Constraints on a Light Singlet Scalar from Combined Exotic Higgs Decays

This paper investigates the phenomenology of a Standard Model extension with a light real gauge-singlet scalar, deriving analytical expressions for exotic Higgs decays to two and three scalars and establishing a global constraint that limits the scalar-Higgs mixing angle to $\cos \theta < 0.12\text{--}0.13$, thereby providing complementary bounds on the model's parameter space.

The Gist

Imagine the universe is built on a set of rules called the Standard Model. For a long time, this rulebook worked perfectly, but scientists knew it was missing a few pages. They suspected there were hidden characters—like dark matter or invisible forces—that the current rules couldn't explain.

One popular idea to fill these gaps is to add a new, invisible character to the story: a light, ghostly particle called a "singlet scalar." Think of this particle as a shy ghost that only interacts with the rest of the universe through a specific "doorway": the Higgs boson.

The Higgs boson is like a famous celebrity at a party. Usually, it interacts with other known particles (like quarks and electrons) in very predictable ways. But if this new "ghost" particle exists, the Higgs might occasionally sneak off the main stage to hang out with the ghost instead. This is called an "exotic decay."

The Big Problem: The Higgs is Too Busy

In this paper, the authors (F. Azari and M. Haghighat) ask a simple question: How often can the Higgs sneak off to visit these ghosts without getting caught?

They know exactly how much "time" the Higgs has to spend at the party. Scientists have measured the total time the Higgs exists before it decays (its "width"). They also know how much time it spends with all the known, standard particles. There is only a tiny sliver of time left over for anything new.

The authors realized that previous studies only looked at one type of "sneak-off" at a time:

1. The Higgs splitting into two ghosts.
2. The Higgs splitting into three ghosts.

They argued that looking at just one type is like checking if a thief stole a watch or a wallet, but not checking if they stole both. To get the real limit, you have to add them up.

The "Budget" Analogy

Think of the Higgs boson's total decay time as a strict monthly budget.

• Standard Expenses: 99% of the budget is already spent on known particles (like money going to rent and groceries).
• The Remaining Budget: There is a tiny, fixed amount of money left for "exotic" spending.

The authors calculated that if the Higgs splits into two ghosts, it costs a certain amount of "money." If it splits into three ghosts, it costs a different amount. They added these costs together and said: "The total cost cannot exceed the remaining budget."

What They Found

By doing the math on this combined budget, they discovered a hard limit on how "connected" the Higgs can be to this new ghost particle.

1. The Mixing Limit: The connection between the Higgs and the ghost is controlled by a number called the "mixing angle" (let's call it cos θ). The authors found that this number must be very small—specifically, less than 0.12 to 0.13.

   • Analogy: Imagine the Higgs and the ghost are two dancers. The "mixing angle" is how close they hold each other. The authors proved they can't hold hands tighter than a very specific, loose grip, or the Higgs would run out of "time" (energy) too fast.

2. The Ghost Mass: This rule applies to ghosts that are very light (between 0 and 40 GeV). If the ghost is too heavy, it's a different story, but for these light ghosts, the rule is strict.

3. The Resulting Limits: Because the mixing has to be so weak, the authors calculated exactly how often these exotic events can happen:

   • The Higgs can turn into two ghosts at most about 0.06 MeV worth of times.
   • The Higgs can turn into three ghosts at most about 0.000005 MeV worth of times.
   • Analogy: It's like saying the Higgs can only throw a secret party with the ghosts once in a blue moon. If it happens more often, the math breaks, and the Higgs wouldn't exist as we see it.

Why This Matters

The authors didn't just look at one channel; they looked at the whole picture. They showed that even if we haven't seen these ghosts directly yet, the fact that the Higgs boson exists and behaves the way it does already tells us that these ghosts must be very shy and very weakly connected to our world.

This provides a new, independent "fence" around where these particles can hide. If future experiments try to find these ghosts, they now know exactly how "loud" the signal can be before it contradicts what we already know about the Higgs boson. It's a way of saying, "We haven't seen you yet, but if you are there, you can't be very loud."

Technical Summary

1. Problem Statement

The Standard Model (SM) is incomplete, particularly regarding dark matter and the nature of the electroweak phase transition. A minimal extension involves adding a real gauge-singlet scalar field ($\Phi$) that couples to the SM Higgs doublet ($H$) via a "Higgs portal" interaction. This extension introduces a new physical scalar particle ($\phi$) with mass $m_\phi$ and a mixing angle ($\theta$) with the SM Higgs ($h$).

While extensive constraints exist for heavy scalars ($m_\phi > 50$ GeV) and light scalars ($m_\phi < 40$ GeV) from direct searches (LEP, LHC) and precision measurements, a comprehensive constraint derived from the combined contribution of all kinematically accessible exotic Higgs decay channels is missing. Specifically, for light scalars where $3m_\phi < m_h$, the Higgs boson can decay into both two scalars ($h \to \phi\phi$) and three scalars ($h \to \phi\phi\phi$). Previous studies often analyzed these channels in isolation, potentially underestimating the total constraint on the model parameters.

2. Methodology

The authors employ a global constraint approach based on the total measured width of the Higgs boson.

• Theoretical Framework:

  • The model extends the SM with a real singlet $\Phi$ possessing a $Z_2$ symmetry ($\Phi \to -\Phi$).
  • The scalar potential includes a portal term: $V_{h\phi} = \lambda_3 H^\dagger H \Phi^2$.
  • After Electroweak Symmetry Breaking (EWSB), the fields mix, resulting in physical mass eigenstates $h$ (125 GeV) and $\phi$. The mixing is parameterized by $\theta$.
  • The free parameters are reduced to physical masses ($m_h, m_\phi$), the mixing angle ($\theta$), and the singlet vacuum expectation value ($v_\phi$).

• Calculation of Decay Rates:

  • Two-body decay ($h \to \phi\phi$): Calculated analytically using the tree-level diagram. The partial width $\Gamma_{h \to 2\phi}$ depends on the effective coupling $\mu'$, which is a function of $\theta, v_h, v_\phi, m_h,$ and $m_\phi$.
  • Three-body decay ($h \to \phi\phi\phi$): Calculated by considering three Feynman diagrams (contact interaction and two intermediate scalar exchanges). The authors demonstrate that in the small-mixing regime ($\cos \theta \ll 1$), the contact diagram dominates, while diagrams involving intermediate exchanges are suppressed by propagator denominators. They derive an analytical expression involving a phase space integral $I(m_\phi)$.

• Global Constraint Formulation:

  • The total experimental Higgs width is $\Gamma_h^{\text{exp}} \approx 3.2^{+2.8}_{-2.2}$ MeV.
  • The SM contribution is suppressed by mixing: $\Gamma_{\text{NMD}} = \sin^2\theta \, \Gamma_{\text{SM}}$.
  • The constraint requires that the sum of exotic decays does not exceed the residual allowed width:
    $$\Gamma_{h \to \phi\phi} + \Gamma_{h \to \phi\phi\phi} \lesssim \Gamma_h^{\text{exp}} - \Gamma_{\text{NMD}}$$
  • Substituting the analytical expressions for the widths leads to a fourth-order polynomial inequality in terms of the singlet VEV ($v_\phi$):
    $$W^4 = a_4 v_\phi^4 + a_3 v_\phi^3 + a_2 v_\phi^2 + a_1 v_\phi + a_0 \lesssim 0$$
  • The coefficients $a_i$ depend on $m_\phi$, $\theta$, and known constants.

3. Key Contributions

1. Combined Channel Analysis: This is the first analysis to derive constraints on the Higgs-singlet mixing by simultaneously considering the sum of $h \to \phi\phi$ and $h \to \phi\phi\phi$ decay rates, rather than treating them separately.
2. Analytical Derivation: The authors provide closed-form analytical expressions for the decay rates and the resulting quartic inequality, allowing for a transparent understanding of the parameter dependencies.
3. Dominance of Contact Term: They rigorously justify neglecting non-contact diagrams for the three-body decay in the small-mixing limit, simplifying the calculation without sacrificing accuracy for the purpose of setting bounds.
4. Novel Mixing Angle Bound: They derive a model-independent upper bound on the mixing angle $\cos \theta$ that applies across the entire light scalar mass range ($0 < m_\phi < 40$ GeV).

4. Key Results

• Upper Bound on Mixing Angle:
  By requiring the quartic inequality to have physical solutions ($v_\phi > 0$), the coefficient of the highest order term ($a_4$) must be negative. This condition yields a strict upper bound on the mixing angle:
  $$\cos \theta < 0.12 - 0.13$$
  This bound is nearly constant across the mass range $0 < m_\phi < 40$ GeV (varying slightly from 0.13 at low masses to 0.12 near 40 GeV).

• Constraints on VEV and Couplings:
  Using a representative mixing angle (e.g., $\cos \theta = 0.1$), the authors solve the inequality to find the minimum allowed singlet VEV:
  $$v_\phi \gtrsim 6 \text{ TeV}$$
  This implies that for the model to be consistent with Higgs width measurements, the singlet field must acquire a very large vacuum expectation value, pushing the couplings $\lambda_3$ and $\lambda_\phi$ to be very small.

• Predicted Exotic Decay Rates:
  Under the assumption of existing independent constraints (e.g., $\cos \theta < 0.1$), the authors predict the maximum allowed partial widths:

  • Two-body: $\Gamma_{h \to \phi\phi} < 0.06 \text{ MeV}$
  • Three-body: $\Gamma_{h \to \phi\phi\phi} < 5 \times 10^{-6} \text{ MeV}$

  Interestingly, the two-body width remains nearly constant ($\approx 0.06$ MeV) across the mass range due to a compensation effect: as $m_\phi$ increases, the phase space factor decreases, but the coupling strength (dependent on $v_\phi$ and mass terms) increases, effectively canceling the suppression.

5. Significance

• Complementary Constraint: The derived bound ($\cos \theta < 0.12$) is competitive with, and in some mass regions (10–20 GeV) comparable to, direct search limits from LEP and LHC. It provides a crucial, independent theoretical constraint that does not rely on specific final-state signatures (like $\phi \to b\bar{b}$ or $\mu^+\mu^-$) but rather on the fundamental conservation of the Higgs total width.
• Global Validity: Unlike direct searches which have "blind spots" or varying sensitivity depending on the decay mode of $\phi$, this width-based constraint applies universally to the entire light scalar mass spectrum below $m_h/2$.
• Future Prospects: The paper highlights that as Higgs width measurements become more precise (e.g., at future lepton colliders), these constraints will tighten further, offering a powerful probe for light scalar portals that are difficult to detect via direct production.

In conclusion, the paper establishes that the total exotic decay width of the Higgs boson imposes a stringent, global limit on light singlet scalars, effectively ruling out large mixing angles and requiring a high scale for the singlet VEV, thereby significantly narrowing the viable parameter space for this class of BSM models.