Connecting Supersymmetry to Non-Supersymmetric theories: the Gross-Neveu-Yukawa example

This paper constructs a unified Lagrangian framework connecting Gross-Neveu-Yukawa, Nambu-Jona-Lasinio-Yukawa, and Wess-Zumino models to elucidate emergent supersymmetry at critical points and leverage supersymmetry Ward identities to simplify loop calculations and reduce computational costs for determining anomalous dimensions in non-supersymmetric theories.

The Gist

Imagine you are trying to solve a massive, incredibly complex jigsaw puzzle. This puzzle represents the fundamental laws of the universe, specifically how tiny particles like electrons (fermions) and force-carrying particles like photons (bosons) interact.

For decades, physicists have had two different puzzle boxes:

1. Box A (The "Normal" Universe): Contains theories like the Gross-Neveu-Yukawa (GNY) model. These describe real-world phenomena but are notoriously difficult to calculate because the pieces are jagged and don't fit together easily.
2. Box B (The "Supersymmetric" Universe): Contains theories like the Wess-Zumino (WZ) model. These are mathematically beautiful and "perfect." In this world, every particle has a "super-partner" (like a boson having a fermion twin), and the math is so symmetrical that the pieces snap together effortlessly.

The Problem:
Box B is beautiful, but it doesn't seem to exist in our real world (we haven't found these super-partners yet). Box A exists, but the math is a nightmare. Physicists wanted to use the "easy math" of Box B to solve the "hard math" of Box A, but they didn't know how to connect the two boxes.

The Solution: The "Universal Adapter"
The authors of this paper, Mrigankamauli Chakraborty and Sven-Olaf Moch, built a Universal Adapter.

Think of it like a universal power strip or a Swiss Army knife. They created a single, generalized mathematical framework (a "Generalized Lagrangian") that can morph into any of these theories.

• Turn a dial, and it becomes the "Normal" GNY model.
• Turn the dial another way, and it becomes the "Perfect" Supersymmetric WZ model.
• Turn it to a specific sweet spot, and it becomes a hybrid where the "Normal" model suddenly starts acting like the "Perfect" one.

The Magic Trick: "Emergent Supersymmetry"
Here is the most fascinating part. The authors discovered that even in the "Normal" universe (Box A), if you look at the system at a very specific, critical point (like the exact moment water turns to ice), the particles start behaving as if they have super-partners.

It's like watching a chaotic crowd of people at a concert. Usually, everyone is moving randomly. But at a specific moment, they all suddenly start dancing in perfect, synchronized formation. The chaos emerges order.

In physics terms, this is called Emergent Supersymmetry. The "Normal" theory secretly contains the "Perfect" symmetry, but it's hidden until you look at it the right way.

Why This Matters: The "Cheat Code"
Why do we care about this hidden symmetry? Because it acts as a Cheat Code for calculations.

Imagine you are trying to calculate the trajectory of a rocket.

• The Old Way: You have to calculate every single tiny air resistance, wind gust, and fuel fluctuation. It takes weeks of supercomputer time.
• The New Way (Using this paper): You realize that at a certain altitude, the physics simplifies because of a hidden symmetry. You use the "Perfect" math from the Supersymmetric world to solve the "Normal" problem.

The authors showed that by using this "Universal Adapter," they could:

1. Unify the theories: Prove that the "Normal" and "Perfect" models are just different versions of the same underlying structure.
2. Save massive amounts of time: They used the hidden symmetry to cancel out thousands of complicated math terms that would otherwise need to be calculated.
3. Speed up discovery: In their specific test case, this method reduced the computing time for complex calculations by about 25%. In the world of high-energy physics, where calculations can take months, saving a quarter of that time is a huge victory.

The Big Picture
This paper is like finding a secret tunnel between a muddy, difficult path and a smooth, paved highway. Even though the highway doesn't exist in the real world (yet), you can use its smooth surface to travel faster along the muddy path.

The authors have given physicists a new tool: a way to borrow the elegance of a hypothetical "perfect" universe to solve the messy, difficult problems of our actual universe. It's a brilliant example of how abstract, theoretical math can become a practical, time-saving tool for understanding reality.

Technical Summary

1. Problem Statement

High-order perturbative calculations in Quantum Field Theory (QFT), particularly for anomalous dimensions of twist-two operators in models like the Gross-Neveu-Yukawa (GNY) and Nambu-Jona-Lasinio-Yukawa (NJLY) models, are computationally expensive. As the loop order increases (e.g., to four loops) and the Mellin moment $N$ grows, the number of independent Feynman integrals grows factorially, making direct computation intractable.

While Supersymmetry (SUSY) offers powerful constraints (Ward identities) that simplify calculations, these theories are often distinct from the non-supersymmetric models of interest. The challenge is to leverage the structural simplicity of SUSY to optimize calculations in non-SUSY theories without losing accuracy. Previous work identified "emergent SUSY" at specific critical points in GNY and NJLY models, but these were treated as separate theories without a unified framework to systematically exploit these symmetries for optimization.

2. Methodology

The authors develop a unified theoretical framework based on a Generalized Lagrangian ($\mathcal{L}_{gen}$) that interpolates between non-supersymmetric models (GNY, NJLY) and supersymmetric ones (Wess-Zumino, WZ).

A. The Generalized Lagrangian

The authors construct $\mathcal{L}_{gen}$ containing:

• $n_s$ real scalar flavors ($\phi^I$) transforming under global $SO(n_s)$.
• $n_f$ Weyl fermion flavors ($\psi^A$) transforming under global $SU(n_f)$.
• General ternary Yukawa interactions and quartic scalar couplings defined by abstract flavor tensors $Z^I_{AB}$ and $M_{IJKL}$.

Key Algebraic Constraint: To ensure the theory is free of "evanescent" flavor terms (terms that vanish in integer dimensions but appear in dimensional regularization and spoil consistency), the flavor tensors must satisfy a modified Clifford algebra:
$$(Z^I)_{AC} (\bar{Z}^J)_{CB} + (Z^J)_{AC} (\bar{Z}^I)_{CB} = 2\delta^{IJ}\delta^B_A$$
This constraint fixes all flavor traces and ensures the Lagrangian behaves consistently under $D = 4-2\epsilon$ dimensional regularization.

B. Emergent Supersymmetry Analysis

The authors analyze the conditions under which $\mathcal{L}_{gen}$ becomes supersymmetric by solving the Ward identity $\gamma_f = \gamma_s$ (equality of fermion and scalar anomalous dimensions) order-by-order in perturbation theory up to four loops.

• They identify two specific points in the $(n_s, n_f)$ parameter space where SUSY emerges:
  1. $(n_s, n_f) = (2, 1)$: Corresponds to the 4D $N=1$ Wess-Zumino model.
  2. $(n_s, n_f) = (1, 1/2)$: Corresponds to the 3D $N=1$ Wess-Zumino model (emergent from the GNY model).
• They verify that at these points, additional Ward identities (e.g., $\beta_a = -3\gamma_\phi$) hold, confirming the emergence of SUSY.

C. Optimization Strategy

The core methodology for optimization involves:

1. Decomposition: Expressing 1PI correlators in $\mathcal{L}_{gen}$ as a sum of flavor-independent loop integrals multiplied by flavor factors.
2. SUSY Substitution: Imposing the specific $(n_s, n_f)$ values and coupling relations ($a=\lambda$) of the emergent SUSY points.
3. Constraint Generation: Using the SUSY Ward identities to derive linear relations between the flavor-independent integrals.
4. Reduction: Using these relations to eliminate the most computationally expensive integrals (often non-planar topologies) before substituting the physical flavor numbers of the target non-SUSY theory (e.g., $n_s=1$ for GNY).

D. Technical Handling of Regularization

The paper addresses subtle issues in dimensional regularization:

• $\gamma_5$ and Levi-Civita Tensors: They demonstrate that for models with two real scalars (like NJLY and 4D WZ), $\gamma_5$ contributions (chiral parts of traces) vanish up to four loops due to kinematic constraints (insufficient independent momenta in 4-point functions). This allows the use of standard trace techniques without complex $\gamma_5$ schemes up to four loops.
• Odd Dimensions: They explicitly handle the $n_s=1$ case (3D SUSY) by accounting for non-vanishing odd traces of $\gamma$-matrices that arise in 3D but not 4D, ensuring the Ward identities hold correctly at the 3D critical point.

3. Key Contributions

1. Unified Framework: The construction of $\mathcal{L}_{gen}$ provides a single mathematical structure encompassing GNY, NJLY, and WZ models, clarifying their interrelations and the mechanism of emergent SUSY.
2. Flavor Algebra Constraints: The derivation of the modified Clifford algebra condition (Eq. 3.6) as the necessary and sufficient condition for non-evanescent flavor behavior in $D$ dimensions.
3. Operator Ward Identities: The derivation of new Ward identities for operator matrix elements (OMEs) in the WZ model, specifically relating singlet fermion and scalar operators ($2\gamma_{qq} + 4\gamma_{pq} = 2\gamma_{pp} + \gamma_{qp}$).
4. Computational Optimization: A proven method to reduce the number of independent integrals required for high-loop calculations in non-SUSY theories by leveraging emergent SUSY relations.

4. Results

• Reproduction of Known Results: The framework successfully reproduces known four-loop results for the GNY and NJLY models by simply setting $n_s=1$ and $n_s=2$ respectively in the general expressions.
• Identification of SUSY Points: Confirmed that emergent SUSY occurs strictly at $(2,1)$ and $(1, 1/2)$ in the $(n_s, n_f)$ plane.
• Optimization Efficiency:
  • Applied to the three-loop renormalization of singlet operators in the GNY model.
  • The method reduces the number of independent integrals by approximately 25%.
  • This reduction is most significant for high Mellin moments ($N \le 30$), where non-planar topologies dominate the computational cost.
  • The authors estimate this could save "several months" of computing time for phenomenologically relevant QCD calculations.
• Resolution of SUSY Violation Claims: The framework clarifies that apparent SUSY violations in $N=2$ SYM at three loops were due to mishandling of flavor Levi-Civita structures, not a true breakdown of SUSY.

5. Significance

• Bridging High-Energy and Condensed Matter: The work provides a rigorous link between high-energy SUSY concepts and condensed matter critical phenomena (GNY/NJLY models), showing how SUSY emerges dynamically at critical points.
• Practical Tool for QCD: While the paper uses GNY as a laboratory, the methodology is directly applicable to Quantum Chromodynamics (QCD). Since QCD singlet operators share structural similarities with these models, this approach offers a viable path to computing high-order anomalous dimensions (e.g., for parton distribution functions) that are currently too expensive to calculate directly.
• Conceptual Advancement: It demonstrates that "emergent" symmetries are not just physical curiosities but powerful mathematical tools that can be systematically imported into non-symmetric theories to simplify perturbative expansions. The work suggests that all-loop SUSY results (like the NSVZ $\beta$-function) could potentially be used to constrain non-SUSY calculations.

In summary, the paper establishes a robust, algebraic framework that unifies diverse QFT models, identifies precise conditions for emergent supersymmetry, and leverages these symmetries to achieve significant computational gains in high-order perturbative calculations.