Proper-time evaluation of the effective action: unequal masses in the loop

The Fock-Schwinger proper-time method is used to derive the effective action in the field theory with the chiral $U(3)\times U(3)$ symmetry explicitly broken by unequal masses of heavy particles. The one-loop effective action is presented as a series in inverse powers of heavy masses. The first two Seeley-DeWitt coefficients of this expansion are explicitly calculated. This powerful technique opens a promising avenue for studying explicit flavor symmetry breaking effects in the effective field theories.


I. INTRODUCTION
The heat kernel technique [1] has been introduced to quantum theory in works of Fock [2,3] and later has been pushed forward by Nambu [4] and Schwinger [5]. In combination with the background field method, this allowed DeWitt to develop the manifestly covariant approach to gauge field theory [6], and quantum gravity [7]. The method allows the nonlocal extension [8,9]. It has been widely used in QCD to construct effective meson Lagrangians [10,11], in chiral gauge theories to study chiral anomalies [12], in cosmology to calculate geometric entropy [13], in QED to find Casimir energies and forces [14], etc.
In all above mentioned cases and in many others [15], it becomes necessary to calculate the determinant of a positive definite elliptic operator that describes quadratic fluctuations of quantum fields in the presence of some external or background fields and which in a compact form contains all the information about the one-loop contribution of quantum fields. The result is an asymptotic expansion for the effective action of the theory in powers of the proper-time with Seeley-DeWitt coefficients a n (x, y). These coefficients are polynomials in the background fields and describe, in the coincidence limit y → x, the local vertices of the corresponding effective Lagrangian. It is remarkable that each term of the expansion is invariant under the action of the group of the internal symmetry, if the theory possesses this symmetry. This follows from the general covariance of the formalism.
When quantum fields have large equal masses m, it is easy to resort to an expansion in the inverse powers of mass, which is valid when all background fields and their derivatives are small compared to the mass of the quantum field. In this case, the asymptotic coefficients a n do not change. Such long wavelengths (λ ≫ 1/m) expansion allows one to obtain an action that takes into account effectively the leading low energy effect of virtual heavy states. This scenario is realized in theories with spontaneously broken symmetry, or in the theories with a large bare mass. A typical example of the first type is the Nambu -Jona-Lasinio (NJL) model [16,17], where ground state in the strong coupling regime is found to be separated by a gap from the excited states (quasiparticles), which are identified with nucleons. Reinterpreting nucleons as quarks, one obtains a low energy meson action from one-loop quark dynamics [18]. The proper-time method is especially useful here [10,11]. Examples of the second type arise under extension of some effective field theory X with symmetry group G to the other effective theory X ′ , when X ′ contains heavy degrees of freedom belonging to some representation of G. At present, such theories are being actively studied in the context of extending the standard model of electroweak interactions [19].
In realistic models one is often confronted with the difficulty that the flavor symmetry is broken by large unequal masses M = diag(m 1 , m 2 , . . . , m f ). In this case the complete factorization of M in the heat kernel is impossible because M does not commute with the rest of an elliptic operator. The consequence is that the Seeley-DeWitt coefficients depend in a complex way on both the fields and the mass-dependent-constants of their interactions. The treatment of such case is known to be an onerous task, especially when both Bose and gauge fields are present in the elliptic operator [20][21][22].
Recently [23,24], a new algorithm based on the propertime method has been proposed for deriving the effective action in a theory with heavy virtual fermions (or bosons) of unequal masses belonging to some representation of the symmetry group G. These short Letters contain the main idea and the final result. Non-trivial calculation details have been omitted due to their complexity. The purpose of the present contribution is to supply all necessary details of such non-trivial calculations, without knowing which it is difficult to be convinced in the validity of the previously stated results.
The new algorithm generalizes the standard large mass expansion of the heat kernel to the case of unequal masses by the formula e −t(M 2 +A) = e −tM 2 1 + ∞ n=1 (−1) n f n (t, A) , (1) where M = diag(m 1 , m 2 , . . . , m f ) is a diagonal mass matrix; t is the proper-time parameter; the expression in the square brackets is the time-ordered exponential OE[−A](t) of A(s) = e sM 2 A e −sM 2 , and A is a positive definite self-adjoint elliptic operator in some background (its explicit form will be clarified later), accordingly If masses are equal, this formula yields the well-known large mass expansion with standard Seeley-DeWitt coefficients a n (x, y) [12]. In fact, formula (1) is an extension of the Schwinger's method used to isolate the divergent aspects of a calculation in integrals with respect to the proper-time [5,7] to the non-commutative algebra.
There is a simple heuristic argument that explains why this formula is also relevant for describing the generalized 1/M series. Indeed, the 1/M expansion is known to be valid when all background fields and their derivatives are small compared to the mass of quantum fields. Therefore, factoring e −tM 2 one separates the leading contribution. The remaining part of the heat kernel may be unambiguously evaluated by expanding it in a power series in t about t = 0. As a consequence, the Seeley-DeWitt coefficients a n receive corrections: a n → b n = a n + ∆a n , where ∆a n vanish in the limit of equal masses.
Currently, there are two methods for deriving quantum corrections induced by virtual states of unequal masses. In [20][21][22], the heat kernel is evaluated on the bases of the modified DeWitt WKB form. This yields a different asymptotic series for the right-hand side of Eq. (1), and, consequently, the different expressions for ∆a n . The approach proposed in [25][26][27] starts from the formula (1), but afterwards an additional resummation of the asymptotic series is applied. This essentially simplifies the calculations, but changes the structure of the 1/M series. As a result, one looses correspondence between a massdependent factor at the effective vertex and a flavor content of the one-loop Feynman diagram which generates the vertex. Here I abandon this resummation.
The utility of the proper-time technique is that it reduces the task of the large mass expansion to a simple algebraic problem which requires less work than one needs for the corresponding Feynman diagrams calculation in momentum space. In the following, we consider a quite non-trivial case of the chiral U (3) × U (3) symmetry broken by the diagonal mass matrix M = diag(m 1 , m 2 , m 3 ) to demonstrate the power of the method. To find the two leading contributions b 1 (x, x) and b 2 (x, x) in the 1/M expansion of the effective action one requires to consider only four terms of the series (1) that results in more than a hundred effective vertices.
The effects of flavor symmetry breaking are currently important in many physical applications: in studies of physics beyond standard model to construct the low energy effective action by integrating out the heavy degrees of freedom [19,28]; in two Higgs doublet models [29] to address the problem of almost degenerate Higgs states at 125 GeV [30,31]; in the low energy QCD to study the SU (3) and isospin symmetry breaking effects [32]. These effects are known to be very important to explore the QCD phase diagram [33], to study a formation of the strange-quark matter [34,35], to study nuclear matter in extreme conditions that arose in nature at the early stages of the evolution of the Universe and in the depths of neutron stars [36,37]. The method described below, in particular, can be used for obtaining low energy meson effective Lagrangian in the framework of the NJL model, as an alternative to the approaches developed in [11,38].
The paper is organized as follows. In Sec. II, we formulate the method and present the basic steps required to construct the desired 1/M expansion. The Sec. III sets out the necessary details related to the calculation of the first two coefficients of the asymptotic series. In Sec. IV, we present the effective Lagrangian describing the self-energy and interactions of zero-spin and one-spin bosons in Minkowski space-time induced by the intermediate one-loop quark diagrams. A short summary and some concluding remarks are given in Sec. V. Many important technical points related to our calculations are collected in six Appendices.

II. PROPER-TIME EXPANSION
In this section we obtain the 1/M series of the effective one-loop action in Euclidian space and explicitly single out the structures necessary for calculating the first two coefficients of this asymptotic expansion.

A. Determinant of the Dirac operator
The logarithm of the formal determinant of the selfadjoint elliptic operator of the second order describes the one-loop radiative corrections to classical theory. In the following, we are interested in the real part of the effective action resulting from the calculation of the determinant of the Dirac operator D in the background scalar s, pseudoscalar p, vector v µ , and axial-vector a µ , fields. The proper-time method cannot be applied directly to fermions, since the Dirac operator D is a first order elliptic operator, and its spectrum is unbounded both above and below. Instead, one should consider the functional representing a real part of the one-loop effective action in Euclidian space as the integral over the proper-time t.
Notice that the Hermitian operator D † E D E is a second order elliptic operator, unbounded above, so we can use the proper-time method to regularize it precisely as for bosons. The integral diverges at the lower limit, therefore, a regulator ρ t,Λ is introduced, where Λ is an ultraviolet cutoff. Since we will carry out calculations in Euclidean space (the subscript E marks this), and the obtained result will be analytically continued to Minkowski space, we will adhere to certain rules of transition from one space to another, which we collect in Appendix A.
For definiteness, suppose that one is dealing with the effective action arising due to integration over light quark degrees of freedom. In this case, the Dirac operator D E in Euclidean four-dimensional space has the form is collected in Y and the covariant derivative d α . In the following we do not need an explicit expression for Y , nevertheless, for completeness, we include this expression in Appendix A. If quarks were massless, the theory would have a global U (3) L × U (3) R chiral symmetry. It is known, however, that the ground state of QCD is not invariant under the action of chiral group. As a result, the entire system undergoes a phase transition accompanied by the appearance of a gap in the fermion spectrum. Quarks get their equal nonzero constituent masses.
Additionally, due to explicit breaking of chiral symmetry, realized through the mass terms of current quarks, one can easily find that the inequality of current masses after spontaneous breaking of chiral symmetry leads to the inequality of constituent quark masses.
Thus, we arrive at a problem in which one needs to study the properties of a system at large scales, i.e., one needs an expansion of the effective action in the inverse powers of large unequal masses of quarks The matrix E i maps the point (u, d, s) of the threedimensional flavor space to the point (u, 0, 0), if i = 1, to the point (0, d, 0), if i = 2, and to the point (0, 0, s), if i = 3. Thus E i is an orthogonal projection onto the flavor space which can be expressed through the λ-matrices: Notice also the following useful relations which are important for our calculations. The first formula is a direct consequence of the fact that the quark mass is given by the diagonal matrix The second formula reflects the projection property of the E i -matrices where (E ij ) mn = δ im δ jn , in particular, E ii = E i . It is true for any flavor matrix A whose entries are given by A ij . Note that here and in the following we sum over repeated flavor indices only when a symbol of the sum is explicitly written out. The orthogonal basis E ij has the properties To advance in the evaluation of expression (3), we use the Schwinger technique of a fictitious Hilbert space [5]. Then a matrix element of a quantum mechanical operator can be treated as The use of a plane wave with Euclidian 4-momenta k, x|k , as a basis greatly simplifies the calculations (details are given in Appendix B) and leads to the representation of the effective action as an integral over the four-momentum k α is a self-adjoint operator in Hilbert space, and the summation over four-vector indices in (15) are implicit. t n a n (x, x) . (16) Here a n (x, x) are the Seeley -DeWitt coefficients a n (x, y) in a coincidence limit x = y, which depend on the background fields and their derivatives, except a 0 (x, x) = 1. Integration over four-momentum and proper-time in (14) is straightforward and we obtain a well-known result where the proper-time integrals J n (m 2 ) are In the case of two subtractions ρ t,Λ = 1 − (1 + tΛ 2 )e −tΛ 2 at the large scale Λ, one finds from (18) The choice of the regularization is closely related to the specific problem under study. Various examples of the proper-time regularization can be found in [12]. The Pauli-Villars regularization we used is usually applied in the NJL model, where the cutoff Λ is a scale of spontaneous chiral symmetry breaking. The functions J n (m 2 ), for n > 1, as m 2 becomes very large are asymptotically equivalent to m −2(n−1) , that is, the expansion (17) is in inverse powers of m 2 . For the series to converge, it is necessary not only that the mass m be large, but also that the background fields change slowly over distances of the order of the fermion field Compton wavelength 1/m. If these criteria are not met, then the production of real quark-antiquark pairs becomes essential, and the expansion is not suitable for applications.

C. The case of unequal masses
Let us return now to the formula (14) and show how one can extend the above tool to the case [M, A] = 0. To make progress in our calculations, we use the formula (1) allowing to factorize the exponent with a non-commuting diagonal mass matrix M . Under a flavor trace, it yields, for instance, The result of calculations for the remaining terms can be represented by the formula where n ≥ 1 and the notation A i ≡ E i A is used. The coefficients c i1i2...in (t) are totally symmetric with respect to any permutation of indices and are easily calculated. The necessary details of such calculations and useful properties of coefficients are collected in Appendix C.
To ensure the fundamental cyclic property of the trace tr(AB) = tr(BA), we define Here we used Eqs. (11) and (12) to calculate the trace. The sum over a cyclic permutation of A ij adds nothing to the standard definition of a trace if A is a matrix. However, in the case when A contains open derivatives, i.e., is a differential operator, cyclic permutation in the trace may change the result. This is why it is necessary to ensure the guaranteed fulfillment of this fundamental property of the trace, which we do with the formula (23). It should be emphasized that here we will restrict ourselves to considering only those terms that survive in the limit M → ∞. In the case of equal masses, this approximation corresponds to considering heat coefficients a 1 and a 2 . To isolate such a contribution, it is necessary to limit ourselves to the terms of order t 2 at most. It means that at the level of A i -dependent expressions one should expand up to and including the t 4 order, since A i has the term ∝ 1/ √ t, Note, that the series is of the mixed type, i.e., it has also exponents depending on t, hidden in the coefficients c i1i2...in (t). In the limit m 1 = m 2 = m 3 = m, these coefficients shrink to c(t) = exp(−tm 2 ) and the massdependent exponent is totally factorized. This way one can recover the standard inverse mass expansion. As it will be shown below, the term with the coefficient c i1i2...in corresponds to the contact contribution of the Feynman diagram with n internal quark lines. It means that c i can be associated with the tadpole contribution, c ij with the self-energy part, c ijk and c ijkl with the triangle and box contributions. The indices contain also information on the flavor content of such one-loop diagrams. For instance, c 123 (t) corresponds to the triangle originated by propagators of up, down, and strange constituent quarks. Now, one can substitute A i = E i A by its expression (15) and integrate over four-momentum k α in (14) by using formula It is evident that the corresponding integral of an odd number of four-momentum k α is zero. Totally symmetric tensor δ α1α2...α2n is determined by the recurrent relation see also Eq. (B8) for details. This yields It remains to put this result into (14) and calculate the integrals over the proper-time. Integration turns the coefficients c i1i2...in (t) into functions J i1i2...in depending on the masses of the fermion fields and the cutoff Λ. These functions describe the leading contributions in the 1/M 2 expansion of the corresponding one-loop Feynman diagrams, to be precise, their local parts which dominate in the limit M → ∞. Here we present the result of such calculations. All necessary details can be found in Appendix D.
where the index n indicates the asymptotic behavior of b n (x, x) at large masses, namely b n ∼ M −2(n−2) . The coefficients b n (x, x) depend on the external fields and quark masses, i.e., they contain information about both the effective meson vertices and corresponding coupling constants. If all masses are equal, the dependence on m is factorized in form of the integral (18) and the fielddependent part takes a standard Seeley-DeWitt form a n (x, x).

III. LEADING TERMS OF THE 1/M EXPANSION
Consider the leading terms b 1 and b 2 . The case n = 0 is of no interest because b 0 contains no fields and can be omitted from the effective action. The coefficients with n ≥ 3 tend to zero in the limit of infinite masses, therefore they are small in comparison with b 1 and b 2 and, at the first stage, can be neglected.
Let us turn to the calculation of the coefficient b 1 . It is given by a part of the expression (28), which is proportional to the proper-time integrals J 0 (m 2 i , m 2 j ). With the use of Eq. (E1), it can be rewritten as Noting that and taking into account that the action of the open derivatives in (29) on the implied unit on the right hand side of Eq. (28) gives zero, we find Since the last term here is a total divergence which can be omitted in the effective action, we conclude where on the last stage we used Eq. (12) and the fact that matrix A may be written in a unique way as a finite linear combination of elements of A in the bases E ij , namely A = m,n A mn E mn . In the same manner, we find It gives for Eq. (29) Let us use now the easily verifiable relation to obtain finally where ∆J 0 (m 2 i , m 2 j ) is given by Eq. (D14). The latter integrals can be collected in the symmetric 3 × 3 matrix ∆J 0 , which has zeros on the main diagonal. In particular, when m i = m j = m, all elements of ∆J 0 vanish. In this specific case, the first term of (36) leads to the well known expression of the Seeley-DeWitt coefficient For convenience of writing the result of our calculations, along with the usual matrix multiplication, we will use the non-standard Hadamard product [39], which is the matrix of elementwise products The Hadamard product is commutative unlike regular matrix multiplication, but the distributive and associative properties are retained. It has previously been proven to be a useful tool when the mass matrices of the type (6) are involved [40]. In terms of Hadamard product the result (36) can be written as where J 0 is considered as a diagonal matrix with elements given by (J 0 ) ij = δ ij J 0 (m 2 i ). This matrix contains contributions of the Feynman one-loop diagram, known as a "tadpole".
Consider the second coefficient b 2 (x, x). In accordance with the general structure of the expression, we distinguish three contributions differing in the degree of Y which is explicitly indicated in the parentheses.
where J is a symmetric 3 × 3 matrix, whose elements J ij = J 1 (m i , m j ) are logarithmically divergent parts (at Λ → ∞) of Feynman self-energy diagrams with masses of virtual particles m i and m j (see Eq. (D6)).

Linear part in
After evaluation of traces it gives b (1) where Next, let us use Eqs. (E1) and (E6) of Appendix E. We obtain To evaluate the first sum in this expression, we employ the explicit form of t-coefficients (43) The first term is now computed as This is a four-divergence of a four-vector. Therefore, it can be omitted from the effective action, because it does not affect the dynamic characteristics of the effective theory. Thus, we find where we used the cyclic permutation of the elements to simplify the right hand side of this expression, that is not forbidden due to the trace, tr D , in (28). In what follows, we will always use the cyclic permutation property when calculating the t-coefficients.
To evaluate the second sum in (44) we need to find the difference 3t where, on the last stage, we took into account that Noticing that expressions vanish under the Dirac matrix trace in (28), we finally obtain where i = j = k.
Since the first term in (51) has the same form as (47), it follows that We need now to calculate the coefficient where the left-right derivative is defined in a standard The coefficients t (123) are required for the evaluation of the last term in (44). They can be obtained with the aid of the easily verifiable relations, As a consequence, we have If we take into account the existence of similar term in the coefficient (51), we obtain Thus, the final result for b where we introduce the notation Note that elements of the matrix Γ α ij commute with each other. In the last line of Eq. (58), R and N are the 3 × 3 matrices with elements In the limit of equal masses, the coefficient b Here the field dependent t-coefficients are given by the following formulas.
Before of calculating (61) directly, we shall first handle the case of equal masses m u = m d = m s = m, which can easily be obtained from (61) after factorizing the common where we used Eq.(49), and To obtain the exact result for (61) it is convenient to consider separately the cases when the sum runs only over the same index values, and then over two and three different values of the flavors The part of b 2 that corresponds to a contribution with coincident indices is Introducing the notation we can rewrite (66) as Now we benefit from the following easily established relations That gives finally To obtain this expression, we, as before, discarded a full divergence of a four-vector ∂ 2 Q ii . Now consider the terms of the sum (61) with two unequal indices. These comprise of the remaining terms of the first sum and the corresponding contributions from the second and third sums where S ij is given by Eq. (D26).
To evaluate the first sum, we note that Here we have omitted a full divergence of a four-vector, and took into account that the action of the derivative that reached the right end of the expression gives zero. In the anticommutator {(d 2 ) ij , (d 2 ) ji }, only the symmetric part of Eq. (72) contributes to the sum. Therefore, the second line of (72) has no influence on the result, and we obtain Recall again that elements of the matrix Γ commute with each other. Combining where the properties of t-coefficients (F12)-(F14) have been used, we learn that this sum does not contain the terms with one derivative. Indeed, We also know from (F10) that Thus, only the coefficientt + iij contains terms with one derivative. The expression for this coefficient is given in Appendix F (see Eq. (F7)).
Before we turn to the full expression for the coefficient β (2) 2 let us first collect terms with two derivatives in β Combined together with similar terms in β 2 they give the kinetic term of spin-1 fields In the case of equal masses, the first term coincides with the standard result (63). It means that the last two terms of (77) after combining them with other contributions to b To see this we need to consider the contribution β 2 . On referring to Eqs. (61) and (E7), we find Using the explicit expression for the coefficients t (iijk) (F19) and Eq. (D22), one can represent the first sum as If we introduce the antisymmetric matrix K with elements then the first part of the expression can be presented in the index-free form where the off-diagonal elements of matrix E γσ are Due to Eq. (79), the diagonal elements of the matrix E γσ do not contribute to (80). It is also clear from (79) that terms with one derivative (80) vanish in the limit of equal masses. Therefore, they cannot cancel the unwanted second term in (77).
Since the coefficientt (123) is known from Eq. (F9), we can give a complete expression for β Now it is necessary to make sure that the terms linear in F αβ vanish in the case of equal fermion masses. To this end let us consider the corresponding contributions to b (71), from the second term of (77) and fromt (123) of Eq. (78). Combined together they give where the flavor matrix H αβ has elements One can see that all elements of this matrix vanish in the limit of equal masses. Our next step is to demonstrate the fulfilment of the limiting condition for terms of the fourth order in Γ. As we already know these terms should also vanish for equalmass case.
In the first step, we transform the following expression Here the first sum is taken from β   (from the first andt + iij terms of (71) correspondingly). This expression can be made into a somewhat more simple form where the matrix H αα is a particular case of (84), and the elements of the matrixĤ αα arê Under the flavor trace in (85), we explicitly singled out the contribution H αα that vanishes in the case of equal masses. The last term in (85) is completely canceled by the corresponding contribution fromĤ αα . Indeed, it is straightforward to see At the second stage, we collect three contributions: the part of β (2) 2 associated with the coefficient (74) the last term in Eq.(77) Together with the terms just considered in (87), they should lead to a result that vanishes in the limit of equal masses. Indeed, it is evident from the direct calculation of these terms that the summation gives thirty three independent combinations of Γ's, which can be collected in the following sums Here, the expression in the parentheses (Γ ij Γ ji ) is understood to be summed over alpha Γ α ij Γ α ji . In the case of equal quark masses (91) is zero, since all J-dependent factors vanish.
It remains to calculate the sums containing the coefficients R ij and S ij in (71). For that we need to know the coefficients 3t − iij − t − iiij and t + iiij − 3t (iijj) . From Eqs. (F8) and (F16) one finds where it is assumed that k = i = j. Then, due to the antisymmetry of R ij , we find Now, from Eqs. (F16) and (F18) one obtains By virtue of the symmetry S ij = S ji , the required sum is Now we have all the base ingredients that remain to be assembled into a single whole.

Full result for b2(x, x)
Here we summarize the above results collecting them in a compact form convenient for further applications. Namely, the result obtained for b 2 (x, x) can be represented as In the limit of equal masses, the first two terms in (96) yield the well-known result [12], and the third one vanishes, that is, it contains only contributions associated with an explicit violation of chiral symmetry. Together with the second term of Eq. (36), this is the main result of our work.
To write down the full expression for ∆b 2 , we distinguish four different contributions Here b (1) 2 is the sum of all terms linear in Y , and Ω (n) is the sum of terms with n derivatives which consists of only spin-one fields, entering into Γ α .
For b (1) The terms with two derivatives are The first term makes an additional contribution to the kinetic part of the effective Lagrangian of spin-1 fields described by the second term in (96). Recall, that the symmetric matrix S is defined by S ij = J ij − J iijj (D26). The shorthand ∂Γ ≡ ∂ α Γ α implies summation over omitted indices α. In applications, one can omit the second term in (99) to ensure that the energy of the massive spin-1 field is positive definite. The terms with one derivative can be collected in the expression which represents the effective three-particle vertices describing the local interaction of vector and axial-vector fields.
The term without derivatives Ω (0) , which is a sum of the terms ∝ Γ 4 , we represent in the form where Ω 1 is given by Eq. (91), and Here, the expression in the parentheses (Γ ij Γ ji ) is understood to be summed over alpha Γ α ij Γ α ji .

IV. CONTINUATION TO MINKOWSKI SPACE
Let us write down the Lagrangian of the effective theory arising from the considered 1/M expansion (28) after continuation to Minkowski space-time The index M explicitly indicates that coefficients b n belong to Minkowski space. The rules of such continuation are given in Appendix A. The coefficient b 1 (x, x) given by Eq. (38) in Minkowski space-time has the following form In particular, Γ µ = v µ + γ 5 a µ , and where The coefficient b 2 (x, x) in Minkowski space-time can be written as where is a representation of (64) in Minkowski space-time. For the contributions to ∆b 2 we have b (1) In a similar way we find where (∂Γ) is understood now as ∂ µ Γ µ , and Ω (1) where g µνμν = g µν gμν + g µμ g νν + g µν g νμ . We will not write out here an expression for Ω (0) , which contains the terms ∝ Γ 4 , since here, to go to the Minkowski space, it is sufficient to replace the indices, while the sign in front of any term remains unchanged.

V. CONCLUSIONS
Above, a new method for obtaining the effective lowenergy Lagrangian in a theory with heavy particles of unequal masses and a non-trivial flavor symmetry group has been presented in a full detail. Unlike [20], where authors use the modified DeWitt WKB form and solve recurrent Schrödinger-like equations to determine the heat coefficients, our calculations are based on the formula (1), which allows us to resum the proper-time series in accordance with the contributions of corresponding one-loop Feynman diagrams with the required number of external fields. As a result, we arrive at an effective action (28), in which each of the heat coefficients can be calculated independently, and corresponds to a certain order in 1/M 2 expansion. We have explicitly calculated two leading coefficients b 1 (x, x) and b 2 (x, x) for the case of the broken U (3) × U (3) chiral symmetry. The result can be used to study the consequences of SU (3) symmetry violations, for instance, in the framework of the NJL-type models, where, until now, rather rough approximations are used to describe this important effect. Although we have limited ourselves to a particular example, the proposed approach can be easily extended to the case of an arbitrary internal symmetry group, for example, to study an explicit and spontaneous symmetry violation within the context of effective field theories beyond the Standard Model. In this Appendix, we collect some useful formulas and conventions made to perform a Wick rotation of the time axis to transit from Minkowski space-time with the metric tensor g µν = diag (1, −1, −1, −1) to Euclidean space with metric δ αβ = diag (1, 1, 1, 1) and back. Here the first two letters of Greek alphabet α, β are used to enumerate the coordinates in Euclidean space α, β = 1, 2, 3, 4, on the contrary, the letters µ, ν, run over the set µ, ν = 0, 1, 2, 3 in Minkowski space-time.
For coordinates x µ and the Lorentz four-vector v µ we apply the following correspondence rules with the Euclidean coordinates x α and the vector components v α It follows then from (A2) and (A3) that For the Dirac γ-matrices we use the following conventions Our choice corresponds to the case when O(4) gamma matrices γ α in the four-dimensional Euclidean space are anti-Hermitian. Indeed, from the standard property γ µ † = (γ 0 , − γ) and definitions made above one finds that γ † α = −γ α . As a consequence, the Euclidean Clifford algebra emerging from {γ µ , γ ν } = 2g µν necessitates a change in the defining equation The other consequence is that γ 5 = −iγ 0 γ 1 γ 2 γ 3 matrix can be defined in Euclidean space as γ 5E = γ 1 γ 2 γ 3 γ 4 . It follows then that γ 5 → γ 5E , and both matrices are Hermitian.
For the Dirac operator where d µ = ∂ µ − iΓ µ , and Γ µ = v µ + γ 5 a µ , one finds where d α = ∂ α + iΓ α , Γ α = v α + γ 5E a α . This yields where and the chiral-covariant objects are We have given these well-known formulas here both for the completeness of our presentation and to clarify notations used in the main text of our work.
where ∆ x is the operator ∆ in the coordinate representation.

Now note that
where Since the operator ∆ x contains derivatives of at most second order, its higher order commutators with ipx are equal to zero. At the last stage we used Eq.(A9), namely As a result we obtain or, for the trace where tr I means the trace over color, isospin, and Dirac indices. Expanding the last exponent in the series in the proper-time, one can take the integrals over fourmomenta with the help of the formulas Note that only even number of internal momenta p α contribute, the odd number, obviously, gives zero; the totally symmetric tensor δ α1α2...α2n is a product of Kronecker's deltas δ α1α2 δ α3α4 . . . δ α2n−1α2n plus all possible permutations of indices; there are (2n − 1)!! terms in total. For example, Finally, combining (3) and (B6) we come to the Eq. (14).
Appendix C: Proof of the formula (22) Here we present some details related with obtaining of Eq. (22). Let us first consider the case when n = 1 Hence, For n = 2 we have Here ∆ ij = m 2 i − m 2 j , and on the last stage we used that trace is symmetric under an exchange i ↔ j. It gives the sum includes two terms obtained by cyclically permuting the indices (i 1 , i 2 ) → (i 2 , i 1 ). For n = 3 Since the trace is symmetric with respect to the cyclic permutation of matrices, one can add to this expression two others, obtained by cyclic permutations of indices (i, j, k) → (j, k, i) and (i, j, k) → (k, i, j), and the result should be divided by three. As a result, only the first term in parentheses will lead to the non zero contribution. This yields the following expression for the coefficient Consider the case n = 4. Here we have By using again the cyclic permutation property of trace, one can easily find out that only the first term in parentheses will contribute with a non zero result. Here we have four cyclic permutations of indices, correspondingly it gives four terms for the coefficient Continuing the above procedure, we can find by induction an arbitrary coefficient in formula (22) Although the sum includes n terms obtained by cyclically permuting the indices i 1 , i 2 , . . . , i n , it is easy to show that the coefficients c i1i2...in are totally symmetric. One can also establish a recurrence formula relating coefficients of order n, c i1i2...in (t), for n > 1, with coefficients of order n − 1 (C10) This relation follows from the following easily verified identities . . . . . . . . . . . . . . . . . . . . .
Recurrent formula (C10), inter alia, simplifies the proof of the relations Indeed, one has, for instance, Using (C10) we can extend this to any c ii...i (t) case.

Appendix D: Integrals over proper-time
If the masses of the Fermi fields are non-degenerate, integration over the proper-time leads to expressions depending on the specific masses of the constituent quarks in the corresponding Feynman diagram. Accordingly, the number of different loop integrals increases. We will classify the integrals in accord with the number of internal quark lines presented in the loop diagram. Since we are dealing with an 1/M expansion, the integrals presented below give only the leading terms of this expansion. Along with separating the main divergences of the Feynman diagrams, such integrals contain finite contributions that vanish in the case of equal quark masses. These finite contributions are uniquely determined by the coefficients c i1i2...in that are completely symmetric with respect to the permutation of any two indices.

Tadpole diagrams
The diagrams with only one internal quark line (tadpole) are associated with the proper-time integral We will use Pauli-Villars regularization corresponding to two subtractions at the mass-scale Λ In this case, it follows that This obviously coincides with (19).

Self-energy diagrams
The quark-loop self-energy graphs are described by the proper-time integrals Due to the symmetry of the coefficients c ij = c ji the result of integration is symmetric with respect to the re-placement m i ↔ m j . Indeed, in the case n = 0, we have where ∆ ij = m 2 i −m 2 j . In the coincidence limit m i → m j , lim mi→mj J 0 (m 2 i , m 2 j ) = J 0 (m 2 i ) .
The other integral for n = 1 is given by the difference of the tadpole contributions Since we also have expressions for the similar J 1 -type contributions coming from the triangle and box diagrams, we will collect them together in one formula From Eq. (C14), it follows that The integrals take this form when all quark masses in the loop are equal.

Triangle diagrams
The proper-time integral dt t ρ t,Λ c ijk (t) = J 1 (m 2 i , m 2 j , m 2 k ) ≡ J ijk (D9) describes a local part of the diagram with three internal quark lines. As before, the full symmetry of the coefficient c ijk allows one to conclude that this integral does not change when any two fermion masses are exchanged.
To verify this, we use the formula (C10), which at the first stage of calculations leads to a seemingly asymmetric result where J 0 (m 2 i , m 2 j ) is given by Eq. (D5). However, the direct calculation of the right hand side of this expression shows explicitly its total symmetry J 1 (m 2 i , m 2 j , m 2 k ) = of equal masses. There are two of them: J iijj and J iiij . As a result, we have From (D6) and (D23) we find the relation where The latter expression is symmetric under exchange of masses and finite at Λ → ∞. Eq. (D24) describes the six possible variants of the distribution of quarks along the inner lines of the box diagram, these are uuud, uuus, dddu, ddds, sssu, and sssd. In particular, it follows that where R ij is given by (D17). On the other hand, the sum of these integrals is J iiij + J jjji = 2J 1 (m 2 i , m 2 j ) + S(m 2 i , m 2 j ) .

(D28)
Appendix E: Mixed sums for bn(x, x) In the main text we make use summations which contain both the mass-dependent remnant, J i1i2...in , of one-loop quark integrals and the corresponding fielddependent coefficients T i1i2...in . The result of such summations depends on the properties of J i1i2...in which were established in Appendix D. Here some important results on these sums are presented.
The simplest case is given by the sum Therefore From (F15), we obtain where δ ++ = 1, and δ −+ = 0. Next, we calculate the coefficient t (iijj) . Using the formulas (E8), (F13) and (F14) it is easy to establish that The formula (F10) gives now Finally, consider the coefficient t (iijk) . In accord with the definition (E8) and the cyclic property (62) we have In conclusion, we recall that the coefficients depending on Y : t ij and t ijk were calculated in the main part of the work.