Higgs Decays to $Z\gamma$ and $\gamma\gamma$ in the Flavor-Gauged Two Higgs Doublet Model

This paper investigates Higgs decays to $Z\gamma$ and $\gamma\gamma$ within the Flavor-Gauged Two Higgs Doublet Model, demonstrating that while charged Higgs loops affect both channels, $h \to Z\gamma$ is uniquely sensitive to fermion-$Z$ vertex corrections that are tightly constrained by $b \to s\ell^+\ell^-$ data, with future High-Luminosity LHC precision expected to significantly probe this model's parameter space.

The Gist

Imagine the universe as a giant, complex machine built by a master engineer. For decades, we thought we understood the blueprints of this machine, a set of rules called the Standard Model. In 2012, we found the final missing piece of the original blueprint: the Higgs boson (often called the "God particle," though it's more like the "mass-giving particle"). It's the glue that gives other particles their weight.

But recently, the engineers noticed something odd. When the Higgs particle decays (breaks apart) into a Z boson and a photon (light), it happens slightly more often than the original blueprints predicted. It's like a clock that usually ticks 100 times a minute, but lately, it's ticking 120 times. Is the clock broken, or is there a hidden gear we didn't know about?

This paper explores a specific theory to explain that "extra ticking." The authors propose a new, slightly more complex blueprint called the Flavor-Gauged Two Higgs Doublet Model (FG2HDM).

Here is the breakdown of their idea, using simple analogies:

1. The New Blueprint: Adding Extra Rooms

The original Standard Model has one "Higgs room" (one Higgs doublet). The authors suggest there is actually a second Higgs room (a second doublet), plus a new storage closet (a singlet), and a new security system (a new force called $U(1)'$).

• The Result: Instead of just one Higgs particle, this new model predicts five extra Higgs particles and a new heavy messenger particle called the $Z'$.
• The Analogy: Imagine you thought a house only had one kitchen. This theory says, "Actually, there's a second kitchen, a pantry, and a secret tunnel connecting them."

2. The Mystery: Why is the Higgs Decaying Differently?

When the Higgs particle decays, it can turn into two photons (light) or a Z boson and a photon.

• The Standard Model says: "This happens because of a loop of particles (like a tiny racetrack) involving heavy particles like the Top quark and the W boson."
• The New Model says: "In our house with two kitchens, there are new racetracks made of Charged Higgs particles (the new particles from the second kitchen) running around the loop."

The authors calculated that these new racetracks could explain why the Higgs is decaying into a Z boson and a photon a bit more often than expected.

3. The Twist: The "Secret Tunnel" Effect

Here is the clever part. The authors found that the new model doesn't just add new racetracks; it also changes the rules of the road for the particles.

• The Analogy: Imagine the Z boson is a delivery truck. In the old model, the truck drives on a smooth highway. In this new model, the "security system" (the new force) builds a secret tunnel under the highway.
• The Effect: This tunnel changes how the truck interacts with the road. Specifically, it changes how the Top quark (the heaviest delivery driver) interacts with the Z boson.
• Why it matters: This "tunnel effect" (called a vertex correction) makes the Higgs decay into a Z boson and a photon even more likely. However, it doesn't change the decay into two photons, because that process doesn't use the Z boson's "tunnel."

4. The Detective Work: Checking the Clues

The authors didn't just guess; they played detective. They had to make sure their new blueprint didn't break other parts of the universe. They checked three main clues:

1. The Light Decay ($\gamma\gamma$): The Higgs decaying into two photons is measured very precisely. It matches the old blueprints perfectly.

   • The Constraint: This means the "new racetracks" (the new Higgs particles) can't be too heavy or too strong, or they would mess up the light decay. The authors found that the new particles must be heavier than 200 GeV (about twice the mass of a proton) to keep the light decay safe.

2. The Top Quark Observables: The "tunnel effect" changes how Top quarks behave.

   • The Constraint: Experiments at the Large Hadron Collider (LHC) measure Top quarks very carefully. The new model must fit these measurements.

3. The "Flavor" Clue ($b \to s\ell^+\ell^-$): This is a rare process where a "bottom" quark turns into a "strange" quark.

   • The Constraint: This is the strictest rule. The new model's "tunnel" affects this process significantly. The authors found that to fit this clue, the "charges" of the new particles must be very specific.

5. The Verdict: A Viable, but Narrow Path

The authors concluded that their new blueprint is possible, but only in a very specific "neighborhood" of parameters:

• The new heavy particles must be heavy (over 200 GeV).
• The interaction between the Higgs and the new particles must be "negative" (a specific mathematical condition).
• The "tunnel" effects must be tuned just right to satisfy the rare bottom-quark decay.

The Future:
Currently, our measurements of the Z-photon decay are a bit fuzzy (like looking at a distant star through fog). The authors predict that in the future, when the High-Luminosity LHC (a super-powered version of the current collider) comes online, it will clear the fog. It will measure the decay with 14% precision. If the "extra ticking" is real, this new machine will definitely hear it, confirming if our "two-kitchen" blueprint is the right one.

Summary

This paper suggests that the Higgs boson might be interacting with a hidden "second family" of particles and a new force. While this explains a slight anomaly in how the Higgs decays, it's a delicate balancing act. The model survives current tests but is tightly constrained by how rare particle decays behave. Future, more precise experiments will tell us if this complex new blueprint is the key to unlocking the universe's deeper secrets.

Technical Summary

1. Problem Statement

The discovery of the Higgs boson ($h$) confirmed the Standard Model (SM), yet fundamental questions regarding matter-antimatter asymmetry, neutrino mass, and dark matter remain. The Flavor-Gauged Two Higgs Doublet Model (FG2HDM) is a Beyond Standard Model (BSM) extension that introduces a second Higgs doublet, a scalar singlet, and a $U(1)'$ flavor gauge symmetry. This model predicts five additional physical scalars and a new neutral gauge boson ($Z'$).

The paper addresses the tension between SM predictions and recent experimental data regarding the rare Higgs decays $h \to Z\gamma$ and $h \to \gamma\gamma$.

• Context: Recent ATLAS and CMS measurements of the signal strength $\mu_{Z\gamma}$ initially showed a deviation from the SM (up to $1.9\sigma$), though combined analyses have since brought it closer to SM expectations ($\mu_{Z\gamma} \approx 1.3$).
• Challenge: The FG2HDM introduces new contributions to these loop-induced processes. Specifically, while both decays are sensitive to charged Higgs loops, $h \to Z\gamma$ is uniquely modified by corrections to the fermion-antifermion-Z ($f\bar{f}Z$) vertex due to $Z-Z'$ mixing. These vertex corrections are also constrained by top-quark observables and Flavor-Changing Neutral Current (FCNC) processes like $b \to s\ell^+\ell^-$.
• Goal: To determine the viable parameter space of the FG2HDM that is consistent with current experimental limits on $\mu_{Z\gamma}$ and $\mu_{\gamma\gamma}$, while satisfying constraints from top-quark physics and FCNCs.

2. Methodology

The authors employed a rigorous theoretical and numerical approach:

• Model Framework: They utilized the FG2HDM Lagrangian, which includes two $SU(2)_L$ scalar doublets ($\Phi_1, \Phi_2$), a singlet ($S$), and a $U(1)'$ gauge symmetry. The model features specific Yukawa textures (BGL-type) to suppress tree-level FCNCs in the up-sector while allowing them in the down-sector to explain $B$-physics anomalies.
• Amplitude Calculation:
  • Calculated one-loop amplitudes for $h \to Z\gamma$ and $h \to \gamma\gamma$ in the unitary gauge.
  • Decomposed the amplitudes into contributions from fermion loops ($T^f$), $W$ boson loops ($T^W$), charged Higgs loops ($T^H$), and mixed $W$-$H$ loops ($T^{WH}$).
  • Used Package-X for the evaluation of scalar loop functions ($C_0$, $\Lambda$).
  • Derived explicit Feynman rules for vertices involving $hZ\gamma$, $h\gamma\gamma$, $Z\bar{f}f$, and $W^\pm H^\mp Z$.
• Signal Strength Definition: Defined the signal strength $\mu_X = |1 + T^{NP}_X / T^{SM}_X|^2$, where $T^{NP}$ represents the new physics contributions.
• Constraints & Scanning:
  • $h \to \gamma\gamma$: Used as a primary constraint because its experimental value ($\mu_{\gamma\gamma} = 1.10 \pm 0.06$) is very close to the SM prediction.
  • $h \to Z\gamma$: Analyzed the impact of charged Higgs loops and $f\bar{f}Z$ vertex corrections.
  • Vertex Corrections: Evaluated the deviation in the top-quark vector coupling ($\Delta C^t_V$) arising from $Z-Z'$ mixing.
  • External Constraints: Imposed limits from:
    1. Top-quark pair production with a $Z$ boson ($t\bar{t}Z$) at the LHC.
    2. The FCNC process $b \to s\ell^+\ell^-$, which is highly sensitive to the same $Z'$ and $Z$-penguin diagrams.

3. Key Contributions

• Identification of Unique Vertex Corrections: The paper highlights that in FG2HDM, the $h \to Z\gamma$ decay receives unique corrections from the $f\bar{f}Z$ vertex (specifically the top-quark coupling) due to $Z-Z'$ mixing. This mechanism does not affect $h \to \gamma\gamma$, creating a distinct signature for this model.
• Negligibility of Mixed Loops: The authors analytically demonstrated that the contribution from mixed $W^\pm$-$H^\pm$ loops ($T^{WH}_{Z\gamma}$) is negligible (order $10^{-8}$ GeV$^{-1}$), comparable to the SM bottom-quark loop, and thus can be ignored in favor of pure charged Higgs contributions.
• Comprehensive Constraint Mapping: The study maps the allowed parameter space by simultaneously satisfying constraints from Higgs decays, top-quark observables, and low-energy flavor physics, a holistic approach often missing in isolated studies.

4. Results

• Charged Higgs Parameter Space:
  • The analysis identifies a viable region where $m_{H^\pm} > 200$ GeV and the trilinear coupling $\lambda_{hH^+H^-} < 0$.
  • In this region, both $\mu_{Z\gamma}$ and $\mu_{\gamma\gamma}$ can be accommodated within the 1$\sigma$ experimental limits.
  • The constraint is primarily driven by $\mu_{\gamma\gamma}$ due to its smaller experimental uncertainty compared to $\mu_{Z\gamma}$.
• $f\bar{f}Z$ Coupling Constraints:
  • The deviation in the top-quark vector coupling, $\Delta C^t_V$, is constrained by three sources: $\mu_{Z\gamma}$, top-quark observables, and $b \to s\ell^+\ell^-$.
  • The $b \to s\ell^+\ell^-$ process imposes the most stringent bounds on the parameters $Q_{tL}$ and $Q_{tR}$ (the $U(1)'$ charges of the top quark), effectively narrowing the allowed region more than the Higgs decay data itself.
  • The $\mu_{Z\gamma}$ constraint is currently weaker due to large experimental uncertainties but provides a complementary check.
• Future Prospects: The paper projects that the High-Luminosity LHC (HL-LHC), with a projected precision of 14% for $\mu_{Z\gamma}$, will significantly enhance sensitivity to the FG2HDM, potentially ruling out or confirming the viable parameter space identified.

5. Significance

• Model Viability: The study confirms that the FG2HDM remains a viable candidate for new physics, capable of explaining $B$-physics anomalies while remaining consistent with current Higgs precision measurements.
• Differentiation Mechanism: It establishes a clear theoretical distinction between $h \to Z\gamma$ and $h \to \gamma\gamma$ in flavor-gauged models. The unique sensitivity of $Z\gamma$ to $Z-Z'$ mixing and vertex corrections offers a powerful probe for distinguishing FG2HDM from other 2HDM variants (like the C2HDM) where such vertex corrections are absent or different.
• Interplay of Scales: The work underscores the critical interplay between high-energy collider data (Higgs and Top) and low-energy flavor physics ($B$-decays). It demonstrates that flavor constraints ($b \to s\ell^+\ell^-$) are currently the dominant factor limiting the model's parameter space, guiding future experimental priorities.
• Guidance for Future Experiments: By quantifying the impact of future HL-LHC precision on $\mu_{Z\gamma}$, the paper provides a roadmap for testing the FG2HDM in the next generation of collider experiments.