Effective fermion-Higgs interactions at an $e^+e^-$ collider with polarized beams

We consider the possibility of new physics giving rise to effective interactions of the form $e^+e^-Hf \bar f$, where $f$ represents a charged lepton $\ell$ or a (light) quark $q$, and $H$ the recently discovered Higgs boson. Such vertices would give contributions beyond the standard model to the Higgs production processes $e^+e^- \to H\ell^+\ell^-$ and $e^+e^- \to H q \bar q$ at a future $e^+e^-$ collider. We write the most general form for these vertices allowed by Lorentz symmetry. Assuming that such interactions contribute in addition to the standard model production processes, where the final-state fermion pair comes from the decay of the $Z$ boson, we obtain the differential cross section for the processes $e^+e^- \to H\ell^+\ell^-$ and $e^+e^- \to Hq\bar q$ to linear order in the effective interactions. We propose several observables with differing CP and T properties which, if measured, can be used to constrain the couplings occurring in interaction vertices. We derive possible limits on these couplings that may be obtained at a collider with centre-of-mass energy of 500 GeV and an integrated luminosity of 500 fb$^{-1}$. We also carry out the analysis assuming that both the electron and positron beams can be longitudinally polarized, and find that the sensitivity to the couplings can be improved by a factor of 2-4 by a specific choice of the signs of the polarizations of both the electron and positron beams for the same integrated luminosity.


Introduction
While the present data from the LHC indicate that the particle of mass around 125 GeV discovered recently may be the standard model (SM) Higgs boson, the accuracy of the present experiments is not sufficient to nail the issue. Many of its couplings to fermions and gauge bosons have been measured and found to be consistent with those expected from the SM [1]. Nevertheless, the data as yet allows for wide deviations from the SM. It is thus an open question whether the SM is the ultimate theory. We need to investigate alternative scenarios for electroweak symmetry breaking, which would be tested at future runs of the LHC, or possibly, at an e + e − collider which has now a reasonable hope of being constructed.
There are a number of scenarios beyond the standard model for spontaneous symmetry breaking, and ascertaining the mass and other properties of the scalar boson or bosons is an important task. This task would prove extremely difficult for the LHC. However, scenarios beyond SM, with more than just one Higgs doublet, as in the case of the minimal supersymmetric standard model (MSSM), would be more amenable to discovery at a linear e + e − collider operating at centre-of-mass (cm) energies of 500-1000 GeV. We are at a stage when the International Linear Collider (ILC) seems poised to become a reality [2].
Scenarios going beyond the SM mechanism of symmetry breaking, and incorporating new mechanisms of CP violation, have also become a necessity in order to understand baryogenesis which has resulted in the present-day baryon-antibaryon asymmetry in the universe. In a theory with an extended Higgs sector and new mechanisms of CP violation, the physical Higgs bosons are not necessarily eigenstates of CP. In such a case, the production of a physical Higgs can proceed through more than one channel, and the interference between two channels can give rise to a CP-violating signal in the production.
There have been a number of studies examining possibilities of measuring couplings of a Higgs boson which may belong to an extension of the standard model [3]- [15]. Here we consider in a general model-independent way the production of a Higgs mass eigenstate H in a possible extension of the SM at an e + e − collider. We restrict ourselves to the case when the Higgs boson is accompanied by a fermion pair. Such a final state can arise in the SM or its extensions through the HiggsStrahlung (HS) process e + e − → HZ, an important mechanism for the production of the Higgs boson, with the final Z decaying into a fermion pair. In case the final-state leptons are e + e − , the final state can also arise in the process of vector boson fusion (VBF), when virtual Z bosons emitted by the e + and e − beams fuse to produce a Higgs boson. However, this does not exhaust all possibilities. We consider here an effective anomalous e + e − Hff vertex, where f represents a charged lepton ( ≡ e, µ, τ ) or a light quark. This vertex is supposed to represent a contribution to the process e + e − → Hff of interactions beyond the SM (BSM), the SM contributions being HS and VBF described above. However, it includes contributions of HS and VBF processes going beyond SM. Not only that, it can include contributions which do not fall under these two categories, as for example, contributions coming from box diagrams for ZH production, or pentagon diagrams for a Hff final state.
We will parametrize the five-particle vertex by means of various Lorentz structures, whose coefficients will be momentum-dependent form factors. We will then propose kinematic observables, whose measurements at an e + e − collider could enable a determination of these form factors, or at least contrain them. We will also estimate 95% confidence-level (C.L.) limits that can be put on these form factors at a collider operating at a centre-of-mass energy of 500 GeV with an integrated luminosity of 500 fb −1 .
One specific practical aspect in which our approach differs from that of the effective Lagrangians is that while the couplings are all taken to be real in the latter approach, we allow the couplings to be complex, and in principle, momentum-dependent form factors.
Polarized beams are likely to be available at a linear collider, and several studies have shown the importance of longitudinal polarization in reducing backgrounds and improving the sensitivity to new effects [16]. In earlier work, it has been observed that polarization does not give any new information about the anomalous ZZH couplings when they are assumed real [4]. However, the sensitivity can be improved by suitable choice of polarization. Moreover, polarization can indeed give information about the imaginary parts of the couplings. A model-independent approach on kinematic observables in one-and two-particle final states when longitudinal or transverse beam polarization is present, which covers those of our present processes without e + e − in the final state, can be found in [17].
In this work, our emphasis has been on estimating limits on couplings which may be measured making use of combination of expectation values of kinematic observables, and/or polarizations. We have also tried to consider rather simple observables, conceptually, as well as from an experimental point of view.
When all couplings are assumed to be independent and nonzero, expectation values are linear combinations of a certain number of anomalous couplings (in our approximation of neglecting terms quadratic in anomalous couplings). By using that many number of observables, for example, different asymmetries, or the same asymmetry measured for different beam polarizations, one can solve simultaneous linear equations to determine the couplings involved. A similar technique of considering combinations of different polarizations was made use of, for example, in [18]. While this is straightforward in principle, we have far too many couplings in the problem. We therefore restrict ourselves to an analysis assuming one coupling nonzero at a time.
The rest of the paper is organized as follows. The next section (Sec. 2) contains a formulation of the effective contact interaction vertices. In Sec. 3 we derive expressions for the differential cross sections including the SM and the contact interaction contributions, the latter to linear order. Sec. 4 contains numerical results for the expectation values of the variables chosen, and the limits on couplings that may be obtained from the measurement of the expectation values. The conclusions and a discussion are contained in Sec. 5.
2 Effective e + e − Hff vertex The effective Lagrangian for a contact e + e − Hff interaction depends on whether f represents an electron or one of the other light fermions (f ≡ = e, q).
An effective Lagrangian which can represent a coupling e + e − He + e − , assuming that the electron vertices conserve chirality as well as that the electron currents are conserved under the assumption that the electron lines will represent on-shell, massless electrons, can be written as An effective Lagrangian for contact e + e − Hff interactions, where f is one of the other light fermions, can be written with the same assumptions as above, and in addition, that the fermion currents conserve flavour, in the form In the above equations, the electron is represented by the field ψ e , f by the field ψ f , and H represents the Higgs field. A, B refer to the chiralities L and R, and P L , P R are the left-and right-chirality projection matrices. M is a common, arbitrary scale introduced to make all the couplings dimensionless. For our analysis, we choose this scale to be equal to 100 GeV, of the order of the mass of the Z boson. The Lagrangian is Hermitian with the couplings λ i AB chosen to be real. The terms with couplings λ 0 AB , λ 1 AB , λ 2 AB , λ 3 AB , λ 7 AB and λ 7 AB conserve CP, whereas the rest of the terms are CP violating. The couplings λ 0 AB , λ 1 AB and λ 2 AB in (1) are symmetric under A and B interchange. The effective Lagrangians in (1) and (2) are restricted to terms upto dimension 9.
An effective five-particle e + e − Hff vertex can be represented in terms of the amplitude for the process which may be parametrized as Eq. (4) is the most general vertex respecting Lorentz invariance and chirality conservation at the e + e − and ff vertices.
are Lorentz scalar form factors which are in general complex. Since no further assumption is made about these form factors, they can be CP-violating. The CP violation could come either from the mixing of the Higgs fields with different CP properties, or from a combination of interaction vertices, some of which violate CP, and contribute to make up the form factors.
Note that since we will neglect all fermion masses, in (4) the terms with i = 1, 2 and j = 3, 4 vanish on using the Dirac equation for the spinors. Since these are complex, there is a total of 88 real form factors. However, as it turns out, only 52 of these contribute to the differential cross section at linear order in the form factors. The form factors of course depend on the actual final state -whether f represents e, = e or q. We will treat these three cases separately.
The vertex in eq. (4) follows from the effective Lagrangian (2), evaluating the process at lowest order, and allowing the real couplings to be replaced by complex, momentum-dependent form factors. In case of f ≡ e, the Lagrangian (1) can give rise to additional terms in order to respect crossing symmetry. These additional terms can be rewritten using Fierz transformations in the same form as in (4), but with additional Dirac matrices, viz., scalar, pseudoscalar and tensor. For simplicity, we neglect these additional terms and assume that the same form of vertex can be used for all light fermions (leptons and quarks) in the final state.
We remind the reader that we do not think of the contact-interaction vertex as a genuine point vertex, but could be made up a combination of three-or four-point vertices and propagators, all of which give the form factors their momentum dependence.
The SM contribution to the process would also take the same form (4). However, we will separately write down the tree-level SM contribution coming from HS and VBF processes (the latter only in case of e + e − in the final state). Thus, the vertex in (4) will be assumed to contain only the anomalous contribution, coming from SM loop contributions or from new physics.
We investigate here how these contact-interaction form factors can be determined, or constrained, at a linear collider, with or without longitudinally polarized beams. Obviously, the number of form factors is large, and one can only constrain one or two of these at a time assuming them to be the only ones nonzero. A more systematic analysis can be carried out for the simultaneous measurement of more than one observable, whose expectation value would be a linear combination of the anomalous couplings, and then solving simultaneous equations to determine individual couplings. Here we make the simplifying assumption that at a time, only one of the several couplings is nonzero, and see how well the measurement of each of the five chosen observables can constrain it. We will see that the possibility of beam polarization enhances the senstivity of the procedure, and by judicious choice of polarization the limits can be much better than those that can be set without polarization.
While the above discussion refers to the process (3) with f ≡ as well as f ≡ q, in what follows, we do not consider the case where the final state has a quark pair. The reason is that since light quark flavours are not possible to distinguish, we would need to add contributions of all flavours, which may all have different contact couplings. The resulting large number of couplings would make the process quite intractable. We thus restrict ourselves to the channels with e + e − and µ + µ − in the final state. The expressions derived below, however, can be easily modified to include a qq in the final state.
In what follows, we shall calculate the differential cross section including the SM amplitudes and the amplitude coming from the contact interaction for the generic process (3), without distinguishing the various final states, it being understood that the VBF contribution will be absent when the final state does not have f ≡ e, and SM couplings and the form factors coming from (4) will be appropriately chosen, depending on the final state.

Differential cross section
In order to obtain the differential cross section for the process in eq. (3), we first obtain the squared matrix element for the process in terms of the contributing processes. We assume that the contact interaction vertex is numerically small, and we include it only to linear order. While we include both HS and VBF contributions for the SM below, it being understood, as mentioned before, that the VBF contribution has to be dropped when the final state does not have an electron-positron pair. We have made use of the software package FORM [19] for algebraic manipulations.
The SM contribution to the process (3) with f ≡ e consists of two sub-processes: Hig-gsStrahlung and vector-boson fusion. The respective amplitudes for these two mechanisms are and In these equations, g V and g A are respectively the vector and axial-vector couplings of the Z to an electron, given by and θ W is the weak mixing angle. In writing the amplitudes (5) and (6), the electron mass is neglected.
For the process (3), with f ≡ = e and f ≡ q, the VBF process does not contribute. The squared matrix element for only the SM contribution, with longitudinally-polarized e − and e + beams, is where Here we have used The squared matrix elements for the interference between the SM and BSM processes for form factors with the various chirality combinations of couplings with longitudinally-polarized e − and e + beams are the following: It is worth noting from the above that the VBF contribution of the SM occurs only for the LL and RR combinations. This is because in the VBF process, chirality conservation in the V , A couplings of the Z bosons to the electrons implies that helicities of the incoming electron (positron) and outgoing electron (positron) are equal.
In terms of the squared matrix elements listed above, the differential cross section in the e + e − cm frame is given by dσ where E 3 and E 4 are the energies of the outgoing f andf , respectively, θ is the polar angle of p 3 , chosen to lie in the xz plane, with the initial e − direction chosen as the z axis, and η is the azimuthal angle of p 4 in a rotated frame with the p 3 direction as the z axis and the same y axis as before. Using the above differential cross section, we study the expectation value of five kinematic observables X i (i = 1 − 5), constructed out of the energy and momenta of the initial and final states, defined by with the z axis chosen along the incoming e − direction. We have used P = p 1 + p 2 and q = p 3 + p 4 . These are some relatively simple observables, with different properties under CP and T as discussed below, and therefore sensitive to different combinations of couplings. These are characterised by well-defined properties under CP and naive time reversal T (i.e., reversal of all spin and momenta, without interchange of initial and final states). X 1 and X 5 are both even under CP. However, while the former is even under T, the latter is odd under T. The remaining observables are odd under CP. Of these, X 2 is even under T, whereas X 3 and X 4 are odd under T. Because of different properties under CP, they would have nonzero expectation values for different combinations of couplings. The behaviour under T decides whether the expectation value depends on the the dispersive or absorptive part of the form factor, since strict CPT (i.e., with genuine, not naive, T) conservation rules out nonzero expectation values of CPT-odd observables in the absence of absorptive parts.
To make these transformation properties clear, we could have chosen appropriate combinations of couplings, derived from momentum-space calculation of amplitudes from the Lagrangian in (1), which would have definite CP properties. In that case, we would have had, apart from the CP-even form factors A, the following combinations: In the above equations, we have suppressed the chirality subscripts. These combinations are to be written for each of the chirality combinations LL, RR, LR and RL. Of these couplings, B 1 , B 2 , C 3 , C 4 , C 5 and C 6 are CP even, and the rest are CP odd. We have however chosen a simpler set of couplings in eq. (4). As a result we find that there are relations between expectation values of our observables which depend on CP and T transformation properties.
In the next section we describe how the expectation values of the chosen variables X i can be used to place limits on various form factors. We have examined the accuracy to which couplings can be determined from a measurement of the correlations of observables X i . The limits which can be placed at the 95% C.L. on a coupling contributing to the correlation of X i is obtained from where the subscript "SM" refers to the value in SM, "σ tot " is the cross section including the contributions of anomalous couplings upto the linear order, and where f is 1.96 when only one coupling is assumed nonzero. As mentioned earlier there are 88 independent form factors for the contact interaction e + e − e + e − H vertex to be constrained, of which only 52 appear in the differential cross section. We categorize all the couplings into four groups, namely, LL, RR, LR and RL based on chiralities. We evaluate the limits on each coupling taking one coupling nonzero at a time.
In Fig. 1 we show the cross sections in the SM for the processes e + e − → He + e − (upper panel) and e + e − → Hµ + µ − (lower panel) as functions of a cut-off θ 0 in the forward and backward directions on the polar angles of the final-state leptons. It is seen that the cross section being in the region of a few femtobarns (or more) for most values of the the cut-off, an integrated luminosity of 500 fb −1 which we consider will give a sizeable number of events.

Limits for polarizations Observable
Coupling

Limits for polarizations Observable
Coupling

Limits for polarizations Observable
Coupling     We can see from Tables 1-2 that the limits on some pairs of couplings are equal. As discussed earlier, this can happen because the CP and T properties of the observables determine a com-

Limits for polarizations Observable
Coupling    The 95 % C.L. limits on the RL contact couplings for f ≡ e, chosen nonzero one at a time, from various observables with unpolarized and longitudinally polarized beams for √ s = 500 GeV and integrated luminosity L dt = 500 fb −1 .   bination of couplings which contributes to the expectation value of that observable. In case of limits on B form factors being equal to the limit on the corresponding C form factor, the reason is merely that both these form factors contribute equally to the differential cross section. In general, the limits obtained using beam polarization are better than those obtained with unpolarized beams. The improvement is by a factor of 2 or 3. However, the sign of the polarization is crucial. Firstly, as observed earlier, only e + and e − polarizations of opposite signs improve the sensitivity. Secondly, the combination P L = −0.8, P L = +0.3 enhances the sensitivity in the cases of the chirality combinations LL and LR, whereas the combination P L = +0.8, P L = −0.3 improves the limits for the combinations RR and RL. In general, the limits can reach the level of a few times 10 −4 , and in some cases, for the right polarization and chirality combinations, even a few times 10 −5 .
We have also evaluated the expectation values of the observables X i (i = 1, 2,. . . ,5) assuming a cut-off on the polar angle θ of the final-state leptons in the forward and backward direction, as required to stay away from the beam pipe, i.e., we restrict the angle according to θ 0 < θ < π − θ 0 . It turns out that the limit on the coupling in such a case can be sensitive to the cut-off θ 0 , and the cut-off, in fact, can be chosen so as the optimize the limit. While this exercise can be done for all choices of beam polarization, we exhibit our results only for the polarization combination which yields the best limits for the respective chirality combinations. We have shown, in Figs. 2-7, the dependence of the limits on the arbitrary scale M and the cut-off angle θ 0 .
The dependence of the limits on the mass scale M may be understood in terms of a simple rescaling. Except for the A terms, which carry a factor M −3 before them, all other terms (B and C) carry a factor M −5 . The limits, therefore, scale in these cases as M 3 and M 5 , respectively. This can be seen to be borne out by the plots.
The dependence of the limits on θ 0 is more complicated. The limit depends on the new physics contribution as well on the SM expectation values, which may have different cut-off dependences. An interesting feature that can be seen from the plots is a peak in the limit on the coupling for some observables. Such a peak arises because the expectation value coming from the contact   Figure 7: The 95 % C.L. limits on the anomalous contact couplings for f ≡ e as a function of the cut-off angle θ 0 chosen nonzero one at a time from observables X 3 , X 4 and X 5 with longitudinally polarized beams for √ s = 500 GeV and integrated luminosity L dt = 500 fb −1 .

Limits for polarizations Observable
Coupling  We now turn to the process e + e − → Hµ + µ − . The analysis carried out in the previous subsection is repeated here, with the same observables. The results are, however, numerically different because of the different SM contribution. It should also be kept in mind that though we use the same symbols B and C for the various form factors, these are different from the ones occurring in the previous subsection in the process e + e − → He + e − .
In this case, there is no contribution from the VBF process in the SM. One consequence is that, since as noted earlier, the VBF contribution came in only for the LL and RR combinations of couplings, the LR and RL combinations of contact couplings do not, therefore, have the corresponding helicity combinations from SM to interfere with. As a result, the results of the previous subsection go through completely for the LR and RL combinations. We do not, therefore, list the corresponding results again in this subsection.
We list below in Tables 5 and 6 the results for limits that can be obtained for unpolarized beams, as well as for the two polarization combinations used earlier. The qualitative conclusions on the dependence of the limits on the polarization drawn in the previous subsection continue to hold also for this process.
As before, we have evaluated the limits in the various cases in the presence of a cut-off θ 0 on the forward and backward angles of the leptons. The plots for the limits as functions of M and θ 0 for the various observables are displayed in Figs. 8, 9 and 10.

Limits for polarizations Observable
Coupling     Figure 10: The 95 % C.L. limits on the anomalous contact LL and RR couplings for f ≡ µ as a function of the scale M (top) and cut-off angle θ 0 (bottom), chosen nonzero one at a time, from the observables X 3 , X 4 and X 5 with longitudinally polarized beams for √ s = 500 GeV and integrated luminosity L dt = 500 fb −1 .

Conclusions and Discussion
We have considered in the foregoing a model-independent way of characterizing the production of a Higgs mass eigenstate H in a possible extension of SM at an e + e − collider. We examine the process in which Higgs boson is accompanied by a fermion pair resulting in a final state which arises in the SM or its extensions through the HiggsStrahlung process e + e − → HZ, with the final Z decaying into a fermion pair or through the process of vector boson fusion, in case the final-state leptons are e + e − . Representing new interactions by an effective anomalous e + e − Hff vertex, we parametrize the vertex by means of various Lorentz structures, whose coefficients will be momentum-dependent form factors.
Choosing certain kinematic observables X i possessing definite CP and T properties, whose expectation values would be measured at the e + e − collider, we have estimated 95% C.L. limits that can be put on these form factors at a centre-of-mass energy of 500 GeV and an integrated luminosity of 500 fb −1 . For simplicity, we assume one coupling to be nonzero at a time, with the remaining couplings set to zero. We find that the limits possible on the couplings range between a few times 10 −2 down to a few times 10 −5 . These are listed in detail in the tables. The analysis has also been carried out assuming that beams can be polarized. It is found that for suitable combinations of e + and e − polarization, the sensitivity can be enhanced. As mentioned earlier, an independent determination of all couplings is not possible by the limited number of observables. However, combining results from different beam polarization combinations can help to determine additional couplings.
We have also determined the projected limits for an experimental situation where a cut-off is put on the forward and backward directions of the fermions. Such a cut-off is needed to remain away from the beam pipe. It can moreover be chosen so as to optimize the efficiency. The corresponding limits are plotted as functions of the cut-off angle θ 0 . Also shown in the plots is the dependence of the limits on the arbitrary scale M introduced to make the coupling constants dimensionless.
We have assumed a detection efficiency 1 for the Higgs boson. While this is an idealized situation, in practice, it should be possible to use a number of different Higgs decay channels and combine the results. It will be possible to utilize even the dominant hadronic (bb) decay channel, since the QCD background is absent. We have not taken into account detector efficiencies or loss of efficiency on imposition of kinematic cuts to eliminate backgrounds in this preliminary work, though we do use a cut on the polar angle of the lepton. While these considerations may change our numerical results somewhat, they are not likely to change drastically.
The limits depend on the arbitrary scale M which we choose. In the standard low-energy effective Lagrangian approach, M is chosen to be the cut-off Λ at which new physics is supposed to become important, and usually taken to be of the order of TeV. However, in our approach, the actual energy dependence is built into the form factors, and we are free to choose any M we like. For a different choice of M , the limits have to be appropriately scaled. The advantage of our scheme is that it does not assume that all new particles are heavier than some scale which is large -in principle, there can be other particles, as for example, other Higgs scalars, not necessarily much heavier than the known one at 125 GeV.
It would be interesting to examine predictions from various popular scenarios for new physics for the various form factors introduced here. This would enable one to determine to what extent these models could be constrained by following the analysis suggested here.