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Two Higgs Models for Large Tan Beta and Heavy Second Higgs



Two Higgs Models for Large tan β and Heavy Second Higgs
Lisa Randall Je?erson Physical Laboratory Harvard University, Cambridge, MA 02138, USA randall@physics.harvard.edu
Abstract We study two Higgs models for large tan β and relatively large second Higgs mass. In this limit the second heavy Higgs should have small vev and therefore couples only weakly to two gauge bosons. Furthermore, the couplings to down type quarks can be signi?cantly modi?ed (so long as the second Higgs is not overly heavy). Both these facts have signi?cant implications for search strategies at the LHC and ILC. We show how an e?ective theory and explicit fundamental two Higgs model approach are related and consider the additional constraints in the presence of supersymmetry or Z2 ?avor symmetries. We argue that the best tests of the two Higgs doublet potential are likely to be measurements of the light Higgs branching fractions. We show how higher dimension operators that have recently been suggested to raise the light Higgs mass are probably best measured and distinguished in this way.

arXiv:0711.4360v2 [hep-ph] 20 Dec 2007

1

Introduction

Two Higgs models are perhaps the simplest alternative to the Standard Model. They are particularly important because they are essential to low-energy supersymmetry but they of course can occur in other models that allow a broader parameter range. The phenomenology of the neutral Higgs sector is slightly subtle since the angles from mass mixing are not in general the same as the angle associated with the relative vevs. However we will see that they generally align to a large extent when one Higgs is somewhat heavier, greatly simplifying the analysis of the implications. In this paper we explore the phenomenology of two Higgs doublet models for large tan β when the lighter Higgs h is light enough so that its decays are dominantly into bs whereas the second Higgs is somewhat heavy. We are motivated in part by the analysis of [1], which performed an operator analysis in the strongly interacting Higgs sector case to elucidate interesting e?ects that can occur when the light Higgs is part of a larger Higgs sector. Similar considerations apply to two Higgs models, since the light Higgs is not exactly the eaten Goldstone boson in this case either.1 For example, we ?nd growth in W W scattering with energy though it corresponds to a higher order operator than in the strongly interacting Higgs models considered in Ref. [1] and is not the most signi?cant deviation from Standard Model predictions. The modi?cation of the light Higgs coupling to bottom type quarks and charged leptons can be signi?cant however for two Higgs doublet models. Although studying deviations in the light Higgs couplings from their Standard Model might not seem to be the best way to study a perturbative theory where the additional states are more likely to be kinematically accessible, we show that for a large parameter range the heavy Higgs will probably elude detection and precise measurements of light ?elds will be the best way to test the Higgs sector. An operator analysis for two Higgses from a purely e?ective theory viewpoint was in fact completed in a recent paper [2] where it was shown that for large Yukawa coupling of the heavy Higgs to the down sector and small Yukawa of the heavy Higgs to the up sector one could ?nd signi?cant deviations in the Higgs partial widths for the light Higgs particle, even when the second heavy Higgs will elude direct detection. In this paper we relate the more conventional two Higgs analyses of Gunion and Haber [3] to the e?ective theory analysis of Mantry, Trott, and Wise [2]. We show that the conclusions reached in that paper (namely large corrections to b Yukawas and di?culties of ?nding a second Higgs) apply quite generally for large tan β. We show that the assumption made there is in some sense less arbitrary than it might seem in that these characterizations apply to the Yukawa couplings for the heavy Higgs in the large tan β limit. We show however that if the dimension-4 operators respect the Z2 symmetry that guarantees a GIM mechanism that the Yukawa modi?cations are expected to be smaller since
Note that we use the conventional notation where the scalar particles in the Higgs sector are called Higgs particles. Purists might restrict this term for the linear combination with a vacuum expectation value but since the ?elds mix it is easier to call them all by this term and to distinguish the light, heavy, and charged Higgses.
1

1

they are no longer enhanced by tan β. However, we will see that the e?ects can still be quite signi?cant. We also consider the relationship between deviations in bottom and tau branching ratios in the various doublet Higgs models that preserve a Z2 symmetry. Finally we are motivated by recent data that point to high supersymmetry breaking or a new physics scale motivating considering a relatively heavy second Higgs and higher e?ective dimension operators in the Higgs sector. We ?nd that the higher e?ective dimension operators of [20] can generate large deviations in Higgs branching ratios and that these deviations in Yukawas are most likely the best way to test for the new operators they suggest. Furthermore, these deviations in Yukawas could distinguish among the di?erent possible higher dimension operators that could in principle correct the light Higgs mass.

2

General Two Higgs Analysis

We will consider a two Higgs theory in the decoupling limit where one of the Higgs is assumed to be light (in the regime in which decays to bs dominate) and the other Higgs is assumed to be relatively heavy. We will use two parameterizations below, and use both mH and M to refer to the mass of the heavy Higgs. Let us ?rst parameterize the two Higgs Lagrangian with the notation of Gunion and Haber [3] (see also [4, 6]) but using the notation H1 and H2 for the two Higgs bosons. We have the gauge invariant scalar potential V 1 1 ? ? ? ? ? = m2 H1 H1 + m2 H2 H2 ? [m2 H1 H2 + h.c.] + λ1 (H1 H1 )2 + λ2 (H2 H2 )2 11 22 12 2 2 1 ? ? ? ? ? + λ3 (H1 H1 )(H2 H2 ) + λ4 (H1 H2 )(H2 H1 ) + ( λ5 (H1 H2 )2 2 ? ? ? + (λ6 (H1 H1 ) + λ7 (H2 H2 ))H1 H2 + h.c.)

(1)

We take all parameters to be real and CP-conserving for simplicity. In a supersymmetric model, these parameters take the values 1 1 λ1 = λ2 = ?λ345 = (g 2 + g′2 ), λ4 = ? g 2 , λ5 = λ6 = λ7 = 0 4 2 (2)

Where λ345 = λ3 + λ4 + λ5 . Notice that the last two parameters are zero in any model that respects a Z2 symmetry in the dimension-4 operators. We might expect this to be approximately the case in any of the standard two Higgs scenarios where an approximate Z2 guarantees a GIM mechanism. However, breaking of the Z2 in the dimension-4 operators above can exist while still not introducing overly-large ?avor changing e?ects [2, 20]. As we will see such cases will lead to particularly interesting deviations from the Standard Model. We now assume an appoximate Z2 symmetry and follow the standard notation and de?ne the ratio of vevs of the two ?elds as tan β = H2 / H1 , where H2 is the ?eld coupling to the top quarks and H1 is the ?eld coupling to the bottom quarks and charged leptons (here we are assuming a Type II model where this is the case but we will explore later other assumptions). The angle α determines the mixing angles of the Higgs ?elds mass eigenstates, so we have 2

1 (3) H1 = √ (cos αH ? sin αL) 2 1 (4) H2 = √ (sin αH + cos αL) 2 where H is the heavy Higgs ?eld and L is the light Higgs ?eld and we are only considering the real parts of H1 and H2 . Notice that with this parameterization the ?elds H and L have nonzero vevs, but this can be subtracted o? as in [3]. The vevs for the two (real) Higgs ?elds (neglecting higher order terms in (v/mH )2 are given by L = v sin(β ? α) (5) H = v cos(β ? α) where cos(β ? α) ? and ? 1 λ = sin 2β(λ1 cos2 β ? λ2 sin2 β ? λ345 cos 2β) ? λ6 cos β cos 3β ? λ7 sin β sin 3β 2 In the large tan β limit, this reduces to ? λ = cos β(?λ2 + λ345 ) + λ7 and in a supersymmetric theory we would have cos β 2 ? λ=? (g + g′2 ) ? ?0.3 cos β 2 (10) (9) (8) ? λv 2 m2 H (6) (7)

Alternatively in the limit that tan β is large one can just solve for sin α (using the mass matrices from [3]) (see Eq. (15) below) to ?nd sin α ? ? cos β + λ7 (v 2 /M 2 ). (11)

Expanding out cos(β ? α) in Eq. (7) gives this same expression. The equations above show that the masses and vevs are aligned up to O(v 2 /M 2 ) corrections. The heavy Higgs gets a vev through its interaction with the light Higgs ?eld that has acquired a larger vev. This would have been more manifest with di?erent notation. For example, the vev of H1 is proportional to v cos β whereas the coe?cient of L is ? sin α, so it might have been natural when the quartic term doesn’t dominate the mixing to have started with the rotated angle ? sin α → cos α in the ?rst place. Notice that the equations above imply that α ? β ? π/2 in the extreme decoupling limit where tan βλv 2 /m2 < 1. Although perhaps not as likely to be physically relevant, we also H consider the opposite limit, where the o?-diagonal term in the mass matrix changes sign. In 3

this case, the above results still hold for α and the vev of the heavy ?eld, although when the λ7 term dominates sin α reverses sign. The answer above su?ces over the entire parameter range but for completeness we compare the result to that of [3] in this limit, noting that the intermediate results can depend on convention.2 Ref. [3] gave α ? π/2 ? β [3], sin α ? cos β, which is the result when the initial conventions for the Lagrangian do not account for cos β > 0. Minimizing the potential with respect to φ1 (substituting in the assumed form for the H1 and H2 vevs) yields the equation [3] 1 m2 = m2 tan β ? v 2 λ1 cos2 β + λ345 sin2 β + 3λ6 sin β cos β + λ7 sin2 β tan β 11 12 2 (12)

When the λ7 -dependent-term dominates, one needs negative cos β to satisfy this equation. However, according to the [3] convention β is always between 0 and π/2. In order to maintain cos β > 0 and m2 > 0 (when λ7 > 0), we need to change the sign of H1 . With this sign 11 change, we can directly solve for sin α (in the large tan β limit to ?nd sin α ? cos β ? λ7 (v/M)2 . In this case we can evaluate cos(β ? α) to ?nd approximately 2 cos β as above, but the more useful quantity would be the quantity that appears in the H vev. Because we have changed the sign of H1 , we see that the vev of H is related to ? cos(α + β), and this again evaluates to λ7 (v/M)2 . Alternatively had we taken a convention where we also changed the sign of sin α, we would have obtained the answers we did for small λ7 above. In either way of proceeding, the vev of the heavy ?eld and sin α (up to an unphysical sign) take the same form, even when λ7 tan β(v/M)2 > 1. These are the physically relevant quantity that enter the heavy Higgs coupling to two gauge bosons and the Yukawas. So our results (6),(11) for the vev and mixing angle apply over the entire parameter range. An alternative approach to a two Higgs model with a heavy Higgs is to take an e?ective theory approach as considered in [2]. Ref. [2] does not assume the existence of a Z2 symmetry (in fact H1 and H2 are never mentioned) so the Yukawa couplings of the heavy Higgs can be taken as free parameters, but the parameters were chosen to be consistent with small FCNC (that is minimal ?avor violation [5], assuming only a single Yukawa matrix structure for up type quarks and another for down type quarks). For simplicity in comparing to their results we will call their light Higgs H and their heavy Higgs S as in Ref. [2] (but note that H is now the light ?eld and S is a doublet). Their Lagrangian is: λ ? v2 2 λS ? 2 V (H, S) = (H H ? ) + M 2 S ? S + (S S) + [g1 (S ? H)(H ?H) + h.c.] 4 2 4 + g2 (S ? S)(H ? H) + g2a (S ? H)(S ? S) + h.c. + g2b (S ? H)(H ?S) + [g3 (S ? S)(S ? H) + h.c.]. (13)

Note that g1 and g2 are couplings completely independent of gauge couplings; we have kept the notation of [2] for simplicity. Secondly, notice that all the same types of terms appear in the non-e?ective theory H1 H2 Lagrangian aside from the quadratic mass mixing
2

We thank Howie Haber for discussions on this limit.

4

term. However, since H2 and H (from [2]) are not identical, the gs would be a function of various couplings in the Lagrangian above. We can expand in terms of cos β to solve for one ?eld in terms of the other. To simply relate couplings we can consider the small cos β limit. In this limit, the heavy Higgs is approximately H1 and the light Higgs is approximately H2 . In this limit we can ? expand to see that g1 = λ. For the more exact result, we can expand H1 and H2 in terms of ? H and L and include the additional Z2 -violating cos β-suppressed terms to ?nd g1 = λ. For simplicity, we concentrate on the λ7 term below. More relevantly for physical consequences, we can relate the vevs and in particular the heavy Higgs vev in the two pictures. Ref. [2] had 3 √ g1 v 3 S 2 =? 2M 2 (14)

(again we are working to leading order in (v/M)2 ). Notice that H and L are real ?elds in √ the ?rst analysis so that the relevant ?eld to compare to is 2S (ignoring the other Higgs components, where H is the heavy Higgs in the fundamental theory). Recall that when cos β is small, λ7 ? g1 . We see that the two values of the expectation value, though having the same parametric dependence, di?er by a factor of -2. The reason for this is that the fundamental Higgs analysis uses the mass eigenstates for the full mass matrix, whereas the e?ective theory analysis did not use mass eigenstates once the g1 -dependent quartic term is included. The physical mass eigenstate is S + 3g1 /2(v/M)2H and has vev ? that agrees with the vev for the fundamental theory when g1 = λ. Eq. (12) tells us that without the m2 term that cos β would agree with Eq. (14) above. However, the g1 quartic 12 (or in Ref. [3] the λ7 -dependent quartic) also contributes to mass mixing, so the the heavy physical mass eigenstate has the vev cited in Eq. (6). To further understand this result, it is of interest to consider the contributions of the quadratic and quartic terms to both mass mixing and vev. Had the only mixing term between H1 and H2 been a mass term, one could in fact simultaneously diagonalize the mass and vev. However, the relative mass squared coming from the quartic is 3/2g1 (v/M)2 , whereas the vev contribution to the linear term is g1 /2(v/M)2 v. So a piece of the quartic 2 can be absorbed into MA as is done in [3]. That is, the mass matrix takes the form
2 M 2 = MA m2

sin2 β ? sin β cos β

? sin β cos β cos2 β

+ B2

(15)

2 where MA = sin β 12 β ? 1 v 2 (2λ5 + λ6 tan β ?1 + λ7 tan β) and the o?-diagonal part of B2 cos 2 contains a term λ7 v 2 sin2 β. After full diagonalization, one is left with the vev of the heavy ? Higgs eignenstate cos(β ? α) = λ(v/M)2 as we found above. Before closing this section, we remark on how small the VEV of the second ?eld is likely to be. This makes the coupling to two W s very suppressed, which is essentially why the heavy Higgs search is quite di?cult as we will discuss further shortly. In the next section we
3

Note the sign correction to [2].

5

discuss the deviation of the light Higgs Yukawa from its Standard Model value. For a large range of parameters, this is the likely to be the best way to search for evidence of a second Higgs.

3

Yukawas

Given the expressions for H1 and H2 in terms of H and L, we can work out the Yukawas for the light and heavy ?elds to the up and down type quarks. In this section we will focus on precision light Higgs measurements and study the deviation of Higgs couplings to fermions from their Standard Model values. We will ?rst consider Type II models (as in the MSSM) in which one Higgs gives mass to charged leptons and down-type quarks and the other Higgs gives mass to up-type quarks. We then have (in relation to the standard Yukawa couplings) [3, 8] ? hDD : ? ? hU U : sin α = sin(β ? α) ? tan β cos(β ? α) cos β (16) (17)

cos α = sin(β ? α) + cot β cos(β ? α) sin β

Note that both of these are of order unity when the second Higgs is heavy and cos(β ? α) is small, as they should be in the decoupling limit. We also see that the corrections term in the down-type Yukawa can grow with tan β and be quite large. Ref. [2] did not assume a Z2 symmetry but did assume minimal ?avor violation. Note that this is more general in that with Z2 symmetry, there are only three distinct possibilities, in which either the same Higgs or orthogonal Higgses couple to up and down type quarks respectively. With only MFV, one can in principle de?ne the Higgs that couples to up-type quarks and the one coupling to down-type quarks as H2 and H1 , but these are not necessarily either the same or orthogonal so there is a continuum of possibilities. However, we will see that only the down-type Yukawa deviations are likely to be signi?cant when tan β is large so the di?erence isn’t necessarily signi?cant. The authors of Ref. [2] de?ned parameters ηd and ηu which when multiplied by the light Higgs Yukawas of the e?ective theory gave the heavy Higgs Yukawas. In terms of the quark masses (and including both the light and heavy Higgs vev contributions), the Yukawa couplings of the heavy Higgses are therefore √ md ? ? ηd 2d √ 1 + 2ηd
S v

d

(18)

and similarly for up quarks, where this expression includes the S vev contribution to the quark masses. Ref. [2] considered the possibility that ηd is large and ηu is small. By integrating out the heavy Higgs (and including its vacuum expectation value contribution to the quark masses),

6

they found a light Higgs Yukawa coupling

4

v 1 ? 3 g1 ηd ( M )2 2 1 v 1 ? 2 g1 ηd ( M )2

(19)

They noted that the correction is large when ηd is big, which is clearly similar to the observation we made above for large tan β. We now show the similarity of these large Yukawa corrections is not a coincidence and that such a scheme is a generic prediction of large tan β.5 This large deviation has signi?cant implications for the search for a second Higgs. The heavy Higgs coupling to down type quarks (again in relation to standard Yukawas) is given by cos α = cos(β ? α) + tan β sin(β ? α) ≈ tan β (20) cos β whereas the coupling to up quarks is given by sin α = cos(β ? α) ? cot β sin(β ? α) << 1 sin β (21)

So we see that large tan β naturally yields a large Yukawa coupling of the heavy Higgs to down quarks and a suppressed coupling to up type quarks. We can see this directly in equation (20) noting that cos α is very close to sin β (which follows from cos(β ? α) being small), so that the value of ηd that this model matches onto is very close to tan β. This follows from the original Z2 symmetry, which favors the heavy Higgs which is approximately H1 coupling to down quarks and the light Higgs which is approximately H2 coupling to light quarks. For completeness and to elucidate the origin of this correction we do the matching for the heavy Higgs down-type Yukawa couplings more exactly in order to compare the two formulations. Again when comparing the results we need to take into account that the [2] analysis based on the e?ective theory doesn’t use the fully diagonalized states. So the Yukawa for the light not quite diagonalized ?eld in the fundamental theory would be approximately 3 ? sin α + 2 λ7 (v/M)2 ? cos β + 1 λ7 (v/M)2 . So we identify 2 ηd = from which we conclude 1 ? 3 g1 ηd (v/M)2 2 = 1 ? λ7 tan β(v/M)2 1 2 1 ? 2 g1 ηd (v/M)
4

tan β 1+
λ7 2

tan β

v 2 M

(22)

(23)

Notice a sign correction from Ref. [2]. This sign has physical consequences since it is the deviation from the Standard Model value. 5 Of course the Lagrangian in [2] is more general, and tan β is not even de?ned in the absence of a Z2 symmetry [9]. Our point is that the particularly interesting case of large tan β is an example of this type of parameter regime.

7

(where we have made the approximate identi?cation λ7 with g1 ) which agrees with Eq. (16). Notice that when integrating out S to determine the Yukawa, one is e?ectively accounting for the mass mixing so in this case the results in the two formulations agree. That is, the Yukawa in Eq. (19) is really the Yukawa for the physical light Higgs. Also note that the di?erent Z2 -violating quartic contribution to the mixing and the vev leads to the correction to the light Higgs Yukawa. We see in either formulation that the correction can be quite large in the large tan β (or large ηd ) regime. For large tan β and not overly heavy Higgs mass, we can have large corrections to the bottom and tau (in type-II models) Yukawa couplings. The sign of the ? correction depends on the sign of λ, which is in general unknown but is determined in supersymmetric models or other models where the physics constraints determine the sign (see below). In Ref. [2], parameters such as g1 ? 2 (note that we have changed the sign of g1 to re?ect the sign correction in the Yukawa and the S vev) and ηd ? 20 were considered, ? corresponding to large tan β and moderate λ. For these parameters the total width could change substantially, being corrected by a factor of 121 for Higgs mass of order a TeV. If, on the other hand parameters were g1 = ?1 and ηD = ?10, the branching ratio was down by 0.008. [2] imagined that the bottom coupling was changed and the ηd -enhanced deviation from the Standard Model may or may not apply to the τ as in Type II models. Note that for either sign of the correction, the rate of decay of the light Higgs into b quarks and hence the total width and the branching fractions into other modes (in the light Higgs regime where decays to bs dominate) will deviate from the Standard Model predictions. In Type II models where leptons and down type quarks both couple to H1 , the best measurement of this Yukawa deviation at the LHC will be the relative branching ratios of photons and taus. In other models in which the tau Yukawa is not changed directly by a large amount but only through the change in total width (as might happen for more general MFV models or in Type III models where the up-type Higgs couples to leptons), one would need to measure the absolute decay rate into τ s or photons since both branching fractions change indirectly through the change in the Higgs total width. In this case, the tau rate would increase or decrease when the photon rate does, unlike Type II models where they would change in opposite directions. The ratio of photon and tau partial branching fractions is likely to be measured at the 15-30 % level [6, 14] and absolute branching fractions might also be measured at reasonable levels [14]. Of course especially for the photon loop e?ects from nonstandard model physics might also be signi?cant. In addition, radiative e?ects involving bs might further suppress this decay [15] as we further discuss below. Whether or not radiative e?ects are signi?cant, tree level e?ects can dominate and give rise to deviations from the expected Standard Model ratios at a potentially measurable level for the LHC and a readily attainable level for the ILC. Notice that the results are very similar to those from [2] since the (large) corrections to the down type Yukawa coupling match. The di?erence would be only in the up type Yukawas, where the [2] Lagrangian has a correction ηu which is in principle independent of ηd . However, since this is small by assumption, it won’t make any measurable di?erence. It is also useful to note what happens to Yukawa modi?cations when a Z2 symmetry is 8

preserved by the quartic interactions that would forbid λ6 and λ7 . In that case, the tan β ? enhancement no longer exists, since λ is proportional to cos β. This is in fact what happens in the MSSM. Although this can decouple more quickly than without tan β enhancement as has been noted in several places (see [3, 7] for example), and is not an enhancement that would allow the sort of large change in branching ratio that was considered in Ref. [2], it still might be measurable. For example, from Eq. (8), we can deduce the tree-level change in Yukawa in a su2 2 v2 persymmetric theory, which is g +g m2 , which is about 0.3 for Higgs mass comparable to 2 H v. The LHC will measure couplings, even for the tau, to an accuracy of at most about 15 % [10, 11, 14]. This means that a 2 sigma measurement might just probe this deviation from the Standard Model. Our calculation would have to be performed more reliably in the limit that the second Higgs is light enough to generate a measurable deviation in Yukawas, but is probably reasonably accurate since the expansion really involves the light Higgs mass squared divided by the heavy Higgs mass squared. We leave more detailed study with light second Higgs mass for future work. At the ILC, both b and τ partial widths will be well measured, with the b partial width particularly accurate. The anticipated experimental accuracy in the b width will be between 1 and 2.4%, that for the τ is between 4.6 and 7.1%, for the photons is between 23 and 35%, and for the c is 8.1-12.3% (see Ref. [19] and references therein). These numbers do not include the theoretical uncertainties estimated to be about 2% for the bs and 12% for the cs for example [6]. Note that the best measured mode at the ILC, the b decay mode, is most likely to have a Yukawa that deviates from its Standard Model prediction. A su?ciently accurate measurement of the total width will also probe deviations of the decay width to bs when that mode dominates. Clearly, by measuring these relative rates at the ILC one can hope to explore much higher masses indirectly through precision light Higgs studies. This could be a very interesting probe of higher-energy physics than will be directly accessible. Notice also that the radiative corrections for very large tan β in supersymmetric theories can take the opposite sign to the tree level corrections we have considered here, as analyzed in [15]. If tan β is indeed very large these radiative corrections need to be taken into account and can end up suppressing the b branching fraction. It is straightforward to extend our analysis to the lepton sector. We consider models that preserve a discrete Z2 ?avor symmetry so that only one type of Higgs ?eld has a tree-level coupling to each of the di?erent fermion types. Clearly, only in Model II, where we expect the leptons to couple to H1 as do the down quarks, do we expect tan β enhancement in the lepton Yukawas. In these models we would expect the τ branching fraction and b branching fraction to change a comparable amount (up to loop e?ects). Radiative corrections to the bottom can be much bigger than those to the tau [12] (see also [13], but unless tan β is very large these are generally smaller than the tree-level corrections but eventually should be accounted for as well. In Model I, where only a single Higgs participates in the Yukawas, we expect H2 to couple to all fermions or else the top quark mass would be too low, which means that no fermions would get large Yukawa corrections. In Type III models as well, the leptons couple to H2 . 9

In both of these latter cases, the correction is suppressed by a factor of cot β and will be too small to matter in the large tan β limit.

4

Gauge Boson Coupling

It is also interesting to consider the light Higgs to two W coupling since the growth with energy isn’t fully stopped until we reach the second Higgs. This is similar to the analysis of [1] where it was argued there would be growth with energy in W W scattering until the strongly interacting scale in composite Higgs models. In practice at the LHC this will probably be a less promising way to search for evidence of a second Higgs because the H → W W won’t be su?ciently precisely measured since the energy reach isn’t big enough to enhance the cross section su?ciently, and because in the case of a doublet Higgs ?eld the corrections to the scattering e?ectively arise from higher-dimension operators than in Ref. [1]. One way to understand the source of the correction to the Higgs W W coupling in the strong coupling case [1] is from a higher order operator of the form cH v 2 ?? (H ? H)? ? (H ? H) 2f 2 f 2 where f is the scale of strong physics, which, after a shift in H ?eld gives a correction h cH m2 W? W ? W v (25) (24)

where h is the light Higgs. In e?ect, a dimension-6 operator could arise only in the presence of a singlet or triplet to be exchanged. In our case, with only a doublet Higgs, our correction is higher order. We expect a correction to hW W of order (v/mH )4 . In practice, we know precisely the coupling of h to a pair of W s. It is proportional to 2 sin(α ? β) = 1 ? cos (α?β) . The correction to unityh is indeed suppressed by (v/mH )4 as we 2 expected and is likely to be too small to measure.

5

Heavy Higgs Direct Searches

The heavy Higgs two vector boson coupling is suppressed by cos(α ? β), since the vev of the ?eld is suppressed by this factor and the vev enters the single Higgs two gauge boson coupling. This means that even when the two gauge boson decay is kinematically allowed, it won’t generally dominate. Similarly, heavy Higgs boson production is suppressed.6 Notice in the coupling to two W s there are no compensating tan β factors as there were for the down and potentially τ Yukawa corrections so the heavy Higgs to two gauge boson coupling is indeed small. CMS recently (2007) [17] studied the heavy Higgs discovery reach in the MSSM with systematic uncertainties taken into account. They found for a relatively light second Higgs
6

Here we are neglecting the other Higgs states but these will also be di?cult to ?nd.

10

(CP even or odd) that to ?nd a Higgs of 150 GeV, tan β must be greater than about 16 and for a Higgs of 250 GeV, tan β must be greater than about 35. This can be compared to the results from the Atlas TDR from 1999 [18] quoted by [10] where it was already noted that for Higgs mass of 250 GeV, tan β greater than 8 was necessary whereas for 500 GeV tan β needs to be at least 17. Clearly the situation has become worse with better understanding of the systematics and a reasonably large value of tan β is required to discover the heavy Higgs. The required large value of tan β is readily understood from our earlier considerations. In Type II two Higgs models preserving a Z2 symmetry, large tan β tells us the coupling of the heavy Higgs to bottom type quarks is enhanced whereas the coupling to top quarks and two gauge bosons is suppressed. Therefore production through bottoms is enhanced when tan β is large. Moreover decays to taus are enhanced in this limit as well and that is likely to be the best search mode. Note that even with the tan β-enhanced coupling to bottom quarks, the amplitude is proportional to the bottom Yukawa as well so only when tan β is su?ciently sizable will the production and decay become visible. Notice that although the analysis was done for the MSSM, the answer can be readily taken over to more general two Higgs models. The bottom and top Yukawas will be determined by tan β at leading order. The more model-dependent coupling is the coupling to two gauge bosons which is suppressed by the heavy Higgs vev (or equivalently cos(β ? α)). Once this is su?ciently small neither production nor decay through this mode is relevant. Note that Ref. [2] considered particular parameters in the two Higgs model to show that a heavy Higgs (of order TeV) can readily elude detection but induce large deviations from the Standard Model in the low-energy e?ective theory. They had large bottom Yukawa and small top Yukawa (to suppress standard Higgs production channel). Our point is that this happens automatically for large tan β (but not so large that the Higgs will be produced directly). Furthermore the CMS analysis shows that even a much lighter heavy Higgs than considered in [2] will not be seen unless tan β is su?ciently large. Of course even if tan β is large and the second Higgs is discovered, it will still be worthwhile to explore the types of deviations in Yukawas we have considered. We conclude that there is a large region of parameter space where precision light Higgs decays will be the best way to search for evidence of a second Higgs. This can also be a way of distinguishing among higher-dimension operator contributions to the Higgs mass squared as we discuss in the following section.

6

Implications for Testing Higher Dimension Operators

Recently Ref. [20] suggested the existence of higher dimension operators involving Higgs ?elds as a way of summarizing all possible models that might raise the Higgs mass without a large stop (or A term) (see also [21, 22]) in models that didn’t contain new light ?elds into which the Higgs could decay and escape observation (see [23–26] and references therein). In this way they hoped to address the little hierarchy problem that seems to require a large 11

stop mass to raise the Higgs mass adequately. It is of interest to ask how to detect such higher dimension operators. The obvious hope would be to ?nd and measure additional Higgs states and study the mass relations. However, as we have discussed, it will be di?cult to ?nd a second Higgs over much of the relevant parameter range and similar considerations apply to other states from the Higgs sector. This leaves the question if the light Higgs does indeed have bigger mass than expected on the basis of the MSSM, are there other ways to distinguish among di?erent possible higher dimension operators that might be contributing to its mass? Here we show that the likely leading operator to a?ect the Higgs mass is also precisely the one that should be best tested in the Higgs partial widths and the Yukawa analysis above readily applies. That means that not only can studying the branching fractions test for these operators, it could help distinguish among them. In Ref. [20], it was demonstrated that at leading order in an e?ective dimension expansion, only one operator contributes to the light Higgs mass in the large tan β (but not so large that higher order mass suppressed terms dominate over cot β suppressed terms) limit. This operator is λ M d2 θ(Hu Hd )2 (26)

(For simplicity, we assume all new parameters are real. We are also retaining the notation of Ref. [20] where Hu and Hd are used for H2 and H1 respectively.) When combined with the supersymmetric operator d2 θ?Hu Hd , we ?nd the quartic term 2λ? ? ? (Hu Hd )(Hu Hu + Hd Hd ) (27) M Such a term can also arise from a D-term type interaction. De?ning ?1 = λ?/M, one ?nds a Higgs mass correction of order 8?1 cot βv 2 (with the v = 246GeV convention we have been using) [20] The authors of Ref. [20] argued that one can get a su?ciently large correction to the Higgs mass (one that replaces the large stop contribution) for parameters such as tan β ? 10 and ?1 ? 0.06 Notice the interesting feature of this operator. Even though the only breaking of the Z2 symmetry in the superpotential was through the lower-dimension ?-term, it feeds into a dimension-4 Z2 -violating operator in the potential. This Z2 -breaking, characterized by ?/M, can be sizable. This will be important below. Alternatively, tan β could be so big (hence cos β so small) that terms suppressed by more powers of 1/M dominate over the leading 1/M correction. Such an unsuppressed ? contribution might arise from an operator (Hu Hu )2 for example. We can now use our previous analysis to consider the e?ect of such operators on a light Higgs coupling to down-type quarks and charged leptons. We see that the ?rst operator, while suppressed by cot β in its impact on the Higgs mass, is in fact exactly the type of operator that gets a tan β enhanced contribution to the Yukawa coupling deviations above. That is because it arises through the Z2 -violating ?-term and contributes directly to λ7 . In particular, λ7 ? 2?1 . If tan β is large, Yukawa couplings receive corrections from tan β2?1 (v/mH )2 12

e?ects. As an example, if mH ? 1.5v, with the parameters given above, these contributions could reduce the h → τ τ rate by a factor of 4, while increasing the h → γγ rate by a similar ? factor (due to the decreased rate to b? Even without discovering the second Higgs, these b). e?ects could be big enough to test for the higher dimension operators indirectly. As an aside we note that recent papers [23–25] have considered the possibility that the light Higgs does not decay into the modes that have been sought for at LEP. In those papers there were alternative beyond the minimal supersymmetric model light modes available into which the Higgs can decay. We have just seen that even without these additional light modes, the Higgs branching ratio into b? and τ τ can be reduced substantially. However even when b ? the branching ratio to bs is so reduced that other modes dominate, the alternative decay modes would have been visible as well, so the Higgs mass bound would not be reduced by more than a few GeV [27] so this doesn’t alter the allowed range of M signi?cantly. Returning to the e?ects on LHC branching fractions, for an operator whose contribution to the squared Higgs mass is suppressed by two powers of M but not by cos β such as ? (Hu Hu )2 , the contribution to the deviation in the Yukawa will nonetheless be suppressed by cos β. Therefore the e?ect on the bottom Yukawa is much smaller than for the Z2 -breaking operator we just considered. We can readily understand the relative signs and magnitudes of Yukawa corrections from the various operators by studying the sign and cos β dependence contributions to both the light Higgs mass squared and to the bottom-type Yukawa couplings of the various operators in the limit that cos β is small. Operator Mass Squared Contribution Down Yukawa Contribution (Hu Hd )2 cos2 β cos β ? (Hu Hu )(Hu Hd ) cos β 1 ? (Hu Hu )2 1 ? cos β

We see that the operators consistent with the Z2 symmetry do indeed give cos β-suppressed contributions to the change in the down-type Yukawas. We also see that the e?ect of the last operator has the opposite sign which is why it increases the branching fraction of the bottom whereas the other operators decrease it. This should be a powerful tool for distinguishing among higher dimension operators should they be present. Therefore if a light Higgs consistent with current experimental constraints and small stop mass is discovered (assuming small A), measuring branching ratios could test which higher dimension operator is the relevant one in raising its mass. In particular the e?ects of the ?rst type of operator can have signi?cant e?ects on the Higgs decay rate and branching ratios which we would not expect for the higher e?ective dimension operators.

7

Conclusion

We conclude that is is very likely that even if there are two Higgs doublet ?elds and the second neutral Higgs scalar is kinematically accessible to the LHC, it is likely that the second Higgs will elude direct detection. This makes the question of indirect evidence for the full 13

Higgs sector very important. We have seen that there is a large parameter range where precision measurements, in particular of the branching fraction of the light Higgs into taus vs. photons, can ?nd indirect evidence for a second Higgs ?eld. If there are Z2 -violating interactions in the Higgs quartic terms, there can be enormous changes to the bottom and tau branching fractions, so large that they will be re?ected in the overall Higgs decay rate and will result in a signi?cant change in the branching fraction to other modes. We have also seen that such measurements can be a powerful way to test for higher dimension operators in a supersymmetric theory and that the operator which is perhaps most likely to a?ect the light Higgs mass will yield signi?cant changes to the decay widths into bottoms and taus. Therefore precision Higgs branching fraction measurements can be extremely important if the world does in fact contain two Higgs ?elds. It will be interesting to do more detailed explorations of parameters, to consider which range is most natural and for how large a parameter range the considerations of this paper apply. It will also be of interest to incorporate the e?ects of CP violation.

8

Acknowledgements

I thank Christophe Grojean and Cedric Delaunay for participating in the early stages of this work and I thank them and Liam Fitzpatrick, Spencer Chang, Howie Haber, Gilad Perez, Massimo Porrati, Giovanni Villadoro, and Mark Wise for reading the manuscript and for very useful discussions. This work was supported in part by NSF grants PHY-0201124 and PHY-0556111. I also thank NYU for their hospitality while this work was being completed.

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