Chiral Invariant Mass in Theoretical Physics

• Chiral invariant mass is a mass component that remains finite in the chiral limit, independent of explicit symmetry breaking.
• In parity-doublet models, the nucleon parameter m₀ represents the chiral invariant mass that survives when the scalar condensate vanishes, affecting neutron-star equations of state.
• This concept also appears in chiral EFT and dynamical mass generation analyses, highlighting its role in symmetry-preserving interactions and invariant-mass methods used in heavy-ion studies.

Searching arXiv for papers on chiral invariant mass across parity-doublet, chiral EFT, and related contexts. arXiv search query: all:"chiral invariant mass" OR title:"chiral invariant mass" Chiral invariant mass is a context-dependent term used for distinct but related ideas in quantum field theory, hadron structure, dense matter, and chiral transport. In its most standard strong-interaction usage, it denotes a mass contribution that remains finite in the chiral limit and is therefore not generated by explicit chiral symmetry breaking from fermion masses. In parity-doublet descriptions of baryons, this quantity is the nucleon parameter $m_0$, which survives when the chiral condensate vanishes; in scale-invariant chiral effective theory for walking dynamics, an analogous role is played by the technidilaton chiral-limit mass $m_\phi$ (Minamikawa et al., 2020, Matsuzaki et al., 2013). In other literatures, the phrase is used more loosely for dynamical mass generation without symmetry-breaking bilinears, or even for invariant-mass analysis variables in searches for the chiral magnetic effect, where it does not denote a chirally invariant mass parameter at all (Zeng et al., 2022, Li et al., 2018).

1. Conceptual scope and definitions

In theories with exact or approximate chiral symmetry, a mass term is “chiral invariant” only if it does not vanish when explicit chiral-symmetry-breaking parameters are sent to zero. The clearest realization in the supplied literature is the parity doublet model, where the nucleon mass contains a part associated with chiral symmetry breaking and a chiral invariant mass $m_0$ which is insensitive to the chiral condensate (Minamikawa et al., 2020). In that setting, $m_0$ is not tied to $\langle \bar q q\rangle$ and remains finite in the chirally restored phase.

A related but distinct usage appears in scale-invariant chiral effective theory for the technidilaton. There the physical dilaton mass satisfies

$$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$

and $m_\phi$ is the dilaton mass in the chiral limit $m_f\to 0$. Because $m_\phi$ survives as $m_\pi^2\to 0$, it functions as a chiral-invariant contribution, although its origin is explicit breaking of scale symmetry through the nonperturbative scale anomaly rather than explicit chiral breaking by fermion masses (Matsuzaki et al., 2013).

The term is not universal across subfields. In chiral kinetic theory for Dirac fermions, the finite fermion mass is explicitly chiral-symmetry breaking; the relevant works state that no separate “chiral invariant mass” is introduced, and that the Dirac mass $m_\phi$0 instead enters as a perturbative source of chirality-violating terms (Wang et al., 2020, Manuel et al., 2021). In heavy-ion CME analyses, “invariant mass” refers to the pair invariant mass $m_\phi$1 used to separate resonance backgrounds from possible chiral-magnetic correlations, not to a chirally invariant mass parameter (Li et al., 2018, Collaboration et al., 2020).

2. Nucleon chiral invariant mass in parity-doublet theory

The parity doublet model implements chiral symmetry with two baryon fields $m_\phi$2 and $m_\phi$3 transforming in a mirror assignment under $m_\phi$4. This permits a chirally invariant mixing term proportional to $m_\phi$5, while Yukawa couplings to the chiral field generate the part of the nucleon mass tied to the scalar condensate (Minamikawa et al., 2020). In mean field, the standard mass eigenvalues are

$m_\phi$6

When $m_\phi$7, one finds

$m_\phi$8

so the positive- and negative-parity partners become degenerate at the chiral invariant mass.

This identification gives $m_\phi$9 a direct physical meaning: it is the mass remaining when the chiral order parameter vanishes. The rest of the nucleon mass is the chiral-symmetry-breaking contribution generated by spontaneous symmetry breaking. In neutron-star applications based on a hadronic parity doublet sector matched to an NJL-type quark sector through a quark–hadron crossover, the value of $m_0$0 strongly affects the low-density hadronic equation of state and hence neutron-star radii (Minamikawa et al., 2020).

The quoted astrophysical constraint is

$m_0$1

obtained by combining the maximum-mass requirement with radius information from LIGO–Virgo and NICER (Minamikawa et al., 2020). A later extended parity doublet analysis including the isovector scalar meson $m_0$2 and the compact object in HESS J1731–347 sharpened this to

$m_0$3

while also emphasizing the sensitivity of $m_0$4 and $m_0$5 to $m_0$6 in the presence of $m_0$7 (Kong et al., 20 Jun 2025). This suggests that, within parity-doublet descriptions, a substantial fraction of the nucleon mass remains in the chirally restored regime.

3. Chiral-limit masses in effective field theory

In baryon chiral perturbation theory, the analogous object is the nucleon mass in the exact chiral limit. A recent manifestly relativistic two-loop calculation writes the nucleon mass as

$m_0$8

where $m_0$9 is the chiral-limit nucleon mass, namely the part that remains when the light-quark masses vanish (Liang et al., 26 Feb 2025). In that sense, $m_0$0 is the chiral-invariant part of the nucleon mass in chiral EFT.

The same work reports a fitted value

$m_0$1

with the $m_0$2 contribution being small, approximately $m_0$3 MeV, and mild up to pion masses of about $m_0$4 MeV (Liang et al., 26 Feb 2025). At the physical point, the decomposition quoted there is

$m_0$5

which isolates a dominant chiral-limit contribution and smaller explicit chiral-breaking corrections.

A scalar-sector counterpart appears in dilaton chiral perturbation theory. The technidilaton mass formula

$m_0$6

separates a chiral-limit term $m_0$7 from terms proportional to $m_0$8 and chiral logarithms (Matsuzaki et al., 2013). The paper states that $m_0$9 is nonzero in the chiral limit and is not tied to explicit chiral symmetry breaking by fermion masses, but instead to explicit breaking of scale invariance through the nonperturbative scale anomaly. The associated PCDC relation,

$\langle \bar q q\rangle$0

makes that origin explicit (Matsuzaki et al., 2013).

These two EFT realizations are structurally similar but conceptually different. In baryon ChPT, the chiral-limit mass $\langle \bar q q\rangle$1 is an LEC of the effective theory. In dilaton ChPT, $\langle \bar q q\rangle$2 is the chiral-limit scalar mass generated by scale-symmetry violation. A plausible implication is that “chiral invariant mass” is best treated as a family of chiral-limit mass parameters rather than a single universal object.

4. Dynamical mass generation and symmetric mass generation

In gauge-invariant analyses of QCD Green’s functions, a nonzero mass can emerge even when the bare fermion mass vanishes. A gauge-invariant treatment of the two-point quark Green’s function with Wilson lines shows that, in the chiral limit, a nonzero scalar part of the propagator equation yields a massless pseudoscalar bound-state solution, corroborated by the axial Ward–Takahashi identity (Sazdjian, 2015). The paper explicitly interprets a nonzero scalar component in the chiral limit as dynamical mass generation.

A closely related Minkowski-space Covariant Spectator Theory analysis defines a momentum-dependent quark mass function

$\langle \bar q q\rangle$3

and in the chiral limit obtains a nonzero dynamical mass function constrained by the existence of a massless pseudoscalar Goldstone boson (Biernat et al., 2013). In the explicit model discussed there,

$\langle \bar q q\rangle$4

with $\langle \bar q q\rangle$5 GeV from comparison to lattice-QCD data (Biernat et al., 2013). These works do not introduce a chirally invariant bilinear mass term in the Lagrangian; instead, they describe a nonzero effective mass generated by a chirally noninvariant vacuum in a chirally symmetric Lagrangian.

A many-body version of symmetry-preserving mass generation appears in the 1+1-dimensional anomaly-free 3-4-5-0 chiral fermion model. There, carefully chosen six-fermion local interactions gap the mirror sector without breaking the chiral $\langle \bar q q\rangle$6 symmetry and without generating any bilinear condensate, realizing symmetric mass generation (Zeng et al., 2022). The paper explicitly characterizes this as a “chiral-invariant mass” in the sense that fermionic excitations become gapped while no symmetry-breaking bilinear mass term appears.

These usages are not identical. In QCD mass-function studies, the mass is dynamical and tied to spontaneous chiral symmetry breaking. In symmetric mass generation, the gap is interaction-driven and symmetry-preserving. This suggests that the phrase can denote either a chiral-limit mass parameter or a symmetry-preserving infrared gap, depending on the degrees of freedom and symmetry realization.

5. Finite mass in chiral kinetic theory

In chiral transport, the supplied literature takes the opposite standpoint: the finite Dirac mass is not chiral invariant. Using equal-time Wigner functions in electromagnetic fields, one analysis shows that for massive fermions the number and spin densities are coupled, whereas in the chiral limit the fermion number density is the only independent distribution for a given chirality (Wang et al., 2020). The classical mass shell remains

$\langle \bar q q\rangle$7

and the paper states explicitly that it does not introduce a mass parameter that is invariant under chiral rotations (Wang et al., 2020).

The same work derives the chiral kinetic equation

$\langle \bar q q\rangle$8

where the left-hand side has exactly the same Berry-curvature structure as the massless theory, and the mass corrections appear only as effective collision terms on the right-hand side (Wang et al., 2020). In a pure magnetic field, $\langle \bar q q\rangle$9 vanishes, so the mass correction is purely quantum, of order $$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$0.

An OSEFT-based derivation reaches a closely related conclusion for $$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$1. There, the first chirality-breaking operator appears only at order $$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$2, tensorial Wigner components become necessary, and vectorial and tensorial components remain decoupled up to second order in the energy expansion while obeying the same dispersion law and transport equation (Manuel et al., 2021). The axial anomaly retains its standard form,

$$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$3

so the mass enters as explicit chiral-symmetry breaking rather than as an invariant mass (Manuel et al., 2021).

Within this transport literature, the phrase “chiral invariant mass” is therefore inappropriate unless explicitly redefined. The consistent statement is that the kinetic structure remains chiral to leading order, while mass acts as a controlled chirality-breaking perturbation.

6. Invariant-mass methods in chiral magnetic effect searches

In heavy-ion phenomenology, the relevant mass concept is pair invariant mass rather than a chirally invariant particle mass. The chiral magnetic effect is probed through the correlator

$$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$4

and the proposed method studies it differentially in

$$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$5

to separate a smooth possible CME contribution from resonance-decay backgrounds that produce peaks in $$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$6 (Li et al., 2018). The core two-component ansatz is

$$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$7

with $$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$8 carrying the resonance structure (Li et al., 2018).

STAR implemented this method in $$M_\phi^2 = m_\phi^2 + (1+\gamma_m)(3-\gamma_m)\frac{N_f F_\pi^2 m_\pi^2}{2F_\phi^2},$$9–$m_\phi$0 Au+Au collisions at $m_\phi$1 GeV and reported strong resonance background contributions to $m_\phi$2 as a function of $m_\phi$3 (Collaboration et al., 2020). Using event-shape engineering, the extracted $m_\phi$4-independent, $m_\phi$5-averaged signal was

$m_\phi$6

or $m_\phi$7 of the inclusive $m_\phi$8, with an upper limit of $m_\phi$9, or $m_f\to 0$0 of the inclusive result, at $m_f\to 0$1 confidence level (Collaboration et al., 2020).

This usage is terminologically separate from the mass parameters discussed above. Here, invariant mass is an analysis coordinate for isolating chiral-magnetic observables from hadronic resonance backgrounds. It does not encode a mass that is invariant under chiral transformations.

7. Related extensions and broader significance

The supplied literature also contains more specialized extensions of the idea. In a gauge-invariant analysis on $m_f\to 0$2, integrating out chiral scalars yields a possible gauge-invariant mass term for gauge fields,

$m_f\to 0$3

together with a finite four-dimensional Wess–Zumino–Witten action (Karabali et al., 2021). The interpretation there is a symmetry-preserving gluonic mass term compatible with gauge symmetry and the chiral structure of the matter sector.

In a BRST formulation of a massive superspin-$m_f\to 0$4 multiplet constructed from a chiral dotted spinor superfield, two mass invariants combine into a single effective mass after imposing the completion-obstruction constraint

$m_f\to 0$5

yielding

$m_f\to 0$6

for all components of the supermultiplet (Dixon, 2015). That construction concerns BRST–SUSY consistency rather than ordinary hadronic chiral symmetry, but it shows that “invariant mass” language can migrate into yet other symmetry contexts.

Across these applications, the common thread is separation between masses generated by explicit chiral breaking and masses or gaps compatible with the underlying symmetry structure. The principal exception is heavy-ion invariant-mass analysis, where the phrase concerns kinematic discrimination rather than symmetry. For that reason, careful specification of framework—parity doublet, chiral EFT, dynamical mass generation, chiral kinetic theory, or CME phenomenology—is essential whenever the term “chiral invariant mass” is used.