Renormalisation Group Invariants (RGIs) Overview

• Renormalisation Group Invariants (RGIs) are combinations of scale-dependent parameters that remain constant along the RG flow, providing clear insights into high-energy physics.
• They are constructed using symmetry-based methods and exhaustive algebraic searches that yield one-loop, two-loop, or all-loop invariant relations depending on the model.
• RGIs play a crucial role in flavor physics, supersymmetry, and effective potential analyses, aiding in tests of unification and parameter extraction in theoretical models.

to=arxiv.search 天天中彩票人工_code {"query":"Renormalization Group Invariants arXiv", "max_results": 10, "sort_by":"relevance"} RTLU to=arxiv.search 天天中彩票买json {"query":"(Feldmann et al., 2015) Renormalization Group Evolution of Flavour Invariants", "max_results": 3, "sort_by":"relevance"} to=arxiv.search 天天中彩票在_code {"query":"(Rystsov et al., 2024) all-loop renormalization group invariants for MSSM", "max_results": 3, "sort_by":"relevance"} Renormalisation Group Invariants (RGIs) are combinations of running parameters that remain constant along the renormalisation-group flow. If the running parameters are $x_i(\mu)$ and $t$ is the logarithmic scale variable, an invariant $I(x(\mu))$ satisfies

$$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$

In practice, RGIs occur in several distinct senses: as exact invariants of truncated perturbative RGEs, as approximate invariants controlled by hierarchies, as basis-independent invariants of flavour structures, and as reparametrisation invariants that are invariant under field redefinitions rather than under scale evolution. Across the literature, RGIs are used to encode UV information in low-energy data, to formulate sum rules, to organize flavour and neutrino sectors without choosing a basis, and to identify cases in which effective potentials or specific operator combinations are exactly scale independent (Verheyen, 2015).

1. Definition, loop order, and the meaning of “invariant”

The defining equation of an RGI is the vanishing of the total derivative with respect to $\ln\mu$. In a Callan–Symanzik formulation this may include explicit scale dependence and field anomalous dimensions,

$$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$

so the relevant notion of invariance depends on the object under study and on the parameter space in which the RG flow is formulated (Pilaftsis et al., 2017).

Perturbative usage is loop-order specific. For one-loop RGIs, $dI/dt=0$ when the $\beta$-functions are truncated at one loop. For two-loop RGIs, one uses an ansatz

$I=I_1+\frac{1}{16\pi^2}I_2$

and imposes

$I_1^{(1)}=0,\qquad I_1^{(2)}+I_2^{(1)}=0,$

with the formal three-loop term neglected at two-loop order (Verheyen, 2015). This distinction is essential: exact at one loop is not the same as exact to all loops.

A second distinction is between basis independence and RG invariance. In flavour physics, quantities built from traces, determinants, adjugates, and commutators of Yukawa spurions are flavour invariants because they are unchanged by flavour-basis transformations. They are not automatically constant under scale evolution. The literature on Standard Model flavour therefore separates flavour invariants from true RG invariants, and explicitly notes that true RG invariants are rare (Feldmann et al., 2015).

A third distinction appears in models with two $t$0 gauge factors. There the central quantities

$t$1

are invariant under generic gauge-field reparametrisations, including rescalings, because $t$2 and $t$3 leave the contraction unchanged. These are reparametrisation invariants, not generally RGIs; they still run with scale through the running of $t$4 (Davidson et al., 19 May 2026).

2. Construction principles and algorithmic searches

One construction strategy is symmetry based. In supersymmetric theories, one-loop scalar-mass $t$5-functions have tightly constrained structures imposed by gauge and global $t$6 symmetries. By contracting scalar masses with suitable linear combinations of preserved charges and combining the result with $t$7 and gaugino-mass terms, one obtains invariants whose one-loop derivatives vanish identically. This logic underlies the MSSM invariants built from $t$8, $t$9, $I(x(\mu))$0, and $I(x(\mu))$1 charges and their analogues in the dMSSM and pMSSM (Beenakker et al., 2015).

A second strategy is exhaustive algebraic search. RGIsearch treats the $I(x(\mu))$2-functions as polynomials in the running parameters with rational coefficients and searches for several classes of invariants. Monomial invariants take the form

$I(x(\mu))$3

and the invariance condition reduces to a linear system for the integer exponents $I(x(\mu))$4. Polynomial invariants take the form

$I(x(\mu))$5

with monomials of common “dimensionality”, and again the condition of term-by-term cancellation produces a sparse linear system for the coefficients $I(x(\mu))$6. Factorized polynomial invariants,

$I(x(\mu))$7

are treated through a generalized-eigenvalue problem in the integer $I(x(\mu))$8 (Verheyen, 2015).

The same work uses automatically discovered additive dimensionalities to block-diagonalize the search and bounds the combinatorics by restricting exponent size and the number of distinct parameters per monomial. The resulting systems are solved with fraction-free Gaussian elimination with Markowitz pivoting, and two-loop searches impose the coupled conditions $I(x(\mu))$9 and $$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$0 on the same sparse-linear-algebra infrastructure (Verheyen, 2015).

A more formal route is the method of characteristics. In the SI2PI analysis, the general condition

$$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$1

is identified as the equation to be solved for coupling-space invariants, although the paper emphasizes exact RG-invariant observables rather than a catalogue of closed-form coupling invariants (Pilaftsis et al., 2017).

3. Flavour and neutrino-sector invariants

In Standard Model flavour physics, the basic hermitian spurions are

$$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$2

For three generations, a convenient generating set consists of $$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$3 CP-even and $$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$4 CP-odd invariants: $$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$5

$$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$6

$$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$7

and

$$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$8

These quantities encode masses, moduli of CKM elements, and CP violation without choosing a flavour basis (Feldmann et al., 2015).

At one loop in the Standard Model with QCD only, the Yukawa RGEs close on a finite octet basis, and the invariant RGEs are explicit. Two exact one-loop RGIs reproduced in this language are the Harrison–Krishnan–Scott invariants,

$$\frac{dI}{dt}=\sum_i \frac{\partial I}{\partial x_i}\,\beta_{x_i}(\{x\})=0.$$9

which hold because the one-loop SM satisfies $\ln\mu$0. The same analysis also states that beyond one loop, or beyond the SM, these combinations are not generally invariant, and that with generic Yukawa structures there are no nontrivial exact RGIs other than those protected by such accidental one-loop identities (Feldmann et al., 2015).

The same framework yields approximate RGIs under realistic hierarchies. With SM-like Wolfenstein/Froggatt–Nielsen scaling and top-Yukawa dominance, seven approximate RG-invariant combinations can be formed, such as

$\ln\mu$1

which encode the multiplicative running patterns of CKM parameters under top-Yukawa dominance (Feldmann et al., 2015).

In the neutrino sector, the one-loop RGE for the coefficient $\ln\mu$2 of the Weinberg operator is multiplicative and flavor-separable in the charged-lepton mass basis. This implies that the phases of all elements of the Majorana mass matrix are one-loop RGIs,

$\ln\mu$3

and that simple ratios of matrix elements are protected. The paper highlights

$\ln\mu$4

as numerically excellent invariants under the approximation $\ln\mu$5, and

$\ln\mu$6

as an exact one-loop RGI. Their invariance is independent of neutrino mass ordering and of the parameterization of the lepton mixing matrix (Haba et al., 2013).

4. Supersymmetric RGIs and high-scale sum rules

In the MSSM and its phenomenological restrictions, one-loop RGIs provide direct probes of high-scale supersymmetry breaking. The most basic examples are

$\ln\mu$7

and the gauge-only combinations

$\ln\mu$8

The hypercharge-trace invariant is

$\ln\mu$9

with $$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$0, and the scalar sector admits flavour-sensitive combinations such as $$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$1, $$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$2, $$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$3, $$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$4, and $$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$5 (Hetzel et al., 2012).

These quantities organize unification tests and mediation diagnostics into algebraic sum rules. Gauge coupling unification implies

$$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$6

Gaugino-mass unification implies

$$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$7

Flavour universality gives

$$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$8

Minimal gauge mediation, general gauge mediation, anomaly mediation, and minimal anomaly mediation each produce characteristic relations among the same invariants, including the fixed AMSB ratios

$$\frac{d I}{d\ln\mu} =\mu\frac{\partial I}{\partial\mu} +\sum_i \beta_i(\lambda)\frac{\partial I}{\partial\lambda_i} -\gamma(\lambda)\phi\frac{\partial I}{\partial\phi}=0,$$9

and, in mAMSB,

$dI/dt=0$0

(Hetzel, 2012).

A complementary pMSSM analysis uses posterior distributions of weak-scale parameters, computes the RGIs point by point, and interprets the resulting distributions as messenger-scale information because the invariants are constant to one-loop accuracy. In that setting, the gauge-only invariants are approximately

$dI/dt=0$1

and the gaugino-unification condition can be written as

$dI/dt=0$2

The same analysis maps RGI measurements into the parameters $dI/dt=0$3 and $dI/dt=0$4 of General Gauge Mediation and into the one-parameter structure of Minimal Gauge Mediation (Carena et al., 2012).

Deflected mirage mediation shows how threshold effects modify the picture. In that scenario the one-loop RGIs are piecewise invariant: below and above the messenger threshold each invariant obeys the usual one-loop relation, but at $dI/dt=0$5 the gaugino invariants jump by

$dI/dt=0$6

while specially constructed D-type invariants remain continuous because messenger-induced scalar thresholds cancel in the corresponding linear combinations (Huitu et al., 2015).

Systematic searches in the MSSM, dMSSM, and pMSSM show that the number of invariants is considerably reduced at two loops. In the full MSSM, a two-loop continuation exists for $dI/dt=0$7,

$dI/dt=0$8

whereas the more constrained dMSSM and pMSSM admit additional two-loop continuations for $dI/dt=0$9 and $\beta$0 (Beenakker et al., 2015).

5. Exact all-loop invariance and scheme dependence

All-loop RGIs arise in special circumstances. In the rigid MSSM, exact all-loop invariants can be constructed from gauge couplings, Yukawa determinants, and the Higgs bilinear parameter $\beta$1 by combining the NSVZ $\beta$2-functions with the nonrenormalisation of the superpotential. The essential ingredients are

$\beta$3

$\beta$4

$\beta$5

together with the exact NSVZ equations for $\beta$6. Eliminating anomalous dimensions yields two independent all-loop RGIs. The paper emphasizes that these invariants hold in the HD+MSL scheme and that in the $\beta$7 scheme the renormalization group invariance does not take place starting from the approximation where the scheme dependence manifests itself (Rystsov et al., 2024).

A different exact statement appears in the Symmetry Improved 2PI formalism. For the $\beta$8 scalar model, the SI2PI effective potential $\beta$9 is proved to be exactly RG invariant in the Hartree–Fock and sunset truncations. The proof combines UV-finite running of the proper 2PI couplings with exact cancellation of the $I=I_1+\frac{1}{16\pi^2}I_2$0 terms in the RG variation of the gap equations, yielding

$I=I_1+\frac{1}{16\pi^2}I_2$1

and therefore

$I=I_1+\frac{1}{16\pi^2}I_2$2

This is explicitly contrasted with ordinary 1PI perturbation theory, where the effective potential is RG invariant only up to higher-order terms at a given truncation (Pilaftsis et al., 2017).

These examples delimit the strongest notion of invariance in the subject. Exact all-loop RGIs typically rely on exact structural inputs: NSVZ relations and nonrenormalisation theorems in supersymmetry, or Ward-identity-based cancellations in SI2PI. Outside such settings, exact invariance is exceptional rather than generic.

6. Multi-scalar, gauge-mixing, and conceptual limits

Recent work generalizes RGIs beyond supersymmetry by exploiting the synergy of scaling and non-overlapping global symmetries in renormalisable multi-scalar theories. The central idea is to identify scale-invariant field directions along which the bilinear part of the scalar potential vanishes identically and to assign spurion charges under symmetries such as CP2 and $I=I_1+\frac{1}{16\pi^2}I_2$3. Under these conditions, the dangerous terms in the $I=I_1+\frac{1}{16\pi^2}I_2$4-functions are forbidden to all orders, so certain bilinear combinations become all-loop RGIs. In the two-scalar $I=I_1+\frac{1}{16\pi^2}I_2$5 example one obtains

$I=I_1+\frac{1}{16\pi^2}I_2$6

on the scale-invariant surface $I=I_1+\frac{1}{16\pi^2}I_2$7, and in the 2HDM one obtains

$I=I_1+\frac{1}{16\pi^2}I_2$8

to all loops under the CP2-symmetric conditions quoted in the paper (Pilaftsis, 18 May 2026).

The same literature stresses the limitations. If symmetries overlap or spurion assignments admit mixing terms, invariants are broken. CP3 invariants require the custodial limit $I=I_1+\frac{1}{16\pi^2}I_2$9; for $I_1^{(1)}=0,\qquad I_1^{(2)}+I_2^{(1)}=0,$0, two-loop hypercharge corrections break $I_1^{(1)}=0,\qquad I_1^{(2)}+I_2^{(1)}=0,$1. Threshold matching must preserve the spurion-charge structure if the invariant is to survive decoupling (Pilaftsis, 18 May 2026).

In models with two $I_1^{(1)}=0,\qquad I_1^{(2)}+I_2^{(1)}=0,$2 gauge symmetries, the non-canonical kinetic-matrix formalism supplies a different class of invariants. The combinations

$I_1^{(1)}=0,\qquad I_1^{(2)}+I_2^{(1)}=0,$3

are invariant under arbitrary $I_1^{(1)}=0,\qquad I_1^{(2)}+I_2^{(1)}=0,$4 field reparametrisations, and the ratio

$I_1^{(1)}=0,\qquad I_1^{(2)}+I_2^{(1)}=0,$5

maps directly onto the effective millicharge. These quantities are observationally useful, but they are not RGIs in general, because $I_1^{(1)}=0,\qquad I_1^{(2)}+I_2^{(1)}=0,$6 obeys one- and two-loop RGEs and kinetic mixing is radiatively generated whenever bi-charged matter is present (Davidson et al., 19 May 2026).

Several recurrent misconceptions are therefore corrected by the literature itself. Basis-independent quantities are not automatically scale independent. Exact one-loop invariance does not imply all-loop invariance. Reparametrisation invariants need not be RGIs. And in generic theories, nontrivial exact RGIs are scarce; outside special symmetry-protected settings, the dominant structures are one-loop exact relations, threshold-sensitive piecewise invariants, or approximate invariants controlled by hierarchies (Feldmann et al., 2015).