A The 3-body phase space

Measuring the Higgs-Vector boson Couplings at Linear Collider


We estimate the accuracy with which the coefficient of the CP even dimension six operators involving Higgs and two vector bosons () can be measured at linear colliders. Using the optimal observables method for the kinematic distributions, our analysis is based on the five different processes. First is the fusion process in the -channel (), where we use the rapidity and the transverse momentum of the Higgs boson as observables. Second is the pair production process in the -channel, where we use the scattering angle of the and the decay angular distributions, reproducing the results of the previous studies. Third is the -channel , fusion processes (), where we use the energy and angular distributions of the tagged and . In the fourth, we consider the rapidity distribution of the untagged events, which can be approximated well as the fusion of the bremsstrahlung photons from and beams. As the last process, we consider the single tagged events, which probe the process. All the results are presented in such a way that statistical errors of the constraints on the effective couplings and their correlations are read off when all of them are allowed to vary simultaneously, for each of the above processes, for GeV, at , and , with and without beam polarization of 80%. We find for instance that the and couplings can be measured with 0.6% and 0.9% accuracy, respectively, for the integrated luminosity of at , and at , , for the luminosity uncertainty of 1% at each energy. We find that the luminosity uncertainty affects only one combination of the non-standard couplings which are proportional to the standard and couplings, while it does not affect the errors of the other independent combinations of the couplings. As a consequence, we observe that a few combinations of the eight dimension six operators can be constrained as accurately as the two operators which have been constrained by the precision measurements of the and boson properties.

Higgs, Vector Bosons, Anomalous couplings, Linear collider
14.80.Cp, 14.70.FM, 14.70.Hp

Pre-Print KEK-TH-1256

I Introduction

The Standard Model (SM) of the elementary particles based on the SU(3) SU(2) U(1) gauge symmetry has proved to be a successful theory to interpret all the precision data available to date. SM predicts a light Higgs boson whose discovery is one of the prime tasks of the upcoming future colliders.

In fact, the present electroweak precision measurements indicate the existence of a light Higgs boson (1); (2). Experiments at the CERN Large Electron Positron collider (LEP) set the lower bound on its mass of at the confidence level (CL) (1). The Fermilab Tevatron, which collides proton and anti-proton at , is currently the only collider which can produce low mass Higgs bosons. Analysis with Run IIb data samples by the CDF and D detectors indicates that the Tevatron experiments can observe the Higgs boson with about 10 total integrated luminosity for the mass of around (3). The Large Hadron Collider (LHC) at CERN will start colliding two protons at in the year 2008, and is geared to detect the Higgs boson in gluon-gluon and vector-boson fusion processes. It will measure ratios of various Higgs boson couplings through variety of decay channels at accuracies of order 10 to 15% with 100 luminosity (4).

Despite the success, SM presents the naturalness problem due to the quadratic sensitivity of the Higgs boson mass to the new physics scale at high energies, which implies that there is a need of subtle fine tuning to keep the electroweak symmetry breaking theory below the  TeV scale. To put it in another way, this may suggest an existence of a new physics scale not far above the TeV scale. The key to probe the new physics beyond the SM theory is to clarify the origin of the electroweak symmetry breaking, the Higgs mechanism. Therefore, it is necessary to measure the Higgs boson properties as precisely as possible, especially the couplings, because they are expected to be sensitive to the symmetry breaking physics that gives rise to the weak boson masses.

With this motivation, we re-examine the potential of the future linear collider, the International Linear Collider (ILC) in the precise measurement of the couplings. Clean experimental environment, well defined initial state, tunable energy, and beam polarization renders ILC to be the best machine to study the Higgs boson properties with high precision. In this paper, we study the sensitivity of the ILC measurements on all the (, , and ) couplings comprehensively and semi-quantitatively by using all the available processes with a light Higgs boson ( 120 GeV); with -channel exchange, with -channel exchange, with -channel exchange, no-tag process from fusion, and single-tagged process that probes via -channel and exchange.

In order to quantify the ILC sensitivity to measure various couplings simultaneously, we adopt the powerful technique of the optimal observables method (5); (6); (7); (8). It allows us to measure several couplings simultaneously as long as the non-standard couplings give rise to different observable kinematic distributions. The results can be summarized in terms of the covariance matrix of the measurement errors, from each process at each energy, that scales inversely as the integrated luminosity.

In order to combine results from different processes and at different energies, we adopt the effective Lagrangian of the SM particles with operators of mass dimension six to parametrize all the couplings (9); (10); (11). This allows us not only to compare the significance of the measurements of various couplings at different energies and at different colliders, but also to study what ILC can add to the precision measurements of the and boson properties in the search for new physics via quantum effects. We therefore parametrize the couplings as linear combination of all the dimension six operators that are allowed by the electroweak gauge symmetry and invariance.

Some of the previous studies based on the optimal observables method are found for -violating effects in via and couplings (8); (12), and also in (8). CP conserving and CP violating effects in process has been studied in ref. (13); (14). In refs. (8); (13); (14) all the relevant couplings are varied simultaneously, and their correlations are studied. More recently, the ILC sensitivity to the and couplings has been studied in refs. (15); (16); (17). Bounds on the coefficients of the Higgs-vector boson dimension-6 operators have been found in refs. (18); (19) based on non-observation of the Higgs boson signal at the Tevatron. Whenever relevant, we compare our results with the previous observations.

This paper is organized as follows. In section II, we describe the low energy effective interactions among the Higgs boson and the electroweak gauge bosons arising from new physics that is parametrized in terms of the effective Lagrangian of the SM particles with operators up to mass dimension six. In section III, we introduce the optimal observables method and explain how we perform the phase space integration when some of the kinematic distributions are unobservable, such as neutrino momenta and a distinction between quark and anti-quark jets. Although we present numerical results for unpolarized beams and for 80% polarized beam only, all the formulas are presented for an arbitrary polarization of and beams. After introducing final state cuts, such as those for the tagging and those for selecting or excluding events, we present the total cross sections for all the five processes at =200 GeV-1 TeV for =120 GeV, and at =250 GeV, 500 GeV, 1 TeV for = 100-200 GeV. Then in section IV we compute the statistical errors of the non-standard couplings extracted from measurements of the -fusion process, . In section V, we study the constraints on the and couplings extracted from production. In section VI, not only the and couplings but also the coupling are studied in the double-tag process via -channel and exchange. In section VII, we obtain the constraints on the coupling from the fusion, in no-tag events, using the equivalent real photon approximation. In section VIII, we consider the single-tag process to constrain the and couplings. In section IX, we address the implication of luminosity uncertainty on the measurement of these couplings. In section X, we summarize all our results, compare them with previous studies, and present our estimates for the ILC constraints on the dimension six operators, which are then compared with the constraints from the precision electroweak measurements of the and boson properties. In Appendices we present our parameterizations of the 3-body phase space (Appendix A), and the explicit forms the -channel and -channel currents and their contractions that appear in the helicity amplitudes (Appendix B).

Ii Generalized vertex with dimension six operators

In our study, we adopt the effective Lagrangian of the Higgs and the gauge bosons with operators up to mass dimension six,


where denotes the renormalizable SM Lagrangian and ’s are the gauge-invariant operators of mass dimension 6. The index runs over all operators of the given mass dimension. The mass scale is set by , and the coefficients are dimensionless parameters, which are determined once the full theory is known. Excluding the dimension 5 operators for the neutrino Majorana masses, and the dimension 6 operators with quark and lepton fields, we are left with the following eight CP even operators that affect the couplings. Notation of the operators are taken from the reference (20).


Here denotes the Higgs doublet field with the hyper-charge , and the covariant derivative is , where the gauge couplings and the gauge fields with a caret represent those of the SM, in the absence of higher dimensional operators. The gauge-covariant and invariant tensors and , respectively, are , and . The coefficients of the operators (2a)-(2h), which are denoted as in the effective Lagrangian of eq.(1), should give us information about physics beyond the SM. So far, the precision measurements of the weak boson properties (2) constrained the operaotrs and , which have been useful in testing some models of the electroweak symmetry breakdown (10); (23). In this report, we explore the accuracy with which the ILC experiments can measure the coefficients of all these eight operators when a light Higgs boson exists.

When the Higgs field acquires the vacuum expectation value , the bilinear part of the effective Lagrangian of eq.(1) is expressed as


After renormalization of gauge fields and their couplings,


and after diagonalization of the mass squared matrices, the effective Lagrangian reads




All the remaining terms in the effective Lagrangian, denoted by dots in eq.(5), are expressed in terms of the renormalized fields, couplings and masses, as defined in eqs.(4) and eqs.(6). The standard gauge interactions are dictated by the covariant derivative


where , and .

Before expressing the interactions of , let us briefly review the observable consequence of new physics in the gauge boson two point functions in eq.(5). First, the ratio of the neutral current and the charged current interactions at low energies deviate (21); (23) from unity,


Second, the extra kinetic mixing between and modifies the and boson propagators


in the notation of ref. (22), which contributes to the parameter (23)


Here the over-lined couplings , and are the effective couplings that contain the gauge-boson propagator corrections at the momentum transfer (22). We will examine the constraints on and from the precision measurements of the weak boson properties in the last section of this report.

The terms describing the couplings in the effective Lagrangian are now expressed as


where the 9 dimensionless couplings, , parametrize all the non-standard interactions:


From the effective Lagrangian of eq.(11), we obtain the Feynman rule for vertex as

Figure 1: The Vertex.

where all three momenta are incoming, , as shown in the fig. 1. and can be , , , , or . The coefficients are


for the couplings,


for the couplings,


for the couplings. It is to be noted that the coupling has the Feynman rule which is not symmetric under an interchange of and . For the couplings,


Although we do not consider off-shell Higgs boson contributions in this report, should be replaced by in the above Feynman rules when the Higgs-boson is off-shell.

Iii Optimal Observables and Phase Space

iii.1 Optimal observables method

The optimal observables method (8) makes use of all the kinematic distributions which are observable in experiments. We therefore summarize our phase-space parameterizations for all the Higgs boson production processes in collisions considered in this study, which can be generically written as


(a) ZH production

(b) vector-boson fusion

Figure 2: Feynman diagrams for .

Here and are the four momenta and helicities, respectively, and and are the four momenta and helicities, respectively, of the produced fermion () and anti-fermion (). For , the processes (18) occur only through the production diagram as shown in fig.2(a), whereas for or , both the diagrams fig.2(a) and fig.2(b) contribute. The effective vertex is depicted by the solid circle in the Feynman diagrams. The production process (a) is sensitive to the and couplings, while the vector-boson fusion processes (b) are sensitive to the coupling for , and the , , couplings for .

The matrix elements for the processes eq.(18) can in general be expressed as


where denotes the SM helicity amplitude, and denotes the non-standard couplings of eq.(12) that contribute to the process. The matrix elements give the helicity amplitudes which are proportional to the coupling . If the and beam polarizations are and () respectively, the differential cross section can be expressed as


where the non-standard couplings are assumed to be real and small, and hence the terms quadratic in couplings are dropped. Here is the 3-body phase space volume of the system, and


gives the differential cross section of the SM. The term proportional to ,


gives the differential distribution which is proportional to .

In the optimal observables method, we make full use of the distribution in order to constrain . For instance, if all have different shapes from each other, then in principle, we can constrain all the coefficients simultaneously. For a given integrated Luminosity , the statistical errors of the measurement can be obtained from a function


where is the number of events in the k’th bin, and is the corresponding prediction of the theory which depends on the parameters of the SM and . In the second line (23b), for to gives the representative phase space point of a bin number with the bin size . Now, if all the coefficients are tiny, the experimental result in the k’th bin should be approximated by the SM prediction as


The function can then be expressed as