QCD Corrections to Jet Correlations in Weak Boson Fusion

Higgs boson production via weak boson fusion is sensitive to the tensor structure of the HVV (V=W,Z) couplings, which distinguishes loop induced vertices from SM expectations. At the CERN Large Hadron Collider this information shows up most clearly in the azimuthal angle correlations of the two forward and backward quark jets which are typical for weak boson fusion. We calculate the next-to-leading order QCD corrections to this process, in the presence of anomalous HVV couplings. Gluon emission does not significantly change the azimuthal jet correlations.

Introduction. The production of Higgs bosons in the weak boson fusion (WBF) process will provide a direct and highly sensitive probe of HW W and HZZ couplings at the CERN Large Hadron Collider (LHC) [1,2,3,4,5]. The determination both of the strength and of the tensor structure of these couplings is crucial for the identification of the produced boson as a remnant of the spontaneous symmetry breaking process which is responsible for W and Z mass generation.
Within spontaneously broken, renormalizable gauge theories like the standard model (SM), this coupling originates from the kinetic energy term, (D µ Φ) † (D µ Φ), of a scalar Higgs field, Φ, whose neutral component obtains a vacuum expectation value (vev), This replacement then leads to a characteristic coupling in the interaction Lagrangian, of the form HV µ V µ (V = W, Z). The existence of the vev is necessary to produce a trilinear HV V coupling at tree level: with v = 0 all couplings to the gauge fields V contain two scalar fields, i.e., only HHV and HHV V couplings would be generated. A trilinear HV V coupling may also be loop-induced, however. The SM Hγγ and Hgg effective couplings are an example: they are induced by W -boson and/or top quark loops.
Gauge invariance dictates a different tensor structure of these loop-induced couplings: the corresponding effective Lagrangian contains the square of the field strength, i.e. the lowest order loop-induced terms are of the form HV µν V µν or HV µνṼ µν , whereṼ µν = 1 2 ǫ µνρσ V ρσ denotes the dual field strength of the gauge field.
The task of future Higgs experiments is, then, twofold: (i) to measure the overall strength of the HV V coupling, and (ii) to identify its tensor structure. One would expect a loopinduced coupling to be much smaller than the expected SM HV V coupling strength. However, the measurement of WBF rates alone will not be sufficient to establish H as being related to spontaneous symmetry breaking: to give just two examples, the loop-induced couplings might be substantially enhanced by additional non-SM particles in the loop or by the existence of multiplets of large weak isospin which couple strongly to H. Or a particular LHC signature may be strongly enhanced by a much larger H decay branching ratio than in the SM. A confirmation that the HV V coupling has tree level strength is, thus, ambiguous: a clear identification of the Higgs boson also requires the identification of the tensor structure of the HV V vertex.
It was pointed out some time ago that the azimuthal angle correlations of the two quark jets in the weak boson fusion process qQ → qQH provide tell-tale signatures for the tensor structure of the HV V couplings [6]: the SM expectation is for a flat distribution, while the loop-induced couplings lead to a pronounced dip at azimuthal separations φ jj of the two tagging jets of 90 degrees for a HV µν V µν coupling and at 0 and 180 degrees for the CP violating HV µνṼ µν vertex. Observation of the tagging jets is crucial for isolating the WBF process from backgrounds and, therefore, their distributions will be available for all WBF samples. Also, signal to background ratios for WBF processes are expected to be very good within the SM, exceeding the 1:1 level for wide ranges of the Higgs boson mass [1,2,3,4,5].
The analysis of Ref. [6] was performed at leading order (LO) in QCD. This means that additional gluon emission, which might lead to a de-correlation of the tagging jets, was ignored in the analysis. Subsequently it was argued [7] that such de-correlation effects play an important role in a related process, gg → Hgg, when the two tagging jets are widely separated in rapidity, which is a typical requirement for WBF studies. In this Letter we analyze this question, by calculating the tagging jet distributions in next-to-leading order (NLO) QCD, for the production of a scalar H via WBF with an arbitrary tensor structure of the HV V vertex. If de-correlation is important, it should show up in the form of large radiative corrections at NLO. We use the term "Higgs boson" as a generic name for the produced scalar in the following.
The NLO calculation. Our calculation is an extension of the NLO QCD corrections for the SM WBF processes qQ → qQH (and crossing related ones) [8,9,10]. For the total cross section these corrections have been known for over a decade [8]. Recently, we have recalculated them by developing a NLO parton level Monte Carlo program [9] which provides the flexibility to calculate arbitrary distributions at NLO, such as the azimuthal angle correlations that we are interested in here.
The calculation of Ref. [9] uses a SM vertex function, Fig. 1. Here we need to generalize this vertex to the most general structure compatible with Lorentz invariance. Taking into account that the quark currents in Fig. 1 and for the corresponding gluon emission processes are conserved, all terms proportional to q µ 1 or q ν 2 may be dropped, and the most general HV V vertex may be written as Here q 1 and q 2 are the four-momenta of the two weak bosons, and the a i (q 1 , q 2 ) are Lorentzinvariant form factors, which might, for example, represent scalar loop integrals in a per-  (1). These currents, and their contractions, are evaluated numerically, using the amplitude formalism of Ref. [11]. All other aspects of the present NLO calculation are handled as in Ref. [9], except that we do not simulate any Higgs boson decays in the following. Factorization and renormalization scales are fixed to µ F = µ R = Q i for QCD corrections to the first or second quark line in Fig. 1. Here Q 1 and Q 2 are the virtualities of the exchanged weak bosons. We use CTEQ6M parton distributions [12] with α s (M Z ) = 0.118 for all NLO results and CTEQ6L1 parton distributions for all leading order cross sections.
Anomalous couplings and form factors. While the g µν -term in the vertex function (1) corresponds to a SM Higgs coupling, the anomalous coupling terms a 2 and a 3 can be related to higher dimensional operators in an effective Lagrangian. They first appear at the dimension-5 level [19] and may be written as where the subscript e or o refers to the CP even or odd nature of the individual operators.
In our discussion we will neglect possible contributions from Hγγ and HγZ couplings which can appear in SU(2) × U(1) invariant formulations [13,14]. The precise mix of HW W , HZZ, HZγ and Hγγ contributions is quite irrelevant for the observable azimuthal angle distributions, as long as we do not consider interference effects between SM and anomalous vertices, and it will not affect our conclusions about the size of NLO corrections. For simplicity we therefore set a 1 = 0 for the anomalous coupling case and choose relative contributions from W W and ZZ fusion as in the SM, by taking g HW W
The effective Lagrangian of Eq. (2) produces couplings for the HW W vertex, and for the HZZ vertex. In general, the a i are form factors which are expected to be suppressed once the momentum transfer, −q 2 i , carried by the virtual gauge boson reaches the typical mass scale, M, of the new physics which is responsible for these anomalous couplings. Below we use the simple ansatz for discussing the consequences of such form factor effects.

Results:
The typical signature of a weak boson fusion event at the LHC consists of the two quark jets (tagging jets) and the Higgs decay products. The tagging jets tend to be widely separated in rapidity, with one quite forward (typical pseudorapidity of 3 to 4) and the second one backward, but frequently still located in the central detector (pseudorapidity below 2.5).
Various Higgs decay modes have been considered in the literature for WBF, H → W W [1], H → τ τ [2], and H → γγ [3] being the most promising ones. While optimized event selection varies, in particular for the decay products, the cuts on the tagging jets are fairly similar in all analyses. Since here we are interested in the QCD features of WBF events, which do not depend on the Higgs decay mode, we perform our NLO analysis without simulating Higgs decays, and we only impose typical WBF cuts on the tagging jets. In order to reconstruct jets from the final-state partons, the k T -algorithm [15] as described in Ref. [16] is used, with resolution parameter D = 0.8. In a given event, the tagging jets are then defined as the two jets with the highest transverse momentum, p T j , with Here y j denotes the rapidity of the (massive) jet momentum which is reconstructed as the four-vector sum of massless partons of pseudorapidity |η| < 5. Backgrounds to weak-boson fusion are significantly suppressed by requiring a large rapidity separation of the two tagging jets. This motivates the final cut ∆y jj = |y j 1 − y j 2 | > 4 , y j 1 · y j 2 < 0 , which includes the requirement that the two tagging jets reside in opposite detector hemispheres.
The structure of the HV V coupling affects the production dynamics of H and we can expect significant deviations in jet observables if, instead of the SM, anomalous couplings describe the vertex of Eq. (1). One example is shown in Fig. 2, where transverse momentum distributions, dσ/dp T j (max), are compared between the SM (solid line) and the CP even coupling a 2 (q 1 , q 2 ), with different form factor scales M in Eq. (5). Here, p T j (max) is the maximum p T of the two tagging jets. Only the shape of the distribution is considered, since the rate can always be adjusted by multiplying the anomalous couplings by a constant factor. Also, we should note that a CP odd coupling leads to very similar curves for a given form factor scale. In all cases we show the LO expectations (dashed lines) together with the NLO results: QCD corrections are of order 10%, typically, and well under control.
One finds that anomalous HV V couplings generally lead to harder p T spectra of the two tagging jets. Since the anomalous Lagrangian in Eq. (2)  A much better observable for distinguishing the different tensor structures of the HV V vertex is the azimuthal angle correlation of the two tagging jets, dσ/dφ jj [6]. Here φ jj is the azimuthal angle between the two tagging jets. The corresponding distributions are shown related to the higher average jet transverse momentum for big form factor scales, M: at small φ jj two high p T jets recoil against the H scalar, resulting in an increased invariant mass of the event compared to the situation with two back-to-back jets. This leads to a more asymmetric φ jj distribution for high form factor scales.
The pronounced dip at 90 degrees, which is characteristic of the CP even coupling, is also found in Hjj production via gluon fusion [17], at LO. This is not surprising because, in the large top mass limit, the Hgg vertex can be described by an effective Lagrangian proportional to HG a µν G aµν , which exhibits the same field strength squared behavior and hence the same tensor structure as the CP even HV V coupling in Eqs. (1,2). Since the two tagging jets are far apart from each other, separated by a large rapidity gap of 4 units of rapidity or more, this LO behavior may be significantly reduced by gluon radiation when higher order QCD corrections are taken into account. Such de-correlation effects have been studied for dijet events at the Tevatron [18]. For Hjj production via gluon fusion, Odagiri [7] has argued that the dip structure is largely washed out by additional gluon emission between the two tagging jets. Our NLO calculations show that such de-correlation effects are irrelevant for weak boson fusion, where t-channel color singlet exchange severely suppresses gluon radiation in the central region. The LO and the NLO curves in Fig. 3 are virtually indistinguishable. In order to better exhibit the size of NLO QCD effects for the WBF case, we show, in Fig. 4(a) the azimuthal angle correlations for a pure CP even anomalous coupling for three different Higgs masses, m H = 120, 200 and 500 GeV. Only small changes are visible when going from LO (dashed lines) to NLO (solid lines). The differences between LO and NLO are smaller than kinematical effects that can be induced by cuts on the Higgs decay products or by variations of the Higgs boson mass.
The small to modest size of the QCD corrections is quantified in Fig. 4(b) where the K factor for the distribution is shown, which is defined as The K-factor is below ≈ 1.4 even in the dip region, where the cross section is severely suppressed. Virtually identical results hold for the CP-odd case. Clearly, the characteristic azimuthal angle distributions of the jets in WBF are not affected in any significant way by NLO QCD corrections.

Conclusions: We have performed a first calculation of the NLO QCD corrections to
Higgs boson production via WBF in the presence of arbitrary anomalous HV V (V = W, Z) couplings. Anomalous couplings lead to characteristic changes in the azimuthal angle correlation of the two tagging jets in weak boson fusion events at the LHC, which provides for a very sensitive test of the tensor structure of the HV V couplings of the Higgs boson or of any other scalar with sufficiently large production cross section in WBF [6]. We have shown by explicit calculation that these azimuthal correlations are not washed out by gluon emission, at NLO QCD, even though the tagging jets are widely separated in rapidity. This behavior can be understood as a consequence of t-channel color singlet exchange in WBF which severely suppresses the central gluon radiation which might cause tagging jet de-correlation.