On the Consistency of a Non-Hermitian Yukawa Interaction ∗ Jean Alexandre , Nick E

Physics Letters B 807 (2020) 135562

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Physics Letters B

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On the consistency of a non-Hermitian Yukawa interaction ∗ Jean Alexandre , Nick E. Mavromatos

Theoretical Particle Physics and Cosmology Group, Department of Physics, King’s College London, Strand, London WC2R 2LS, UK a r t i c l e i n f o a b s t r a c t

Article history: We study different properties of an anti-Hermitian Yukawa interaction, motivated by a scenario of Received 9 April 2020 radiative anomalous generation of masses for the right-handed sterile neutrinos. The model, involving Received in revised form 12 June 2020 either a pseudo-scalar or a scalar, is consistent both at the classical and quantum levels, and particular Accepted 12 June 2020 attention is given to its properties under improper Lorentz transformations. The path integral is Available online 18 June 2020 consistently deﬁned with a Euclidean signature, and we discuss the energetics of the model, including Editor: N. Lambert the options for dynamical mass generation. © 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

1. Introduction and motivation C − C = M 5 M iλ a(x) ψR ψR ψψR i λ a(x) ψ γ ψ , (1)

Contrary to popular belief, quantum mechanical systems with M = C + where C denotes the charge conjugate, and ψ ψR ψR denotes non-Hermitian Hamiltonians have been argued to be consistent, the Majorana field. In the model of [12]the Yukawa couplings λ ∈ in the sense of being characterised by real energy eigenvalues, R are real and the interaction (1)is Hermitian. thus acquiring potential physical significance, in the framework The model involves a kinetic mixing of the axion field a(x), of PT symmetry [1], which by now has found many experimen- which could be one (or more) of the axions that characterise string tal and theoretical applications. Although the quantum-mechanical theory effective models, stemming from string moduli [13]with a PT symmetric non-Hermitian Hamiltonian are well understood by fundamental axion b(x) that, in four (uncompactified) space-time now, the corresponding field theoretical analogues [2]have only dimensions, stems from the spin-one (Kalb-Ramond) field of the recently started becoming the object of systematic studies [3–9], massless (bosonic) gravitational multiplet of the underlying string with potential phenomenological relevance to particle physics. theory [14]: In general it is enough for a non-Hermitian model to have an anti-linear symmetry, in order to allow for a regime of real ∂ b(x)∂μa(x), (2) energies [10], and not necessarily PT. In our note we shall in- γ μ stead consider a CPT-symmetric anti-Hermitian Yukawa interac- where the parameter γ ∈ R has been taken to be real in [12]. In tion (AHY). Comments on non-Hermitian Yukawa neutrino sector [12]it was assumed that the field b(x) couples to global gravi- with real energies can already be found in [5], which studies a tational anomalies. Upon diagonalisation of the kinetic terms (2) non-Hermitian extension of Quantum Electrodynamics. We note by appropriate field redefinitions, the effective action contains a that a non-Hermitian Yukawa model is considered in [11], with kinetic term for the axion field 1 (1 − γ 2)∂ a ∂μa, which thus ne- complex energies though, in order to give an effective descrip- 2 μ cessitates the restriction tion of particle decay, more specifically Higgs decay to a pair of opposite-sign leptons. |γ | < 1 , (3) An AHY interaction as the one studied here could be motivated by the scenario described in [12]. Let us explain why. Such a to avoid problems with unitarity. model is inspired by string theory considerations, and describes Moreover, the axion couples now to the Gravitational anomaly a radiative anomalous generation of (Majorana) masses for the with a γ -dependent coupling [12] right-handed sterile neutrinos, ψR , as a result of shift-symmetry breaking interactions of the latter with axions of Yukawa type γ c1 a(x) μνρσ ˜ − R Rμνρσ (4) 2 2 192π MPl 1 − γ μνρσ Corresponding author. in a standard notation, with MPl the reduced Planck mass, R * ˜ E-mail address: [email protected] (J. Alexandre). the Riemann tensor and Rμνρσ its dual, and c1 ∝ N f , a constant https://doi.org/10.1016/j.physletb.2020.135562 0370-2693/© 2020 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 J. Alexandre, N.E. Mavromatos / Physics Letters B 807 (2020) 135562

2 depending on the details of the chiral fermion sector of the under- 1 μ M 2 ¯ ¯ ¯ 5 L = ∂μφ∂ φ − φ + ψi∂ψ/ − mψψ + λφψγ ψ. (9) lying model, consisting of N f fermionic species that circulate in 2 2 the anomalous graviton loop [15]. For real λ, the Yukawa interaction is imaginary, since ψ¯ γ 5ψ is However, if one is prepared to use non-Hermitian Hamiltonian, anti-Hermitian: (ψ¯ γ 5ψ)† =−ψ¯ γ 5ψ. Under discrete transforma- with an imaginary mixing term γ = iγ˜ , with γ˜ real, then the re- tion we have striction (3)in the range of γ˜ is lifted, at the cost that, upon ∗ canonically normalising the axion-a(x) kinetic term, one obtains charge conjugation C : ψ(t,r) →−iγ 2ψ (t,r) (10) an effective Yukawa coupling 0 parity P : ψ(t, r) → γ ψ(t, −r) iλ iλ C − C = M 5 M : → 1 3 − a(x) ψR ψR ψψR a(x) ψ γ ψ . (5) time-reversal T ψ(t, r) γ γ ψ( t, r), 1 + γ˜ 2 1 + γ˜ 2 such that the anti-Hermitian interaction φψ¯ γ 5ψ is CPT-even if φ The radiative mechanism of [12], entails the generation of a is a pseudo-scalar, which gives an extra motivation for consider- (Majorana) mass for the sterile neutrino ing axions. We note that a study of discrete symmetries and their λ relation to Hermiticity is provided in [17]. ∝ γ −8 9 M R c1 MPl , (6) 1 − γ 2 2.1. Equations of motion where is the Ultra Violet (UV) momentum cutoff used for the regularisation of the effective gravitational field theory [16], which The equation of motion for the (pseudo) scalar field is could in general be lower than MPl, although in most cases the + 2 = ¯ 5 two scales are identified. Notably, the mass M R is independent of φ M φ λψγ ψ, (11) the details of the axion potential and, thus, the way its mass arises where the left hand side is real since, motivated by the physical in the model. model of [12], we assume a real mass M for the real scalar fields, We now remark that, in the extended model in which γ = iγ˜ , and the right hand side is imaginary. Hence we obtain the follow- γ˜ ∈ R, one observes that the radiative fermion mass (6)can re- ing equation of motion for the scalar field main real provided one considers AHY interactions (5)with purely imaginary coupling λ = iλ˜ , where λ˜ ∈ R: φ + M2φ = 0 , (12) λ˜ λ˜ − C − C =− M 5 M and the on-shell constraint for fermions a(x) ψR ψR ψψR a(x) ψ γ ψ . 1 + γ˜ 2 1 + γ˜ 2 ¯ 5 (7) ψγ ψ = 0 , (13) In such a case the gravitational anomaly term (4)would also be which means that no chiral condensate is allowed. We note that anti-Hermitian. Such a term though vanishes for flat (or Robertson- the conditions (12) and (13)are valid on-shell only, and they de- Walker) metric backgrounds, which we restrict our attention to in fine a saddle point of the path integral. Fluctuations do not satisfy this work.1 The point of the current note is to discuss some prop- these conditions though: since ψ and ψ¯ fluctuate independently, erties of such AHY interactions and demonstrate the consistency of the quantity ψ¯ γ 5ψ is not purely imaginary in the path integral. the approach. Fluctuations in the scalar field do not respect the equation (12), The structure of the article is the following: in the next sec- but they remain real. tion 2, we discuss the most important properties of the model with The constraint (13)also allows consistent properties under im- anti-Hermitian Yukawa interactions, and demonstrate its consis- proper Lorentz transformations. Indeed, if φ is a pseudo-scalar, tency, from the point of view of unitarity and Lorentz covariance, then both left- and right-hand sides of the equation of motion including improper Lorentz transformations. We also point out that (11)are P -odd, since ψ¯ γ 5ψ is a pseudo-scalar. But if φ is a scalar, the anti-Hermitian Yukawa interactions is CPT even for pseudo- then the only consistent interpretation of eq. (11)is to have both scalar particles, of interest in the model of [12]. In section 3, we left- and right-hand sides vanishing, in order to respect improper- discuss the energetics of the model, demonstrating that dynam- Lorentz-transformations covariance properties. ical mass generation for the fermions cannot occur in this model The equation of motion for the fermion is more subtle to inter- for quite generic reasons, unless extra interactions are included. Fi- pret, due to the anti-Hermitian interaction. Indeed, the variation of nally, section 4 contains our conclusions and some open questions the action with respect to ψ¯ gives to be studied in the future. 5 ¯ μ ¯ ¯ 5 i∂ψ/ − mψ + λφγ ψ = 0 ⇔ i∂μψγ + mψ + λφψγ = 0 . 2. Anti-Hermitian Yukawa model (14) We consider a Dirac fermion and a real scalar field, with the On the other hand, the variation of the action with respect to ψ Lagrangian leads to

i∂ ψ¯ γ μ + mψ¯ − λφψ¯ γ 5 = 0 , (15) 1 On the other hand, in theories where the fermions couple to gauge ﬁelds, μ even if the gravitational anomalies vanish in background space-times, the axions showing a mismatch with eq. (14)in the sign of the coupling couple to gauge global anomalies. In general, therefore, the anomalous terms will constant λ →−λ. A ﬁrst consistent set of equations of motion be proportional to the non-zero divergence of the axial (chiral) fermionic current. ¯ Hence, in the extended model of [12]with anti-Hermitian anomaly terms (4), with is therefore obtained by the variation with respect to ψ, together γ = iγ˜ , γ˜ ∈ R, we will have shift-symmetry preserving interactions of the form: with the Hermitian conjugation of this equation. A second set of consistent equations is obtained by the variations with respect ˜ γ 5μ i a(x)∂μ J , γ˜ ∈ R . (8) to ψ, together with the Hermitian conjugation of this equation, + ˜ 2 1 γ which feature the opposite sign for λ. These two sets of equations In the current work we ignore the role of such anomalous interactions. are physically equivalent though, since quantum corrections in this J. Alexandre, N.E. Mavromatos / Physics Letters B 807 (2020) 135562 3 model depend on λ2 only, as well as physical processes such as loop,2 leading to the non-conservation of the classical chiral cur- fermion-fermion or scalar-scalar scattering. We note that an alter- rent: native approach is possible, based on similarity transformations, μ 1 μνρσ ˜ which does not feature the above ambiguity in the derivation of ∂μ ψR γ ψR =− R Rμνρσ (21) 192 2 the equations of motion [9]. π As remarked above, such anomalies play a crucial role on the ra- 2.2. Conserved current diative fermion mass generation of the model of [12], and its non- Hermitian extension, and in the latter case will lead to additional

In the presence of a constant background scalar field φ0, the non-Hermitian terms of the form (8). But the question of grav- effective mass term for fermions is ity/gauge anomalies is not covered in this letter, where we focus on the minimal model (9). ¯ 5 ψ(m − λφ0γ )ψ , (16) 2.3. Path integral and it is shown in [2] that the corresponding energies are real as long as In order to define the path integral without potential problems related to convergence, due to an imaginary interaction, we switch |λφ |≤|m|, (17) 0 here to the Euclidean metric, where this interaction behaves as which is assumed in this section. From the equations of motion a pure phase term. Introducing the sources j, η¯, η for the fields (14)one can easily obtain φ, ψ, ψ¯ respectively, we have then μ ¯ 5 μ ¯ μ ∂μ j = 2iλφ0 ψγ ψ where j ≡ ψγ ψ (18) ¯ Zλ[ j, η¯, η]= D[φ,ψ,ψ] exp (−S Herm − SantiHerm − Ssources) , 5μ ¯ 5 5μ ¯ μ 5 ∂μ j = 2im ψγ ψ where j ≡ ψγ γ ψ, (22) which shows consistent properties under improper Lorentz trans- μ 5μ where formations if φ is P -odd: ∂μ j is then indeed a scalar and ∂μ j is a pseudo-scalar. From eqs. (18), the conserved probability cur- 1 M2 = 4 μ + 2 + ¯ + ¯ rent is S Herm d x ∂μφ∂ φ φ ψi∂ψ/ mψψ (23) 2 2 μ μ φ0 5μ J = j − λ j , (19) = 4 + ¯ + ¯ m Ssources d x( jφ ηψ ψη) which was already derived in [4], requiring unitarity of the theory. =− 4 ¯ 5 The probability is indeed less than one, and hence the approach is SantiHerm λ d x φψγ ψ self consistent, if and only if eq. (17)is valid, which, as mentioned 4 ¯ 5 above, is the same requirement that leads to real energy eigenval- = iλ d x φ with ≡ iψγ ψ. ues. There is a subtle point here, however, which deserves careful μ discussion. The vanishing of ∂μ J would imply the identity Because of the anti-Hermitian term in the Lagrangian, one cannot in general cancel δS/δψ and δS/δψ¯ simultaneously, where μ φ0 5μ S = S + S , which can potentially lead to an ambigu- ∂μ j = λ ∂μ j . (20) Herm anti Herm m ity in defining the saddle point in the path integral [7]. In our ¯ 5 = In both cases, whether φ is a scalar or a pseudo-scalar, the con- case though, because of the on-shell constraint (13), ψγ ψ 0, the saddle point is uniquely defined by the Hermitian part of the stant φ0 is P -even, such that the right-hand side of eq. (20)is a pseudo-scalar, whereas the left-hand side is a scalar, therefore Lagrangian, and in this sense the quantity plays the role of a violating covariance properties under improper Lorentz transfor- real phase. mations. With the above in mind, we note here few relevant properties Nevertheless, because of the additional constraint (13), one can of the Hermitian conjugate of the partition function: see from eqs. (18) that the currents jμ and j5μ are individually μ • ∗[ ¯] = [ ¯] conserved, such that conservation of the current J is actually It is straightforward to see that Zλ j, η, η Z−λ j, η, η . consistent with improper Lorentz transformations independent of Given the dependence of quantum corrections on λ2, the par- the parity properties of the field φ. Therefore, for the specific tition function is then of the form Z = ρ(λ2) exp iλθ(λ2) ; model (9), because the (pseudo)scalar field is real, the vector and ∗ • The change of functional variable φ →−φ leads to Z [ j, η, η¯] the axial currents are individually conserved, independently of the λ = Zλ[− j, η, η¯]; parity of the scalar field. ∗ • The change of functional variable ψ → γ 5ψ leads to Z [ j, η, η¯] We now remark that, in a Hermitian theory, conservation of the λ = Z [ j, ξ,¯ ξ], where ξ ≡−γ 5η and ξ¯ = ηγ¯ 5. vector current would have been a direct consequence of its role as λ the Noether current associated with the global U (1) symmetry of Using the above properties, one can check that the background the model. However, Noether’s theorem does not apply in a general non-Hermitian theory, as discussed in [6], but in the present scalar field φb is real for vanishing fermion sources, since situation, the constraint (13)leads to the expected conservation − − μ ∗ 1 δ ∗ 1 δ law ∂μ j = 0. φ = ∗ Z [ j, 0, 0]= Zλ[ j, 0, 0]=φb , b Z [ j, 0, 0] δ j λ Z [ j, 0, 0] δ j We finally comment on chiral fermions. For concreteness, we λ λ consider one species right-handed neutrinos, ψR with vanishing (24) bare mass m = 0, which is relevant in [12]. In this case, for which 5 the anti-Hermitian mass λφ0γ must also vanish if we impose real 2 Sterile neutrinos do not couple to gauge fields, otherwise, i.e. if we have other energies (cf. (17)), there are gravitational anomalies [15], due to (charged) types of chiral fermions, like left-handed leptons and quarks in the Stan- the circulation of the right-handed chiral fermion in the graviton dard Model sector, there are also gauge anomalies. 4 J. Alexandre, N.E. Mavromatos / Physics Letters B 807 (2020) 135562 ¯ and the condensate < ψψ > is real for vanishing scalar source, = D[φ] exp − ψ¯ i∂ψ/ + mψψ¯ + η¯ψ + ψ¯ η since x 2 ∗ ∗ 1 δ Z [0, η, η¯] 1 < ψψ¯ > = λ + −1 ∗[ ¯] ¯ φG φ , Zλ 0, η, η δηδη 2 2 [ ¯] ferm 1 δ Zλ 0, η, η ¯ such that the Euclidean S , which plays the role of vacuum = =< ψψ > . (25) ef f Zλ[0, η, η¯] δηδη¯ energy functional, is larger than that for the free theory, and one Next we proceed to discuss the Energetics of the AHY interac- cannot expect fermion dynamical mass generation, in contrast to tions. the usual Hermitian case, where such a dynamical mass lowers the energy of the system (in fact, in the Hermitian case, integrating out 3. Energetics of the non-Hermitian interaction the massive (pseudo) scalar field would produce an attractive four fermion term, capable of inducing dynamical mass generation). The In this section we discuss some energetic arguments associ- situation changes, of course, if one has additional attractive, and ated with the AHY interactions, which could be of relevance if one sufficiently strong, four-fermion interaction terms in the model (9), is interested in establishing whether such interactions alone can e.g. of the type lead to dynamical mass generation, as an alternative to the afore- 1 2 mentioned radiative mass mechanism (6) of [12]. As we shall see, − ψ¯ 5ψ , (29) 2 γ based on quite generic arguments, within the model (9) dynamical 2 f4 mass for the fermions is not possible. where f4 has dimensions of mass. A detailed analysis in such a case [18] confirms the possibility of dynamical fermion and 3.1. Fermionic effective theory (pseudo)scalar mass generation, for sufficiently strong four-fermion couplings of the order of the UV cutoff , used for the regularisa- To discuss fermion dynamical mass generation, we assume a tion of the UV divergences of the effective theory. We note that for real non-zero bare (pseudo) scalar mass M = 0. The fermionic ef- the above energetics arguments, it suffices to employ only the in- ferm tegrand (26)of the path integral over fermions, and thus one does fective action Sef f is obtained after integrating out the scalar field ⎛ ⎞ not need to linearise the four fermion interaction, as in [17]. Such a linearisation, in terms of auxiliary scalar felds, is empoloyed in ferm exp(−S ) ≡ exp ⎝− ψ¯ i∂ψ/ + mψψ¯ + η¯ψ + ψ¯ η⎠ the full Schwinger-Dyson treatment of dynamical mass generation ef f in such non hermitian theories [18]. ⎛ x ⎞ 3.2. Scalar effective theory 1 − × D[φ] exp ⎝− φG 1φ + iλφ ⎠ (26) 2 The scalar effective action Sscal is obtained after integrating out ⎛ x ⎞ ef f massive fermions 2 ⎛ ⎞ ⎝ ¯ ¯ ¯ λ ⎠ = exp − ψi∂ψ/ + mψψ + η¯ψ + ψη + G , 2 2 scal ⎝ 1 μ M 2 ⎠ ¯ exp(−S ) ≡ exp − ∂μφ∂ φ + φ + jφ D[ψ,ψ] x ef f 2 2 x −1 =− + 2 ⎛ ⎞ where G M and is defined in eqs. (23) and, as ex- plained above, it is treated as a real phase factor. If we neglect × exp ⎝− ψ(¯ i∂/ + m − λφγ 5)ψ⎠ . higher order derivatives, we have then x 2 ferm ¯ ¯ λ ¯ 5 2 S ψi∂ψ/ + mψψ + |ψ ψ| , (27) For a constant scalar field configuration φ0, the effective potential ef f 2 γ 2M is then x M2 which, because of the original imaginary Yukawa interaction, in- = 2 − + − 5 Uef f (φ0) φ0 Tr ln(p/ m λφ0γ ) , (30) cludes a repulsive 4-fermion interaction, and thus increases the 2 energy of the system. This also implies that, on setting the bare such that fermion mass to zero, m = 0, in (9), we cannot have dynamical − 5 mass generation for the fermions, since the corresponding process dUef f 2 λγ = M φ0 − Tr (31) dφ p/ + m − λφ 5 is related to the formation of a condensate. It also implies that 0 0γ one cannot generate dynamically a chiral fermion mass μ (equiv- 2 2 + m2 − λ2φ2 ¯ 5 2 λ φ0 2 2 2 2 0 alently, a non vanishing chiral condensate ψ γ ψ ). The above = M φ0 + − (m − λ φ ) ln , 4 2 0 2 − 2 2 conclusions are confirmed in [18]by a Schwinger-Dyson analysis. π m λ φ0 This result can be expected from a general argument [19], used where is the UV cut off, and we note that (/p)2 =−p2 with the to study the energetics of parity violation in gauge theories, such Euclidean metric. The energies are therefore real for m2 ≥ λ2φ2, as QCD: due to the pure phase nature of the AHY interaction (cf. 0 as expected from [2]. We remind here that a discrete antilinear (23)), the fermionic effective action satisfies symmetry is necessary to ensure a regime of real energies [10], which in our case is CPT. − ferm ≤ D[ ] − ¯ + ¯ + ¯ + ¯ 2 2 → 2 exp( Sef f ) φ exp ψi∂ψ/ mψψ ηψ ψη It is interesting to note that, in the limit λ φ0 m , the effec- x tive potential consists in a mass term only 1 − 1 λ2 + G 1 + i (28) (1) 2 2 (1) 2 2 2 φ φ λφ Uef f → (M ) φ with (M ) = M + . (32) 2 2 0 4π 2 J. Alexandre, N.E. Mavromatos / Physics Letters B 807 (2020) 135562 5

If one keeps in mind the anti-Hermitian fermion mass term Declaration of competing interest ¯ 5 λφ0ψγ ψ, obtained for a constant scalar field configuration in the original action (9), the limits λφ0 →±m correspond to the so- The authors declare that they have no known competing finan- called exceptional points, where the number of fermionic degrees of cial interests or personal relationships that could have appeared to freedom is halved [5], since one of the chiralities has a vanishing influence the work reported in this paper. probability density [4]. We note that these exceptional points are specific to non-Hermitian models, and have no analogue in Hermi- Acknowledgements tian models. From the point of view of the scalar effective model, the result (32)shows that these exceptional points corresponds to The work of NEM and JA is supported in part by the UK Science a non-interacting theory: quantum corrections suppress all inter- and Technology Facilities research Council (STFC) under the re- 2 2 2 search grants ST/P000258/1 and ST/T000759/1. NEM also acknowl- actions and lead to a free system. In the regime m <λ φ0 , the effective potential features a complex dressed mass and an imagi- edges a scientific associateship (“Doctor Vinculado”) at IFIC-CSIC- nary self-interaction Valencia University, Valencia, Spain. 2 4 1 (1) 2 λ 2 2 λ 4 References Uef f = (M ) − i m φ + i φ + real , (33) 2 4π 0 16π 0 [1] C.M. Bender, Contemp. Phys. 46 (2005) 277, https://doi .org /10 .1080 / where we note that the imaginary parts of the effective potential 00107500072632, arXiv:quant -ph /0501052 [quant -ph]. are finite. As a consequence of eq. (33), decay rates are generated [2] C.M. 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