Antenna subtraction at NNLO with hadronic initial states: double real initial-initial configurations

Abstract

The antenna subtraction method handles real radiation contributions in higher
order corrections to jet observables. The method is based on antenna functions,
which encapsulate all unresolved radiation between a pair of hard radiator
partons. To apply this method to compute hadron collider observables,
initial-initial antenna functions with both radiators in the initial state are
required in unintegrated and integrated forms. In view of extending the antenna
subtraction method to next-to-next-to-leading order (NNLO) calculations at
hadron colliders, we derive the full set of initial-initial double real
radiation antenna functions in integrated form.

Full text

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arXiv:1107.4037v1 [hep-ph] 20 Jul 2011
Preprint typeset in JHEP style - PAPER VERSION
ZU-TH 15/11, NSF-KITP-11-126
Antenna subtraction at NNLO with hadronic initial
states: real-virtual initial-initial configurations
Thomas Gehrmanna,b, Pier Francesco Monnia
aInstitut f¨ ur Theoretische Physik, Universit¨ at Z¨ urich, Wintherturerstrasse 190,
CH-8057 Z¨ urich, Switzerland
bKavli Institute for Theoretical Physics, University of California,
Santa Barbara, CA 93106, USA
Abstract: The antenna subtraction method handles real radiation contributions in higher
order corrections to jet observables. The method is based on antenna functions, which
encapsulate all unresolved radiation between a pair of hard radiator partons. To apply this
method to compute hadron collider observables, initial-initial antenna functions with both
radiators in the initial state are required. In view of extending the antenna subtraction
method to next-to-next-to-leading order (NNLO) calculations at hadron colliders, we derive
the one-loop initial-initial antenna functions in unintegrated and integrated form.
Keywords: QCD, Jets, Collider Physics, NLO and NNLO Calculations.

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Contents
1.Introduction1
2. Initial-intial antenna subtraction at NNLO3
3.Integration of one-loop antenna functions6
4.Conclusions10
A. Master integrals in initital-initial configuration11
B. Example of an integrated antenna function:ˆD1
qg,gantenna 18
1. Introduction
Jet production observables are studied extensively at hadron colliders. Since the distribu-
tion of final state jets relates directly to the parton-level dynamics, jet observables can be
used for precision studies of QCD [1], especially in view of determinations of the strong
coupling constant and the parton distribution functions in the proton. Experimental mea-
surements of these observables at the Tevatron [2,3] attained an accuracy of a few per cent
(or even better in certain kinematical ranges), and first results from the LHC [4,5] already
show the potential for precision jet physics. Consequently, meaningful precision studies
must rely on theoretical predictions accurate to the same level. In perturbative QCD, this
precision usually requires corrections at next-to-next-to-leading order (NNLO).
NNLO calculations of observables with n jets in the final state require several ingre-
dients: the two-loop corrected n-parton matrix elements, the one-loop corrected (n + 1)-
parton matrix elements, and the tree-level (n + 2)-parton matrix elements.
massless jet observables of phenomenological interest, these matrix elements are available
for some time already.
The (n+1)-parton and (n+2)-parton matrix elements contribute to n jet observables
at NNLO if the extra partons are unresolved or are clustered to form an n-jet final state.
Consequently, these extra partons are unconstrained in the soft and collinear regions, and
yield infrared divergences. In these cases, the infrared singular parts of the matrix elements
need to be extracted and integrated over the phase space appropriate to the unresolved
configuration to make the infrared pole structure explicit. The single soft and collinear
limits of one-loop matrix elements [6–10] and the double unresolved limits of tree-level
matrix elements [11–14] are process-independent, and result in a factorization into an
unresolved factor times a matrix element of lower multiplicity.
For most
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To determine the contribution to NNLO jet observables from these configurations,
one has to find subtraction terms which coincide with the full matrix element and are
still sufficiently simple to be integrated analytically in order to cancel their infrared pole
structure with the two-loop virtual contribution. Often starting from systematic methods
for subtraction at NLO [15–18], several NNLO subtraction methods have been proposed
in the literature [19–26], and are worked out to a varying level of sophistication.
For observables with partons only in the final state, an NNLO subtraction formalism,
antenna subtraction, has been derived in [27]. The antenna subtraction formalism con-
structs the subtraction terms from antenna functions. Each antenna function encapsulates
all singular limits due to the emission of one or two unresolved partons between two colour-
connected hard radiator partons. This construction exploits the universal factorization of
matrix elements and phase space in all unresolved limits. The antenna functions are de-
rived systematically from physical matrix elements [28]. This formalism has been applied
in the derivation of NNLO corrections to three-jet production in electron-positron annihi-
lation [29,30] and related event shapes [31,32], which were used subsequently in precision
determinations of the strong coupling constant [33–38]. The formalism can be extended
to include parton showers at higher orders [39], thereby offering a process-independent
matching of fixed-order calculations and logarithmic resummations [34,35,40,41], which is
done on a case-by-case basis for individual observables [42] up to now. The formalism can
be extended to include massive fermions [43].
For processes with initial-state partons, antenna subtraction has been fully worked out
only to NLO so far [44]. In this case, one encounters two new types of antenna functions,
initial-final antenna functions with one radiator parton in the initial state, and initial-initial
antenna functions with both radiator partons in the initial state. The framework for the
construction of NNLO antenna subtraction terms involving one or two partons in the initial
state has been set up in [45] in the context of a proof-of-principle implementation of the
contribution of the gg → 4g tree-level subprocess to di-jet production at hadron colliders.
The initial-final and initial-initial antenna functions appearing in the NNLO subtraction
terms are obtained from crossing the final-final antennae. Their integration has to be
performed over the appropriate phase space. In the case of the initial-final antennae, this
has been accomplished in [46]. For the initial-initial tree-level double real radiation antenna
functions, partial results have been obtained in [47]. It is the aim of the present paper to
derive the setup for NNLO antenna subtraction for single unresolved singularities at one-
loop and to compute the integrated one-loop initial-initial antenna functions required in
this context.
Other approaches to perform NNLO calculations of exclusive observables with initial
state partons are the use of sector decomposition and a subtraction method based on the
transverse momentum structure of the final state. The sector decomposition algorithm [48]
analytically decomposes both phase space and loop integrals into their Laurent expansion
in dimensional regularization, and performs a subsequent numerical computation of the
coefficients of this expansion. Using this formalism, NNLO results were obtained for Higgs
production [49] and vector boson production [50] at hadron colliders. Both reactions were
equally computed independently [51] using an NNLO subtraction formalism exploiting the
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specific transverse momentum structure of these observables [24], which was also applied
most recently to compute NNLO corrections to associated WH production [52].
This paper is structured as follows: in Section 2, we construct the subtraction terms
required at NNLO for initial-initial configurations with one unresolved parton at one loop.
They require one-loop 2 → 2 antenna functions with two partons in the initial state and
one parton and one off-shell neutral current in the final state. The analytic integration of
the initial-initial one-loop antenna functions is described in Section 3. Finally, we conclude
with an outlook in Section 4.
2. Initial-intial antenna subtraction at NNLO
Antenna subtraction of initial-initial configurations at NLO is derived in detail in [44].
Subtraction terms with two hard partons in the initial state are built along the same lines
as in the final-final and initial-final case. The NLO antenna subtraction term for an m-jet
production process, to be convoluted with the appropriate parton distribution functions
for the initial state partons, for a configuration with the two hard emitters in the initial
state (partons i and k with momenta p1and p2), reads:
dˆ σS,(II)= N
?
X0
m+1
dΦm+1(k1,...,kj−1,kj,kj+1,...,km+1;p1,p2)
1
Sm+1
?
×J(m)
j
ik,j(p1,p2,kj)
???Mm(˜k1,...,˜kj−1,˜kj+1,...,˜km+1;x1p1,x2p2)
???
2
m (˜k1,...,˜kj−1,˜kj+1,...,˜km+1). (2.1)
All the momenta in the arguments of the reduced matrix elements and the jet functions
are redefined, which is a consequence of requiring the correct collinear factorization prop-
erties in both initial-state collinear limits. Its should be noted that the jet function J(m)
(constructing m jets from m partons) requires all redefined momenta to be resolved. The
two hard radiators are simply rescaled by factors x1and x2respectively. The spectator
momenta are boosted by a Lorentz transformation onto the new set of momenta {˜kl, l ?= j}.
The mapping must be based on a factorization of the (m + 1)-particle phase space, must
satisfy overall momentum conservation and keep the mapped momenta on the mass shell.
In this case, this turns out to severely restrict the possible mappings.
The tree-level antenna function X0
ik,jdepends only on the incoming momenta p1,p2
and on the outgoing momentum kj. It accounts for all singular configurations where parton
j is unresolved and colour-connected to partons i (incoming with momentum p1) and k
(incoming with momentum p2). The jet function J(m)
(2.1) depend only on the redefined momenta. With a suitable factorization of the phase
space [44], one can perform the integration of the antenna function analytically.
The factorization of the phase space is obtained by requiring that the two mapped
initial state momenta should be of the form
m
m
and the reduced matrix element in
P1= x1p1
P2= x2p2,(2.2)
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so that
˜ q ≡ P1+ P2
is in the beam axis. Since the vector component of q ≡ p1+p2−kjis in general not along
the p1−p2axis we need to boost all the momenta so that ˜ q = Λq and˜kl= Λklin order to
restore momentum conservation. By requiring this boost to be only transverse, the phase
space mapping is determined uniquely, resulting in the factorization
dΦm+1(k1,...,km+1;p1,p2) = dΦm(˜k1,...,˜kj−1,˜kj+1,...,˜km+1;x1p1,x2p2)
×δ(x1− ˆ x1)δ(x2− ˆ x2)[dkj]dx1dx2.(2.3)
where [dk] = ddkδ(+)(k2)/(2π)d−1and
ˆ x1=
?s12− sj2
?s12− s1j
s12
s12− s1j− sj2
s12− s1j
?1
?1
2
,
ˆ x2=
s12
s12− s1j− sj2
s12− sj2
2
.(2.4)
Inserting the factorized expression for the phase space measure in (2.1), the subtrac-
tion terms can be integrated over the antenna phase space. In the case of initial-initial
subtraction terms, the antenna phase space is trivial: the two remaining Dirac delta func-
tions can be combined with the one particle phase space, such that there are no integrals
left. We define the initial-initial integrated antenna functions as follows:
Xik,j(x1,x2) =
1
C(ǫ)
?
[dkj]x1x2δ(x1− ˆ x1)δ(x2− ˆ x2)Xik,j, (2.5)
where we introduced C(ǫ) = (4π)ǫ/(8π2)e−γEǫ.
Substituting the one-particle phase space, and carrying out the integrations over the Dirac
delta functions, we have,
Xik,j(x1,x2) = (Q2)−ǫ
eǫγE
Γ(1 − ǫ)J(x1,x2)Q2Xik,j,(2.6)
with Q2= q2= (p1+ p2− kj)2. The Jacobian factor, J(x1,x2) is given by
J(x1,x2) =x1x2(1 + x1x2)
(x1+ x2)2
(1 − x1)−ǫ(1 − x2)−ǫ
?(1 + x1)(1 + x2)
(x1+ x2)2
?−ǫ
,(2.7)
and the two-particle invariants are given by:
s1j= −s12x1(1 − x2
2)
x1+ x2
,sj2= −s12x2(1 − x2
1)
x1+ x2
. (2.8)
The integrated subtraction term is then,
dˆ σS,(II)=
?
×dΦm(k1,...,kj−1,kj+1,...,km+1;x1p1,x2p2)
m+1
?
j
N
Sm+1
?
dx1
x1
dx2
x2
C(ǫ)Xik,j(x1,x2)
– 4 –