Entropy constraints on effective field theory

In effective field theory, the positivity bounds of higher derivative operators are derived from analyticity, causality, and unitarity. We show that the positivity bounds on some operators of the effective field theory, e.g., dimension-eight term of a single massless scalar field, the Standard Model Effective Field Theory dimension-eight $SU(N)$ gauge bosonic operators, and higher-derivative operators in the Einstein-Maxwell theory, generated by interactions between heavy and light degrees of freedom can be derived by the non-negativity of relative entropy. For such effective field theories, we prove that the interactions increase thermodynamic entropy at a fixed charge and an extremal point of energy, which is intimately connected with the extremality relations of black holes exhibiting Weak-Gravity-Conjecture. These arguments are applicable when corrections from the interactions involving higher-derivative operators of light fields are not dominant in the effective field theories. The entropy constraint is a consequence of the Hermiticity of Hamiltonian, and any theory violating the non-negativity of entropy would not respect the second law of thermodynamics.


I. INTRODUCTION
Relative entropy [1][2][3] is a fundamental quantity in probability theory and information theory.The relative entropy, which is non-negative, depicts a distance between two probability distributions and plays important roles in statistical mechanics [4][5][6] and quantum information theory [7][8][9].In the context of information-thermodynamics, the distance between two probability distributions is an essential concept to derive a non-negativity of difference in von-Neumann entropy between initial and final states [4,5,10], so-called second law of thermodynamics.
The WGC states that the U (1) charge-to-mass ratio of extremally charged black holes is larger than unity in any gravitational effective field theory (EFT) that admits a consistent UV completion [16].Some proofs for this statement have been made using black holes and entropy consideration [12,13], or positivity bounds from unitarity and causality [19,20].In particular, Refs.[12,13] are based on a positivity of entropy difference between Einstein-Maxwell theories with and without perturbative corrections that are described by higher-derivative operators.
The crucial role of relative entropy in informationthermodynamics suggests that the positivity of entropy difference in the WGC is intimately connected to the distance between two theories, which has been studied in different contexts [21][22][23][24].That inspires us to establish a connection between relative entropy and positivity bounds in the EFTs.In the work, we provide lower and upper bounds on perturbative corrections from interactions between heavy and light degrees of freedom to the Euclidean effective action.From the upper bound, we obtain the same bounds on some operators of EFTs, e.g., dimension-eight term of a single massless scalar field, the Standard Model EFT (SMEFT) dimensioneight SU (N ) gauge bosonic operators, and higherderivative operators in the Einstein-Maxwell theory, as those positivity bounds achieved in conventional EFT studies [17,25,26] when the higher-derivative operators are generated by the interactions between heavy and light fields.The constraints on such EFTs are applicable when the perturbative corrections from the interactions involving higher-derivative operators of the light fields are not dominant in the EFTs.
Ref. [16] implies a possibility that the WGC-like behavior in the perturbative correction to extremality relations of black hole [12] can be generalized to a broad class of thermodynamic systems on the condition that the correction to entropy is non-negative.We prove that the corrections to the entropy at a fixed charge and an extremal point of energy from the operators, such as the dimension-eight term of a single massless scalar field, the SMEFT dimension-eight SU (N ) gauge bosonic operators and higher-derivative operators in the Einstein-Maxwell theory, are non-negative when the corrections from the interactions involving higher-derivative operators of the light fields are not dominant in the EFTs.

II. DISTANCE BETWEEN TWO THEORIES
Consider a field theory that contains a set of light fields φ's and that of heavy fields Φ's; see Fig. 1.We introduce a thermodynamic system A described by the Euclidean action I 0 [φ, Φ], which does not involve interactions between φ's and Φ's.See Appendix A for the detailed The distance, i.e. the relative entropy between P0 and Pg, yields lower and upper bounds on perturbative correction from the interaction between heavy and light degrees of freedom to the Euclidean effective action. .definition of I 0 #1 .We define a probability distribution function for the system A as P 0 ≡ e −I0 /Z 0 [β, φ], where β is an inverse temperature of the system and φ denotes a background field corresponding to the light field, which is held fixed while the path integral over Φ's is performed.Even if the path integral over φ's is performed, following explanations do not change much; see Appendix B. Note that heavy background fields are expressed by the light ones using the equation of motions.The partition function is given as the Euclidean path integral Z 0 [β, φ] ≡ β d[Φ]e −I0 , which is determined by the Wick-rotated Lagrangian and boundary conditions.The effective action of the system is W 0 [β, φ] ≡ − ln Z 0 [β, φ].
The relative entropy between P 0 and P g is defined as It is greater than or equal to zero, with the equality holding if and only if P g = P 0 .Thanks to this nonnegativity, the relative entropy is often used as a distance between P 0 and P g even though it is not a symmetric function of the two sets of probabilities S(P 0 P g ) = S(P g P 0 ).A simple algebra yields S(P 0 P g ) = β d[Φ] (P 0 ln P 0 − P 0 ln P g ) #1 The theory A is a reference theory to obtain constraints on the low energy theory generated by Ig.Note that we discuss the constraints on the theory described by Ig, not I 0 .
where I I g=0 ≡ β d[Φ]P 0 • I I is an expectation value of the interaction, which satisfies (dW g /dg) g=0 = β d[Φ]P 0 • I I .Note here that the derivation of Eq. ( 2) does not rely on the expansion in g.In Eq. ( 2), the path integral is performed only over the heavy degrees of freedom, and the self-interacting term of the light degrees of freedom cancels in I I g=0 .The path integral over the light degrees of freedom does not change Eq. (2); see Appendix B. It follows from the non-negativity of the relative entropy that where ∆W (E) g denotes the difference between the effective actions of the two systems in the Euclidean space.Another choice of relative entropy, is related to the renormalization group [21].It provides a lower bound ∆W (E) g ≥ g• I I g with I I g ≡ β d[Φ]P g •I I .We end up with the inequalities which implies that the sign of the interaction controls the sign of perturbative corrections to the Euclidean effective action.For example, the Euclidean effective action is increased in the theory with g • I I g ≥ 0 but decreased in the theory with g • I I g=0 ≤ 0. We emphasize that the inequalities (5) are applicable to the UV theories in which one-loop contributions from one-light-particle-irreducible diagrams with heavy-and-light-field mixing appear; then, we perform the path integral over φ's, i.e., focusing on the case of Even for such a case, Eq. ( 5) holds; see Appendix B.
Here, we provide another explanation of the meaning of the upper bound of Eq. (5).By expanding W g [β, φ] with respect to g, the upper bound of Eq. ( 5) yields where Note here that g • I I g=0 cancels in Eq. (6).Therefore, the upper bound of Eq. ( 5) means that the Euclidean effective action decrease by the perturbative corrections of the second or higher order corrections for g.As the inequalities (5) do not rely on either Lorentz symmetry or gauge symmetry, it works for a wide class of quantum theories that consist of both light and heavy degrees of freedom.Consider thermodynamic systems described by quantum mechanics, which generally do not respect the Lorentz symmetry.Define the Hamiltonian of the system as H ≡ H 0 + H I , where H I denotes the interaction between light and heavy degrees of freedom, and H 0 does not involve the interactions.Define the theory A as the Hamiltonian H 0 .By introducing the auxiliary parameter g, also define the theory B as a Hamiltonian H g ≡ H 0 + g • H I .The density operators of the theory A and B are respectively defined as follows: with the partition functions The non-negativity of relative entropy between ρ 0 and ρ g yields where the effective actions, and the expectation value of the interaction are defined as Therefore, Eq. ( 9) yields This inequality means that the Euclidean action decreases for the non-positive interacting theory defined by g • H I g=0 ≤ 0. Also, consider another choice of the relative entropy S(ρ g ||ρ 0 ) as follows: where . Thus, Eq. ( 14) yields Combining Eq. ( 13) and (15), we obtain This inequality corresponds to Eq. ( 5), and it is clear that the UV properties such as symmetry is not necessary to obtain the entropy constraints of Eq. ( 5).

III. EXAMPLES
Equipped with the distance between the two theories, we are ready to discuss the entropy constraints on various EFTs.In this section, under the setup of the previous section, i.e., the Euclidean path integral method is valid, and perturbative corrections are generated from the interacting term, we take two different approaches, i.e., the top-down approach and the bottom-up approach.In the top-down approach, the relative entropy is evaluated in the UV theories with the light and heavy degrees of freedom to check the validity of the inequalities of (5).In the bottom-up approach, it is supposed that the UV theory is not specified for a given EFT, and the higher-dimensional operators of the EFT are generated by integrating out the heavy fields.We focus on the EFTs, where the perturbative corrections to the leading terms, such as renormalizable terms , can be eliminated by the field redefinition and study the constraints on the EFTs.

A. Top-down approach
We adopt the top-down approach and check the consistency of the entropy constraints by evaluating the effective action of the UV theories.The temperature of the system is assumed to be zero in the first four examples.
(a) A tree level UV completion of the single massless scalar field theory: Consider a theory in Minkowski space: where φ denotes a massless scalar field, Φ is a heavy scalar field with mass m, α and β are dimensionless parameters, and Λ is some mass scale.Define the actions I 0 and I I in Minkowski space as follows: The theory B is defined as I g ≡ I 0 + g • I I with the parameter g.In this example, higher dimensional operators are generated at tree level, and the interaction I I does not contribute to the Euclidean effective action at the first order for g.At tree level, the Euclidean effective actions are calculated as follows: From Eq. ( 20) and ( 21), the shift of the Euclidean effective action is obtained as Then, the expectation value of the interaction is calculated as Combining the upper bound of Eq. ( 5), (22), and ( 23), we obtain the positivity bound as follows: Therefore, the coefficient of the dimension-eight operator of Eq. ( 20) is positive because of the non-negativity of relative entropy.(b) A tree level UV completion of the single mass less scalar field theory with a linear term: We discuss the effects of the linear term of Φ in I 0 , which generally generates a non-zero expectation value of the interaction I I g=0 .As shown later, the constraints on EFTs can arise even if I I g=0 takes a non-zero value.Consider an action in Minkowski space defined as where v is a dimensionful parameter.Define I 0 and I I as follows: By introducing the parameter g, define the theory B as I g ≡ I 0 + g • I I .The Euclidean effective actions are calculated as follows: From Eq. ( 28) and ( 29), the shift of the Euclidean effective action is given by At tree level, the expectation value of I I in Euclidean space is calculated as where we used It is clear that the expectation value g • I I g=0 generally takes a non-zero value even in the tree-level UV completion.For v = 0, both linear term of Φ in Eq. ( 25) and expectation value g • I I g=0 vanish.This fact holds in general UV theory involving the linear term of Φ.Here, it should be noted that the constraint on the higherdimensional operators can be derived even if g • I I g=0 takes a non-zero value.Combining Eq. ( 30), (31), and the upper bound of Eq. ( 5), the expectation value g • I I g=0 cancels, and we obtain Consequently, the relative entropy yields the constraint on the coefficient of the dimension-eight operator of Eq. ( 28).The reason why the expectation value I I g=0 cancels in Eq. ( 32) is the same as that it cancels in Eq. ( 6).
Here, we show that the expectation value g• I I g=0 can be removed by a redefinition of Φ.By defining Φ ≡ η +v, the action of Eq. ( 25) is expressed as follows: Note here that the liner term of η does not arise in Eq. (33).Then, we define, where I I denotes the interaction, and I 0 does not involve it.By introducing the parameter g, define the theory B as I g ≡ I 0 +g•I I .Then, the Euclidean effective actions are calculated as follows: The shift of the Euclidean effective action is calculated as Since the linear term of η does not arises in Eq. (33), the expectation value of the interaction I I in Euclidean space takes a zero value as follows: Then, from Eq. ( 38), (39) and the upper bound of Eq. ( 5), we obtain This result is the same as Eq.(32) because the expectation value of the interaction I I cancels in the relative entropy, i.e., the relative entropy is invariant under the redefinition to eliminate the linear term of Φ.Therefore, we found that the constraint on the EFT does not depend on the condition of vanishing the linear term.
The above explanations are based on the theory of Eq. ( 25), but the invariance of the inequality of Eq. ( 3) under the field redefinition to eliminate the linear term of Φ hold in general UV theories.Similar to the above explanations, take two different approaches.
First, consider the UV theory with the linear term as follows: where I lin 0 involves the linear term of Φ, and I I is the interacting term.Consider the classical solution v of I lin 0 , where indices of the classical solution, such as Lorentz indices, are omitted.Also, the classical solution of I for Φ is assumed to be v +f (φ), where f depends on the light field φ because of the interacting term I I .Note here that f (φ) vanishes in the limit of I I → 0. By introducing the parameter g, we define I g ≡ I lin 0 + g • I I .At tree level, the Euclidean effective actions of I g=1 and I g=0 are respectively calculated as follows: The shift of the Euclidean effective action is calculated as The expectation value of the interaction I I in the Euclidean space is also calculated as where 44), (45), and the inequality of Eq. (3), we obtain This inequality corresponds to Eq. (32).Next, consider the field redefinition Φ ≡ η + v.Then, Eq. ( 41) is expressed as For convenience, define where I 0 does not include the linear term of η.By introducing the parameter g, we also define I g ≡ I 0 +g•I I .At tree level, the Euclidean effective actions of I g and I 0 are respectively calculated as follows: Note here that the classical solution of I for η is f (φ), and that of Similarly, the classical solution of I 0 for η is zero, and that of I 0 for Φ = v + η is v.Then, the shift of the Euclidean effective action is calculated as The expectation value of the interaction I I in the Euclidean space is calculated as where [φ,η] .Combining Eq. ( 52), (53), and the inequality of Eq. (3), we obtain This result is the same as Eq. ( 46).Consequently, it is found that the inequality of Eq. ( 3) is invariant under the field redefinition to remove the linear term of Φ.We often define the heavy fields such that the linear term vanishes for ease of calculation of the relative entropy.We mention it in the following calculations when such a definition is used.(c) Euler-Heisenberg theory: The action of quantum electrodynamics of electron field (ψ) in Minkowski space is where D µ = ∂ µ + ieA µ is the covariant derivative, m is the mass of ψ, and is the field strength of photon.Define I 0 and I I as follows: By introducing the parameter g, the theory B is defined as I g ≡ I 0 + g • I I .The Euclidean effective actions of theories A and B are respectively calculated as follows: where (d 4 x) E is the volume of Euclidean space-time, , the Wilson coefficents γ 1 = 1/2 and γ 2 = 7/8 [27], A µ is the background field satisfing ∂ µ F µν = 0 with constant F µν , and the vacuum energy is omitted because it cancels in relative entropy.The details of the wave function renormalizations are explained in Appendix D. From Eq. ( 58) and ( 59), the difference of the Euclidean effective action at the one loop level is From Eq. ( 59), the expectation value of the interaction I I in the Euclidean space is also calculated as follows: Combining the inequality (5), Eq. ( 61) and (60), the shift of the Euclidean effective action is given by The left-hand side of Eq. ( 62) denotes the linear combination of dimension-eight operators of Eq. ( 59), and it is found that the constraints on the EFTs arise from the relative entropy.Consequently, the Euler-Heisenberg theory satisfies the non-negativity of relative entropy because γ 1 and γ 2 are positive values.(d) Massive, gravitationally coupled scalar field at tree level [12]: To explain how to define the interaction I I in gravitational theories, consider a simple theory in Minkowski space: where R µνρσ is the Riemann tensor, R is the scalar curvature, and a Φ , b Φ are dimensionful coupling constants.Define the non-interacting and interacting terms as follows: It should be noted that the theory A does not include the interaction between Φ and A µ , R µνρσ , but the interaction between g µν and Φ.The higher-derivative operators generally arise from the interaction between g µν and Φ, but such effects are discussed later in (h).The theory B is defined as I g = I 0 + g • I I with the parameter g.In this example, I 0 and I I are obtained as The Euclidean effective actions of theories A and B are respectively calculated as follows: where g µν and A µ denote the background fields.From Eq. ( 68) and ( 69), the shift of the Euclidean effective action is calculated as From Eq. (69), the expectation value of the interaction I I at the tree level is calculated as Eqs. ( 5), (70), and (71) yield The left-hand side of Eq. ( 72) denotes the linear combination of higher-dimensional operators of Eq. ( 69), and the constraints on the EFT arise from the relative entropy.As explained later, the entropy constraints by the relative entropy is a generalization of Ref. [12], which includes the result of Ref. [12].(e) A spin system in one dimension: Consider a spin system in one dimension defined by a Hamiltonian where σ i = ±1 is a spin on site i, J is a coupling constant characterizing exchange interactions, N is the number of sites, µ is a magnetic moment, and M is an external magnetic field.Then, define H 0 and H I as follows: By introducing the parameter g, the theory B is defined as Then, density operators are given by with the partition functions, = 2 e βJ + e −βJ N/2 , (76) For each of the theories, the effective actions are defined as W 0 (β) = − ln Z 0 (β), and W g (β) = − ln Z g (β).The expectation value of the interaction is calculated as Tr[ρ 0 H I ] = 0, and the shift of the Euclidean effective action is given by This result is consistent with Eq. ( 13) because cosh(2βgµM ) ≥ 1.The entropy constraints explain why the free energy of the spin system decreases by the external magnetic field.

B. Bottom-up approach
We adopt the bottom-up approach and derive the constraints on a class of EFTs, where corrections to the leading terms, such as the kinetic term and the Einstein-Hilbert term, can be eliminated by the redefinition of the light field.For such a class of EFTs, consider the higher-dimensional operators generated by integrating out Φ.The interaction of the UV theory is generally expressed as Throughout the bottom-up approach, we suppose this general form of interaction for a given EFT.Here, assume J[φ] does not include the higher-dimensional operators.In other words, we assume corrections from the interactions involving higher-derivative operators of the light fields are not dominant in the EFTs.The assumption is quantitatively reasonable because the higher-dimensional operator J[φ] is suppressed by a heavier mass than Φ.The expectation value of the interaction is calculated as follows: When J[φ] preserves the symmetries of the EFT, J[φ] can be proportional to the leading term, such as the kinetic term and the Einstein-Hilbert term, and generally takes a non-zero value.If J[φ] is the higher-dimensional operator, the EFT includes terms proportional to J[φ] generated from degrees of freedom other than Φ.Therefore, it would be quantitatively and qualitatively reasonable to impose the above assumption.As explained later, I I g=0 can take zero value by a suitable field redefinition when J[φ] does not preserve the symmetries of the EFT, such as the gauge symmetry.We focus on two cases: tree-level UV completion and loop-level UV completion.In the tree-level UV completion, we assume the tree-level effects dominate the perturbative corrections from the heavy degrees of freedom to the Euclidean effective action.On the other hand, in the loop-level UV completion, we assume the loop-level effects dominate the perturbative corrections to the Euclidean effective action.For each EFT, we evaluate the relative entropy as follows: (f) Single massless scalar field with dimension-eight term: Consider an effective action in Minkowski space defined by where we used a metric signature convention, g µν = diag(+1, −1, −1, −1), and the second term is induced by integrating out heavy fields.Because of the shift symmetry: φ → φ + const., Eq. ( 80) involves only the kinetic term as the renormalizable term, and corrections to the kinetic term can be removed by the field redefinition of φ.We suppose that the dimension-six operators are eliminated by demanding ∂ µ ∂ µ φ = 0 with constant ∂ µ φ.Because of the assumption, i.e., J[φ] does not include the higher-derivative operators, J[φ] can be ∂ µ φ or ∂ µ φ∂ µ φ, which preserve the shift symmetry, but ∂ µ φ effects on I I g=0 vanish because I I g=0 preserves the Lorentz symmetry.When we suppose that the EFT arises from integrating out heavy degrees of freedom, the first order corrections for g to the Euclidean effective action are expressed as For each tree and loop-level UV completions, we evaluate the constraint from the relative entropy as follows: • Tree-level UV completion -First, consider the EFT generated by the tree-level UV completion.Not depending on details of the UV theory, up to the dimension-eight operator, the Euclidean effective action of the theory B is calculated as follows: where α tree 2 and β tree 2 denote the second or higher order corrections for g.Note here that β tree 2 does not include the first order correction for g because of the assumption, i.e., J[φ] does not include the higher-dimensional operators.It is assumed that α tree 2 and β tree 2 are generated at the tree level.Also, according to the procedure in Eq. ( 47), (48), and (49), the first order correction for g is removed in α tree 2 .We choose the background fields as ∂ µ φ = const.to remove the dimension-six operators.The Euclidean effective actions of the theory B and A are respectively obtained as where the wave function renormalization is performed in Eq. (83); see Appendix D. Note here that φ is also a classical solution of Then, the shift of the Euclidean effective action is calculated as (85) Also, from Eq. ( 83), we obtain The detail derivation of Eq. ( 86) is provided in Appendix D. From Eq. ( 5) or (B12), combining Eq. ( 85) and (86) yields Equation ( 87) denotes the constraint on the coefficient of dimension-eight operator of Eq. ( 83).
• Loop-level UV completion -Next, consider the EFT generated by the loop-level UV completion.The Euclidean effective action of the theory B is calculated as follows: where α loop 1 is the first order correction for g, α loop 2 and β loop 2 are the second or higher order correction for g, and E vac is the vacuum energy coming from Φ and φ.It is assumed that α loop 1 , α loop 2 , and β loop 2 are generated from the loop corrections of Φ.We choose ∂ µ φ = const.to remove the dimension-six operators.Since the background field φ is also a classical solution of W 0 [φ], the Euclidean effective action for the theory B and A are respectively obtained as Eq.(D15), Then, the shift of the Euclidean effective action is obtained as (91) Also, from Eq. ( 81) and (89), we obtain where α loop 1 denotes the first order correction for g and satisfies a relation of the form g • (dα loop 1 /dg) = α loop 1 .From Eq. ( 5) or (B12), combining Eq. ( 91) and (92) yields Equation ( 93) yields the constraint on the dimension-eight operator generated at the loop level.
For both tree and loop-level UV completion, demanding ∂ µ ∂ µ φ = 0 with constant ∂ µ φ, after Wickrotation the inequality (5) gives rise to Consequently, the coefficient c must be positive to respect the entropy constraints, when it arises from integrating out the heavy fields.This result is the same as the positivity bound from the unitarity and causality.(g) Standard Model EFT (SMEFT) dimension-eight SU (N ) gauge bosonic operators: Consider an effective action in Minkowski space defined by where the dimensional-eight operators O i 's are [26] where ν is the field strength of the gauge field A a µ and g denotes the gauge coupling of SU (N ).
The Greek letters stand for Lorentz indices, the Italic letters represent SU (N ) color indices, and totally antisymmetric and symmetric structure constants are defined by [T a , T b ] = if abc T c and {T a , T b } = δ ab 1/N + d abc T c with T a the generator of SU (N ) Lie algebra.To avoid the effect from the dimension-six operators we follow [26] to choose a background field satisfying the leading-order equation of motion, , where u 1,2 is a constant real vector in SU (N ) color space, 1,2 is a constant four-vector, and w 1,2 is an arbitrary Cartesian coordinate in spacetime satisfying ∂ µ w 1 = l µ and ∂ µ w 2 = k µ with l µ and k µ being constant four-vectors.
When J[A a µ ] does not include the higher-dimensional operators, there are two cases: (i) J[A a µ ] preserves the gauge symmetry or (ii) not.For case (i), J[A a µ ] ∝ F a µν F a,µν holds.The CP violating term generally arises, but we assume such a term is removed by axion-like degrees of freedom in the UV theory.Then, from Eq. ( 79), the first order corrections for g to the Euclidean effective action are expressed as For case (ii), J[A a µ ] can be proportional to A a µ , or A a µ A a,µ because of the covariant derivative of the kinetic term.Since corrections from the interacting terms of the higher-dimensional operators would not be dominant effects, we focus on corrections from the kinetic terms.However, J[A a µ ] ∝ A a µ vanishes because I I g=0 keeps the Lorentz symmetry.Although J[A a µ ] ∝ A a µ A a,µ generally remains, it can be eliminated by the gauge fixing condition, which is called a nonlinear gauge; see Ref. [28,29].Therefore, we focus on the case of Eq. ( 108) below.For each tree and loop-level UV completions, the constraints on the SMEFT from the relative entropy are evaluated as follows: • Tree-level UV completion -Consider the EFT generated by the tree-level UV completion.The Euclidean effective action of the theory B is generally calculated as follows: where α tree 2 and β tree i,2 denote the second or higher order corrections for g, and β tree i,2 does not include the first order correction for g because of Eq. ( 108).The corrections α tree 2 and β tree i,2 are assumed to be generated at the tree-level.According to the procedure in Eq. ( 47), (48), and (49), the first order correction for g is eliminated in α tree 2 .The background fields are chosen to hold F µν = const.. Since A a µ is also a classical solution of W 0 [A], the Euclidean effective actions of the theory B and A are respectively obtained as follows: where the wave function renormalization is performed in Eq. ( 110); see Eq. (D24).Then, the shift of the Euclidean effective action is calculated as follows: From Eq. (110), the first order correction for g is calculated as, From Eq. ( 5) or (B12), combining Eq. ( 112) and (113) yields The left-hand side of Eq. ( 114) denotes the coefficients of the dimension-eight operators of Eq. (110).Therefore, the relative entropy yields the constraints on the linear combination of the the dimension-eight operators.
• Loop-level UV completion -Consider the SMEFT generated by the loop-level UV completion.The Euclidean effective action of the theory B is generally calculated as follows: where α loop 1 is the first order correction for g, α loop 2 and β loop 2,i are the second or higher order correction for g, and E vac is the vacuum energy coming from Φ and A a µ .It is assumed that α loop where the wave function renormalization is performed in Eq. ( 117); see Eq. (D29).Then, the shift of the Euclidean effective action is calculated as follows: Also, from Eq. (117), the first order corrections for g is calculated as where g . From Eq. ( 5) or (B12), combining Eq. ( 119) and (120) yields Equation (121) yields the constraint on the dimension-eight operator generated at the looplevel.
It is found that, for both tree and loop-level UV completion, the inequality (5) gives rise to After Wick-rotation, Eq. (122) yields, where with ∝ (0, 1, 0, 0), and µ 2 ∝ (0, 0, 0, 1).We end up with positivity bounds as follows: which are completely consistent with the positivity bounds from unitarity and causality [26,29].More comprehensive constraints are studied in Ref. [29] by considering more general solutions, which yield additional constraints on the Wilson coefficients of SU (3) gauge bosonic operators.
(h) Einstein-Maxwell theory with higher-derivative operators: Consider a gravitational effective action in Minkowski space defined by where other operators up to four-derivative are eliminated by the field redefinition of g µν .Also, the Gauss-Bonnet combination, i.e., R µνρσ R µνρσ − 4R µν R µν + R 2 , is a total derivative and vanishes in four dimensions.
Consider the higher-derivative operators generated from the UV theory defined by I[g µν ; R µνρσ , A, Φ], where g µν is the metric of space-time, R µνρσ is the Riemann tensor, A µ is the U (1) gauge boson, and Φ is the heavy degrees of freedom.Then, the noninteracting and interacting terms are defined as Eq. ( 66) and (67).It should be noted that the theory of I 0 does not include the interaction between A µ , R µνρσ and Φ, but the interaction between g µν and Φ.The gravitational operators up to four-derivative such as R 2 µν are generated from I 0 and can contribute to α 1 and α 2 by the field redefinition of g µν .Our entropy consideration does not constrain such effects because the relative entropy constrains only the higher-derivative operators generated from the interaction I I .In the following explanations, especially for loop-level UV theory, we suppose that the R 2 µν operator effects are not dominant by assuming a large charge-to-mass ratio of the particle integrated out.
Similar to the SMEFT, when J[g µν ; R µνρσ , A µ ] does not include the higher-derivative operators, there are two cases: (i) J[g µν ; R µνρσ , A µ ] ∝ F µν F µν or R, and (ii) J[g µν ; R µνρσ , A µ ] ∝ A µ , or A µ A µ .Because of the same reason as the SMEFT, we focus on the following case, For each tree and loop-level UV completions, the constraints on the EFT from the relative entropy are evaluated as follows: • Tree-level UV completion -Consider the EFT generated by the tree-level UV completion.Then, by integrating out the heavy fields, the Euclidean effective action is generally calculated as follows: where α tree 2,R , α tree 2,F , β tree 2,1 , β tree 2,2 and β tree 2,3 denote the second or higher order corrections for g.Note here that β tree 2,1 , β tree 2,2 and β tree 2,3 do not include the first order correction for g because of Eq. (127).According to the procedure in Eq. ( 47), (48), and (49), the first order correction for g is eliminated in α tree 2,R and α tree 2,F .Since the gravitational higher-derivative operators involving the Riemann tensor can be removed by field redefinition, and the Riemann-squared operator vanishes in four dimensions, we omit such terms.The effective actions of the theory B and A are respectively obtained as Eq.(D44), where A µ and g µν include the effects of the higher-derivative terms.It should be noted that the first order correction for the higher-derivative terms vanishes in W 0 by using the equation of motion.Then, the shift of the Euclidean effective action ∆W (E) g denotes corrections from the higherderivative terms.Also, from Eq. ( 129), the first order correction for g is calculated as From Eq. ( 5) or (B12), combining Eq. ( 129) and (131) yields This inequality means that the relative entropy yields the negative shift of the Euclidean effective action by the higher-derivative operators generated at tree level.
• Loop-level UV completion -Next, consider the EFT generated by the loop-level UV completion.
The Euclidean effective actions of the theory B and A are respectively obtained as Eq. ( D53) and (D54), where and β loop 2,3 are the second or higher order corrections for g, α loop  1,R and α loop 1,F are the first order corrections for g, and Λ loop 0,Φ is the vacuum energy coming from Φ.The last terms of Eq. ( 133) and (134) arise from the one-loop correction of light fields in M 2 Pl R/2 and F µν F µν /4.Since such a correction does not depend on g, they cancel in relative entropy.From Eq. (133), the first order correction for g is calculated as where g • (dα loop 1,R /dg) = α loop 1,R and g • (dα loop 1,F /dg) = α loop 1,F .From Eq. ( 5) or (B12), Eq. ( 133), ( 134) and (135) yields Here, we defined the effective action without the first order corrections for g as follows: It should be noted that the one-loop corrections from R and F µν F µν cancel in Eq. ( 136).Therefore, ] denotes the shift of the Euclidean effective action by the higherderivative operators.Consequently, even for the loop-level UV completion, the relative entropy yields the negative shift of the Euclidean effective action by the higher-derivative operators.
For both tree and loop-level UV completion, it is found that the non-negativity of relative entropy yields the negative shift of the Euclidean effective action by the higher-derivative operators.As explained in the next section, this result is closely related to the WGC-like behavior.
Here, we consider the relative entropy when additional higher derivative operators are added to theory A. In Eq. ( 136), the loop effects from light fields cancel in the relative entropy, and the relative entropy does not depend on whether the higher derivative operators are added to the theory A or not.Consider the action of theory A with the additional higher derivative operators as follows: where I c denotes the additional higher derivative operators consisting of light fields.Then, the Euclidean effective action of theory A of Eq. ( 134) is modified as follows: where I c eliminates the divergences of loop effects from the light fields and would make the probability distribution function well-defined.Note here that I c does not depend on the parameter g because the theory A is defined from the action I g by taking the limit of g = 0.
In other words, the action of theory B is also modified as follows: Then, the Euclidean effective action of the theory B of Eq. ( 133) is also rewritten as follows: where I c also eliminates the divergences coming from the loop effects from the light fields in the effective action of theory B. Then, the relative entropy of Eq. ( 2) is modified as follows: where we used I I g=0 = β d[Φ]P 0 • I I = (dW g /dg) g=0 in the first line.Substituting Eqs. ( 139) and (141) into Eq.( 142), we obtain where I c cancels in W 0 − W g , and (dW g /dg) g=0 = (dW g /dg) g=0 holds because I c does not depend on the parameter g.Therefore, the relative entropy of Eq. ( 2) does not depend on whether the higher derivative operators consisting of the light fields are added to the theory A.
(i) Weak Gravity Conjecture: Last but not the least, we discuss the close connection between the entropy inequality ( 5) and the WGC.The WGC states that quantum gravity theories have to contain a charged particle with the charge-to-mass ratio larger than unity, which is motivated by a gedanken experiment of the decay of an extremal black hole.The extremality bound, M ≥ M ext = Q where M and Q denote the mass and charge of the black hole described by the Einstein-Maxwell theory and M ext represents the minimum mass, would indicate existence of a particle with the chargeto-mass ratio larger than unity.The extremality bound is modified by a perturbative correction in the Einstein-Maxwell theory; however, the conclusion of the above gedanken experiment remains for an extremal BH of arbitrary large size if the perturbative correction does reduce M ext at fixed charge.Based on thermodynamic, Ref. [16] generalizes a relation between the perturbative corrections to the black hole entropy and the extremality bound [12] to a wide class of thermodynamic system as where is the parameter introduced to characterize the perturbative corrections in the system, and Q is the charge.Note here that the extremal limit is taken in Eq. ( 144).From Eq. ( 144), if • (∂S/∂ ) M, Q > 0, then a perturbed extremal system is less massive than its unperturbed counterpart at fixed charge.Consider the effective action including the perturbative correction as where ≤ g • I I g=0 in accord to the inequality (3).Note that • (∂W /∂ ) =0 contains the higher order correction of O(g 2 ).Again, the parameter characterizes the perturbative corrections and we consider the leading term of hereafter.The free energy of the thermodynamic system, where φ is a local minimum of W , β is the inverse temperature, S is the thermodynamic entropy, and µ is the chemical potential.Therefore, the difference in the free energy between the two theories is where 2) is used because φ is a local minimum of W .In gravitational EFTs, this point has been mentioned in Ref. [16] with special attention to contributions from boundary terms.From the relation (∂S/∂ ) M, Q = −β(∂G/∂ ) β, µ in Refs.[14,16], we obtain Combining Eqs. ( 5) and ( 148), lower and upper bounds on the perturbative correction to entropy are given by For the EFTs discussed in III B, under the assumption that J does not include the higher-derivative operators, the shift of the Euclidean effective action by the higher-derivative operators becomes non-positive at zero temperature.When we substitute such non-positive perturbative corrections from the higher-derivative operators into • (∂W /∂ ) =0 in Eq. ( 145), the righthand side of Eq. ( 148) takes a non-negative value up to the first order of the higher-derivative operators, and the WGC-like behavior arises in the EFTs discussed in III B. In particular, to derive the above arguments for the Einstein-Maxwell theory with higher-derivative operators, it is also supposed that the R 2 µν operator effects are not dominant because of a large charge-tomass ratio of the particle integrated out.Note here that the exception is possible because the entropy constraints rely on the Euclidean path integral method.Some conditions to apply the entropy constraints are explained in Appendix E. Although the entropy constraint is a generalization of Ref. [12], investigations of the adaption range of the entropy constraint on the WGC is one of our future directions.
We comment on a connection between this work and Ref. [12].In Ref. [12], it is demonstrated that the Euclidean effective action decreases by higher-derivative operators generated at tree level.For convenience, we briefly review it.At finite temperature β, consider the actions I 0 and I g .The saddle point approximation yields where φ 0 is the classical solution of I 0 , φ g and Φ g are those of I g , and I g [ φ 0 , 0] = I 0 [ φ 0 , 0] holds because the interacting term of I g vanishes for Φ = 0.It should be noted that the relation is derived by taking the limit of g = 0 in this work.Thus, I 0 [ φ 0 , 0] and I g [ φ g , Φ g ] are the Euclidean effective action of the theory A and B, respectively.Since Φ g denotes the local minimum of I g and would take a small value because of heavy field mass suppressions, the inequality of (150) arises by the saddle point approximation.The action I 0 does not generate the higher-dimensional operators, but the action I g yields them through the interacting term between φ and Φ.Therefore, the inequality (150) means that the Euclidean effective action decreases by higher-dimensional operators generated at tree level.In other words, at fixed temperature β, the free energy decreases by higher-dimensional operators generated at tree level.Note here that the inequality of (150) does not need the extremal limit to be valid.Although the origin of the inequality is slightly different, Ref. [12] is essentially the same as this work at the tree-level.

IV. IMPLICATION OF ENTROPY CONSTRAINT
The entropy constraint is intimately connected to the unitarity of time evolution.In the study, the canonical distributions are adopted as the density operator, which is a positive semidefinite (Hermitian) operator with trace one.In other words, the Hamiltonians of the two theories are Hermitian to ensure the non-negativity of relative entropy.Therefore, the entropy constraint on the EFTs is consistent with the positivity bound obtained from unitarity considerations.
So far we have studied the constraints on theories from the non-negativity of relative entropy, however, the second law of thermodynamics is also intimately connected with the non-negativity of relative entropy [6].For example, consider a thermodynamic system consisting of a system, and an external heat bath system described by the Hamiltonian H B .We assume that the initial state of the entire system is ρ S ⊗ e −βH B /Z B , where ρ S is a quantum state of the system, β is an inverse temperature of the external heat bath system, and Z B ≡ Tr B [e −βH B ] is obtained by tracing out the heat bath system degrees of freedom.After the time evolution by the unitary operator U , the final state of the entire system becomes U ρ g ⊗ e −βH B /Z B U † .Then, the final state of the system is obtained as ρS ≡ Tr B [U ρ g ⊗ e −βH B /Z B U † ] by tracing out the heat bath system.The definition of relative entropy Eq. ( 1) yields [6]  Therefore, the non-negativity of relative entropy yields the second law of thermodynamics, and any theory violating the non-negativity of relative entropy does not respect the second law of thermodynamics.It is remarkable that the non-negativity of relative entropy yields a unified understanding of various phenomena, e.g., the positivity bounds on EFTs, the WGC-like behavior in thermodynamics, and the second law of thermodynamics.

V. CONCLUDING REMARKS
In this Letter, we have studied the positivity bounds on EFTs, and the WGC-like behavior in thermodynamics in terms of the non-negativity of relative entropy.Form the relative entropy, we obtained the lower and upper bounds on perturbative corrections from the interaction between heavy and light degrees of freedom to the Euclidean effective action.We argued that the bounds are applicable in both field theoretical systems and quantum mechanical systems.Focusing on the class of EFTs, e.g., the single massless scalar field with dimension-eight term, SMEFT SU (N ) gauge bosonic operators, and Einstein-Maxwell theory with higherderivative operators, generated by the interactions, we found that the upper bound yields the positivity bounds as the same as those derived by unitarity and causality in the conventional EFT study [26].This argument holds when the corrections from the interactions involving higher-derivative operators of the light fields are not dominant in the EFTs.By combining the entropy constraints and pure thermodynamics, it is also shown that the WGC-like behavior arises in some EFTs, e.g., the single massless scalar field with dimension-eight term, SMEFT SU (N ) gauge bosonic operators, and Einstein-Maxwell theory with higher-derivative operators, up to the first order of the higher-derivative operators.Finally, we remark that the positivity bounds on EFTs, the WGC-like behavior in thermodynamics, and the second law of thermodynamics are intimately connected by the non-negativity of relative entropy.
where the probability distributions are defined as The first line denotes the definition of the relative entropy.In the second line, we used following relations, In the fourth line, we used the following definitions, (B6) In the last line, we used the non-negativity of relative entropy and ∆W From Eq. (B1), the upper bound of the shift of the Euclidean effective action is expressed as Similarly, another choice of the relative entropy is calculated as follows, In the fourth line, we used The last line yields the lower bound of the shift of the Euclidean effective action as follows, Combining Eq. (B8) and (B11), we get Eq.( 5).Note here that the derivation of Eq. ( 5) does not depend on the detail form of I g .Since, however, the relative entropy is calculated based on the Euclidean path integral method, Eq. ( 5) may be broken when the Euclidean path integral method does not work, see Appendix E.
For the dynamical light fields, similar to Eq. ( 2), the relative entropy is calculated as follows, where , and the partition functions are defined as The expectation value of the interaction is expressed as where the partial derivative is performed with the fixed classical solution φ g .Also, the another choice of relative entropy of Eq. ( 4) is calculated as follows, where the expectation value of the interaction is expressed as Here, similar to Eq. (B15), the partial derivative is performed with the fixed classical solution.
Appendix C: Relative entropy under field redefinition It is clear that entropy constraint is satisfied in systems described by the Gaussian distributions.Note here that the relative entropy is invariant under the field redefinition of x h .Although the definition of the interaction of Eq. (C1) is not invariant under the redefinition of x h , the definition of the relative entropy of Eq. ( 1) and the integral of the Gaussian distributions do not change under the field redefinition.
To see the invariant formulation under the field redefinition, let us consider a tree level UV completion described by the following action in Euclidean space: where φ A is an auxiliary field.We define the theory B as I g = I 0 + g • I I with the parameter g, and At tree level, the expectation value of the interaction I I is calculated as Therefore, the definition of the relative entropy yields where we used Here, consider a field redefinition: Under this field redefinition, the actions are transformed as Similarly, the relative entropy is transformed as where α tree 2 and β tree 2 denote the second or higher order corrections for g.Note here that β tree 2 does not include the first order correction for g because of Eq. (D1).It is assumed that α tree 2 , and β tree 2 are generated at the treelevel.Also, in the second line, according to the procedure in Eq. ( 47), (48), and (49), the first order correction for g is eliminated in α tree 2 .The background field φ denotes the classical solution of the effective action of The equation of motion of W g [φ] is expressed as To remove the dimension-six operators, we choose the background fields as follows, where ∂ µ φ = const..Note here that the background field φ is also a classical solution of W 0 [φ].Therefore, the Euclidean effective actions of theories B and A are respectively obtained as Then, the shift of the Euclidean effective action is calculated as Also, from Eq. (D6), we obtain the following relation where (d φ /dg) g=0 = 0 because α tree 2 denotes the second or higher order corrections for g.From Eq. ( 5) or (B12), combining Eq. (D8) and (D9) yields Equation (D10) denotes the constraint on the coefficient of dimension-eight operator of Eq. (D6).
• Loop-level UV completion -Next, consider the EFT generated by the loop-level UV completion.Then, the partition function is calculated as follows, where α loop 1 is the first order correction for g, α loop 2 and β loop 2 are the second or higher order correction for g, E Φ vac is the vacuum energy coming from the loop-level correction of Φ, and E vac is the vacuum energy of Φ and φ.It is assumed that α loop 1 , α loop 2 , and β loop 2 are generated from the loop corrections of Φ.The background field φ denotes the classical solution of the effective action of The equation of motion of Eq. ( D12) is expressed as follows, We choose the background field as follows, where ∂ µ φ = const.to remove the dimension-six operators.Since the background field φ is also a classical solution of W 0 [φ], the Euclidean effective actions of theories B and A are respectively obtained as Then, the shift of the Euclidean effective action is calculated as Also, from Eq. (D15), we obtain where (d φ /dg) g=0 = 0 is used.Note here that α loop 1 denotes the first order correction for g and satisfies a relation of the form g • (dα loop 1 /dg) = α loop 1 .From Eq. ( 5) or (B12), combining Eq. (D17) and (D18) yields In the loop-level UV completions, Eq. (D19) yields the constraint on the dimension-eight operator generated at the loop-level.
2. SMEFT dimension-eight SU (N ) gauge bosonic operators When J[A a µ ] does not include the higher-dimensional operators, there are two cases: (i) J[A a µ ] preserves the gauge symmetry or (ii) not.For case (i), J[A a µ ] ∝ F a µν F a,µν holds.In general, the CP violating term arises, but we supposed that such a term is removed by axion-like degrees of freedom in the UV theory.Then, from Eq. (79), the first order corrections for g to the Euclidean effective action are expressed as For case (ii), J[A a µ ] can be proportional to A a µ , and A a µ A a,µ because of the covariant derivative of the kinetic term.Since corrections from the interacting terms of the higher-dimensional operators would not be dominant effects, we focus on corrections from the kinetic terms.Then, J[A a µ ] ∝ A a µ vanishes because I I g=0 keeps the Lorentz symmetry.Although J[A a µ ] ∝ A a µ A a,µ generally remains, it can be eliminated by the gauge fixing condition.Therefore, we focus on the case of Eq. (D20) below.
• Tree-level UV completion -Consider the EFT generated by the tree-level UV completion.The partition function is generally calculated as follows, where α tree 2 and β tree i,2 denote the second or higher order corrections for g, and β tree i,2 does not include the first order correction for g because of Eq. (D20).The corrections α tree 2 and β tree i,2 are assumed to be generated at the tree-level.According to the procedure in Eq. ( 47), (48), and (49), the first order correction for g is eliminated in α tree 2 .The background field A a µ denotes the classical solution of the effective action of The background fields are chosen as follows, where F µν = const.Since A a µ is also a classical solution of W 0 [A], the Euclidean effective actions of theories B and A are respectively obtained as follows, Then, the shift of the Euclidean effective action is calculated as follows, Also, the first order corrections for g is calculated as where (dA /dg) g=0 = 0 is used.From Eq. ( 5) or (B12), combining Eq. (D26) and (D27) yields The left-hand side of Eq. (D28) denotes a linear combination of coefficients of the dimension-eight operators of Eq. (D24).
• Loop-level UV completion -Consider the SMEFT generated by the loop-level UV completion.The partition function is generally calculated as follows, where α loop 1 is the first order correction for g, α loop 2 and β loop 2,i are the second or higher order correction for g, E Φ vac is the vacuum energy coming from the loop-level correction of Φ, and E vac is the vacuum energy of Φ and A a µ .It is assumed that α loop 1 , α loop 2 , and β loop 2,i are generated from the loop corrections of Φ.The background field A a µ denotes the classical solution of the effective action of We choose the background field as follows, where F µν = const.to remove the dimension-six operators.A a µ is also a classical solution of W 0 [A], and the Euclidean effective actions of theories B and A are respectively obtained as follows, Then, the shift of the Euclidean effective action is calculated as follows, Also, the first order corrections for g is calculated as where (dA /dg) g=0 = 0 is used.From Eq. ( 5) or (B12), combining Eq. ( D34) and (D35) yields where g • (dα loop 1 /dg) = α loop 1 is used.In the loop-level UV completion, Eq. (D36) yields the constraint on the dimension-eight operator generated at the loop-level.

Einstein-Maxwell theory with higher-dimensional operators
Consider the Einstein-Maxwell theory with higher-dimensional operators generated from the UV theory defined by I[g µν ; R µνρσ , A, Φ], where g µν is the metric of space-time, R µνρσ is the Riemann tensor, A µ is the U (1) gauge boson, and Φ is the heavy degrees of freedom.Define the non-interacting and interacting terms as follows, where the cosmological constant is omitted because it cancels in the relative entropy.It should be noted that the theory of I 0 does not include the interaction between Φ and A µ , R µνρσ , but the interaction between g µν and Φ.Note that gravitational operators such as R 2 µν can be generated from I 0 .Also, the Gauss-Bonnet combination, i.e., R µνρσ R µνρσ − 4R µν R µν + R 2 , is a total derivative and vanishes in four dimensions.In this work, we focus on the higher-dimensional operators generated from the interaction between Φ and A µ , R µνρσ .
Similar to the SMEFT, when J[g µν ; R µνρσ , A µ ] does not include the higher-derivative operators, there are two cases: (i) J[g µν ; R µνρσ , A µ ] ∝ F µν F µν or R, and (ii) J[g µν ; R µνρσ , A µ ] ∝ A µ or A µ A µ .Because of the same reason as the SMEFT, we focus on the following case, For each of the tree and loop level UV completion, the constraints on the EFTs are evaluated as follows, • Tree-level UV completion -Consider the EFT generated at the tree-level UV completion.Then, the partition function is generally calculated as follows, where α tree 2,R , α tree 2,F , β tree 2,1 , β tree 2,2 and β tree 2,3 denote the second or higher order corrections for g.Note here that β tree 2,1 , β tree 2,2 and β tree 2,3 do not include the first order correction for g because of Eq. (D39).According to the procedure in Eq. ( 47), (48), and (49), the first order correction for g is eliminated in α tree 2,R and α tree 2,F .Since the gravitational operators only involving the Riemann tensors can be removed by field redefinition, and the Riemann-squared operator can be dropped in four dimensions, we omit such terms.The background fields A µ and g µν denote the classical solutions of the effective action of We choose the background field as follows, The effective actions of theories B and A are respectively obtained as follows, where A µ and g µν include the effects of the higher-derivative terms.It should be noted that the first order correction for the higher-derivative terms vanishes in W 0 by using the equation of motion.Then, ∆W ] denotes the shift of the Euclidean effective action by the higher-derivative terms.Also, from Eq. (D44), the first order correction for g is calculated as where (dA /dg) g=0 = 0 and (dg µν /dg) g=0 = 0 are used.From Eq. ( 5) or (B12), Eq. (D46) yields Consequently, it is found that the relative entropy yields the negative shift of the effective action by the higher derivative terms generated at the tree-level.where α loop 2,R , α loop 2,F , β loop 2,1 , β loop 2,2 and β loop 2,3 are the second or higher order corrections for g, α loop 1,R and α loop 1,F are the first order corrections for g, and Λ loop 0,Φ is the vacuum energy coming from Φ.The last term of Eq. (D48) arises from loop corrections of light fields in M 2 Pl R/2 and F µν F µν /4.Since these corrections do not depend on g, they cancel in relative entropy.The background fields A µ and g µν denote the classical solution of the effective action of Similar to the case of the tree-level UV completion, the first order correction for the higher-derivative terms vanish in W 0 by using the equation of motion.Also, from Eq. (D53), the first order correction for g is calculated as where (dA µ /dg) g=0 = 0 and (dg µν /dg) g=0 = 0 are used.Note here that the last term of Eq. (D53) does not depend on g.From Eq. ( 5 (D57) Therefore, W non-lin g [g µν , A] − W 0 [g 0,µν , A 0 ] denotes the corrections from the higher-derivative terms to the Euclidean effective action.It should be noted that the one-loop correction from R and F µν F µν cancels in Eq. (D56).

Appendix E: Loophole of entropy constraints
We discuss the loophole of the entropy constraints.As discussed in Ref. [20] and [30], positive perturvative corrections to the Euclidean action can arise in some examples.We show that the loophole arises because the entropy constraints are based on the saddle point approximation in the Euclidean path integral method.First, we consider the entropy constraints on tree-level UV completions, and clarify a relation between this work and Ref. [12].The relative entropy of Eq. ( 1) is calculated as S(P 0 ||P g ) = where we used I g [φ, 0] = I 0 [φ, 0] similar to Ref. [12].This inequality has been provided in Ref. [12], and it is clear that the entropy constraints by the relative entropy is a generalization of Ref. [12].The key point of derivation of Eq. (E2) is that the relative entropy must be evaluated around the local minimum of heavy degrees of freedom.Otherwise, the saddle point approximation does not work well, and the perturbative corrections to the Euclidean effective action can be positive.
To see the loophole, let us consider following action in Minkowski space: where φ A is an auxiliary field.The solution of the equation of motion of φ A is calculated as • (∂S/∂ ) M, Q ≥ 0 To derive the positive perturbative corrections from the higher-derivative terms to thermodynamic entropy at a fixed energy and charge, in addition to the conditions (a), (b), (c) and (d), we impose conditions: (e) thermodynamics relations hold in the system, and (f) the system is the weak-dynamics theory, where O( 2 ) terms are negligible.

andFIG. 1 .
FIG. 1. Schematic illustration of the distance between theoryA and theory B, defined by the action I0 and Ig, respectively.The distance, i.e. the relative entropy between P0 and Pg, yields lower and upper bounds on perturbative correction from the interaction between heavy and light degrees of freedom to the Euclidean effective action..

1 , α loop 2 ,
and β loop 2,i arise from the loop corrections of Φ.We choose the background field satisfying F a µν = const.to remove the dimension-six operators.A a µ is also a classical solution of W 0 [A], and the Euclidean effective actions of the theory B and A are respectively obtained as Eq.(D32), ) where ∆s ≡ −Tr S [ρ S ln ρS ] + Tr S [ρ S ln ρ S ] denotes the difference in the thermodynamic entropy of the system, ∆q ≡ Tr[ρ S ⊗ e −βH B /Z B H B ] − Tr[ρ S ⊗ e −βH B /Z B U † H B U ] is a heat exchange between the system and the external heat bath system, and the second term −β • ∆q represents the difference in the thermodynamic entropy of external heat bath systems at inverse temperature β.

( 0 NOFIG. 2 .
FIG.2.A flow chart for conditions of applicability of entropy constraints: Each step explain which conditions are necessary to use the entropy constraints.