Covariant Feynman Rules: Automated Vertex Extraction

• Covariant Feynman rules are a systematic framework that derives perturbative interaction vertices directly from Lorentz-covariant and gauge-invariant Lagrangians.
• They employ automated index restoration and contraction algorithms to maintain the proper Lorentz and internal symmetry structures during the derivation process.
• FeynRules automates the extraction and translation of vertices into various diagram calculation platforms, ensuring multi-platform model portability and precise numerical computations.

Covariant Feynman rules provide a systematic, algorithmic route to extracting perturbative interaction vertices from quantum field-theoretic Lagrangians while rigorously tracking their Lorentz and internal symmetry properties. The concept refers to both the functional structure of diagrammatic rules—ensuring explicit Lorentz and gauge covariance—and to practical computational frameworks that automate or encode these properties for arbitrary particle physics models, including the Standard Model and its extensions. The FeynRules package implements this paradigm by automating the derivation of Feynman rules from a covariant Lagrangian input, preserving all index structures, and providing robust translation interfaces into a range of diagram-calculation platforms (0806.4194).

1. Foundation: Manifestly Covariant Lagrangian Input

Covariant Feynman rules are rooted in a model definition at the Lagrangian level expressed in terms of Lorentz-covariant and gauge-covariant building blocks—fields with explicitly declared transformation properties and associated indices (Lorentz, spin, gauge), covariant derivatives $D_\mu$, field-strength tensors (e.g., $F^{a}_{\mu\nu}$), Dirac matrices, and group generators.

For example:

• Gauge-boson fields $G_\mu^a$ in QCD are defined with explicit Lorentz index $\mu$ and adjoint index $a$.
• Field-strength tensors are written manifestly covariantly:

$$F_{\mu\nu}^a = \partial_\mu G_\nu^a - \partial_\nu G_\mu^a + g_s f^{abc} G_\mu^b G_\nu^c.$$

• A typical covariant Lagrangian term for a gauge theory:

$$\mathcal{L}_{\text{QCD}} = -\frac{1}{4} F_{\mu\nu}^a F^{a\,\mu\nu},$$

and fermion-gauge interaction terms:

$$\mathcal{L}_{\text{int}} = g_s\, \bar{\psi}_f\, \gamma^\mu\, T^a\, \psi_f\, G_\mu^a.$$

The construction requires each parameter and field to be assigned its symmetry transformation properties and index structure.

2. Algorithmic Index Restoration and Contraction

A central technical feature of the covariant rules is the algorithmic reconstruction of all suppressed indices and guaranteed contraction consistent with the symmetry group and Lorentz invariance. FeynRules, for example, builds each Lagrangian term internally as a tensor, explicitly restoring:

• Lorentz indices,
• Dirac spinor indices,
• Internal symmetry indices such as color or flavor.

Templates like Ga[μ, s, r] encode Dirac matrices $\gamma^\mu_{sr}$, and internal contraction operations (ExpandIndices, TensDot) symmetrize and order the contraction, ensuring correctness.

For instance, a fermion-gauge interaction is converted from

$$\bar{\psi}_{s,f}\, \gamma^{\mu}_{ss'}\, \psi_{s',f}\, G_{\mu}^a\, T^a$$

by expanding all indices and yielding a manifestly covariant representation suitable for input into any Feynman-diagram program.

3. Canonical Quantization and Extraction of Vertices

The extraction of Feynman rules from the Lagrangian proceeds through a canonical quantization prescription:

• Each monomial in the Lagrangian, generically

$$\mathcal{L} = g_{\alpha_1 \cdots \alpha_n}\, (\partial\cdots\partial)\, \phi_{\alpha_1}\phi_{\alpha_2}\cdots\phi_{\alpha_n}$$

(with multi-indices $\alpha_i$),

• Is sandwiched between appropriate creation and annihilation operators according to bosonic or fermionic statistics. Terms are commuted using (anti)commutation relations, creating overall signs as required for fermions.
• Derivative operators act to pull down external momenta at the amplitude level.

A standard fermion-vertex extraction, e.g., for QED yields:

$-ie\,\delta_{ff'}\,\gamma^\mu_{ss'}$

preserving full covariant structure and ensuring gauge invariance.

4. Translation to Feynman Diagram Calculation Programs

The derived vertices, with explicit symmetry and Lorentz index structure, are stored in a generic internal form and then translated into the specific input format of Feynman diagram calculators (CalcHEP/CompHEP, FeynArts/FormCalc, MadGraph/MadEvent, Sherpa, etc.). This requires mapping:

• Indices and symbols to the conventions of each tool,
• Coupling constants and parameters identified by reference names (PDG codes, generator names),
• Preservation of index and symmetry structure down to the low-level code.

Suppression of indices for compact display is permitted, but, crucially, all contraction information is retained in the translation layer, allowing restoration for downstream calculations or output.

Step	Covariant Treatment	Tool Support
Input Lagrangian	Manifestly Lorentz/gauge covariant	User-written in FeynRules
Index restoration	Automated index reinsertion and contraction	ExpandIndices, TensDot
Canonical quantization	Symbolic commutation, extraction of vertices	Built-in to FeynRules
Diagram translation	Preservation of all index, symmetry structure	WriteCHOutput, WriteMGOutput, etc.

The interfaces guarantee that covariance, Lorentz/gauge invariance, and the structure required for unitarity/high-energy consistency are not lost at any stage.

5. Methodologies and User Workflow in Model Implementation

The process of model implementation for deriving covariant Feynman rules is as follows:

1. Declare particles and parameters: Specify all fields, grouping into classes as appropriate, define transformation properties, and index structure.
2. Write the Lagrangian: Using notation such as Ga[μ] for Dirac matrices, FS for field-strength tensors, and del for derivatives, encode interaction and kinetic terms in manifestly covariant fashion.
3. Automated extraction: FeynRules expands, contracts, and symmetrizes indices, follows the canonical quantization prescription, and extracts the list of interaction vertices in full covariant form.
4. Export and interface: Use translation functions to convert model output into the specific format (symbolic or numerical) required by various Feynman diagram calculators.

This approach eliminates ambiguities arising from implicit or inconsistent index contraction and ensures that any custom model (including non-standard gauge symmetries or exotic field content) can be handled on the same rigorous covariant footing.

6. Technical Features Securing Lorentz and Gauge Covariance

• Model format: All fields are declared with their indices and symmetry transformation properties; the underlying format extends that of FeynArts.
• Special symbols: Dedicated symbols (e.g., Ga, FS, del) automatically encode covariant transformation properties.
• Restoration/Contraction Algorithms: ExpandIndices and TensDot reconstruct all contractions, making all internal structure explicit.
• Extraction algorithm: Canonical quantization steps are coded to maintain the original covariant structure—multiplying by $i$ at the end and discarding unphysical normalization factors.
• Translation conventions: Consistent prescription for mapping coupling constants, generator names, and PDG codes across tools.

7. Significance for Multi-platform Model Portability and Physical Consistency

The preservation of the full covariant structure in both generic and platform-specific forms of the Feynman rules is essential for:

• Guaranteeing physical correctness: Gauge and Lorentz invariance are required for unitarity and correct high-energy amplitude behavior.
• Multi-platform model portability: The ability to write a new model once and have it work across all major Feynman diagram calculation tools without loss of generality.
• Unambiguous numerical calculations: Explicit contraction and covariant forms prevent ambiguity in sign conventions and parameter mapping.

The result is that, regardless of gauge fixing scheme, calculation program, or underlying model complexity, the derived perturbative expansion is both unambiguous and explicitly consistent with the symmetry structure required by quantum field theory (0806.4194).

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In summary, covariant Feynman rules as implemented in FeynRules comprise an automated, index-explicit framework for extracting interaction vertices directly from covariant Lagrangians, supporting full portability and physical consistency across particle physics computation platforms, and systematically preserving Lorentz and gauge invariance at every stage of the symbolic and numerical workflow.