Spontaneously Breaking Non-Abelian Gauge Symmetry in Non-Hermitian Field Theories

We generalise our previous formulation of gauge-invariant PT-symmetric field theories to include models with non-Abelian symmetries and discuss the extension to such models of the Englert-Brout-Higgs-Kibble mechanism for generating masses for vector bosons. As in the Abelian case, the non-Abelian gauge fields are coupled to non-conserved currents. We present a consistent scheme for gauge fixing, demonstrating Becchi-Rouet-Stora-Tyutin invariance, and show that the particle spectrum and interactions are gauge invariant. We exhibit the masses that gauge bosons in the simplest two-doublet SU(2)xU(1) model acquire when certain scalar fields develop vacuum expectation values: they and scalar masses depend quartically on the non-Hermitian mass parameter mu. The bosonic mass spectrum differs substantially from that in a Hermitian two-doublet model. This non-Hermitian extension of the Standard Model opens a new direction for particle model building, with distinctive predictions to be explored further.


I. INTRODUCTION
manner and demonstrate Becchi-Rouet-Stora-Tyutin (BRST) invariance [34]. We explore the scalar vacuum expectation values (vev's) and tree-level spectra of the gauge and scalar boson masses in a simple version of the model with a single quartic coupling. They depend quartically on the non-Hermitian coupling µ 2 and differ significantly from the masses in the conventional Hermitian two-Higgs-doublet model (2HDM, see Ref. [35]). Thus, this non-Hermitian extension of the Standard Model offers prospects for distinctive experimental predictions that may be explored further in a systematic programme of PT -symmetric phenomenology.

II. SCALAR LAGRANGIAN
In this Section, we extend the non-Hermitian model of Ref. [22] to include two complex scalar doublets, giving the non-Hermitian 2HDM on which we base the discussion of non-Abelian gauge symmetry and its breaking in the next Section.

A. Lagrangian
We follow here similar steps to those described in Ref. [22], starting with the Lagrangian where Φ i are complex doublets and µ is a non-Hermitian mass parameter. This system is invariant under the PT -symmetry, acting on the c-number fields as under which Φ 1 is a scalar doublet whereas Φ 2 is a pseudoscalar doublet. The eigenvalues of the squared mass matrix are real provided the following inequality holds: which is assumed throughout the first two Sections of this work. Note that the eigenvalues become degenerate at |µ 2 | = |m 2 1 − m 2 2 |/2. This marks the exceptional point, which lies at the boundary between the regions of unbroken and broken PT symmetry. At this point, the squared mass matrix becomes defective and we lose an eigenvector. We discuss these exceptional points further in Section IV D.
Because of the non-Hermitian mass term proportional to µ 2 , the equations of motion one obtains by varying the action with respect to Φ i or to Φ † i are not equivalent for non-trivial solutions, i.e.
These two sets of equations of motion are related by PT -symmetry though or, equivalently, by a change in the sign of µ 2 . As can be seen from the eigenvalues (4), observables depend on µ 4 only, so these two sets of equations of motion are physically equivalent. This is also valid at the quantum level, see Ref. [29], as can be derived from the reality of the partition function, provided the sources for the scalar fields satisfy appropriate PT properties.
We choose here the equations of motion provided by the variation of the action with respect to Φ † i : together with their Hermitian conjugates We note that this formulation differs from that suggested in Ref. [27], where the author introduces a similarity transformation that transforms the non-Hermitian Lagrangian L to a Hermitian one L . The difference in approach is reflected in differences in the masses of the gauge fields, which we discuss in Section IV D.

B. Conserved currents
The Lagrangian (1) is invariant under the U(1) transformations which correspond to the current and also invariant under the SU(2) transformations which correspond to the current where τ = (τ 1 , τ 2 , τ 3 ) is composed of the Pauli matrices.
The equations of motion (7) show, however, that these currents are not conserved: except at the Hermitian point µ 2 = 0. The fact that symmetries of the Lagrangian do not correspond to conserved currents for non-Hermitian theories is a direct consequence of the fact that the two functional variations in Eq. (6) cannot vanish simultaneously for non-trivial solutions. Instead, a careful treatment of Noether's original derivation [36] shows that there still exist conserved currents for non-Hermitian theories, but these correspond to transformations that do not leave the Lagrangian invariant [22] (see also Ref. [37] for a summary).
In the present model, we find that the conserved currents are, in fact, which correspond to the following transformations: and The relative sign between the charge assignments of the two fields reflects the usual interpretation of viable PT -symmetric theories as systems with coupled gain and loss.

III. GAUGING THE SCALAR MODEL
Since the conserved currents do not correspond to the usual Noether currents, gauging the model (1) is non-trivial, as we describe in this Section, generalising the approach taken in Ref. [29] to the non-Abelian case. We refer to the non-conserved currents corresponding to symmetries of the Lagrangian as Noether currents, but note that the conserved currents are in fact those consistent with Noether's original derivation (see Ref. [22]).

A. Coupling to the Noether currents
We introduce an Abelian gauge field B α and an SU (2) gauge field W α , together with the SU(2)×U(1) gauge transformations where D α β = ∂ α β − g( W α × β). In order to write a gauge-invariant theory, one should couple the gauge fields to the Noether currents, such that the scalar kinetic terms are given by where D α is given by the usual minimal-coupling prescription, i.e., As in the Standard Model, we rotate the gauge fields as where θ W is the weak mixing angle, to obtain Also as in the Standard Model, the Lagrangian for the gauge fields is with

B. Consistent field equations
Since the gauge fields are coupled to currents that are not conserved, additional terms need to be added to the Lagrangian in order to have consistent field equations [29]. For this, it is enough to consider the usual gauge-fixing terms, which must be added to the classical equations of motion in the non-Hermitian case (not just at the quantum level in order to define the path integral, as in the Hermitian case). The gauge-fixing terms in the Lagrangian involve ghost fields η and η, taking the form whereχ The equations of motion for the full Lagrangian are then given by together with their Hermitian conjugates, where Taking into account the current divergences (13), the derivatives of the above equations of motion lead to the constraints which must be satisfied in order for the field equations to be consistent. As explained in the next Subsection, BRST symmetry allows one to write the latter constraints independently of the ghost fields, as We can summarise our approach as follows. In order to respect gauge invariance, we need to couple the gauge fields to the Noether currents. However, because these currents are not conserved, we need to introduce gauge-fixing terms, which restrict gauge invariance, but imply consistent field equations. The residual gauge invariance is enough to ensure that gauge fields remain massless in the absence of spontaneous symmetry breaking (SSB), and it is defined by the gauge functions β 0 , β satisfying We therefore obtain a consistent gauge theory with a non-Hermitian scalar sector, as in the Abelian case [29].

C. BRST Transformation
In this Subsection, we derive the gauge constraint (29) for W β using the BRST transformation, which is a residual symmetry of the Lagrangian after gauge fixing. In order to define it, one can introduce an auxiliary field T to write the gauge-fixing Lagrangian (24) in the alternative form and the original Lagrangian (24) can be recovered after integrating out T . The BRST transformations are defined as where θ is an infinitesimal Grassmann parameter. The gauge-invariant terms (18) and (22) in the Lagrangian are invariant under the BRST transformation, and the gauge-fixing Lagrangian (31) transforms as a total derivative, so the action is invariant under this BRST transformation. Using the auxiliary field T , the equation of motion (26c) for the gauge field W α can be written in the form and a covariant derivative leads to A BRST transformation of Eq. (26f) leads then to the relation so that which, together with Eq. (34), leads to Since, from the equations of motion for T , one finds one finally obtains the expected constraint which, unlike Eq. (28a), is independent of the ghost fields.
For further discussions of BRST (and anti-BRST) symmetries in the context of non-Hermitian field theories, see Ref. [38].

IV. SPONTANEOUS SYMMETRY BREAKING
Spontaneous symmetry breaking (SSB) is possible if the sign of m 2 1 in the Lagrangian (1) is changed, and we study here the corresponding scalar vacuum expectation values and vector masses.

A. Vacuum expectation value
With this change of sign, the Lagrangian (1) has a symmetry-breaking vacuum that is given by which is physical as long as The vacuum is defined up to a SU(2)×U(1) transformation, and it can be chosen so that with With this choice, the vacuum expectation value is unbroken by the transformation such that the Abelian subgroup of SU(2)×U(1) generated by σ = I + τ 3 remains unbroken.
This subgroup corresponds to the electromagnetic interaction, with Noether current From Eq. (21), we see that the gauge field A µ couples to the current I α + cos θ W + J α +,3 sin θ W , which can be identified with the current (45) if e = g cos θ W = g sin θ W .
The U(1) EM charge is conserved at the tree level, although the Noether current is in general not conserved. Exploration of the possibility of charge non-conservation beyond the tree level lies beyond the scope of this paper. Its existence and observability would in principle depend upon the completion of the bosonic model considered here to include fermions, which is also a topic for future work.
We can then express the scalar Lagrangian in terms of fluctuations around the vacuum (42) as where We note that the terms linear in fluctuations are a consequence of the non-Hermitian nature of the system. However, they do not play a role in the equations of motion δS/δΦ † i ≡ 0, since they depend onΦ i only. These equations of motion are The massless Goldstone modes consist of charged and neutral fields: The remaining fields consist of a charged field and three neutral fields. The charged fields are given by and one neutral field is given by with degenerate squared mass Finally, we can express the last two neutral fields as with masses where and It is not obvious that M 2 is positive or that M 2 H and M 2 h are real, and we derive the corresponding conditions on µ 2 in the next Section.
The eigenvectors of non-Hermitian matrices are not orthogonal with respect to the Hermitian inner product In the case of PT -symmetric theories, however, the eigenmodes of the non-Hermitian Hamiltonian are orthogonal with respect to the PT inner product and we have normalised the fields G ± , G, H ± , D, H and h accordingly. These eigenmodes are non-trivial linear combinations of the scalar components of Φ 1 and the pseudoscalar components of Φ 2 and, as such, they cannot be eigenstates of P. Instead, the P transformation relates the left and right eigenmodes, which are distinct for a non-Hermitian Hamiltonian.
We remark that the PT norm used for the modes G, G ± , D and H ± in Eqs. • If m 2 1 < m 2 2 < 3m 2 1 then µ 4 < T h (Condition III) or T H < µ 4 < m 2 1 m 2 2 (Conditions I and III); • If 3m 2 1 < m 2 2 then µ 4 < m 2 1 m 2 2 (Condition I). and the equations of motion read as follows: A α gauge field From these equations, we can see that the gauge field masses are as in the Hermitian Standard Model.

D. Comments on the exceptional points
At the zero exceptional points µ 2 = ±m 2 2 , the vevs become which vanish in the degenerate limit m 2 1 = m 2 2 . For m 2 1 = m 2 2 , though, the gauge boson masses at the exceptional points are remaining physical and non-zero.
In order to make sense of this, in spite of the divergence of the PT norm and the apparent non-normalisability of the Goldstone modes (see Subsection IV A), it is helpful to reconsider the behaviour of the non-Hermitian theory at the exceptional point. As an example, let us consider the following 2 × 2 squared mass matrix of the non-interacting theory [22]: For m 2 1 > m 2 2 , the eigenvectors of this squared mass matrix are where (not to be confused with the ghost field appearing earlier). The eigenvectors are not orthogonal with respect to the usual Hermitian inner product: except in the Hermitian limit µ → 0 (η → 0). They are, however, orthogonal with respect to the PT inner product, and orthonormality fixes The exceptional point of this mass matrix occurs when η → 1, at which point the normalisation of the eigenvectors diverges. This signals that the mass matrix has become defective, having the Jordan normal form and we lose an eigenvector. In fact, we see that in the limit η → 1 the eigenvectors e + and e − become parallel to one another. However, the issue of the non-orthogonality of these eigenvectors is then moot, and we can normalise them with respect to the Hermitian inner product, fixing In other words, at the exceptional point the system behaves like a Hermitian theory with one fewer degree of freedom.
Returning to the case of spontaneously-broken gauge symmetries at the zero exceptional point, the explanation for the non-vanishing masses of the gauge bosons is that the Goldstone modes must be normalised with respect to Hermitian conjugation and not PT conjugation (which has become ill-defined). The discontinuity in the behaviour of the system as we approach exceptional points means that we must treat separately these particular points in parameter space.
Thus, our conclusion is that it is possible to give masses to gauge bosons in a gaugeinvariant way through SSB also for non-Hermitian theories, even at the exceptional points.
At these points, however, the counting of eigendirections must allow for the fact that the Hamiltonian has become defective.
We note that different results were derived in Ref. [27], which is based on an alternative interpretation of a similar (Abelian) version of this non-Hermitian theory, and where the gauge boson masses are zero at the zero exceptional point. The difference in our results can be traced back to differing interpretations of the complex conjugate: we take complex conjugation to act linearly on the fields, whereas in Ref. [27] it is taken to act antilinearly on one of the fields (as motivated by a similarity transformation to a Hermitian theory). This has the effect of interchanging v 2 2 → −v 2 2 in the expression for the gauge boson masses, such that they then vanish at the zero exceptional point, when v 2 1 = v 2 2 . It is then argued that this is consistent with the fact that the Goldstone modes cannot be normalised with respect to the PT norm, which diverges at exceptional points, and these modes therefore cannot be "eaten" by the gauge field. This then leads Ref. [27] to conclude that it is possible to break the gauge symmetry of a non-Hermitian model spontaneously without giving a mass to the gauge bosons. Our conclusion is the opposite: the gauge boson remains massive in the symmetry-broken phase, even at the zero exceptional point. We note the following features from each panel of Fig. 2: • In the region m 2 1 > 3m 2 2 , the mass M 2 goes to zero at the exceptional point µ 2 = m 2 2 . If µ 2 were to become larger then m 2 2 then M 2 would become negative and we would enter the phase of broken PT symmetry.
• In the region m 2 1 /3 < m 2 2 < m 2 1 , the masses M 2 H and M 2 h become equal at the point • For m 2 2 > 3m 2 1 , the masses are all real and positive as long as tanh 2 β < m 2 1 /m 2 2 .
We note in the lower right panel of Fig. 3 that the gauge-boson masses vanish at the point where the symmetry is restored, as we would expect. It is interesting to compare the masses in this PT -symmetric non-Hermitian model with those in a similar Hermitian 2HDM with the following Lagrangian, involving a Hermitian mass-mixing term: The vacuum expectation values for this Lagrangian are After expressing the Lagrangian in terms of the shifted fieldΦ i where we can calculate the eigenvalues. As in the non-Hermitian model, the massless states consist of massless charged scalar and pseudoscalar Goldstone fields The normalisations of the eigenmodes should be compared with those in Subsection IV A.
We remark that this Hermitian model is not PT symmetric if Φ 1 and Φ 2 transform as a scalar and a pseudoscalar, respectively. It is, however, PT symmetric if both Φ 1 and Φ 2 transform as scalars or pseudoscalars, and the Hermitian and PT norms coincide, as is expected for a Hermitian, PT -symmetric theory.
The remaining massive fields include a charged scalar, a neutral pseudoscalar and two neutral scalar fields. The charged scalars are and the pseudoscalar is with degenerate squared mass Lastly, we can express the neutral scalar boson fields as with squared masses where and λ = κ cos 4 β , (91a) The squared masses for this Hermitian model are plotted in Fig. 4 in the parameter ranges 2m 2 1 > m 2 2 (left panel) and 2m 2 1 < m 2 2 (right panel). We see that the mass spectra are completely different from the non-Hermitian, PT -symmetric case, offering distinctive phenomenological possibilities.
Before concluding, we remark that, by comparing the expressions above with those in Subsection IV A, we can see that the non-Hermitian 2HDM that we have considered in this work is an analytic continuation of the Hermitian 2HDM, obtained by taking m 4 12 → −µ 4 . In other words, the Hermitian 2HDM lies in the fourth quadrant of the (m 2 2 /m 2 1 , µ 4 /m 4 1 ) plane, not shown in Fig. 1.

VI. CONCLUSION
In this paper, we have exhibited a consistent description of a non-Abelian two-Higgsdoublet model with a non-Hermitian scalar mass-mixing term, which generalises the non-Hermitian extension of the Abelian Higgs model given in Refs. [26,29]. As in Ref. [29], the main point that leads to a consistent model in the present article consists of restricting gauge invariance to a sub-class of gauge field configurations. The corresponding constraint plays the role of a conventional gauge-fixing condition, but which must be taken into account at the classical level already, in order to find consistent field equations. Within this framework, we have described the realisation of SSB and compared its features with the Hermitian case.
An interesting question is the significance of the exceptional points. As explained in this article, the number of eigendirections is reduced there, so that this limit is not continuous.
It is indeed easy to see that, in the non-interacting model, one can write a unique equation of motion for Φ 1 + Φ 2 only, with mass (m 2 1 + m 2 2 )/2, when taking the exceptional limit |µ 2 | → |m 2 1 − m 2 2 |/2, cf. Eq. (78). The introduction of gauge or self-interactions does not allow this though, and one can therefore question the stability of the exceptional points under quantum corrections, which appear as soon as interactions are switched on. However, the treatment of radiative corrections and further study of the exceptional points goes beyond the scope of the present article.
We have noted that physical observables depend on µ 4 , and thus not on the set of equations of motion we choose. This can be checked also with the masses of scalar excitations and gauge bosons: the transformation µ 2 → −µ 2 leads to changes in the signs of α and β [see Eq. (56)], such that the masses obtained after SSB are not modified. It was shown in Ref. [29] that the quantum theory also depends on µ 4 only, and we expect the same to be valid here, since this feature is based on the scalar sector properties of the partition function, which is very similar here.
Finally, we note that the scalar boson mass spectrum in the non-Abelian non-Hermitian model differs significantly from that in the Hermitian version. This shows that the non-Hermitian model opens up new phenomenological perspectives, which merit a subsequent more detailed discussion.