Reproducing sterile neutrinos and the behavior of flavor oscillations with superconducting-magnetic proximity effects Thomas E. Baker Department of Physics & Astronomy, California State University, Long Beach, CA 90840 and Department of Physics & Astronomy, University of California, Irvine, CA 92697 (Dated: June 18, 2021) The physics of a superconductor subjected to a magnetic field is known to be equivalent to neutrino oscillations. Examining the properties of singlet-triplet oscillations in the magnetic field, a sterile neutrino{shown to be a Majorana fermion{is suggested to be represented by singlet Cooper pairs and moderates flavor oscillations between three flavor neutrinos (triplet Cooper pairs). A superconductor-exchange spring system's rotating magnetization profile is used to simulate the mass-flavor oscillations in the neutrino case and the physics of neutrino oscillations are discussed. Symmetry protected triplet components are presented as weak process states. Phases acquired due to the Fulde-Ferrell-Larkin-Ovchinnikov effect produce a complex phase that may be responsible for charge-parity violation in flavor oscillations. PACS numbers: 14.60.Pq, 13.35.Hb, 14.60.St, 14.60.Lm, 74.45.+c, 74.90.+n I. INTRODUCTION superconducting state depending on its strength. Weak magnetic fields are expelled from a superconductor [38] Neutrinos [1, 2] exist in three known flavors [3, 4] corre- since the photons of the applied field acquire a finite sponding to each generation in the Standard Model [5{7] mass in the symmetry broken superconducting state [15]. and have very small masses with respect to other particles At high magnetic fields, the superconducting state is de- [8{10]. These flavor states may oscillate [11, 12] between stroyed completely or forms an Abrikosov lattice of flux each other. Measuring the chirality of the neutrino shows vortices in a type-II superconductor [39]. that they are always left-handed [13, 14]. Despite these An intermediate regime of moderate magnetic fields experimental facts, a complete physical picture is difficult exists where the magnetic field is not strong enough to to determine due to the small cross section for measuring break the paired electrons and hybridizes the up and neutrino events. The conventionally accepted Standard down electron bands. Effectively, the electron with a Model does not account for oscillations of neutrinos [15]. spin collinear to the magnetic field has an increased mo- In certain situations, a condensed matter system may mentum (and decreased momentum for the anti-parallel possess properties that allow it to mimic a particle spin). This momentum splitting gives an oscillation in physics system. For example, dispersion relations caus- the superconducting order parameter between the sin- ing electrons to obey the Dirac equation [16], proper- glet and triplet Cooper pairs and was demonstrated ties of Weyl fermions in semi-metals [17{21], topologi- nearly simultaneously by Fulde-Ferrell [40] and Larkin- cal Majorana modes in gapped proximity systems [22], Ovchinnikov [41] (FFLO). and Anderson-Higgs modes in superconductors [23{28] all mimic the physics found in large particle experiments at high energies [29{33]. Studying the condensed matter physics may allow for details unavailable in the particle physics case to be analyzed in greater detail and with direct experimental verification. A model was constructed by Pehlivan-Balantehkin- arXiv:1601.00913v2 [hep-ph] 14 Feb 2016 Kajino-Yoshida (PBKY) in Ref. 34 which shows that a neutrino gas is analogous to a magnetic field ap- plied to a superconductor. Interestingly, conventional superconductivity, characterized by the Bardeen-Cooper- Schrieffer (BCS) [35] model, is a competing order to mag- netism because an electron's spin tends to align with a magnetic field. This breaks Cooper pairing{which are electrons of opposite spin forming a quasi-particle respon- sible for the superconducting state [35{37]. The PBKY model establishes that neutrino oscillations are equiva- lent to Cooper pairs in the analogy. Investigating the coexistence of these competing FIG. 1. A sterile neutrino (0) can transition to a flavored phases of matter is an worthwhile topic of its own; apply- state (νµ; ντ ; νe) just as singlets transition to triplets in su- ing a magnetic field can have a variety of effects on the perconducting proximity effects. 2 One system to study the FFLO effect, and therefore is summarized in Sec. II A. The PBKY model is phrased neutrino oscillations, is a superconductor in proximity to in Sec. II B as a mean field of another field and it is shown a ferromagnet [42{46]. The superfluid can tunnel into that the flavor basis has a source field. the adjacent material, and cause the entire system to become superconducting [47] with measurable results at nanoscale distances [48{51]. A. Overview of the The motivating feature of the FFLO effect that begs Pehlivan-Balantekin-Kajino-Yoshido model for comparison with neutrinos is that three triplet Cooper neutrinos pairs are connected to the singlet Cooper pair, just as three flavored neutrinos are connected to a sterile The most general Hamiltonian we can write for the neutrino{a simple extension of the Standard Model to two-flavor neutrino oscillation is [63] include flavor oscillations [52, 53]{and is summarized in Fig. 1. A sterile neutrino may couple to the flavor states y y y ∗ y H = Ω1a^1a^1 + Ω2a^2a^2 + Ωma^1a^2 + Ωma^2a^1 (1) via the mass term as though it were a right handed par- ticle. These particles have been suggested to be a candi- where the prefactors Ω are arbitrary complex coefficients. date for dark matter [52, 54{56]. The particular sterile The operatora ^y (^a) represents the creation (destruction) neutrino appearing here strongly satisfies the effects re- of a particle. quired of a dark matter candidate [57]. Assuming that particles 1 and 2 correspond to the mass In this paper, the connection between BCS supercon- basis, the Hamiltonian becomes ductivity with an applied magnetic field and the PBKY model from Ref. 34 is reviewed in Sec. II A. It is shown X m2 m2 H = 1 a^y(p)^a (p) + 2 a^y(p)^a (p) : (2) that a real external field is applied to the superconduct- ν 2p 1 1 2p 2 2 ing state in Sec. II B by examining the model for flavor p states. since the mass states propagate in a given system Section III A provides an overview of superconducting [64]. The energies of the two particles given by E = proximity effects by first discussing transport equations p 2 2 2 of the superfluid. The physics of those equations is dis- p + m ≈ const: + m =(2p). cussed in Sec. III B. Had we written this Hamiltonian in the flavor basis, for flavors α and β, the unitary transformation The connection between quantities in the particle physics case and the proximity system are covered in a^ (p) cos θ sin θ a^ (p) Sec. IV. Section IV A identifies neutrino type based α = 1 (3) a^ (p) − sin θ cos θ a^ (p) on the expansion of the Gor'kov function in the con- β 2 densed matter case. Section IV B uses the symmetry can be used where θ is a mixing angle. of the singlet state to identify the sterile neutrino as a Note that the term Majorana fermion. The minimally extended Standard Model (MESM) is shown to be analogous to Ginzburg- X m2 + m2 1 2 a^y(p)^a (p) +a ^y(p)^a (p) (4) Landau (GL) theory [58] in Sec. IV C. The physics of 4p 1 1 2 2 flavor oscillations are investigated by comparison with p superconductor-exchange spring systems (discussed in Sec. IV D and further discussed with particular regard is a constant assuming a constant number of neutrinos. to sterile neutrinos in Sec. IV E). Symmetry protected Subtracting it from Eq. (2) gives triplet components introduced in Refs. 59 and 60 are in- X ! y y terpreted as weak process states allowing for the con- Hν = a^ (p)^a1(p) − a^ (p)^a2(p) : (5) 2 1 2 servation of energy and momentum in Sec. IV F. The p angular momentum quantum number in the supercon- ductor is discussed in the neutrino case in Sec. IV G. The where Mikheyev-Smirnoff-Wolfenstein (MSW) effect is related 2 to polarization effects in Sec. IV H by noticing similar ! = δm =(2p) (6) behavior in the condensed matter case. A possible mech- 2 2 2 anism for charge-parity (CP) violation from the FFLO and δm = m1 − m2. Isospin operators in the mass basis phase is discussed in Sec. IV I. Other possibilities are cov- can be defined based on Eq. (5) as ered in Sec. IV J. 1 J z = (^ay(p)^a (p) − a^y(p)^a (p)); (7) p 2 1 1 2 2 J + =a ^y(p)a (p); J − =a ^y(p)a (p): (8) II. NEUTRINO OSCILLATIONS p 1 2 p 2 1 which satisfy the SU(2) algebra The PBKY model presented in Ref. 34 equates a neu- + − z z ± ± trino gas with a superconductor in a magnetic field. This [Jp ; Jq ] = 2δpqJp ; [Jp ; Jq ] = ±δpqJp (9) 3 TABLE I. A correspondence table between neutrinos and Anderson's reformulation of BCS superconductivity [61] derived from the mathematical analogy of the PBKY model in Ref. 34. Items below the divide are introduced in this paper based on the PBKY model. Note that p, the energy of the Cooper pair, is 2! in the neutrino case. The vector Gor'kov function possesses an extra angular momentum, `, whose physical significance for the neutrino case is discussed in Sec. IV G. All other connections are justified in Sec. IV as well. Condensed Matter Superconductor Neutrino Particle Physics Pairing potential ∆(r) GF =V Vacuum symmetry breaking per gas volume Cooper pair momentum p p Momentum Time reversed counterpart p < 0 or ! < 0 jν¯e,µ,τ i Anti-neutrinos Matsubara frequency (T 6= 0) !n ! Vacuum oscillation Temperature T T Temperature Gor'kov Function F = f0 + ^v · f jνα $ νβ i Neutrino oscillations Vector Gor'kov function f jνe,µ,τ i Flavored neutrino oscillations Singlet Gor'kov function f0 jν0i Sterile Neutrino (Majorana) Magnetic field B B Source field Perpendicular, parallel component [62] f?; fk jν1i; jναi mass, flavor eigenstate Acquired angular momentum ` ` Flavor-Angular Momentum Cartesian coordinates ^r e; τ; µ Lepton flavors ~ y ~ Rewriting in terms of the vector J , and defining a vector operator as Jp = UpJpUp gives the same functional form B = (0; 0; −1), for the Hamiltonian, Eqs.