## The Ideal Higgs Scenario and Its Ramifications

Jack Gunion U.C. Davis

NTU-Davis Workshop, December 16, 2008

### Outline

1. The “ideal” Higgs boson motivation for a light a with m_{a} < 2m_{b}.
2. Constraints from LEP and Upsilon Decays.

3. Constraints from Tevatron and LHC.

4. Relation to a_{µ}.

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, m_{h}_{SM} < 160 GeV and the ∆χ^{2} minimum is near 85 GeV when
all data are included.

The latest m_{W} and m_{t} measurements also prefer m_{h}_{SM} ∼ 100 GeV.

The blue-band plot may be misleading due to the discrepancy between
the ”leptonic” and ”hadronic” measurements of sin^{2} θ_{W}^{ef f}, which yield
sin^{2} θ_{W}^{ef f} = 0.23113(21) and sin^{2} θ_{W}^{ef f} = 0.23222(27), respectively. The
SM has a CL of only 0.14 when all data are included.

If only the leptonic sin^{2} θ_{W}^{ef f} measurements are included, the SM gives a fit
with CL near 0.78. However, the central value of m_{h}_{SM} 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 m_{H} 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 m_{H} ≤ 105 GeV requires one of the final three modes.

• Perhaps the ideal Higgs should be such as to predict the 2.3σ excess at
M_{bb} ∼ 98 GeV seen in the Z + bb final state.

The simplest possibility for explaining the excess is to have m_{H} ∼ 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 (m_{a} < 2m_{b}
to avoid LEP Z + b^{0}s 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:

hH_{u}i = h_{u} gives mass to up-type quarks and hH_{d}i = h_{d} gives mass to
down-type quarks and leptons. v^{2} = h^{2}_{u} + h^{2}_{d} is fixed by the value of m^{2}_{Z}.
Given this, it is useful to define the remaining free parameter of the Higgs
sector as

tan β = h_{u}

h_{d} . (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 h_{1}, h_{2}, h_{3}, a_{1}, a_{2} 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 m_{a} < 2m_{b}.

• Of particular importance are the constraints on C_{abb}, where the generic
C_{af f} is defined by

L_{af f} ≡ iC_{af f}ig_{2}m_{f}

2m_{W} f γ_{5}f a . (2)
We will only discuss models in which C_{abb} = C_{aµ}^{−}_{µ}^{+}. (To escape, requires
3 or more doublets.)

The most useful current limits on C_{abb} 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 m_{a}, 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 C_{abb} 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 C_{abb} appear in Fig. 2,

Figure 2: ^{Limits on} ^{C}_{abb}^{.}

The most unconstrained region is that with m_{a} > 8 GeV, especially
9 GeV < m_{a} < 12 GeV.

In the ∼ 9 GeV <∼ m^{a} <∼ 12 GeV region only the OPAL limits are relevant.

Those presented depend upon how the a ↔ η_{b} states mixing is modeled.

A particular model is employed, but there has been little recent work on this.

Perhaps now that the first η_{b} 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 C_{abb} = tan β and C_{att} = 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 C_{abb} = tan β = 1/C_{att} =1, 2, 3 in the 2HDM-II. Also shown
(’s) are the predictions for the NMSSM with tan β = 10 and cosθ_{A} = 0.1 for which
C_{abb} = tanβ cos θ_{A} = 1 and C_{att} = cot β cos θ_{A} = 1/100 — not much different from
the C_{abb} = tan β = 1/C_{att} = 1 case.

• Translating the 630 pb^{−1} results into limits on C_{abb} gives the dotted
histogram in the 6 − 9 GeV region in Fig. 4 (below):

Figure 4: ^{Limits on} ^{C}_{abb} including those from the Tevatron analysis.

The Tevatron limits are the best for ∼ 8 GeV < m_{a} < ∼ 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 m_{a} 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 a

_{µ}

• Given C_{abb} limits, an interesting question is whether there is any possibility
that a light a could be responsible for the observed a_{µ} discrepancy which
is of order ∆a_{µ} ∼ 30 × 10^{−10}.

• The maximum possible value of δa_{µ} from the a occurs for the maximum
allowed C_{abb} regardless of the value R_{b/t}^{2} = C_{abb}/C_{att} (for the 2HDM-
II, R^{2}_{b/t} = tan^{2} β = C^{2}

abb, but for more complicated Higgs sectors, very
different values for this ratio are possible). Note: C_{att} enters at the two
loop-level.

Figure 5 (next page) shows that it is quite improbable that a light a could
explain ∆a_{µ}, regardless of the R_{b/t} choice.

Only in the small window in m_{a} from about 8 GeV (9.5 GeV for 2HDM-II)
up to ∼ 12 GeV, where C_{abb} limits are the weakest (C_{abb} <∼ 15 − 60),
might it be possible.

Figure 5: Results for δa^{max}_{µ} from a CP-odd a for various R^{2}_{b/t} = C_{abb}/C_{att} models are
plotted after incorporating the C_{abb} experimental limits. Curves are for R_{b/t} = 1, 3, 10, 50
and for the 2HDM-II prediction of R_{b/t} = tan β = C_{abb} (which looks like R_{b/t} = 50 for
m_{a} >∼ 9 GeV and is the isolated red curve at lower m_{a}.)

### 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 W_{L}W_{L} → W_{L}W_{L} unitarity) is avoided if m

te <∼ 1 TeV.

• But, the Minimal Supersymmetric Model requires a term in the superpotential of form

W 3 µ cH_{u}Hc_{d} , (3)

where Hc_{u} and Hc_{d} 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 ∼ M_{P} 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 cH_{u}Hc_{d} + κ

3Sb^{3} . (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 m^{2}_{Z} after RGE evolution to low-energies.

• In any supersymmetric model, the value of m^{2}_{Z} 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 h_{1} 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 m_{h} > 114 GeV. A high level of GUT-scale
parameter fine-tuning is required to get m^{2}_{Z} correct if m_{h} > 114 GeV.

In the NMSSM , an h_{1} with m_{h}_{1} <∼ 100 GeV escapes LEP limits if

h_{1} → a_{1}a_{1} is large and m_{a}_{1} < 2m_{b}. Fine-tuning of GUT-scale parameters
to get the observed m^{2}_{Z} is not required.

• An important question: Is fine-tuning of GUT-scale parameters (namely
the A_{λ} and A_{κ} soft-SUSY-breaking parameters associated with the λ and
κ superpotential terms) required to achieve the above a_{1} 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 A_{λ} and A_{κ} close to zero (the
U (1)_{R} symmetry limit) and evolving gives low-scale A_{λ} and A_{κ} values that
will typically yield a light a_{1}.

The real question is will B(h_{1} → a_{1}a_{1}) be large enough (>∼ 0.7).

• In the NMSSM context, a crucial quantity for the latter is cos θ_{A}, the
coefficient of the MSSM-like doublet Higgs component of the a_{1}:

a_{1} = cos θ_{A}A_{M SSM} + sin θ_{A}A_{S} . (5)

• One finds that to achieve B(h_{1} → a_{1}a_{1}) > 0.7 for m_{a}_{1} < 2m_{b} will not
require fine-tuning, provided m_{a}_{1} > 7.5 GeV (implying a_{1} → τ^{+}τ^{−}) and
C_{abb} = cos θ_{A} 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 θ_{A}| required to
get B(h_{1} → a_{1}a_{1}) > 0.7. In the end, |C_{abb}| >∼ 0.35 is required.

• As a result, The a_{1} 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 |C_{abb}| < 1 level in the 7.5 GeV <∼ m^{a}^{1} <∼ 10 − 11 GeV
region.

Typically one must gain a factor of 2 to 3 improvement in |C_{abb}| 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, S_{1} and S_{2}.

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 H_{u} − H_{d} sector.

Pull a_{1} mainly out of S — i.e. mainly S-singlet.

We assume m_{a}_{1} >∼ 2m^{τ} is needed to avoid light-a_{1}-fine-tuning.

As in the ideal Higgs scenario, m_{h} <∼ 100 GeV is wanted for precision
electroweak and a_{1} must have some minimal amount of doublet in order
for B(h → a_{1}a_{1}) to be large enough for h to evade LEP limits.

Ignore the CP-even partner of a_{1}, which we assume is heavy (and is
mainly S-singlet).

2. Pull h_{1}and h_{2} out of the S_{1} and S_{2} 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 → h_{1}h_{1}
with h_{1} → 2h_{2} → 4a_{1} → 8τ .

Each τ then decays to µ and so each side of the event has 8µ.

4. To describe the observed events, m_{H} ∼ 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 a_{1} since B(h → a_{1}a_{1}) would then, as described earlier, not be large

enough for h to escape the LEP limit.

Probably can make h_{1} or h_{2} long lived by making one of them nearly
entirely singlet. However, this is uncertain without a real calculation
since, say, h_{1} → γγ is mediated by chargino/higgsino loops and this
decay might take over, as it does in the case of a pure singlet a_{1} in the
simple NMSSM (K. Cheung et al).

(Might also be a problem for the Ellwanger et. al. model where a_{1} is
supposed to have a long lifetime. Did they look at the one-loop induced
a_{1} → γγ decay?)

6. Of course, simultaneously one could have the Ellwanger et. al. process
gg → A production with A → h_{1}a_{2} and a_{2} → h_{1}a_{1} (my labeling
switches a_{1} and a_{2} of Ellwanger et. al.), with subsequent decays of h_{1}
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?

### Conclusions

• A light a with m_{a} < 2m_{b} of the ”ideal” Higgs scenario with m_{h} < 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
C_{abb} 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 θ_{A},
m_{a} <∼ 2m^{b} high-tan β NMSSM models.

Already, the less preferred, larger | cos θ_{A}| 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 m_{a} 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 m_{a} <∼ 2m^{B} 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 < m_{a} < 12 GeV
window left for which the a might explain ∆a_{µ} and this is possible only if
C_{abb} = tan β is large.

It would appear that extending the hadron colliders to high enough m_{a} to
rule this out is possible.

• In the NMSSM:

The preferred NMSSM models do not have large C_{abb} = cos θ_{A} tan β
coupling.

Instead, small light-a_{1}-fine-tuning models with high tan β have small cos θ_{A}
for which C_{abb} = cos θ_{A} tan β <∼ 1.

At low tan β, cos θ_{A} 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 a_{1} → µ^{+}µ^{−} peak would be observable since
B(a_{1} → µ^{+}µ^{−})/B(a_{1} → τ^{+}τ^{−}) >∼ 0.0035 is not that small, especially if
m_{a}_{1} is so close to 2m_{τ} as to make the a_{1} 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)