Measuring the HiggsVector boson Couplings at Linear Collider
Abstract
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 nonstandard 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.
pacs:
14.80.Cp, 14.70.FM, 14.70.HpPrePrint KEKTH1256
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 lepbound ; pdg2006 . Experiments at the CERN Large Electron Positron collider (LEP) set the lower bound on its mass of at the confidence level (CL) lepbound . The Fermilab Tevatron, which collides proton and antiproton 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 tevreport . 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 gluongluon and vectorboson 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 zeppkinPRD62013009 .
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 reexamine 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 semiquantitatively by using all the available processes with a light Higgs boson ( 120 GeV); with channel exchange, with channel exchange, with channel exchange, notag process from fusion, and singletagged 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 atwood ; davier ; diehl ; gunion . It allows us to measure several couplings simultaneously as long as the nonstandard 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 linear ; Grinstein ; STpram . 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 gunion ; ma , and also in gunion . CP conserving and CP violating effects in process has been studied in ref. kniehl ; StongHagi . In refs. gunion ; kniehl ; StongHagi 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. debchou ; mamta ; TaoHan . Bounds on the coefficients of the Higgsvector boson dimension6 operators have been found in refs. tevatron ; photonhiggs based on nonobservation 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 antiquark 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 GeV1 TeV for =120 GeV, and at =250 GeV, 500 GeV, 1 TeV for = 100200 GeV. Then in section IV we compute the statistical errors of the nonstandard 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 doubletag process via channel and exchange. In section VII, we obtain the constraints on the coupling from the fusion, in notag events, using the equivalent real photon approximation. In section VIII, we consider the singletag 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 3body 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,
(1) 
where denotes the renormalizable SM Lagrangian and ’s are the gaugeinvariant 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 LowEnegyEff .
(2a)  
(2b)  
(2c)  
(2d)  
(2e)  
(2f)  
(2g)  
(2h) 
Here denotes the Higgs doublet field with the hypercharge , 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 gaugecovariant 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 pdg2006 constrained the operaotrs and , which have been useful in testing some models of the electroweak symmetry breakdown Grinstein ; Oblique1 . 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
(3)  
After renormalization of gauge fields and their couplings,
(4a)  
(4b) 
and after diagonalization of the mass squared matrices, the effective Lagrangian reads
(5)  
where
(6a)  
(6b)  
(6c)  
(6d) 
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
(7) 
where
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 Oblique0 ; Oblique1 from unity,
(8) 
Second, the extra kinetic mixing between and modifies the and boson propagators
(9) 
in the notation of ref. Hagiwara:1998 , which contributes to the parameter Oblique1
(10) 
Here the overlined couplings , and are the effective couplings that contain the gaugeboson propagator corrections at the momentum transfer Hagiwara:1998 . 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
(11)  
where the 9 dimensionless couplings, , parametrize all the nonstandard interactions:
(12a)  
(12c)  
(12d)  
(12e)  
(12g)  
(12h)  
(12i) 
From the effective Lagrangian of eq.(11), we obtain the Feynman rule for vertex as
(13) 
where all three momenta are incoming, , as shown in the fig. 1. and can be , , , , or . The coefficients are
(14a)  
(14b) 
for the couplings,
(15a)  
(15b) 
for the couplings,
(16a)  
(16b)  
(16c) 
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,
(17a)  
(17b)  
(17c) 
Although we do not consider offshell Higgs boson contributions in this report, should be replaced by in the above Feynman rules when the Higgsboson is offshell.
Iii Optimal Observables and Phase Space
iii.1 Optimal observables method
The optimal observables method gunion makes use of all the kinematic distributions which are observable in experiments. We therefore summarize our phasespace parameterizations for all the Higgs boson production processes in collisions considered in this study, which can be generically written as
(18) 
Here and are the four momenta and helicities, respectively, and and are the four momenta and helicities, respectively, of the produced fermion () and antifermion (). 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 vectorboson 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
(19) 
where denotes the SM helicity amplitude, and denotes the nonstandard 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
(20) 
where the nonstandard couplings are assumed to be real and small, and hence the terms quadratic in couplings are dropped. Here is the 3body phase space volume of the system, and
(21) 
gives the differential cross section of the SM. The term proportional to ,
(22) 
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
(23a)  
(23b) 
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
(24) 
The function can then be expressed as
(25) 
where
(26a)  
(26b) 
where we take as a nominal integrated luminosity through out this report. If the total number of events is sufficiently large, the integral representation in eq.(26b) gives a good approximation for the matrix. The value of and the mean value depend on the actual experimental results, or the small deviation from the equality in eq.(24). If the SM prediction gives a reasonably good description of the data in most of the phase space region, then the statistical errors of and their correlations are determined solely in terms of the covariance matrices , which is the inverse of the matrix given in eq.(26);
(27) 
In practice, however, we should address the following subtleties:

If the statistical error becomes small, systematic errors need to be considered.

The results depend on how we split the total Luminosity to different beam polarizations.

Not all the 3body phase space points are observable in experiments.
As for the first issue, we assume that the energy and angular resolutions of ILC detectors are good enough to justify our integral approximation of eq.(26), and consider only the impacts of the luminosity uncertainty as a source of the systematic error which is discussed in section IX. We leave the difficult problem of background contaminations and the spectrum distribution due to bremsstrahlung and beamstrahlung photon emissions to future studies. In short, our results should be regarded as an ultimate accuracy of the couplings measurement for a perfect detector in a backgroundfree environment, when the SM predictions are accurately known.
On the second point, we provide numerical results for the two very simple cases only:

Unpolarized beam : The total integrated luminosity is given for collisions with at each collider energy . However, in order to save the length of this article, we provide the unpolarized results specifically only for channel production at . They are calculated for all the processes at all energy choices and are used to evaluate the significance of the beam polarization after all the channels and energies are combined in section X.

80% polarized beam : Exactly half of the total luminosity is given for collisions with , and the remaining half with .
In general, the covariance matrix depends on the partition of the total luminosity into experiments with different sets of and beam polarizations. If the and