Suppression of Quantum Corrections by Classical Backgrounds

We use heat-kernel techniques in order to compute the one-loop effective action in the cubic Galileon theory for a background that realizes the Vainshtein mechanism. We find that the UV divergences are suppressed relative to the predictions of standard perturbation theory at length scales below the Vainshtein radius.


Introduction
Higher-derivative theories are perturbatively nonrenormalizable. As a result, their predictivity is limited by the necessity to introduce an infinite number of counterterms in order to cancel the ultraviolet (UV) divergences appearing in the quantum corrections. Such theories can still be treated as effective below an energy scale Λ suppressing the couplings in the nonrenormalizable terms. If the UV completion of the theory at the scale Λ is not known, one must include all the effective terms allowed by the low-energy symmetries. Despite these general expectations, it is still possible that the UV behavior of the theory may be improved through a rearrangement of the perturbative expansion, or at the nonperturbative level. For example, one could incorporate some of the higher-derivative terms in an effective propagator. In Fourier space the propagator would then fall much faster than the standard one for increasing momenta, so that the UV divergences could be reduced or eliminated. However, this approach does not have internal consistency [1]. The additional terms incorporated in the propagator become relevant near the UV scale Λ. It is impossible to justify the exclusion of terms with even more derivatives, which could give larger contributions near Λ.
We are interested in a different aspect of the quantum theory: The possibility that the classical background around which the fields are expanded can reduce the magnitude of quantum corrections. This scenario makes sense only for inhomogeneous backgrounds, as in the opposite case the effect amounts to a simple redefinition of scales. A specific example we have in mind involves the cubic Galileon theory, which describes the dynamics of the scalar mode that survives in the decoupling limit of the DGP model [2]. The action contains a higher-derivative term, cubic in the field π(x), with a dimensionful coupling that sets the scale Λ at which the theory becomes strongly coupled. The tree-level action in Euclidean space is with ν ∼ 1/Λ 3 . The action is invariant under the Galilean transformation π(x) → π(x)+b µ x µ +c, up to surface terms. Despite the presence of four derivatives in the second term, the equation of motion is a second-order partial differential equation. This property, which guarantees the absence of ghosts in the spectrum in the trivial vacuum, is also preserved within the Galileon theory, which includes a finite number of higher-order terms [3]. The Galileon theory can provide a realization of the Vainshtein mechanism, which has been introduced in order to suppress the propagation of the physical mode of the massive graviton that survives in the limit of vanishing mass [4]. The cubic theory of eq. (1) has a spherically symmetric solution π cl = π cl (w), with w = r 2 , given by where the prime denotes a derivative with respect to w and we have assumed that c, ν > 0. For w ≫ w V , with w V = r 2 V ∼ (νc) 2/3 the square of the Vainshtein radius, the solution is π ′ cl ∼ c w −3/2 , so that π cl ∼ c/r. On the other hand, for w ≪ w V , we have π ′ cl ∼ c/ν w −3/4 , so that π cl ∼ c/ν √ r. This solution requires the presence of a large point-like source at the origin, with strength depending on c. The classical fluctuations δπ of the field around a general background π cl obey the linearized equation ∆ cl δπ = 0, with the operator evaluated for π = π cl . (We employ covariant notation, even though we work in Euclidean space.) For the background (2), the first term dominates at distances much larger than the Vainshtein radius, so that the fluctuations δπ propagate as free waves. On the other hand, the dominance of the last two terms at distances smaller than the Vainshtein radius, where ν π cl ≫ 1, results in the suppression of the classical fluctuations. In ref. [5] it was argued that the same mechanism can lead to the suppression of quantum fluctuations as well, thus reducing the effect of quantum corrections at the scales at which the Vainshtein mechanism operates. The essence of the argument is that the higher-derivative terms generate a large effective wavefunction renormalization Z for the fluctuation δπ. If this can be absorbed in the definition of a canonically normalized field, the couplings of the theory are reduced by powers of Z. Even though this intuitive argument seems reasonable, it is not rigorous because of the position dependence of Z in the background (2). In this work we introduce an appropriate modification of the heat-kernel calculation of the one-loop corrections in order to examine the issue through a more rigorous approach.

Perturbation theory
Our task is to evaluate the one-loop effective action where the operator ∆ is given by eq. (3). Following ref. [6], we calculate the heat kernel of ∆ through the relation The effective action can be obtained from the diagonal part of the heat kernel as A lower limit has been introduced for the ǫ-integration in order to regulate the possible UV divergences.
The higher-derivative terms in the effective action are generated through the expansion of the exponential in eq. (5). The operators act either on functions, such as π, appearing in ∆, or on exp(ikx). An efficient way of carrying out the expansion is implied by the analysis of ref. [6]. The integrant of eq. (5) can be viewed as an operator acting on an arbitrary function f (x). After the expansion of the exponential is performed, one sets f (x) = 1 in order to retain only the terms that are relevant for the evaluation of the heat kernel. In this process we employ the operator identities In order to determine the UV divergences, which appear for ǫ → 0, it is useful to rescale k in eq. (17) by √ ǫ, as was done in ref. [6]. The diagonal part of the heat kernel becomes with the implicit assumption that it will be evaluated through its action on f (x) = 1.
The standard procedure is to isolate the term exp(−k 2 ) and expand the rest of the exponential. The k-integration can be performed with the help of formulae such as The results of perturbation theory are obtained through a double expansion in ν and ǫ. For a given power of ν, the lower-order terms in ǫ, up to ǫ 2 , reproduce the UV divergences of the effective action. For example, at order ν 2 the leading divergence is associated with a term ∼ ( π) 2 . The corresponding diagonal part of the heat kernel is and the contribution to the effective action The heat-kernel analysis is consistent with the expectations for the quantum corrections in the Galileon theory. The structure of the divergent terms in the one-loop effective action is, schematically, [5,7,8] The result (12) reproduces the leading divergence in eq. (13) for m = 2. An explicit calculation, carried out in ref. [9] for the theory of eq. (1) through dimensional regularization, reproduced the logarithmic term. A similar calculation of all the terms with m = 2 was performed in ref. [10]. The problem with the effective action (13) is that it cannot be trusted in the region below the Vainshtein radius, where ν π cl ≫ 1. If we expand the field as π = π cl (x) + δπ(x) in eq. (13), with the perturbation δπ assumed to be small, ν π cl would act as an effective expansion parameter. For example, a series of interaction terms ∼ ν 2 Λ 4 (ν π cl ) n ( δπ) l would be generated. For the series in n to converge, ν π cl should be smaller than 1. In the opposite case, the UV divergences of the theory seem to be enhanced by the presence of the background. We shall reformulate the calculation of the heat kernel in a way that accounts more efficiently for the influence of the background for large values of ν π cl . As we shall see, the weak point of this approach is the loss of explicit invariance under the Euclidean group. This is not a fundamental issue. For example, for π = π cl + δπ the correction (12) would result in a term linear in δπ, of the form ∼ Λ 4 ( π cl )( δπ). The original symmetry is not apparent for a general background π cl , even though it is obvious in eq. (12).

The effect of the background
In order to investigate the effect of the background on the UV divergences, we split the field in eq. (9) as π = π cl + δπ and consider the correlation functions of δπ. We define a "metric" and the operators The exponent in eq. (9) becomes We have seen that the expansion of the heat kernel in powers of ǫ reproduces the UV divergences of the theory. The combination ν π cl in eq. (17) is the classical expansion parameter, visible also in eq. (13). This parameter becomes large below the Vainshtein radius [5], which means that it does not generate a convergent series. On the other hand, δπ can be viewed as a second expansion parameter, apart from ǫ, with the term ν δπ assumed to be small. Within this scheme, the "metric" G µν includes the terms of zeroth order both in √ ǫ and δπ. All such terms must be treated on equal footing, and this is accomplished by our way of evaluating the heat kernel. The main technical difficulty is that the momentum integration in eq. (9) cannot be performed easily for general G µν . In calculations of the heat kernel on gravitational backgrounds, one usually employs Gauss normal coordinates that simplify the quadratic term. The perturbative expansion of the heat kernel results in corrections invariant under diffeomorphisms, so that the choice of coordinate system does not play any role. Our problem is of a different nature. The theory we are considering does not possess an invariance under transformations of the "metric" G µν . Thus we are forced to diagonalize the quadratic term explicitly. This is not feasible for a general G µν , but is possible for the one resulting from a spherically symmetric background π cl (r 2 ). The drawback is the loss of explicit invariance of the final result under the Euclidean group.
In eq. (17) we can render the "metric" G µν trivial by rescaling the momenta as Through differentiation of this relation, x-derivatives of G µν can be expressed in terms of derivatives of S µ ν . The first term of eq. (17) now takes the simple form −k ′2 . It is not possible, however, to isolate trivially a term exp(−k ′2 ) in the heat kernel and expand the rest of the exponential. The reason is that k ′µ does not commute with the derivative operators in must be employed, with X = −G µν k µ k ν and Y consisting of the remaining terms in eq. (17). Then, each of the exponentials, apart from the first one, must be expanded, the momenta k rescaled and the k ′ -integrations carried out. This procedure is very cumbersome and we postpone a detailed presentation for a future publication. We focus here on the leading divergence in the effective action, which can be obtained by observing that the ǫ-independent terms in eq. (17) do not include derivative operators and commute with −G µν k µ k ν . As a result, the contribution to the diagonal part of the heat kernel which is quadratic in δπ and contains the quartic divergence is where we have dropped the prime on k.
For a spherically symmetric background π cl = f (r 2 ), we consider a Cartesian system of coordinates with one of its axes along the radial direction. We obtain where the first entry corresponds to the time component, the second to the radial, and the last two to the components perpendicular to the radial. We easily find that The Jacobian determinant of the transformation is det S. After performing the momentum integration, the contribution to the diagonal part of the heat kernel can be put in the form where Using eq. (6), we find the contribution to the effective action It is apparent from eq. (27) that the invariance under the Euclidean group is broken by the background. For a homogeneous background, for which S is the four-dimensional unit matrix, eq. (27) reproduces eq. (12). On the other hand, if the effective action is evaluated around the background of eq. (2), the effective Lagrangian density has a very strong radial dependence. In order to obtain a pictorial representation of the r-dependence, we observe that the functions P , V µν , W µνρσ involve fourth powers of the matrix S and are also proportional to its determinant. In fig. (1) we display the product of the determinant of S and the fourth power of each of its diagonal elements for a background given by eq. (2) with ν = 1 and c = 10 6 . All dimensionful quantities are measured in units of the fundamental scale Λ. The Vainshtein radius is r V ∼ (νc) 1/3 = 100. It is apparent that the quantum corrections are suppressed below r V . We estimate that (det S) S i i 4 , with i = 0, 1, 2 or 3, scales as r 6 /(νc) 2 ∼ (r/r V ) 6 , a behavior that is verified by fig. (1). A substantial suppression, by several orders of magnitude, is expected for small r V > ∼ r > ∼ 1/Λ.
Apart from the term we considered, there is an infinite number of higher-derivative terms with possible UV divergences. These result from the expansion of the exponential in eq. (9) in the way we described in the beginning of this section. As we mentioned earlier, the calculation of the effective action is very complicated because of the breaking of Euclidean invariance by the background. However, certain features are apparent: • The momentum integration factor in the heat kernel generates a factor ǫ −2 after the rescaling, while the relation to the effective action involves the integration factor dǫ/ǫ. This means that the divergences in the effective action result from terms in the expansion of the exponential of (17) with powers of ǫ up to 2.
• The suppression of the quantum corrections by the background arises through the matrix S that rescales the momenta. In the region where the Vainshtein mechanism operates, the elements of S have typical values ∼ |ν π cl | −1/2 ∼ (r/r V ) 3/4 ≪ 1. The Lagrangian density is also multiplied by an overall suppression factor det S ∼ |ν π cl | −2 ∼ (r/r V ) 3 , arising from the Jacobian determinant.
• Any power of k 2 δπ, resulting from the expansion of the exponential of (17) is multiplied by the same power of S 2 after the rescaling of k is performed. In the context of standard perturbation theory, UV divergent terms ( δπ) l would be enhanced by the presence of the background, as we discussed at the end of section 2. Within our scheme, they are suppressed by powers of S.
• A possible enhancement is generated by the factor ∼ |ν π cl | ∼ (r V /r) 3/2 ≫ 1 in eq. (17). However, terms involving this factor also include powers of ǫ from the operators D ǫ and L µν ǫ . As the power of ǫ cannot exceed 2 (for a logarithmic divergence), the maximal enhancement is limited to a multiplicative factor ∼ |ν π cl | 4 ∼ (r V /r) 6 . This is always overcompensated by the powers of S.
• A further enhancement can be possibly generated in the expansion of the exponential when derivative operators incorporated in D ǫ and L µν ǫ act on G µν . Again, this enhancement is limited by the requirement that ǫ do not exceed 2.
The general conclusion that can be reached by the above considerations is that the one-loop divergences of the theory are not enhanced by arbitrary powers of ν π cl as the perturbative result (13) would imply. Moreover, suppression factors are generated through the modified calculation of the heat kernel that we employed. They can be viewed as a result of the strong wavefunction renormalization induced by the background. We demonstrated the strong suppression of the quartically divergent term, quadratic in the field, through an explicit calculation. In the following section we sketch an efficient approach for studying all the other divergent terms.

Discussion
Our analysis leads to the remarkable conclusion that the one-loop quantum corrections can be suppressed in certain regions of an inhomogeneous background. This is a feature not encountered in renormalizable theories. For example, one may consider a domain-wall background in a renormalizable scalar theory with a double-well potential. The background will influence the quantum corrections through the effective mass term of the fluctuations m(π cl ). In order to repeat our calculation, we would redefine the momenta in the heat kernel as k ′2 = k 2 + m 2 (π cl ). This change of integration variable would have no significant effect on the UV divergences, as long as Λ ≫ m(π cl ).
The question of whether the quantum effects may be suppressed on the background of classical configurations has also been addressed in the context of classicalization [11,12]. This proposal concerns the nature of high-energy scattering in certain classes of nonrenormalizable scalar field theories. It advocates that scattering can take place at length scales much larger than the typical scale associated with the nonrenormalizable terms in the Lagrangian. Quantum corrections are expected to be subleading at such scales, so that a semiclassical description should be sufficient. The inspiration is taken from ultra-Planckian scattering in gravitational theories, during which a black hole is expected to start forming at distances comparable to the Schwarzschild radius. The analogue of the black hole is a semiclassical configuration, the classicalon, generated by a point-like source. In ref. [13] it was argued that quantum fluctuations in δπ can be suppressed for theories, such as the "wrong-sign" DBI theory, that admit classicalons. The application of our approach to classicalizing theories is a direction for future research.
In view of broader applications, we sketch a general framework which permits the discussion of all theories for which the fluctuation operator ∆ involves up to second derivatives and can be parametrized as with G µν , F µ , B functionals of the scalar field π and its derivatives. Following the steps of section 2, the one-loop effective action (4) can be obtained from the heat kernel, whose diagonal part takes the form with the implicit assumption that it acts on f (x) = 1. The term exp(−G µν k µ k ν ) can be isolated through use of the Baker-Campbell-Hausdorff formula (19). The terms up to order ǫ 2 in the expansion of the remaining exponentials incorporate all possible UV divergences. The "metric" G µν can be diagonalized as in eq. (18), while the momentum integrations can be carried out through use of formulae such as (10). Finally, the effect of a nontrivial background can be studied by writing the field as π = π cl + δπ and expanding in powers of δπ.
For a spherically symmetric background, with (G 0 ) µν given by eq. (21), the effective action (27) is reproduced. It is straighforward to extend the expansion in eq. (31) in order to obtain terms of higher order in δπ. Such terms will be suppressed by additional powers of ν π cl . This should be contrasted with the expectations from standard perturbation theory for the quartically divergent terms, as given by eq. (13) and discussed at the end of section 2.
We conclude by noting that both classicalization and the Vainshtein mechanism rely on strong nonlinear effects associated with the background. The combined picture arising through our work and ref. [13] supports the speculation that the suppression of quantum effects by the background may be a usual phenomenon in higher-derivative theories.