â² (v₃·k)â³ (v₄·k)â⁴(k²−m²)â⁵((n·k) (¯n·k)−m²−1)â⁶

I IBPs →reduction→DE→solutions →R

dm²→Ijk(β, θ)

(43)

Bubble part of the soft function from differential equations

∝

!

µν

where

!

µν

=

I Topology:

Z d^dk

a+2α a a a

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Validation with the bubble: results

1. Cancellation of αpoles

I Analytically

w^q¯₁₃^q

− 8 3α+ 8

3α−8(5+3γE+2 ln 2)

9α +8(5+3γE+2 ln 2) 9α +. . .

I Numerically

w^q¯₁₃^q

−2.66597

α +2.66597

α −4.13986

α +4.13986 α +. . .

2. Agreement with thenf part of RG prediction

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Validation with the bubble: results

3. Agreement with analytic result

−G¹³+¹₂G³³

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Complete Soft Function at NNLO

I In momentum space

S⁽²⁾(qT, β, θ) = 1 q_T^p

S_bubble⁽²⁾ (β, θ) +S_1-cut⁽²⁾ (β, θ) +S_2-cut⁽²⁾ (β, θ)

I In position space

S⁽²⁾(L⊥, β, θ) = 1

+L⊥+L²_⊥+. . .

×

= 1

²S^(2,−2)+1 S^(2,−1)

| {z }

+S^(2,ren)

can be cross-checked against RG, fixes many terms inS^(2,ren)

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Complete Soft Function at NNLO

I In momentum space

S⁽²⁾(qT, β, θ) = 1 q_T^p

I In position space

S⁽²⁾(L⊥, β, θ) = 1

+L⊥+L²_⊥+. . .

×

= 1

S^(2,−2)+1

S^(2,−1)+S^(2,ren)

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Results: higher order poles

Even though the NNLO Soft Function exhibits at most 1

² singularity, higher order poles appear in individual contributions.

I All αpoles, as well as 1

⁴ pole cancel within each colour structure, for example

1 ⁴

I 1

³ pole cancels between 1-cut and 2-cut contributions

1 ³

†We usedβ=0.4,θ=0.5.

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Results: higher order poles

I Allαpoles, as well as 1

1 ⁴

I 1

1 ³

(50)

Results: higher order poles

I Allαpoles, as well as 1

1 ⁴

I 1

1 ³

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Results: comparison with Renormalization Group

I Double pole

h

S_direct^(2,−2)+S_RGE^(2,−2)i

β=0.4,θ=0.5=

I Single pole

h

S_direct^(2,−1)+S_RGE^(2,−1)i

β=0.0,θ=0.0

=

(52)

Conclusions

I Our goal: Use small-qT factorization andqT slicing to perform independent calculation of top pair production at NNLO

I The only missing component: the NNLO soft function

I We have developed a sector decomposition-based framework for calculation of the NNLO soft function integrals

I The framework has been extensively validated and cross-checked:

1. Cancellation ofαandpoles beyond 1/²

2. Perfect agreement with analytic calculation for bubble graphs 3. RG result for the complete NNLO soft function recovered

I Next and final step: the missing bit of the NNLO Soft Function for top pair production:theqT-independent part of the ⁰term.

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Acknowledgements

The project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement NO. 665778.