⚛️ phenomenology

Calculation for Electric Dipole Moments of Lepton and Neutron in the N-B-LSSM via the Mass Insertion Approximation

This paper calculates the one-loop electric dipole moments of leptons and the neutron within the N-B-LSSM framework using the Mass Insertion Approximation, deriving analytical expressions that reveal their dependence on specific CP-violating phases and demonstrating that the model's predictions can satisfy current experimental limits across a reasonable parameter space.

Original authors: Shuang Di, Wei-Hang Zhang, Rong-Zhi Sun, Xing-Xing Dong, Guo-Zhu Ning, Shu-Min Zhao

Published 2026-03-17

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CC BY 4.0

Original authors: Shuang Di, Wei-Hang Zhang, Rong-Zhi Sun, Xing-Xing Dong, Guo-Zhu Ning, Shu-Min Zhao

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 universe as a giant,精密ly tuned machine. For decades, physicists have had a "User Manual" for this machine called the Standard Model. It explains how particles like electrons and neutrons behave. But there's a problem: the manual says the machine should be perfectly symmetrical in a specific way (like a mirror image), but we know the real universe isn't. It has a slight "tilt" or preference, known as CP Violation.

In the Standard Model, this tilt is tiny—so small that it's like trying to hear a whisper in a hurricane. Yet, experiments are getting better at listening. They are looking for a specific kind of "whisper" in particles called Electric Dipole Moments (EDMs).

Think of an EDM like a tiny, internal magnet inside a particle. If a particle has an EDM, it means its positive and negative charges are slightly separated, like a tiny bar magnet. If we find a particle with a strong EDM, it's a smoking gun that proves there is New Physics hiding beyond our current manual.

The New Model: The "N-B-LSSM"

The authors of this paper are exploring a new, expanded version of the universe's manual called the N-B-LSSM.

The Old Model (MSSM): Imagine a house with a standard number of rooms.
The New Model (N-B-LSSM): The authors added a new wing to the house. They introduced Right-Handed Neutrinos (ghostly particles that usually don't interact) and Singlet Higgs fields (extra energy fields).
The Twist: They also added a new type of "plumbing" called Gauge Kinetic Mixing. Imagine two separate water pipes (forces) that were thought to be independent, but in this new model, they are slightly connected. This connection changes how water (forces) flows through the house.

The Detective Work: Mass Insertion Approximation (MIA)

Calculating how these new rooms and pipes affect the tiny magnets (EDMs) is incredibly hard. It's like trying to predict how a single drop of ink will spread through a complex maze of pipes without actually pouring it in.

To solve this, the authors used a clever shortcut called the Mass Insertion Approximation (MIA).

The Analogy: Imagine you are trying to figure out how a car's engine works, but the engine is too complex to take apart. Instead, you pretend that one specific bolt is slightly loose (a "mass insertion"). You calculate how that one loose bolt changes the engine's hum. Then you do it for another bolt.
By doing this, the authors could write down simple formulas that show exactly which new parts of their model (like the new pipes or the extra rooms) are responsible for making the EDMs bigger or smaller.

What They Found: The "Goldilocks" Zone

The team ran thousands of simulations, tweaking the "knobs" of their new model (changing the strength of forces, the mass of new particles, and the angles of their "tilt").

Here is what they discovered, translated into everyday terms:

The "Tilt" Matters Most: The most important factor is the CP-violating phase (let's call it the "Angle of Tilt"). If you turn this angle, the EDMs wiggle up and down like a sine wave. It's like turning a radio dial; sometimes you get static (no signal), and sometimes you get a loud song (a big EDM).
The "Volume" Knob (tan β): There is a parameter called tan β that acts like a volume knob. Turning it up makes the EDMs louder (bigger) without changing the pattern of the song.
The "New Pipe" Effect (gYB): The new connection between the forces (gauge mixing) acts like a filter. Depending on how you set it, it can either amplify the signal or cancel it out.
The Electron Problem: The electron is the most sensitive detector. The Standard Model predicts its EDM is almost zero. The new model predicts it could be bigger, but not too big. If it's too big, it breaks the rules set by real-world experiments.
- The Solution: The authors found a "Goldilocks" zone. By carefully balancing the "Angle of Tilt" and the "Volume," they found a sweet spot where the model predicts a detectable signal for the electron and neutron, but it's still small enough to not have been caught by current experiments yet.
The Neutron: The neutron is a composite particle (made of quarks), so it's like a choir of singers. The new model shows that the new "pipes" and "rooms" can make the choir sing much louder, potentially making the neutron's EDM detectable in future experiments.

The Big Picture

This paper is essentially a theoretical blueprint. It tells us:

"If you build the universe this way (N-B-LSSM), here is exactly how the tiny magnets (EDMs) should behave."
"We found a way to make the model work without breaking the rules of current experiments."
"Future experiments, which are becoming incredibly precise, might finally catch a glimpse of these new particles and forces."

In summary: The authors built a more complex, interesting version of the universe's rulebook. They used a clever math trick to predict how this new universe would look to our most sensitive detectors. They found that while the new universe is hidden right now, it's hiding in a very specific, testable way. If we keep looking with better microscopes (experiments), we might finally see the new "rooms" and "pipes" they added.

1. Problem Statement

The Standard Model (SM) of particle physics successfully describes fundamental interactions but fails to explain the observed matter-antimatter asymmetry of the universe. The only source of CP violation in the SM is the single phase in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, which predicts Electric Dipole Moments (EDMs) for fermions (leptons and neutrons) that are orders of magnitude smaller than current experimental upper limits.

Consequently, EDMs serve as highly sensitive probes for Beyond the Standard Model (BSM) physics. Supersymmetry (SUSY) introduces new CP-violating phases (e.g., in gaugino masses and trilinear couplings) that can significantly enhance EDMs. However, this creates a tension: models must generate sufficient CP violation for phenomena like Electroweak Baryogenesis (EWB) while simultaneously suppressing EDMs to satisfy strict experimental constraints (e.g., $|d_e| < 4.1 \times 10^{-30} e\cdot\text{cm}$ ).

The Next-to-Minimal Supersymmetric Standard Model with a local $U(1)_{B-L}$ gauge symmetry (N-B-LSSM) offers a richer structure than the MSSM. It includes right-handed neutrino superfields, additional singlet Higgs superfields, and gauge kinetic mixing between $U(1)_Y$ and $U(1)_{B-L}$ . The specific problem addressed is to systematically calculate the one-loop EDMs for leptons and the neutron within this extended framework to determine viable parameter spaces that satisfy current experimental bounds.

2. Methodology

The authors employ the Mass Insertion Approximation (MIA) to derive analytical expressions for the EDMs. This method is chosen for its ability to provide intuitive insight into the dependence of EDMs on specific model parameters and CP-violating phases, avoiding the complexity of full numerical diagonalization of the entire mass matrix.

Model Framework: The N-B-LSSM extends the MSSM gauge group to $SU(3)_C \times SU(2)_L \times U(1)_Y \times U(1)_{B-L}$ $S U (3)_{C} \times S U (2)_{L} \times U (1)_{Y} \times U (1)_{B - L}$ . It introduces:
- Right-handed neutrino superfields ( $\hat{\nu}$ ).
- Three singlet Higgs superfields ( $\hat{\chi}_1, \hat{\chi}_2, \hat{S}$ ).
- An additional gaugino ( $\tilde{B}'$ ).
- Gauge kinetic mixing characterized by the parameter $g_{YB}$ .
Analytical Calculation:
- Lepton EDMs: Calculated via one-loop diagrams involving neutralino-slepton and chargino-sneutrino loops. The MIA treats flavor and chirality violations as insertions in the propagators.
- Neutron EDM: Calculated as a composite of quark EDMs ( $d_q$ ), quark Chromo-EDMs ( $d^C_q$ ), and the Weinberg three-gluon operator ( $C_5$ ). The calculation includes contributions from gluino-squark, neutralino-squark, and chargino-squark loops.
- Renormalization: The Wilson coefficients calculated at the SUSY matching scale ( $\Lambda$ ) are evolved down to the chiral symmetry breaking scale ( $\Lambda_\chi$ ) using Renormalization Group Equations (RGEs) to obtain physical EDM values.
Numerical Analysis: A systematic scan of the parameter space is performed, varying key parameters such as:
- Gauge couplings: $g_{YB}$ , $g_B$ .
- CP-violating phases: $\theta_1, \theta_2, \theta_{\mu H}, \theta'_1, \theta_{BB'}, \theta_3$ .
- Mass scales: $M_{SUSY}$ , gaugino masses, and sfermion masses.
- $\tan \beta$ (ratio of Higgs VEVs).

3. Key Contributions

Derivation of Analytical Formulas: The paper provides explicit one-loop analytical expressions for lepton and quark EDMs in the N-B-LSSM using MIA. These formulas explicitly show the dependence on the new gauge mixing parameter $g_{YB}$ , the new CP phases ( $\theta'_1, \theta_{BB'}$ ), and the singlet sector parameters.
Identification of New CP Sources: The study isolates the impact of new CP-violating phases introduced by the $U(1)_{B-L}$ sector (specifically $\theta'_1$ associated with the $\tilde{B}'$ mass and $\theta_{BB'}$ associated with the mixing mass $m_{BB'}$ ) on EDMs.
Suppression Mechanisms for Electron EDM: The paper demonstrates three distinct mechanisms to satisfy the extremely tight electron EDM constraints without requiring unnaturally heavy superpartners:
- Small Phases: Setting CP phases to very small values.
- Heavy Masses: Pushing sfermion masses to the multi-TeV range (though this challenges naturalness).
- Phase Cancellation: Utilizing destructive interference between different loop contributions (e.g., between $\theta_1$ and $\theta'_1$ ) to suppress the net imaginary part while keeping phases and masses at natural TeV scales.
Neutron EDM Parameter Space: The authors map the viable parameter space for the neutron EDM, showing how the interplay between the gluino mass ( $m_{\tilde{g}}$ ), the strong phase $\theta_3$ , and the new $B'$ boson mass ( $m_{B'}$ ) affects the final result.

4. Key Results

Lepton EDMs:
- Tau ( $d_\tau$ ) and Muon ( $d_\mu$ ): The EDMs exhibit sinusoidal oscillations with respect to CP phases ( $\theta_{\mu H}, \theta_1$ ) with a period of $2\pi$ . The amplitude scales significantly with $\tan \beta$ . The parameter $g_{YB}$ can induce constructive or destructive interference, modifying the magnitude of $d_\tau$ and $d_\mu$ .
- Electron ( $d_e$ ): The electron EDM is the most constrained. The study finds that the phase cancellation mechanism is the most viable solution, allowing for $O(1)$ CP phases and TeV-scale masses while keeping $d_e$ below $4.1 \times 10^{-30} e\cdot\text{cm}$ .
Neutron EDM ( $d_n$ ):
- $d_n$ is dominated by the gluino contribution involving the phase $\theta_3$ .
- The neutron EDM shows a strong dependence on the $B'$ boson mass ( $m_{B'}$ ) and the phase $\theta'_1$ . As $m_{B'}$ increases, the loop contribution is suppressed.
- A synergistic effect is observed between $\theta_1$ and the bino mass $m_{\tilde{B}}$ ; larger phases can yield larger EDMs even with moderate mass scales.
- The competition between left-handed ( $m_{\tilde{d}_L}$ ) and right-handed ( $m_{\tilde{d}_R}$ ) down-squark masses is significant: increasing $m_{\tilde{d}_L}$ strongly suppresses $d_n$ , whereas increasing $m_{\tilde{d}_R}$ enhances it.
Viable Parameter Space: The numerical analysis confirms that there exists a region in the N-B-LSSM parameter space where the EDMs of the electron, muon, tau, and neutron simultaneously satisfy current experimental upper limits.

5. Significance

Theoretical Tool: This work establishes a complete analytical framework using MIA for studying EDMs in $U(1)_{B-L}$ extended supersymmetric models, serving as a benchmark for future theoretical investigations.
Experimental Guidance: The results provide specific constraints on the kinetic mixing parameter $g_{YB}$ and the new CP-violating phases. This guides experimentalists in interpreting future high-precision EDM measurements (e.g., ACME, nEDM experiments) as potential signals for the N-B-LSSM.
Naturalness: By demonstrating that the phase cancellation mechanism can satisfy electron EDM constraints with TeV-scale particles, the paper alleviates the "SUSY CP problem" to some extent, suggesting that the N-B-LSSM remains a natural candidate for new physics.
Model Validation: The study highlights that the unique features of the N-B-LSSM (gauge kinetic mixing and singlet sectors) leave distinct imprints on EDMs, distinguishing it from the MSSM and offering a clear observational window for testing $B-L$ extended models.