Dynamics of Finite-Temperature CFTs from OPE Inversion Formulas

We apply the OPE inversion formula to thermal two-point functions of bosonic and fermionic CFTs in general odd dimensions. This allows us to analyze in detail the operator spectrum of these theories. We find that nontrivial thermal CFTs arise when the thermal mass satisfies an algebraic transcendental equation that ensures the absence of an infinite set of operators from the spectrum. The solutions of these gap equations for general odd dimensions are in general complex numbers and follow a particular pattern. We argue that this pattern unveils the large-$N$ vacuum structure of the corresponding theories at zero temperature.

Introduction.-The description of critical systems in nontrivial backgrounds requires data not present in the plane geometry. Perhaps the simplest example is that of conformal field theories (CFTs) on S 1 β × R d−1 , with β the radius of the circle, that describe finite-size or finitetemperature critical systems. In such a case, the twopoint function of a scalar operator φ(x) will in principle depend on the one-point functions of all operators that appear in its operator product expansion (OPE) with itself, since the latter can be nonzero. In particular, for an In d = 2 the plane is conformally related to the cylinder and, although one-point functions of conformal primaries vanish on the latter, there exist operators such as the energy-momentum tensor which transform anomalously under a conformal map. This fixes their one-point functions on the cylinder and therefore the CFT data on R 2 determine the finite-size/finite-temperature corrections to correlation functions on S 1 β × R [1,2]. For d > 2 there is no conformal transformation between R d and S 1 β × R d−1 , and generically one needs to find other ways to determine the additional data b O . A first step in this direction was described in [3], where the leading anisotropic finite-size corrections to the two-point function of scalars in R d were connected to the ratio of the thermal free-energy density of the system and the normalization C T of the energy-momentum tensor two-point function. An extension of these ideas to the nontrivial 3d O(N ) vector model was performed in [4,5], where the relevance of the planar OPE to the description of the finite-size/finitetemperature CFTs was demonstrated. In the latter works the crucial point was that parameters such as b O were independently determined by the gap equation of the vector model. In particular, once the bosonic thermal mass was determined, all one-point functions could be evaluated and hence the full finite-temperature two-point function could be reconstructed.
In more recent developments, the improved understanding of CFTs on R d using numerical and analytic bootstrap methods (see [6] for a recent review) calls for an extension of these advances to finite-size/finite-temperature critical systems. In this context an interesting work has recently appeared [7], whose main result is a Lorentzian inversion formula for the thermal two-point function of a scalar φ(x) with dimension ∆ φ . Using the OPE one can show that the Euclidean position-space [24] thermal two-point function takes the generic form is a polar angle when R d−1 is written in spherical coordinates. C ν s (cos θ) are Gegenbauer polynomials with ν = d/2 − 1. The sum in (1) runs over all operators O s in the OPE φ × φ with spin s and dimension ∆ Os . The coefficients a Os are given by (following the conventions of [7]) with C Os and g φφOs the corresponding two-and threepoint function coefficients, and (a) n the Pochhammer symbol. The unit operator 1 is the unique operator with dimension zero, and here so that the momentum-space two-point function is unitnormalized. Complexifying ∆ one defines the spectral function a(∆, s) via whose poles at ∆ = ∆ Os with residues −a Os yield the physical spectrum. Assuming that the physical poles lie on the right of the imaginary axis one can close the contour clockwise for r < 1 (we set β = 1 from now on) if a(∆, s) does not grow exponentially at infinity. One can arXiv:1806.02340v2 [hep-th] 11 Jun 2018 then use the orthogonality of Gegenbauer polynomials (see e.g. [8, 7.313]) to project the right-hand side of (4) on a spin-s state and then integrate with a suitable power in the region of convergence r ∈ [0, 1] to obtain a(∆, s) as where This is termed Euclidean inversion formula in [7]. Writing x = cos θ = (w + 1/w)/2 with w = e iθ one can transform (5) into a contour integral over the unit circle in the complex-w plane. To exploit the analytic structure of the two-point function g(r, cos θ) one would like to allow w to explore the full complex plane. This can be done by a suitable complexification of the Euclidean variables r, θ, defining z = rw andz = r/w which are now independent real variables. As a function of w, g(r, w) is assumed to have the cuts (−∞, −1/r), (−r, 0), (0, r) and (1/r, ∞), and to grow not faster than w s0 (resp. 1/w s0 ) for large (resp. small) w for some constant s 0 . Moreover, one needs to use the analytic extension of the Gegenbauer polynomials to the whole complex plane as Then, the integral giving a(∆, s) will receive contributions from the discontinuities across the cuts of g(r, w) as well as from the arcs at infinity. The final result is with The discontinuity relevant for the evaluation of (10) is the one across the cut (1/r, ∞), since all others are related to it.
Gap equations from the inversion formula.-The OPE inversion formulas are powerful tools when they are applied to already known correlation functions. In this context, one needs an ansatz for the thermal two-point function before applying OPE inversion. For bosons, one of the simplest choices is to consider the momentum-space two-point function where ω n = 2πn, n = 0, ±1, ±2, . . . , are the bosonic Matsubara frequencies along the finite direction. Clearly, (12) is motivated by known work on thermal field theory which shows that fields develop generically a thermal mass m th at finite temperature. From our point of view we are asking whether the simple ansatz (12) can define a thermal CFT. We make no reference to a Lagrangian, although it is known that (12) can be obtained, for example, in the large-N limit of the O(N ) model. In arbitrary d (12) can be Fourier-transformed to where K α (x) is the modified Bessel function of the second kind. Defining z = τ + i|x| we have |X n | = (n − z)(n −z). From now on we focus on odd d = 2k + 1, k = 1, 2, . . . , and in that case we may write [8, 8.468] [25] with These coefficients also appear in the Bessel polynomials [9] The relevant discontinuity Disc(G (d) ) now follows simply from understanding the discontinuity of the function across the cut due to the square-root branch point at x = 0. Assuming that the cut goes from x = 0 to x = ∞ it can be verified that where with θ n (x) = x n y n (1/x) the so-called reverse Bessel polynomials [10].
Using the results (18), (19) we can now calculate (9). For the discontinuity part we find a (k) in the conventions of [7], where Li α (z) = ∞ n=1 z n /n α is the polylogarithm. The result (20) only pertains to the leading term in az-expansion of the quantity under the integral in (9) [26], reproducing contributions of operators with ∆ = d − 2 + s. These are higher-spin conserved currents saturating the unitarity bound. Subleading terms in thez-expansion can also be considered and would lead to expressions that could be denoted by a arcs (∆, s) is nonzero only for s = 0 and in that case it needs to be taken into account carefully. We find For the latter operators we will first focus on the l = 0 case, corresponding to the φ 2 operator, which appears both from (21) and (20). If we demand the absence of this operator from the spectrum, as required by the fact that it is substituted by the σ operator, then the residue of the ∆ = d − 2 arc contribution should cancel the s = 0 contribution in (20). This turns out to give rise to a condition that determines m th , namely This is called the gap equation and it is here presented for any d = 2k + 1, k = 1, 2, . . . . Higher poles in (21) at ∆ = d − 2 + 2l, l = 1, 2, . . . , correspond to scalar operators of the form φ ∂ 2l φ. Such operators also arise from subleading terms in thez expansion of the quantity under the integral in (9), from expressions we previously referred to as a The arc contribution of the identity operator provides a quick consistency check of our computations. Since the identity operator has ∆ = 0 we see that the pole associated with it appears due to Γ(− ∆ 2 ) in (21). For the residue of that pole we find exactly as required to reproduce the correct normalization of the identity operator in our conventions-for this we need to take into account a 1 from (3) and recall that we are working in conventions where the 1/2 k+1 π k in (14) has been rescaled away. It is also possible to study finite-temperature fermionic two-point functions using the inversion formula. The simplest case to consider is the singlet projection of the two-point functions of Dirac fermions ψ i (x),ψ i (x) in odd dimensions, with ∆ ψ = ∆ φ + 1/2. We denote by i, j = 1, 2, . . . , 2 d−1 2 the spinor indices. Notice that (24) vanishes at zero temperature which means that the unit operator is absent in the finite-temperature OPE. The corresponding unitnormalized momentum-space two-point function is where the fermionic Matsubara frequencies are ω n = 2π(n + 1/2), n = 0, ±1, ±2, . . . . The propagator (25) vanishes form th = 0 so we will only considerm th = 0 in the fermionic case from now on. The calculations follow closely the bosonic case-e.g. it is known that fermionic Matsubara sums reduce to a linear combination of bosonic ones. We then notice that by virtue of the relationship ∆ ψ = ∆ φ +1/2, the fermionic formulas can all be obtained from the bosonic ones by the simple shift ∆ → ∆ − 1. The arc contributions in the fermionic case are thus given bỹ relevant for operators of dimension ∆ = 2m + 1 and ∆ = d − 1 + 2m, m = 0, 1, 2, . . . . The former are contributions that do not arise from the discontinuity part, having the formσ m withσ the shadow field ofψψ. Note that, as expected, there is no contribution from the unit operator. The latter provide contributions from operators of the formψ∂ 2m ψ that coincide with those coming from the discontinuity. The fermionic gap equation is the condition for the cancellation of the latter operators from the spectrum and it reads Discussion.-One of the messages of this work is that OPE inversion formulas can reveal the nontrivial dynamics of finite-temperature CFTs. In the simple examples we have studied, the dynamics effect a rearrangement in the operator spectrum which is ensured by the gap equations (22) and (27). An analysis of the gap equations shows that their solutions follow a pattern which, as we will argue below, is intimately related to the vacuum structure of scalar and fermionic theories near even dimensions.
In the bosonic case the gap equation (22) with the well-known solution In d = 5 the gap equation becomes [11] This has a complex conjugate pair of solutions given numerically by In fact, we find that for d = 3, 7, 11, . . . the bosonic gap equation (22)  The above pattern for the solutions of bosonic and fermionic gap equations for all odd d fits nicely with a renormalization-group understanding of universality classes of scalars and fermions in general dimensions. In the bosonic case the standard lore is that the large-N universality class for scalars in d = 2k + 1, k = 1, 2, . . . , is accessible via the ε expansion starting from d = 2k + 2. Using the general-d large-N results of [12][13][14] and [15][16][17][18][19], this has been verified in specific cases in [20,21] and [22]. The key ingredient in such studies is the Hubbard-Stratonovich transformation which introduces a field σ via the classically marginal interaction σφ 2 . This way σ has dimension ∆ σ = 2 in all d, and the scalars φ can be integrated out resulting in an effective potential for σ of the general form where g * is some critical dimensionless coupling. For general d the effective potential can also receive contributions from terms involving derivatives of σ, but the term σ to an unbounded potential, and hence to the absence of a global minimum, regardless of the sign of the Tr d log contribution. This matches exactly the pattern we see for m th . A real m th implies a global minimum, while a complex m th signals unstable local extrema with nonzero decay width.
In the fermionic case our results are consistent with the understanding that the corresponding large-N universal-ity classes in d = 2k + 1, k = 1, 2, . . . are also accessible via the ε expansion starting from a generalization of the Gross-Neveu-Yukawa model to d = 2k + 2 [23]. The corresponding Hubbard-Stratonovich transformation introduces the fieldσ via the classically marginal interactioñ σψψ. Hereσ has dimension ∆σ = 1 in all d, and integrating out the fermions leads to an effective potential of the form Notice that the Tr d log term enters with the opposite sign compared to the bosonic case. In this case the universal termσ d gives always a bounded from below contribution (recall d is even). However, the Tr d log term alters the form of the effective potential in d − ε. More specifically, for d = 4, 8, 12, . . . this term gives a negative contribution that dominates at infinity leading to an unstable vacuum structure, while for d = 6, 10, 14, . . . it gives a positive contribution that guarantees the presence of a global minimum. In either case there can be a number of unstable extrema. This matches exactly the pattern for them th solutions to the fermionic gap equations.
To summarize, OPE inversion formulas applied to CFTs in nontrivial geometries reveal crucial dynamical properties of critical systems at the level of the operator spectrum. The consistency of the lift to the nontrivial geometry requires that CFTs develop thermal masses that solve a gap equation. Remarkably, these thermal masses also encode information about the vacuum structure of CFTs even at zero temperature.
We would like to thank E. Perlmutter, J. Plefka, K. Siampos, and T. N. Tomaras for useful discussions and communications. ACP wishes to acknowledge the hospitality of the CERN Theory Division throughout the completion of this work.