Kinematical Constraints on QCD Factorization in the Drell-Yan Process

Abstract

We study factorization schemes for parton shower models in hadron-hadron collisions. As an example, we calculate lepton pair production mediated by a virtual photon in quark--anti-quark annihilation, and we compare factorized cross sections obtained in the conventional $\bar{\rm MS}$ scheme with those obtained in a factorization scheme in which a kinematical constraint due to parton radiation is taken into account. We discuss some properties of factorized cross sections.

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arXiv:0708.0105v2 [hep-ph] 31 Oct 2007
1
RUP-07-2
October, 2007
Kinematical Constraints on QCD Factorization
in the Drell-Yan Process
Hidekazu Tanaka, Yu Matsuda and Hirokazu Kobayashi
Department of Physics, Rikkyo University,
Nishi-ikebukuro, Toshima-ku Tokyo, Japan, 171
ABSTRACT
We study factorization schemes for parton shower models in hadron-hadron colli-
sions. As an example, we calculate lepton pair production mediated by a virtual pho-
ton in quark–anti-quark annihilation, and we compare factorized cross sections ob-
tained in the conventional MS scheme with those obtained in a factorization scheme
in which a kinematical constraint due to parton radiation is taken into account. We
discuss some properties of factorized cross sections.
§1.Introduction
Hadron-hadron scattering processes are useful for investigating various prop-
erties of high-energy reactions, as well as to search for exotic particles. However,
the leading-logarithmic (LL) order of quantum chromodynamics (QCD) is insuffi-
cient to evaluate these processes. Thus the next-to-leading logarithmic (NLL) order
contributions should be taken into account.
In the evaluation of hadron scattering cross sections, logarithmic contributions
due to collinear parton production are subtracted from a hard scattering cross sec-
tion, and they are absorbed into the parton distributions of hadrons. The next-to-
leading-order (NLO) calculation is necessary in order to remove theoretical ambi-
guities due to the factorization procedure as well as the choice of the factorization
scale.
In actual calculations, Monte Carlo methods are powerful tools for the evalua-
tion of exclusive processes. The initial state parton radiation can be generated by
using the parton shower models based on perturbative QCD. One such algorithm is

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proposed in Ref.1), in which the parton showers are generated according to an algo-
rithm consisting of a model based on the evolution of momentum distributions with
the modified minimal subtraction (MS) scheme.2)In this model, the scaling viola-
tion of the parton distributions is generated with only information from the splitting
functions of the parton branching vertices and input distributions at a given low
energy. It has been found that this method reproduces the scaling violation of the
flavor singlet parton distributions, up to their normalizations, to an accuracy of the
NLL order of QCD. Furthermore, exact NLL-order vertex functions for the decay
branching processes are taken into account in this model.3)
In this paper, we study factorization schemes for the parton shower models in
hadron-hadron collisions, in which the collinear singularities are subtracted using the
MS scheme. In order to implement factorization schemes that are appropriate for
the Monte Carlo methods using the parton shower models to an accuracy of the NLL
order of QCD, we take into account a kinematical constraint due to virtual partons.
The momenta of partons in the scattering processes are conserved in this method.
At NLO accuracy, the processes q¯ q → γ∗g, qg → γ∗q and ¯ qg → γ∗¯ q contribute
to the hard scattering cross section in actual physical systems. As an example, we
calculate factorized cross sections for the process q¯ q → γ∗g using our method. Other
processes can be calculated with similar method presented in this paper.
In §2, we summarize the calculation of the cross section of the q¯ q → γ∗g process
using the dimensional regularization method. Factorization schemes for the collinear
singularity are explained in §3. Some properties of the factorized cross sections are
presented in §4. Section 5 contains a summary and some comments.
§2.Cross section of the Drell-Yan process
In this section, we summarize the calculation of the cross section of the Drell-Yan
lepton-pair production in quark (q)–anti-quark (¯ q) annihilation,
q(pq) + ¯ q(p¯ q) → γ∗(q) + g(pg) → l−(p−) + l+(p+) + g(pg),
mediated by a photon γ∗with virtuality (p−+p+)2= q2= Q2, where a gluon (g) is
radiated in the final state. Here, pi(i = q, ¯ q,g) and p±denote the momenta of the
corresponding particles.
Ignoring the particle masses, the cross section of the Born term,
(2.1)
q(pq) + ¯ q(p¯ q) → γ∗(q) → l−(p−) + l+(p+),(2.2)
is given by
dˆ σ(B)
q¯ q
dQ2= ˆ σ0(Q2,ǫ)δ(ˆ s − Q2),(2.3)
with
ˆ σ0(Q2,ǫ) =
πα
NCQ2ˆ e2
q(2 − 2ǫ)
2α
3 − 2ǫ
Γ(2 − ǫ)
Γ(2 − 2ǫ)
?
Q2
4πµ2
?−ǫ
,(2.4)

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in 4 − 2ǫ dimensions. Here NC= 3 and the electric coupling constant of the quark
is defined by ˆ e2
The differential cross section for real gluon radiation is given by
qαµ2ǫ, with the mass parameter µ and the dimensionless coupling α.
dˆ σ(r)
dˆ τd(−ˆt)=
q¯ q
αs
2πˆ sˆ σ0(Q2,ǫ)
?(−ˆt)(−ˆ u)
4πˆ sµ2
?−ǫ
1
Γ(1 − ǫ)
˜K(r)
q¯ q(ˆ τ,−ˆt,ǫ),(2.5)
with
˜K(r)
q¯ q(ˆ τ,−ˆt,ǫ) = CF
?
ˆ s
?1
−ˆt+
1
−ˆ u
?
1
CF
ˆPqq(ˆ τ,ǫ) − 2
?
,(2.6)
where ˆ τ = Q2/ˆ s and
ˆPqq(ˆ τ,ǫ) =ˆP(0)
qq(ˆ τ) + ǫˆP′
qq(ˆ τ),(2.7)
with
ˆP(0)
qq(ˆ τ) = CF1 + ˆ τ2
1 − ˆ τ,
ˆP′
qq(ˆ τ) = −CF(1 − ˆ τ).(2.8)
Here, CF = 4/3 and αsare the color factor and the strong coupling constant, re-
spectively. The Mandelstam variables are defined by
ˆ s = (pq+ p¯ q)2, ˆt = (pq− pg)2, ˆ u = (p¯ q− pg)2,
which satisfy ˆ s +ˆt + ˆ u = Q2.
Integrating over the range 0 ≤ −ˆt ≤ ˆ s(1 − ˆ τ) and adding virtual loop contribu-
tions, we obtain the cross section4)as
(2.9)
dˆ σq¯ q
dˆ τ
=αs
2πˆ σ0(Q2,ǫ)Γ(1 − ǫ)
Γ(1 − 2ǫ)
?
ˆ s
4πµ2
?−ǫ
Kq¯ q(ˆ τ,ǫ), (2.10)
with
Kq¯ q(ˆ τ,ǫ) = CF
?2
−8 +2
−ǫ
1
CF
?ˆP(0)
?
qq(ˆ τ)
?
++ 4(1 + ˆ τ2)
?log(1 − ˆ τ)
1 − ˆ τ
?
+
+
?
3π2
δ(1 − ˆ τ)
?
.(2.11)
Here, we have
?ˆf(ˆ τ)
?
+=ˆf(ˆ τ) − δ(1 − ˆ τ)
?1
0
dyˆf(y)(2.12)
for a functionˆf(ˆ τ) unregulated at ˆ τ = 1.

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§3.Factorization schemes
In actual Monte-Carlo simulations with the parton shower models, the virtu-
alities and longitudinal momentum fractions of the partons in the initial state are
generated according to non-branching probabilities (form factors) and splitting func-
tions of parton branching processes, respectively.1)The four-momenta of the partons
are constructed from these values.
In order to obtain a finite cross section for the process q¯ q → γ∗g, we subtract the
collinear contributions corresponding to the branching processes q → qg and ¯ q → ¯ qg
from the hard scattering cross section.
The momentum of the quark rq, defined by rq= pq−pgfor the branching process
q(pq) → q(rq) + g(pg),
is described by using the momentum fraction zq, the virtuality r2
momentum rqT, with pq· rqT= p¯ q· rqT= 0, as
rq= pq− pg= zqpq+ αqp¯ q+ rqT,
where αq= r2
because the relations −p2
The virtuality of a photon Q2is expressed as
(3.1)
q, and the transverse
(3.2)
q/ˆ s for on-shell gluon radiation (p2
q,−p2
g= 0). Here, we set p2
qare expected in parton shower generation.
q= p2
¯ q= 0,
¯ q≪ −r2
Q2= (rq+ p¯ q)2,(3.3)
to O(αs) accuracy of QCD. Using Eqs. (3·2) and (3·3), the quantity ˆ τ is given by
ˆ τ = zq+ αq.(3.4)
According to Eq. (3·4), the subtraction term divided by ˆ σ0(Q2,ǫ) is defined by
dS[F]
qq
dˆ τd(−r2
with
q)=
?1
0
dzq
d˜S[F]
dzqd(−r2
qq
q)δ(zq− ˆ τ − (−r2
q)/ˆ s),(3.5)
d˜S[F]
dzqd(−r2
qq
q)=αs
2π
1
Γ(1 − ǫ)
?
−r2
4πµ2
q
?−ǫP[F]
qq (zq,ǫ)
−r2
q
,(3.6)
where the splitting function P[F]
The conventional MS subtraction scheme5)corresponds to the approximation
δ(zq− ˆ τ − (−r2
contribution is subtracted from the hard scattering cross section in the range 0 ≤
−ˆt ≤ M2, with ˆ τ = zq and −ˆt = −r2
to a factorization scale for the separation between the initial state radiation and the
hard scattering process. In the case of the conventional MS scheme, M2is larger
than the kinematical boundary for −ˆt, given by ˆ s(1 − ˆ τ) in 1 − ˆ τM≤ ˆ τ ≤ 1, where
qq (zq,ǫ) depends on the factorization scheme F.
q)/ˆ s) ≃ δ(zq− ˆ τ) in Eq. (3·5). In this approximation, the collinear
q. Here, the mass parameter M corresponds

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we define ˆ τM≡ M2/ˆ s. Taking the condition zq= ˆ τ + (−r2
collinear contributions are subtracted within the phase space for the hard scattering
process, namely 0 ≤ −ˆt ≤ ˆ s(1 − ˆ τ).
We define the integrated contribution as
q)/ˆ s ≤ 1 into account, the
?ˆ sw[F](I)
0
d(−r2
q)
dS[F]
dˆ τd(−r2
qq
q)=αs
2π
1
Γ(1 − ǫ)
?
ˆ s
4πµ2
?−ǫ
˜F[F](I)
qq
(ǫ, ˆ τ), (3.7)
where I denotes the region of phase space for the hard scattering process to be
considered. Here, ˆ sw[F](I)is a boundary of the −r2
splitting function with the MS scheme is defined by
qintegration. For example, the
P[MS]
qq
(zq,ǫ) =
?ˆP(0)
qq(zq)
?
+, (3.8)
for the branching process presented in Eq. (3·1).
With zq = ˆ τ(F = MS) and w[F](S)= ˆ τM, the subtraction term for collinear
gluon radiation is given by
˜F[MS](S)
qq
(ǫ, ˆ τ) =
?1
−ǫ+ log ˆ τM
?
P[MS]
qq
(ˆ τ,ǫ).(3.9)
Similarly, with zq= ˆ τ + (−r2
˜F[MS′](S)
qq
q)/ˆ s (F = MS′) and w[MS′](S)= 1 − ˆ τ, we obtain
?1
+3
2(1 − ˆ τ)+
which is valid for 1 − ˆ τM≤ ˆ τ ≤ 1 (I = S).
The calculational method employed in the soft gluon region is explained in Ap-
pendix A. By replacing q with ¯ q in the above equations, we obtain the subtraction
terms for the anti-quark legs, which are the same as those for the quark legs.
(ǫ, ˆ τ) = CF
−ǫ
1
CF
1
?ˆP(0)
− (1 − ˆ τ) −1
qq(ˆ τ)
?
++ (3 + ˆ τ2)
?log(1 − ˆ τ)
?
1 − ˆ τ
,
?
+
3π2δ(1 − ˆ τ) (3.10)
§4.Factorized cross sections
The factorized cross section in the factorization scheme F is defined by
dˆ σ[F](S)
q¯ q
dˆ τ
=αs
2πˆ σ0(Q2,0)K[F](S)
q¯ q
(ˆ τ), (4.1)
with
K[F](S)
q¯ q
(ˆ τ) = Kq¯ q(ˆ τ,ǫ) −˜F[F](S)
qq
(ǫ, ˆ τ) −˜F[F](S)
¯ q¯ q
(ǫ, ˆ τ),(4.2)
for 1 − ˆ τM≤ ˆ τ ≤ 1. Here, we obtain
K[MS](S)
q¯ q
(ˆ τ) = CF
?
4(1 + ˆ τ2)
?log(1 − ˆ τ)
?
1 − ˆ τ
?
−8 +2
+
−
2
CF
?ˆP(0)
qq(ˆ τ)
+log ˆ τM+
?
3π2
?
δ(1 − ˆ τ)
?
(4.3)