Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theories

# Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theories

Andreas Fring and Takanobu Taira
Department of Mathematics, City University London,
Northampton Square, London EC1V 0HB, UK
E-mail: a.fring@city.ac.uk, takanobu.taira@city.ac.uk
###### Abstract

We study the interplay between spontaneously breaking global continuous and discrete antilinear symmetries in a newly proposed general class of non-Hermitian quantum field theories containing a mixture of complex and real scalar fields. We analyse the model for different types of global symmetry preserving and breaking vacua. In addition, the models are symmetric under various types of discrete antilinear symmetries composed out of nonstandard simultaneous charge conjugations, time-reversals and parity transformations; CPT. While the global symmetry governs the existence of massless Goldstone bosons, the discrete one controls the precise expression of the Goldstone bosons in terms of the original fields in the model and its physical regimes. We show that even when the CPT-symmetries are broken on the level of the action expanded around different types of vacua, the mass spectra might still be real when the symmetry is preserved at the tree approximation and the breaking only occurs at higher order. We discuss the parameter space of some of the models in the proposed class and identify physical regimes in which massless Goldstone bosons emerge when the vacuum spontaneously breaks the global symmetry or equivalently when the corresponding Noether currents are conserved. The physical regions are bounded by exceptional points in different ways. There exist special points in parameter space for which massless bosons may occur already before breaking the global symmetry. However, when the global symmetry is broken at these points they can no longer be distinguished from genuine Goldstone bosons.

Goldstone bosons in different PT-regimes of non-Hermitian scalar QFT

Goldstone bosons in different PT-regimes of non-Hermitian scalar quantum field theories

Andreas Fring and Takanobu Taira

Department of Mathematics, City University London,

Northampton Square, London EC1V 0HB, UK

E-mail: a.fring@city.ac.uk, takanobu.taira@city.ac.uk

\abstract@cs

## 1 Introduction

It is quite well understood how to extend the conventional framework of Hermitian classical and quantum mechanics [1, 2, 3] to allow for the inclusion of non-Hermitian systems. When the latter systems admit an antilinear symmetry [4], such as for instance being invariant under a simultaneous reflection in time and space, referred to as -symmetry, this can be achieved in a self-consistent manner. In these circumstances one encounters three types of regimes with qualitatively different behaviour, a -symmetric phase, a spontaneously broken -symmetric phase and a completely -symmetry broken phase. Based on the formal analogy between the Schrödinger equation and the propagation of light in the paraxial approximation described by the Helmholtz equation many of the findings obtained in the quantum mechanical description have been confirmed experimentally and further developed in classical optical settings with the refractive index playing the role of a complex potential [5, 6, 7, 8, 9].

When implementing and extending these idea and principles to quantum field theories there is less consensus, and for some aspects alternative resolutions have been proposed. Naturally, as a direct extension of the well studied purely complex cubic potential in quantum mechanics the scalar field theory with imaginary cubic self-interaction term has been investigated at first [10, 11] and also the more generally deformed harmonic oscillator has been generalised to a field theoretical interaction term more recently [12]. Non-Hermitian versions with a field theoretic Yukawa interaction [13, 14, 15, 16] have been investigated in regard to Higgs boson decay. Besides bosonic theories also generalizations to non-Hermitian fermion theories such as a free fermion theory with a -mass term or the massive Thirring model have been proposed [17]. -symmetric versions of quantum electrodynamics have been studied [18, 19] as well.

Here we will focus on a feature that is very central to standard Hermitian quantum field theory, the Goldstone theorem, and investigate further how it extends to non-Hermitian theories. We recall that in the Hermitian case the theorem states that the number of massless Goldstone bosons in a quantum field theory is equal to the dimension of the coset , with denoting a global continuous symmetry group of the action and the symmetry group that is left when the theory is expanded around a specific vacuum [20, 21]. The question of extension was recently addressed by Alexandre, Ellis, Millington and Seynaeve [22] and separately by Mannheim [23]. Interestingly, both groups found that the theorem appears to hold for non-Hermitian theories as well, but they proposed two alternative variants for it to be implemented. In addition, Mannheim suggests that the non-Hermitian theory possess the new feature of an unobervable Goldstone boson at a special point. Here we find that the Goldstone bosons takes on different forms depending on whether the theory is in the -symmetric regime, at standard exceptional point or what we refer to as the zero-exceptional point. We distinguish here between a standard exceptional point, corresponding to two nonzero eigenvalues coalescing, and a zero-exceptional point defined as the point when a zero eigenvalue coalesces with a nonzero eigenvalue.

The problem that both groups have tried to overcome at first is the feature that the equations of motion obtained from functionally varying the action with respect to the scalar fields on one hand and on the other separately with respect to its complex conjugate field are not compatible. This is a well known conundrum for non-Hermitian quantum field theories and has for instance been pointed out previously and elaborated on in [24, 25] for a non-Hermitian fermionic theory. Hence, without any modifications the proposed non-Hermitian quantum field theories appear to be inconsistent. To resolve this problem the authors of [22] proposed to use a non-standard variational principle by keeping some non-vanishing surface terms. In contrast, Mannheim [23] utilizes the fact that the action of a theory can be altered without changing the content of the theory as long as the equal time commutation relations are preserved, see e.g. [17]. Utilizing that principle he investigates a model based on a similarity transformed action of the previous one in which the entire set of equations of motion have consistent properties. Remarkably, it was found for both versions that the theory expanded around the global -symmetry breaking vacuum contains a massless Goldstone boson. Moreover, while in the approach that only validates half of the standard set of equations of motion non-standard currents are conserved and Noether’s theorem seems to be evaded, the approach proposed in [23] is based on the standard variational principle leading to standard Noether currents.

Here we largely adopt the latter approach and analyse theories expanded about different types of vacua, global symmetry breaking and also preserving ones, for a class of models containing a mixture of several types of complex scalar of fields and also real self-conjugate fields. In particular, we identify the physical regions in parameter space by demanding the masses to be non-negative real-valued in order to be physically meaningful. This has not been considered previously, but is in fact quite essential as potentially the theory might be entirely unphysical. As is turns out, in many scenarios we are able to identify some physical regimes that are, however, quite isolated in parameter space. We find some vacua that break the -symmetries on the level of the action, but still possess physically meaningful mass spectra, as the symmetry breaking occurs at higher order couplings than at the tree approximation. Moreover, we derive the explicit forms of the Goldstone boson in all three -regimes, the symmetric and spontaneously broken phases, as well as at the exceptional point.

Our manuscript is organised as follows: In section 2 we introduce a general model with scalar field that might be genuinely complex but in some versions also contain real self-conjugate fields. In section 3 and 4 we investigate two specific examples of this general class of models in more detail and identify the physical regions in which Goldstone bosons may or may not occur. We investigate different types of vacua that may break the global -symmetry and also several variants of discrete -symmetries that might be broken separately. Starting from a complex squared mass matrix we construct the -operator that together with -operator can be used to identify the real eigenvalue regime and show how these operators, that can be thought off as quantum mechanical analogues, are related to the quantum field theoretical -operator. We identify the explicit form of the Goldstone boson in terms of the original fields in the action in different -regimes. In section 5 we investigate how the interaction term may be generalised so that the action still respects a discrete -symmetry and a continuous global -symmetry. We state our conclusions and present an outlook in section 6.

## 2 A non-Hermitian model with n complex scalar fields

We consider here generalizations of the model originally proposed in [22] and further studied in [23]. To be a suitable candidate for the investigation of the non-Hermitian version of Goldstone’s theorem the model should be not invariant under complex conjugation, possess a discrete -transformation symmetry and crucially be invariant under a global continuous symmetry. The actions involving the Lagrangian densities functional of the general form

 Ln=n∑i=1(∂μϕi∂μϕ∗i+cim2iϕiϕ∗i)+n−1∑i=1κiμ2i(ϕ∗iϕi+1−ϕ∗i+1ϕi)−n∑i=1gi4(ϕiϕ∗i)2\specialhtml:\specialhtml: (2.0)

possess all of these three properties. The parameter space is spanned by the real parameters and . The latter constants might be absorbed into the mass and the couplings when allowing them to be purely imaginary or real. However, we keep these constants separately since their values distinguish between different types of qualitative behaviour as we shall see below. When fixing those constants to specific values the action reduces to the model discussed in [22, 23]. In order to keep matters as simple as possible in our detailed analysis, we will set here for , but in section 5 we argue that the interaction term may be chosen in a more complicated way with all three properties still preserved.

Functionally varying the action separately with respect to and gives rise to the two sets of equations of motion

 δInδϕi=∂Ln∂ϕi−∂μ[∂Ln∂(∂μϕi)]=0,δInδϕ∗i=∂Ln∂ϕ∗i−∂μ[∂Ln∂(∂μϕ∗i)]=0.\specialhtml:\specialhtml: (2.0)

We comment below on the compatibility of these equations. Evidently, the action is not Hermitian when for some . However, it is invariant under two types of -transformations

 CPT1:ϕi(xμ)→(−1)i+1ϕ∗i(−xμ),CPT2:ϕi(xμ)→(−1)iϕ∗i(−xμ),    i=1,…,n.\specialhtml:\specialhtml: (2.0)

As pointed out in [26] these types of symmetries are not the standard transformations as some of the fields are not simply conjugated and does not simply act on the argument of the fields, but also acquire an additional minus sign as a factor under the transformation. Such type of symmetries were studied in the quantum field theory context in more detail in [26] and as argued therein make the non-Hermitian versions good candidates for meaningful and self-consistent quantum field theories, in analogy to their quantum mechanical versions, despite being non-Hermitian.

In addition, the action related to (2) is left invariant under the continuous global -symmetry

 ϕi→eiαϕi,  ϕ∗i→e−iαϕ∗i,     i=1,…,n, α∈R,\specialhtml:\specialhtml: (2.0)

when none of the fields in the theory is real, that is when for all . Applying Noether’s theorem and using the standard variational principle for this symmetry one obtains

 δLn=∂μ[n∑i=1∂Ln∂(∂μϕi)δϕi+∂Ln∂(∂μϕ∗i)δϕ∗i]+n∑i=1[δInδϕiδϕi+δInδϕ∗iδϕ∗i]. (2.0)

Thus provided the equations of motion in (2) hold, and when using the global -symmetry in the variation with and , we derive the Noether current associated to this symmetry as

 jμ=iα∑i(ϕi∂μϕ∗i−ϕ∗i∂μϕi).\specialhtml:\specialhtml: (2.0)

Below we discuss in more detail under which circumstances this current is conserved. We will argue that Noether’s theorem holds in its standard form and is not evaded as concluded by some authors. Next we are mainly interested in the study of mass spectra resulting by expanding the potentials around different vacua as this probes the Goldstone theorem.

## 3 Discrete antilinear and continuous global symmetry

We now discuss the model in more detail with all fields being genuinely complex scalar fields, i.e. , . Then the action for (2) takes on the form

 I3(ϕi,ϕ∗i,∂μϕi,∂μϕ∗i)=∫d4xL3,\specialhtml:\specialhtml: (3.0)

with Lagrangian density functional

 L3=3∑i=1∂μϕi∂μϕ∗i−V3, (3.0)

and potential

 V3=−3∑i=1cim2iϕiϕ∗i+cμμ2(ϕ∗1ϕ2−ϕ∗2ϕ1)+cνν2(ϕ2ϕ∗3−ϕ3ϕ∗2)+g4(ϕ1ϕ∗1)2. (3.0)

Compared to (2) we have simplified here the interaction term by taking and . The model contains the real parameters and . While this action is not Hermitian, that is invariant under complex conjugation, it respects various discrete and continuous symmetries. It is invariant under two types of -transformations (2)

which are both discrete antilinear transformations. Moreover, the action (3) is left invariant under the continuous global -symmetry (2), which gives rise to the Noether current (2)

 jμ=iα∑3i=1(ϕi∂μϕ∗i−ϕ∗i∂μϕi). (3.0)

With the dimension of the global symmetry group being just , we may only encounter two possibilities for the Hermitian case, that is the model contains one or no massless Goldstone boson when the symmetry group for the expanded theory is or , respectively, after a specific vacuum has been selected [20, 21]. As we shall see, breaking in our model the global -symmetry for the vacuum will give rise to the massless Goldstone bosons in the standard fashion, albeit with some modifications and novel features for a non-Hermitian setting. The six equations of motion in (2) read in this case

 □ϕ1−c1m21ϕ1−cμμ2ϕ2+g2ϕ21ϕ∗1 = 0, (3.0) □ϕ2−c2m22ϕ2+cμμ2ϕ1+cνν2ϕ3 = 0, (3.0) □ϕ3−c3m23ϕ3−cνν2ϕ2 = 0, (3.0) □ϕ∗1−c1m21ϕ∗1+cμμ2ϕ∗2+g2ϕ1(ϕ∗1)2 = 0, (3.0) □ϕ∗2−c2m22ϕ∗2−cμμ2ϕ∗1−cνν2ϕ∗3 = 0, (3.0) □ϕ∗3−c3m23ϕ∗3+cνν2ϕ∗2 = 0, (3.0)

with d’Alembert operator and metric . We encounter here the same problem as pointed out for with four scalar fields investigated in [22, 23], namely that as a consequence of the non-Hermiticity of the action the equations of motions obtained from the variation with regard to the fields , (3)-(3), are not the complex conjugates of the equations obtained from the variation with respect to the fields , (3)-(3). Hence, the two sets of equations appear to be incompatible and therefore the quantum field theory related to the action (3) seems to be inconsistent.

An unconventional solution to this conundrum was proposed in [22], by suggesting to omit the variation with respect to one set of fields and also taking non-vanishing surface terms into account. Even though this proposal appears to lead to a consistent model, it remains somewhat unclear as to why one should abandon a well established principle from standard complex scalar field theory. Here we adopt the proposal made by Mannheim [23], which is more elegant and, from the point of view of extending the well established framework of non-Hermitian quantum mechanics to quantum field theory, also more natural. It consists of seeking a similarity transformation for the action that achieves compatibility between the two sets of equations of motion. It is easy to see that any transformation of the form , that leaves all the other fields invariant will achieve compatibility between the two sets of equations (3)-(3) and (3)-(3)

The analysis to achieve this is most conveniently carried out when reparameterising the complex fields in terms of real component fields. Parameterising therefore the complex scalar field as with , the action in (3) acquires the form

 I3 = ∫d4x{3∑i=112[∂μφi∂μφi+∂μχi∂μχi+cim2i(φ2i+χ2i)]+icμμ2(φ1χ2−φ2χ1) +icνν2(φ3χ2−φ2χ3)−g16(φ21+χ21)2}.

This approach differs slightly from Mannheim’s, who took the component fields to be complex as well. The continuous global -symmetry (2) of the action is realised for the real fields as , , that is and for small. The symmetries in (3) manifests on these fields as

In this form also the antilinear symmetry

leaves the action invariant. Let us now transform the action in the form (3) to an equivalent Hermitian one.

### 3.1 A CPT equivalent action, different types of vacua

We define now the analogue to the Dyson map [27] in quantum mechanics as

 η=exp[π2∫d3xΠφ2(x,t)φ2(x,t)]exp[π2∫d3xΠχ2(x,t)χ2(x,t)],\specialhtml:\specialhtml: (3.0)

involving the canonical momenta and , . Using the Baker-Campbell-Haussdorf formula we compute the adjoint actions of on the scalar fields as

 ηφiη−1=(−i)δ2iφi,   ηχiη−1=(−i)δ2iχi,     ηϕiη−1=(−i)δ2iϕi,    ηϕ∗iη−1=(−i)δ2iϕ∗i. (3.0)

The equal time commutation relations , , for are preserved under these transformations. Applying them to in (3), we obtain the new equivalent action

 ^I3=ηI3η−1= +cμμ2(φ1χ2−φ2χ1)+cνν2(φ3χ2−φ2χ3)−g16(φ21+χ21)2.

The -symmetry is still realised in the same way as for , but the -symmetries for are now modified to

accommodating the fact that no explicit imaginary unit is left in the action. Notice that these symmetries are, however, no longer antilinear and therefore lack the constraining power of predicting the reality of non-Hermitian quantities. The equations of motion resulting from functionally varying with respect to the real fields are

 −□φ1 = ∂V∂φ1=−c1m21φ1−cμμ2χ2+g4φ1(φ21+χ21), (3.0) −□χ2 = −∂V∂χ2=−c2m22χ2+cμμ2φ1+cνν2φ3, (3.0) −□φ3 = ∂V∂φ3=−c3m23φ3−cνν2χ2, (3.0) −□χ1 = ∂V∂χ1=−c1m21χ1+cμμ2φ2+g4χ1(φ21+χ21), (3.0) −□φ2 = −∂V∂φ2=−c2m22φ2−cμμ2χ1−cνν2χ3, (3.0) −□χ3 = ∂V∂χ3=−c3m23χ3+cνν2φ2. (3.0)

We may write the action and the corresponding equation of motions more compactly. Introducing the column vector field , the action acquires the concise form

 ^I3=12∫d4x[∂μΦTI∂μΦ−ΦTHtΦ−g8(ΦTEΦ)2].\specialhtml:\specialhtml: (3.0)

Here we employed the Hessian matrix which for our potential reads

 H(Φ)=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝g4(3φ21+χ21)−c1m21−cμμ20g2φ1χ100−cμμ2c2m22−cνν20000−cνν2−c3m23000g2φ1χ100g4(φ21+3χ21)−c1m21cμμ20000cμμ2c2m22cνν20000cνν2−c3m23⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (3.0)

In (3.0) we use , and the -matrices , with and . The equation of motion resulting from (3.0) reads

 −□Φ−IHtΦ−g4I(ΦTEΦ)EΦ=0.\specialhtml:\specialhtml: (3.0)

We find different types of vacua by solving , amounting to setting simultaneously the right hand sides of the equations (3.0)-(3.0) to zero and solving for the fields . Denoting the solutions by , we find the vacua

 Φ01 = (0,0,0,0,0,0), (3.0) Φ02 = K(0)(1,c3cμm23μ2κ,−c3cμm23μ2κ,0,0,0), (3.0) Φ03 = K(0)(0,0,0,−1,c3cμm23μ2κ,cνcμν2μ2κ), (3.0) Φ04 = (φ01,c3cμm23μ2φ01κ,−cνcμν2μ2φ01κ,−K(φ01),c3cμm23μ2K(φ01)κ,cνcμν2μ2K(φ01)κ), (3.0)

where for convenience we introduced the function and constant

Notice, that in the vacuum the field is generic and not fixed. When varied it interpolates between the vacua and . For and we obtain and , respectively. We also note that at the special value of the coupling so that . Next we probe Goldstone’s theorem by computing the masses resulting by expanding around the different vacua in the tree approximation.

### 3.2 The mass spectra, PT-symmetries

Defining the column vector field with vacuum component as defined above and , we expand the potential about the vacua (3.0)-(3.0) as

 V(Φ)=V(Φ0+^Φ)=V(Φ0)+∇V(Φ0)T^Φ+12^ΦTH(Φ0)^Φ+…. (3.0)

The linear term is of course vanishing, as by design . The squared mass matrix is read off from (3.0) as

 (M2)ij=[IH(Φ0)]ij.\specialhtml:\specialhtml: (3.0)

The somewhat unusual emergence of the matrix is due to the fact that as a consequence of the similarity transformation we now have negative signs in front of some of the kinetic energy terms, see also (3.0) and (3.0).

In general this matrix is not diagonal, but in the -symmetric regime we may diagonalise it and express the fields related to these masses in terms of the original fields in the action. Denoting the eigenvectors of the squared mass matrix by , , the matrix , containing the eigenvectors as column vectors, diagonalizes as with as long as is invertible. The latter property holds in general only in the -symmetric regime. Rewriting

 ^ΦTM2^Φ=∑im2iψ2i=∑im2i(^ΦTIU)i(U−1^Φ)i,\specialhtml:\specialhtml: (3.0)

we may therefore introduce the masses for the fields

 ψi:=√(^ΦTIU)i(U−1^Φ)i\specialhtml:\specialhtml: (3.0)

as the positive square roots of the eigenvalues of the squared mass matrix , that is . Naturally this means the fields in the specific form (3.0) are absent when is not invertible and since physical masses are non-negative we must also discard scenarios in which or as unphysical.

Since the squared mass matrix is not Hermitian, but may have real eigenvalues in some regime, we can employ the standard framework from -symmetric quantum mechanics with playing the role of the non-Hermitian Hamiltonian [1, 3]. We can then identify the antilinear -operator that ensures the reality of the spectrum in that particular regime. The time-reversal operator  simply corresponds to a complex conjugation, but one needs to establish that the -operator obtained from the quantum mechanical description is the same as the one employed at the level of the action. In order to identify that connection let us first see which properties the -operator must satisfy at the level of the action. Expressing in the form

 I3[Φ]=IM3[Φ]+Iint3[Φ]=12∫d4x[ΦT(□+M2)Φ]+Iint3[Φ]. (3.0)

with real field vector , the action of the -operator on is

 CPT:IM3[Φ]→12∫d4x[ΦT[PTP□+PT(M2)∗P]Φ]. (3.0)

Hence for this part of the action to be invariant we require the -operator to obey the two relations

This is in fact the same property needs to satisfy in the -quantum mechanical framework. Let us see how to construct when given the non-Hermitian matrix . We start by constructing a biorthonormal basis from the left and right eigenvectors and , respectively, of

 M2vn=εnvn,(M2)†un=εnun\specialhtml:\specialhtml: (3.0)

satisfying

 ⟨un|vn⟩=δnm,∑n|un⟩⟨vn|=∑n|vn⟩⟨un|=I.\specialhtml:\specialhtml: (3.0)

The left and right eigenvectors are related by the -operator as

 |un⟩=snP|vn⟩.\specialhtml:\specialhtml: (3.0)

with defining the signature. Combining (3.0), (3.0) and the first relation in (3.0) we can express the -operator and its transpose in terms of the left and right eigenvectors as

The biorthonormal basis can also be used to construct an operator, often denoted with the symbol , that is closely related to the metric used in non-Hermitian quantum mechanics

 (3.0)

Despite its notation, this operator is not to be confused with the charge conjugation operator employed on the level of the action. The operator satisfies the algebraic properties [28]

 [C,M2]=0,[C,PT]=0,C2=I.\specialhtml:\specialhtml: (3.0)

When compared to the quantum mechanical setting the operator plays here the analogue to the Dyson map and the combination is the analogue to the metric operator . However, constructing with as a starting point does of course not guarantee that also will be invariant under when using this particular -operator. In fact, we shall see below that there are many solutions to the two relations in (3.0) that do not leave invariant. Thus for these -operators the symmetry is broken on the level of the action, but the mass spectra would still be real as the symmetry is preserved at the tree approximation and the breaking only occurs at higher order.

### 3.3 U(1) and CPT invariant vacuum, absence of Goldstone bosons

We investigate now in more detail the theory expanded about the vacuum in (3.0). According to our discussion at the end of the last section the theory expanded about this vacuum is invariant under the global -symmetry and all four -symmetries. As the dimension of the coset equals the standard field theoretical arguments on Goldstone’s theorem suggest that we do not expect a Goldstone boson to emerge when expanding around this vacuum. We confirm this by considering the squared mass matrix as defined in (3.0), which for this vacuum decomposes into Jordan block form as

 M21=⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝−c1m21−cμμ20000cμμ2−c2m22cνν20000−cνν2−c3m23000000−c1m21cμμ20000−cμμ2−c2m22−cνν20000cνν2−c3m23⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (3.0)

where we label the entries of the matrix by the fields in the order as defined for the vector field . The two blocks are simply related as . We find that the eigenvalues of each block only depend on the combination , so that we have three degenerate eigenvalues with linear independent eigenvectors and it therefore suffices to consider one block only and subsequently implement the degeneracy. Evaluating the constant term of the third order characteristic equations we obtain