Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Subtraction of Infrared Singularities in Perturbative QCD:a new scheme

Chiara Signorile-Signorile

Università degli Studi di Torino

II year seminar, 20 September 2019

in collaboration with:L. Magnea, E. Maina, G. Pelliccioli, P. Torrielli and S. Uccirati

based on:Magnea, Maina, Pelliccioli, CS, Torrielli, Uccirati JHEP1812(2018)062, arXiv:1809.05444[hep-ph]Magnea, Maina, Pelliccioli, CS, Torrielli, Uccirati JHEP1812(2018)107, arXiv:1806.09570[hep-ph]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 1 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Environment

Natural environment:

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 2 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Environmentp-p collision:

DO NOT PANIC! General theorems ensure the factorisation of short-distancessubprocess from long-distances ones.

→ short distances described in Perturbative QCD.

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 3 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Environmentp-p collision:

DO NOT PANIC! General theorems ensure the factorisation of short-distancessubprocess from long-distances ones.

→ short distances described in Perturbative QCD.Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 3 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Introduction

Framework: QCD

quantum field theory with unbroken SU(3) non-abelian gauge invariance.strong interactions between quarks and gluonasymptotically free in the ultraviolet regime → Perturbation Theory

At hight energy quarks masses are negligible and two kinds of divergences ariseTEXT

Mpp + k

k

µ, a

hp: m = 0

−igs ū(p) /�(k) tai(/p + /k)

(p + k)2 + iηM

(p + k)2 = p0 k0(1− cos θpk) = 0 →{k0 = 0 Softcos θpk = 1 Collinear

Covariant ApproachSoft and Collinear configurations produce on-shell intermediated states thatpropagate indefinitely before emission

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 4 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

IR singularities cancellation

KLN Theorem

IR divergences from real radiation phase space and from virtual corrections.order-by-order cancellation in perturbation theory for a generic IR-safeobservable.

At NLO, the cancellation occurs between the following diagram sample

ex. e+e− → qq̄

2e−

e+

q

q̄

+

2

e+

e− q

q̄

g +

e−

e+ q̄

q

g

e−

e+

q

q̄

LO : α2 [born] NLO : α2αs [real] NLO : α2αs [virtual]

Real and Virtual contributions

hp: d = 4− 2� , � < 0{Real contr.→ implicit poles in �→ unresolved radiation PS.Virtual contr.→ explicit poles in �.

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 5 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Outline

2

e+

e− q

q̄

g +

e−

e+ q̄

q

g

e−

e+

q

q̄

Real contributioncomplicated phase space integration

↓

Subtraction

↓

counterterm

Virtual contributiondivide singularities according to their nature

↓

Factorisation

↓

completeness

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 6 / 20

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Subtraction Pattern

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 7 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Subtraction pattern

Given a generic amplitude with n massless particles in the final state [e+e− → jets]

An(pi ) = A(0)n (pi ) +A(1)n (pi ) +A

(2)n (pi ) + . . .

An IR-safe observable X receives contribution at NLO according to

dσNLO

dX= lim

d→4

{∫dΦn Vn δn +

∫dΦn+1 Rn+1 δn+1

}where δi = δ(X − Xi ), Xi the i-particle configuration, and

Vn = 2Re[A(0)†n A

(1)n]

Rn+1 =∣∣A(0)n+1∣∣2.

ProblemNumerical implementation requires to handle finite quantities → radiation IR poleshave to be subtracted before performing the phase space integration.

Subtraction ideamake the real contribution finite before performing the PS integration by adding andsubtracting a counterterm. [Magnea, Maina, Pelliccioli, CS, Torrielli, Uccirati 1806.09570]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 8 / 20

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Subtraction pattern

Given a generic amplitude with n massless particles in the final state [e+e− → jets]

An(pi ) = A(0)n (pi ) +A(1)n (pi ) +A

(2)n (pi ) + . . .

An IR-safe observable X receives contribution at NLO according to

dσNLO

dX= lim

d→4

{∫dΦn Vn δn +

∫dΦn+1 Rn+1 δn+1

}where δi = δ(X − Xi ), Xi the i-particle configuration, and

Vn = 2Re[A(0)†n A

(1)n]

Rn+1 =∣∣A(0)n+1∣∣2.

ProblemNumerical implementation requires to handle finite quantities → radiation IR poleshave to be subtracted before performing the phase space integration.

Subtraction ideamake the real contribution finite before performing the PS integration by adding andsubtracting a counterterm. [Magnea, Maina, Pelliccioli, CS, Torrielli, Uccirati 1806.09570]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 8 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Subtraction pattern

Given a generic amplitude with n massless particles in the final state [e+e− → jets]

An(pi ) = A(0)n (pi ) +A(1)n (pi ) +A

(2)n (pi ) + . . .

An IR-safe observable X receives contribution at NLO according to

dσNLO

dX= lim

d→4

{∫dΦn Vn δn +

∫dΦn+1 Rn+1 δn+1

}where δi = δ(X − Xi ), Xi the i-particle configuration, and

Vn = 2Re[A(0)†n A

(1)n]

Rn+1 =∣∣A(0)n+1∣∣2.

ProblemNumerical implementation requires to handle finite quantities → radiation IR poleshave to be subtracted before performing the phase space integration.

Subtraction ideamake the real contribution finite before performing the PS integration by adding andsubtracting a counterterm. [Magnea, Maina, Pelliccioli, CS, Torrielli, Uccirati 1806.09570]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 8 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Subtraction pattern

Counterterms and Subtraction at NLO

Counterterm properties{same singular limits as the realanalytically integrable in d dim

dσNLOctdX

=∫

Φn+1 Kn+1 , In =∫

dΦrad Kn+1

dσNLO

dX=∫

dΦn(Vn + In

)δn︸ ︷︷ ︸

finite in d=4

+∫

dΦn+1(Rn+1 δn+1−Kn+1 δn

)︸ ︷︷ ︸

finite in d=4

→ Several different schemes are available:Antenna, Nested soft-collinear, ColorfulNNLO, N-jettiness, Unsubtraction, Q⊥scheme, Projection to Born, . . .

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 9 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Counterterms construction at NLO

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 10 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Practical implementation of the Subtraction methodIngredients

fundamental limits Si , Cijsingular structure of R

partition of the phase space Wijmomentum mapping {k} → {k̄}

Sector Wij → Si ,Cij

Goal: RWij − K ij → finite

Counterterm: K ij =[Si + Cij (1− Si )

]RWij

S̄i

C̄ij

S̄i C̄ij

The IR limits of R in the mapped kinematics featureuniversal kernelBorn matrix element

SiR({k}) = −N∑c,d

δfi gscd

sic sidBcd ({k̄}(icd))

CijR({k}) = N1sij

Pµνij (sir , sjr )Bµν({k̄}(ijr))

SiCijR({k}) = 2N Cfj δfi gsjr

sij sirB({k̄}(ijr))

Born kinematics: mass-shell condition+momenta conservationChiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 11 / 20

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Factorisation approach

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 12 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Factorised virtual amplitude

Factorisation ideamake the virtual contribution finite by adding and subtracting a counterterm,identified through completeness relations. [Magnea, Maina, Pelliccioli, CS, Torrielli, Uccirati 1809.05444]

Factorisation formula for massless virtual amplitude [Sterman 9606312] [Gardi, Magnea 0908.3273]

An(piµ

)=

n∏i=1

[Ji(

(pi · ni )2/(n2i µ2))

Ji,E(

(βi · ni )2/n2i) ] Sn(βi · βj ) Hn(pi · pj

µ2,

(pi · ni )2

n2i µ2

)where pµi = Qβ

µi , β

2i = 0, and n2i 6= 0 auxiliary vector, µ renormalisation scale.

Ji Collinear functionJi,E Soft-Collinear functionSn Soft functionHn Hard region

Functions’ properties:universalitygauge invariancesimple operator definition

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 13 / 20

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Factorised virtual amplitude

Factorisation ideamake the virtual contribution finite by adding and subtracting a counterterm,identified through completeness relations. [Magnea, Maina, Pelliccioli, CS, Torrielli, Uccirati 1809.05444]

Factorisation formula for massless virtual amplitude [Sterman 9606312] [Gardi, Magnea 0908.3273]

An(piµ

)=

n∏i=1

[Ji(

(pi · ni )2/(n2i µ2))

Ji,E(

(βi · ni )2/n2i) ] Sn(βi · βj ) Hn(pi · pj

µ2,

(pi · ni )2

n2i µ2

)where pµi = Qβ

µi , β

2i = 0, and n2i 6= 0 auxiliary vector, µ renormalisation scale.

Ji Collinear functionJi,E Soft-Collinear functionSn Soft functionHn Hard region

Functions’ properties:universalitygauge invariancesimple operator definition

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 13 / 20

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Radiative functions at cross-section level

Goal1 build cross sections quantities2 find a relation between radiative quantities and virtual functions

Taking inspiration from the virtual functions we define cross-section level operatorsincluding real radiations

Sn,m({ki};βi ) ≡∑{λi}

〈0|n∏

i=1

Φβi (0,∞)|k1, λ1...km, λm〉 〈k1, λ1...km, λm|n∏

i=1

Φβi (∞, 0) |0〉

Jq,m({ki}; p, n) ≡∫

ddx eil·x∑{λi}

〈0|Φn(∞, x)ψ(x)|p, s; k1, λ1 . . . km, λm〉 ×

×〈p, s; k1, λ1 . . . km, λm|ψ̄(0)Φn(0,∞) |0〉

m = 0 −→ virtual cross section functions

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 14 / 20

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Counterterms at NLO

Recall:dσNLO

dX=∫

dΦn[Vn + In

]δn︸ ︷︷ ︸

finite in d=4

+∫

dΦn+1[Rn+1 δn+1 − Kn+1 δn

]︸ ︷︷ ︸

finite in d=4

Factorisation → virtual contribution decomposed in functions

Vn = H(0)†n (pi ) S(1)n,0 H

(0)n (pi ) +

∑ni=1H(0)†n (pi )

[J(1)i,0 (pi )− J

(1)i,E,0(βi )

]H(0)n (pi ) .

Completeness → virtual functions linked to real functions

S(1)n,0(βi ) +∫

dΦ1 S(0)n,1(k, βi ) = fin. = + = finite...

...

...

...

β1

βi

βn

k

β1

βi

βn

β1

βi

βn βn

βi

β1

...

......

... k

J(1)i,0 (l , p, n) +∫

dΦ1 J(0)i,1 (k; l , p, n) = fin. =+

pi

k

pi

k

pi pi= finite

=⇒ Starting from the virtual structure we can identify real emission counterterms.

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 15 / 20

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Subtraction at NNLO

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 16 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

NNLO Subtraction pattern@NNLO:

more configurations contribute

dσNNLO

dX=∫

dΦn VVn δn(X) +∫

dΦn+1 RVn+1 δn+1(X)+∫

dΦn+2 RRn+2 δn+2(X)

RRn+2 =∣∣∣A(0)n+2∣∣∣2 VVn = ∣∣∣A(1)n ∣∣∣2 + 2Re[A(0)†n A(2)n ] RVn+1 = 2Re[A(0)†n+1A(1)n+1]

more counterterms to add and subtract∫dΦn+2 K (1) δn+1 : K (1) → same 1-unr. singularities as RR∫dΦn+2

(K (12) + K (2)

)δn : K (12) + K (2) → same 2-unr. singularities as RR.

[1-unr.(2-unr.), pure 2-unr.]∫dΦn+1 K (RV) δn : K (RV) → same 1-unr. singularities as RV

and integrate in the radiative phase space

I(i) =∫

dΦrad,i K (i) , I(12) =∫

dΦrad,1 K (12) , I(RV) =∫

dΦrad K (RV)

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 17 / 20

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Subtraction pattern at NNLO

dσNNLO

dX=∫

dΦn[

VVn︸︷︷︸singular in d=4 , finite in Φn

]δn

+∫

dΦn+1[ (

RVn+1)︸ ︷︷ ︸

singular in d=4, singular in Φn+1

δn+1

]

+∫

dΦn+2[

RRn+2︸ ︷︷ ︸finite in d=4, singular in Φn+2

δn+2

]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 18 / 20

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Subtraction pattern at NNLO

dσNNLO

dX=∫

dΦn[

VVn︸︷︷︸singular in d=4 , finite in Φn

]δn

+∫

dΦn+1[ (

RVn+1)︸ ︷︷ ︸

singular in d=4, singular in Φn+1

δn+1

]

+∫

dΦn+2[RRn+2 δn+2 − K (1)δn+1 −

(K (12) + K (2)

)δn︸ ︷︷ ︸

finite in d=4 and in Φn+2

]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 18 / 20

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Subtraction pattern at NNLO

dσNNLO

dX=∫

dΦn[

VVn︸︷︷︸singular in d=4 , finite in Φn

]δn

+∫

dΦn+1[ (

RVn+1)δn+1 −

(K (RV )

)δn︸ ︷︷ ︸

singular in d=4, finite in Φn+1

]

+∫

dΦn+2[RRn+2 δn+2 − K (1)δn+1 −

(K (12) + K (2)

)δn︸ ︷︷ ︸

finite in d=4 and in Φn+2

]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 18 / 20

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Subtraction pattern at NNLO

dσNNLO

dX=∫

dΦn[

VVn︸︷︷︸singular in d=4 , finite in Φn

]δn

+∫

dΦn+1[ (

RVn+1 + I(1))︸ ︷︷ ︸

finite in d=4, singular in Φn+1

δn+1 −(K (RV ) − I(12)

)︸ ︷︷ ︸finite in d=4, singular in Φn+1

δn

︸ ︷︷ ︸finite in d=4 and in Φn+1

]

+∫

dΦn+2[RRn+2 δn+2 − K (1)δn+1 −

(K (12) + K (2)

)δn︸ ︷︷ ︸

finite in d=4 and in Φn+2

]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 18 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Subtraction pattern at NNLO

dσNNLO

dX=∫

dΦn[VVn + I(2) + I(RV)︸ ︷︷ ︸finite in d=4 and in Φn

]δn

+∫

dΦn+1[ (

RVn+1 + I(1))︸ ︷︷ ︸

finite in d=4, singular in Φn+1

δn+1 −(K (RV ) − I(12)

)︸ ︷︷ ︸finite in d=4, singular in Φn+1

δn

︸ ︷︷ ︸finite in d=4 and in Φn+1

]

+∫

dΦn+2[RRn+2 δn+2 − K (1)δn+1 −

(K (12) + K (2)

)δn︸ ︷︷ ︸

finite in d=4 and in Φn+2

]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 18 / 20

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Check list

DoneProvide a complete and efficient algorithm to subtract the IR poles at NLO.Investigate the factorisation properties of a virtual amplitude to definecounterterms at NLO.Find a map to link the two approached at NLO.Generalised the main aspects of the Subtraction procedure and theFactorisation analysis at NNLO.

Factorisation and Subtraction beyond NLOMagnea, Maina, Pelliccioli, CS, Torrielli, Uccirati JHEP 1812(2018) 062, ArXiv 1809.05444[hep-ph].

Local analytic sector subtraction at NNLOMagnea, Maina, Pelliccioli, CS, Torrielli, Uccirati JHEP 1812(2018) 107, ArXiv 1806.09570 [hep-ph].

Factorisation and Local Subtraction of Infrared Divergences for QCD processesCS, to appear on JPCS.

Analytic tools for IR subtraction beyond NLOMagnea, Maina, Pelliccioli, CS, Torrielli, Uccirati PoS LL2018 (2018) 013.

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 19 / 20

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Wish list

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 20 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Backup

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 21 / 20

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Collinear splitting at NLO

Single-radiative jet function at the lowest perturbative order in the coupling constant∑s

Jq, 1 (k; l , p, n) =4παsCF

(l2)2(2π)dδd (l − p − k)

[−/l γµ/pγµ/l +

l2

k · n(/l /n/p + /p/n/l

)]=⇒ Sudakov parametrisation for momenta pµ and kµ

pµ = zlµ +O (l⊥) , kµ = (1− z)lµ +O (l⊥) , n2 = 0 .

=⇒ Leading behaviour in the l⊥ → 0 limit∑s

Jq, 1 (k; l , p, n) =8παsCF

l2(2π)d δd (l − p − k)

[1 + z2

1− z− � (1− z)

],

The leading order unpolarised DGLAP splitting function Pq→qg .

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 22 / 20

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Collinear counterterm

Final-state collinear radiation is modelled by a gauge-invariant extension of the jetfunction, the radiative jet function (RJF). For outgoing quark

ūs(p)Jq,m(k1 . . . km; p, n) ≡ 〈p, s; k1, λ1; . . . ; km, λm| ψ̄(0)Φn(0,∞) |0〉

≡ ūs(p)∑∞

p=0J (p)q,m(k1, . . . , km; p, n)

where Jq,0 is the virtual function, with J (0)q,0 = 1, and J(p)q,m proportional to g2p+ms .

At cross-section level in the unpolarised case

Jq,m(k1, . . . , km; p, n) ≡∑∞

p=oJ(p)q,m(k1, . . . , km; p, n)

≡∫

ddx eil·x∑

{λi}〈0|Φn(∞, x)φ(x) |p, s; kj , λj 〉 〈p, s; kj , λj | ψ̄(0)Φn(0,∞) |0〉

where lµ = pµi +∑m

i kµi fixes the total momentum flowing in the final state.

Completeness relation∞∑m=0

∫dΦm+1Jq,m(k; l , p, n) = Disc

[∫ddxeil·x 〈0|Φn(∞, x)ψ(x)ψ̄(0)Φn(0,∞) |0〉

]

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 23 / 20

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Practical implementation of the Subtraction methodIngredients of our method:

fundamental limits Si , Cij selecting only the leading behaviour in the specificconfiguration

Si O({kn}

)⇒ lim

kµi →0(ei→0)

O({kn}

) ∣∣∣∣leading terms

Cij O({kn}

)⇒ lim

kµ⊥→0(ωij→0)

O({kn}

) ∣∣∣∣leading terms

k⊥ki

kj

Singular structure of R under the fundamental limits [9908523]- universal kernel- Born matrix element

SiR({k}) = −N∑c,d

δfi gscd

sic sidBcd ({k}/i )

CijR({k}) = N1sij

Pµνij (sir , sjr )Bµν({k}/i /j , k)

SiCijR({k}) = 2N Cfj δfi gsjr

sij sidB({k}/i )

Born kinem.: mass-shell condition and momenta conservation just in the limits

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 24 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Practical implementation of the Subtraction methodIngredients of our method:

fundamental limits Si , Cij selecting only the leading behaviour in the specificconfiguration

Si O({kn}

)⇒ lim

kµi →0(ei→0)

O({kn}

) ∣∣∣∣leading terms

Cij O({kn}

)⇒ lim

kµ⊥→0(ωij→0)

O({kn}

) ∣∣∣∣leading terms

k⊥ki

kj

Singular structure of R under the fundamental limits [9908523]- universal kernel- Born matrix element

SiR({k}) = −N∑c,d

δfi gscd

sic sidBcd ({k}/i )

CijR({k}) = N1sij

Pµνij (sir , sjr )Bµν({k}/i /j , k)

SiCijR({k}) = 2N Cfj δfi gsjr

sij sidB({k}/i )

Born kinem.: mass-shell condition and momenta conservation just in the limits

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 24 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Practical implementation of the Subtraction methodIngredients of our method:

fundamental limits Si , Cij selecting only the leading behaviour in the specificconfiguration

Si O({kn}

)⇒ lim

kµi →0(ei→0)

O({kn}

) ∣∣∣∣leading terms

Cij O({kn}

)⇒ lim

kµ⊥→0(ωij→0)

O({kn}

) ∣∣∣∣leading terms

k⊥ki

kj

Singular structure of R under the fundamental limits [9908523]- universal kernel- Born matrix element

SiR({k}) = −N∑c,d

δfi gscd

sic sidBcd ({k}/i )

CijR({k}) = N1sij

Pµνij (sir , sjr )Bµν({k}/i /j , k)

SiCijR({k}) = 2N Cfj δfi gsjr

sij sidB({k}/i )

Born kinem.: mass-shell condition and momenta conservation just in the limits

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 24 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Practical implementation of the Subtraction methodIngredients of our method:

fundamental limits Si , Cij selecting only the leading behaviour in the specificconfiguration

Si O({kn}

)⇒ lim

kµi →0(ei→0)

O({kn}

) ∣∣∣∣leading terms

Cij O({kn}

)⇒ lim

kµ⊥→0(ωij→0)

O({kn}

) ∣∣∣∣leading terms

k⊥ki

kj

Singular structure of R under the fundamental limits [9908523]- universal kernel- Born matrix element

SiR({k}) = −N∑c,d

δfi gscd

sic sidBcd ({k}/i )

CijR({k}) = N1sij

Pµνij (sir , sjr )Bµν({k}/i /j , k)

SiCijR({k}) = 2N Cfj δfi gsjr

sij sidB({k}/i )

Born kinem.: mass-shell condition and momenta conservation just in the limits

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 24 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Practical implementation of the Subtraction method

partition of the phase space Φn+1 with sector functions Wij =∑

k,l 6=kσijσkl

,where σij = 1/ei ωij (based on FKS subtraction):

- minimum amount of singularities

SiWij =1/wij∑j′ 6=i 1/wij′

CijWij =ej

ei + ej

SiWab = 0 , ∀i 6= a CijWab = 0 , ∀a, b /∈ π(i , j)

- sum to unity∑i,j 6=i

Wij = 1 , Si∑j 6=i

Wij = 1 , Cij∑

a,b∈perm(ij)

Wab = 1

momentum mapping: {k1, . . . , kn+1} → {k̄1, . . . , k̄n} (based on CS subtraction):

- phase space factorisation dΦn+1 = dΦ̄n dΦ̄rad- n on-shell particles conserving momentum.

{k̄}(abc) ={{k}/a/b/c , k̄

(abc)b , k̄

(abc)c

}k̄(abc)b + k̄

(abc)c = ka + kb + kc

ka

kb

kc

k̄b

k̄c

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 25 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Practical implementation of the Subtraction method

partition of the phase space Φn+1 with sector functions Wij =∑

k,l 6=kσijσkl

,where σij = 1/ei ωij (based on FKS subtraction):

- minimum amount of singularities

SiWij =1/wij∑j′ 6=i 1/wij′

CijWij =ej

ei + ej

SiWab = 0 , ∀i 6= a CijWab = 0 , ∀a, b /∈ π(i , j)

- sum to unity∑i,j 6=i

Wij = 1 , Si∑j 6=i

Wij = 1 , Cij∑

a,b∈perm(ij)

Wab = 1

momentum mapping: {k1, . . . , kn+1} → {k̄1, . . . , k̄n} (based on CS subtraction):

- phase space factorisation dΦn+1 = dΦ̄n dΦ̄rad- n on-shell particles conserving momentum.

{k̄}(abc) ={{k}/a/b/c , k̄

(abc)b , k̄

(abc)c

}k̄(abc)b + k̄

(abc)c = ka + kb + kc

ka

kb

kc

k̄b

k̄c

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 25 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Practical implementation of the Subtraction method

partition of the phase space Φn+1 with sector functions Wij =∑

k,l 6=kσijσkl

,where σij = 1/ei ωij (based on FKS subtraction):

- minimum amount of singularities

SiWij =1/wij∑j′ 6=i 1/wij′

CijWij =ej

ei + ej

SiWab = 0 , ∀i 6= a CijWab = 0 , ∀a, b /∈ π(i , j)

- sum to unity∑i,j 6=i

Wij = 1 , Si∑j 6=i

Wij = 1 , Cij∑

a,b∈perm(ij)

Wab = 1

momentum mapping: {k1, . . . , kn+1} → {k̄1, . . . , k̄n} (based on CS subtraction):

- phase space factorisation dΦn+1 = dΦ̄n dΦ̄rad- n on-shell particles conserving momentum.

{k̄}(abc) ={{k}/a/b/c , k̄

(abc)b , k̄

(abc)c

}k̄(abc)b + k̄

(abc)c = ka + kb + kc

ka

kb

kc

k̄b

k̄c

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 25 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Practical implementation of the Subtraction method

partition of the phase space Φn+1 with sector functions Wij =∑

k,l 6=kσijσkl

,where σij = 1/ei ωij (based on FKS subtraction):

- minimum amount of singularities

SiWij =1/wij∑j′ 6=i 1/wij′

CijWij =ej

ei + ej

SiWab = 0 , ∀i 6= a CijWab = 0 , ∀a, b /∈ π(i , j)

- sum to unity∑i,j 6=i

Wij = 1 , Si∑j 6=i

Wij = 1 , Cij∑

a,b∈perm(ij)

Wab = 1

momentum mapping: {k1, . . . , kn+1} → {k̄1, . . . , k̄n} (based on CS subtraction):

- phase space factorisation dΦn+1 = dΦ̄n dΦ̄rad- n on-shell particles conserving momentum.

{k̄}(abc) ={{k}/a/b/c , k̄

(abc)b , k̄

(abc)c

}k̄(abc)b + k̄

(abc)c = ka + kb + kc

ka

kb

kc

k̄b

k̄c

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 25 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

Practical implementation of the Subtraction method

partition of the phase space Φn+1 with sector functions Wij =∑

k,l 6=kσijσkl

,where σij = 1/ei ωij (based on FKS subtraction):

- minimum amount of singularities

SiWij =1/wij∑j′ 6=i 1/wij′

CijWij =ej

ei + ej

SiWab = 0 , ∀i 6= a CijWab = 0 , ∀a, b /∈ π(i , j)

- sum to unity∑i,j 6=i

Wij = 1 , Si∑j 6=i

Wij = 1 , Cij∑

a,b∈perm(ij)

Wab = 1

momentum mapping: {k1, . . . , kn+1} → {k̄1, . . . , k̄n} (based on CS subtraction):

- phase space factorisation dΦn+1 = dΦ̄n dΦ̄rad- n on-shell particles conserving momentum.

{k̄}(abc) ={{k}/a/b/c , k̄

(abc)b , k̄

(abc)c

}k̄(abc)b + k̄

(abc)c = ka + kb + kc

ka

kb

kc

k̄b

k̄c

Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 25 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

IR divergences anatomyFramework: QCD’90s: experimental evidences pointed out the necessity to add some color in the theory.

Ω−

s = 3/23 strange quarkswave function symmetric inspace, space and flavour.

Spin-Statistic Theorem: a newquantum number is needed, the color.

Number of colors verified to be 3:

R =σ(e+e− → h)

σ(e+e− → µ+µ−)∗= 3

[2(23

)2+ 3(−13

)2 ]=

113

*Ecm ≥ 10GeV [L. Montanet et al. Phys. Rev. D50, 1173 (1994)]Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 26 / 20

Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup

enviroment

The new theory of colored particles describes the strong interactions between quarksand gluons. It is modelled by

Lqcd = ψ̄i[i /D(A)ij −mδij

]ψj −

14(F aµν)2−λ

2(∂ · Aa)2 + ∂µη̄a

[δab ∂

µ + gs fabc Aµc]ηb

(Dµ(A))ij≡δij ∂µ+ig(Aµ)ij

F aµν =∂µAaν−∂νA

aµ−gs f

abc A

bµA

cν

The QCD coupling constant runs with the energy scale: two opposite regimes

αs(µ) =α(µ0)

1− β02π α(µ0) logµµ0

µ→∞β0