arXiv:hep-ph/0604200v1 24 Apr 2006
Next-to-leading order QCD corrections
to Z boson pair production via
Barbara J¨ ager1, Carlo Oleari2and Dieter Zeppenfeld1
1Institut f¨ ur Theoretische Physik, Universit¨ at Karlsruhe, P.O.Box 6980, 76128 Karlsruhe,
2Dipartimento di Fisica ”G. Occhialini”, Universit` a di Milano-Bicocca, 20126 Milano,
Vector-boson fusion processes are an important tool for the study of electroweak
symmetry breaking at hadron colliders, since they allow to distinguish a light Higgs
boson scenario from strong weak boson scattering. We here consider the channels
WW →ZZ and ZZ →ZZ as part of electroweak Z boson pair production in associ-
ation with two tagging jets. We present the calculation of the NLO QCD corrections
to the cross sections for pp→e+e−µ+µ−+ 2 jets and pp→e+e−νµ¯ νµ+ 2 jets via
vector-boson fusion at order αsα6, which is performed in the form a NLO parton-level
Monte Carlo program. The corrections to the integrated cross sections are found to
be modest, while the shapes of some kinematical distributions change appreciably at
NLO. Residual scale uncertainties typically are at the few percent level.
One of the primary goals of the CERN Large Hadron Collider (LHC) is the discovery
of the Higgs boson and a thorough investigation of the mechanism of electroweak (EW)
symmetry breaking [1, 2]. In this context, vector-boson fusion (VBF) processes have emerged
as a particularly interesting class of processes. Higgs boson production in VBF, i.e. the
reaction qq→qqH, where the Higgs decay products are detected in association with two
tagging jets, offers a promising discovery channel  and, once its existence has been verified,
will help to constrain the couplings of the Higgs boson to gauge bosons and fermions .
In order to distinguish possible signatures of strong weak-boson scattering from those of a
light Higgs boson, a good understanding of WW →ZZ and ZZ →ZZ scattering processes,
which are part of the VBF reaction qq→qqZZ, is needed. This requires the computation
of next-to-leading order (NLO) QCD corrections to the qq →qqZZ cross section, including
the leptonic decays of the Z bosons. Experimentally, very clean signatures are expected
from the ZZ →ℓ+ℓ−ℓ′+ℓ′−decays in VBF with four charged leptons in the final state, the
disadvantage of this channel being a rather small Z →e+e−or Z →µ+µ−branching ratio of
about 3%. The ZZ →ℓ+ℓ−ν¯ ν channel, with two undetected neutrinos, on the other hand,
results in a larger number of events due to the larger Z →ν¯ ν branching ratio .
LO results for EW ZZ jj production in VBF have been available for more than two
decades. The first calculations  were performed employing the effective W approxima-
tion , where the vector bosons radiated off the scattering quarks are treated as on-shell
particles and, therefore, kinematical distributions characterizing the tagging jets cannot be
predicted reliably. In the following years, exact calculations for qq →qqZZ have been com-
pleted, first without Z boson decay , and then including leptonic decays of the Z bosons
within the narrow width approximation .
We go beyond these approximations and develop a fully-flexible parton level Monte Carlo
program, which allows for the calculation of cross sections and kinematical distributions for
EW ZZ jj production via VBF at NLO QCD accuracy. The program is structured in com-
plete analogy to the respective code for EW W+W−jj production presented in Ref. .
Here, we calculate the t-channel weak-boson exchange contributions to the full matrix ele-
ments for processes like qq →qqe+e−µ+µ−and qq→qqe+e−νµ¯ νµat O(α6αs). We consider
all resonant and non-resonant contributions giving rise to a four charged-lepton and a two
charged-lepton plus two neutrino final state, respectively. Contributions from weak-boson
exchange in the s-channel are strongly suppressed in the phase-space regions where VBF
can be observed experimentally and therefore disregarded throughout. We do not specifi-
cally require the leptons and neutrinos to stem from a genuine VBF-like production process,
but also include diagrams where one or two of the Z bosons are emitted from either quark
line. Diagrams, where the final state leptons stem from a γ →ℓ+ℓ−decay or non-resonant
production modes, are also taken into account. Finite-width effects are fully considered.
For simplicity, we nonetheless refer to the qq→qq ℓ+ℓ−ℓ′+ℓ′−and qq→qqℓ+ℓ−ν¯ ν processes
computed this way generically as “EW ZZ jj” production.
The outline of the paper is as follows. In Sec. 2 we briefly summarize the calculation
of the LO and NLO matrix elements for EW ZZ jj production making use of the helicity
techniques of Ref. . Section 3 deals with phenomenological applications of the parton-
level Monte Carlo program which we have developed. Conclusions are given in Sec. 4.
2 Elements of the calculation
The calculation of NLO QCD corrections to EW ZZ jj production closely resembles
our earlier work for EW W+W−production in association with two jets . The main
differences lie in the electroweak aspects of the processes, while the QCD structure of the
NLO corrections is very similar. The techniques developed in Ref.  can therefore be
adapted readily and only need a brief recollection here. For simplicity, we focus on the
e+e−µ+µ−decay channel in the following. The application of the basic features discussed
for this case to the e+e−νµ¯ νµleptonic final state is then straightforward.
The Feynman graphs contributing to pp→e+e−µ+µ−jj can be grouped in six topologies,
respectively, for the 579 t-channel neutral-current (NC) and the 241 charged-current (CC)
exchange diagrams which appear at tree level. These groups are sketched in Fig. 1 for
the specific NC subprocess uc→uce+e−µ+µ−. The first two of these correspond to the
emission of two external vector bosons V from the same (a) or different (b) quark lines.
The remaining topologies are characterized by the vector-boson sub-amplitudes Lαβ
V →e+e−µ+µ−, V V →µ+µ−and V V →e+e−. In each case, V stands for a virtual γ or Z
boson, and α and β are the tensor indices carried by these vector bosons. The propagator
call “leptonic tensors” in the following. Graphs for CC processes such as us→dce+e−µ+µ−
are obtained by replacing the t-channel γ or Z bosons in Fig. 1 with W bosons. They
give rise to the new lepton tensors Lαβ
W+W−→e+e−µ+µ−, W+W−→e+e−and W+W−→µ+µ−.
V V, Γα
V V,µand Tαβ
V V,e, which describe the tree-level amplitudes for the processes V V →e+e−µ+µ−,
V+imVΓV) are included in the definitions of the sub-amplitudes, which we
W+W−,µfor the sub-amplitudes
Contributions from anti-quark initiated t-channel processes such as ¯ uc→ ¯ uce+e−µ+µ−,
which emerge from crossing the above processes, are fully taken into account. On the other
hand, s-channel exchange diagrams, where all vector bosons are time-like, contain vector-
boson production with subsequent decay of one of the bosons into a pair of jets. These
contributions can be safely neglected in the phase-space region where VBF can be ob-
served experimentally, with widely-separated quark jets of large invariant mass. In the
uc→uce+e−µ+µ−. Diagrams analogous to (a), (d), (e) and (f), with vector-boson
emission off the lower quark line, are not shown.
The six Feynman-graph topologies contributing to the Born process
same way, u-channel exchange diagrams are obtained by the interchange of identical final-
state (anti)quarks. Their interference with the t-channel diagrams is strongly suppressed for
typical VBF cuts and therefore completely neglected in our calculation.
For the treatment of finite-width effects in massive vector-boson propagators we resort
to a modified version of the complex-mass scheme , which has already been employed
in Refs. [13, 10]. We globally replace vector-boson masses m2
changing the real value of sin2θW. This procedure respects electromagnetic gauge invariance.
The amplitudes for all NC and CC subprocesses are calculated and squared separately for
each combination of external quark and lepton helicities. To save computer time, only the
summation over the various quark helicities is done explicitely, while the four distinct lepton
helicity states are considered by means of a random summation procedure.
V− imVΓV, without
The computation of NLO corrections is performed in complete analogy to Ref. . For
the real-emission contributions we encounter 2892 diagrams for the NC and 1236 for the CC
processes, which are evaluated using the amplitude techniques of Ref.  and the leptonic
tensors introduced above. Singularities in the soft and collinear regions of phase space are
regularized in the dimensional-reduction scheme  with space-time dimension d = 4 − 2ǫ.
The cancellation of these divergences with the respective poles from the virtual contributions
is performed by introducing the counter terms of the dipole subtraction method . Since
the color and flavor structure of our processes are the same as for Higgs boson production
in VBF, the analytical form of subtraction terms and finite collinear pieces is identical
to the ones given in Ref. . The finite parts of the virtual contributions are evaluated
by Passarino-Veltman tensor reduction , which is implemented numerically. Here, the
fast and stable computation of pentagon tensor integrals is a major issue, which is tackled
by making use of Ward identities and mapping a large fraction of the pentagon diagrams
onto box-type contributions with the methods developed in . The residual pentagon
contributions amount only to about 1 ? of the cross sections presented below.
The results obtained for the Born amplitude, the real emission and the virtual corrections
have been tested extensively. For the tree-level amplitude, we have performed a compari-
son to the fully automatically generated results provided by MadGraph , and we found
agreement at the 10−13level. In the same way, the real emission contributions have been
checked. For the latter, also QCD gauge invariance has been tested, which turned out to be
fulfilled within the numerical accuracy of the program. The numerical stability of the finite
parts of the pentagon contributions is monitored by checking numerically that they satisfy
electroweak Ward identities with a relative error less than δ = 1.0. This criterion is violated
by about 3% of the generated events. The contributions from these phase-space points to
the finite parts of the pentagon diagrams are disregarded and the remaining pentagon parts
are corrected by a global factor for this loss. In Ref.  we found that this procedure gives
a stable result for the pentagon contributions when varying the accuracy parameter δ.