The Ideal Higgs Scenario and Its Ramifications
Jack Gunion U.C. Davis
NTU-Davis Workshop, December 16, 2008
1. The “ideal” Higgs boson motivation for a light a with ma < 2mb. 2. Constraints from LEP and Upsilon Decays.
3. Constraints from Tevatron and LHC.
4. Relation to aµ.
5. The NMSSM Context.
6. Anything to do with CDF Multi-muon events?
Criteria for an ideal Higgs theory
• The theory should predict a Higgs with SM coupling-squared to W W, ZZ and with mass in the range preferred by precision electroweak data. The latest plot is:
At 95% CL, mhSM < 160 GeV and the ∆χ2 minimum is near 85 GeV when all data are included.
The latest mW and mt measurements also prefer mhSM ∼ 100 GeV.
The blue-band plot may be misleading due to the discrepancy between the ”leptonic” and ”hadronic” measurements of sin2 θWef f, which yield sin2 θWef f = 0.23113(21) and sin2 θWef f = 0.23222(27), respectively. The SM has a CL of only 0.14 when all data are included.
If only the leptonic sin2 θWef f measurements are included, the SM gives a fit with CL near 0.78. However, the central value of mhSM is then near 50 GeV with a 95% CL upper limit of ∼ 105 GeV (Chanowitz, xarXiv:0806.0890).
• Thus, in an ideal model, a Higgs with SM-like ZZ coupling should have mass no larger than 105 GeV. Our generic notation will be H.
But, at the same time, It should avoid the LEP limits on such a light Higgs.
One generic possibility is for its decays to be non-SM-like.
Table 1: LEP mH Limits for a H with SM-like ZZ coupling, but varying decays.
Mode SM modes 2τ or 2b only 2j W W∗ + ZZ∗ γγ /E 4e, 4µ, 4γ
Limit (GeV) 114.4 115 113 100.7 117 114 114?
Mode 4b 4τ any (e.g. 4j) 2f + /E
Limit (GeV) 110 86 82 90?
To have mH ≤ 105 GeV requires one of the final three modes.
• Perhaps the ideal Higgs should be such as to predict the 2.3σ excess at Mbb ∼ 98 GeV seen in the Z + bb final state.
The simplest possibility for explaining the excess is to have mH ∼ 100 GeV and B(H → bb) ∼ 0.1B(H → bb)SM (assuming H has SM ZZ coupling).
• All of this can be accomplished in the NMSSM with no fine-tuning, ...., but for now I wish to be more general and only look at the generic possibility of suppressing the H → bb branching ratio by having a light a (ma < 2mb to avoid LEP Z + b0s limits) with B(H → aa) > 0.7.
Since the Hbb coupling is so small, very modest Haa coupling suffices.
The scenario (a) is easy to achieve in general 2HDM-II models, (b) is not possible in the MSSM, but (c) is a very natural possibility in the NMSSM where a light a corresponds to a U (1)R symmetry limit.
• Two-Higgs Doublet (2HDM) type-II and related reminders:
hHui = hu gives mass to up-type quarks and hHdi = hd gives mass to down-type quarks and leptons. v2 = h2u + h2d is fixed by the value of m2Z. Given this, it is useful to define the remaining free parameter of the Higgs sector as
tan β = hu
hd . (1)
In the 2HDM, the physical Higgs particles are the CP-even h, H, the CP-odd A, and the charged Higgs pair H±.
The MSSM Higgs sector is a constrained 2HDM-II model.
The NMSSM Higgs sector has an additional (complex) singlet Higgs field and Higgs particles h1, h2, h3, a1, a2 and h±.
Constraints on a from LEP and Upsilon Decays
To fit with the Ideal Higgs scenario, we are especially interested in an a with ma < 2mb.
• Of particular importance are the constraints on Cabb, where the generic Caf f is defined by
Laf f ≡ iCaf fig2mf
2mW f γ5f a . (2) We will only discuss models in which Cabb = Caµ−µ+. (To escape, requires 3 or more doublets.)
The most useful current limits on Cabb for a light a come from CUSB-II (old 90% CL) limits on B(Υ → γX) (where X is assumed to be visible), recent CLEO-III limits on B(Υ → γa) assuming a → 2τ, OPAL limits on e+e− → bba → bb2τ and DELPHI limits on e+e− → bba → bbbb.
(The Tevatron limits on bba → bb2τ apply for quite high ma, beyond the region we wish to focus on.)
The CLEO-III limits are now particularly strong.
Figure 1: Limits on B(Υ → γτ+τ−).
• For the most part the extracted Cabb limits (JFG, arXiv:0808.2509) are quite model-independent other than weak dependence on up-quark couplings (mostly via the top gg coupling loop, but also through B(a → τ τ ) and B(a → bb)). The extracted limits on Cabb appear in Fig. 2,
Figure 2: Limits on Cabb.
The most unconstrained region is that with ma > 8 GeV, especially 9 GeV < ma < 12 GeV.
In the ∼ 9 GeV <∼ ma <∼ 12 GeV region only the OPAL limits are relevant.
Those presented depend upon how the a ↔ ηb states mixing is modeled.
A particular model is employed, but there has been little recent work on this.
Perhaps now that the first ηb state has been observed, this region can be better pinned down.
Constraints from Tevatron and LHC
• However, we (JFG+Dermisek) have recently discovered that Tevatron data on the di-muon spectrum also has an impact.
In particular, a recent CDF analysis has been directly employed to place a 90% CL upper limit on σ() × B( → µ+µ−), where the is some narrow resonance, relative to the measured σ(Υ) × B(Υ → µ+µ−).
The histogram shown in the following figure is the CDF 630 pb−1 result.
In the figure, the predictions for the cross section ratio for the a are (for Cabb = tan β and Catt = cot β)): +’s=tan β = 1, ’s=tan β = 2,
×’s=tan β = 3. Fortunately, the a and Υ cross sections are quite flat in y and only small |y| production is kept in the experimental analysis.
Figure 3: 90% CL limits on σ(a)B(a→µ+µ−)
σ(Υ)B(Υ→µ+µ−) at small |y| for L = 630 pb−1, compared to expectations for the a for Cabb = tan β = 1/Catt =1, 2, 3 in the 2HDM-II. Also shown (’s) are the predictions for the NMSSM with tan β = 10 and cosθA = 0.1 for which Cabb = tanβ cos θA = 1 and Catt = cot β cos θA = 1/100 — not much different from the Cabb = tan β = 1/Catt = 1 case.
• Translating the 630 pb−1 results into limits on Cabb gives the dotted histogram in the 6 − 9 GeV region in Fig. 4 (below):
Figure 4: Limits on Cabb including those from the Tevatron analysis.
The Tevatron limits are the best for ∼ 8 GeV < ma < ∼ 9 GeV.
CDF should push analysis above 9 GeV to at least the BB threshold (and perhaps a bit beyond since the threshold region is complex and LEP limits on a light h → aa might still be obeyed for ma somewhat above threshold).
Did multi-µ events prevent this? (more later)
Limits will improve as more integrated luminosity is accumulated/analyzed.
• What about the LHC? A careful analysis is required. New issues include:
1. Triggering on soft muons.
Probably a recoiling jet is required to boost the µ momenta.
2. bb backgrounds will be bigger than at the Tevatron.
3. Muon isolation is clearly trickier, especially at higher luminosity.
4. Low L running might provide the optimal situation since you can simply take all data and then work on it.
5. Is LHCb better than CMS/ATLAS?
• Typical string models predict a plethora of light a’s and light h’s that have fermionic couplings, even if not W W couplings. ⇒ very important to pursue strongest possible Υ → γa and gg → a → µ+µ− limits.
Implications for aµ
• Given Cabb limits, an interesting question is whether there is any possibility that a light a could be responsible for the observed aµ discrepancy which is of order ∆aµ ∼ 30 × 10−10.
• The maximum possible value of δaµ from the a occurs for the maximum allowed Cabb regardless of the value Rb/t2 = Cabb/Catt (for the 2HDM- II, R2b/t = tan2 β = C2
abb, but for more complicated Higgs sectors, very different values for this ratio are possible). Note: Catt enters at the two loop-level.
Figure 5 (next page) shows that it is quite improbable that a light a could explain ∆aµ, regardless of the Rb/t choice.
Only in the small window in ma from about 8 GeV (9.5 GeV for 2HDM-II) up to ∼ 12 GeV, where Cabb limits are the weakest (Cabb <∼ 15 − 60), might it be possible.
Figure 5: Results for δamaxµ from a CP-odd a for various R2b/t = Cabb/Catt models are plotted after incorporating the Cabb experimental limits. Curves are for Rb/t = 1, 3, 10, 50 and for the 2HDM-II prediction of Rb/t = tan β = Cabb (which looks like Rb/t = 50 for ma >∼ 9 GeV and is the isolated red curve at lower ma.)
The NMSSM Context
The ideal Higgs scenario is naturally realized in the very attractive Next-to- Minimal Supersymmetric Model (JFG+Dermisek):
Motivations for SUSY and for the NMSSM version thereof
• SUSY cures the hierarchy problem coming from the top-quark loop correction to the Higgs mass-squared by introducing a canceling stop- squark loop. Fine-tuning of the Higgs mass to be below ∼ 700 GeV (required for WLWL → WLWL unitarity) is avoided if m
te <∼ 1 TeV.
• But, the Minimal Supersymmetric Model requires a term in the superpotential of form
W 3 µ cHuHcd , (3)
where Hcu and Hcd are the Higgs superfields whose scalar components acquire vevs that gives rise to the up-type and down-type quark masses, respectively.
Phenomenologically, µ ∼ f ew × 100 GeV is needed to avoid various experimental limits and yet have a SM-like Higgs that is good for precision electroweak data.
However, theoretically, µ ∼ MP is expected (or µ = 0 because of a discrete symmetry). This is called the µ problem.
• The NMSSM is obtained from the MSSM by adding a superfield Sb that is a SM singlet which provides a beautiful solution to the µ problem.
Starting from the superpotential
W 3 λ bS cHuHcd + κ
3Sb3 . (4)
one gets µ ∼ λhSi and hSi will be of order 1 TeV.
Indeed, since λ and κ are dimensionless, all dimensionful quantities, including hSi are set by the scale of supersymmetry breaking ( and above we noted that the scale of supersymmetry breaking as represented by m
te must be <∼ 1 TeV in order to avoid fine-tuning in getting a Higgs mass-squared that is in the right balk park).
• Meanwhile, the NMSSM preserves the ”good” MSSM features: coupling constant unification and RGE generation of EWSB.
• The NMSSM also allows a solution to the ”other” fine-tuning problem of the MSSM, namely, how precisely must the GUT-scale parameters of the model (e.g. soft-SUSY-breaking masses) be tuned to obtain the observed value of m2Z after RGE evolution to low-energies.
• In any supersymmetric model, the value of m2Z is least sensitive to the GUT-scale parameters if the stops have m
et <∼ 350 GeV.
For such stop masses, the lightest CP-even Higgs (whether h in the MSSM or h1 in the NMSSM) will have mass <∼ 100 GeV.
In the MSSM , this is a problem since the h has SM-like couplings and decays so that LEP requires mh > 114 GeV. A high level of GUT-scale parameter fine-tuning is required to get m2Z correct if mh > 114 GeV.
In the NMSSM , an h1 with mh1 <∼ 100 GeV escapes LEP limits if
h1 → a1a1 is large and ma1 < 2mb. Fine-tuning of GUT-scale parameters to get the observed m2Z is not required.
• An important question: Is fine-tuning of GUT-scale parameters (namely the Aλ and Aκ soft-SUSY-breaking parameters associated with the λ and κ superpotential terms) required to achieve the above a1 properties.
The answer is not necessarily. To understand this statement we need to learn a bit more about the NMSSM.
• First, starting from GUT-scale parameters Aλ and Aκ close to zero (the U (1)R symmetry limit) and evolving gives low-scale Aλ and Aκ values that will typically yield a light a1.
The real question is will B(h1 → a1a1) be large enough (>∼ 0.7).
• In the NMSSM context, a crucial quantity for the latter is cos θA, the coefficient of the MSSM-like doublet Higgs component of the a1:
a1 = cos θAAM SSM + sin θAAS . (5)
• One finds that to achieve B(h1 → a1a1) > 0.7 for ma1 < 2mb will not require fine-tuning, provided ma1 > 7.5 GeV (implying a1 → τ+τ−) and Cabb = cos θA tan β has absolute value <∼ 1! (This is relaxed in certain scenarios with tan β ≤ 3.)
• Further, for any tan β value there is a lower bound on | cos θA| required to get B(h1 → a1a1) > 0.7. In the end, |Cabb| >∼ 0.35 is required.
• As a result, The a1 of the NMSSM Ideal Higgs scenario might in fact be observed if Υ decays and the Tevatron di-muon spectrum can both be pushed to the |Cabb| < 1 level in the 7.5 GeV <∼ ma1 <∼ 10 − 11 GeV region.
Typically one must gain a factor of 2 to 3 improvement in |Cabb| limits relative to current limits, which statistically means a factor of about 10 in luminosity.
The CDF multi-muon events and NMSSM extensions.
• There is nothing sacred about having just one additional singlet superfield.
A large fraction of string models have multiple singlets.
• Suppose we have S, S1 and S2.
We then have 5 CP-even states and 4 CP-odd states.
• hSi provides µ as before.
• One scenario (a variant of Ellwanger et. al.): I try to preserve the ideal scenario, but also get multi-muon events.
1. Pull an A, h, H mainly from the Hu − Hd sector.
Pull a1 mainly out of S — i.e. mainly S-singlet.
We assume ma1 >∼ 2mτ is needed to avoid light-a1-fine-tuning.
As in the ideal Higgs scenario, mh <∼ 100 GeV is wanted for precision electroweak and a1 must have some minimal amount of doublet in order for B(h → a1a1) to be large enough for h to evade LEP limits.
Ignore the CP-even partner of a1, which we assume is heavy (and is mainly S-singlet).
2. Pull h1and h2 out of the S1 and S2 singlets. These are quite light states with masses in the 10 GeV to 20 GeV range and they have very small Yukawa couplings (implying that higgs to higgs pair chain decays are probable).
3. The envisioned decay sequence for the multi-muon events is H → h1h1 with h1 → 2h2 → 4a1 → 8τ .
Each τ then decays to µ and so each side of the event has 8µ.
4. To describe the observed events, mH ∼ 100 GeV is needed and σ(gg → H) >∼ 100 pb is required.
Since H is largely doublet, σ(gg → H) can be enhanced to needed
>∼ 100 pb level for tan β > 30.
5. Finally, one of the Higgs in the chain must be long-lived. This cannot be a1 since B(h → a1a1) would then, as described earlier, not be large
enough for h to escape the LEP limit.
Probably can make h1 or h2 long lived by making one of them nearly entirely singlet. However, this is uncertain without a real calculation since, say, h1 → γγ is mediated by chargino/higgsino loops and this decay might take over, as it does in the case of a pure singlet a1 in the simple NMSSM (K. Cheung et al).
(Might also be a problem for the Ellwanger et. al. model where a1 is supposed to have a long lifetime. Did they look at the one-loop induced a1 → γγ decay?)
6. Of course, simultaneously one could have the Ellwanger et. al. process gg → A production with A → h1a2 and a2 → h1a1 (my labeling switches a1 and a2 of Ellwanger et. al.), with subsequent decays of h1 as before.
One side of the event has 8µ and the other side then has 10µ.
7. Where are the pure e and mixed e, µ events?
Are electron efficiencies so much worse?
• A light a with ma < 2mb of the ”ideal” Higgs scenario with mh < 105 GeV (escaping LEP limits because B(h → aa → 4τ ) is large) might be discoverable in the di-muon spectrum at the Tevatron or LHC.
• Alternatively, the Tevatron and LHC might be able to place limits on the Cabb of a light a that would be difficult to reconcile with a specific model.
This appears to be within reach even for the most preferred small-cos θA, ma <∼ 2mb high-tan β NMSSM models.
Already, the less preferred, larger | cos θA| models in the high-tan β NMSSM scenarios are being ruled out over part of the relevant mass region beyond that accessible in Υ decays.
Potentially, the hadron colliders could go to higher di-muon masses and they definitely should.
• Having both Υ decay and hadron collider data appears to be crucial.
The former covers the low ma region (where the di-muon Drell-Yan background overwhelms the hadron collider a → µ+µ− signal and muon triggering becomes hard).
The latter is the only way (and apparently a viable way) to access the higher ma <∼ 2mB and above threshold regions.
• If we were to see an a with the right properties, this would give enormous impetus to focusing on the pp → pph and W W → h with h → aa → 4τ search modes.
• For a generic 2HDM-II model, there is only a small 10 GeV < ma < 12 GeV window left for which the a might explain ∆aµ and this is possible only if Cabb = tan β is large.
It would appear that extending the hadron colliders to high enough ma to rule this out is possible.
• In the NMSSM:
The preferred NMSSM models do not have large Cabb = cos θA tan β coupling.
Instead, small light-a1-fine-tuning models with high tan β have small cos θA for which Cabb = cos θA tan β <∼ 1.
At low tan β, cos θA is larger than 1 for an attractive class of models and Tevatron data might be able to rule out such scenarios for somewhat lower L.
The multi-muon CDF events might have an extended NMSSM explanation.
But, one must explain absence of similar multi-e events.
And, it is likely that the narrow a1 → µ+µ− peak would be observable since B(a1 → µ+µ−)/B(a1 → τ+τ−) >∼ 0.0035 is not that small, especially if ma1 is so close to 2mτ as to make the a1 long-lived (mostly singlet also required of course).
Why didn’t they do a narrow peak analysis along the lines of what I discussed earlier? (Strassler)