Dynamical generation of field mixing via flavor vacuum condensate

In this paper we study dynamical chiral symmetry breaking of a generic quantum field theoretical model with global chiral flavor symmetry. By purely algebraic means we analyze the vacuum structure for different symmetry breaking schemes and show explicitly how the ensuing non-trivial flavor vacuum condensate characterizes the phenomenon of field mixing. In addition, with the help of Ward-Takahashi identities we demonstrate the emergence of a correct number of Nambu-Goldstone modes in the physical spectrum.


Introduction
Particle mixing [1], has undergone rapid development, both theoretically and experimentally. The Standard Model [2], as it stands now, does not accommodate non-zero neutrino masses or mixing in its Lagrangian. Despite immense experimental and theoretic effort, the true origin of the fermion mixing is still rather elusive, though it is generally believed that some, yet unknown, nonperturbative quantum-field theoretical (QFT) mechanism should be responsible for its emergence. The basic features of fermion mixing were firstly examined with genuine QFT methodology in the late 90's [3]. There, it was observed that field mixing is not the same as wave-function (i.e., first-quantized) mixing and, in fact, corrections to neutrino oscillation formula were found in the QFT setting [4,5]. The origin of these discrepancies can be retraced to a non-trivial nature of the mixing transformation [6], which implies a rich vacuum structure, characterized by the fermion-antifermion pair condensate and unitarily inequivalent vacuum sectors [3,7].
In this Letter, we consider the QFT treatment of generation of both masses and (field) mixing, from an algebraic (and hence manifestly non-perturbative) point of view. In fact, the only requirement we enforce in our considerations is the invariance of our model under the global SU (2) V × SU (2) A × U (1) V symmetry. The aim is to analyze symmetry-breaking schemes that are pertinent for a dynamical generation of masses and field mixing and show that, in this context, the aforementioned structure of the vacuum for mixed fields, previously studied in connection with neutrino oscillations, appears in a natural way.
We envisage that a particular breaking scheme is realized via spontaneous symmetry breaking (SSB) phase transition whose pattern is dictated by the potential-energy term in the original or effective-action Lagrangian. In fact, the specific form of the potential is immaterial for our description. Apart from discussing explicit structural forms of broken-phase vacua we also show how the correct number of Nambu-Goldstone (NG) bosons appears in the physical spectrum of the full theory. A similar attempt was done in Ref. [8], where the Nambu-Jona Lasinio model [9] was studied. Note that the present literature on dynamical generation of mixing, mostly concentrates on phenomenological aspects [10].
The paper is organized as follows: in Section 2 we classify all possible ways in which chiral-charge conservation can be broken when a generic mass-matrix term is added to a chirally symmetric action. In Section 3 we formulate the Ward-Takahashi (WT) identities for three quintessential chiral SSB schemes. This is done by employing Umezawa's ε-term prescription [11,12]. In Section 4, we show how the vacua that are pertinent for dynamical generation of mixing are algebraically related to masses and mixing generators. Various remarks and generalizations are addressed in the concluding Section 5.

Explicit chiral symmetry breaking and fermion mixing
Let us consider the Langrangian density L which is invariant under the global chiral-flavor group G = SU (2) A × SU (2) V × U (1) V . Let the fermion field be a flavor doublet Under a generic chiral-group transformation g, we have where σ j , j = 1, 2, 3 are the Pauli matrices and φ, ω, ω 5 are real-valued transformation parameters of G. Noether's theorem implies the conserved currents and the ensuing conserved charges From these we recover the Lie algebra of the chiral-flavor group G, i.e.
Here i, j, k = 1, 2, 3 and ε ijk is the Levi-Civita pseudo-tensor. For massless fermions the Lagrangian is invariant under both the flavor and axial flavor transformations. The chiral symmetry is explicitly broken when a mass term L M = −ψ M ψ is added to L. In fact, one can easily verify, that Note, that these relations do not presuppose any specific form of the original chirally symmetric action. We now demonstrate how particular choices of the mass matrix M can affect the structure of the residual (unbroken) subgroup. i) Let M = m 0 1I where 1I is the identity matrix, then (6) reduces to i.e., the scalar and vector currents remain conserved and the broken-phase sym-

then Noether currents satisfy
thus reducing the residual symmetry to The residual symmetry is then H = U (1) V . The rôle of this symmetry can be understood as follows: We first introduce the flavor-charges [5] where the total flavor-charge Q = Q I + Q II . In absence of mixing (i.e., for diagonal M ), the charges Q I and Q II are separately conserved. In the case of mixing among the two generations, the residual symmetry is just the phase symmetry connected with the conservation of the total flavor charge Q. Note, that we have not included term m 2 σ 2 into expansion of M . This is because we are not interested in an additional CP violation. Clearly, should we studied extension to three-flavor chiral models, we should include such a term.

Dynamical generation of masses and mixing
In the last Section we have seen how the mass formation and mixing phenomenon are characterized by the residual symmetry. For instance, the field mixing can be dynamically generated via the SSB scheme To proceed, let us recall [13,14,15] that the key signature of SSB is the existence of some local operator(s) φ so that on the vacuum |Ω .
Here v i are the order parameters and N i represent group generators from the quotient space G/H. In our case N i will be given by Q and Q 5 according to the SSB scheme chosen. By analogy with quark condensation in QCD [16], we will limit our following considerations to order parameters that are condensates of fermion-antifermion pairs. To this end we introduce the following composite operators Let us now look in some detail at the three SSB schemes G → H from Section 2.

i) SSB sequence corresponding to a single mass generation [situation i)] is
The broken-phase symmetry (which corresponds to dynamically generated mass matrix M = m 0 1I) is characterized by the order parameter One can easily check that this is invariant under the residual group H but not under the full chiral group G. In order to discuss the NG modes it is convenient to introduce in the Lagrangian density a symmetry-breaking term via the so-called ε-term prescription [11,12] as L ε = i ε i φ i where φ i are orderparameter operators characterizing a particular SSB scheme. Whenever some φ i 's do not mix under the action of G/H they carry independent ε i . Otherwise, the associated ε-terms must be identical. At the end of calculations the limit ε i → 0 has to be taken. This allows to avoid technical difficulties related to infinite-range correlations due to NG field(s).
To proceed, we employ the WT identity in the form [11,15] i which is valid for any local operator ψ(x). Here δX = i[N, X] and N are group generators from G/H. For the SSB scheme (14) we take N k = Q 5,k , φ i = Φ 0 δ i0 With this the WT identity (16) acquires the form Because the LHS differs from zero, the Källén-Lehmann spectral representation of RHS implies that quantity ε 0 ρ(k = 0, m k )/m 2 k (ρ is spectral distribution) is non-vanishing for ε 0 → 0 and thus masses m 2 k ∝ ε 0 (k = 1, 2, 3) due to positive definiteness of ρ. This is the NG theorem, which states that the expression of Φ 5 k in the physical states representation, called dynamical map or Haag expansion, will contain the gapless NG fields B 5 k as linear terms [11,15]. Note that we have three NG fields which equals to dim(G/H).
ii) As a second case we consider the breaking scheme which, as we saw is Section 2, is responsible for the dynamical generation of different masses. In this case the order parameters take the form The corresponding ε-term prescription has now the form , and δ 2 1(2) Φ 3 = −Φ 3 and hence the WT identities (16) boil down to This gives two NG modes. Case with v 0 coincides with Eq. (17) for Φ 5 1 and Φ 5 2 and hence it yields another two NG modes. Since, δ 5,1(2) Φ 3 = δ 1(2) Φ 0 = 0, no new NG modes are provided by related WT identities. Note that the WT identities related to δ 5,3 Φ 0 = −iΦ 5 3 and δ 5,3 Φ 3 = −iΦ 5 0 have also extra contributions from mixed correlators and hence no direct statement about NG excitations related to Φ 5 3 and Φ 5 0 is possible. Mixed correlators can be removed when δ 5,3 is applied to ψ = Φ 0 + Φ 3 rather than to Φ 0 and Φ 3 separately. This provides one additional NG mode. Hence, the dynamical maps of Φ 1 , Φ 2 , Φ 5 1 , Φ 5 2 and Φ 5 3 +Φ 5 0 will in their Haag expansions contain NG fields as linear terms. The number of NG fields is now five which coincides with dim( . iii) Finally, we consider the breaking scheme which, according to Section 2 is responsible for the dynamical generation of field mixing. The SSB scheme (21) is characterized by the order parameters Therefore, this case requires the presence of an exotic kind of pairs in the vacuum, which mixes fermion and antifermions with different flavors. In other words, in order to have field mixing, we need a mixing at the level of the vacuum condensate structure. The ε-term prescription assumes now the form L ε = ε 0 Φ 0 + ε 1 Φ 1 + ε 3 Φ 3 . The generators belonging to G/H are Q 1 , Q 2 , Q 3 and Q 5 and hence all ε i ≡ ε.

Mixing and Bogoliubov transformations
It is well known [13,15] that the manifold of degenerate ground states in the broken phase -ordered-phase vacuum manifold, is isomorphic to the quotient space G/H. We might, thus employ the Perelomov group-related coherent states (CS) [17] to find an explicit representation of the ensuing vacuum manifold. The form of the interaction part of the Lagrangian would be then reflected in the way the renormalized parameters and fields in the CS run with the renormalization scale. This complicated model-dependent task can often be conveniently bypassed by the mean-field approximation (MFA). In MFA, only quadratic terms are relevant after dynamical symmetry breaking [13,18] and so the resulting quasi-fieldsψ 1 ,ψ 2 have a simple mode expansioñ where j = 1, 2 anũ r k,j ,ṽ r −k,j are massless spinors.α r k,j ,β r k,j annihilate the corresponding (fiducial) vacuum |0 . By assuming the validity of MFA we can now employ existing results from theory of Bogoliubov transformations to discuss the structure of vacuum manifolds in our three SSB schemes.
Above |0 m together with (15) yield the order parameter ii) Dynamical generation of different masses follows from a simple generalization of (25). In fact, it is easy to see that the mass vacuum has the form Here Θ k,j = 1 2 cot −1 (|k|/m j ) , j = 1, 2. B(m 1 , m 2 ), once more, factorizes as the product of Bogoliubov transformations The ladder operators of massive fields can now be defined as (cf. e.g. Refs. [6]) α r k,j = cos Θ k,jα r k,j + ǫ r sin Θ k,jβ with j = 1, 2. In terms of these, we can expand the mass fields where j = 1, 2 and ω k,j = |k| 2 + m 2 j . The order parameters (19) iii) The broken-phase MFA vacuum for the dynamical mixing generation is more delicate. To this end, we first start with the chiral charge Exponentiation yields the generator of a rotation around 2nd-axis: Because, classically, [Q 2 , H] = 0, we can consider the degenerate set of groundstates (vacuum manifold): These, for each θ, have the same structure as the flavor vacuum from Ref. [3].
In fact, it can be shown [6] that [B,R −1 ] = 0 and in terms of mass fields (33) one has which can be recognized as the mixing generator [3] at the reference time t = 0. The |0 eµ , for θ = 0 is actually the vacuum state when mixing is dynamically generated, and can be explicitly written as; where U k = cos (Θ k,1 − Θ k,2 ) and V k = sin (Θ k,1 − Θ k,2 ). The order parameters (22) assume now the form Notice the presence of exotic fermion-antifermion pairs which mix fields of different types.

Conclusions and Discussion
In this Letter we have discussed three types of (dynamical) symmetrybreaking schemes for a generic Lagrangian density with SU (2) A × SU (2) V × U (1) V chiral symmetry, namely cases where the broken-phase symmetries were We demonstrated that the three breaking schemes lead to: i) dynamically generated single mass for a doublet field, ii) dynamically generated two different masses for the doublet fields (without mixing) and iii) dynamically generated mixing among the doublet fields.
Our analysis is based on an algebraic (and hence manifestly non-perturbative) point of view. In particular, we employed Umezawa's ε-term prescription alongside with WT identities for SSB to gain information about NG bosons and ensuing (infinitely degenerate) set of ground states. The explicit form of the ground states that are responsible for dynamical generation of masses and field mixing was obtained in MFA where they were phrased in terms of generators of Bogoliubov transformations acting on fiducial vacuum states. We showed that the vacuum sate, when mixing is dynamically generated, yelds the same condensate structure of the flavor vacuum which was introduced to study neutrino oscillations [3]. In particular, we have shown that order parameters that correspond to fermion bilinear operators that mix fermions of different types are true signatures of the mixing phenomenon.
Let us finally add some comments. A common feature of SSB is the appearance of topological defects [15]. The number of such defects is related to the quench time of SSB in which they are formed via the Kibble-Zurek mechanism [19]. On the other hand, type of defects in 3-D configuration space is determined by a non-trivial homotopy group π n (G/H) (n = 0, 1, 2). By analogy with condensed matter systems we might expect that defects formed will provide an important observational handle on the dynamics of the mixing-related SSB transition.
It is known [20] that Lorentz symmetry may be spontaneously broken by the flavour vacuum, in the sense that the corresponding dispersion relations of states constructed as Fock excitations of the flavour vacuum are modified as compared to the standard Lorentz covariant ones. In this sense, one can discuss flavour mixing in a fixed frame, such as finite temperature situations, which break Lorentz symmetry. In such a context, the considerations in this work could imply in principle dynamical generation of a vacuum expectation value of chiral currents. This is nothing other than a chiral chemical potential, whose presence in the case of quarks in finite temperature QCD (which is a field theory characterised by mixing) can lead to interesting effects, such as the chiral magnetic effect [21].