Dynamical Majorana Neutrino Masses and Axions II: Inclusion of Axial Background and Anomaly Terms

We extend the study of a previous publication \cite{amsot} on Schwinger-Dyson dynamical mass generation for fermions and pseudoscalar fields (axion-like particles (ALP)), in field theories containing Yukawa type interactions between the fermions and ALPs, by incorporating anomaly terms and/or (constant) axial background fields. The latter are linked to some Lorentz (and CPT) violating scenarios for leptogenesis in the early Universe. We discuss both Hermitian and non-Hermitian Yukawa interactions, anomaly terms and axial backgrounds, which are all motivated in the context of some scenarios for radiative (anomalous) Majorana sterile neutrino masses in string-isnpired, low-energy, effective field theories, including attractive four-fermion interactions. We show that, for a Hermitian Yukawa interaction, there is no (pseudo)scalar dynamical mass generation, but there is fermion dynamical mass generation, provided one adds a bare (pseudo)scalar mass. For this case, the hermitian anomaly terms play a similar role in inducing dynamical mass generation for fermions as the four-fermion attractive interactions. For antihermitian Yukawa interactions, an antihermitian anomaly resists mass generation. The axial background terms assist dynamical mass generation induced by antihermitian Yukawa interactions, in the sense that the larger the magnitude of the background, the larger the dynamical mass. For hermitian Yukawa interactions, however, the situation is the opposite, in the sense that the larger the background the smaller the dynamical mass. We also compare the anomaly-induced dynamical mass with the radiative fermion mass in models of sterile neutrinos, and find that in cases where the dynamical mass occurs, the latter dominates over the anomalously generated radiative sterile-neutrino mass.


I. INTRODUCTION AND MOTIVATION
In a previous article [1], we have discussed dynamical mass generation in field theory models with weak Yukawa interactions, in the presence of attractive four fermion contact interactions. The work was motivated by some stringinspired models involving anomalously generated Majorana fermion masses [2], where the small Yukawa couplings of pseudoscalar axions with Majorana neutrinos are assumed to be induced by shift-symmetry breaking non-perturbative effects, such as string instantons.
The effective action, describing the interaction of (canonically normalised) axion and sterile neutrino fields reads [2] S a = d 4 x √ −g 1 2 (∂ µ a(x)) 2 − γ c 1 a(x) where a(x) denotes an axion field (which in the context of the string-effective model of [2] could be different from the QCD axion, e.g., associated with string moduli fields [3], ψ C R = (ψ R ) C is the (Dirac) charge-conjugate of the right-handed fermion ψ R , which in [2] is considered to be a sterile neutrino, ∇ µ denotes the (torsion-free) gravitational covariant derivative, and the index j in ψ j runs over fermion species ψ j , including the right-handed (sterile) fermions ψ R . The quantity (2) with M Pl the reduced Planck constant, plays the rôle of the axion constant in this model. The . . . in (1) denote repulsive four-fermion self-interaction terms involving the (square of the) axial fermion current, J 5 µ ≡ j ψ j γ µ γ 5 ψ j , where the sum is over all fermion species, including the right-handed fermions. Such repulsive four fermion terms arise after integrating out the (quantum) torsion fields of the effective field theory provided by the field strength of the antisymmetric spin-1 tensor field that exist in the massless gravitational string multiplet of the underlying string theory [2]. They will not be of relevance to us in this work. The coefficient c 1 in the gravitational anomaly [4] term depends on the number of chiral fermions that circulate in the anomaly loop. We remind the reader that, in terms of the anomalous axial current for massless chiral fermions, the gravitational anomaly terms are given by [4] d 4 x √ −g γ c 1 a(x) where the last equality is obtained by partial intergration, assuming that the fields and their derivatives vanish at infinity. For the purposes of our work we ignore gauge anomalies, as we restrict ourselves to sterile right-handed fermions (neutrinos), which in our concrete examples discussed here and in [1], are assumed to be the only chiral fermion species circulating in the anomaly loop diagram. In this case one may set c 1 = 1. The Yukawa coupling λ in (1) is assumed to be due to non perturbative string instanton effects, and as such is expected to be small. These terms break explicitly the shift symmetry a(x) → a(x) + c, in the same spirit as the instanton-generated potential for the axion fields. The coefficient γ expresses the strength of a kinetic mixing [2] between the axion field and the corresponding gravitational axion field, which, in four space-time dimensions, is dual to the aforementioned field strength of the antisymmetric tensor field of the string gravitational multiplet. For real γ, the approach is only valid for The mechanism for the anomalous Majorana mass generation proposed in [2] leads to a two-loop Majorana righthanded neutrino ψ R mass: where Λ is an Ultra-Violet (UV) momentum cutoff. In an UV complete theory, such as strings, Λ and κ −1 are related [2] via the string mass scale and compactification radii of the extra-dimensional spatial manifold. For a generic quantum gravity model, independent of string theory, one may use simply Λ ∼ κ −1 . As explained in [2], the sterile fermion mass (5) is independent of the axion-a(x) potential, and thus its mass.
In [1] we have suggested that the constraint (4) can be evaded, upon the simultaneous complexification of the parameters γ, λ, which now become purely imaginary In such a case one is effectively working with non-Hermitean Hamiltonians but connected to the so-called PT symmetric framework [5]. We remark that in the case of axion (pseudoscalar) fields, the Yukawa interaction term is PT-symmetry odd. On the other hand, if we had a scalar field in such interactions, one would obtain PT symmetric Hamiltonians [5] of the type discussed in [6], which have been argued to be consistent field theories describing phenomenologically relevant neutrino oscillations. Nonetheless, as discussed in [7], such interactions can lead to real energies in a certain regime of their parameters, thus making a connection with PT symmetric systems [8]. In fact, according to the discussion in [9], the existence of real energies is guaranteed if one has an underlying anti-linear symmetry, which could be more general than PT. In our case such an antilinear symmetry is the CPT symmetry [7]. In [7] we also demonstrated the consistency of such non-Hermitian models with Lorentz invariance (including also improper Lorentz transformations), as well as unitarity [10]. It is important to note that, upon (6), the anomalous fermion mass (5) remains real: Notice that the sign of the fermion mass depends on the sign of the productλγ, and can always be chosen to be positive, although for fermions, unlike bosons, such a sign is not physically relevant.
In [1] we presented an alternative way to generate a sterile-neutrino mass, dynamically induced by the Yukawa axion-sterile-fermion interaction via a study of the pertinent Schwinger-Dyson (SD) equations. We examined the generation of a dynamical mass for both fermion and (pseudo)scalar fields, in cases with hermitian and antihermitian Yukawa interactions. In [1] we ignore anomalies, setting γ =γ = 0. In the context of the action (1), then, we have shown that dynamical masses for the fermion and axion fields, m and M respectively, of approximately equal magnitude, is possible, provide there is a bare axion mass M 0 : which indicates non-perturbative (in the Yukawa coupling λ) small dynamical fermion and (pseudo)scalar masses. For the case of antihermitian Yukawa couplings (6) (withγ = 0), dynamical mass for the fermions is not possible, due to energetics [7,11], but for (pseudo)scalar fields one can have a one-loop induced mass in the absence of a bare (pseudo)scalar mass M 0 = 0.
In the presence of additional attractive four-fermion interactions in the Lagrangian of the model (1), say of the with f 4 a dimensionful coupling with mass dimension +1, the situation changes drastically [1]. For hermitian Yukawa couplings, dynamical masses for fermions and scalars of order can be generated, which are much larger than the masses (8) in the pure Yukawa case where f 4 → ∞. As follows from (11), we observe that, for consistency, the four-fermion coupling has to be proportional to the UV cutoff [1], In the antihermitian Yukawa interaction case, in the presence of the interactions (10), one can obtain dynamical fermion and (pseudo) scalar mass generation, in the absence of bare (pseudo)scalar mass M 0 = 0, which are of the same order as in the corresponding hermitian-Yukawa case (11): with the four fermion coupling given by (12), as in the hermitian Yukawa interactions case. The above results can be straightforwardly extended to the Majorana spinor case [1], of direct relevance to the model of [2], up to some numerical factors of 2. Specifically, for the corresponding case (8), one has for the Majorana fermion dynamical mass: In the presence of four-fermion interactions (10), the solution for dynamical fermion and scalar masses m M is qualitatively similar as in (11), but with a different value of f 4 : In this article we shall examine the role of the anomaly coefficients γ,γ in the appropriate hermitian or non-hermitian models. In the presence of such coefficients, radiative masses for the fermions are generated anomalously, as discussed in [2]. The point of the current work is to compare the dynamically generated mass with these radiative masses.
To ensure the reality of the radiative masses, and thus avoid quantum instabilities in the non-hermitian models, we impose the simultaneous complexification (6), which will determine the relevant cases.
In addition to the anomaly terms, we also include a (constant) axial background B µ term (where B µ has mass dimension +1). The inclusion of such a term is motivated by CPT Violating models for leptogenesis involving heavy sterile neutrinos [12]. Indeed, the decay of the latter in the presence of constant axial backgrounds of the form (violating spontaneously CPT and Lorentz symmetries) with δ µ 0 a Kronecker delta, can lead to lepton asymmetries at tree level. In the models of [12], the lagrangians are hermitian and bare masses for the sterile neutrinos have been assumed. It is the purpose of this section to examine whether the axial background itself can induce dynamical masses or the fermions and/or axions, which in turn could have implications for leptogenesis, as per the conclusions of [12]. However, we shall also go beyond the hermitian cases. In particular we shall include antihermitian axial backgrounds B µ . Such a case (but considering antihermitian axial backgrounds in a Nambu-Jona-Lasinio fermionic only model [13], in the absence of Yukawa interactions, following a one-loop and not a complete SD treatment) has also been considered independently in [14] with the conclusion that the background enhances the dynamical mass generation induced by the four fermion interactions of the model. We shall confirm this result in our model as well, using a SD analysis [1].
We therefore consider the prototype Lagrangian (using (3) for the gravitational anomaly terms to express them in terms of axial fermion currents): where, in the context of the model [2], f b is given by (2), but of course one can treat it more generally as an arbitrary mass scale to be determined self consistently in terms of the UV cutoff Λ in the phase where dynamical mass generation occurs [1]. For concreteness we restrict our discussion to Dirac fermions, given that extension to the Majorana fermion case is straightforward and does not change qualitatively the conclusions. We shall return to the Majorana case when we compare the dynamical and radiatively anomalous fermion masses. The reader should have noticed that in (17), we have introduced the generic notation γ (λ) for the anomaly coefficients (Yukawa couplings) for both hermitian and antihermitian cases, the latter obtained upon using (6). Moreover we used g 2 /f 2 b instead of 1/f 2 4 (cf. (10)) for the coupling of the attractive four-fermion-interaction terms. This is for notational convenience, to make a more direct contact with the model (1) [2], especially when we compare the contributions to the dynamical fermion mass coming from the anomaly terms with those due to the four-fermion interactions, as well as when we make a comparison between the radiative (cf. (5), (7)) and dynamical fermion masses. At this point we stress that the axial current is classically conserved only in the case of massless chiral fermions, and in such a case its conservation might be spoiled by anomalies [4], cf. (3). In the case of massive fermions, however, of mass m, one has classically that ∇ µ J 5 µ = 2 i m ψ γ 5 ψ. Thus, when one considers the full quantum fermionic path integral, as necessary in SD treatments of dynamical fermion mass generation, the anomaly terms play in general a non-trivial role in mass generation, as we shall discuss in this work.
Below we shall examine various cases involving both hermitian and antihermitian interactions, as per the initial motivations of this work, outlined in the introduction and in [1]. We shall not deal with the most general solutions of the SD equations, but instead we shall extend appropriately our previous considerations to include axial background and anomaly terms. In particular, as far as the anomaly terms are concerned, we shall insist on maintaining the reality of the radiatively induced fermion masses (5), (7), otherwise there would be instabilities in our system. This means that we shall consider cases in which the product γλ always remains real that is, one should examine models with either γ, λ ∈ R or iγ, iλ ∈ R. As already mentioned, for brevity, we shall use the same symbols γ and λ in both hermitian and antihermitian cases, with the understanding though of the above restriction (18). Regarding the quantum SD treatment of antihermitian interactions, we should mention that in [1] and here, we do not attempt a rigorous definition of the path integral measure of the various fields. Such an approach has been undertaken recently in [15] in the context of a simplified PT symmetric theory [5] of a pseudoscalar field. In our case, the situation is more complicated because of the Yukawa interactions of the pseudoscalar fields a(x) with fermions, cf. (1). In our construction of the SD equations in [1] and here, we use only formally the appropriate non-hermitian interactions, by working in a Euclidean path-integral, and search for consistent solutions for mass generation for both (pseudo)scalar and fermion fields.
In general, there are two types of masses for the fermions that can be generated dynamically, a Dirac mass, m, corresponding to a term in the effective action in the massive phase of the form mψ ψ, and a chiral mass µ corresponding to a term µψ γ 5 ψ. Despite the presence of antihermitian Yukawa interactions, such theories have real energies [8], provided |m| > |µ| (we stress again that, in the presence of antihermitian field-theoretic Yukawa interactions, the underlying anti-linear symmetry [9] that guarantees the reality of the energies, under the above condition for the (dynamical) masses, is CPT symmetry [7]). This motivated the inclusion of such non-hermitian models in our studies of dynamical mass generation in [1,7], which was not considered in [16]. For our purposes here, we note that we seek solutions for dynamical mass generation for which the chiral mass, or equivalently a chiral condensate <ψ γ 5 ψ >, vanishes µ = 0. Such solutions are consistent with considering real pseudoscalar fields in the path integral, as done in [15], for which case one would obtain from (1) a classical equation of motion for the scalar fields of the form (including a real mass M for the scalars, in general) corresponding to a saddle point in the path integral. We next renark that, in the absence of anomalies, the γ term on the right-hand side of (19) vanishes for massless fermions. In the case of massive fermions, of mass m, this term reads simply 2i γ m f bψ γ 5 ψ. For real (pseudo)scalar fields, in the antihermitian case (6), this equation would imply, for the general case 2 γ m f b = −λ we consider here [1,2], the constraint [1,7]:ψ γ 5 ψ = 0. This is associated with the fact that, for the saddle point, where quantum fluctuations of the fields are ignored, the left hand side of (19) is real, for real φ fields we consider here, while the right-hand side is imaginary (it goes without saying, of course, that (19) is not satisfied by the quantum fields away from the saddle point, for which there are no restrictions). For our purposes, it is sufficient that we find consistent solutions with the above constraint implemented dynamically, as a vanishing condensate for the fermion fields, ψ γ 5 ψ = 0, or equivalently µ = 0. Energetics arguments [7,11] support this point of view in the case of antihermitian Yukawa interaction models. In [1] we extended this constraint also to the models with hermitian Yukawa interactions, and this will also characterise the SD analysis in the present article, in the sense of setting the chiral mass µ = 0 in all our SD equations, looking for non-trivial solutions for dynamically generated Dirac mass m = 0.
The structure of the article is as follows: in the next section II, we discuss SD dynamical mass generation in the presence of hermitian axial background and hermitian Yukawa interactions, in the absence of anomaly terms, both in the presence and absence of four fermion attractive interactions. This is because we want to study the effects of the axial background prior to inclusion of non-trivkal anomaly coefficients. In section III, we repeat the analysis for hermitian axial backgrounds but antihermitian Yukawa interactions, again in the absence of anomalies. In section IV we study non hermitian axial backgrounds and hermitian Yukawa interactions, while in section V we complete our study on the role of axial backgrounds in mass generation for non-anomalous models by examining the case of antihermitian axial backgrounds in the presence of antihermitian Yukawa interactions. In section VI, we discuss the role of hermitian anomaly terms in dynamical mass generation induced by hermitian Yukawa interactions, in the absence of axial backgrounds, while in section VII we repeat the study for the case where the anomaly and the Yukawa interaction terms are both antihermitian. Finally, section VIII contains our conclusions and outlook, as well as a comparison between the dynamically generated fermion masses and the radiative (anomalously-generated) ones (5), (7), in the context of the sterile neutrino model of [2].

II. HERMITIAN AXIAL BACKGROUNDS AND HERMITIAN YUKAWA INTERACTIONS -NO ANOMALIES
In this and the subsequent sections we shall be brief with technical details on the SD formalism, given that those have been provided in [1], where we refer the interested reader for details. We shall concentrate instead in merely giving the final SD equations for mass generation and describing the pertinent solutions.
Let us first ignore the anomaly coefficient, by setting γ = 0. In this case, the Lagrangian (17) becomes: When the background term B µ is constant, one can incorporate it in the fermionic propagator. Upon linearising the four fermion interaction using the auxiliary field σ, we write for the generating functional We note, for the benefit of the reader, that the source terms, including that (K) for the auxiliary field σ, are introduced for the consistent derivation of the SD equations [1], which is not given here, for brevity. In this way, the interaction part is the same as in the hermitian Yukawa interaction model, in the presence of attractive four-fermion interactions, discussed in [1]. Therefore, the corresponding SD equations are formally the same as in that case, with the difference that now the fermionic propagator contains the additional background B µ terms. Following the argumentation in [1], which was also reviewed briefly at the end of the introductory section I, we look for dynamical mass scenarios in which the chiral fermion mass vanishes, µ = 0 and only a Dirac mass m is generated. This will characterise all the cases considered in the current work.
In the rainbow approximation sufficient for weak Yukawa interactions (with |λ| 1) we are restricting ourselves here and in [1], 2 the pertinent SD equations, in the presence of a bare scalar mass M 0 , whose introduction is necessary in this case [1] in order to have dynamical mass for the scalar field, read: where the SD dressed propagators for the fermion field in the (constant) axial background (G f ), the (pseudo)scalar field (G s ) and the auxiliary scalar σ field that linearises the four-fermion interactions (G σ ), are given by [1]: , where M and m denote the dynamical scalar and Diracfermion masses, respectively, and we use the notation / p = γ µ p µ , with γ µ the Dirac γ-matrices, and p 2 = p µ p µ . To solve the SD equations we follow a similar treatment as in [14], by writing the fermion propagator as After Wick rotation in (22) and (23), use of the spherical coordinates (p, θ, φ 1 , φ 2 ), so as to write B · p = |B||p| cos θ, and using an UV cutoff Λ in the "radial" integral, we obtain the following system of algebraic equations: and Where we have defined α = m − |B| and β = m + |B|.

A. Dynamical (pseudo)scalar mass
To see whether the axial background B µ is capable of inducing a dynamical mass for the (pseudo)scalar field, we set the bare mass to zero M 0 = 0. We shall consider the case m M . On substituting in (25), then, and defininḡ M = M/Λ andB = B/Λ we obtain: We plot the right hand side of (27) for different values of B in fig. 1. As we can readily see from the figure, this function never intersects 1, which means that there is no solution, that is the background cannot induce dynamically a scalar mass in the absence of a bare mass M 0 .
Once a non zero bare mass M 0 = 0 is introduced, the latter can be determined by the fermion-mass SD equation (26), which depends only on the renormalised masses m, M . The bare mass M 0 is easily found from the small |B| limit of (25), after setting m M , under the condition Λ 2 M 2 : where M can be determined from the solution to the fermionic SD equation, which we now proceed to discuss for the case m M .
In fig. 2 we plot the curves corresponding to the right hand side of (29) for different values ofB. We notice that there are curves intersecting the constant solid line at 1, which corresponds to the left hand side of (29), for specific choices of λ and g. For these cases we will have solutions for the equation (29) with m M . We observe that there are solutions when the background field B becomes sufficiently small, |B| < Λ, corresponding to m M < Λ.
At this stage we remark, that in the absence of the attractive four-fermion interactions, i.e. when g =ḡ = 0, there is dynamical fermion mass, for sufficiently small |B| < m, which matches smoothly the non-perturbative solution found in [1], for |B| → 0. This can be seen analytically by considering the small-|B| limit of the SD equation for the fermion, (29), which, up to and including order |B| 2 terms, reads for g = 0: where the last equality is valid upon making the approximation Λ M |B|. Equation (30) matches smoothly the situation for the hermitian Yukawa interaction described in [1] in the limit |B| → 0. For small but finite values of

|B|
M one obtains slightly modified non perturbative solutions, which are obtained from The latter equation cannot be solved analytically but it is easy to see that there are solutions, sufficiently close to those found in the corresponding case studied in [1]. Indeed, if we represent Λ 2 /M 2 = e ξ , ξ 1 > 0, then we obtain from (31) the consistency condition ξ − 16π 2 λ 2 = |B| 2 2 M 2 1, for |λ| 1; this has solutions for ξ, which turns out to take on a slightly higher value than the corresponding one in the absence of the axial background, leading to a further suppression of the dynamical fermion mass. The larger the magnitude of the background the bigger the suppression of the dynamical mass.

III. HERMITIAN AXIAL BACKGROUNDS AND NON-HERMITIAN YUKAWA INTERACTIONS -NO ANOMALIES
Next we study the case in which the Yukawa coupling is non-hermitian. For brevity, we shall omit the details of the derivation of the SD equations, and give, instead, directly the solutions, which are and where again α = m − |B| and β = m + |B|. We have not incorporated a bare mass here, because, as we shall see below, a scalar mass can be dynamically generated by the Yukawa coupling in this case. This is the main difference from the hermitian Yukawa coupling case. As we shall see below, the fermion mass exhibits a similar behavior with the background |B| as in the hermitian case.

A. Dynamical (pseudo)scalar mass
We are interested in solutions for m M . On setting m M , and definingM = M/Λ andB = B/Λ we obtain from (32) We plot in fig. 3 the right hand side of the above equation. The reader can readily observe that the curves always intersect the solid line corresponding to the value 1, representing the left hand side of Eq. (34), which implies the existence of non-trivial solutions to this equation. Thus, both fermion and scalar masses, of equal magnitude, are dynamically generated in this case, in a similar spirit to the situation when the axial background is absent. The presence of the background affects the magnitude of the mass. The larger the background, the smaller the dynamical mass.

B. Dynamical fermion mass
As before, using m M and definingM = M/Λ andB = B/Λ, one obtains from (33) :  FIG. 3: The Plot shows different values of the right hand side (r.h.s.) of (34) for the fixed value λ 2 = 0.02. All the curves intersect with the constant curve at 1, representing the left-hand side of (34), which means that, for non-Hermitian Yukawa coupling and hermitian constant axial background, there is dynamical mass generation for the scalar, without the need to introduce a bare mass.
Inserting in (35) the value for M obtained in (34), we can get the value of the four fermion interaction coupling g/f b for which we have dynamical mass generation. In figure 4 we show one example withB = 0.003 and λ 2 = 0.02. The dashed line, corresponding to the right hand side Eq. (34), gives a value of M/Λ 0.023 for which (34) is satisfied. Using this value in (35) we observe that there is a consistent solution, provided that the four-fermion coupling assumes the valueḡ which is very close to the value (12) (upon the correspondence f 4 = f b /g) in the absence of the axial background. The dotted-dashed line in fig. 4 represents the right hand side of (35) using these values forB, λ andḡ. We observe that the dashed and dotted-dashed lines intersect the dotted line, corresponding to the fixed value 1, at the same point. This implies that there is a consistent solution for the equations (34) and (35), that is, there is dynamical mass generation for this case, for the fixed value (36) of the four-fermion interaction coupling.

IV. ANTIHERMITIAN BACKGROUND IN HERMITIAN YUKAWA INTERACTIONS-NO ANOMALIES
We now consider the case when the axial background B µ is antihermitian [14] and the Yukawa coupling is hermitian. The results in this case are obtained by replacing |B| → i|B| in (25) and (26). The pertinent SD equations become where here α = m − i|B| and β = m + i|B|.
A. The limit λ → 0, g = 0 We notice that in the limit λ 2 → 0, g = 0, where only the (attractive) four-fermion interaction is present, we recover the result of [14]. In fact, in this limit, Eqs. (37) and (38) read and The scalar mass is only the bare mass because, since there is no interaction in this limit. The fermionic equation (40) is essentially the same as equation (34) in ref. [14], where the Nambu Jona Lasinio four-fermion model was studied in the presence of a constant non-hermitian axial background. The conclusion in [14] points to the fact that inclusion of this background increases the dynamical fermion mass. In this case, of course, the condition m M is not valid, in view of (39).
Fig . 5 shows the curves corresponding to the right hand side of (41) with M 0 = 0 for different values ofB. We notice that there is no intersection between these curves and the left hand side of (41) of value 1, which implies that (41) has no solution if M 0 = 0. On the other hand, for non-zero bare scalar mass M 0 = 0, one can obtain consistent mass generation, similarly to the case of hermitian background. Indeed, in the small |B| M Λ limit, the value of M 0 , can be obtained by expanding (37) in powers of the background, and using m M Λ: where M can be found from the fermionic equation, which we now proceed to study.

C. Analysis of the fermionic equation
On considering m M and usingM = M/Λ withB = B/Λ we obtain from (38): In figure 6 we plot the right hand side of (43) for specific values of λ and g. We observe that, for different values of B, the various curves always intersect the value 1, corresponding to the left hand side of (43). This implies the    The above results imply that irrespective of whether the axial background is hermitian or antihermitian, in the case of Hermitian Yukawa interactions, one needs to give a bare mass to the (pseudo) scalar field in order to obtain non-trivial dynamical mass generation for both the fermion and the scalar fields. As we observe from fig. 6, for the case of anti-hermitian axial backgrounds and hermitian Yukawa interactions, the larger the background, the bigger the mass, so the presence of the antihermitian background assists dynamical mass generation, in similar spirit to the results of [14] for the NJL model in the absence of Yukawa interactions.

V. ANTIHERMITIAN BACKGROUND IN ANTIHERMITIAN YUKAWA INTERACTIONS-NO ANOMALIES
For the case of both antihermitian background and Yukawa interactions, one may obtain the pertinent solutions for mass generation by substituing |B| → i|B| in (32) and (33). This gives and where here α = m − i|B| and β = m + i|B|.
A. Dynamical (pseudo)scalar mass As before, for solutions m M and redefiningM = M/Λ andB = B/Λ we get The situation is similar to the non-hermitian Yukawa coupling and hermitian background, for which one does not need a bare mass for the scalar to generate dynamical mass. Figure 7 shows the plot of the right hand side of (46) as a function ofM for different values of B. All the curves intersect the solid line, corresponding to the fixed value 1, which represents the left hand side of (46).

FIG. 7:
The Plot shows different values of the right hand side of (46) for the fixed value λ 2 = 0.02. All the curves intersect the solid line, which represents the left hand side of (46). It means that, for non-Hermitian Yukawa coupling and non-hermitian background, there is dynamical mass generation for the scalar, without the need to introduce a bare mass.
Comparing figures 3 and 7, where in both cases the Yukawa interaction is non-hermitian, we observe that, for a non-hermitian axial background, the larger the background, the bigger the dynamical mass. This is to be contrasted with the case of a hermitian axial background, for which, as we have seen in section III A, the larger the background, the smaller the dynamical mass.
From (46) one obtains a value for M , and then, upon inserting it in (47), one determines the value of the fourfermion coupling g/f b for which there is a consistent dynamical mass, as in the hermitian background case. As a concrete example, in fig. 8 we plot the right hand side of Eqs. (46) and (47), forB = 0.0004 and λ 2 = 0.01. We observe from the figure that the dashed line, corresponding to the right hand side of (46), intersects the constant dotted line at 1, representing the left hand side of the equation, which implies the existence of a non-trivial solution for M/Λ = 0.159. Using this value in (47), we then obtain a consistent solution forḡ = 12.58, in the sense that for this value ofḡ the three curves have a common intersection at M/Λ = 0.159. This demonstrates that there is dynamical mass generation with m M < Λ in this case. 3

FIG. 8:
The dashed line corresponds to the right hand side of (46) for λ 2 = 0.01 andB = 0.0004. The dotted-dashed line represents the right hand side of (47) for the above values of λ 2 andB, andḡ = 12.58. The constant dotted line at 1 represents the left hand side of the equations (46) and (47). The existence of a common intersection point for all three curves demonstrates the existence of dynamical mass generation.

VI. HERMITIAN ANOMALY TERM AND HERMITIAN YUKAWA INTERACTION -NO BACKGROUND
In this and the next sections we shall consider the effects of the anomaly terms in (17) (γ = 0) in the absence of a constant axial background. This is the situation encountered in the model of [2] (cf. (1), (3)). The Lagrangian for this case reads Motivated by the physics of the model [2], in which the anomaly term coefficient is related to a shift-symmetryrespecting kinetic mixing between the gravitational and standard axion fields, while the Yukawa interactions are associated with shift-symmetry breaking non-perturbative (e.g. string instanton) effects, it is natural to assume the following regime of the various couplings appearing in (48): which we restrict our attention to in what follows (except when we discuss the g = 0 limit, in which only 1 |γ| > |λ| > 0 will be assumed).
After the standard linearization of the four fermion interactions by means of the auxiliary scalar field σ, the generating functional, is given by gravitational anomalies (and, of course, more generally, of Dirac fermions in the presence of non-hermitian mixed -gravitational and gauge-anomalies), and whether such topological properties persist in such cases.
We hope to tackle such issues in future publications.