Subtraction of Infrared Singularities in Perturbative QCD...
Transcript of Subtraction of Infrared Singularities in Perturbative QCD...
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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]
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Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup
Environment
Natural environment:
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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
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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
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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
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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 �.
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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
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Subtraction Pattern
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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]
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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
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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]
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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, . . .
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Counterterms construction at NLO
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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
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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
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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
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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
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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.
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Subtraction at NNLO
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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)
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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
]
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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
]
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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
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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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Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup
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.
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Wish list
Chiara Signorile-Signorile Subtraction of Infrared Singularities in Perturbative QCD: a new scheme 20 / 20
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Introduction Subtraction NLO Factorisation NLO NNLO Conclusions Backup
Backup
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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 .
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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〉
]
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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
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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
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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
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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