Long-Range Minimal Models in 2D CFT

• Long-range minimal models are nonlocal deformations of Virasoro minimal models where a primary field couples with a generalized free field to yield an operator of dimension 2-δ flowing to an infrared fixed point.
• The (m,2,2) and (m,1,2) families feature complementary weak-coupling regimes with distinct beta function coefficients, anomalous dimensions, and dual perturbative descriptions.
• Advanced methods like Coulomb-gas and Mellin-Barnes techniques enable analytic large-m expansions, crucial for extracting asymptotic behavior of beta functions and Virasoro blocks.

Searching arXiv for papers directly related to long-range minimal models and adjacent constructions. Long-range minimal models are a class of nonlocal conformal field theories in two dimensions obtained as deformations of Virasoro minimal models by coupling a relevant primary $\phi_{r,s}$ to a generalized free field $\chi$ so that the composite operator $O(x)=\phi_{r,s}(x)\chi(x)$ has scaling dimension $2-\delta$; the deformation

$$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$

can then flow to an infrared fixed point at $g_*^2=\delta/\beta_3$ when the cubic coefficient of the beta function is real and positive (Behan et al., 30 Sep 2025). In this usage, the long-range minimal model (LRMM) of type $(m,r,s)$ is the infrared fixed point associated with the unitary, diagonal Virasoro minimal model $\mathcal M_{m+1,m}$ and a bosonic generalized free field chosen so that the perturbation is weakly relevant (Behan et al., 30 Sep 2025).

1. Definition as deformations of Virasoro minimal models

The construction starts from the unitary, diagonal Virasoro minimal model $\mathcal M_{m+1,m}$, $m\ge 3$, with central charge

$\chi$0

and primary fields $\chi$1 of weight

$\chi$2

A bosonic generalized free field $\chi$3 is then introduced with scaling dimension

$\chi$4

so that

$\chi$5

The ultraviolet theory at $\chi$6 is therefore the nonlocal CFT given by “minimal model $\chi$7 GFF,” deformed by $\chi$8 (Behan et al., 30 Sep 2025).

Under the assumptions stated for standard conformal perturbation theory—specifically, that no relevant or marginal composite operators mix with $\chi$9, provided $O(x)=\phi_{r,s}(x)\chi(x)$0 and no other OPE channel is near marginality—the beta function truncates to

$O(x)=\phi_{r,s}(x)\chi(x)$1

A real zero at

$O(x)=\phi_{r,s}(x)\chi(x)$2

defines the infrared fixed point. This criterion is the operational definition of the LRMM in the 2025 construction (Behan et al., 30 Sep 2025).

The construction is explicitly nonlocal because the generalized free field sector is not generated by a local two-dimensional CFT. A plausible implication is that the adjective “long-range” refers not to a lattice interaction written directly in real space, but to the nonlocal continuum sector that is coupled to the minimal-model primary.

2. The $O(x)=\phi_{r,s}(x)\chi(x)$3 family and the two perturbative regimes

The family based on $O(x)=\phi_{r,s}(x)\chi(x)$4 is singled out because it admits two complementary weak-coupling descriptions. In this case,

$O(x)=\phi_{r,s}(x)\chi(x)$5

The first description is a near-mean-field regime. One begins with a generalized free field $O(x)=\phi_{r,s}(x)\chi(x)$6 in $O(x)=\phi_{r,s}(x)\chi(x)$7 and adds a local $O(x)=\phi_{r,s}(x)\chi(x)$8-point vertex. Equivalently, for $O(x)=\phi_{r,s}(x)\chi(x)$9 one chooses

$2-\delta$0

which gives a weakly relevant coupling $2-\delta$1 for $2-\delta$2. The renormalized beta function is

$2-\delta$3

with

$2-\delta$4

The infrared fixed point obeys $2-\delta$5, and the weak anomalous dimension of the interaction operator is

$2-\delta$6

(Behan et al., 30 Sep 2025).

The second description is a near-short-range, or Sak, regime obtained by conformal perturbation around $2-\delta$7. One turns on

$2-\delta$8

and finds again

$2-\delta$9

with

$$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$0

For integer $$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$1, the integral is evaluated by radial and block expansions, and $$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$2 is positive for all $$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$3. Its large-$$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$4 asymptotic is

$$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$5

The leading anomalous dimensions of Virasoro primaries satisfy

$$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$6

while higher-spin currents in the identity multiplet acquire nonzero anomalous dimensions, for example

$$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$7

These two expansions are proposed to be dual descriptions of the same line of fixed points, with the shadow relation

$$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$8

as the basic matching condition (Behan et al., 30 Sep 2025).

The large-$$S_{\rm int}=g_0\int d^2x\,O(x)=g_0\int d^2x\,\phi_{r,s}\chi$$9 limit is problematic in both perturbative regimes. The stated consequence is that nonperturbative methods are required in the intermediate range for all values of $g_*^2=\delta/\beta_3$0 (Behan et al., 30 Sep 2025).

3. Models based on $g_*^2=\delta/\beta_3$1

The families based on $g_*^2=\delta/\beta_3$2 and $g_*^2=\delta/\beta_3$3 differ qualitatively from the $g_*^2=\delta/\beta_3$4 case. Here there is only one known weakly relevant perturbative expansion, and it is reported to be well behaved at large $g_*^2=\delta/\beta_3$5. The same conformal-perturbative method applies, but the OPE now contains only two blocks (Behan et al., 30 Sep 2025).

At one loop, the beta function again has the form

$g_*^2=\delta/\beta_3$6

with $g_*^2=\delta/\beta_3$7 extracted from the same basic four-point integral. Numerically, for finite $g_*^2=\delta/\beta_3$8, one finds $g_*^2=\delta/\beta_3$9, while the large-$(m,r,s)$0 asymptotic is

$(m,r,s)$1

The leading scaling dimensions obey

$(m,r,s)$2

and, for operators that do not mix, one has at $(m,r,s)$3

$(m,r,s)$4

The construction also exhibits operator mixing: $(m,r,s)$5 with odd $(m,r,s)$6 mix with $(m,r,s)$7, producing one-loop splittings; a representative example is

$(m,r,s)$8

for $(m,r,s)$9 (Behan et al., 30 Sep 2025).

A systematic two-loop computation for infinitely many primaries $\mathcal M_{m+1,m}$0 in type $\mathcal M_{m+1,m}$1 was carried out, and the large-$\mathcal M_{m+1,m}$2 limits were obtained both numerically and through Mellin-amplitude methods. In the terminology of the source, this confirms that the $\mathcal M_{m+1,m}$3 family is comparatively well behaved at large $\mathcal M_{m+1,m}$4 (Behan et al., 30 Sep 2025).

4. Coulomb-gas and Mellin-Barnes techniques

A central technical issue is that standard large-$\mathcal M_{m+1,m}$5 perturbation theory around minimal models becomes difficult because $\mathcal M_{m+1,m}$6 and $\mathcal M_{m+1,m}$7 grow rapidly with $\mathcal M_{m+1,m}$8. To address this, the construction introduces a “chiral first” Coulomb-gas approach together with Mellin-Barnes methods (Behan et al., 30 Sep 2025).

In the chiral approach, one expands in $\mathcal M_{m+1,m}$9 before integrating, using the Mellin representation

$\mathcal M_{m+1,m}$0

Contour deformation then yields explicit large-$\mathcal M_{m+1,m}$1 expansions of Virasoro blocks. The first two examples quoted are

$\mathcal M_{m+1,m}$2

and

$\mathcal M_{m+1,m}$3

For the full four-point function needed in $\mathcal M_{m+1,m}$4, two successive Symanzik star integrals lead to a seven-fold Mellin integral. The analytic continuations are then performed with the Mathematica package MB, and the resulting expansion reproduces

$\mathcal M_{m+1,m}$5

for the $\mathcal M_{m+1,m}$6 family (Behan et al., 30 Sep 2025).

The stated significance of these methods is twofold. First, they provide analytic access to integrals that had previously been known only numerically. Second, they produce large-$\mathcal M_{m+1,m}$7 asymptotics for infinitely many anomalous dimensions, especially in the $\mathcal M_{m+1,m}$8 family, where the perturbative expansion remains controlled (Behan et al., 30 Sep 2025).

5. Relation to long-range deformations of non-unitary minimal series

A distinct but closely related line of work studies long-range deformations of the non-unitary minimal series $\mathcal M_{m+1,m}$9 through a nonlocal kinetic term with propagator $m\ge 3$0 and through deformations of the short-range minimal model by a long-range field $m\ge 3$1 (Eustachon, 27 Mar 2026). In that setting, two constructions are compared: a Landau-Ginzburg action with interaction $m\ge 3$2 and a direct deformation

$m\ge 3$3

The perturbative result is sharply different from the unitary LRMM story: for $m\ge 3$4 one finds $m\ge 3$5, real nontrivial fixed points, and a spectrum match between the two constructions, whereas for every $m\ge 3$6 one finds $m\ge 3$7, so $m\ge 3$8 is negative, $m\ge 3$9 is pure imaginary, and the stress tensor becomes slightly relevant (Eustachon, 27 Mar 2026).

The $\chi$00 case is the long-range Lee-Yang model. In this special case, the source states that the long-range Lee-Yang fixed point coincides with the known long-range Ising universality class after appropriate analytic continuation of exponents, and both flows produce the same two low-lying operator dimensions and the same marginality crossover (Eustachon, 27 Mar 2026). Independent support for this broader perspective comes from one-dimensional bootstrap sum rules: “Super Sum rules for Long-Range Models” tests long-range versions of the Ising, $\chi$01, and Lee-Yang models and finds that the sum rules correctly predict the CFT data characterising these theories, while also substantially reducing the allowed parameter space in numerical bootstrap studies (Ghosh et al., 23 Mar 2026).

A plausible implication is that “long-range minimal models” is not a uniformly stable category across minimal-series constructions: its viability depends strongly on which minimal series is deformed and on whether one works in a unitary or non-unitary setting.

6. Open problems, crossover structure, and neighboring meanings

Several unresolved issues define the current state of the subject. In the unitary LRMM construction, the $\chi$02 family has two complementary perturbative descriptions but a problematic large-$\chi$03 limit in both regimes, so nonperturbative methods are required in the intermediate range for all values of $\chi$04. In the non-unitary $\chi$05 constructions, the short-range limit $\chi$06 is described as subtle because perturbation theory around nonlocality breaks down as wave-function renormalization develops poles at $\chi$07; functional RG or Hamiltonian truncation is identified as needed to connect the long- and short-range regimes smoothly (Behan et al., 30 Sep 2025, Eustachon, 27 Mar 2026).

The subject also sits near, but is not identical to, other long-range theories in mathematical physics. In long-range Ising models on $\chi$08, for instance, there are rigorous results on planelike interfaces, class A minimal sets, and the reciprocal approximation of lattice ground states and nonlocal minimal surfaces. In that literature, if $\chi$09 satisfies

$\chi$10

one obtains nonlocal perimeter minimizers whose boundaries stay in a strip of bounded width, together with $\chi$11-convergence from discrete long-range Ising energies to continuum nonlocal perimeter functionals (Cozzi et al., 2016). These objects are mathematically “minimal,” but they are not minimal models in the Virasoro-CFT sense.

Within conformal field theory, therefore, long-range minimal models designate a specific program: constructing nonlocal two-dimensional fixed points by coupling minimal-model primaries to generalized free fields, analyzing their beta functions and anomalous dimensions, and comparing complementary long-range and short-range descriptions. The current picture is most developed for the unitary $\chi$12 construction and for the special Lee-Yang case among non-unitary series, while the general nonperturbative structure remains open (Behan et al., 30 Sep 2025).