collinear limit of qcd amplitudeslldesy/talks/rodrigo.pdf · germÆn rodrigo, loops and legs 2004...

Collinear limit of QCDamplitudes

Germán Rodrigo

IFIC Valencia

[email protected]

Loops and Legs 2004, Zinnowitz, April 2004

* S. Catani, D. de Florian, GR, Phys. Lett. B586 (2004) 323

S. Catani, D. de Florian, GR, W. Vogelsang, hep-ph/0404xxx

Outline

Motivation

Multiple collinear limit

Factorization in colour-space:

splitting matrix

Examples

Conclusions and outlook

Germán Rodrigo, Loops and Legs 2004 Collinear limit of QCD amplitudes, 1

Collinear limits in QCD

high precision experiments (LEP, HERA,

Tevatron, LHC, LC) demands QCD predictions

beyond NLO

understanding better IR singular behaviour of

multiparton QCD amplitudes



1. evaluate IR finite cross-sections ->

substraction terms




substraction terms

2. IR properties of amplitudes exploited to

compute logarithmic enhanced perturbative terms ->

resummations




substraction terms



resummations

3. improve physics content of

Monte Carlo event generators




substraction terms



resummations

3. improve physics content of

Monte Carlo event generators

4. beyond QCD: hints on the structure of highly

symmetric gauge theories (e.g. N=4 QCD)


PDF and fragmentation func.NNLO anomalous dimensions viaMellin Moments [Moch, Vermaseren, Vogt]

Alternative method of calculating AP kernelbased on factorization of splitting amplitudes[Kosower, Uwer]

NLO evolution requires one-loop 1 → 2,and tree 1 → 3 splitting amplitudes

NNLO evolution requires two-loop 1 → 2,one-loop 1 → 3, and tree 1 → 4

N3LO evolution, up to 1 → 5


Multiple collinear limit

momenta p1, . . . , pm of m partons become parallel

subenergies sij = (pi + pj)2

of the same order, and vanish simultaneously

(p1 + . . . + pm)ν = P̃ ν +s1...m nν

2 n · P̃

back-to-back light-like momenta (P̃ 2 = 0, n2 = 0)

P̃ ν: collinear direction

zi = n · pi

n · P̃: longitudinal momentum fraction,

∑mi=1 zi = 1


Singular behaviourMatrix element

M = (gS)q[M(0) +

αS

2πM(1) +O(α2

S)]

at tree-level (s = sij, sijk, . . .)

M(0)(p1, . . . , pm, pm+1, . . . ) ∼(

1√s

)m−1

at one-loop (scaling violation)

M(1)(p1, . . . , pm, pm+1, . . . ) ∼(

1√s

)m−1(s

µ2

)−ε


singular behaviour captured by

universal (process-independent)factorization properties (splitting amplitudes)

[Catani, Grazzini, Bern, Dixon, Kosower, Glover, Campbell, Del Duca . . .]

factorization directly in colour space

[Catani, de Florian, GR]


Factorization formulaColour-space factorization, multiple collinear limit:

external legs on-shell with physical polarizations

|M(0)a1,...,am,am+1,...(p1, . . . , pm, pm+1, . . . )〉 '

Sp(0)a1...am

(p1, . . . , pm) |M(0)a,am+1,...(P̃ , pm+1, . . . )〉


at One-loop

|M(1)a1,...,am,am+1,...(p1, . . . , pm, pm+1 . . . )〉

' Sp(1)a1...am

(p1, . . . , pm) |M(0)a,am+1...(P̃ , pm+1, . . . )〉

+Sp(0)a1...am

(p1, . . . , pm) |M(1)a,am+1,...(P̃ , pm+1, . . . )〉


Splitting matrixMatrix in colour+spin space

Sp(c1,...,cm;s1,...,sm)(ca,sa)a1...am

=(〈c1, . . . , cm| ⊗ 〈s1, . . . , sm|

)Spa1...am

(|ca〉 ⊗ |sa〉

)Colour conservation

m∑i=1

T i Spa1...am= Spa1...am

T a

colour charge: Tij = taij(−taji) for a quark (antiquark),

and Tbc = ifabc for a gluon


Example ITree-level splitting matrix g → q1 + q̄2

Sp(0) (β1,β2)(c)q1q̄2 (p1, p2) = µε tcβ1β2

1s12

u(p1) /ε∗(P̃ ) v(p2)

ε physical polarization vector of the parent gluon

Splitting amplitude

• Split(0)q1q̄2 by simply removing colour factor tcβ1β2

• m ≥ 3: splitting matrices and splitting amplitudes not

simply proportional, but related by gauge invariance and

colour algebraGermán Rodrigo, Loops and Legs 2004 Collinear limit of QCD amplitudes, 11

Generalized splitting functionSquare of the splitting matrix Spa1...am

, summed over

final-state colours and spins and averaged over colours

and spins of the parent parton, defines the m-parton

splitting function

〈P̂ (0)a1···am

〉 =(

s1...m

2 µ2ε

)m−1

|Sp(0)a1...am

|2

〈P̂ (1)a1···am

〉 =(

s1...m

2 µ2ε

)m−1 [(Sp(0))†Sp(1) + (Sp(1))†Sp(0)

]generalization of the customary (i.e. with m = 2)Altarelli-Parisi splitting functions


Divergent structureOne-loop splitting matrix

Sp(1) = Sp(1) div. + Sp(1) fin.

where (unrenormalized)

Sp(1) div.(p1, . . . , pm) =Sε

2

1ε2

m∑i,j=1(i 6=j)

T i · T j

(−sij − i0

µ2

)−ε

+(−s1...m − i0

µ2

)−ε 1

ε2

m∑i,j=1

T i · T j

(2− (zi)

−ε − (zj)−ε)

−1ε

(m∑

i=1

(γi − εγ̃R.S.

i

)−(γa − εγ̃R.S.

a

)− m− 1

2(β0 − εβ̃R.S.

0 )

)]}×Sp(0)(p1, . . . , pm)


Divergent structureOne-loop splitting matrix

Sp(1) = Sp(1) div. + Sp(1) fin.

where (renormalized)

Sp(1) div.(p1, . . . , pm) =Sε

2

1ε2

m∑i,j=1(i 6=j)

T i · T j

(−sij − i0

µ2

)−ε

+(−s1...m − i0

µ2

)−ε 1

ε2

m∑i,j=1

T i · T j

(2− (zi)

−ε − (zj)−ε)

−1ε

(m∑

i=1

(γi − εγ̃R.S.

i

)−(γa − εγ̃R.S.

a

))]}×Sp(0)(p1, . . . , pm)


Scheme dependence

Flavour coefficients [Catani, Kunszt]

γq = γq̄ γg = β0/23CF/2 (11CA − 2Nf)/6

and

γ̃q = γ̃q̄ γ̃g = β̃0

C.D.R. 0 0D.R. CF/2 CA/6


Divergent structure

✔ Tested with all 1 → 2 processes

beyond 1 → 2 ?


Example II

Tree-level splitting matrix q → q1 + Q̄2 + Q3

Sp(0) (β1,β2,β3)(β)q1Q̄2Q3

(p1, p2, p3) = µ2ε∑

c

tcβ3β2tcβ1β

× 1s123s23

u(p3) γµ v(p2) u(p1) γν u(P̃ ) dµν(p2 + p3, n)

where

dµν(k, n) = −gµν +kµnν + nµkν

n · kphysical polarization tensor of the gluon with momentum

k, and n2 = 0. Physical gauge: only diagrams where the

parent parton emitted and absorbed collinear radiation


Example II

Splitting function [Campell-Glover, Catani-Grazzini]

〈P̂ (0)q1Q̄2Q3

〉 =12CFTR

s123

s23

[−

t223,1

s23s123+

4z1 + (z2 − z3)2

z2 + z3

+(1− 2εδR.S.)(

z2 + z3 −s23

s123

)]where

tij,k ≡ 2zi sjk − zj sik

zi + zj+

zi − zj

zi + zjsij

The parameter δR.S. depends on the RS: δC.D.R. = 1 and

δD.R. = 0.


Example II: One-loop

One-loop splitting matrix

Sp(1) (β1,β2,β3)(β)

q1Q̄2Q3(p1, p2, p3) = µ4ε 8π2

s123

{∑c

tcβ3β2tcβ1β S(p1, p2, p3)

+∑b,c

(tbtc + tctb)β3β2(tctb)β1β A(p1, p2, p3)

First term: same structure as tree-level

Second term: new colour structure, new one-loop

(quantum) effect


A(p1, p2, p3) =

−12

i

∫ddq

(2π)du(p3)

[γσ(q/+ p/2)γµ

(s2q + i0)− γµ(q/+ p/3)γσ

(s3q + i0)

]v(p2)

×dµν(q, n) dσρ(q + p2 + p3, n)u(p1) γν(p/1 − q/)γρ u(P̃ )

(q2 + i0)(t1q + i0)(s23q + i0)

where

t1q = (p1− q)2 , siq = (pi + q)2 , s23q = (p2 +p3 + q)2


New loop integrals

Box scalar one-loop integrals

D0 = −i

∫ddq

(2π)d

1(q2 + i0)(t1q + i0)(s2q + i0)(s23q + i0)

but also pentagon-like, with gauge propagators

D0,n = −i

∫ddq

(2π)d

1(q2 + i0)(t1q + i0)(s2q + i0)(s23q + i0)

× 1n · (q + p23)

reduction to box-like


• Basic one-loop integrals to high orders in ε

• Explicit expressions for A(p1, p2, p3)and 〈P̂ (1)

q1Q̄2Q3〉




q1Q̄2Q3〉

? Divergent part ✔




q1Q̄2Q3〉


? RS dependence embodied in 〈P̂ (0)q1Q̄2Q3

〉




q1Q̄2Q3〉



〉? No subdivergences in any subregion of

phase-space




q1Q̄2Q3〉



〉? No subdivergences in any subregion of

phase-space

? Checked with triple collinear limit of

one-loop helicity ME for e+e− → q̄qQ̄Q

from [Bern-Dixon-Kosower,Glover-Miller] (only D.R.)


Asymmetries in the nucleon sea[Catani, de Florian, GR, Vogelsang, hep-ph/0404xxx]

Flavour NS, diagonalized evolution eqs. (Mellin space)

f (V ) ≡Nf∑i=1

(fqi− fq̄i

) , f (±)qi

≡ fqi± fq̄i

− 1Nf

Nf∑j=1

(fqj

± fq̄j

)

P (V ) = PVqq − PV

qq̄ + Nf

(PS

qq − PSqq̄

), P (±) = PV

qq ± PVqq̄

evolution operator

U (A)(Q,Q0) = exp

{∫ Q2

Q20

dq2

q2P (A)(αS(q2))

}, A = V,±


Asymmetries in the nucleon sea

(q − q̄) (Q2) = U (−)(Q,Q0)

×

[(q − q̄) (Q2

0) +1

Nf

(U (V )(Q,Q0)U (−)(Q,Q0)

− 1

)f (V )(Q2

0)

]

At LO and NLO: U (V ) = U (−), asymetries produced only

if asymetry at Q0. At NNLO

U (V )(Q,Q0)U (−)(Q,Q0)

−1 = −P(2)Sns

8πb0

[(αS(Q2)

4π

)2

−(

αS(Q20)

4π

)2]+O(N3LO)

where P(2)Sns ∝ dabcd

abc


strange asymmetry

Total strangeness

∫ 1

0

dx[s(x)− s̄(x)] = 0

but s(x) 6= s̄(x) in

general

second moment

〈x(s − s̄)〉 ≈ −5 × 10−4

(Q2 = 20GeV2)


heavy flavours

assumming

charm and bottom

asymmetries vanish

at threshold



❦ Factorization directly in colour-space:

splitting matrix




splitting matrix

❦ General structure of infrared and

ultraviolet divergences of one-loop splitting

matrices




splitting matrix

❦ General structure of infrared and

ultraviolet divergences of one-loop splitting

matrices

❦ Method and tools (one-loop integrals)

sufficient to evaluate any one-loop splitting

matrix of any splitting process a →a1 + a2 + a3


collinear limit of qcd amplitudeslldesy/talks/rodrigo.pdf · germÆn rodrigo, loops and legs 2004...

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