PHYSICAL REVIEW D 98, 0509 (08) Quantum field theory of article oscillations: Neutron-antineutron conversion Anca Tureanu Deartment of Physics, Univer

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2 PHYSICAL REVIEW D 98, 0509 (08) Quantum field theory of article oscillations: Neutron-antineutron conversion Anca Tureanu Deartment of Physics, University of Helsinki, P.O. Box 64, FIN-0004 Helsinki, Finland (Received 7 Aril 08; ublished 6 July 08) We formulate the quantum field theory descrition of neutron-antineutron oscillations in the framework of canonical quantization, in analogy with the Bardeen Cooer Schrieffer theory and the Nambu Jona- Lasinio model. The hysical vacuum of the theory is a condensate of airs of would-be neutrons and antineutrons in the absence of the baryon-number violating interaction. The quantization rocedure defines uniquely the mixing of massive Bogoliubov quasiarticle states that reresent the neutron. In site of not being mass eigenstates, neutron and antineutron states are defined on the hysical vacuum and the oscillation formulated in asymtotic states. The exchange of the baryonic number with the vacuum condensate engenders what may be observed as neutron-antineutron oscillation. The convergence between the resent canonical aroach and the Lagrangian/ath integral aroach to neutron oscillations is shown by the calculation of the anomalous (baryon-number violating) roagators. The quantization rocedure roosed here can be extended to neutrino oscillations and, in general, to any article oscillations. DOI: 0.03/PhysRevD I. INTRODUCTION The Bardeen Cooer Schrieffer (BCS) theory of suerconductivity [], esecially in Bogoliubov s treatment [], became well known to article hysicists by the work of Nambu and Jona-Lasinio [3], who exlored the analogy between the equations of motion governing the electrons in a suerconductor near the Fermi level and the free Dirac equation of a massive fermion in Weyl reresentation. The sontaneous breaking of the U symmetry associated with the electric charge in the BCS theory is analogous to the sontaneous chiral symmetry breaking in the Nambu Jona-Lasinio (NJL) model. Ever since, the BCS theory has influenced article hysics in a way that is hard to overestimate (see, e.g., Ref. [4] ). Another analogy that can be established, this time in the language of Majorana fermions, is between the BCS Lagrangian and the effective Lagrangian of neutronantineutron oscillations, which breaks baryon-number symmetry [5 ] (for a recent review, see, e.g., Ref. [3] ). Neutron oscillations are a toical issue in resent-day article hysics, mainly as a otentially observable window into the baryon-number violating henomena that led to baryogenesis. Exerimental searches for neutron-antineutron conversion have been erformed both with free neutron beams and Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the ublished article s title, journal citation, and DOI. Funded by SCOAP 3. within nuclei [4]. At the Euroean Sallation Source (ESS), new exeriments are being lanned, aiming at imroving by 3 orders of magnitude [5] the best bound on the oscillation time ( s), obtained at ILL-Grenoble. Searches for neutron-mirror neutron oscillations [6] are also under consideration [7]. Recently, theoretical models have been roosed in which neutron-antineutron oscillations could occur moderately raidly, at levels near to current limits and within reach of an imroved search, around s. Such models may be suersymmetric [8], or involve large extra dimensions [9], or rely on constraints from ostshaleron baryogenesis [0]. Without seculating about the ossible source of the baryon-number violation (which in rincile can be achieved by sontaneous breaking of symmetry connected to a baryonic majoron [] ), the violation is usually considered to be exlicit and not sontaneous. Nevertheless, Bogoliubov s formalism for the BCS theory can be adated to the descrition of the neutron-antineutron oscillations, taking lace via Bogoliubov quasiarticles of Majorana tye, which reresent the rimary fermionic excitations of the system. The analogy between fermion oscillations with Majorana seudoscalar mass term and BCS theory has been noted in the Lagrangian icture in [,3], where a relativistic equivalent of the Bogoliubov transformation [to which we shall return in Sec. II, Eq. (.5) ] was used for the diagonalization of the Lagrangian. This intuitive connection is develoed in this work into a general canonical quantization formulation. The neutron and antineutron states are no more definite states in the hysical Fock sace of the system; however, they can be defined as a suerosition of (hysical) = 08 = 98() = 0509(3) Published by the American Physical Society

3 ANCA TUREANU PHYS. REV. D 98, 0509 (08) mass eigenstates, by extension of the would-be neutron/ antineutron states in the absence of the baryon-violating interaction. The collective excitations of quasiarticles, secific to BCS theory and the NJL model, aear also in the baryon-number violating ground state of neutron oscillations. The condensate structure of vacuum for neutron and neutrino oscillations was first analyzed and exlored in []. For comactness of terminology, we shall call bare neutrons the would-be neutrons in the absence of the baryon-number violating interaction. This is in analogy with the term bare electron, naming an electron without interaction with the lattice, in the BCS theory. In this language, the ground state of the baryon-number violating Hamiltonian is a condensate of airs of bare neutrons and antineutrons, with oosite momenta and sins the analogues of Cooer airs in the theory of suerconductivity. The aroach described in this work can be alied as the quantum field theory of the free oscillations of any tye of articles. Any oscillation henomenon is a result of the fact that the fields that interact and aear in the relativistic construction of the Lagrangian are not fields with definite mass, but mixings thereof. On the other hand, the massive states are not observed individually, but only in timedeendent suerositions, reresenting the oscillating articles. The mixing is caused by some additional interactions (baryon-number violating interaction in the case of neutrons, leton number violating in the case of neutrinos, strangeness violating weak interaction in the case of K 0 mesons, etc.), which are not taken into account when the oscillating articles are roduced. The vacuum condensate is then the reservoir of fermionic number, strangeness, etc., as well as of the extra chirality, which gives their definite masses but indefinite quantum numbers to the quasiarticles associated with the mass eigenfields. The difference in the masses of the quasiarticles roduces the oscillation, which is essentially the oscillation of a quantum number, realized through the exchange with the vacuum condensate. We may say that the quasiarticles as mass eigenstates create the observable kinematical effects, while the oscillating articles are subject to the dynamical effects included in the Lagrangian. The interaction revents the mass discrimination, and the lack of a mass analyzer leads to what is erceived as article oscillations. The neutrino oscillations are the rototyical oscillation henomena that have been studied extensively over a long time (for a review and an historical account, see Ref. [4] ). A quantum field theoretical framework has been develoed, and the revailing icture consists in viewing the neutrino oscillations as a single rocess encomassing roduction, roagation, and detection, with the neutrino in the intermediate (virtual) state. This aroach was ioneered in Ref. [5] (see also the reviews [6] and references therein for an udated status). In site of the concentrated effort and several ingenious theoretical solutions, there are aradoxical features [7] and questions for whose answer there is still no consensus, such as the following: How are the flavor neutrino states suosed to be defined? Are the flavor states momentum eigenstates? Is it necessary that the massive neutrinos which mix have equal energies? Are the flavor states hysical and in which sense? These are fundamental issues arising about any oscillation rocess. The quantization method described in this work rovides a natural and unequivocal answer to a crucial question: If we know that a certain field is exressed as a definite mixing of other fields, how do we define the states corresonding to the former field in terms of the states of the latter fields? The answer will be given in Sec. it will turn out to have been imossible to guess without invoking the ower of the resent quantization rocedure. II. LAGRANGIAN DESCRIPTION OF NEUTRON-ANTINEUTRON OSCILLATIONS The free neutron-antineutron oscillations are analyzed by the quadratic effective Hermitian Lagrangian with general B terms added: L fl x i x m fl x x T x C x fl x C fl T x i 5 e i T x C 5 x e i fl IV, and x C 5 fl T x ; : where m,, 5, and are real arameters, x is the neutron field, and C is the charge-conjugation matrix. The Standard Model Lagrangian is invariant under the global U baryonic number transformations, under which the neutron field transforms as x e i x. Clearly, under such a transformation the terms roortional to and 5 in the Lagrangian (.) are noninvariant. They are the only Lorentz-invariant baryon-number violating terms that can be written, and they are Majorana mass terms of scalar and seudoscalar tyes. A seudoscalar mass term 0 im fl x 5 x can in rincile be added as well, but we shall omit it, as its role in this B Lagrangian is sulanted by the 5 term. The baryon-number violating terms are quadratic, but it is assumed that they are the effective exression of an interaction whose details are unknown. For this reason, when referring to those terms we may use the term baryon-number violating interaction. We shall adot a charge conjugation invariant version of the Lagrangian (.). With the traditional convention for defining the charge conjugated sinor as C x C c x C fl T x ; : the Lagrangian (.) is invariant under charge conjugation only when 0 ; therefore, we fix the hase in this way. Irresective of the hase, the Lagrangian (.) is arity violating. This can easily be seen if we adot the convention P x ;t P 0 x ;t : :3 There is no hase convention for the definition of arity that can render the Lagrangian invariant. Thus, the Lagrangian 0509-

4 QUANTUM FIELD THEORY OF OSCILLATIONS: PHYS. REV. D 98, 0509 (08) L fl x i x m fl x x T x C x fl i 5 T x C 5 x fl x C fl T x x C 5 fl T x :4 is C invariant and P and CP violating. The P and CP violation are in line with the exected electric diole moment for the neutron and cannot be eliminated from the Lagrangian by any field redefinition. The effect is due to the interlay of the vector couling in the term and the axial vector couling in the 5 term. Incidentally, if in the Lagrangian (.) one takes either 0 or 5 0,the remaining Lagrangian can always be shown to be both P and C invariant, by a redefinition of the hase of the corresonding oerations. The diagonalization of the Lagrangian (.) and the analysis of neutron-antineutron oscillations were erformed in detail in Ref. []. The P and CP violation of the Lagrangian was shown not to have observable effects in the free n fln transition robability (for recent discussions of the discrete symmetries, esecially CP, in neutronantineutron oscillations, see [ 3] ). The Lagrangian analysis in [] involved the introduction of a relativistic analogue of the Bogoliubov transformation, which mixes the fields x and c x, and diagonalizes the 5 term: i m x i 5 5 c x 0 ; i m c x i 5 5 x 0 ; :6 with c C fl T. We rewrite them as i m i 5 5 x 0 ; i m i 5 5 x 0 ; :7 in terms of the Majorana fields x x c x ; :8 which satisfy Dirac equations with different masses and thus diagonalize the Lagrangian (.4). The mass eigenvalues are easily obtained by setting x e ix : For we note that which is rewritten as = m i ; = m i : :9 = m i ; c x x c cos N x i 5 sin N x c cos N x i 5 sin N x ; :5 = i 5 5 m : From here we find, for, where the fields q N and c N are of Dirac tye and sin 4 = m 5. The role of this transformation is crucial in the exact Lagrangian analysis of neutron oscillation. Also, in the context of the seesaw mechanism, which is described by the Lagrangian (.) with =, a similar (but C -noninvariant) relativistic Bogoliubov transformation roves essential in absorbing the C violation and rendering the Majorana neutrino a roer eigenfield of the charge conjugation oerator on a new vacuum [3,3].We shall return to the details of the relativistic Bogoliubov transformation in Sec. V, when comaring the results of the Lagrangian and Hamiltonian formulations in the calculation of the anomalous (baryon-number violating) roagators. Here, we collect only a few necessary formulas ertaining to the Lagrangian formalism, which will be needed later on. The equations of motion derived from the Lagrangian (.4) are It was shown in [] that the artial diagonalization of the Lagrangian (.4), which removes the baryon-number violating 5 0 term, leads to a term of the tye im fl x 5 x, which may reflect the effect of the QCD vacuum. while for we obtain M m 5 ; :0 M m 5 : : Uon diagonalization, the Lagrangian (.4) becomes L fl x i x M fl x x fl x i x M fl x x : : Thus, the Lagrangian descrition leads to the exression of the neutron field x as mixing of the free massive Majorana fields x : x x x : :3 The rest of this aer will be concerned with finding the suerosition of states of which reresent the states corresonding to the field. For this urose, we shall have to ass to the Hamiltonian descrition of the model

5 ANCA TUREANU PHYS. REV. D 98, 0509 (08) In anticiation, let us still recall that a roer Dirac field x of mass m can always be written as the sum of two Majorana fields with the same mass m, which are constructed as x x c x. The neutron field x, however, is the mixture of two mass-nondegenerate Majorana fields, therefore not a Dirac field. The meaning of the neutron and antineutron as article states associated with the field x becomes more subtle. III. HAMILTONIAN DESCRIPTION AND VACUUM STRUCTURE The direct way to canonically quantize the model described by the Lagrangian (.4) is by solving the equations of motion (.9), for the Majorana fields with definite masses, and alying equal-time canonical anticommutators, which would lead to the algebra of the creation and annihilation oerators. The system is exactly solvable. However, such a straightforward method would obscure the baryon-number violation, as well as the associated dynamical mass generation. For this reason, we shall adot a different method of canonical quantization, which has the benefit of uncovering a more telling intuitive icture of the oscillation henomenon. A. Canonical quantization and Bogoliubov quasiarticles The method that we are going to use is analogous to the one develoed by Bogoliubov for the treatment of the BCS model [] (for a edagogical resentation, see, e.g., Ref. [33] ) and by Nambu and Jona-Lasinio in their BCSinsired theory of dynamical generation of nucleon masses [3] (see also [34] ). In current arlance, it is based on the unitarily inequivalent reresentations of canonical (anti) commutators. An amle exosition thereof can be found in the monograh [35]. Unitarily inequivalent reresentations can exist only in systems with an infinite number of degrees of freedom, in other words, in quantum field theory. In contrast, in quantum mechanics, the Stone von Neumann theorem ensures that all reresentations of the canonical commutators are unitarily equivalent. The existence of unitarily inequivalent reresentations in quantum field theory is an essential ingredient of Haag s theorem [36]. Here, we shall summarize the main asects needed for the alication to the neutron-antineutron oscillation model. In the Heisenberg icture, the evolution of a fermionic system is exressed by the time-deendent Heisenberg fields, satisfying the equation of motion i t x x ;H ; 3 : where H is the Hamiltonian of the system and the fields x satisfy equal-time anticommutation relations. The Heisenberg fields act on the Fock sace of hysical states, i.e., those states that corresond to observable free articles. They are obtained by the alication of creation oerators to the hysical vacuum of the model. Consequently, the Fock sace has to satisfy the requirement that the Heisenberg fields are exressed in terms of creation and annihilation oerators of the hysical free articles. (It is erhas more familiar to think that the Heisenberg fields are exressed by a Dyson exansion in terms of incoming or outgoing hysical fields.) When this condition is fulfilled, the total Hamiltonian of the system takes the form of a free Hamiltonian. This is one of the essential features of the Heisenberg icture and will rovide us the basis for solving the baryon-number violating model defined by the Lagrangian (.4). The method described below is called the self-consistent method, in the sense that it relies on the self-consistency between the Hamiltonian and the choice of the Fock sace of hysical articles [35]. In ractice, we obtain first the classical Hamiltonian of the system starting from the Lagrangian. Then we choose a candidate for the hysical field (i.e., a field that satisfies a certain free field equation of motion) and quantize it canonically. We go to the Schr dinger icture and exress the Hamiltonian in terms of the creation and annihilation oerators of the candidate field. It is clear that there are infinitely many free fields to choose from, each defined by a different mass. Tyically, one makes a meaningful selection by considering the solution of the free art H 0 of the total Hamiltonian H (see, for examle, Ref. [37] ). If the Hamiltonian is not diagonal, then the candidate field is not the hysical field. What we need to do is to diagonalize the Hamiltonian. Uon diagonalization, the Hamiltonian will be exressed in terms of the creation and annihilation oerators of the true hysical fields, acting on the true ground state of the model. The hysical Fock sace can be constructed and the roblem is solved (the return to the Heisenberg icture being then straightforward). The diagonalization is achieved by establishing certain relations between the creation and annihilation oerators of the candidate field and those of the true hysical field. These turn out to be Bogoliubov transformations, reserving the canonicity of the algebra. The quanta of the hysical fields will therefore be Bogoliubov quasiarticles. The two sets of canonical oerators are related by a transformation that seemingly is unitary. However, it turns out that they act as creation and annihilation oerators in two orthogonal Fock saces, constructed on orthogonal vacua, and therefore the transformation is not unitarily imlementable [35]. For this reason, the two Fock saces are said to be unitarily inequivalent reresentations of the canonical algebra. Although there are in rincile an infinity of unitarily inequivalent reresentations, it should be stressed once more that only one reresentation is hysical, and that is the one in which the Hamiltonian is diagonal. The corresonding vacuum is the one and only vacuum of the theory (excet the case when there is sontaneous breaking of symmetry)

6 QUANTUM FIELD THEORY OF OSCILLATIONS: PHYS. REV. D 98, 0509 (08) All the assertions in the above summary will be substantiated below on the concrete model defined by the Lagrangian (.4). We shall also draw some arallels with the BCS theory or the NJL model whenever these arallels may rove illuminating.. Hamiltonian We start by writing the Hamiltonian corresonding to the baryon-number violating Lagrangian (.4) : H d 3 x fl x i k k x m fl x x d 3 x T x C x fl x C fl T x d 3 xi 5 T x C 5 x fl x C 5 fl T x ; H 0 H =B ; 3 : where H 0 stands for the Dirac Hamiltonian of a field with the mass m, while H reresents the baryon-number =B violating art.. Choice of a candidate hysical field The next ste is to ick u a candidate x for the role of hysical field. The meaningful choice out of the arbitrary ossibilities is to take x as the would-be neutron field in the absence of the baryon-number violating interaction. This is the solution of the free Dirac equation with mass m, i m x 0 ; 3 :3 in other words, the eigenfield of the free Dirac Hamiltonian H 0 in (3.). Hence, we roceed by going to the Schr dinger icture, at t 0, and making the identification [,34,37] x ; 0 x ; 0 ; 3 :4 in the Hamiltonian (3.). In this way, we can naturally assign baryonic quantum numbers to the quanta of the field x, which will be called bare neutrons and antineutrons. Moreover, in the limit, 5 0, the states associated with the field x coincide with the states associated with x. Consequently, we shall have a handle to define what is meant by neutron and antineutron when the baryonic number is violated. We exand the Hamiltonian (3.) in terms of the modes of the bare neutron field, x ; 0 d 3 X 3 = a u i x e b v e i x ; 3 :5 which is written in helicity basis (see Aendix A ), with m. The charge conjugated sinor c C fl T, with the conventions from Aendix A,is c d 3 X x ; 0 i x sgn b 3 = u e a v e i x : 3 :6 The oerators a; a ;b;b are creation and annihilation oerators on a vacuum j0 i, which we may call article vacuum, and satisfy ordinary anticommutation relations: a j 0 i b j 0 i 0 ; 3 :7 f a ;a k g 0 0 k ; f b ;b k g 0 0 k ; 3 :8 all the other anticommutators being zero. The states a j 0 i and b j 0 i 3 :9 reresent bare neutron and antineutron states, resectively, of mass m and definite momentum and helicity. We assign baryonic number to the bare neutron states and to the bare antineutron states. In analogy with the theory of neutrino oscillations, we may think about the Fock sace of states built on the vacuum j0 i as a sace of flavor states. 3. Mode exansion of the Hamiltonian Using (3.5) and (3.6) in (3.), we find, with the hel of the relations (A4), H 0 d 3 x X a a b b 3 :0 and H =B i d 3 X m sgn a b b a a a a a b b b b sgn 5 a a a a b b b b : 3 :

7 ANCA TUREANU PHYS. REV. D 98, 0509 (08) The baryon-number violating art of the Hamiltonian is, as exected, nondiagonal. The terms containing a b b a indicate the neutron-antineutron transition. The rest of the nondiagonal terms suggest the airing of neutrons and antineutrons, in the manner of the Cooer airs in the BCS theory. We omit the vacuum energy and resent throughout the Hamiltonian in normal form. We may roceed from here to the diagonalization, but it is technically advantageous to take into account the hint rovided by the equations of motion (.7), namely that the fields which diagonalize the Lagrangian are Majorana fields. Therefore, we shall reexress the Hamiltonian (3.) in terms of the creation and annihilation oerators associated with the degenerate Majorana fields of mass m into which the Dirac field x can be slit. We note that the convention adoted for the charge conjugation transformation leads to the following action on the creation and annihilation oerators: C a C sgn b ; C b C sgn a : 3 : As a result, we obtain the creation and annihilation oerators of the Majorana fields x defined by identified by the subscrit a M b M x The inverse of the above transformation reads a sgn b x c x ; 3 :3 M, in the form a sgn b ; a sgn b : 3 :4 a M b M ; a M b M : 3 :5 At t 0, the Majorana fields x ; 0 are exressed as x ; 0 x ; 0 d 3 3 = d 3 3 = X X a M u e i x sgn a M v e i x ; b M u e i x sgn b M v e i x : 3 :6 Using the formulas (3.5) we recast the Hamiltonian (3.) in terms of the Majorana oerators: H d 3 X m a m M a M b M b M 5 sgn i a M a M b M b M 5 sgn i a M a M b M b M : 3 :7 In this form, the a M - and b M -tye oerators are disentangled and we can diagonalize each set searately. Incidentally, in the BCS language the exression 5 sgn i 3 :8 H 4. Diagonalization of the Hamiltonian and Bogoliubov transformations We diagonalize the Hamiltonian as d 3 X A A B B ; is the analogue of the ga function [33]. 3 :9 In the BCS theory, the Hamiltonian is written in terms of the creation and annihilation oerators of the bare electrons, the interaction with the lattice roviding the nondiagonal terms. by adoting the following Bogoliubov transformations, suggested by the form of the Hamiltonian (3.7) :

8 QUANTUM FIELD THEORY OF OSCILLATIONS: PHYS. REV. D 98, 0509 (08) A a M i e i a M ; B b M i e i b M ; 3 :0 where and are comlex coefficients and are real. They all deend in rincile on the helicity, but we omit the helicity index. The quantities and are real, having the meaning of energies to be determined. In order for the new oerators to satisfy the canonical anticommutation relations f A ;A k g 0 0 k ; f B ;B k g 0 0 k ; 3 : with all the other anticommutators being zero, the coefficients in (3.0) have to satisfy the conditions j j j j ; j j j j : 3 : In other words, the conditions (3.) ensure that the transformations (3.0) are canonical. Tyically, conditions (3.) suggest that the Bogoliubov transformations are rotations in the sace of creation and annihilation oerators, for which a customary notation [35,37] is cos ; sin ; cos ; sin : 3 :3 We shall diagonalize the art of the Hamiltonian deending on a M, a M. Introducing the Ansatz (3.0) into (3.9), we find d 3 X A A d 3 X j j a M a M j j a M a M i e i a M a M i e i a M a M : 3 :4 Identifying the coefficients with those in (3.7), we arrive at the following equations: j j j j m ; cos i sin i 5 sgn ; cos i sin i 5 sgn : 3 :5 From the last two relations in (3.5) we infer that and can be taken to be real, leading to Thus, we obtain from where sin cos cos ; sin tan 5 sgn : sgn 5 ; 3 :6 q tan sgn q 5 ; tan 5 q q : tan 5 3 :7 With these results we return to (3.5) and find s 5 ; 3 :8 which we insert into the first equation of (3.5) : 4 5 m : 3 :9 Equations (3.8) and (3.9), together with the requirement of canonicity (3.), are satisfied by the real exressions where q s m ; s m ; 3 :30 M ; with M m 5 : 3 :3 Insecting the Hamiltonian (3.7), we notice that the art deending on b M, b M is identical to the art deending on a M, a M, u to the substitution. As a result, we infer immediately the form of the corresonding coefficients:

9 ANCA TUREANU PHYS. REV. D 98, 0509 (08) where q s m ; s m ; 3 :3 ; with M m 5 : 3 :33 M 5. The hysical vacuum and the Fock sace of the quasiarticles The set of oerators that diagonalize the Hamiltonian act on a new vacuum j 0 i, which satisfies A j 0 i B j 0 i 0 ; 3 :34 and reresents the hysical vacuum of the model. The hysical article states are Bogoliubov quasiarticles, of Majorana tye, with the definite masses M m 5. The relation between j 0 i and the bare articles vacuum j0 i is derived by assuming that the vacuum of quasiarticles is written as an arbitrary suerosition of airs of Majorana articles associated with the fields x : j 0 i N ; e R a M a M e R b M b M j0 i ; 3 :35 where N is a normalization constant. Using (3.34) and (3.0), one finds that R i e i =. Pauli s rincile imlies that therefore, Recalling (3.), we note that leading to the normalized quasiarticle vacuum in the form a M a M n b M b M n 0 ; for n> ; j 0 i N ; R a M a M R b M b M j 0 i : 3 :36 h 0 j i e i a M a M i e i a M a M j 0 i h 0 j j j j j a M a M a M a M j 0 i h 0 j j j j j a M a M a M a M j 0 i h 0 j j j j j j 0 i ; j 0 i ; i e i a M a M i e i b M b M j 0 i : 3 :37 Just as in the BCS theory, the hase of the Cooer airs of bare Majorana articles is given by the hase of the ga function (3.8). This hase, in the resent case, is fixed by the choice of the arameters m; ; 5 in the Lagrangian (.4) and for each air deends on the momentum of its constituents. The hysical vacuum is therefore unique. 3 The bare Majorana articles comosing the airs have oosite momenta and sins, consistent with the Poincar e invariance that imlies energy momentum and angular momentum conservation. 3 In contrast, in the BCS theory or NJL model, the hase of the ga function is arbitrary due to the U symmetry of the Lagrangian, and its variation leads to an infinity of degenerate vacua, which is the essence of the sontaneous breaking of symmetry. The Fock sace built on the vacuum j 0 i consists of Majorana article states with two different masses, M and M, given by (3.3) and (3.33) : HA HB j 0 i A j 0 i ; j 0 i B j 0 i : 3 :38 These quasiarticles, with an indefinite baryon number, are the only hysical articles in the model. Neutron and antineutron do not exist as article states. 6. Vacuum condensate and baryon-number violation Coleman s theorem states that the invariance of the vacuum is the invariance of the world [38]. We therefore

10 QUANTUM FIELD THEORY OF OSCILLATIONS: PHYS. REV. D 98, 0509 (08) exect to see violation of the baryonic number in the vacuum condensate. As mentioned earlier, the bare neutron and antineutron states have definite baryonic numbers. On the other hand, the baryonic number is undefined for the states of bare Majorana articles, a M j 0 i and b M j 0 i. We may attemt to rewrite the vacuum condensate as suerosition of airs of bare neutrons and antineutrons, a a j 0 i and b b j 0 i. To this end, we insert (3.4) into the Bogoliubov transformations (3.0) and find A B a i e i a sgn b i e i b ; a i e i a sgn b i e i b : 3 :39 The requirement on the hysical vacuum (3.34) then imlies simultaneously with a i e i a j 0 i 0 ; b i e i b j 0 i 0 ; 3 :40 a i e i a j 0 i 0 ; b i e i b j 0 i 0 : 3 :4 As long as and, relations (3.40) and (3.4) are in conflict. Consequently, for the general case with arbitrary and 5 arameters, we have to content ourselves with the exression (3.37) for the vacuum condensate. In the secific case when 0, we notice that q M M m 5 ; s ; s ; sin sgn ; cos 0 : 3 :4 This is the only instance when the relations (3.40) and (3.4) are comatible and the hysical vacuum can be written as j 0 ij 0 ; sgn a a sgn b b j 0 i ; 3 :43 with the airs or bare neutrons and antineutrons carrying baryon number, and thus exlicitly exhibiting the baryon-number violation Unitary inequivalence of reresentations Let us calculate the inner roduct of the two vacua, using (3.37) and taking into account (3.7), (3.30), and (3.3) : h 0 j 0 i ; j jj j m = ; m d 3 X ex 3 ln m m : 3 :44 In the large momentum limit, m 5, and the exonential diverges R as ex 5 d, which leads to the orthogonality of the two vacua, m = h 0 j 0 i 0 : 3 :45 The Fock saces built on the bare vacuum j0 i and on the quasiarticle vacuum j 0 i are, consequently, also orthogonal. (This can easily be confirmed by taking the inner roduct of two arbitrary states belonging to the two saces.) The latter is the hysical one, while the former is an auxiliary sace, an artifact of the quantization method. Although the bare article states cannot be found among the hysical states, this does not mean that the bare oerators cannot act on the hysical vacuum. The oerators a; a ;b;b act on j 0 i through their relations to the quasiarticle oerators, i.e., the inverse Bogoliubov transformations (3.49) together with (3.5), always creating and annihilating articles with masses M and never bare articles of mass m. This feature will be used further in defining neutron and antineutron states in Sec. IV. 4 Incidentally, if, 5 m and we exand the coefficients of the general Bogoliubov transformation (3.0) to second order in and 5 [see (4.6) below], we find again and, and the vacuum condensate can be recast in a form similar to (3.43). In Ref. [], for examle, the hysical vacuum was derived in this aroximation, for

11 ANCA TUREANU PHYS. REV. D 98, 0509 (08) B. Heisenberg fields Having diagonalized the Hamiltonian (3.7) in the Schr dinger icture, we can now easily move to the Heisenberg icture. We have obtained the solutions of the Hamiltonian (3.) as two nondegenerate Majorana fields. Their time evolution is given by e iht x ; 0 e iht x ;t : 3 :46 The corresonding creation and annihilation oerators evolve as A ;t e iht A e iht A e i t ; A ;t e iht A e iht A e i t ; B ;t e iht B e iht B e i t ; B ;t e iht B e iht B e i t ; 3 :47 where we used H in the form (3.9). Thus, the rimary time-deendent Majorana fields will read x ;t x ;t d 3 3 = X A U e i t x sgn A V e i t x ; d 3 X 3 = B U e i t x sgn B V e i t x ; 3 :48 with the sinors U ;V and U ; V satisfying the equations of motion (.7) in momentum sace. Inverting the Bogoliubov transformations (3.0), namely a M A i e i A ; b M B i e i B ; 3 :49 we obtain the time evolution of the oerators a M ; b M : a M ;t A e i t i e i A e i t ; b M ;t B e i t i e i B e i t : 3 :50 Using (3.5) and (3.50), we can exress the time-deendent a ;t ;b ;t as well. C. Diagonalization of Hamiltonian and rimary Majorana fields In the tyical cases of a mass shift of Dirac fermions by vacuum condensate encountered in the NJL model [3,34], the Bogoliubov transformations relating the creation and annihilation oerators of different masses can be obtained by two equivalent rocedures. One of them is what we have described above: having derived the Hamiltonian of the system, H H 0 H int,thefield D x is relaced in the Hamiltonian, at t 0 (Schr dinger icture), by the solution D x of the equation of motion i t D x D x ;H 0, i.e., D x ; 0 D x ; 0 ; 3 :5 and the Hamiltonian is subsequently diagonalized by using Bogoliubov transformations. The quasiarticle oerators that diagonalize the total Hamiltonian will be the creation and annihilation oerators of the field D x, which satisfies the equation of motion i t D x D x ;H. Tyically, the bare field D x and the quasiarticle field D x are free Dirac fields with different masses. In this way, one finds the solution D x without solving directly its equation of motion. This method is essentially a relativistic extension of Bogoliubov s aroach to the theories of suerfluidity and suerconductivity []. The second rocedure is the one used in the work of Nambu and Jona-Lasinio [3] : knowing the Hamiltonian H, one solves the equation of motion i t D x D x ;H, and subsequently identifies its solution, at t 0, with the solution of i t D x D x ;H 0. In other words, one imoses the boundary condition (3.5) to the two known solutions. In this case, the urose is strictly to find the Bogoliubov transformations and the relation between the bare article vacuum and the quasiarticle vacuum. The results are the same as those obtained by the Hamiltonian diagonalization method. The mixing of fields in the baryon-number violating model that we have been analyzing requires more care in the alication of the rocedures outlined above. We have seen that the Hamiltonian diagonalization rocedure succeeds when using the identification [3], x ; 0 x ; 0 : Recall that the field x is a Dirac field of mass m, while x is not a Dirac, nor a Majorana, field. In effect, the field x does not satisfy a simle equation of motion, but an equation in which it is mixed with its charge conjugate c x, Eq. (.6).A rotation of the creation and annihilation oerators of x does not take us to new creation and annihilation oerators, because there are no such oerators for the field x. This is an indication that the second rocedure outlined above cannot work with the boundary condition [3]. In hindsight, we realize that the actual identification of fields for which [3] was standing was where x satisfy x ; 0 x ; 0 ; 3 :

12 QUANTUM FIELD THEORY OF OSCILLATIONS: PHYS. REV. D 98, 0509 (08) i m x 0 and x satisfy Eqs. (.7), i m i 5 5 x 0 ; i m i 5 5 x 0 : We call the fields rimary Majorana fields, as they are the simlest combinations of the neutron field and its charge conjugate c, which satisfy uncouled equations of motion. The two nondegenerate rimary Majorana fields can be related to the two mass-degenerate bare Majorana fields by different rotations of the creation and annihilation oerators. In Aendix B we shall rove that the Bogoliubov transformations (3.0), with the coefficients secified by (3.30) and (3.3), can be found also by the second rocedure outlined above, starting from the boundary condition (3.5). We emhasize secifically the role of the rimary Majorana fields, because in certain situations one can choose other combinations of Majorana fields that diagonalize the Lagrangian as well. For examle, when 0, 5 0, the Lagrangian is diagonal in terms of x,but also in terms of the Dirac-tye fields N x which satisfy the Dirac equation (5.4) (see the discussion in Sec. VA ) and are related to and c by the relativistic Bogoliubov transformation (.5). Because of the simlicity of the equation of motion for N x, it may be temting to use the relativistic Bogoliubov transformation as the basis for the boundary condition at t 0, namely to make the identification x ; 0 c x ; 0 x ; 0 c x ; 0 cos N x ; 0 i 5 sin N c x ; 0 cos N c x ; 0 i 5 sin N x ; 0 In this case, the resulting transformations between the oerators of x and those of N x are essentially incomatible, in the sense that the two annihilation oerators of N x, say A N and B N, do not destroy the same vacuum condensate j N 0 i. This inconsistency does not aear if one adheres to rimary fields and formula (3.5). The identification of rimary fields is an essential ste in treating any quantum systems with mixings of fields, such as the seesaw mechanism Lagrangian or various models of neutrino mixing and oscillation. : 3 :53 IV. NEUTRON STATES IN PHYSICAL FOCK SPACE AND THE PROBABILITY n nfl TRANSITION When we embed the quadratic Lagrangian (.4) into the Standard Model, the field x lays the role of neutron field and takes art in the neutron interactions already resent there. At the same time, we have to give u the icture of the neutron as a article with definite mass and flavor. It is then necessary to redefine the notion of neutron and antineutron, when there are no creation and annihilation oerators for them. The only ossibility for a consistent definition is to associate the neutron and antineutron with the field x,in other words, to define these states by their dynamical relations with the other articles with which they interact. The natural rocedure is to use the Schr dinger icture identification [3], x ; 0 x ; 0 ; together with the consistency requirement that, in the limit when the baryon-number violating interaction vanishes (i.e.,, 5 0 ), one recovers the bare, or flavor, neutron state defined on the vacuum j0 i. In ractice, we start by Fourier transforming the field x ; 0 : d 3 x e i x fl x ; 0 3 = X a flu 0 0 b 0 flv : 0 Uon multilication by 0 u and the use of relations (A3), we find d 3 x 3 = e i x fl x ; 0 0 u a : 4 : The oerator in the left-hand side of (4.), acting on the vacuum j0 i, roduces the bare neutron state a j 0 i.we shall therefore adot it as the definition of the neutron 5 creation oerator on the hysical vacuum, in which case it is referable to also relace x ; 0 by x ; 0, 5 In Ref. [], the definition ofq the neutron state is (with our R notations and conventions) a i m sgn b. However, this exression does not give sensible results when alied to multiarticle (antineutron) states in the limit when the baryon-number violating interaction vanishes; therefore, it cannot be a roer neutron creation oerator in any setu. Moreover, it is not alicable to the case when bare articles are massless. m d 3 x 3 = e i x fl x ; 0 u 0509-

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