Non-Relativistic Effective Field Theories

• Non-Relativistic Effective Field Theories are controlled expansions that integrate out high-energy modes to capture low-energy dynamics using Wilson coefficients.
• They simplify complex systems by matching relativistic theories to effective nonrelativistic descriptions through systematic 1/M or velocity expansions.
• NREFTs are applied in heavy-ion physics, condensed matter, and cosmology, enabling precise predictions of bound states, thermal widths, and scattering phenomena.

Searching arXiv for recent and foundational papers on non-relativistic effective field theories to ground the article. Non-Relativistic Effective Field Theories (NREFTs) are effective descriptions of systems whose relevant degrees of freedom are characterized by energies close to rest mass, small spatial momenta, or long-wavelength collective behavior compared with a higher microscopic scale. Across quantum field theory, condensed matter, cosmology, heavy-ion physics, and hydrodynamics, they exploit a separation of scales to organize observables in expansions such as $1/M$, $T/M$, $v$, or $\nabla^2/m^2$, while encoding short-distance physics in Wilson coefficients and low-energy dynamics in local operators or effective potentials. The literature considered here presents NREFTs in several complementary forms: as contractions of relativistic theories onto Newton–Cartan backgrounds (Bergshoeff et al., 2015), as heavy-particle EFTs for Majorana fermions in thermal media (Biondini et al., 2013), as systematic nonrelativistic limits of scalar and Dirac theories with relativistic corrections (Namjoo et al., 2017, Santos et al., 2018), as few-body EFTs for resonant short-range interactions (Hammer et al., 2016), as parity-violating geometric EFTs in $2+1$ dimensions (Wu et al., 2014), and as Schwinger–Keldysh EFTs for Galilean hydrodynamics (Jain, 2020).

1. Scale separation and the logic of non-relativistic expansion

NREFTs are organized by a hierarchy between a hard scale and softer dynamical scales. In thermal leptogenesis with heavy Majorana neutrinos, the relevant regime is $M \gg T \gg M_W$, with typical nonrelativistic momenta $|\mathbf{p}| \sim \sqrt{MT} \ll M$; this justifies an expansion in $1/M$ and the treatment of thermal physics entirely within the low-energy theory (Biondini, 2014, Biondini et al., 2013). In scalar Yukawa systems for heavy fermions, the hierarchy is $M \gg Mv \gg Mv^2$ with mediator mass $m \ll M$, motivating a sequence from a non-relativistic Yukawa EFT to a potential EFT (Biondini et al., 2021). In scalar-field NREFTs, the nonrelativistic regime is encoded by small parameters

$T/M$0

supplemented in some formulations by a weak-coupling parameter $T/M$1 (Namjoo et al., 2017), or more generally by

$T/M$2

and, in cosmology, $T/M$3 (Modirzadeh et al., 11 Jul 2025).

This scale separation is the defining structural principle. Short-distance effects at the heavy scale are integrated out and stored in Wilson coefficients, while low-energy fields describe particle propagation, medium effects, bound states, or collective modes. In heavy Majorana neutrino EFT, the generic structure is

$T/M$4

with $T/M$5 the nonrelativistic heavy field and $T/M$6 the Wilson coefficients obtained by matching (Biondini, 2014). In scalar few-body EFTs with short-range interactions, the derivative expansion takes the form

$T/M$7

with power counting set by the ratio $T/M$8 and modified when fine tuning generates an unnaturally large scattering length (Hammer et al., 2016).

A recurring theme is that “nonrelativistic” need not mean weakly interacting or small field amplitude in an absolute sense. One construction for general scalar potentials explicitly relaxes small-amplitude assumptions provided the mass term remains dominant (Modirzadeh et al., 11 Jul 2025). This suggests that NREFTs are best understood as controlled expansions around a dominant frequency or mass scale, rather than around small occupation number or weak classical field values.

2. From relativistic theories to non-relativistic fields

A major route to NREFT is to start from a relativistic parent theory and isolate the slow modes. For a real scalar with relativistic Lagrangian

$T/M$9

one may define a complex nonrelativistic field by a nonlocal transformation involving

$v$0

so that

$v$1

The resulting free theory has a manifest $v$2 symmetry, canonical commutator

$v$3

and exact equation of motion

$v$4

(Namjoo et al., 2017). Expanding $v$5 yields the relativistic kinetic corrections

$v$6

For Dirac theory, the nonrelativistic limit can be constructed by decomposing the relativistic spinor into Weyl components, recombining them into “large” and “small” fields,

$v$7

factori​ng out the rest-energy phase $v$8, and integrating out the high-energy field $v$9 in the path integral (Santos et al., 2018). The effective action for the low-energy field $\nabla^2/m^2$0 begins as

$\nabla^2/m^2$1

and reproduces the Pauli Hamiltonian

$\nabla^2/m^2$2

with $\nabla^2/m^2$3 at tree level (Santos et al., 2018). After normalization of the wave function via

$\nabla^2/m^2$4

one obtains the Pauli–Schrödinger equation plus $\nabla^2/m^2$5 corrections (Santos et al., 2018).

For heavy Majorana fermions, the relativistic field satisfies $\nabla^2/m^2$6, so the nonrelativistic EFT contains a single projected field $\nabla^2/m^2$7 obeying

$\nabla^2/m^2$8

Its free propagator is

$\nabla^2/m^2$9

generated by the leading Lagrangian $2+1$0 (Biondini et al., 2013). This construction is explicitly analogous to HQET, but adapted to a Majorana degree of freedom.

A related but geometrically distinct relativistic-to-nonrelativistic limit is the contraction onto Newton–Cartan backgrounds. There, one introduces a contraction parameter $2+1$1 and rescales the relativistic vielbein and a flat $2+1$2 background $2+1$3 as

$2+1$4

The result is a nonrelativistic field theory on torsionless Newton–Cartan geometry with temporal vielbein $2+1$5, spatial vielbein $2+1$6, and central charge gauge field $2+1$7 (Bergshoeff et al., 2015).

3. Effective actions, Wilson coefficients, and power counting

The operator organization of NREFT depends on the physical regime. In heavy-particle EFTs, the expansion is typically in inverse powers of a large mass. For heavy Majorana neutrinos, the EFT Lagrangian is written as

$2+1$8

with the leading dimension-five interaction

$2+1$9

(Biondini, 2014). Matching at $M \gg T \gg M_W$0 gives

$M \gg T \gg M_W$1

which then controls the leading thermal width (Biondini, 2014). A closely related Majorana EFT develops a larger operator basis at dimension seven, including Higgs, fermion, and gauge operators, with explicitly matched imaginary Wilson coefficients such as

$M \gg T \gg M_W$2

(Biondini et al., 2013).

In the scalar Yukawa NREFT denoted NRY, the bilinear and four-fermion sectors are organized simultaneously in $M \gg T \gg M_W$3 and in velocity power counting. Up to $M \gg T \gg M_W$4 in bilinears and $M \gg T \gg M_W$5 in four-fermion operators, the Lagrangian contains terms such as

$M \gg T \gg M_W$6

and the analogous antiparticle terms (Biondini et al., 2021). Tree-level matching yields

$M \gg T \gg M_W$7

while the scalar sector obeys

$M \gg T \gg M_W$8

(Biondini et al., 2021). The sign difference between particle and antiparticle Yukawa couplings is a distinctive feature of scalar mediation and later drives the multipole structure in the potential EFT.

For few-body NREFT with short-range interactions, the derivative expansion is accompanied by a distinction between natural and resonant scaling. In the natural case,

$M \gg T \gg M_W$9

whereas resonant systems with a large scattering length require

$|\mathbf{p}| \sim \sqrt{MT} \ll M$0

(Hammer et al., 2016). The need to resum the leading two-body contact $|\mathbf{p}| \sim \sqrt{MT} \ll M$1 nonperturbatively then follows directly from power counting.

Power counting may also require anisotropic scaling. In the EFT for the nonrelativistic limit of Dirac theory, relativistic mass dimensions incorrectly classify the spatial kinetic term as irrelevant, so the appropriate scaling is instead $|\mathbf{p}| \sim \sqrt{MT} \ll M$2,

$|\mathbf{p}| \sim \sqrt{MT} \ll M$3

Under this scaling, both $|\mathbf{p}| \sim \sqrt{MT} \ll M$4 and $|\mathbf{p}| \sim \sqrt{MT} \ll M$5 are marginal, whereas $|\mathbf{p}| \sim \sqrt{MT} \ll M$6 is irrelevant (Santos et al., 2018). This suggests that nonrelativistic power counting is often most transparent when time and space are assigned different engineering dimensions.

4. Potentials, bound states, and nonperturbative sectors

Potential NREFTs arise when the soft scale associated with relative momentum is itself integrated out, leaving pair degrees of freedom interacting through a Schrödinger potential and ultrasoft fields. In scalar-mediated dark sectors, pNRY promotes the two-body wave function to a bilocal field $|\mathbf{p}| \sim \sqrt{MT} \ll M$7 and yields the Lagrangian

$|\mathbf{p}| \sim \sqrt{MT} \ll M$8

The leading pair equation is

$|\mathbf{p}| \sim \sqrt{MT} \ll M$9

which reduces to a Coulomb Schrödinger equation when $1/M$0 (Biondini et al., 2021). The corresponding levels are

$1/M$1

The matching of the potential from NRY to pNRY gives, in momentum space,

$1/M$2

and in coordinate space

$1/M$3

(Biondini et al., 2021). For mediator mass $1/M$4, this becomes a Yukawa potential with $1/M$5 and correspondingly modified relativistic corrections (Biondini et al., 2021).

The heavy-pair, long-range regime also underlies Sommerfeld enhancement and its unitarization. In a Keldysh–Schwinger NREFT for a heavy complex scalar pair with static potential $1/M$6 and local annihilation kernel $1/M$7, the relative Green’s function satisfies

$1/M$8

in the absence of annihilation (Binder et al., 13 Apr 2026). The spectral function at the origin is

$1/M$9

showing both bound-state poles and continuum Sommerfeld enhancement (Binder et al., 13 Apr 2026). Including the short-distance annihilation potential self-consistently modifies the retarded Green’s function to

$M \gg Mv \gg Mv^2$0

which unitarizes the enhancement near resonances (Binder et al., 13 Apr 2026).

Few-body NREFTs provide the complementary zero-range limit. For resonant short-range bosonic interactions, summing the bubble chain generated by $M \gg Mv \gg Mv^2$1 yields

$M \gg Mv \gg Mv^2$2

with $M \gg Mv \gg Mv^2$3 renormalized according to

$M \gg Mv \gg Mv^2$4

(Hammer et al., 2016). This is the canonical nonperturbative sector of short-range NREFT: the shallow bound or virtual state appears only after resummation.

5. Geometry, symmetries, and background-field formulations

A broad class of NREFTs is most naturally coupled to geometric background fields rather than formulated solely in flat space. In the Newton–Cartan contraction of relativistic theories, the nonrelativistic fields

$M \gg Mv \gg Mv^2$5

satisfy

$M \gg Mv \gg Mv^2$6

and define torsionless Newton–Cartan geometry (Bergshoeff et al., 2015). The gauge field $M \gg Mv \gg Mv^2$7 descends from the relativistic flat $M \gg Mv \gg Mv^2$8 connection $M \gg Mv \gg Mv^2$9 and couples to particle number. For the scalar,

$m \ll M$0

and the nonrelativistic Lagrangian is

$m \ll M$1

with the mass current

$m \ll M$2

(Bergshoeff et al., 2015).

This background interpretation is especially important in parity-violating $m \ll M$3-dimensional NREFTs for quantum Hall systems and chiral superfluids. There one begins from a general nonrelativistic microscopic action in curved space,

$m \ll M$4

with

$m \ll M$5

(Wu et al., 2014). A covariant map relates a relativistic gauge field $m \ll M$6 to the nonrelativistic variables $m \ll M$7, and a diffeomorphism-invariant constraint determines the shift vector $m \ll M$8 (Wu et al., 2014). Applying this map to a relativistic Chern–Simons term yields the leading nonrelativistic effective action

$m \ll M$9

The Wen–Zee parameter obeys

$T/M$00

the Hall viscosity is

$T/M$01

and the Landau orbital angular momentum density is

$T/M$02

with $T/M$03 (Wu et al., 2014).

An even more general background-field formulation appears in nonrelativistic hydrodynamics. There, Galilean hydrodynamics is recast as relativistic hydrodynamics on a null background with metric

$T/M$04

and null Killing vector $T/M$05 (Jain, 2020). Null reduction gives Newton–Cartan data $T/M$06, and the hydrodynamic constitutive relations are encoded in a Schwinger–Keldysh EFT. The flat-space one-derivative SK action is

$T/M$07

which generates the conservation laws, dissipative transport, and stochastic noise subject to KMS constraints (Jain, 2020).

These constructions show that “nonrelativistic symmetry” is not a single concept. Depending on the application it may mean Bargmann symmetry on Newton–Cartan backgrounds (Bergshoeff et al., 2015), Galilean covariance with Wen–Zee couplings (Wu et al., 2014), or Galilean hydrodynamics formulated via null reduction and Schwinger–Keldysh doubling (Jain, 2020).

6. Applications across fields and recurring structures

The range of NREFT applications is unusually broad, but several structural motifs recur.

In leptogenesis and heavy-ion analogies, heavy Majorana neutrino EFT isolates hard physics in vacuum Wilson coefficients and computes thermal widths through low-energy condensates. The leading thermal correction from the operator $T/M$08 is

$T/M$09

(Biondini, 2014). The more complete Majorana EFT yields

$T/M$10

(Biondini et al., 2013). This is a direct example of “matching at zero temperature, medium effects in EFT loops.”

In scalar cosmology, a real scalar with general potential

$T/M$11

is rewritten exactly in terms of a slow complex field $T/M$12 and then coarse-grained over the fast phase. The leading effective Lagrangian is

$T/M$13

with conserved particle number

$T/M$14

(Modirzadeh et al., 11 Jul 2025). For a quartic relativistic interaction, one recovers

$T/M$15

while nonanalytic and nonpolynomial potentials map to coarse-grained functions involving Bessel functions, such as

$T/M$16

for an axion-like cosine and

$T/M$17

for a dilaton-like exponential (Modirzadeh et al., 11 Jul 2025). The same framework provides an effective fluid description with

$T/M$18

and sound speed

$T/M$19

(Modirzadeh et al., 11 Jul 2025).

In few-body and nuclear contexts, the central application is the EFT of resonant short-range interactions and its descendants such as pionless EFT. The paper on general EFT aspects emphasizes that nonrelativistic systems may require nonperturbative treatment of selected operators while others remain perturbative, with contact interactions resummed through Lippmann–Schwinger equations (Hammer et al., 2016). A plausible implication is that the distinction between perturbative and nonperturbative sectors is not a special pathology but an ordinary feature of NREFT whenever low scales emerge dynamically.

In dark-sector bound-state physics, scalar-mediated NREFTs and pNREFTs give analytic control over annihilation and bound-state formation. A particularly distinctive result is that all S-wave annihilation coefficients vanish at $T/M$20, while P-wave coefficients obey

$T/M$21

leading to the nonrelativistic cross section

$T/M$22

(Biondini et al., 2021). Bound-state formation proceeds through quadrupole and relativistic operators rather than a dipole, because the dipole term cancels exactly in the multipole expansion: $T/M$23 (Biondini et al., 2021).

7. Relativistic corrections, fast modes, and field redefinitions

A common misconception is that the nonrelativistic limit is obtained simply by dropping fast oscillatory terms. Several of the papers explicitly show this is incomplete. In the scalar formalism with nonlocal field redefinition, fast harmonics

$T/M$24

are sourced by nonlinear interactions, and integrating them out generates local operators in the effective slow-mode theory (Namjoo et al., 2017). The resulting equation of motion for $T/M$25 includes both relativistic dispersion corrections and nonlinear backreaction,

$T/M$26

and the corresponding effective Lagrangian contains the $T/M$27 operator

$T/M$28

(Namjoo et al., 2017). This term comes entirely from integrating out fast oscillatory modes.

The classical NREFT for a real scalar reaches a similar conclusion by comparing two EFT constructions. The effective potential

$T/M$29

has coefficients

$T/M$30

but obtaining the corrected $T/M$31 requires including gradient interactions in the matching (Braaten et al., 2018). The Mukaida–Takimoto–Yamada form must also be supplemented by the missing time-derivative interaction

$T/M$32

before it becomes equivalent to the Braaten–Mohapatra–Zhang form (Braaten et al., 2018). The equivalence is then established through a sequence of field redefinitions, including

$T/M$33

This case study shows that operator bases with or without time derivatives may be equivalent, but only after a complete accounting of all terms contributing at the target order (Braaten et al., 2018).

In the Dirac EFT, field redefinition also plays a central role because the low-energy field $T/M$34 obtained directly from integrating out $T/M$35 is not canonically normalized. The properly normalized Pauli wave function is

$T/M$36

(Santos et al., 2018). This suggests that canonical normalization and operator organization are not merely aesthetic choices; they determine the clarity of power counting and the direct identification of physical observables.

8. Renormalization-group flows and anisotropic scaling

Not all NREFTs are expansions around free Schrödinger particles. Lifshitz-type NREFTs, especially in holographic contexts, possess anisotropic scaling and two distinct ultraviolet cutoff scales, one for energy and one for momentum (Koutentakis, 2019). In weakly coupled Lifshitz scalar theory, the one-loop beta functions for a dimensionful $T/M$37 coupling $T/M$38 are

$T/M$39

$T/M$40

(Koutentakis, 2019). In the marginal case $T/M$41,

$T/M$42

(Koutentakis, 2019).

In the holographic Einstein–Maxwell–Dilaton realization, the momentum- and energy-scale beta functions for a scalar coupling $T/M$43 are

$T/M$44

with $T/M$45 the superpotential and $T/M$46 the blackness function (Koutentakis, 2019). Near a Lifshitz fixed point, they become

$T/M$47

$T/M$48

For hyperscaling-violating backgrounds,

$T/M$49

(Koutentakis, 2019).

These constructions are technically distinct from particle NREFTs, but they belong to the same family insofar as they are low-energy, symmetry-constrained theories with anisotropic scaling and nonrelativistic dispersion. A plausible implication is that “NREFT” should not be restricted to $T/M$50 expansions alone; it also includes long-distance descriptions organized by Lifshitz scaling and separate energy/momentum renormalization.

9. Conceptual synthesis and common pitfalls

Several broad lessons emerge from these treatments.

First, NREFT is defined by controlled scale separation, not by any single derivation. A relativistic parent theory may be contracted geometrically to Newton–Cartan space (Bergshoeff et al., 2015), reduced by integrating out heavy spinor components (Santos et al., 2018), rewritten via a nonlocal canonical transformation (Namjoo et al., 2017), coarse-grained over fast oscillations (Modirzadeh et al., 11 Jul 2025), or matched through on-shell amplitudes (Braaten et al., 2018, Hammer et al., 2016). The resulting EFTs differ in field variables and operator bases, but the physical content is the same when matching is done consistently.

Second, Wilson coefficients need not be real. In thermal and open-system applications, imaginary coefficients encode dissipation and reaction rates. The heavy Majorana coefficient $T/M$51 acquires $T/M$52 and thereby generates a thermal width (Biondini, 2014, Biondini et al., 2013). The annihilation kernel $T/M$53 in the Keldysh–Schwinger formulation plays the same role for heavy-pair systems (Binder et al., 13 Apr 2026).

Third, background geometry is not ancillary in nonrelativistic physics. The central charge gauge field $T/M$54 in Newton–Cartan geometry is the source for particle number (Bergshoeff et al., 2015). The gauge field $T/M$55, spin connection $T/M$56, and shift vector $T/M$57 encode transport and geometric response in quantum Hall EFTs (Wu et al., 2014). Null backgrounds and Newton–Cartan data provide the natural covariant language for stochastic Galilean hydrodynamics (Jain, 2020).

Fourth, fast modes and higher-derivative operators are often indispensable. The backreaction of fast oscillations generates the $T/M$58 term in scalar NREFT (Namjoo et al., 2017). Gradient operators correct the $T/M$59 coefficient in classical scalar EFT (Braaten et al., 2018). Higher-dimensional operators in heavy-particle EFTs determine $T/M$60 thermal corrections (Biondini et al., 2013). A common misconception is that such terms are optional refinements; the explicit matching calculations show they are required for correctness at the claimed order.

Fifth, some sectors must be treated nonperturbatively. Large scattering length forces resummation of $T/M$61 bubbles in short-range EFT (Hammer et al., 2016). Near-threshold bound states require potential resummation and self-consistent treatment of annihilation to avoid apparent unitarity violation in Sommerfeld enhancement (Binder et al., 13 Apr 2026). This suggests that the perturbative operator expansion and the nonperturbative resummation of selected operators are complementary, not contradictory, aspects of NREFT.

Finally, field redefinitions are integral to the subject. They relate local and nonlocal scalar constructions (Namjoo et al., 2017), connect MTY and BMZ classical scalar EFTs (Braaten et al., 2018), and normalize the Pauli field in the Dirac limit (Santos et al., 2018). They do not change on-shell observables but can make symmetries, power counting, or particle-number conservation manifest.

Taken together, these results present NREFTs as a family of precision frameworks for low-energy nonrelativistic phenomena: they derive from relativistic theories when appropriate, couple naturally to Newton–Cartan or null backgrounds when geometry matters, accommodate both perturbative and nonperturbative dynamics, and organize medium effects, bound states, and transport within a unified effective-theory logic (Bergshoeff et al., 2015, Biondini et al., 2013, Namjoo et al., 2017, Wu et al., 2014, Santos et al., 2018, Biondini et al., 2021, Modirzadeh et al., 11 Jul 2025, Jain, 2020, Hammer et al., 2016, Binder et al., 13 Apr 2026, Koutentakis, 2019, Braaten et al., 2018).