NNLO soft function for top pair production at small qT

NNLO soft function for top pair production at small q

Sebastian Sapeta

IFJ PAN Kraków

In collaboration with Ren´e ´Angeles-Martinez and Michał Czakon based on arXiv:1809.01459

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 resumma- tion at NNLL [Czakon, Ferroglia, Heymes, Mitov, Pecjak, Scott, Wang, Yang ‘18]

I Small-qT resummation at NNLL

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

The q

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

The q

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

The q

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 approximation known

Soft Collinear Effective Theory (SCET)

SCET'QCD _{IR limit}

I Hard degrees of freedom are integrated out into Wilson coefficients, which are then used to adjust new couplings of the (effective) theory.

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

I The separation of fields in the Lagrangian into collinear, anti-collinear and soft sectors, facilitates proofs of factorization theorems

Soft Collinear Effective Theory (SCET)

SCET'QCD _{IR limit}

I Hard degrees of freedom are integrated out into Wilson coefficients, which are then used to adjust new couplings of the (effective) theory.

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

I The separation of fields in the Lagrangian into collinear, anti-collinear and soft sectors, facilitates proofs of factorization theorems

Soft Collinear Effective Theory (SCET)

SCET'QCD _{IR limit}

I Hard degrees of freedom are integrated out into Wilson coefficients, which are then used to adjust new couplings of the (effective) theory.

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

I The separation of fields in the Lagrangian into collinear, anti-collinear and soft sectors, facilitates proofs of factorization theorems

Small-q

factorization in SCET

whereF=H,Z,W,ZZ,WW,t¯t, . . .

dσ^F

dΦ =B1⊗ B2⊗ H ⊗ S+O q²_T

q²

Small-q

factorization in SCET

whereF=H,Z,W,ZZ,WW,t¯t, . . .

dσ^F

dΦ =B1⊗ B2⊗ H ⊗ S+O q²_T

q²

Small-q

factorization in SCET

Gluons’ momenta in light-cone coordinates

k_i^µ= k_i⁺,k_i⁻,k^⊥_i

where k^± =k⁰±k³

Expansion parameter

λ=q_T² q² 1

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

Small-q

factorization in SCET

Gluons’ momenta in light-cone coordinates

k_i^µ= k_i⁺,k_i⁻,k^⊥_i

where k^± =k⁰±k³

Expansion parameter

λ=q_T² q² 1 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

Top pair production at small-q

through NNLO

dσ^NNLO

dq_Tdy dM dcosθ =X

i,¯i

B_i/h1⊗ B¯i/h₂⊗Tr[H_i¯i⊗ S_i¯i]

where

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

βt =

r

1−4m_t² M²

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](and up to NNLO in the threshold limit [Ferroglia, Pecjak, Yang ‘12; Wang, Xu, Yang and Zhu ‘18])

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

Top pair production at small-q

through NNLO

dσ^NNLO

dq_Tdy dM dcosθ =X

i,¯i

B_i/h1⊗ B¯i/h₂⊗Tr[H_i¯i⊗ S_i¯i]

where

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

βt =

r

1−4m_t² M²

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-qT limit[Li, Li, Shao, Yan, Zhu ‘13;

Catani, Grazzini, Torre ‘14](and up to NNLO in the threshold limit [Ferroglia, Pecjak, Yang ‘12; Wang, Xu, Yang and Zhu ‘18])

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

Top pair production at small-q

through NNLO

dσ^NNLO

dq_Tdy dM dcosθ =X

i,¯i

B_i/h1⊗ B¯i/h₂⊗Tr[H_i¯i⊗ S_i¯i]

where

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

βt =

r

1−4m_t² M²

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-qT limit[Li, Li, Shao, Yan, Zhu ‘13;

Catani, Grazzini, Torre ‘14](and up to NNLO in the threshold limit

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.

Rapidity divergences and analytic regulator

Modification of the measure[Becher, Bell ‘12]

Z

d^dkδ⁺(k²)→ Z

d^dk ν

k₊ ^α

δ⁺(k²)

Kinematics and notation

q(p1) + ¯q(p2) → t(p3) + ¯t(p4) +P

ig(ki) g(p1) + ¯g(p2) → t(p3) + ¯t(p4) +P

ig(ki)

I Invariants ˆ

s= (p₁+p₂)² M²= (p₃+p₄)² t1= (p1−p3)²−m²_t u1= (p1−p4)−m²_t

I Small-q_T limit

ˆs,M²,|t1|,|u1|,m²_t q_T² = (p3+p4)²_T Λ²_QCD

I 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

p₁^µ=

√ ˆ s

2 n, p₂^µ=

√ ˆ s

2 ¯n, p_3,4^µ =mtv_3,4^µ +λ^µ_3,4

Kinematics and notation

q(p1) + ¯q(p2) → t(p3) + ¯t(p4) +P

ig(ki) g(p1) + ¯g(p2) → t(p3) + ¯t(p4) +P

ig(ki)

I Invariants ˆ

s= (p₁+p₂)² M²= (p₃+p₄)² t1= (p1−p3)²−m²_t u1= (p1−p4)−m²_t

I Small-q_T limit

ˆs,M²,|t1|,|u1|,m²_t q_T² = (p3+p4)²_T Λ²_QCD

I 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

p₁^µ=

√ ˆ s

2 n, p₂^µ=

√ ˆ s

2 ¯n, p_3,4^µ =mtv_3,4^µ +λ^µ_3,4

Kinematics and notation

q(p1) + ¯q(p2) → t(p3) + ¯t(p4) +P

ig(ki) g(p1) + ¯g(p2) → t(p3) + ¯t(p4) +P

ig(ki)

I Invariants ˆ

s= (p₁+p₂)² M²= (p₃+p₄)² t1= (p1−p3)²−m²_t u1= (p1−p4)−m²_t

I Small-q_T limit

ˆs,M²,|t1|,|u1|,m²_t q_T² = (p3+p4)²_T Λ²_QCD

I 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

p₁^µ=

√ ˆ s

2 n, p₂^µ=

√ ˆ s

2 ¯n, p_3,4^µ =mtv_3,4^µ +λ^µ_3,4

Kinematics and notation

q(p1) + ¯q(p2) → t(p3) + ¯t(p4) +P

ig(ki) g(p1) + ¯g(p2) → t(p3) + ¯t(p4) +P

ig(ki)

I Invariants ˆ

s= (p₁+p₂)² M²= (p₃+p₄)² t1= (p1−p3)²−m²_t u1= (p1−p4)−m²_t

I Small-q_T limit

ˆs,M²,|t1|,|u1|,m²_t q_T² = (p3+p4)²_T Λ²_QCD

I Momenta

n= (1,0,0,1), ¯n= (1,0,0,−1)

Soft function

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

S_bare(q_T,v₃,v₄)∝X

δ(q_T− |P

ik_i⊥|)Q

iδ⁺(k_i²)

I external momenta →Wilson Lines (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

Soft function

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

S_bare(q_T,v₃,v₄)∝X

δ(q_T− |P

ik_i⊥|)Q

iδ⁺(k_i²)

I external momenta →Wilson Lines (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

Soft function

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

S_bare(q_T,v₃,v₄)∝X

δ(q_T− |P

ik_i⊥|)Q

iδ⁺(k_i²)

I external momenta →Wilson Lines (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

Renormalization

dσ

dΦ =B^(bare)₁ ⊗ B₂^(bare)⊗Trh

H^(bare)⊗ S^(bare)i

=ZBB₁^(bare)⊗ZBB^(bare)₂ ⊗Trh

Z^†_HH^(bare)ZH⊗Z^†_SS^(bare)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

Renormalization

dσ

dΦ =B^(bare)₁ ⊗ B₂^(bare)⊗Trh

H^(bare)⊗ S^(bare)i

=ZBB₁^(bare)⊗ZBB^(bare)₂ ⊗Trh

Z^†_HH^(bare)ZH⊗Z^†_SS^(bare)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

Renormalization

dσ

dΦ =B^(bare)₁ ⊗ B₂^(bare)⊗Trh

H^(bare)⊗ S^(bare)i

=ZBB₁^(bare)⊗ZBB^(bare)₂ ⊗Trh

Z^†_HH^(bare)ZH⊗Z^†_SS^(bare)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

Renormalization

dσ

dΦ =B^(bare)₁ ⊗ B₂^(bare)⊗Trh

H^(bare)⊗ S^(bare)i

=ZBB₁^(bare)⊗ZBB^(bare)₂ ⊗Trh

Z^†_HH^(bare)ZH⊗Z^†_SS^(bare)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

Renormalization

dσ

dΦ =B^(bare)₁ ⊗ B₂^(bare)⊗Trh

H^(bare)⊗ S^(bare)i

=ZBB₁^(bare)⊗ZBB^(bare)₂ ⊗Trh

Z^†_HH^(bare)ZH⊗Z^†_SS^(bare)ZS

i

=B₁(µ)⊗ B₂(µ)⊗Tr[H(µ)⊗ S(µ)]

finite

separately divergent

separately finite

d dσ

Renormalization

I RG equation d

dlnµS_i¯i(µ) =−γ^s†

i¯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

Renormalization

I RG equation d

dlnµS_i¯i(µ) =−γ^s†

i¯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

β

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^−2γE

I We had to calculate ordercorrection

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

−2w³⁴_i¯i

1+β_t² β

h

L⊥lnxs−Li2

−xstg²θ 2

+Li2

−1 x tg²θ

2

Soft function at NNLO

Three distinct groups of diagrams:

I Bubble

I Single-cut

I Double-cut

DIFFERENTIAL EQUATIONS

DIRECT INTEGRATION

SECTOR DECOMPOSITION

Soft function at NNLO

Three distinct groups of diagrams:

I Bubble ^I Single-cut

I Double-cut

DIFFERENTIAL EQUATIONS

DIRECT INTEGRATION

SECTOR DECOMPOSITION

Soft function at NNLO

Three distinct groups of diagrams:

I Bubble ^I Single-cut

I Double-cut

DIFFERENTIAL EQUATIONS

DIRECT INTEGRATION

SECTOR DECOMPOSITION

Soft function at NNLO

Three distinct groups of diagrams:

I Bubble ^I Single-cut

I Double-cut

DIFFERENTIAL EQUATIONS

DIRECT INTEGRATION

SECTOR DECOMPOSITION

Soft function at NNLO

Three distinct groups of diagrams:

I Bubble ^I Single-cut

I Double-cut

DIFFERENTIAL EQUATIONS

DIRECT INTEGRATION

SECTOR DECOMPOSITION

Soft function at NNLO

Three distinct groups of diagrams:

I Bubble ^I Single-cut

I Double-cut

DIFFERENTIAL EQUATIONS

DIRECT INTEGRATION

SECTOR DECOMPOSITION

Double-cut part

|M⁽⁰⁾_g,g,a1,...(k,l,p1, . . .)|²'

1 2

X

ijkl

Sij(k)Skl(l)hM⁽⁰⁾_a1,...(p1, . . .)| {Ti·Tj,Tk·Tl} |M⁽⁰⁾_a1,...(p1, . . .)i

−C_AX

ij

Sij(k,l)hM⁽⁰⁾_a1,...(p₁, . . .)|Ti·T_j|M⁽⁰⁾_a1,...(p₁, . . .)i

S^(2),ff¯

i¯i (q_⊥) = 1 Sd−3

Z

dΩ_d−3d^dk d^dlδ₊(k²)δ₊(l²) δ^(d−2)(q_⊥−k_⊥−l_⊥)

× hc_I^i¯i|M^∗(0)

f,¯f,a1,...(k,l,p₁, . . .)M⁽⁰⁾

f,¯f,a1,...(k,l,p₁, . . .)|c_J^i¯ii

Double-cut part

|M⁽⁰⁾_g,g,a1,...(k,l,p1, . . .)|²'

1 2

X

ijkl

Sij(k)Skl(l)hM⁽⁰⁾_a1,...(p1, . . .)| {Ti·Tj,Tk·Tl} |M⁽⁰⁾_a1,...(p1, . . .)i

−C_AX

ij

Sij(k,l)hM⁽⁰⁾_a1,...(p₁, . . .)|Ti·T_j|M⁽⁰⁾_a1,...(p₁, . . .)i

S^(2),f_i¯i ^¯f(q⊥) = 1 S_d−3

Z

dΩd−3d^dk d^dlδ+(k²)δ+(l²) δ^(d−2)(q⊥−k⊥−l⊥)

i¯i ∗(0) (0) i¯i

Double-cut NNLO integrals

Example:

˜I3gv,ij =

Z d^dk1d^dk2δ⁺(k₁²)δ⁺(k₂²)δ((k1+k2)²_T−q²_T) (n·k₁)^α(n·k₂)^α(n_i·k₁) (n_j·(k₁+k₂)) (k₁+k₂)²

I divergent in the limits→0 andα→0

I a range of overlapping singularities

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

To disentangle overlapping singularities and calculate regularized integrals we use the method ofsector decomposition[Binoth, Heinrich, ‘00; Borowka, Heinrich, Jahn, Jones, Kerner, Schlenk, Zirke ‘17].

Double-cut NNLO integrals

Example:

˜I3gv,ij =

Z d^dk1d^dk2δ⁺(k₁²)δ⁺(k₂²)δ((k1+k2)²_T−q²_T) (n·k₁)^α(n·k₂)^α(n_i·k₁) (n_j·(k₁+k₂)) (k₁+k₂)²

I divergent in the limits→0 andα→0

I a range of overlapping singularities

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

To disentangle overlapping singularities and calculate regularized integrals we use the method ofsector decomposition[Binoth, Heinrich, ‘00; Borowka, Heinrich, Jahn, Jones, Kerner, Schlenk, Zirke ‘17].

Double-cut NNLO integrals

Example:

˜I3gv,ij =

Z d^dk1d^dk2δ⁺(k₁²)δ⁺(k₂²)δ((k1+k2)²_T−q²_T) (n·k₁)^α(n·k₂)^α(n_i·k₁) (n_j·(k₁+k₂)) (k₁+k₂)²

I divergent in the limits→0 andα→0

I a range of overlapping singularities

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

To disentangle overlapping singularities and calculate regularized integrals we use the method ofsector decomposition[Binoth, Heinrich, ‘00; Borowka, Heinrich, Jahn, Jones, Kerner, Schlenk, Zirke ‘17].

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

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

The strategy

Given the integral:

boundary integralencodes allα, →0 singularities

finiteweight

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 singula- rities

I Expand the result in Laurent series inandα

I Numerically integrate series coefficients

IG = Z

d^dk1d^dk2IG × WG

=X

j

Z 1

0

Q2d−3

i=1 dxi(IG× WG)_j

=X

j

X

k∈sectors

Z 1

0

Q2d−3

i=1 dxi(IG× WG)_jk

= X

j,m,n

X

k∈sectors

Z 1

0

a_mn× W_G

α^mn

= X

j,m,n

X

k∈sectors

c_mn^jk α^mn

The strategy

Given the integral:

boundary integralencodes allα, →0 singularities

finiteweight

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 singula- rities

I Expand the result in Laurent series inandα

I Numerically integrate series coefficients

IG = Z

d^dk1d^dk2IG × WG

=X

j

Z 1

0

Q2d−3

i=1 dxi(IG× WG)_j

=X

j

X

k∈sectors

Z 1

0

Q2d−3

i=1 dxi(IG× WG)_jk

= X

j,m,n

X

k∈sectors

Z 1

0

a_mn× W_G

α^mn

= X

j,m,n

X

k∈sectors

c_mn^jk α^mn

The strategy

Given the integral:

boundary integralencodes allα, →0 singularities

finiteweight

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 singula- rities

I Expand the result in Laurent series inandα

I Numerically integrate series coefficients

IG = Z

d^dk1d^dk2IG × WG

=X

j

Z 1

0

Q2d−3

i=1 dxi(IG× WG)_j

=X

j

X

k∈sectors

Z 1

0

Q2d−3

i=1 dxi(IG× WG)_jk

= X

j,m,n

X

k∈sectors

Z 1

0

a_mn× W_G

α^mn

= X

j,m,n

X

k∈sectors

c_mn^jk α^mn

The strategy

Given the integral:

boundary integralencodes allα, →0 singularities

finiteweight

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 singula- rities

I Expand the result in Laurent series inandα

I Numerically integrate series coefficients

IG = Z

d^dk1d^dk2IG × WG

=X

j

Z 1

0

Q2d−3

i=1 dxi(IG× WG)_j

=X

j

X

k∈sectors

Z 1

0

Q2d−3

i=1 dxi(IG × WG)_jk

= X

j,m,n

X

k∈sectors

Z 1

0

a_mn× W_G

α^mn

= X

j,m,n

X

k∈sectors

c_mn^jk α^mn

The strategy

Given the integral:

boundary integralencodes allα, →0 singularities

finiteweight

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 singula- rities

I Expand the result in Laurent series inandα

I Numerically integrate series coefficients

IG = Z

d^dk1d^dk2IG × WG

=X

j

Z 1

0

Q2d−3

i=1 dxi(IG× WG)_j

=X

j

X

k∈sectors

Z 1

0

Q2d−3

i=1 dxi(IG × WG)_jk

= X X Z 1

0

a_mn× WG

α^mn

= X

j,m,n

X

k∈sectors

c_mn^jk α^mn

The strategy

Given the integral:

boundary integralencodes allα, →0 singularities

finiteweight

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 singula- rities

I Expand the result in