The Standard Model partial unification scale as a guide to new physics model building

This paper proposes a general parametrization for new physics corrections to gauge couplings, demonstrating that full unification typically occurs near the Standard Model's partial unification scale unless corrections are highly asymmetric, while also revealing a phenomenologically significant possibility for unification at approximately 100 TeV via string-inspired corrections.

The Gist

Imagine the universe is built on three fundamental forces that act like three different teams of runners: the Strong Force (holding atoms together), the Weak Force (responsible for radioactive decay), and the Electromagnetic Force (light and electricity).

In the Standard Model of physics (our current best rulebook), these three teams run at different speeds. As they run higher up in energy (like climbing a mountain), their speeds change. Scientists have long hoped that at the very top of the mountain, all three teams would meet at the exact same speed. This is called Grand Unification. It would mean that at high energies, these three forces are actually just one single "Super-Force."

The Problem: They Miss Each Other

In our current rulebook (the Standard Model), if you calculate where these runners are going, the Strong and Weak teams actually cross paths at a very high altitude (about $10^{16}$ GeV). But the Electromagnetic team is running a different race and misses them by a wide margin. It's like two runners high-fiving on a trail, but the third runner is on a completely different path.

However, if you add Supersymmetry (a popular theory suggesting every particle has a heavy "shadow twin"), the paths change, and all three runners meet perfectly at the top. This has been the main reason physicists have been excited about Supersymmetry for decades.

The Surprise: A "Mirage"

The authors of this paper, Isabella Masina and Mariano Quirós, noticed something strange. Even without Supersymmetry, the Standard Model's Strong and Weak runners meet at a specific altitude ($\mu_{SM}^{32}$).

They asked: Is it a coincidence that the "Supersymmetry meeting point" is almost exactly the same height as the "Standard Model partial meeting point"?

They found the answer is no, it's not a coincidence. It's because of how the runners' speeds change (their "beta functions").

The New Tool: The "Correction Glasses"

To explain how different theories could fix the race, the authors invented a simple way to look at the problem using three pairs of "correction glasses" (parameters $\epsilon_1, \epsilon_2, \epsilon_3$). These glasses represent how new, undiscovered physics might tweak the runners' speeds.

They discovered two main scenarios:

1. The "Mirage" Scenario (The Illusion of Supersymmetry)

If the new physics tweaks the Strong and Weak runners' speeds by the same amount, the meeting point stays exactly where the Standard Model's partial meeting point is ($\approx 10^{16}$ GeV).

• The Analogy: Imagine two runners are wearing identical heavy boots. They slow down together, but they still meet at the same spot on the trail as before.
• The Result: Many theories (like Split-SUSY or 2-Higgs-Doublet models) act like this. They make it look like low-energy Supersymmetry is real because the meeting point is the same. The authors call this "Mirage SUSY." It's a mirage because it looks like the famous theory, but it might be something else entirely.

2. The "Divergence" Scenario (Breaking the Illusion)

If the new physics tweaks the Strong and Weak runners' speeds by different amounts, the meeting point moves.

• The Analogy: Imagine one runner gets a jetpack (speeds up) and the other gets a heavy anchor (slows down). They will meet at a completely different spot on the mountain.
• The Result: This allows the unification scale to be much lower or much higher than the standard $10^{16}$ GeV.

The Exciting Discovery: A Low-Altitude Meeting

The paper explores a specific "String Theory" inspired scenario where the runners meet at a surprisingly low altitude: 100 TeV (100,000 GeV).

• Why is this exciting? The current Large Hadron Collider (LHC) operates around 13-14 TeV. A 100 TeV scale is within reach of future colliders (like the proposed Future Circular Collider).
• The Metaphor: Instead of the runners meeting at the peak of Mount Everest (which is too high to reach), they might meet at the top of a tall skyscraper we can actually climb. If this is true, we might be able to see the new physics in our lifetime.

The "Bulk" Connection

The paper also looks at a theory where our universe is like a 3D slice of a higher-dimensional space (like a slice of bread in a loaf). If particles can run through the "crust" (the extra dimensions), it changes their speed dramatically.

• They found that if exactly three families of particles (like our three generations of quarks and leptons) run through this extra space, the math works out perfectly for unification at that accessible 100 TeV scale. It's as if the universe has a hidden "secret code" involving the number three that makes the forces unify at a reachable height.

Summary

1. The Coincidence: The fact that the Standard Model's partial unification is close to the Supersymmetry scale isn't random; it's a mathematical necessity if the corrections to the forces are similar.
2. The Mirage: Many theories that aren't actually Supersymmetry can still "fake" it by meeting at the same high altitude.
3. The New Hope: By tweaking the rules differently (specifically using ideas from String Theory and extra dimensions), the forces might unify at a much lower energy (100 TeV).
4. The Takeaway: The "Partial Unification Scale" of the Standard Model is a useful compass. It tells us that if we find new physics, it will likely either look like a "Mirage" of Supersymmetry at high energies, or it could be a "Low-Hanging Fruit" scenario at 100 TeV, which would be a massive discovery for future experiments.

Technical Summary

1. Problem Statement

The Standard Model (SM) of particle physics fails to achieve full Gauge Coupling Unification (GCU) at high energies; the three gauge couplings ($\alpha_1, \alpha_2, \alpha_3$) do not meet at a single point. However, the non-Abelian couplings ($\alpha_2$ and $\alpha_3$) exhibit a "partial unification" at a specific scale, denoted as $\mu_{32}^{SM} \approx 2.8 \times 10^{16}$ GeV.

Interestingly, this scale is remarkably close to the unification scale ($M_X^{SUSY} \sim 2 \times 10^{16}$ GeV) predicted by the Minimal Supersymmetric Standard Model (MSSM) with TeV-scale superpartners. The authors investigate whether this proximity is a mere coincidence or a deeper feature of the beta functions. Furthermore, they seek a general framework to understand how various Beyond the Standard Model (BSM) scenarios (both with and without a "desert" between the electroweak and unification scales) can achieve full unification and how their unification scales relate to the SM partial unification scale.

2. Methodology

The authors employ a combination of Renormalization Group Equation (RGE) analysis and a novel general parametrization:

• Beta Function Analysis: They calculate the one-loop and two-loop running of gauge couplings for the SM, MSSM, 2-Higgs-Doublet Model (2HDM), and Split-SUSY. They explicitly analyze the differences in beta function coefficients ($b_i$) between these models to explain the proximity of their partial unification scales.
• General Parametrization ($\epsilon_i$): The core methodological innovation is the introduction of parameters $\epsilon_i$ ($i=1,2,3$) to encode BSM corrections at a unification scale $M_X$. The unified coupling $\alpha_G$ is defined as:
  $$\alpha_G = (1+\epsilon_1)\alpha_1^{SM}(M_X) = (1+\epsilon_2)\alpha_2^{SM}(M_X) = (1+\epsilon_3)\alpha_3^{SM}(M_X)$$
  Here, $\alpha_i^{SM}$ are the SM couplings evolved to $M_X$. The parameters $\epsilon_i$ encapsulate effects from modified beta functions, threshold corrections, or stringy/Kac-Moody level modifications.
• Scenario Classification: The authors distinguish between:
  • Non-Desert Models: New physics exists below $M_X$, modifying beta functions (e.g., SUSY, 2HDM).
  • Desert Models: No new physics until $M_X$, where unification is driven by UV effects (e.g., string theory corrections, power-law running in extra dimensions).

3. Key Contributions

A. The "Mirage SUSY" Phenomenon

The paper demonstrates that the closeness of the MSSM unification scale to the SM partial unification scale ($\mu_{32}^{SM}$) is not unique to SUSY. It arises because the difference in beta function coefficients for the non-Abelian groups ($b_2 - b_3$) is nearly identical in the SM, MSSM, 2HDM, and Split-SUSY.

• Condition: If the corrections to the non-Abelian couplings are equal ($\epsilon_2 \approx \epsilon_3$), the unification scale $M_X$ is forced to be close to $\mu_{32}^{SM}$.
• Terminology: The authors coin the term "Mirage SUSY" for any BSM model where $\epsilon_2 \approx \epsilon_3$. These models mimic the unification scale of low-energy SUSY but may have different unified coupling strengths ($\alpha_G$) or different underlying physics (e.g., 2HDM or Split-SUSY).

B. Disentangling $M_X$ from $\mu_{32}^{SM}$

The authors derive an analytical relationship showing that the deviation of the unification scale from the SM partial scale depends exponentially on the difference between non-Abelian corrections:
$$\frac{M_X}{\mu_{32}^{SM}} = \exp\left[ \frac{2\pi}{\alpha_{32}^{SM}} \frac{\epsilon_3 - \epsilon_2}{(1+\epsilon_3)b_2^{SM} - (1+\epsilon_2)b_3^{SM}} \right]$$

• If $\epsilon_3 \neq \epsilon_2$, $M_X$ can be significantly different from $\mu_{32}^{SM}$.
• Specifically, $\epsilon_3 > \epsilon_2$ pushes $M_X$ higher, while $\epsilon_3 < \epsilon_2$ lowers it.

C. String-Inspired Unification and Low Scales

Applying the parametrization to string-inspired scenarios (Kac-Moody levels $k_i$), the authors identify a phenomenologically remarkable possibility:

• Low Scale Unification: By choosing specific Kac-Moody levels (e.g., $k_2=2, k_3=1$), they find a solution where full unification occurs at $M_X \approx 100$ TeV.
• This suggests a low string scale scenario, potentially accessible to future colliders (like FCC-hh), distinct from the traditional GUT scale.

D. Power-Law Running and Bulk Fermions

For models with extra dimensions where gauge bosons and fermions propagate in the bulk, the authors analyze "power-law" running.

• They establish a direct connection between the unification scale and the number of fermion generations ($\eta$) propagating in the bulk.
• They find that for $\eta = 3$ (matching the observed number of families), unification can occur at scales as low as a few TeV to 100 TeV, depending on the compactification scale $M_c$.

4. Results

• Partial Unification Coincidence: The SM partial unification scale $\mu_{32}^{SM} \approx 2.8 \times 10^{16}$ GeV is robust. Models with $\epsilon_2 \approx \epsilon_3$ (MSSM, 2HDM, Split-SUSY) naturally predict $M_X \approx \mu_{32}^{SM}$.
• Unified Coupling Strengths: While the scale is similar for "Mirage SUSY" models, the unified coupling $\alpha_G$ varies significantly:
  • MSSM: $\alpha_G \approx 0.036$
  • 2HDM: $\alpha_G \approx 0.022$
  • Split-SUSY: $\alpha_G \approx 0.027$
• Low Scale Unification: String-inspired corrections with $k_2=2, k_3=1$ allow for unification at $M_X \approx 10^5$ GeV (100 TeV) with $\alpha_G \approx 0.06$.
• Hypercharge Constraint: The paper notes that achieving unification at $\mu_{32}^{SM}$ by adding only hypercharged matter (which would require $\epsilon_1 < 0$) is inconsistent with standard BSM matter content (which usually yields $\epsilon_1 > 0$). Thus, unification at this scale requires specific non-Abelian corrections or UV threshold effects.

5. Significance

• Model Building Guide: The paper provides a powerful diagnostic tool. By calculating the required $\epsilon_i$ parameters for a proposed model, physicists can immediately determine if the model predicts a GUT scale near $10^{16}$ GeV (Mirage SUSY) or a drastically different scale.
• Re-evaluating SUSY: It suggests that the observation of a unification scale near $10^{16}$ GeV does not uniquely confirm low-energy SUSY; other "Mirage" models could produce the same scale.
• Phenomenological Implications: The identification of a viable unification scenario at $\sim 100$ TeV is significant. It shifts the search for Grand Unification from the unreachable GUT scale to the energy range of future high-energy colliders, potentially allowing for the direct detection of string states or Kaluza-Klein excitations.
• Connection to Extra Dimensions: The work highlights a non-trivial link between the number of fermion families and the unification scale in extra-dimensional theories, offering a potential explanation for the number of generations based on unification requirements.

In conclusion, the authors establish the SM partial unification scale as a critical anchor point for BSM physics, demonstrating that the nature of corrections to non-Abelian couplings ($\epsilon_2$ vs $\epsilon_3$) dictates whether new physics manifests at the traditional GUT scale or at experimentally accessible lower scales.