⚛️ phenomenology

Dark hyperCharge Symmetry

This paper introduces a new "dark hyperCharge" $U(1)_X$ symmetry where Standard Model fermions are chiral and anomaly cancellation is achieved via three dark singlet fermions, identifying the lightest of these as a viable dark matter candidate that interacts with the visible sector through a heavy $Z'$ boson while satisfying current collider and dark matter constraints.

Original authors: Hemant Prajapati, Rahul Srivastava

Published 2026-03-19

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Hemant Prajapati, Rahul Srivastava

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the Standard Model of physics as a giant, incredibly successful orchestra. For decades, this orchestra has played the music of the universe perfectly, predicting how particles interact with stunning accuracy. But, like any great symphony, there are a few missing notes. We know there are "dark matter" (invisible stuff holding galaxies together) and "neutrino oscillations" (ghostly particles changing flavors) that the orchestra's sheet music doesn't explain.

The authors of this paper, Hemant Prajapati and Rahul Srivastava, propose adding a new instrument to the orchestra: a new force called "Dark Hypercharge."

Here is the story of their discovery, broken down into simple concepts.

1. The Problem: The "Chiral" Mess

In the current orchestra, some instruments (particles) are "chiral." This is a fancy way of saying they have a left-handed version and a right-handed version, and they behave very differently.

The Analogy: Imagine a dance where the Left-Handed dancers wear red shoes and the Right-Handed dancers wear blue shoes.
The Issue: In physics, when you mix these different dancers, you sometimes create a "glitch" in the system called an anomaly. Think of an anomaly as a mathematical error that would cause the entire universe to fall apart or the laws of physics to break.

Usually, physicists fix these glitches by adding new dancers who are "vector-like" (they wear both red and blue shoes, so they cancel out the errors easily). But the authors asked: What if we don't take the easy way out? What if we create a new system where the dancers are still chiral, but we balance the books in a completely new way?

2. The Solution: The "Dark Hypercharge" Team

The authors propose a new symmetry called Dark Hypercharge (DHC).

The New Rule: They assign specific, unique "charges" (like dance permissions) to the Standard Model particles. Crucially, they make the left-handed and right-handed versions of these particles have different charges under this new rule.
The Fix: To stop the universe from glitching (canceling the anomalies), they introduce three brand-new, invisible dancers called Dark Fermions.
- These new dancers live in the "Dark Sector" (a hidden room in the orchestra hall).
- They are the only ones who can balance the math.
- The lightest of these three new dancers becomes a perfect candidate for Dark Matter.

3. The Messenger: The $Z'$ Boson

How do we know about this hidden room? There is a new messenger particle called the $Z'$ boson.

The Analogy: Imagine the visible world (us, atoms, stars) and the dark world (dark matter) are two separate islands. The $Z'$ boson is a bridge connecting them.
Because the bridge exists, the visible particles can talk to the dark particles. This allows the dark matter to interact with our world, but only very weakly, which is why we haven't seen it yet.

4. The Detective Work: Hunting at the LHC

The authors didn't just write equations; they played detective to see if this idea could survive real-world testing. They looked at data from the Large Hadron Collider (LHC), the world's biggest particle accelerator.

The Scenario: They imagined a heavy $Z'$ boson (heavier than the known Z boson).
The Prediction: When the LHC smashes protons together, it might create this heavy $Z'$ $Z^{'}$ .
- Old Models: Usually, these heavy particles decay into visible things like pairs of electrons or muons (leaving a loud "crash" in the detectors).
- The DHC Twist: In this new model, the $Z'$ loves to decay into the Dark Fermions (the invisible dancers).
- The Result: The $Z'$ would disappear into the dark sector about 90% of the time. To the detectors, it would look like a particle was created and then vanished without a trace, leaving behind "missing energy."

5. The Verdict: It Works!

The authors ran the numbers and found that this model is surprisingly robust:

It fits the math: The anomalies cancel out perfectly.
It fits the data: The model survives all current tests from the LHC. Even though the $Z'$ is hard to find because it hides so often, the model doesn't break any rules.
It explains Dark Matter: The lightest dark fermion acts as a stable Dark Matter particle. It has the right mass and interaction strength to explain why the universe has the amount of dark matter we observe today.

Summary

Think of the Standard Model as a puzzle that's almost complete but missing a few pieces.

The Authors' Idea: They found a new way to arrange the existing pieces (making them "chiral" in a new way) and added three new, invisible pieces (Dark Fermions) to fill the gaps.
The Connection: A new bridge ( $Z'$ ) connects our visible world to this new invisible world.
The Outcome: The lightest invisible piece is our Dark Matter, and the bridge is a new force we might be able to detect by looking for "missing energy" in particle collisions.

It's a fresh, creative approach that turns a mathematical headache (anomalies) into a solution for the universe's biggest mystery (Dark Matter).

1. Problem Statement

The Standard Model (SM) is successful but incomplete, failing to explain neutrino oscillations and the nature of Dark Matter (DM). A common Beyond Standard Model (BSM) approach involves extending the gauge group with an extra $U(1)_X$ symmetry.

The Challenge: Introducing a new $U(1)_X$ gauge symmetry often leads to gauge anomalies (quantum inconsistencies that violate unitarity and renormalizability).
The Limitation of Current Models: Most existing $U(1)_X$ models assume vector-like charge assignments for SM fermions (where left- and right-handed components have identical charges). This simplifies anomaly cancellation but restricts the model space and often requires specific flavor structures.
The Gap: There is a lack of systematic exploration of chiral $U(1)_X$ solutions where SM fermions have different charges for their left- and right-handed components. Such chiral assignments are theoretically more complex because they can forbid standard Yukawa couplings (preventing mass generation) and require careful anomaly cancellation.

2. Methodology

The authors systematically investigate a new class of chiral $U(1)_X$ symmetries, termed "Dark HyperCharge" (DHC) symmetries. Their methodology involves:

Anomaly Cancellation Analysis: They derive the conditions for canceling all perturbative gauge anomalies ( $[SU(3)_C]^2 U(1)_X$ , $[SU(2)_L]^2 U(1)_X$ , $[U(1)_Y]^2 U(1)_X$ , etc.) and mixed gravitational anomalies.
Chiral Scenarios: They analyze three distinct scenarios based on how many generations of SM fermions carry the new charge:
1. SI: Only one generation is charged.
2. SII: Two generations share the same charge; one is uncharged.
3. SIII: All three generations are charged with identical charges (flavor-universal).
Introduction of Dark Fermions: To cancel the anomalies induced by chiral SM fermions, they introduce three SM gauge singlet Dark Fermions (DFs) ( $f_i$ ). The charges of these DFs are uniquely determined by the anomaly cancellation equations.
Mass Generation Mechanism: They impose constraints to ensure SM fermions can acquire mass. This requires the $U(1)_X$ charges to allow Yukawa couplings with the Higgs (potentially requiring additional Higgs doublets for non-universal scenarios) or specific scalar singlets.
Phenomenological Benchmarking: They select a specific benchmark model (from the SIII flavor-universal case) to perform detailed phenomenological analysis. This model features:
- SM fermion charges resembling hypercharge but scaled.
- Dark fermions with significantly higher $U(1)_X$ charges.
- A $Z'$ boson mass generated via a scalar singlet $\chi$ .
Simulation and Constraints: Using tools like SARAH, SPheno, micrOMEGAS, and MadGraph, they calculate:
- $Z'$ production cross-sections at the LHC.
- Branching ratios (visible vs. invisible).
- Dark Matter relic density.
- Direct detection cross-sections.

3. Key Contributions

A. Theoretical Framework: Dark HyperCharge (DHC)

The paper defines a new class of chiral $U(1)_X$ symmetries where:

SM fermions are chiral under $U(1)_X$ .
Anomaly cancellation is achieved by adding three chiral SM-singlet dark fermions.
The lightest dark fermion serves as a Dark Matter candidate.
The $Z'$ boson acts as the mediator between the visible and dark sectors.

B. Systematic Classification of Solutions

The authors provide explicit charge assignments for SM fermions and the three dark fermions for the SI, SII, and SIII scenarios. They demonstrate that:

In the SIII (flavor-universal) case, the model avoids tree-level Flavor Changing Neutral Currents (FCNCs).
The solutions allow for a wide range of charge ratios, offering more flexibility than vector-like models.

C. Phenomenological Distinctiveness

The benchmark model reveals unique signatures compared to standard vector-like $Z'$ models (like $B-L$ ):

Invisible Decays: Because the dark fermions carry large $U(1)_X$ charges, the $Z'$ boson has a very large invisible branching fraction (up to ~90%), decaying predominantly into dark sector fermions.
Collider Constraints: While the production cross-section is high (due to large quark charges), the large invisible width relaxes constraints from dilepton resonance searches (ATLAS/CMS) in certain mass regimes, while tightening them in others depending on the dark fermion mass.

D. Dark Matter Viability

The authors prove that the lightest dark fermion ( $F_1$ ) is a viable Majorana Dark Matter candidate:

Relic Density: It satisfies the Planck satellite constraints ( $\Omega h^2 \approx 0.12$ ) for masses $M_{F1} \gtrsim 150$ GeV, primarily through $Z'$ -mediated annihilation.
Direct Detection: The model predicts spin-dependent scattering with nucleons (since Majorana fermions have no vector coupling). The predicted cross-sections satisfy current limits from LUX-ZEPLIN (LZ) and IceCube.

4. Key Results

Anomaly-Free Chiral Solutions: The paper successfully constructs anomaly-free chiral $U(1)_X$ models without requiring vector-like SM fermions, a significant departure from standard literature.
$Z'$ Mass and Mixing: The $Z'$ mass is generated via spontaneous symmetry breaking of a scalar singlet. The mixing angle $\alpha$ between the $Z$ and $Z'$ is shown to be small for heavy $Z'$ masses, satisfying the $\rho$ -parameter constraints ( $\rho \approx 1$ ).
LHC Constraints:
- For the benchmark model, the $Z'$ production cross-section is higher than in $B-L$ models due to larger quark charges.
- However, the large invisible branching ratio means that dilepton resonance searches (the standard LHC probe) are less constraining for DHC models when $M_{Z'} > 2 M_{F1}$ , allowing for a wider viable parameter space.
Dark Matter Constraints:
- The model is consistent with the observed DM relic abundance for $M_{F1} \gtrsim 150$ GeV.
- The spin-dependent DM-nucleon scattering cross-section is well below the current exclusion limits set by LZ and IceCube.
- The $\rho$ -parameter imposes an upper bound on the gauge coupling $g_X$ , ruling out high-coupling, low-mass regions.

5. Significance

New Model Building Paradigm: This work challenges the assumption that $U(1)_X$ extensions must be vector-like. It opens a vast, previously unexplored landscape of chiral solutions that can address flavor anomalies and dark matter simultaneously.
Dark Sector Connection: It provides a concrete mechanism where the "dark" sector is not just an add-on but is intrinsically linked to the anomaly cancellation of the visible sector's chiral charges.
Experimental Guidance: The paper highlights that standard "dilepton resonance" searches might miss these models due to their high invisible decay rates. It suggests that future searches should focus on mono-X signatures (missing energy) or specific kinematic regions where the invisible branching ratio is maximal.
Addressing Anomalies: The flexibility of the charge assignments makes these models promising candidates for explaining various experimental anomalies (e.g., ATOMKI, $B \to K \nu \nu$ , electron $g-2$ ) that are difficult to reconcile with standard vector $Z'$ models.

In conclusion, the paper establishes "Dark HyperCharge" as a robust, anomaly-free, and phenomenologically rich extension of the Standard Model, offering a compelling link between the chiral structure of the SM, the existence of a new gauge boson, and the nature of Dark Matter.