Valid Mass Regularization Techniques

• Valid Mass Regularization is a framework ensuring that mass-dependent schemes yield finite, symmetry-compatible results across quantum field theory, lattice methods, stochastic PDEs, and machine learning.
• It encompasses various methods, including torsional regularization in lepton self-energy, careful Pauli–Villars subtraction schemes, and mass transport approaches in robust learning, each addressing specific divergence and consistency challenges.
• Practical applications range from eliminating ultraviolet divergences in QED to preserving continuum limits in lattice QCD and providing statistically justified regularization in distributionally robust machine learning models.

Searching arXiv for recent and foundational papers relevant to mass regularization across QFT, lattice, PDE, and learning. Search 1: "mass regularization regularization mass arXiv" Search 2: "torsional regularization self-energy bare mass electron arXiv (Grosso et al., 2021)" Search 3: "Ambiguities in Pauli-Villars regularization arXiv (Kleiss et al., 2014)" In the literature surveyed here, “valid mass regularization” refers not to a single standardized construction but to several related questions about whether a mass-dependent or mass-related prescription is mathematically convergent, symmetry-compatible, and faithful to the target observable. In perturbative quantum field theory, the issue is typically whether self-energies, determinants, or multi-loop amplitudes become finite without spoiling gauge symmetry, supersymmetry, anomaly structure, or the intended relation between bare and renormalized mass. In lattice field theory, it concerns whether a discretization changes only the regularization and not the continuum theory. In stochastic PDE, it concerns regularization of mass-critical dynamics, while in distributionally robust learning it concerns bounded transportation of probability mass (Grosso et al., 2021, Frezzotti et al., 2021, Spitz et al., 8 May 2025, Shafieezadeh-Abadeh et al., 2017).

1. Criteria of validity

Across the field-theoretic examples, validity is not exhausted by mere finiteness. Torsional regularization is presented as valid “in the sense of convergence” because the relevant loop integrals become convergent sums and the regulator parameter $U$ reproduces the standard logarithmic divergence as $U\to 0$ (Grosso et al., 2021). Zeta-function regularization with a complex fermion mass is presented as valid because the anomaly equation and Green functions retain the correct phase dependence even though the zeta-regularized determinant itself is independent of the mass phase (Mitra, 2011). In the MSSM Higgs sector, dimensional reduction is valid only if the relevant Slavnov–Taylor identities hold so that the required counterterms are exactly the usual symmetric counterterms generated by multiplicative renormalization, with no additional finite SUSY-restoring counterterms (Stöckinger et al., 2018).

The same literature also emphasizes failure modes. Naive finite-$k$ Pauli–Villars regularization is ambiguous because divergent one-loop scalar integrals retain $k$-dependent finite parts after renormalization, so the regularization is underdetermined unless one supplements it with an additional prescription (Kleiss et al., 2014). A distinct caution arises at two loops: dimensional regularization combined with on-mass-shell renormalization can fail to locally cancel the ultraviolet subdivergence in a certain class of self-energy diagrams, even though a genuine two-loop counterterm can still render the full renormalized sum finite (Kato, 2024). Fujikawa’s analysis frames dimensional regularization as “generic” precisely because cutoff-induced quadratic mass shifts are treated as kinematical and unphysical, leaving intact the physical logarithmic scaling structure of the theory (Fujikawa, 2016).

2. Torsional regularization of lepton self-energy

A particularly explicit mass regularization scheme is developed in Einstein–Cartan gravity, where torsion is coupled algebraically to spin. The torsion tensor is the antisymmetric part of the affine connection,

$$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$

and for Dirac matter the spin tensor is completely antisymmetric, so torsion is effectively axial. The central input is that in the presence of torsion the momentum components fail to commute,

$[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$

Choosing a frame in which the dual axial torsion pseudovector has only a temporal component gives

$$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$

After defining $\mathbf n=\mathbf p/Q$, this becomes the $su(2)$ algebra, with discrete spectrum

$$n=\sqrt{j(j+1)},\qquad n_z=m,\qquad j=0,1,2,\dots,\quad m=-j,\dots,j.$$

Because the paper further takes $U\to 0$0, with $U\to 0$1 a positive constant of order $U\to 0$2, the spacing between momentum eigenvalues increases with momentum magnitude. The continuum loop integral is then replaced by a convergent density-of-states sum, and the master formula for

$U\to 0$3

becomes a discrete sum whose large-$U\to 0$4 behavior is $U\to 0$5, independently of $U\to 0$6, which removes ultraviolet divergences (Grosso et al., 2021).

Applied to the one-loop QED self-energy, this prescription yields a finite mass shift and eliminates the ultraviolet divergence of the standard self-energy. The paper also states that the infrared divergence is absent. The resulting relation is not mass generation from zero, but a finite equation

$U\to 0$7

from which one solves for the bare mass $U\to 0$8. For $U\to 0$9, the reported charged-lepton bare masses are

$k$0

which are about $k$1 of the observed masses. The paper is explicit, however, that this establishes at most a physically motivated regulator for the one-loop self-energy; Lorentz covariance is argued rather than made manifest, gauge consistency is not proved at the level of a full Ward-identity analysis, $k$2 is heuristic, and the calculation is only one-loop (Grosso et al., 2021).

3. Conventional mass regulators, generic masses, and dimensional schemes

Pauli–Villars regularization provides a contrasting notion of mass regularization. In the scalar one-loop setting studied in “Ambiguities in Pauli-Villars regularization,” the regulated propagator is obtained by subtracting heavy regulator propagators, and a degenerate $k$3-fold subtraction gives

$k$4

For divergent integrals, the resulting expressions depend on the number of subtractions $k$5; the logarithmic divergence is robust, but power divergences and finite terms depend on $k$6, so naive finite-$k$7 Pauli–Villars is ambiguous. The proposed resolution is an asymptotically large number of subtractions together with the scaling

$k$8

which produces finite, $k$9-independent limits and is claimed to yield a prescription “automatically valid in any number of dimensions” (Kleiss et al., 2014).

A different role is played by nonzero masses in the analysis of generic-mass banana integrals. There the integrals are already defined in dimensional regularization, and the masses are physical parameters entering the Symanzik polynomial $k$0, not a standalone substitute for dimensional regularization. The paper constructs $k$1 differential operators annihilating the $k$2-loop generic-mass banana integral, proves that the singular locus of the generated ideal is contained in the set of Landau singularities of the first and second type, and computes holonomic rank $k$3 up to $k$4, matching the number of master integrals. This supports the narrower statement that generic nonzero masses are mathematically controlled parameters inside a dimensionally regulated framework, not that masses alone replace dimensional regularization (Flieger, 6 Aug 2025).

Fujikawa’s “Dimensional regularization is generic” gives the corresponding interpretive claim for scalar mass divergences. In $k$5 theory, the quadratically divergent induced mass is treated as independent of scale changes of the physical mass and therefore as kinematical and unphysical. In a higher-derivative cutoff scheme, one introduces a subtraction $k$6 constrained by

$k$7

removes the $k$8-type mass shift subtractively, and then renormalizes the physical mass multiplicatively. The paper’s conclusion is that a regulator with an explicit mass scale is valid if, after this subtraction, it reproduces the logarithmic renormalization structure and homogeneous renormalization-group behavior exposed by dimensional regularization (Fujikawa, 2016).

4. Complex mass terms and phase-sensitive observables

For fermions with a complex chiral mass phase, the issue is not ultraviolet convergence alone but whether the regularization tracks physically relevant phase dependence. The Euclidean Dirac operator is

$k$9

and zeta-function regularization defines the determinant through the positive operator

$$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$0

The mass phase $$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$1 therefore drops out exactly from the zeta-regularized determinant. The paper argues that this is not a defect of the regularization. The phase reappears in the inverse Dirac operator entering Green functions and, crucially, in the anomaly equation once one introduces sources for the axial current and the rotated pseudoscalar density (Mitra, 2011).

The resulting Euclidean anomaly equation is

$$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$2

The paper’s central misconception correction is therefore precise: phase independence of the determinant does not imply phase blindness of the theory. Zeta regularization is presented as faithful to the physics of a complex fermion mass provided one does not over-interpret $$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$3 in isolation (Mitra, 2011).

5. Lattice twisted-mass regularization for isospin breaking

In lattice QCD and QCD+QED, mass regularization is tied to discretization rather than loop divergence subtraction. The rotated twisted-mass (RTM) proposal is designed for the RM123 approach, where isospin-breaking terms are treated perturbatively. Starting from

$$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$4

the continuum isospin-breaking term

$$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$5

is rotated in flavor space by

$$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$6

so that

$$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$7

The lattice action in the rotated basis keeps the twisted Wilson term diagonal, with opposite Wilson parameters for the rotated flavors,

$$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$8

Rotating back to the original basis changes only the flavor orientation of the Wilson term, from $$S^{k}{}_{ij}=\frac12(\Gamma^{k}_{ij}-\Gamma^{k}_{ji}),$$9 to $[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$0 (Frezzotti et al., 2021).

The paper’s validity claim is that RTM changes the lattice realization, not the target continuum theory. In RM123, the ensemble averages are computed in the isosymmetric theory, and the paper states that the action with which these averages are computed is exactly the same in RTM and standard twisted-mass schemes. RTM therefore preserves the standard twisted-mass virtues, “including primarily the non-perturbative improvement at $[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$1 of parity conserving observables.” Its practical advantage is statistical: because the rotated isospin-breaking insertion is flavor-changing, the relevant mesonic correlators can be expressed with quark–antiquark pairs carrying opposite values of the Wilson parameter $[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$2, which are found to have much smaller statistical fluctuations than the same-$[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$3 correlators. For the pion channel,

$[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$4

so the physical pion splitting can be extracted from a statistically favorable rotated correlator. In the QCD+QED example, the difference $[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$5 is stated to be UV finite after proper current and external-field renormalization, and the contact-term counterterms cancel in that difference (Frezzotti et al., 2021).

6. Mass-critical and transport-based notions of regularization

Outside relativistic QFT, “mass regularization” acquires different meanings. For the stochastic nonlinear Schrödinger equation,

$[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$6

the mass-critical case is

$[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$7

After the rescaling

$[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$8

the equation becomes

$[p_i,p_j]=2i\hbar\, S^k{}_{ij}p_k.$9

with

$$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$0

The main theorem states that for arbitrary deterministic $$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$1 initial data,

$$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$2

The paper is explicit that this is a regularization of mass-critical dynamics, not a restoration of deterministic mass conservation: the noise is non-conservative, and $$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$3 is a continuous martingale rather than a conserved quantity (Spitz et al., 8 May 2025).

In supervised learning, “Regularization via Mass Transportation” uses the mass of probability measures rather than particle or field mass. The empirical distribution

$$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$4

is surrounded by a Wasserstein ball

$$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$5

and one minimizes the worst-case expected loss

$$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$6

For many linear regression and classification problems this is exactly equivalent to empirical risk minimization plus a dual-norm penalty, for example

$$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$7

in the classification case with $$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$8. The paper further proves that, for an appropriate choice of $$[p_x,p_y]=iQp_z,\qquad [p_y,p_z]=iQp_x,\qquad [p_z,p_x]=iQp_y.$$9, the worst-case expected loss provides a finite-sample upper confidence bound on the out-of-sample loss. This suggests a broader usage of the term: validity can mean that a mass-transport regularizer is not only tractable but also statistically justified (Shafieezadeh-Abadeh et al., 2017).

In this broader literature, validity therefore has domain-specific content. In perturbative QFT it means convergence together with faithful symmetry and renormalization structure; in lattice theory it means continuum-limit equivalence with improved cutoff behavior; in stochastic PDE it means suppression of blow-up mechanisms in the mass-critical regime; and in distributionally robust learning it means robustness under bounded transportation of probability mass together with explicit generalization guarantees.