Heavy-Light Vacuum Block Approximation

• Heavy–light vacuum block approximation is a semiclassical framework in 2D CFT that isolates the identity Virasoro family using heavy and light operator insertions.
• It employs monodromy methods and accessory-parameter techniques to derive exponential forms and incorporate quadratic and 1/c corrections in four-point correlators.
• Its holographic interpretation connects heavy operator backreactions with conical defects or BTZ backgrounds, enabling light probes to reveal classical geodesic lengths.

The heavy–light vacuum block approximation is the semiclassical large-$c$ description of the Virasoro vacuum conformal block in a two-dimensional CFT when two external primaries are “heavy,” with dimensions scaling as $h_H\sim O(c)$, and two are “light,” with $h_L/c\ll 1$. In this regime the vacuum block exponentiates,

$$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$

and the semiclassical function $f$ may be expanded in powers of the light ratio, $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$ (Beccaria et al., 2015). The approximation organizes the vacuum-sector contribution to heavy–light four-point functions, admits a monodromy formulation through an accessory-parameter problem, and has a holographic interpretation in which the heavy insertions create a conical defect or BTZ background probed by light operators (Beccaria et al., 2015).

1. Heavy–light scaling and semiclassical form

The defining limit is $c\to\infty$ with the heavy ratio fixed and the light ratio small. One convention uses $\eta=h_H/c=O(1)$ and $\epsilon=h_L/c\ll 1$ (Beccaria et al., 2015). Another uses

$$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$

with

$h_H\sim O(c)$0

which is the parameter controlling the heavy background (Banerjee et al., 2016). In the thermal regime one also writes

$h_H\sim O(c)$1

so that the block takes a thermal form on the cylinder (Fitzpatrick et al., 2015).

At leading large $h_H\sim O(c)$2, the heavy–light vacuum block can be written in several equivalent kinematic parametrizations. In cylinder time $h_H\sim O(c)$3, defined by $h_H\sim O(c)$4, one has

$h_H\sim O(c)$5

while in the uniformizing coordinate

$h_H\sim O(c)$6

the same object becomes

$h_H\sim O(c)$7

These expressions make explicit the Euclidean-time periodicity inherited from the heavy state background (Fitzpatrick et al., 2015).

This approximation is specifically a vacuum-block approximation: it isolates the identity Virasoro family in a heavy–light correlator. A plausible implication is that its physical content is strongest when the vacuum channel dominates the relevant OPE channel, although the summarized sources formulate the approximation itself independently of any global dominance claim.

2. Monodromy construction of the four-point block

The standard derivation inserts a formal level-2 null state and reduces the conformal block problem to a Fuchsian equation,

$h_H\sim O(c)$8

with a semiclassical stress tensor containing a single accessory parameter $h_H\sim O(c)$9 in the four-point gauge $h_L/c\ll 1$0, $h_L/c\ll 1$1, $h_L/c\ll 1$2, $h_L/c\ll 1$3: $h_L/c\ll 1$4 The vacuum exchange is imposed by requiring trivial $h_L/c\ll 1$5 monodromy around the contour linking $h_L/c\ll 1$6 and $h_L/c\ll 1$7, equivalently $h_L/c\ll 1$8 (Banerjee et al., 2016).

Solving the monodromy condition perturbatively in $h_L/c\ll 1$9 gives the leading accessory parameter

$$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$0

and since $$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$1, integration yields

$$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$2

Exponentiating then gives

$$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$3

For $$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$4, this reproduces the expected periodicity $$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$5 in the thermal regime (Banerjee et al., 2016).

In the $$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$6 notation, the same leading non-trivial coefficient is

$$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$7

obtained by fixing the accessory parameter through trivial monodromy and integrating $$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$8 (Beccaria et al., 2015). The difference between these closed forms reflects differing coordinate and normalization conventions rather than a distinct approximation.

3. Quadratic light corrections and $$F_{\rm vac}(c;h_H,h_H,h_L,h_L;z)=\exp\!\left[-\frac{c}{6}\,f(\eta,\epsilon;z)+O(c^0)\right], \qquad \eta\equiv \frac{h_H}{c},\quad \epsilon\equiv \frac{h_L}{c},$$9 refinements

The heavy–light semiclassical expansion extends beyond the linear probe term. The first next-to-leading contribution is the quadratic correction $f$0, or equivalently the $f$1 term in

$f$2

A fully closed form is available for $f$3: $f$4 where $f$5, $f$6, and $f$7 are given explicitly in terms of rational functions, logarithms, $f$8, the digamma $f$9, Euler’s constant $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$0, and several hypergeometric functions $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$1. The function $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$2 is then recovered by integrating with the boundary condition $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$3 as $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$4, and the result agrees with high-order series extracted from the Zamolodchikov recursion (Beccaria et al., 2015).

Several structural properties are explicit at this order. The expansion in $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$5 converges for $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$6 in the $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$7-channel; for real cross-ratios $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$8 the block is manifestly real; and crossing symmetry under $$f(\eta,\epsilon;z)=\epsilon f_1(\eta;z)+\epsilon^2 f_2(\eta;z)+O(\epsilon^3)$$9 is maintained order by order (Beccaria et al., 2015). These statements delimit the regime in which the quadratic heavy–light vacuum block is directly controlled.

A distinct but related refinement is the perturbative $c\to\infty$0 expansion of the vacuum block beyond the strict semiclassical exponent. One writes

$c\to\infty$1

where the new functions furnish a “quantum” correction $c\to\infty$2 and a “semi-classical” correction $c\to\infty$3. Their explicit closed forms involve incomplete Beta functions

$c\to\infty$4

and generalized harmonic numbers $c\to\infty$5, and can also be represented by contour integrals built from heavy-pair stress-tensor correlators and rational kernels (Fitzpatrick et al., 2015). The same projector-plus-kernel method extends to general Virasoro blocks, not only the vacuum block (Fitzpatrick et al., 2015).

4. AdS$c\to\infty$6 interpretation: conical defects, BTZ backgrounds, and worldlines

The approximation admits a bulk description in which the heavy operators backreact to produce a classical geometry and the light operators propagate as probes. At leading order, the heavy background is described either as a conical defect or, in the thermal regime, as a static BTZ geometry with “mass” $c\to\infty$7; the light operators are represented by geodesics in that background, and $c\to\infty$8 is proportional to the regulated geodesic length (Beccaria et al., 2015).

In the conical-defect worldline construction, two heavy operators of weight $c\to\infty$9 at $\eta=h_H/c=O(1)$0 produce the metric

$\eta=h_H/c=O(1)$1

with $\eta=h_H/c=O(1)$2. A light probe of dimension $\eta=h_H/c=O(1)$3 follows a spacelike geodesic $\eta=h_H/c=O(1)$4 connecting the boundary insertion points, with action $\eta=h_H/c=O(1)$5. After solving the turning-point condition and renormalizing the proper length, one finds

$\eta=h_H/c=O(1)$6

and the corresponding classical block is quoted as

$\eta=h_H/c=O(1)$7

This agrees with the large-$\eta=h_H/c=O(1)$8, heavy–light Virasoro block obtained by CFT recursion or monodromy methods (Belavin et al., 2017).

Beyond the probe limit, the $\eta=h_H/c=O(1)$9 term $\epsilon=h_L/c\ll 1$0 encodes the first backreaction of the geodesic on the BTZ background and supplies a more refined AdS$\epsilon=h_L/c\ll 1$1/CFT$\epsilon=h_L/c\ll 1$2 matching problem than the leading geodesic picture (Beccaria et al., 2015). In the $\epsilon=h_L/c\ll 1$3 expansion, the leading block is associated with a geodesic in the backreacted BTZ geometry, whereas the subleading corrections are interpreted as one-loop effects such as graviton exchange between the light-operator geodesic and the heavy geodesic or self-energy diagrams on the light geodesic (Fitzpatrick et al., 2015). The same framework explains how monodromies in Euclidean time arise from geodesic crossing singularities in geodesic Witten diagrams (Fitzpatrick et al., 2015).

5. Higher-point factorization, degenerate insertions, and Heun theory

In higher-point heavy–light correlators, the four-point vacuum block reappears as a universal building block for a class of OPE channels. When $\epsilon=h_L/c\ll 1$4 light operators are paired so that each pair fuses into the vacuum in the vertical channels, the saddle-point action factorizes at leading order in $\epsilon=h_L/c\ll 1$5 as

$\epsilon=h_L/c\ll 1$6

and hence

$\epsilon=h_L/c\ll 1$7

This factorization is derived by repeating the same monodromy analysis used for the four-point block in each pairwise channel, and it underlies calculations of the entanglement entropy of an arbitrary number of disjoint intervals for heavy states (Banerjee et al., 2016).

A related semiclassical mechanism appears in degenerate five-point blocks. With four heavy primaries of dimensions $\epsilon=h_L/c\ll 1$8 and one light degenerate insertion $\epsilon=h_L/c\ll 1$9, the chiral block factorizes in the classical limit $$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$0 as

$$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$1

Here the exponential carries the heavy classical four-point block, while $$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$2 is a finite light fluctuation (Piatek et al., 2017). After dividing out the heavy exponential, the BPZ null-vector equation reduces to the normal-form Heun equation,

$$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$3

with accessory parameter

$$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$4

In the vacuum intermediate channel $$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$5, the classical block reduces to the classical vacuum block,

$$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$6

which is the heavy–light vacuum block approximation in this degenerate-block setting (Piatek et al., 2017). The same factorization also yields Floquet-type solutions of Heun’s equation with prescribed monodromy (Piatek et al., 2017).

A separate route to higher-point heavy–light blocks is the global-to-heavy–light map. In the five-point case, one uses the nonlinear reparametrization

$$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$7

together with Jacobian factors $$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$8 to construct the heavy–light block from the global block. In this way the heavy–light vacuum specialization reproduces the perturbative classical block previously obtained by monodromy (Alkalaev et al., 2015).

6. Semiclassical breakdown, nonperturbative structure, and related extensions

The approximation has controlled but finite validity. In Lorentzian time, analytic and numerical studies show that the semiclassical heavy–light vacuum block initially exhibits exponential decay,

$$\epsilon_H\equiv \frac{6h_H}{c}=O(1),\qquad \epsilon_L\equiv \frac{6h_L}{c}\ll 1,$$9

but this behavior crosses over at

$h_H\sim O(c)$00

to a universal late-time power law,

$h_H\sim O(c)$01

For the vacuum block the transition occurs at $h_H\sim O(c)$02, and the semiclassical approximation is described as excellent for $h_H\sim O(c)$03 but parametrically unreliable once $h_H\sim O(c)$04 (Chen et al., 2017).

In Euclidean time, the strict semiclassical block exhibits thermal image singularities at

$h_H\sim O(c)$05

the “forbidden singularities.” At finite $h_H\sim O(c)$06, however, the exact Virasoro block is analytic for all $h_H\sim O(c)$07 and smooth at each would-be $h_H\sim O(c)$08. Near those points, non-perturbative $h_H\sim O(c)$09 effects become unsuppressed, a different semiclassical saddle takes over, and a finite Euclidean bubble of radius $h_H\sim O(c)$10 forms around each forbidden singularity. Crossing the boundary of this bubble is crossing an anti-Stokes line (Chen et al., 2017). Closely related observations about Euclidean-time monodromies and their cancellation in the full correlator appear in the perturbative $h_H\sim O(c)$11 analysis, which emphasizes that only non-perturbative corrections in $h_H\sim O(c)$12 can resolve the singularities associated with the information paradox (Fitzpatrick et al., 2015).

The same heavy–light logic has been extended beyond the Virasoro setting. In $h_H\sim O(c)$13 theory, semi-classical blocks exponentiate when $h_H\sim O(c)$14, and perturbative heavy and heavy–light vacuum $h_H\sim O(c)$15-blocks can be computed from an oscillator construction, with heavy insertions creating a flat-space cosmology labelled by $h_H\sim O(c)$16 and $h_H\sim O(c)$17 (Ammon et al., 2020). In large-$h_H\sim O(c)$18 Virasoro $h_H\sim O(c)$19-point blocks with superlight insertions, the logarithm of the block is related to the weighted length of a holographic Steiner tree on the Poincaré disk, and the resulting $h_H\sim O(c)$20- and $h_H\sim O(c)$21-point constructions solve the monodromy conditions to first order in the superlight weights (Pavlov, 2021). This suggests that the heavy–light vacuum block approximation is not merely a four-point probe formula, but part of a broader semiclassical architecture linking monodromy problems, worldline geometries, and block exponentiation.