M N Chernodub

Conformal anomaly and gravitational pair production

Conformal anomaly and gravitational pair production

M. N.

Introduction. Signatures of vacuum instability in a strong electric field were first found in work by Sauter on the Klein paradox [START_REF] Sauter | Uber das Verhalten eines Elektrons im homogenen elektrischen Feld nach der relativistischen Theorie Diracs[END_REF]. This effect has been recognized and developed further by Heisenberg and Euler [START_REF] Heisenberg | Consequences of Dirac's theory of positrons[END_REF] and later formalized in terms of a pair production process in QED by Schwinger [START_REF] Schwinger | On gauge invariance and vacuum polarization[END_REF][START_REF] Schwinger | The Theory of Quantized Fields. 5[END_REF].

The physical interpretation of this instability, often called the Schwinger effect, is linked to the quantum vacuum fluctuations in which virtual pairs of electrons e - and positrons e + are constantly created to be annihilated shortly later. In a sufficiently strong background electric field, the created e + e -particles are taken away in opposite directions by the field. As they are spatially separated, they cannot annihilate and become real particles. Thus, a sufficiently strong electric field creates matter (e + e -pairs) from the vacuum.

A similar phenomenon exists in gravitational fields near black holes. A black hole emits the Hawking radiation [START_REF] Hawking | Black hole explosions?[END_REF][START_REF] Hawking | Particle creation by black holes[END_REF], which can be associated with the particle tunneling process [START_REF] Maulik | Hawking radiation as tunneling[END_REF] in which one particle from the pair, created in the vicinity of the event horizon, gets swallowed by the hole while another particle has sufficient energy escape to infinity. The escaping particles form the outgoing energy flux, which diminishes the mass of the black hole and, therefore, leads to the black hole evaporation. Due to a nonlocality of the tunneling process, this effect operates in an extended vicinity above the black hole event horizon, thus creating the notion of the quantum atmosphere [START_REF] Steven | Hawking radiation, the Stefan-Boltzmann law, and unitarization[END_REF] (see also [START_REF] Eune | Proper temperature of the Schwarzschild AdS black hole revisited[END_REF]). It was recently suggested that such quantum atmospheres could possess nontrivial thermodynamic features that can be probed in condensed matter experiments [START_REF] Bermond | Anomalous Luttinger equivalence between temperature and curved spacetime: From black hole's atmosphere to thermal quenches[END_REF].

In 1+1 spacetime dimensions, the Hawking radiation can be related to a gravitational (Einstein) anomaly which implies a non-conservation of energy-momentum of a chiral particle in a curved spacetime [START_REF] Robinson | A Relationship between Hawking radiation and gravitational anomalies[END_REF]. The anomaly appears due to quantum fluctuations when classical symmetries are inconsistent with the quantization procedure [START_REF] Reinhold A Bertlmann | Anomalies in quantum field theory[END_REF]. In 1+1 and 3+1 spacetime dimensions, the Hawking effect can also be interpreted [START_REF] Christensen | Trace Anomalies and the Hawking Effect[END_REF] in terms of conformal (or trace) anomalies [START_REF] Duff | Twenty years of the Weyl anomaly[END_REF][START_REF] Capper | The one loop neutrino contribution to the graviton propagator[END_REF][START_REF] Capper | Photon corrections to the graviton propagator[END_REF] 1 .

In addition, one can argue that the particle creation in a static gravitational field can also produce particles even without the event horizon [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF]. In this scenario, the virtual pairs of particles are separated by local tidal forces and become real particles, similar to what happens in the Schwinger effect. Some of these real particles will fall to the gravitating body and will later be recaptured, while other particles will escape to infinity and create, similarly to the Hawking effect, an outgoing flux of matter [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF].

In our article, we argue that in the off-event-horizon mechanism of Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] of particle pair production, the creation rate in the background gravitational field can be directly related to the conformal anomaly. We will show that our approach works for the Schwinger pairproduction mechanism in QED and is consistent with the Savvidi vacuum instability in non-Abelian gauge theories [START_REF] Savvidy | Infrared Instability of the Vacuum State of Gauge Theories and Asymptotic Freedom[END_REF].

We set = c = 1 everywhere in the article and work in the mostly-plus metric convention.

Particle production and effective action. The rate density of particle production events dN/dt = Γ is determined by the imaginary part, Γ = 2 Im L eff , of the Lagrangian L eff associated with the effective action [START_REF] Schwinger | On gauge invariance and vacuum polarization[END_REF],

W = d 4 x √ -gL eff , (1)
which takes into account quantum corrections. The rate Γ has a sense of non-persistence of vacuum due to pair 1 It is worth mentioning about terminology used in the paper. The terms "scale" and "conformal" in relation to the symmetries of the system and the quantum anomalies are often used interchangeably in the literature. Mathematically, these concepts correspond to different symmetries as the requirement of local conformal invariance is much stronger than the condition of global scale invariance. Physically, the distinction between the scale and conformal concepts is frequently ignored because all physically relevant scale-invariant field theories in four spacetime dimensions also exhibit conformal invariance [START_REF] Nakayama | Scale invariance vs conformal invariance[END_REF]. Moreover, since these anomalies are seen as a non-zero trace of the energy-momentum tensor, they are also called "trace" anomalies.

creation [START_REF] Schwinger | On gauge invariance and vacuum polarization[END_REF][START_REF] Schwinger | The Theory of Quantized Fields. 5[END_REF].

In our paper, we argue that in conformal field theories, the pair-creation rate Γ in a background gravitational and gauge (electromagnetic or gluon) fields can be related to the conformal (trace) anomaly:

Γ = π 2 T µ µ , (2)
where T µ µ ≡ T µ µ an is the anomalous trace of the energy-momentum tensor T µν . Relation (2) is quantum because, in classical conformal theories in an even number of spacetime dimensions, the trace of the stressenergy tensor vanishes identically, (T µ µ ) cl ≡ 0. Quantum fluctuations can violate this identity, T µ µ = 0, hence the term "trace anomaly" or "conformal anomaly".

In order to keep Eq. ( 2) as simple as possible, we used the convention that this equation has a relation to the pair production if and only if T µ µ 0. Otherwise, Γ < 0 is equivalent to Γ ≡ 0 because a negative pair production rate does not lead to the production of pairs.

The particle production rate of N massless scalar degrees of freedom in the curved d = 3 + 1 dimensional spacetime (described by the metric g µν ) in the presence of the classical electromagnetic field (characterized by the field strength F µν ) has been found in the recent work [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] 2 :

Γ N sc = N 32π 1 180 R µναβ R µναβ -R µν R µν (3)
+ 1 2 1 6 -ξ 2 R 2 - q 2 12 F µν F µν ,
where the curved background is expressed via the Riemann tensor R µναβ , the Ricci tensor R µν = R α µαν , and the scalar curvature R ≡ R µ µ . The subscript "N sc" in Eq. (3) stands for N scalar degrees of freedom (for example, N = 1 for a neutral scalar field and N = 2 for a complex scalar field).

The quantity q in Eq. ( 3) is the electric charge of the scalar particle minimally coupled to electromagnetism. A neutral (q = 0) scalar field carrying one degrees of freedom (N = 1) is described by the following Lagrangian:

L = - 1 2 ∂ µ φ∂ µ φ - 1 2 ξRφ 2 - 1 2 m 2 φ 2 , (4)
where the parameter ξ controls the local coupling of the Ricci curvature scalar R to the scalar field. The conformally invariant massless theory corresponds to ξ = 1/6. For consistency with previous studies, we also add to Eq. (4) the mass term, which will be set to zero at the end of our considerations, m = 0. The charged (complex) scalar field carrying the elementary electric charge q = e has N = 2 degrees of freedom with corresponding modifications of Eq. ( 4). The remarkable feature of the pair-production effect ( 3) is that it can take place in static gravitational fields, which immediately suggests that this effect is a Hawking-type of radiation associated with the presence of an event horizon [START_REF] Hawking | Black hole explosions?[END_REF][START_REF] Hawking | Particle creation by black holes[END_REF]. However, the pair production (3) takes place even in the absence of an event horizon (that is, not only for a black hole), thus indicating that this phenomenon is either an addition or a generalization of Hawking radiation [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF].

Effective action, trace anomaly, and pair production. It is instructive first to start from the simplest case of the scalar field for which the trace anomaly has been elaborated in great detail in Ref. [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF]. Our relation (2) between the conformal (trace) anomaly and the off-event-horizon particle production can be deduced by matching the anomalous term in the one-loop effective action W represented as an integral over the proper time s of Ref. [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF] with the representation of the same action in terms of the spectral parameter s in the heat-kernel approach of Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] based on the Barvinsky-Vilkovisky expansion [START_REF] Barvinsky | Covariant perturbation theory. 2: Second order in the curvature. General algorithms[END_REF].

The one-loop action functional W is given by a formal divergent expression, W = (i/2) ln det G -1 , where G(x, x ) ≡ iT φ(x)φ(x ) represents the Green function associated with the quadratic Lagrangian (4):

- 1 √ -g ∂ µ √ -gg µν ∂ ν + ξR + m 2 G(x, x ) (5)
= δ(x -x )/ √ -g .

The functional W has a close relation to the expectation value of the energy-momentum tensor, T µν , in its response, W → W + δW to the metric variation, g µµ → g µν + δg µν in (even) D spacetime dimensions:

δW = i 2 Tr GδG -1 = d D x √ -g T µν 1 2 δg µν , (6)
thus giving access to the evaluation of the trace T µ µ , allowing us to uncover an eventual conformal (trace) anomaly.

The variation of the effective action ( 6) can be expressed in the proper-time representation of Schwinger and DeWitt [START_REF] Schwinger | On gauge invariance and vacuum polarization[END_REF][START_REF] Dewitt | Quantum field theory in curved spacetime[END_REF] (in notations of [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF]):

δW = - i 2 δTr ∞ 0 ids is e -isH , (7)
via a relativistic Hamiltonian-like operator H = ∆ + ξR + m 2 , where a second-order differential operator ∆ represents the kinetic term, the coupling to the curvature R plays a role of an external potential, and m 2 gives the mass term. The correct analytical properties of Eq. ( 7) and similar subsequent relations are maintained by an appropriate complex continuation of the mass term, m 2 → m 2 (1 -i0 + ), silently assumed here. The effective Lagrangian (1) takes the following form:

L eff = 1 2 1 (4π) D 2 ∞ 0 ids (is) 1+ D 2 e -im 2 s F (x, x; is; D) , (8)
where F (x, x ; is; D) is the weight bi-scalar in the propertime Green's function x, s|x , 0 = x|e -isH |x defined in a manner similar to Eq. ( 8). The Green's function satisfies the Schrödinger-like equation: -∂ ∂is x, s|x , 0 = H x, s|x , 0 , which gives a quantum-mechanical flavor to the whole proper-time formalism.

We will not dwell on the precise definition of the biscalar F , which can be found in detail in Refs. [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF][START_REF] Dewitt | Quantum field theory in curved spacetime[END_REF].

The key mathematical point of our arguments is that the function F allows for the power-series expansion in terms of the proper time s (omitting other arguments):

F = 1 + is f 1 + (is) 2 f 2 + . . . , (9)
where, in four space-time dimensions, the O(s 2 ) term captures the trace anomaly [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF]. On the other hand, the O(s 2 ) term in an identical 3 expansion of the same 1-loop effective action over the proper time s has been shown in Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] to be associated with the (off-event-horizon) pair-production rate Γ. The mentioned equivalence of the O(s 2 ) terms allows us to identify the trace-anomalous origin of the pair production and eventually leads us to Eq. (2) as we discuss below. The renormalized energy-momentum tensor,

T µν ren = 1 4 A 4 g µν + non-anomalous part, (10)
contains the anomalous part given by an A 4 function and a non-anomalous part (not shown explicitly). In D = 4 spacetime dimensions, the A 4 function in stress-energy tensor [START_REF] Bermond | Anomalous Luttinger equivalence between temperature and curved spacetime: From black hole's atmosphere to thermal quenches[END_REF] is related to the O(s 2 ) prefactor in the power series expansion (9) of the bi-scalar F [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF]:

A 4 = 1 2 1 (4π) 2 ∂ ∂is 2 e -im 2 s F (x, x; is, 4) s=0 . ( 11
)
3 Taking into account the signs and i-th prefactors arising from the difference between Minkowski/Euclidean spacetimes employed in Refs. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF][START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF] one finds that f 1 = ( 1 6 -ξ)R term in Eq. (A20) of the proper-time approach of Ref. [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF] coincides precisely with the second, O(s) term under the integral in Eq. (S.17) of the heatkernel expansion of Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF]. Analogously, f 2 in Eq. (A24) of Ref. [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF] coincides precisely with purely gravitational contribution to the third, O(s 2 ) term under the integral in Eq. (S.17) of Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF]. The f 2 term in series ( 9) is also reproduced, up to irrelevant contact term R, by the m = 0 expression in the square brackets of our Eq. ( 13) below. Notice that our functions fa in Eq. ( 9) correspond to fa of Ref. [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF] and not to fa of Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF].

In the massless theory (m = 0), the last term in Eq. ( 10) reduces to a traceless tensor, and the trace of the energymomentum tensor [START_REF] Bermond | Anomalous Luttinger equivalence between temperature and curved spacetime: From black hole's atmosphere to thermal quenches[END_REF] is fully determined by the trace (scale) anomaly [START_REF] Robinson | A Relationship between Hawking radiation and gravitational anomalies[END_REF]:

T µ µ ≡ g µν T µν ren = A 4 , [for m = 0] , (12)
where the short-hand notation T µ µ is used for representational convenience (see also the discussion of Ref. [START_REF] Duff | Twenty years of the Weyl anomaly[END_REF] on non-commutativity of the regularization operation and the trace operation).

Finally, combining Eqs. ( 9), ( 11), [START_REF] Reinhold A Bertlmann | Anomalies in quantum field theory[END_REF] and matching them with the O(s 2 ) term in the effective action of Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] leads us to our main result [START_REF] Heisenberg | Consequences of Dirac's theory of positrons[END_REF]. In the rest of the paper, we ensure that Eq. ( 2) is valid for physical environments where both sides of this equation are known. We also discuss photon and neutrino pair production.

A neutral scalar field in curved spacetime. As the first check of our result (2), we consider a singlecomponent neutral scalar field of mass m and generic non-conformal coupling ξ to gravity described by Lagrangian (4). It is well known that quantum fluctuations in this theory produce the following trace anomaly [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF][START_REF] Deser | Non-local conformal anomalies[END_REF]:

T µ µ 1sc = 1 (4π) 2 1 180 R µναβ R µναβ - 1 180 R µν R µν (13) + 1 6 1 5 -ξ R + 1 2 1 6 -ξ 2 R 2 + 1 2 m 4 .
We substitute Eq. ( 13) to our formula (2) and recover exactly the result of Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] given in Eq. ( 3) for the offhorizon pair creation rate for a single (N = 1) neutral (q = 0) massless (m = 0) scalar field 4 . The subscript "1sc" in Eq. [START_REF] Christensen | Trace Anomalies and the Hawking Effect[END_REF] stresses that this expression is derived for a one-component scalar field. General case. A quantum field theory of N S scalar degrees of freedom, N F Dirac fermions (a single Majorana or Weyl fermion contributes half of a Dirac fermion, N F = 1/2) and N V species of massless vector fields, the trace anomaly gets the following form [START_REF] Duff | Twenty years of the Weyl anomaly[END_REF][START_REF] David Birrell | Quantum fields in curved space[END_REF][START_REF] Iosif | Introduction to quantum field theory with applications to quantum gravity[END_REF]:

T µ µ = bC 2 + b E 4 + cF µν F µν , (14)
where

C 2 = R µναβ R µναβ -2R µν R µν + R 2 3 , (15)
is the Weyl tensor squared and

E 4 = R µναβ R µναβ -4R µν R µν + R 2 , ( 16
)
4 These equations correspond to the same physical result since the R term, present in Eq. ( 13) and absent in Eq. ( 3), can be removed by a finite local counterterm during the renormalization procedure and is, therefore, physically irrelevant.

is the Euler density (the integrand of the topological, Gauss-Bonnet term) in D = 4 dimensions. In Eq. ( 14), the physically irrelevant R term is omitted 5 and the conformal coupling (ξ = 1/6 for scalars) is assumed. The parameters are as follows [13-16, 20, 26]:

b = 1 120 1 (4π) 2 (N S + 6N F + 12N V ) , (17) b
= - 1 360 1 (4π) 2 (N S + 11N F + 62N V ) . (18)
As an immediate check, one finds that in a pure gravitational background (F µν = 0), N S = N species of neutral scalar fields (with N D = N V = 0), Eq. ( 14) expectedly reduces to Eq. ( 13) with a factor N and leads us, via Eq. ( 2), to the recent result (3). The last term ( 14) represents a non-universal ("matter") part which accounts for renormalization effects related to the scale dependence of the couplings of the theory. While Eq. ( 14) gives the matter term for a vectorfield background, a nontrivial scalar background in an interacting scalar field theory can also generate a mattertype contribution to the conformal anomaly which can be found in Ref. [START_REF] Asorey | Trace anomaly and induced action for a metric-scalar background[END_REF].

For gauge vector fields coupled minimally with matter fields via the electric coupling e, the prefactor

c = - β(e) 2e , (19)
of the last term in Eq. ( 14) depends the beta function β(e) = µ de/dµ associated with the running of the coupling e. A nonvanishing beta function expresses the fact that radiative corrections make the electric charge e = e(µ) dependent on the renormalization energy scale µ. This effect appears as a result of vacuum polarization which implies, for example, that the electric charges of a particle probed at a large distance (by a low-energy photon) does not match the charge of the same particle at a short distance (probed by a high-energy photon). Therefore, radiative corrections can break the scale invariance of the system and naturally contribute to the scale anomaly [START_REF] Duff | Twenty years of the Weyl anomaly[END_REF]. Notice that the gravitational part of the anomaly represented by the first two terms in Eq. ( 14) is exact in one loop implying that the higher-order corrections to this expression vanish. This statement is not true for the third, matter term since radiative corrections exist, generally, in all loops [START_REF] Shifman | Anomalies and Low-Energy Theorems of Quantum Chromodynamics[END_REF].

Scalar QED in flat spacetime. A similarity between the gravitational particle production and the Schwinger pair production in flat spacetime in the background electric field has been noticed in Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] on the 5 See also a relevant discussion on R in Ref. [START_REF] Duff | Twenty years of the Weyl anomaly[END_REF].

basis of equation ( 3). Here we show that the conformal anomaly plays an essential role in this relation.

Consider a theory of N S species of massless complex scalars coupled to electromagnetism with the same electric coupling e (a "scalar Quantum Electrodynamics" or sQED). Since we already established the relation with the gravitational part of the trace anomaly in this theory, we consider below a flat spacetime where first two (gravitational) terms in the trace ( 14) vanish. However, this theory still possesses the trace anomaly because its one-loop beta function is non-zero [START_REF] Weisskopf | The electrodynamics of the vacuum based on the quantum theory of the electron[END_REF][START_REF] Dunne | Heisenberg-Euler effective Lagrangians: Basics and extensions[END_REF],

β 1loop sQED = N S e 3 48π 2 . ( 20
)
Equations ( 19) and ( 20) imply that the coefficient in the last term of the trace anomaly ( 14) is c = N S e 2 /(96π 2 ). Then Eq. ( 2) gives us Γ sQED = -N S e 2 F µν F µν /(192π), which exactly coincides with the pair production rate (3) of Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] if one takes into account that each complex field carries two degrees of freedom: N = 2N S . According to our convention in Eq. ( 2), there is no particle production for a negative production rate. Since F µν F µν = 2(B 2 -E 2 ), implies the absence of particle creation in a pure magnetic field because Γ sQED < 0. However, in the electric-field background, one gets the following well-known result the complex scalar field (reproduced also in Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] for N S = 1):

Γ sQED = N S e 2 E 2 96π . (21)
This equivalence further supports the validity of Eq. ( 2) in one loop.

Spinor QED. The pair creation rate in the (spinor) QED with a single flavor of massless Dirac fermion and a single gauge (electromagnetic) field can be derived with the use of correspondence (2) together with the anomaly relations ( 14)-( 19) by setting N S = 0, N F = 1, and N V = 1. Taking into account that the one-loop beta function of the spinor QED is four times bigger (per particle) than its scalar analogue [START_REF] Dewitt | Quantum field theory in curved spacetime[END_REF] [START_REF] Dunne | Heisenberg-Euler effective Lagrangians: Basics and extensions[END_REF]:

β 1loop QED = e 3 12π 2 , (22)
one gets the following prediction for the particle production rate:

Γ (m=0) QED = 1 11 520π -19R µναβ R µναβ + 184R µν R µν (23)
-55R 2 - e 2 48 F µν F µν ,
Notice that in flat background, the particle production rate in the massless QED [START_REF] Deser | Non-local conformal anomalies[END_REF] reduces exactly to the well-known QED result [START_REF] Schwinger | On gauge invariance and vacuum polarization[END_REF][START_REF] Dunne | Heisenberg-Euler effective Lagrangians: Basics and extensions[END_REF] in the limit m → 0:

Γ (m=0) QED = e 2 E 2 24π [electromagnetic] . (24)
To estimate the contribution of the gravitational part to the production rate, consider now a purely gravitational background given by the static Schwarzschild spacetime of a body with the mass M :

ds 2 = -1 - 2M G r dt 2 + 1 - 2M G r -1 dr 2 + r 2 (dθ 2 + sin 2 θdϕ 2 ) (25)
Given Ricci flatness (R µν = 0) of this metric, the gravitational contribution to the pair-creation rate ( 23) is provided only by the Kretschmann scalar:

K = R µναβ R µναβ = 48 G 2 M 2 r 6 . (26)
Thus, it appears that the purely gravitational contribution to the pair production rate ( 23) is always negative

δΓ (m=0) QED = - 19 240π G 2 M 2 r 6 [gravitational] , (27)
implying that our conformal anomaly mechanism alone cannot create particles outside of the event horizon even in the presence of the strong gravitational field and even for massless QED. Moreover, our result imply that the ordinary Schwinger pair production due to background electric field (24) will be inhibited by the gravitational contribution ( 27) in curved spacetime.

It is worth here mentioning that the considerations of Ref. [START_REF] Duff | Twenty years of the Weyl anomaly[END_REF] on the sense of the conformal anomaly in the context of renormalization of quantum field theories suggest that in conformally-non-invariant theories (for example, for massive fields), the right-hand-side of Eq. ( 2) should be modified: T µ µ → g µν T µν ren -g µν T µν ren . This conjecture implies, in particular, that the explicitly non-conformal mass term m 4 will not enter Eq. [START_REF] Heisenberg | Consequences of Dirac's theory of positrons[END_REF].

Coming back to the massless case in flat spacetime, the proportionality of the pair-creation rates for scalar [START_REF] Barvinsky | Covariant perturbation theory. 2: Second order in the curvature. General algorithms[END_REF] and spinor [START_REF] Deser | Non-local conformal anomalies[END_REF] QED to their beta functions, Eqs. [START_REF] Brown | Stress-tensor trace anomaly in a gravitational metric: Scalar fields[END_REF] and [START_REF] Dewitt | Quantum field theory in curved spacetime[END_REF], respectively, is not surprising given an intimate relation between the effective Euler-Heisenberg Lagrangian and the beta function (for an excellent review, see Ref. [START_REF] Dunne | Heisenberg-Euler effective Lagrangians: Basics and extensions[END_REF]). Since the beta function also contributes to the trace anomaly, the relation of the trace anomaly to the pair-creation rate closes the logical triangle, thus qualitatively supporting Eq. ( 2) on physical grounds.

Our results [START_REF] Heisenberg | Consequences of Dirac's theory of positrons[END_REF] suggest that in the flat spacetime, the creation rate of pairs of massless particles in classical (electromagnetic) background is related to the respective beta function:

Γ flat = - πβ(e) 4e F µν F µν . ( 28
)
This result should be valid at least in one-loop order with the already mentioned reservation that a negative production rate implies no production.

No photon production. In realistic QED in weak background electromagnetic fields with the strength below the Schwinger limit, the four-photon scattering can be neglected [START_REF] Dunne | Heisenberg-Euler effective Lagrangians: Basics and extensions[END_REF], so that photon propagation can be described by free Maxwell theory with simple Lagrangian, L ph = -(1/4)F µν F µν . The particle creation rate corresponds to N F = 0, N S = 0 and N V = 1 and gives us the following discouraging result Γ ph = -13G 2 M 2 /(120πr 4 ) < 0 implying that no photons can created in the gravitational field due to this conformal anomaly mechanism.

Neutrino-anti-neutrino pairs. Similar considerations can also be applied to neutrino-anti-neutrino pair creation with the appropriate replacement of the spinor degrees of freedom by the sum of Dirac and Majorana/Weyl neutrino species: N D → N ν = N D +(1/2)N M and taking N V = N S = 0 in the above expressions. One gets Γ ν = 7G 2 M 2 /(240πr 4 ) > 0, so that the pairs of light neutrino can potentially be created by sufficiently strong background gravitational field. For a Dirac neutrino, the pair production rate Γ ν is 7/2 times bigger than the rate of pair production for scalar particles which is about two times bigger than the one for the Hawking radiation. The relevant estimations for scalar particles in physically interesting gravitational fields can be found in Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF].

Finally, one could ask whether these results, derived for massless spinors, are applicable to particles with mass m ν . For the pure electromagnetic contribution to the pair creation rate ( 23), the condition is well known [START_REF] Dunne | Heisenberg-Euler effective Lagrangians: Basics and extensions[END_REF]: the electric field strength should substantially exceed the critical Schwinger field, E E Sch c = m 2 e /e 1.3 × 10 18 V/m. Likewise, the same condition can be obtained by demanding that the gravitational contribution should exceed the anticipated6 ∝ m 4 e term generated by the explicit breaking of the conformal symmetry. For neutrinos in the field of a gravitating body with mass M , the applicability condition then reads as r r c with the critical field r c = √ GM /m ν up to a O(1) factor. Savvidi magnetic instability in QCD. In the pure magnetic field, both in scalar QED and spinor QED, the right-hand of Eq. ( 2) is a negative quantity and, therefore, no instability associated with the particle production can occur. Of course, this natural conclusion is supported by the fact that their beta functions, Eqs. ( 20) and [START_REF] Dewitt | Quantum field theory in curved spacetime[END_REF], are positively defined. But what happens if the beta function is negative?

Consider, for example, Yang-Mills (YM) theory which determines non-perturbative properties of Quantum Chromodynamics (QCD). The beta function of N c -color YM theory, β YM (g) = -11N c g 3 /(48π 2 ), is a negative function of the strong coupling constant g. Adopting Eq. ( 28) to non-Abelian fields possessing the field-strengths F a µν , one gets the following formal perturbative expression for the gluon production rate:

Γ pert YM = 11N c g 2 192π F a µν F a,µν ≡ 11N c 96π (gB a ) 2 -(gE a ) 2 , (29)
where the sum over the gluon species, a = 1, . . . , N 2 c -1, is implicitly assumed.

Equation ( 29) represents a formal expression which is not applicable to the ground state of YM theory because Eq. ( 29) corresponds to the anomalous breaking of scale symmetry associated with the perturbative renormalization of couplings -hence the superscript "pert" in Eq. ( 29) -while in YM theory, the conformal symmetry is broken dynamically and non-perturbatively 7 . Despite this fact, Eq. ( 29) still allows us to make another interesting relationship with already known effect: the instability of the perturbative gluonic vacuum. Indeed, since Γ YM = 11N c (gB a ) 2 /(96π) > 0, even the weakest background gluomagnetic field leads to the creation of gluon pairs and makes the gluonic vacuum unstable. This observation matches well with the instability of the perturbative gluon vacuum [START_REF] Nielsen | An Unstable Yang-Mills Field Mode[END_REF] which drives creation of the magnetic condensate (the Savvidi vacuum [START_REF] Savvidy | Infrared Instability of the Vacuum State of Gauge Theories and Asymptotic Freedom[END_REF]) and the formation of the suggested magnetic-spaghetti vacuum state [START_REF] Bech Nielsen | A Quantum Liquid Model for the QCD Vacuum: Gauge and Rotational Invariance of Domained and Quantized Homogeneous Color Fields[END_REF][START_REF] Ambjorn | On the Formation of a Random Color Magnetic Quantum Liquid in QCD[END_REF] precisely due to the negativeness of the YM beta function, β YM (g) < 0 (see also a related critical discussion in Ref. [START_REF] Yildiz | Vacuum behavior in quantum chromodynamics[END_REF]).

Conclusions. We suggested the simple formula (2) for the off-horizon particle production rate in curved spacetime proposed in Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] and argued that its underlying mechanism is based on the anomalous breaking of the conformal symmetry. The anomalous particle production can occur in static gravitational fields and can operate, in particular, above the event horizons of black holes. These two properties discriminate the anomalous production from the dynamical Casimir effect in timedependent backgrounds and the Hawking mechanism of particle production, which occurs near the black-hole horizons.

Our formula (2) agrees with known results for the pair production rate in the gravitational background for a scalar field presented recently in Ref. [START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] where the offhorizon pair production has been suggested first. We also reproduce the known expressions for the Schwinger pair production in QCD with scalar and spinor particles. Our approach also supports instability in the perturbative gluonic vacuum in the chromomagnetic field, thus suggesting the formation of the magnetic condensate in 7 In QCD, the magnitude of the dynamical breaking of conformal symmetry, T µ µ Λ 4 QCD , is determined by an intrinsic mass scale Λ QCD of the order of a few hundred MeV.

accordance with widely accepted ideas about the nature of the QCD vacuum.

Our mechanism suggests that the photon pairs cannot be produced in a static Schwarzschild spacetime. However, our estimations show that a sufficiently strong gravitational field can create pairs of neutrinos and antineutrinos (as well as other light spinors), thus providing us with another channel for the evaporation of black holes and other gravitating objects.

We have slightly re-arranged and combined the original expressions of Ref.[START_REF] Michael | Gravitational pair production and black hole evaporation[END_REF] for further convenience.

Cf. the last term in Eq. (13).