Page 1

arXiv:hep-ph/0604200v1 24 Apr 2006

Bicocca-FT-06-7

KA–TP–04–2006

SFB/CPP–06–20

hep-ph/0604200

Next-to-leading order QCD corrections

to Z boson pair production via

vector-boson fusion

Barbara J¨ ager1, Carlo Oleari2and Dieter Zeppenfeld1

1Institut f¨ ur Theoretische Physik, Universit¨ at Karlsruhe, P.O.Box 6980, 76128 Karlsruhe,

Germany

2Dipartimento di Fisica ”G. Occhialini”, Universit` a di Milano-Bicocca, 20126 Milano,

Italy

Abstract

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.

Page 2

1 Introduction

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 [3] and, once its existence has been verified,

will help to constrain the couplings of the Higgs boson to gauge bosons and fermions [4].

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 [5].

LO results for EW ZZ jj production in VBF have been available for more than two

decades. The first calculations [6] were performed employing the effective W approxima-

tion [7], 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 [8], and then including leptonic decays of the Z bosons

within the narrow width approximation [9].

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. [10].

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

2

Page 3

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. [11]. 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 [10]. 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. [10] 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

Tαβ

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

factors 1/(q2−m2

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 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−, Tαβ

W+W−,eand Tαβ

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

3

Page 4

uu

cc

V

VV

e+

e−

µ+

µ−

uu

cc

V

V

V

e+

e−

µ+

µ−

uu

cc

V

V

Lαβ

V V

e+e−

µ+

µ−

uu

cc

V

V

Γα

V

e+

e−

µ+

µ−

uu

cc

V

V

V

Tαβ

V V,µ

e+

e−

µ+

µ−

uu

cc

V

V

V

Tαβ

V V,e

µ+

µ−

e+

e−

(a)(b)

(c) (d)

(e)(f)

Figure 1:

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

4

Page 5

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 [12], 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.

Vwith m2

V− imVΓV, without

The computation of NLO corrections is performed in complete analogy to Ref. [10]. 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. [11] and the leptonic

tensors introduced above. Singularities in the soft and collinear regions of phase space are

regularized in the dimensional-reduction scheme [14] 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 [15]. 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. [16]. The finite parts of the virtual contributions are evaluated

by Passarino-Veltman tensor reduction [17], 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 [10]. 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 [18], 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. [10] we found that this procedure gives

a stable result for the pentagon contributions when varying the accuracy parameter δ.

5