Neutron-Antineutron Oscillations: Discrete Symmetries and Quark Operators

PACIFIC-2018.9 Gump Station, Moorea, French Polynesia Aug 31 — Sept 6, 2018 Neutron-Antineutron Oscillations: Discrete Symmetries and Quark Operators Arkady Vainshtein William Fine Theoretical Physics Institute University of Minnesota, KITP, University of California, Santa Barbara With Zurab Berezhiani Search for the neutron-antineutron oscillations was suggested by Vadim Kuzmin in 1970, and such experiments are under active discussion now, see D. G. Phillips, II et al., Phys. Rept. 612, 1 (2016) This is a transition where the baryon charge B is changed by two units. The observation of the transition besides demonstration of the baryon charge non-conservation could be also important for explanation of baryogengesis. Of course, following Sakharov conditions, it should be also accompanied by CP non-conservation. Thus, discrete symmetries associated with neutron-antineutron mixing are of real interest. C, P and T symmetries in | Δ B| =2 transitions In our 2015 text Zurab Berezhiani, AV, arXiv:1506.05096 we noted that the parity P , defined in such a way that P 2 =1 , is broken in n-nbar transition as well as CP .. Indeed, eigenvalues of parity P are ± 1 and opposite for neutron and antineutron. So, n-nbar mixing breaks P . We noted, however, that it does not automatically imply an existence of CP breaking in absence of interaction. In September of the same 2015 we presented at the INT workshop in Seattle a modified definition of parity P z , 2 such that P z = − 1 , and parities P z are i for both, neutron and antineutron. With this modification all discrete symmetries are preserved in n-nbar transition. The issue of discrete symmetries was later discussed in few papers in 2016: K. Fujikawa and A. Tureanu D. McKeen and A. E. Nelson S. Gardner and X. Yan, In particular, Nelson and MaKeen insisted, incorrectly, that one can keep P 2 =1 . Fujikawa and Tereanu also used it, 2 while Gardner and Yan did it right with P z = − 1 , following Wolfenstein’s and Kayser’s applications to neutrino case. I will present some details of an interesting history of the subject which goes back to Majorana and Racah’s papers of 1937. Dirac Lagrangian for neutron μ LD = inγ¯ ∂μn − m nn¯ describes free neutron and antineutron and preserves the baryon charge, B =1 for n , B = − 1 for n¯ . Continuous U(1) B symmetry: n → eiαn, n¯ → e−iαn¯ Another term − im nγ¯ 5 n consistent with B conservation can be rotated away by the chiral rotation, n → e iβγ 5 n . Four degenerate states: two spin doublets differ byhh B . How does baryon number non-conservation shows up? At the level of free particles it could be only bilinear |Δ B| =2 mass terms: C = iγ2γ0 T T T T n Cn, n Cγ5n, nC¯ n¯ , nCγ¯ 5n¯ At these bilinear in fields the most generic Lorentz invariant modifications reduce by field redefinitions to the only one term, breaking baryon charge by two units, 1 ΔL = − nTCn +¯nCn¯T C = iγ2γ0 B 2 d where is a real positive parameter. Redefinitions are due μ to U(2) symmetry of the kinetic term inγ¯ ∂μn ( What is the status of C , P and T discrete symmetries? Let us start with the charge conjugation C ,: C: n ←→ nc = Cn¯T 2 Kind of Z 2 symmetry, C =1 . Most simple in the Majorana representation nc = n∗ . Lagrangians can be rewritten as i m L = nγ¯ μ∂ n + nc γμ∂ nc − nn¯ + nc nc , D 2 μ μ 2 1 ΔL = − nc n +¯nnc , B 2 what makes C-invariance explicit. Lagrangians are diagonalized in terms of Majorana fields n1,2 n ± nc n1,2 = √ , Cn1,2 = ± n1,2. 2 1 L = n¯ γμ∂ n − m n¯ n , D 2 k μ k k k k=1,2 1 ΔL = − n¯ n − n¯ n . B 2 1 1 2 2 Splitting into two Majorana spin doublets with masses 1 M1 = m + M2 = m − . The parity transformation P involves, besides reflection of space coordinates, the substitution P: n → γ0n, nc →−γ0nc . We use γ 0 Cγ 0 = − C . The opposite signs reflect the opposite parities of fermion and antifermion C.N. Yang ’50 V.B. Berestetsky ’51 The definition satisfies P 2 =1 so eigenvalues of P are ± 1 , opposite parities for fermion and antifermion. Different parities of neutron and antineutron implies that their mixing breaks P parity. Indeed, P -transformation changes p y, , p Δ L B to( − Δ L B ). WithC -invariance it implies that Δ L B is also CP odd. This CP-oddness, however, does not translates immediately into observable CP-breaking effects. To get them one needs an interference of amplitudes provided only by interaction. This subtlety is discussed in number of textbooks, see e.g. V. B. Berestetsky, E. M. Lifshitz and L. P. Pitaevsky, Let’s remind it. When B is conserved there is there is no transition between sectors with different .B . One can combine P with a U(1) B phase rotation and define Pα iBα iα 0 c −iα 0 c Pα =Pe : n → e γ n, n →−e γ n P2 =e2iBα =1 Of course, then α but the phase is unobservable while B is is conserved. When B is its not conserved the only remnant of U(1) B rotations is Z 2 symmetry, n →− n . It means that we can P 2 = −1 consider a different parity definition P z , such that z . Thus, choosing α = π / 2 we come to iBπ/2 0 c 0 c Pz =Pe : n → iγ n, n → iγ n . Moreover, in case of Majorana fermions it is the only possible choice. Indeed, in Majorana representation 0 σ where γ0 = 2 σ2 0 0 only iγ preserves reality of the Majorana spinor. Also P z n ± nc n1,2 n1,2 = √ , preserves the Majorana structure of fields, 2 This was derived by Ettore Majorana and Giulio Racah in 1937. Now P z parities of n and n¯ are the same− i m, so their Pz mixing does not break the parity. It means that all p discrete symmetries, C , P z and T are preserved by Δ L B . A few comments. First, preservation of T follows from CPT theorem provided by Lorentz invariance and locality. Second, it is amusing that the same parity for n and nc equal to− i mis consistent with the notion of the opposite parities for fermion and antifermion: the product of their parities is (-1). Third, P z commutes with C , i.e. CPz=PzC, in contrast with P which anticommutes, CP = − PC Similar effects for neutrino were noted by Wolfenstein ’81. as well as by Kayser ’82. Weyl spinor description Two right-handed Weyl spinors, forming a flavor doublet, ψiα,i=1, 2,α=1, 2 , together with with their two complex conjugates, left- handed spinors, α˙ iα ∗ ψi =(ψ ) ,i=1, 2, α˙ =1, 2 In the chiral basis four-component neutron spinor is ψ1 ψ1α ψ2α n = = ,nc = 2 2 ∗ ¯ ¯ −iσ (ψ ) ψ2˙α ψ1˙α The most general Lorentz invariant Lagrangian, quadratic in fields is 1 L = ψα˙ i∂ ψiα− m ψiαψk + mki ψ ψα˙ i αα˙ 2 ik α k α˙ i μ μ m = m ik ∗ ∂αα˙ = σαα˙ ∂μ , σ = {1, σ} ik ki m =(mik) ti d The symmetry of the kinetic term is apparently U(2). It is broken by mass terms. Mass matrix m ik .can be viewed as transforming under U(2). Its overall phase rotation related to U(1). Under SU(2) it is the adjoint, i.e., isovector, μa, a =1, 2, 3, i ij a a i mk = ε mjk = μ (τ )k ,a=1, 2, 3 a μ is complex, two real isovectors {Re μa, Im μa} We can can orient mass matrix −μ1 − iμ2 μ3 m = ik μ3 μ1 − iμ2 μ1 =0. Re μ2 =0, Im μ3 =0. in the convenient way: , 3 Im μ2 Re μ3 m m = = 0 Re μ3 Im μ2 m It finishes the proof of generality. Thus, we show that a generic mass matrix can be rotated to the standard form m 0 by a certain U(2) transformation V , T m 0 = V mV How discrete transformations look in the Weyl description? Let us start with the charge conjugation C : 1 α 2 α C: ψ ←→ ψ , ψ1˙α ←→ ψ2˙α In terms of U(2) transformations it can written as −iπ/2 iπτ 1/2 1 C: ψ → UC ψ, UC =e e = τ 1 † In generic basis U C becomes U C = Vτ V . Thus, C is a component of Z 2 - the discrete survivor of broken U(2), T UC mU C = m Parity transformation, besides inversion of spatial coordinates, acts as 1α 2α Pz : ψ → i ψ2˙α ,ψ → i ψ1˙α, 2α 1α ψ1˙α → iψ , ψ2˙α → iψ . Again, it can be written as † Pz : ψ → i ψUP , ψ → iUP ψ, 1 1 T with U P = τ for the special basis and U P = Vτ V for bit b i (32) l l P2 = −1 4 arbitrary one. Clearly, z . CP For z, T , CP z T we get † T ψ → U †U ψ =iV ∗V †ψ. CPz : ψ → i ψUC UP =i ψVV , P C T: ψ → ψVV T , ψ →−V ∗V †ψiα , 1α iα CPzT: ψ → iψ , ψiα →−i ψiα . with V =1 for the special basis of the mass matrix. Six-quarks operators: discrete symmetries New physics beyond the Standard Model, leading to | Δ B | =2 transitions, induces the effective six-quark interaction, 1 L (ΔB = −2) = c Oi , M 5 i Oi = T i qA1 qA2 qA3 qA4 qA5 qA6 , A1A2A3A4A5A6 where coefficients T i account for color, flavor and spinor structures.

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