Nuclear Physics B (Pros . Suppl.) 16 (1990) 628-631 North-Holland
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Higgs Bosons in the Standard Model: The Minimal One Doublet and Extended Alternatives J.F. Gunion University of California at Davis, Davis CA 95616 I present a brief review of the theory and phenomenology of several alternatives for the Higgs boson sector of the Standard Model. Despite the remarkable success of the Standard Model (SM) to date, there is still no evidence regarding the nature ofelectroweak symmetry breaking (EWSB) . Even if EWSB arises via elementary scalar field vacuum expectation values, leading to a physical spectrum of elementary Hi bosons, the nature of the Hi sector is largely unconstrained by experiment. In this very brief review, I focus on Higgs sector possibilities in the context of the SM. Certainly, there is sector is more a significant possibility that the Hi complicated than the one complex doublet field of the minimal version of the model. Thus, it is useful to consider non-minimal Higgs sectors for the SM and their theoretical and experimental features. In particular, I will pay special attention to features which arise when one employs Higgs representations beyond doublets and singlets under SU(2)L . I will not consider the alterations and restrictions to these extended scenarios that would arise if they were part of a new physics scenario such as supersymmetry or the lowenergy sector of a superstring theory. A more complete discussion of many of the issues considered here can be found in ref. 1. Because of limited space in these Proceedings, I will give very few references here, and ask the interested reader to refer to this lengthier review for references to the original work.
1. The Standard Model Higgs Boson The simplest choice of Higgs sector is that of the minimal Standard Model, a single complex Higgs doublet field with isospin T = 1/2 and hypercharge Y =1, leading to a single observable Higgs boson, 4e . Constraints on the q° are few. Its mass is not determined by the theory, although if WW scattering is to remain perturbative at all energies moo must lie below about 1 TeV . Other bounds can be derived on the basis of perturbative grand unification . If there is no new physics up to very large energy scales, then a strong upper bound on moo can be derived using renormalization group techniques, ranging from about 150 GeV for mt < mW to about 220 GeV for mt - 200 GeV . However, if the scale of new physics is relatively low (in the * Work supported, in part, by the Department of Energy. 0920-5632/90/$3.50 © Elsevier Science Publishers B.V . North-Holland
TeV region), then the Higgs boson mass can be much larger, and non-perturbative analysis becomes necessary. Investigations of 04 theory on the lattice are sensitive to the possible triviality of pure #4 theory, and appear to require moo S 0.7 TeV, assuming no new physics appears below moo . In particular, they suggest that a strongly-interacting WW sector, with no other new physics, is not consistent . A prudent experimentalist will pay little attention to all these theoretical bounds, and simply attempt to develop experiments that probe all possible values for moo . The search is only now beginning. Low-energy experiments almost certainly rule out moo < 2m,,. As LEP/SLC accumulate Z decays, we can look forward to probing the region moo S 40 GeV, while LEP-II will probe up to about 60 GeV . Beyond that we must turn to the next generation of colliders, the LHC and SSC and a possible TeV scale linear e+e - collider. Search techniques have been thoroughly explored in the literature; for a recent update on the SSC techniques see a companion contribution to these Proceedings .1', For the remainder of this talk I wish to focus on Higgs sectors that go beyond the minimal one doublet field. Aside from simplicity, the only technical ad vantage of the one-doublet model is that spontaneous symmetry breaking automatically preserves U(1)E&M. (In more complicated sectors, some constraints on Higgs potential parameters are necessary in order to avoid breaking the electromagnetic symmetry when EWSB occurs.) Certainly, new physics models almost inevitably require a non-minimal Higgs sector. For instance, a supersymmetric model requires at least two Higgs doublet fields.
2 . Two-Doublet Higgs Sector
Theory The simplest extensions of the minimal SM Higgs sector are those obtained by the introduction of one or more additional doublets and some singlet Higgs: fields. Such extensions have the advantage of retaining PEW = MW/(mi cost ow) equal to 1 at tree-level
J.F. Gunion/Higgs Bosons in the standard mode! automatically. Here I consider the simplest such extension of just one additional doublet field. The Higgs boson spectrum now includes two neutral scalars, h° and H° (with mho < mHo by convention), a neutral pseudoscalar, AO, and a pair of charged Higgs bosons H-* . Here, the words scalar and pseudoscalar refer to quark couplings proportional to 1 and 7s, respectively. There are now six parameters to the Higgs sector: mho, mHo, mAo, mgt, a and #, where a (-w < a < 0) is a mixing angle that arises in diagonalizing the hO-H° mass matrix, and tan# = vz/vl (0 < P :5 r/2) is the ratio of vacuum expectation values of the neutral real fields of the two complex doublets. Of the physical Higgs bosons, only H° and h° couple to vector boson pairs . In order to guarantee perturbative unitarity in VV scattering (V = W, Z) we must now require that ~h=ho~go 9lW+R , - = 9,'O.W+W(and similarly for the Z couplings), and that this sum rule be saturated by h's with mass below about 1 TeV. (Thus, for instance, H° could be arbitrarily heavy so long as it decouples from the VV channels.) In terms of the parameters a and #, these requirements are realized via 9how+W 900W+W-
= sin(# - a)
9HOW+W- = 900W+W -
cos(# - a) (2.1)
with cos(# - a) -+ 0 if mHo becomes larger than - 1 TeV. We also find that there is no longer any strict lower bound on mho (regardless of mt) so long as MHO is not too small. In order to avoid flavorchanging neutral currents, fermion couplings cannot be arbitrary. Only two possibilities are allowed : Model I, in which all quarks couple to the same Higgs field (presumed to be 02 by convention) ; and Model II, in which up quarks couple to 02 and down quarks couple to ¢l. It is conventional to suppose that leptons couple to the same field as the down quarks, although this is not required in the SM context . (It is worth noting that the minimal supersymmetric model (MSSM) requires that the couplings be Model II, and that down quarks and leptons couple to the same field.) In Model I, all quark and lepton couplings are proportional to 1/ sin # and are enhanced if tan # < 1 over SM values (with some dependence on a in the case of H° and hO), while in Model II the down quark and lepton couplings are proportional to 1/ cos #. One can check that the fermion couplings obtained satisfy unitarity requirements in the VV --+ ff scattering channel. Phenomenologically important new couplings arise for the first time at the two-doublet level : namely couplings of Higgs pairs to vector bosons . In the twodoublet model we have (writing cos 9w = cw)
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cos(# - a), AOAOZ = Few-
9H°AOZ
9hOH+W-
9H°H+W- = Z sin(# - a) .
= 2 Cos(# - ac),
= 2c sin(# - a), W (2.2)
From eqs . (2.1) and (2.2) we note that if, for example, the H°W+W- coupling is small, then the HOA°Z coupling is large. This point will have phenomenological importance.
Finally, we note that the minimal SM one-doublet Higgs sector is effectively obtained in the limit where mgt, mAo, and mHo are all much larger than mho, and a -+ # - r/2. In this case, the h° has SM couplings to both fermions and vector bosons . (This is the limit that is automatic in the MSSM - where all Higgs masses and a are determined by choosing just two par rameters, e.g. mH+ and tan# - when mH+ --+ oo.) Phenomenology As for the minimal SM one-doublet sector, we find that experimental constraints are only just beg' to emerge. Recent analysis's' shows that a very precise accidental correlation between mt and mH+ would be required in order that any light neutral H' with mass < 2m, not have been observed in low-en rare decays . Charged Higgs exchange contributions to Bd - Wd mixing, e, cl, etc. become more and more restrictive as mt lower bounds increase . Typically, these contributions become too large unless either mH+ is large or tan # is large; there is an excluded range in MH+ - tan # space at small mH+ and small tan # that depends on CKM matrix parameters and expands with increasing mt. Moving up in energy scale, Tristan and other lowenergy e+e- collider experiments rule out the case where h° is very light (so it decays outside the detector) and mAo IS Va-12, so long as cost (# - a) is near 1 - smaller values yield progressively weaker limits . For the charged Higgs, Tristan experiments restrict mH+ to lie above about 25 GeV. Turning to direct detection at larger masses, we first consider H*. Since P(Z ~ H+H -)/T (Z -+ mD) - 0.15Bs/s (where B is the appropriate phase space factor), the SLC/LEP Z factories will be able to probe the region mgt S 40 TeV with ® (5000) produced Z's . At higher energy e+e - colliders, including LEP-II, the relevant cross section is a(e+c H+H -) - 0.21a(e+e - -+ I&+is- )B 3 12 . Thus, an integrated luminosity of order 500 units of R will allow any such machine to exclude charged Higgs bosons up to very near the kinematical limit of V,9-/2. At a hadron collider, the rate for pp -+ tt followed by t -+ H+b and/or f --> H'b is expected to be substantial if mt > MHf +mb, even if t -+ +b is also allowed . But, if mHt is larger, the main production mechanism becomes gb --+ [H+ (and charge conjugate) and the
J.F. Gunion/Higgs Bosons in the standard model
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primary decay channel H+ -_t tb has impossibly large backgrounds. Exotic decays, such as H+ - h0W+ (or, in a SUSY context, H+ -o- jt+O) can have a large branching ratio, if allowed, and would offer significant possibilities . Further study is required to assess the exact range of parameter space over which they would allow detection. Returning to the neutral Higgs bosons, 107 Z decays can probe the full range mho Z 40 GeV only if sine (,6 - a) is not suppressed, while a similar range of MHO can be probed if c062 (Q - a) is not suppressed. In fact, all SM techniques work for both H° and ho so long as they share fairly equally the VV coupling strength, see eq . (2 .1) . For example, at a higher energy e+e collider or hadron collider, VV fusion processes would have the largest cross sections . However, if the H° is much heavier than the h°, we have seen that perturbative unitarity suggests that the hO saturates the V V coupling (indeed, this is predicted in the MSSM) in which case the H° becomes substantially more difficult to produce and detect . The A°, which has no VV couplings, is always problematical . At an e+e collider the best possibilities emerge from Z or Z* production of Higgs pairs : T(Z - H°A°) -1 . 2 -~ sin (Q - a)B3/2 r (z --, vL)
R(Z* -- H°A0) -0.1 sin2 (a
- a)
B3/2 ,
(2.3)
where B is an appropriate phase space factor . (For the h° the corresponding results are obtained by replacing sine (Q - a) by C082 (.8 - a) .) Note that if the he saturates the VV coupling strength, i .e . sin(# - a) -+ 1 (see eq. (2 .1)), we are at least guaranteed that the Z, Z* -+ H°A° processes will be maximal in strength . Thus, an e+e collider with sufficient energy is guaranteed to find all the Hi bosons of the two doublet model . At a pp collider the situation is more difficult. For instance, while the AO is still produced with a high rate through gg -+ A° (via a triangle loop large rrat is good), if A° decays primarily to tt or 66, backgrounds will be impossibly large. However, the decay A° -+ Zh° followed by h° -+ 66 would be very important if phase space allowed and the appropriate coupling, see eq . (2 .2), is not suppressed. Detection in this decay mode is quite feasible . (As for the H*, in the SUSY context one should also be alert to the fact that decays of the AO to neutralinos and charginos might dominate .) 3
eyond Two Doublets
Clearly we can continue the process of adding doublets and ringlets indefinitely. Indeed, many superstring scenarios yield Higgs sectors that contain more
than two doublets and several singlet Higgs fields. The simplest such extension is, of course, a Higgs sector with two doublets and one complex singlet field . We then have a Higgs spectrum that includes three neutral scalars and two pseudoscalars . Clearly, the danger of such additions, as far as experimental discovery of the Higgs bosons is concerned, is the possibility of multiple dilution of the V VH° and ZH°Aj couplings (in general, Ei 9vvH° = 92ovv and Ei,f 9iH°A~ _ 2 g /( 2cw) 2) so important to the discovery of the neutral Higgs bosons of such models. (In contrast, the H+H,r*, Z* -+ processes are not affected by multiple copies and would allow access to the charged Higgs sector of such a model at an e+e collider of sufficient energy.) The only constraint on such extensions of which I am aware is the possibility that they might lead to too large a value for the neutron electric dipole 1 1 moment (EDM) . ` Once, there are more than two doublets in the Higgs sector the possibility of CP violating phases in the Higgs potential becomes natural, and these can yield three glr.on operators that give rise to a large dipole moment . To the extent that one regards the requirement (perhaps implemented by some symmetry) of purely real Higgs potential parameters as a fine tuning, models with a Higgs sector containing more than just two doublets can be regarded as unpalatable. Of course, in the more complicated models, if only the Higgs bosons that would appear in a twodoublet model are light, and the others substantially heavier, the CP violation could be confined to a sector that would yield a relatively small contribution to the neutron's EDM.
4. Triplet Higgs Representations Theory A SM Higgs sector that includes triplet fields while retaining pEW - 1 at tree level is not straightforward to construct . A realistic Higgs sector containing triplet fields must contain at least one doublet field in order to allow for quark masses (since triplet fields do not have the appropriate quantum numbers to couple to qq). If one arbitrarily adds a single triplet field then 1PEW -11 > 0 .01 (the approximate experimental upper limit) unless 1v1,o1/1v1/2,11 < 0.047 for a Y = 0 triplet and JV1,t21/1v1/2,11 :5 0 .081 for a Y = f2 triplet, where vT,y is the vev of the neutral member of the representation . For such small vev's, the Higgs triplet does not really participate in EWSB . The simplest case that has triplets with significant vev's contains one doublet field, one real triplet field (Y = 0), and one complex triplet field (Y = 2), such that vl,o = v1,2 = 6. Defining v1 X2 .1 = a, we find that pEW = 1 with gauge boson masses deriving entirely from the combination v2 = a2 -t- 862 ; both the triplets and the doublets can contribute significantly to EWSB . We shall return to a few remarks concerning the phenomenology of this model shortly.
J.F. Gunion/Higgs Bosons in the standard model There are two very characteristic features of a Higgs sector containing triplets (with neutral members having a non-zero vacuum expecataton value) and having per = 1 . The first is a non-zero H - W+Z vertex. Indeed, to avoid such a vertex one must satisfy one of two conditions : a) there can be no representations with Y 96 0; or b) there can be only complex representations with the same Y and T = Y/2. It is easy to show that a) yields pmv = 1 while b) only yields pEW -=1 for doublets (plus an arbitrary number of singlets) . Thus, a Higgs sector with triplets can only yield pEw = 1 for arbitrary triplet vev'e there is a nonzero H- W+Z vertex. (It tu=ns out that this leads to a new naturalness problem, 1a1 but we have no space for discussion here.) The second important general characteristic is the existence of a doubly-charged Higgs boson pair (H++--- ) ; these are automatically present in the Y = 2 complex triplet (s) required in order to have per = 1 . Further, when the triplet vev's are nonzero (as we assume), the H++W- W - coupling will be non-zero . Thus, both the H+ and H++ will influence vector boson scattering processes .
if
Finally, it is perhaps worth nothing that no superstring model with a reasonable low-energy group (close to that of the SM) has yet been constructed which yields low-energy matter representations that are large enough to contain triplet Higgs fields . However, there is currently no theorem that all possible compactifications with acceptable phenomenology have this feature . Phenomenology Returning to the minimal one-doublet, two-tripjets (Y = 0 and Y = 2) model mentioned above, we note just a few of the most crucial phenomenological features. °~ First, there are now many physical Higgs bosons. There is a five-plet (under the custodial SU(2)R group of the model), H6+, Hs,+, H6, Hb , H6 , a triplet, H3+, Hs, Hs , and two singlets H1 and H11. Only the Hs's and H1 have tree-level quark couplings, and only the H6's, the Hi' and the Hi have tree-level V V couplings . In the limit of b/a -~ 0, the Hi plays the role of the single neutral Higgs of the minimal one-doublet model, and has all the same couplings . The other Higgs decouple from EWSB . But if a and b are comparable, all Higgs bosons play an important role in EWSB and in the satisfaction of V V scattering unitarity. For example, in W+W+ scattering, there are 3 t-channel graphs (with H50 , and Hi' exchange), 3 u-channel graphs (same exchanged Higgs bosons), and one s-channel graph (Hr.+,+ exchange) . Together, they maintain good
H1,
high energy behavior for W+W+ -+ W+W+ . Most interestingly, the presence of the H5++ graphs means that the neutral Higgs boson couplings are no longer constrained by the type of sum rule that appeared in two-doublet models . In general, all can be strongly coupled to the W +W - and ZZ channels, and there is
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no obvious tendency toward the coupling constant dilution that makes detection possibly problematical in the many-doublet case. Clearly, the key signature for a triplet model is the detection of a doubly-charged ' boson . The two obvious techniques for this are e+e - -* HS - H6++ and pp -i~ W +W + -+ H6+, followed by decay, e .g. H6++ -* y y+yy+ either as a real 2-body decay or with one or both W+'s virtual . The like-sign dileptons that emerge when both W's decay leptonically provide a highly exotic signature, so that not many events are required. The SSC should easily be able to probe Mgg masses up to order 1 TeV in this way, so long as b/a is not suppressed (thereby suppressing the required couplings for this mode of production and decay) .
5. Conclusions Clearly, even within the context of the SM, there is a plethora of possible Higgs sector scenarios . The experimental program required to either detect or eliminate all possibilities is extensive,. Only by comb' information from all existing machines (ranging from Tristan to SLC/LEP) with future results from the SSC and a future e+e TeV scale linear collider can we be sure of complete coverage . However, each machine should have a good chance of revealing some portion of the Higgs spectrum of a complex Higgs sector . REFERENCES 1 . For a review and references, see J.F . Gunion, G .L . Kane, H.E. Haber, and S. Dawson, preprint UCD-89-4 (1989), to be published by AddisonWesley. 2. J .F . Gunion, "The SSC: Status and Physics Update, these Proceedings. 3 . J .F . Gunion and W .S. Hou, preprint in preparation . 4 . S . Weinberg, preprint UTTG-30-89 (1989) . 5 . M.S. Chanowitz and M . Golden, Phys. Lett. 165 (1985) 105; J.F. Gunion, R. Vega, and J . Wudka, preprint in preparation .