

 19 Oct 1995

EXTENDED GAUGE MODELS AND e\Gamma e\Gamma ! W \Gamma W \Gamma AT THE NLC

THOMAS G. RIZZO Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA

ABSTRACT

We briefly discuss two possible manifestations of the lepton number violating reaction e

\Gamma e\Gamma ! W \Gamma

i W

(\Lambda )\Gamma j , which probes the masses and mixings of heavy

Majorana neutrinos, at the Next Linear Collider(NLC). Cross sections for this

process are shown to be potentially quite large at center of mass energies of order 1-1.5 TeV.

The possibility that some version of the generic process e\Gamma e\Gamma ! W \Gamma i W (

\Lambda )\Gamma

j mightbe observable at high energy colliders, such as the NLC, has been a subject of much

discussion for almost 15 years1. From this choice of notation we want to make it clear from the beginning that this generic reaction represents a rather large number of distinct processes depending on the model under consideration, the most obvious being the production of an on-shell pair of `left-handed' Standard Model(SM) W 's within the SU (2)L \Omega U (1)Y framework. Another class of reactions is based on the Left-Right Symmetric Model.

Of course, no matter what the specific process is, a number of model independent requirements must be satisfied. First, since we are dealing with the weakly interacting sector of a renormalizable gauge theory, we require that partial wave unitarity be satisfied for center of mass energies much larger than all other mass scales in the calculation. This is relatively straightforward to enforce when constructing various gauge theory models but some care is still required. Secondly, and just as important, we must

remember that the reaction e\Gamma e\Gamma ! W \Gamma i W (

\Lambda )\Gamma

j is lepton number violating, i.e., it corre-sponds to a j\Delta Lj = 2 process in complete analogy with neutrinoless double beta decay,

fifi0*. (In fact, e\Gamma e\Gamma ! W \Gamma i W (

\Lambda )\Gamma

j is often referred to as inverse neutrinoless doublebeta decay.) This means that whatever specific model is being considered must not only

have a mechanism to induce j\Delta Lj = 2 reactions but that their form and magnitude must be consistent with existing experimental constraints2 arising from fifi0*. The most common way of introducing j\Delta Lj = 2 operators is through Majorana mass terms in the neutrino mass matrix which requires an extension to (at least) the fermion sector of the SM. e\Gamma e\Gamma ! W \Gamma W \Gamma then takes place via t- and u-channel Majorana neutrino exchanges where the couplings of the neutrinos to the W and electron are model dependent. Clearly, in the limit of vanishing Majorana neutrino masses, the e\Gamma e\Gamma ! W \Gamma W \Gamma amplitude vanishes.

It is easy to see why the simplest two neutrino scheme of this kind which includes the see-saw mass generation pattern cannot yield observable cross sections at the NLC.

Fig. 1. Discovery limits for e

\Gamma e\Gamma ! W \Gamma W \Gamma at the NLC with various center of mass energies(dashed lines). Mi is the heavy neutrino mass while U 2ei is the square of the relevant mixing angle. Unpolarized beams are assumed together with a luminosity of 80(s=1 T eV 2)f b

\Gamma 1. The

dotted curve in the 2 TeV case corresponds to polarized beams. The region being probed is above the curves in all cases and 10 events are required for discovery. The diagonal solid line is the fifi0* constraint of B'elanger et al. while the horizontal solid line arises from universality. Here, the allowed region is below the line.

If D and M represent the Dirac and Majorana terms in the 2x2 mass matrix then, in the limit D=M !! 1, the two eigenvalues are m1 ' \Gamma D2=M and m2 ' M with a mixing angle OE ' \Gamma D=M . OE is thus forced to be small by the same mechanism which induces the hierarchy in the see-saw mass relation. The amplitude for the ith neutrino exchange behaves as Ai , miF (m2i =s; M 2W =s), with F ! 1 in the limit that s ! 1, thus vanishing as mi ! 0. If we sum over both neutrino exchanges and use the mass and mixing relations above, we find that the total amplitude behaves in the s ! 1 limit as A , M [F (m2) \Gamma F (m1)]OE2. Here, unitarity is clearly satisfied as the difference in the brackets vanishes in the high energy limit. We see, however, that the cross section is now proportional to OE4 and we recall that OE must be small. To see how devastating this mixing angle suppression is, let us assume that M = 500 GeV, OE = 10\Gamma 2 andp

s = 1 TeV. A complete calculation then yields a cross section of ' 0:015 fb, which is far too small to be observable at the NLC with an integrated luminosity of 100 f b\Gamma 1.

Clearly, within the SM framework, the only way around this disappointing result is to add additional massive neutral fermion fields and to search for a fine-tuning solution to maximize the cross section (i.e., up to the few fb level) while trying to satisfy all the other constraints. As mentioned above, the fifi0* constraint is particularly important in the case of heavy neutrino exchange and is a source of controversy amongst the various groups who disagree about the size of the uncertainty in the nuclear matrix elements. B'elanger et al.1 argue that these uncertainties are no more than a factor of 2-3 while Heusch and Minkowski1 claim that they are substantially larger. The impact of this one constraint can be crucial as shown in Fig.1 from the work of B'elanger et al... We see that a 2 TeV NLC with polarized beams will be barely able to produce an observable rate and that generally higher energy machines are required. Heusch and Minkowski argue that the fifi0* bound is set far too low in this plot and that a 1 TeV machine is sufficient to obtain a reasonable cross section. It is clear that a thorough examination of the disputed nuclear matrix elements is in order so that this situation can be rectified in the near future.

If we go beyond the SM gauge group, the W 's need not be associated with the SU (2)L group factor. The quintessential example of this situation is provided by the Left-Right Symmetric Model3 based on the gauge group SU (2)L \Omega SU (2)R \Omega U (1). In this model we can imagine not just the SM W \Gamma W \Gamma production process but also W \Gamma W \Gamma R and W \Gamma R W \Gamma R as well. But aren't WR's too massive to be pair produced at the NLC? There are many published lower bounds on the WR mass, MR, from a number of different sources but all of them are easily avoided by the great parameter freedom of the LRM. In fact, the LRM owes much of its survival over the last two decades to the plethora of free `parameters' it contains: (a) the ratio of the SU (2)R and SU (2)L gauge couplings, 0:55 ^ ^ = gR=gL ^ 2, with the lower limit being forced upon us by the internal consistency of the model and the upper bound is from naturalness arguments; (b) the masses of the right-handed(RH) neutrinos, (c) the elements of the RH CKM mixing matrix, VR, which are a priori different than the conventional VL of the SM, and (d) the existence of other additional particles, e.g., charged Higgs, in the spectrum which may participate in loop graphs for rare processes and may also appear as on-shell final states in WR decay.

Before we turn to the production of WR's in e\Gamma e\Gamma collisions we first must justify

Fig. 2. (a)Tevatron WR search reach as a function of ^eff as described in the text for the luminosities of (from bottom to top) Run Ia, 100, 250, 500, and 1000 pb

\Gamma 1. (b) Percentage of

the VR parameter space allowing the mass of WR below a given value from the CDF run Ia search.

the possibility that they can indeed be light. So let us briefly discuss how the conventional limits are evaded. First, in this model the magnitude of the mass(MN ) of the right-handed neutrino, *R, is a priori unknown. If the usual see-saw mechanism is invoked, *R is expected to appear as a heavy Majorana neutrino with a mass set by the RH breaking scale. One of the well-known low energy limits on MR arises from the polarized _ decay experiments4 which yield bounds of order 480 GeV. Of course if the *R's are more massive than a few hundred MeV then they cannot be produced as final states in either K, ss, or _ decay thus avoiding this low energy bound in a rather trivial way. Most analyses of the LRM assume that the elements of VR and the conventional CKM matrix, VL, differ at most by phase factors. If VR = VL, then it has been known for some time that considerations of the KL \Gamma KS mass difference result in a rather strong lower bound5,6 on the mass of WR of order 1.6 TeV(or more), thus placing it outside the search capabilities of existing colliders as well as the NLC. However, if we remove the constraint of VR = VL and allow VR to be arbitrary, even in the absence of fine-tuning, we find that WR can be as light as 280 GeV for a top quark mass of 180 GeV. (Recall that VR has nine free parameters: three mixing angles and six phases.)

The last well-known constraint on MR arises from direct searches at the Tevatron. Unlike the corresponding Z0 searches at hadron colliders, W 0 searches via the DrellYan process have many subtleties; this is most certainly the case within the LRM. The standard CDF/D0 W 0 search7 assumes that the q0 _qW 0 production vertex has SM strength (i.e., (i) ^ = 1 and (ii) jVLij j = jVRij j), that the RH neutrino is (iii) `light' and `stable', appearing as missing ET in the detector, and that the WR's leptonic

branching fraction(Bl) is the same as the SM value apart from contributions due to open top(i.e., (iv) no exotic decay channels are open). If any of these assumptions are invalid, what happens to the search reach? Assumptions (i) and (iv) are easily

accounted for by the introduction of an effective ^ parameter, ^eff = ^qBl=BSMl which simply adjusts the overall cross-section normalization with the resulting reach shown in Fig.2a. If assumption (ii) is invalid, a significant search reach degradation occurs as is shown in Fig.2b for the CDF run Ia result; e.g., one finds via a Monte Carlo study8 that for 50(10)% of the VR parameter space, the CDF run Ia WR limit is weakened to less than 550(400) GeV. This reduction is a result of modifying the weight of the various parton luminosities which enter into the calculation of the production cross-section. Life gets much harder if *R does not appear as missing ET . A massive *R will most likely decay within the detector to `\Sigma + jj, with either charge sign equally likely if *R is a Majorana fermion. D07 has looked for WR's decaying in this manner and has ruled out a reasonable portion of the MR \Gamma MN plane under the assumption that ^ = 1 and VL = VR. For MN ! MR, the approximate region MR ! 550 GeV and MN ! 270 GeV is found to be excluded. Perhaps the worst case scenario is when *R is more massive than WR so that WR has only hadronic (or exotic) decay channels open. Can WR be seen as a bump in dijets? Clearly the chances are somewhat better at the Tevatron, where S=B is perhaps manageable given reasonable statistics, than at the LHC. CDF has already performed such an analysis with run Ia data7 with somewhat limited results in that only a narrow MR range is presently excluded even when ^ = 1 and VL = VR is assumed. All of these searches will improve dramatically, however, as the Tevatron luminosity increases.

At the NLC, the amplitude for e\Gamma e\Gamma ! W \Gamma R W \Gamma R (or W \Gamma R W \Lambda \Gamma R ) receives both t\Gamma and u\Gamma channel contributions from the RH neutrino in a manner similar to what we saw in the case of the SM discussion above. Apart from a factor of ^ this coupling strength is unity and no small mixing angles are involved. However, there is also a new s\Gamma channel contribution from the exchange of a doubly-charged Higgs boson(\Delta ), with mass M\Delta . This doubly-charged Higgs field is a natural consequence of both SU (2)R breaking as well as the generation of the Majorana mass term for the neutrino and the see-saw mechanism. Since the e\Gamma e\Gamma \Delta coupling is proportional to MN and the e\Gamma N WR coupling is chiral, the total amplitude is found to be proportional to MN . The combination of exchanges in all three channels preserves unitarity and can be easily made to satisfy the fifi0* constraints since WR is so massive and we have complete freedom in the VR elements. At NLC energies, i.e., ps = 0:5 \Gamma 1:5 TeV, the cross section for e\Gamma e\Gamma ! W \Gamma R W \Gamma R is quite large (of order a few pb) provided it is kinematically accessible. The production rates are fairly sensitive to the values of MN and M\Delta , and the cross section is found to have a rather flat angular distribution due to the dominance of the longitudinal-longitudinal helicity amplitude. Of particular interest is the possibility that the s\Gamma channel \Delta may appear as a resonance depending upon the value of ps.

Unfortunately, the `reach' of the NLC for WR pair production is rather limited since we are constrained to WR masses less than ps=2. It is thus reasonable to contemplate that like-sign WR pair production may not be kinematically accessible at these center of mass energies. This forces us to consider the possibility of singly producing

Fig. 3. Cross section for like sign WR production at a 1 TeV NLC assuming ^ = 1 and MR = 480 GeV as a function of (a)MN and (b)M\Delta . In(a)[(b)], the curves on the right(left)- hand side correspond, from top to bottom, to M\Delta =800,1200,500,1500,200, and 2000 GeV [MN =1500,1200,800, 500, 200 GeV].

WR's via the reaction e\Gamma e\Gamma ! W \Gamma R (W \Gamma R )\Lambda ! W \Gamma R jj. We limit ourselves to this jj mode to allow for the possibility that MN ? MR in which case WR can only decay to jj (barring the existence of exotic decay modes). Of course, allowing one of the WR's to be off-shell we are forced to pay the price of an additional gauge coupling as well as three-body phase space. This results in a substantial reduction in the cross section from the on-shell case to the level of a few f b. This implies that machine luminosities in the range of L = 100 \Gamma 200f b\Gamma 1 are required to make use of this channel. The total event rates for e\Gamma e\Gamma ! W \Gamma R (W \Gamma R )\Lambda ! W \Gamma R jj are found in Figs. 4 and 5, in which we have set ^ = 1 and have scaled by an integrated luminosity of 100f b\Gamma 1. Here we see that the reach for WR's in this channel can be as large as MR ' 0:8ps.

Fig. 4a(b) shows the number of expected WR + jj events, as a function of MR, at a ps = 1(1:5) TeV e\Gamma e\Gamma collider for different choices of MN and M\Delta . The results are seen to be quite sensitive to the values of these mass parameters even when MR is fixed. Setting MR = 1 TeV in Figs. 5a and 5b, we again see reasonable event rates for most choices of MN and M\Delta assuming ps = 1:5 TeV. The exact rate is, however, a sensitive probe of both the N and \Delta masses. For most choices of the input masses we obtain extremely flat distributions in cos` so that detector acceptance is not a problem. However, when N is light a significant angular dependence is observed. This is simply a result of the t\Gamma and u\Gamma channel poles which develop as MN tends to zero. Of course, small MN also leads to a small cross section since the matrix element vanishes in this massless limit.

Potential backgrounds to the process e\Gamma e\Gamma ! W \Gamma R (W \Gamma R )\Lambda ! W \Gamma R + jj at the NLC

Fig. 4. Event rates per 100f b

\Gamma 1 for WR + jj production at a 1 TeV e\Gamma e\Gamma collider assuming

^ = 1 (a) as a function of MR for MN = M\Delta = 1 TeV (dots), M\Delta = 1:2 TeV and MN = 0:4 TeV (dashes), M\Delta = 0:3 and MN = 0:1 TeV (dash-dots), M\Delta = 2; MN = 0:6 TeV (solid), or M\Delta = 1:8 and MN = 0:6 TeV (square dots). (b)Same as (a) but for a 1.5 TeV collider.

are easily controlled and/or removed. For example, there may be some contamination from the SM process e\Gamma e\Gamma ! W \Gamma L W \Gamma L **, but this can be easily eliminated by using missing energy cuts and demanding that the WR final state be reconstructed from either the jj or eN ! eejj decay modes. (Since the on-shell WR decays to either jj or eN ! eejj there is no missing energy in the signal process.) In addition, with polarized beams, we can take advantage of the fact that WR couples via right-handed currents while any SM background must arise only via left-handed currents. Within the LRM itself a possible background could arise from a similar lepton-number conserving processes such as e\Gamma e\Gamma ! W \Gamma R W \Gamma R N N . Even if such a final state could be produced, in comparison to the process we are considering, the subsequent N decays would lead to a final state with too many charged leptons and/or jets.

In summary, the process e\Gamma e\Gamma ! W \Gamma i W (

\Lambda )\Gamma

j offers an interesting opportunity toexplore the masses and mixings of heavy Majorana neutrinos at the NLC. In the case

where a pair of conventional left-handed W 's is produced there remains some controversy as to whether large cross sections can be made consistent with the experimental bounds from fifi0*. This issue must be cleared up soon. We have noted that there are a number of existing constraints which point to WR being heavy but that they can be evaded by taking full advantage of the parameter freedom of the LRM. In the LRM, it is also easy to satisfy fifi0* bounds due to the flexibility of the right-handed CKM mixing matrix and the large value of the WR mass with the result that very large cross sections are obtained at the NLC up to the kinematic limit. To go to larger masses,

we have considered the production of one on-shell, one off-shell WR with the off-shell W \Lambda R ! jj. We pay a big price to do this, i.e., an additional power of the coupling constant as well as three-body phase space, but the resulting rates are still significant out to MR ' 0:8ps for integrated luminosities in excess of 100 f b\Gamma 1.

References

1. There has been a huge amount of work on this subject; see, for example, T.G. Rizzo,

Phys. Lett. B116, 23 (1982) and Phys. Rev. D50, 5602 (1994); D. London, G. Belanger, and J.N. Ng, Phys. Lett. B188, 155 (1987); J. Maalampi, A. Pietil"a, and J. Vuori, Phys. Lett. B297, 327 (1992) and Nucl. Phys. B381, 544 (1992); M.P. Worah, Enrico Fermi Institute report EFI 92-65, 1992; C.A. Heusch and P. Minkowski, Nucl. Phys. B416, 3 (1994); J. Gluza and M. Zralek, University of Silesia reports SU-TP-1/95 and TP-US1/95/04, 1995; M. Raidal, Nucl. Phys. B441, 49 (1995); P. Helde et al., Nucl. Phys. B437, 305 (1995); G. B'elanger et al., Annecy report ENSLAPP-A-537/95, 1995. 2. For a recent review, see M. Moe and P. Vogel, Ann. Rev. Nuc. Part. Sci. 44, 247

(1994). 3. For an overview of the LRM and original references, see R.N. Mohapatra, Unification and Supersymmetry, (Springer, New York, 1986). 4. A. Jodiddo et al., Phys. Rev. D34, 1967 (1986) and Phys. Rev. D37, 237 (1988);

J. Imazoto et al., Phys. Rev. Lett. 69, 877 (1992). 5. See, for example, G. Beall, M. Bander, and A. Soni, Phys. Rev. Lett. 48, 848 (1982). 6. See also P. Langacker and S.U. Sankar, Phys. Rev. D40, 1569 (1989) for a review

of all constraints. 7. F. Abe et al., CDF Collaboration, Phys. Rev. Lett. 74,

2900 (1995), Phys. Rev. Lett. 74, 3538 (1995), Fermilab-CONF-95/155-E, 1995; R. Harris, CDF Collaboration, Fermilab-CONF-95/152-E, 1995; S. Abachi et al., D0 Collaboration, Fermilab-Pub-95-283-E and Fermilab-CONF-95-210-E, 1995. 8. T.G. Rizzo, Phys. Rev. D50, 325 (1994). Fig. 5. Event rates per 100 f b

\Gamma 1 for WR + jj production at a 1.5 TeV e\Gamma e\Gamma collider assuming

^ = 1 and MR=1 TeV (a)as a function of MN for M\Delta =0.3(0.6,1.2,1.5,2) TeV corresponding to the dotted(dashed, dash-dotted, solid, square-dotted) curve; (b)as a function of M\Delta for MN =0.2(0.5,0.8,1.2,1.5) TeV corresponding to the dotted(dashed, dash-dotted, solid, squaredotted) curve.

