A general framework to diagonalize vector–scalar and axial-vector–pseudoscalar transitions in the eﬀective meson Lagrangian

A new mathematical framework for the diagonalization of the nondiagonal vector–scalar and axial-vector–pseudoscalar mixing in the eﬀective meson Lagrangian is described. This procedure has unexpected connections with the Hadamard product of n × n matrices describing the couplings, masses, and ﬁelds involved. The approach is argued to be much more eﬃcient as compared with the standard methods employed in the literature. The diﬀerence is especially noticeable if the chiral and ﬂavor symmetry is broken explicitly. The paper ends with an illustrative application to the chiral model with broken U (3) L × U (3) R symmetry.


I. INTRODUCTION
The QCD Lagrangian with n massless flavors is known to possess a large global symmetry, namely the symmetry under U(n) V × U(n) A chiral transformations of quark fields. It has been shown by Coleman and Witten [1] that, in the limit of a large number of colors N c , under reasonable assumptions, this symmetry group must spontaneously break down to the diagonal U(n) V . Consequently, the massless quarks get their constituent masses M 0 , and massless Goldstone bosons appear in the spectrum [2][3][4]. These non-perturbative features of the QCD vacuum can be modeled in analogy with the phenomenon of superconductivity [5,6]. For that, one should regard the constituent quarks as quasiparticle excitations and the mesons as the bound states of quark-antiquark pairs. The dynamics of such bound states is described by the chiral effective Lagrangian [7][8][9][10][11].
Were the axial U(n) A symmetry exact, one would observe parity degeneracy of all states with otherwise the same quantum numbers. Due to the mechanism of spontaneous symmetry breaking, described by Nambu and Jona-Lasinio (NJL), the mass splitting occurs between chiral partners, e.g. m a 1 /m ρ = √ Z, where Z ≃ 2 in accordance with a celebrated Weinberg result [12][13][14]. In fact, the splitting between J P = 1 − and 1 + states is a result of the partial Higgs mechanism: the A µ ∂ µ φ mixing term which appears in the free meson Lagrangian [15] after spontaneous symmetry breaking must be canceled by an appropriate redefinition of the longitudinal component of the massive axial-vector field A µ = A ′ µ + κ∂ µ φ. The result is that the axial field A µ "eats a piece" of the Goldstone boson φ and "gets fat": m 2 a 1 − m 2 ρ = 6M 2 0 . Here the quark mass may be expressed in terms of observable values: f π (the pion decay constant) and g ρ (the ρ → ππ decay constant), namely 6M 2 0 = Zg 2 ρ f 2 π .
It is commonly believed that axial-vector fields A µ , defined in the symmetric vacuum, and A ′ µ , defined in the non-symmetric vacuum, should have the same chiral transformations [16]. As a consequence of that rather natural idea one must use the covariant derivative ∇ µ φ in the replacement above and write A µ = A ′ µ + κ∇ µ φ. Such derivative contains non-linear field combinations. Thus, upon substitution in the quadratic form to be diagonalized, new meson interaction terms emerge which are not present in the original Lagrangian. Although, in principle, there is no reason to object to new interactions as long as they fulfill the symmetry requirements, they are subleading in large N c counting as compared to the mixing which occurs at leading order. If one restricts the analysis to leading order in N c , i.e. to the level of the free meson Lagrangian, chiral symmetry will not be supported.
Recently [17,18], it has been shown that the linear replacement A µ = A ′ µ + κ∂ µ φ changes the chiral transformation laws of the axial-vector field A ′ µ as compared to the A µ ones, though in a way that leaves the group properties intact. The special thing about these new transformations is their dependence on the classical parameter M ′ which is a diagonal n × n matrix in the n-flavor space; in the non-symmetric ground state its eigenvalues are all equal and nonzero M ′ = diag(M 0 , M 0 , . . . , M 0 ); in the symmetric vacuum, M 0 = 0 and the transformations coincide with the standard ones.
In nature, chiral symmetry is also broken explicitly by the current quark masses m = diag(m u , m d , m s ) (it has been shown in [19] for QCD with two flavors that isospin is not spontaneously broken; in the following, we will consider mainly the case n = 3, although our discussion is valid for an arbitrary number of flavors n). Due to the current quark masses, the U(n) V symmetry breaks down to U(1) n V . Then, it follows from the gap equation that the constituent quark mass matrix M is diagonal but its eigenvalues are all unequal and nonzero M = diag(M u , M d , M s ). This leads to a new mixing between the vector, V µ , and the scalar, σ, fields and, as a result, to the redefinition of the longitudinal component of the vector field: V µ = V ′ µ + κ ′ ∂ µ σ. This makes the vector field V ′ µ heavier. It is the purpose of this letter to point out that the result [17,18]  We also demonstrate that the linear replacement of variables that we found has unexpected connections with the Hadamard product of n × n matrices describing the couplings, masses, and fields. That allows us to reformulate the diagonalization procedure entirely in terms of the Hadamard product, in contrast to the conventional methods used in the literature which we refer to as being standard.
To find the chiral transformations of the meson fields in the non-symmetric vacuum we use the NJL Lagrangian which includes both spin-0 and spin-1 U(3)×U(3) symmetric four-quark interactions. It is known that this Lagrangian undergoes dynamical symmetry breaking [26].
It also reproduces the qualitative features of the large-N c limit. The model describes both phases and gives a solid framework for the study of the transformation laws of the qq The source of interest regarding the chiral transformations of spin-1 fields resides in the use of these fields in the effective Lagrangians describing the strong interactions of hadrons at energies of ∼ 1 GeV [23,24,27]. Presently these theories are actively used in studies of τ -lepton decay modes [28], e + e − hadron production [29], and the QCD phase diagram [30,31]. The chiral transformations we suggest allow for greater freedom in the construction of such Lagrangians by noting that one may use different representations for these transformations as long as they obey the same group structure. In this context it is important that transformations include the flavor symmetry breaking effects while not spoiling the symmetry breaking pattern of the theory. The latter is very important for controllable calculations.
In Section 2 we study the NJL model of quarks with the U in hot/dense/magnetized matter [32,33], or apply it in the study of hybrid stars [34], where eight-quark interactions seem to have an important impact.

II. CHIRAL TRANSFORMATIONS OF MESON FIELDS
We start from the quark version of the original NJL model [5,6] with nonlinear fourquark spin-0 and spin-1 interactions which allow a chiral group and where there is considerable freedom in the choice of auxiliary fields in the vector and axial-vector channels. The Lagrangian density of the model is G S and G V are universal four-quark coupling constants with dimensions (length) 2 and m is the current quark mass matrix. The symmetry group G acts on the quark fields q (in this notation the color and flavor indices are suppressed) as follows Here α = α a λ a /2, β = β a λ a /2, a = 0, 1, . . . , 8 and α a , β a ∈ R; λ 0 = 2 3 1 with 1 being a unit 3 × 3 matrix and λ a (a > 0) are the usual SU (3) Gell-Mann matrices with the following basic trace property tr(λ a λ b ) = 2δ ab ; the resulting infinitesimal transformations are This symmetry is explicitly broken due to non-zero values of the current quark masses Following [7], one may wish to introduce auxiliary fields σ a = G S (qλ a q), φ a = G S (qiγ 5 λ a q), , and resort equivalently (in the functional integral sense) to the theory with the Lagrangian density with D being a Dirac operator in the presence of mesonic fields where m a is defined by m a = 1 2 tr(mλ a ) with m = m a λ a = diag(m u , m d , m s ). The latter implies that or equivalently The symmetry is now These transformation laws are valid in the symmetric Wigner-Weyl realization of chiral symmetry, where σ = 0. The generators possess a Lie algebra structure δ [1,2] φ = i α [1,2] , φ − β [1,2] , σ = [δ 1 , δ 2 ]φ, δ [1,2] V µ = i α [1,2] , V µ + i β [1,2] , where iα [1,2] III. V −σ AND A−φ MIXINGS AND GENERAL LINEAR SHIFT For further progress we have to bosonize the theory to get profit from the 1/N c expansion.
In the large-N c limit the meson functional integral of the theory (5) where i = u, d, s and Λ is an intrinsic cutoff of the NJL model. The current quark masses m i affect (through the gap equation) the constituent quark masses M i which accumulate the explicit and flavor symmetry breaking effects enhancing them. In particular, in the chiral Since a dynamically broken symmetry is not spoiled in the Lagrangian, we can expand around a non-zero vacuum expectation value of σ without breaking the symmetry. For that we perform the shift σ → σ + M, leading to The equation (8) must still hold, now in the form because otherwise the symmetry breaking pattern (7) will be not preserved. This gives us the modified transformation laws of spin-0 fields in the non-symmetric phase We can still associate the infinitesimal transformations (19,20) with the chiral group G, because M does not ruin the symmetry algebra of G given by (14,15). This can be easily checked. On the other hand, the existence of such a symmetry shows that the part of the effective meson Lagrangian following fromqD M q (after integrating over the quark degrees of freedom in the generating functional of the theory) includes vertices with explicit and flavor symmetry breaking effects which are still altogether G-invariant as expressed in (18).
The results above have no consequences for the transformations of spin-1 fields in the non-symmetric phase. The equations (12) and (13) agree with the requirement (18).
Let us now turn to the vacuum-to-vacuum transition amplitude of the model beyond the mean-field approximation To obtain the effective meson Lagrangian one should consider the long wavelength expansion of the quark determinant, detD M , the formal expression for the path integral over quarks.
The appropriate tool here is the Schwinger-DeWitt method [35]. This yields the local lowenergy effective meson Lagrangian. Renormalizing the meson fields by bringing their kinetic terms to the standard form (e.g., φ R = gφ, where g ∼ 1/ √ N c ), one arrives at the picture corresponding to the large N c limit: the free parts of the meson Lagrangian count as g 2 N c ∼ N 0 c , the three-meson interactions as g 3 N c ∼ 1/ √ N c , the four-meson amplitudes as g 4 N c ∼ 1/N c [8,9].
Obviously, the V−σ and A−φ mixing arises at N 0 c order: the mixing is described by vertices proportional to tr (A µ {M, ∂ µ φ}) and tr (V µ [M, ∂ µ σ]). The first one is a result of spontaneous symmetry breaking, while the second one is a direct consequence of the flavor symmetry breaking enforced in the broken phase. Both lead to additional contributions to the kinetic terms of pseudoscalar (through the transitions ∂φ → A → ∂φ) and scalar (through the transitions ∂σ → V → ∂σ) states. Consequently, these fields must be renormalized again to the standard form. In this case one gets correct expressions for the masses of spin-0 states.
Alternatively, one may wish to eliminate the mixing and diagonalize the free Lagrangian by the replacements where the entries of X µ and Y µ are appropriate combinations of spin-0 fields. They should also depend on M, as it is required by the mixing terms. These replacements introduce further mixing terms through the V µ and A µ mass terms which must add up to zero in the end, fixing the coefficients of the combinations used in (22). In principle, there are an infinity of possible physically equivalent replacements in the form of (possibly infinite) sums of field products. According to Chisholm's theorem [36,37], all such redefinitions yield the same result when computing observables as long as they preserve the form of the free part of the Lagrangian. This reasoning ensures us that we may always restrict to the minimal necessary terms for our intended purposes, i.e. to linear field combinations.
From the point of view of the 1/N c expansion, the minimal replacement in (22) is unique.
All others include additional nonlinear combinations in fields, but must coincide with (22) in their linear part, i.e. at N 0 c order (see also discussion around eq.26). This is a direct consequence of the fact that the mixing terms have their origin at the level of the free Lagrangian.
We may now require that the replacement (22) does not violate the symmetry condition (18). This is ensured if the spin-1 states transform like Gathering separately the factors multiplying γ µ and γ µ γ 5 we conclude that (23) is equivalent These transformations must preserve the algebraic structure of the chiral group G, i.e. the composition properties of the group which are specified in (15).
It seems natural to require that V ′ µ and A ′ µ have the same transformation properties as V µ and A µ . From (24)- (25), it follows then that X µ and Y µ need to be chiral partners and should also transform like V µ and A µ . This would disable the linear solution X µ ∝ ∂ µ σ and Y µ ∝ ∂ µ φ due to the transformation properties of spin-0 fields (19) and (20). Instead, one would look for nonlinear combinations of fields for X µ and Y µ which respect the symmetry transformations (12) and (13) and contain the linear terms necessary for the diagonalization.
For instance, if we assume that flavor symmetry is unbroken, we may use the solution [25], where the constant κ is fixed by a diagonalization condition. The nonlinear terms essentially modify the interaction part of an effective meson Lagrangian without physical consequences [17,36]. However, if the flavor symmetry is broken the replacement (26) does not solve the problem. This is why such natural replacements are not efficient in the direct calculations.
Now, let us consider the minimal replacement in (22). In this case, we are faced with the opposite situation: it makes calculations as simple as possible, but one should take care in justifying such a replacement. Indeed, in this case, as it follows from (24)-(25), the fields V ′ µ and A ′ µ cannot transform like V µ and A µ . Can this introduce some spurious symmetry breaking and change the physics of spin-1 states? To answer the question we compute the commutators δ [1,2] V ′ µ and δ [1,2] A ′ µ . The symmetry will be respected if the commutators depend only on the parameters of the infinitesimal chiral transformations α [1,2] and β [1,2] , i.e., leaving the group composition properties (15) unchanged. We have One can see that the chiral group structure will be preserved overall as long as X µ and Y µ can be chosen in such a way that their transformation laws obey the algebraic structure of G as well This clearly defines the freedom one may have in the choices of X µ and Y µ . In particular, it is not forbidden for them to transform like spin-0 chiral partners X µ ∼ ∂ µ σ and Y µ ∼ ∂ µ φ as it follows from the diagonalization procedure of the considered NJL model. Another interesting case has been considered in [17,18], where X µ = 0, but Y µ = 0 (the case without flavor symmetry breaking).
Any point made so far on the chiral transformation properties of fields in matrix form may be carried over to a formulation based on individual matrix entries. If the Lagrangian contains mixing terms in the form tr (V µ [M, ∂ µ σ]) = tr (V µ (i∆ M • ∂ µ σ)) and , it suffices to define X µ and Y µ as Here, k and k ′ are symmetric coefficient matrices in a flavor space whose entries should be fixed from the Lagrangian diagonalization requirements; ∆ M and Σ M are mass-dependent matrices defined as and the symbol • stands for the Hadamard (or Schur) product (see e.g. [38]) defined as without summation over repeated indices. This product is commutative unlike regular matrix multiplication, but the associative property is retained, as well as the distributive property over matrix addition, i.e.

IV. APPLICATION TO AN SU (3) CHIRAL MODEL
Now, let us consider a physical application of the method presented above. For that we chose a recently proposed effective U(3) × U(3) chiral model [39]. It extends the model [9] presented in the text by including the U (1) A breaking 't Hooft interaction [40], eight-quark interactions and systematically taking into account the explicit and flavor symmetry breaking effects. Our choice is motivated by the growing interest in the eight-quark interactions in hadronic matter, including the physics of stars, and by the importance of the axial anomaly and the flavor symmetry breaking effects in the study of the QCD phase diagram.
In both models [9,39] we arrive, after bosonization, to the same mixing terms Here, the trace is to be taken in flavor space; ̺ 2 is a constant cutoff dependent factor related with the evaluation of the quark determinant, and M is the constituent quark mass matrix, group. This gives with ∆ M , Σ M as defined in (31). For simplifying this expression we have used the fact that, for any U(3) symmetric matrix S (e.g. Σ M ) or antisymmetric matrix Ω (e.g. ∆ M ) and any other B, C ∈ U(3), it is always true that This form for the mixing terms provides a direct hint to the adequacy of the forms (30) for diagonalizing the Lagrangian. Carrying on such replacements will induce from V µ and A µ new similarly shaped mixing terms with k, k ′ appearing as adjustable coefficients.
The model's mass terms in the unshifted Lagrangian may be expressed as for V µ and for A µ . Here, H (1) and H (2) are symmetric U(3) matrices. In the model [9] H (1) = H (2) ∝ 1.
The model [39] leads to a more general form of H (1) and H (2) which makes the standard diagonalization procedure algebraically heavier.
After replacing the fields according to (30), we get the following additional mixing terms: The Hadamard power is used here and it stands for The cancellation of V −σ mixing requires Since the latter sum must vanish, if we equate to zero the coefficients of the independent combination ∂ µ σ ij V µ ji we obtain for any i, j. Due to the antisymmetry of ∆ M , this condition is always satisfied for i = j independently of k values. For i = j, we have This expression defines the values of k entries which diagonalize the Lagrangian, and show us that k is a symmetric matrix coinciding (after some renormalizations of fields) with the known result [9], for H (1) ∝ 1.
A convenient way to write these results in matrix form is where the Hadamard inverse has been used; its definition may be given as (A •−1 ) ij = (A ij ) −1 .
It may be checked that these results are in complete agreement with the previously obtained values for k, k ′ coefficients in [39].
We remark that the standard treatment of the problem in [39] requires the analytic manipulation of expressions involving something like 10 or more flavor indices which are contracted among themselves in non-trivial ways. This can easily become a cumbersome and error-prone calculation. Furthermore, the previous form for X µ and Y µ obscures the fact that each matrix entry of the spin-1 fields is effectively shifted by a single entry of the spin-0 matrix field; this is made explicit within the formalism presented here. The present formulation yields all the results in an efficient and closed-form way.

V. CONCLUSION
Resorting to arguments pertaining to the Lie algebra associated with chiral transformations and to Chisholm's theorem, we have shown that one may always use the most general linear shifts of V µ and A µ fields (22) for dealing with V − σ and A − φ mixing in chiral models without compromising the chiral symmetry properties of the Lagrangian, as long as one admits the corresponding new transformation laws (24,25) for the shifted fields. This result is independent of the number of flavors and works even when the U(n) × U(n) chiral symmetry is explicitly broken to U(1) n .
Although our arguments have been presented using an NJL-type quark model as a starting point, we expect that our proposed strategy for dealing with the mixing terms is fully applicable to other kinds of chiral models such as the linear sigma model [41], massive Yang-Mills models and so on [23]. This is justified if one understands that the form of these mixing terms is fundamentally constrained by the symmetry requirements which should, in principle, be the same in any effective chiral model for mesons. A particular shifting scheme