Reproducing Sterile Neutrinos and the Behavior of Flavor Oscillations With

Home , Frederick Reines, Toshihide Maskawa

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 applied to a superconductor. Interestingly, conventional superconductivity, characterized by the Bardeen-Cooper- Schrieffer (BCS) [35] model, is a competing order to magnetism 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 responsible for the superconducting state [35–37]. The PBKY model establishes that neutrino oscillations are equivalent 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 † † † ∗ † H = Ω1aˆ1aˆ1 + Ω2aˆ2aˆ2 + Ωmaˆ1aˆ2 + Ωmaˆ2aˆ1 (1) via the mass term as though it were a right handed particle. 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 ˆ† (â) 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 aˆ†(p)â (p) + 2 aˆ†(p)â (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 aˆ (p) cos θ sin θ aˆ (p) Sec. IV. Section IV A identifies neutrino type based α = 1 (3) aˆ (p) − sin θ cos θ aˆ (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 aˆ†(p)â (p) +a ˆ†(p)â (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 ω † † terpreted as weak process states allowing for the con- Hν = aˆ (p)â1(p) − aˆ (p)â2(p) . (5) 2 1 2 servation of energy and momentum in Sec. IV F. The p angular momentum quantum number in the superconductor 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 ω = 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 = (â†(p)â (p) − aˆ†(p)â (p)), (7) p 2 1 1 2 2 J + =a ˆ†(p)a (p), J − =a ˆ†(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 2ω 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 ω < 0 |ν¯e,µ,τ i Anti-neutrinos Matsubara frequency (T 6= 0) ωn ω Vacuum oscillation Temperature T T Temperature Gor’kov Function F = f0 + ˆv · f |να ↔ νβ i Neutrino oscillations

Vector Gor’kov function f |νe,µ,τ i Flavored neutrino oscillations Singlet Gor’kov function f0 |ν0i Sterile Neutrino (Majorana) Magnetic field B B Source field Perpendicular, parallel component [62] f⊥, fk |ν1i, |ναi mass, flavor eigenstate Acquired angular momentum ` ` Flavor-Angular Momentum Cartesian coordinates ˆr e, τ, µ Lepton flavors

~ † ~ 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. (10) and (11), in flavor space but with B = (sin 2θ, 0, − cos 2θ) . X flavor Hν = ωB · J~p (10) The form of Eqs. (10) and (11) are identical to An- p derson’s single particle reformulation of BCS theory [61], which is discussed in Appendix A. This term resembles the Zeeman term [65, 66] which cou- To obtain BCS theory, the replacements ω → 2 ~ p ples an external magnetic field, B, to a spin, Jp. where p is the single-particle energy of an electron and In the dense neutrino gas, there are also self-refractions of a Cooper pair if ∆ = 0 (i.e., a pair is created and then of the neutrinos in addition to the flavor oscillations out- sent into a region where there is no pair potential), lined above. The necessary terms for the Hamiltonian involve four particle terms where α scatters from β (also  z 1 † † 1 † † Jp = 2 (aαaα − aβaβ) ⇒ 2 (ck↑ck↑ + ck↓ck↓) α = β) [34]  J − = a† a ⇒ c c (14) √ p β α p↓ −p↑  + † † † 2GF X J = a aβ ⇒ c c , H = J~ · J~ (11) p α p↓ −p↑ νν V p q p,q for fermionic operators c, Eqs. (10) and (11) are the BCS if the single angle approximation [34] is assumed so that Hamiltonian up to an arbitrary constant. This provides scattering is isotropic where GF is Fermi’s constant and all necessary information required for the first half of Ta- V is the quantization volume. This term provides an in- ble IV which summarizes the PBKY model. teraction between spins. While the above analysis is for Arguably the most important quantity in the PBKY states of the neutrino gas, it is supposed that these are model’s correspondence (Table I) is the connection be- the same states available to a single neutrino (see argu- tween the paired electron states, captured by the Gor’kov ment in Ref. 34). Therefore, any analysis of the behavior function (see Sec. III A) [68], and neutrino oscillations. of the states in the gas corresponds to the states available The connection between neutrinos and Gor’kov func- for the particle. tion implies that formulating the superconducting case in So far, the analysis has used exclusively the mass basis. terms of transport equations for the Green and Gor’kov To obtain the suitable form of the unitary transformation functions can be used as an analysis tool for neutrino for the flavor states instead, we solve the time dependent oscillations. Schrödingerequation [67] ∂U i p = H U (12) B. Mass and flavor PBKY models as a mean field ∂t ν p where t is the time. The ansatz for Up is [34, 67] Before discussing the transport theory in depth, however, it is useful to show that the external magnetic field P + P 2 z P − p zpJp p ln(1+|zp| )Jp − p zpJp derived in the PBKY model is of interest. Keep in mind Up = e e e (13) that for the condensed matter system, BCS theory is the iδ where zp = e tan θ. Performing the unitary rotation physical theory and that Anderson’s reformulation is an † (i.e., U aˆ1U) gives Eq. (3). Defining the flavor isospin abstraction. But the neutrino case uses the mass or flavor 4 vector as the physical quantity, describing a neutrino’s oscillation, and is real. The appearance of a magnetic field on the superconducting state is of importance since the physics of the superconducting state changes greatly if a field is applied versus not applied. The magnetic field presented so far is not the same as a magnetic field applied on an individual electron. The PBKY model recognizes the external field as physical for the neutrino oscillations. For the mass basis, the field provides no source terms. The flavor basis contains an extra source term and implies the model is written in the mean field limit. The mass basis Hamiltonian appears as Eqs. (10) and (11). This can be simplified to

X z X x H = t ωPp + H ωPp (15) p p √ FIG. 2. The ladder of approximations used throughout this 2GF X 1 paper is summarized in this flow chart. The PBKY model is + P zP z + (P +P − + P −P +)) V p q 2 p q p q the most fundamental to the arguments presented in the text. p,q The MESM presented in the text is chosen for its consistency x + with GL, though it could assume any other form that reduces where P is either J (mass) or J (flavor), P = (P + to GL in the appropriate limit. P −)/2, t = −1 for mass and − cos 2θ for flavor, and H = 0 for mass and sin 2θ for flavor. One can view the appearance of a source term applied for the Gor’kov function in the PBKY model to say that the flavor oscil- III. SUPERCONDUCTING–MAGNETIC PROXIMITY EFFECTS lations themselves are not conserved while the number of neutrinos is allowed to be. Earlier in writing Eq. (4), the field was ensured to have Transport equations for superconductivity are summa- a constant number of particles. This works for the mass rized in Sec. III A with the resulting physics described in basis, but the flavor basis has an external source, H, Sec. III B. in Eq. (15) which may not allow for the number conservation on its own. Thus, this implies that the de- + − − + rived Hamiltonian, specifically the term Pp Pq +Pp Pq , satisfies the number conservation by explicitly being A. Transport equations for superconducting + − − + proximity effects written in the mean-field of a full term, Pp Pq Pp Pq + − + − within a constant. Substituting Pp Pq → Pp Pq + + − The utility of having identified the PBKY model as hPp Pq i returns the mean field for a well-defined con- + − equivalent to theories of superconductivity is that this stant, hPp Pq i. The connection with BCS theory and the flavor oscillations themselves is more straightforward determines the structure of the Green and Gor’kov func- in this form, especially since interchanging operators to tions [68] as well as the differential operator (from the + + − − GL Lagrangian). From these elements, non-relativistic give −Pp Pq Pp Pq gives the correct negative sign for the BCS interaction term. Note that the problem can be transport equations can be derived for the superfluid. phrased in one dimension and the Jordan-Wigner trans- A full derivation is contained in Ref. 70 (see article by formation [69] to turn the spin vectors into fermion op- V. Chandrasekharan), and a summary is provided here. + † − z † erators (i.e., Pp → cp, Pp → cp, and Pp → cpcp − 1) The effect on the superconducting state by the exter- making the connection to BCS theory clear. nal magnetic field is best exposed from the field theory The main points of the analysis of the PBKY model is by following a string of approximations summarized in that the Gor’kov function in the BCS superconductivity Fig. 2. The simplifications begin with BCS and GL to describes neutrino oscillations and that an external mag- reduce the equations and fields to the minimum number netic field is applied for the flavor basis, implying the of variables needed to describe each Cooper pair. The flavor oscillations themselves have a source term. This resulting Usadel’s equations depend only on the center produces very different physics for the BCS ground state of mass position and energy of each pair [71]. than if there were no external field. The absence of a Gor’kov connected the microscopic BCS theory and magnetic field for the mass basis will be recovered below macroscopic GL theory to form what is known as and the external magnetic field will be introduced more Gor’kov’s equations [72]. These equations parameterize easily in scalar field theory and the Standard Model after Cooper pairs in terms of the momentum of each elec- discussing the transport equations for superconductivity. tron, their positions, and their energies. The quantity of 5 interest is called the Gor’kov function, F, [68] the pairing potential. The form of the mass term gives the oscillations discussed in this section. F = −ih0|T cˆk,↑cˆ−k,↓|0i (16) It may be desirable to keep the expansion to second order, but the resulting term is not renormalizable in with time ordering operator T andc ˆ is a fermionic low- the particle physics model and would imply extra physics ering operator with subscripts for momentum and spin, (perhaps another variety of ϕ particle). However, for respectively. Since the superconductor pairs electrons to- 0 simplicity, we will limit ourselves to one extra particle. gether, it is expected that this correlator creating two paired holes (the time-reversal creates two paired electrons) has a non-zero expectation value in the symme- B. Singlet-triplet oscillations, symmetry protected try broken superconducting phase. A simplification of triplets, and cascading effects Gor’kov’s equations by Eilenberger [73] rewrites F in terms of the center of mass momentum, position, and The FFLO effect pairs electrons with momenta k + q energy. Further, a fast oscillation in the Green’s function and −k + q instead of k and −k as in traditional BCS is integrated out so the final result is in the quasi-classical superconductivity. Two particles (for a simpler example, limit. The final simplification from Usadel [71] considers consider two free particles) with these momenta relations Eilenberger’s equations in the diffusive limit where many imply an oscillation of the order parameter proportional non-magnetic impurities sufficiently randomize the cen- to v , the Fermi-velocity [43]. The oscillation in the order ter of mass momentum on a length scale that is smaller F parameter corresponds to singlet and triplet phases [79]. than the characteristic pairing length of the Cooper pair. One may describe each pairing state in the |`, si basis, This generates a set of equations in the center of mass where s = s +s , and note that ` = 1 (=0) for the triplet coordinate and energy for the field as well as the energy. 1 2 (singlet). Note that applying the interaction term on a The key step to arriving at Usadel’s equation is to expand singlet pair (|0, 0i) gives the Green and Gor’kov function as Lüder’shas done to give [74–77] B · S|0, 0i = (Lˆ− + Lˆ+)(Sˆz)|0, 0i ⇒ |1, 0i (19)

F = f0 + ˆv · f + ... (17) since the external magnetic field carries an angular momentum changed by the raising and lowering operators, where higher order terms are not necessary, f denotes 0 Lˆ±. Note that the spin operator controls which direction the Gor’kov function for singlet pairing only, ˆv is the the angular momentum is applied. normalized Fermi velocity, f are the triplet Gor’kov func- As pointed out nearly simultaneously by Bergeret- tions, and an average over all momentum is taken as the Volkov-Efetov [59] and Kadigrobov-Shekhter-Jonson [60], final step in Usadel’s derivation [70, 71]. Effectively, this when the angular momentum of a triplet Cooper pair is expansion expands the field in spherical harmonics to first perpendicular to the applied magnetic field, the triplet order [44]. A similar expression can be found for the clean components possess s = ±1 and are symmetry protected limit for Eilenberger’s equations where one must also in- from the magnetic field. Perpendicular magnetic con- clude the center of mass momentum of the Cooper pair figurations are often used in condensed matter systems and an extra matrix structure [78]. For simplicity, the fol- so that the Cooper pairs may propagate deep into the lowing arguments are constructed in the dirty limit [71]. magnetic material [62]. Usadel’s equations are useful since they often allow for a So far, the discussion has focused around promoting a simple parameterization of the Green and Gor’kov func- singlet pair to a so-called ‘long-ranged’ triplet pair with tions and the resulting solution of the parameters clearly s = ±1. This is what is generally meant when discussing displays the behavior of the fully interacting functions, the (forward) FFLO effect since the s = ±1 pairs can be though the same argument can be found from any level used to control supercurrents for spintronic application of approximation. [80]. Note that the Green’s function, G, of the Cooper pairs However, the reverse FFLO effects also apply. For ex- also may be expanded as in Eq. (17) [71]. Since the field ample, applying a field in the direction of an s = ±1 ϕ(x) = R G(x, x0)q(x0)dx0 in the most general mathemat- component reverts it back to an s = 0 component which ical sense for a forcing function q, the expansion of G may undergo singlet-triplet oscillations [81]. In a rotat- shows that the field can be expanded in a similar expan- ing magnetic field, the tree-level (see Eq. (18)) transition sion as Eq. (17) (i.e., ϕ = ϕ + ˆv · ~ϕ). Thus, expanding 0 between a triplet in one of three cartesian directions to m2 2 the mass term (= − 2 ϕ ) in relativistic GL (scalar field) a triplet in another cartesian direction produces singlets theory to first order with Eq. (17) gives in the system and can introduce ‘short-range’ (s = 0) 1 m2 components as though they were ‘long-ranged’ (s = ±1) LGL = ϕ ϕ − ϕ2 + ϕ (ˆv · ~ϕ) + (ˆv · ~ϕ)ϕ [62, 81–83]. In general, any direction the magnetic field 2 2 0 0 0 points can create a triplet component, and this is sum- λ + ϕ4 + Bϕ (18) marized in Fig. 3 for a Bloch domain wall where the ex- 4! ternal field only points in two directions, generating only where is the D’Alembertian operator and λ captures two components of f. This implies that any rotation 6 anywhere in the system will cause a cascade between all sian directions. To be more specific, the Eq. (17) expands available pairing types allowed by the magnetic field [81]. in the quasi-particle velocity defined generally as [84] In general, local and abrupt changes in the magnetiza- δm2 tion produces high concentrations of f0 components as vp ≡ ∇pp = − 2 pˆ [neutrinos]. (20) well as transitions between components of f [81]. p To first order, this analysis describes what is expected where the last equality uses the form in the PBKY at tree level in Eq. (28) expanded with Eq. (27) and the model for the energy, p (= 2ω for neutrinos outside interpretation of Fig. 1. Figure 3 may be regarded as the the gas). Formally defining this quantity in the diffu- two-flavor version of Fig. 1 (justified in Sec. IV A). sive limit where the momentum goes to zero requires the replacement p → p2 + β2, where β is a small parameter on the order of δm. This allows for an expansion around IV. CORRESPONDENCE BETWEEN p = 0 givingv ˆ = −(δm/β)2pˆ. In the limit where δm NEUTRINOS AND SUPERCONDUCTING and β, which is seen experimentally for neutrinos, go to PROXIMITY SYSTEMS zero together, Eq. (20) is well defined. The oscillation of the superconducting order parameter is proportional to Many statements can be made about the connections vF , the Fermi-velocity and maximum of vp, [43] which made in the previous sections. The list of topics cov- is dependent on δm2 here. That Eq. (20) depends on ered in each subsection are listed here: sterile and fla- the mass difference implies that the expansion is for two vored neutrinos from Lüder’sexpansion (IV A), identi- flavors represented by δm2. fication of sterile neutrinos as Majorana from symme- Equation (6) also implies that Eq. (20) is related to a try relations (IV B), the physics of flavor oscillations similar expression in the mass difference basis. Connect- by comparison with superconductor-exchange spring sys- ing Eq. (20) to Eq. (27) can be accomplished simply by tems (IV D), discussion with particular regard to sterile summing Eq. (17) over the three possible mass difference neutrinos in (IV E), symmetry protected triplet compo- δm2. The coordinate system for the problem is also al- nents as weak process states (IV F), angular momentum tered to one where the neutrino’s momentum is always in the neutrino case (Sec. IV G), MSW effects (IV H), CP pointing in one direction for simplicity (this is satisfied if violation (IV I), and other possibilities (IV J). the neutrino is restricted to one dimension, for example). Another route to changing the basis is to use Eq. (3) for each of the three mass differences and corresponding A. Lüder’sexpansion and neutrino flavor momenta. Note that summing Eq. (17) seems to imply the need The PBKY model establishes that a single neutrino for one f0 (equivalently χ0) component for each genera- oscillation is equivalent to a Cooper pair represented by tion in LSM as was mentioned earlier in a different con- a Gor’kov function, F. To establish each components text in Sec. IV C. The present analysis will leave the f0 (χ ) component as a single field. Summing over mass f0, f in Eq. (17) it is first noted that the expansions of 0 Eqs. (27) (see Sec. IV C)–which expands in neutrino fla- differences allows for one to replace Eq. (17) by Eq. (27), vor instead–and (17) are in different bases. Equation (27) effectively expanding the field in flavor going forward. expands in flavor or mass while Eq. (17) expands in carte- Repeating the derivation of Usadel’s equation for an expansion in mass differences has no effect on the final form of Usadel’s equations [70]. So, the vector components, f, can be regarded as representing the three neu- f trino flavors up to a unitary transformation. z Having identified the correspondence between the vector f and the three flavors of neutrino, the component f0 is a scalar in the condensed matter case and is invariant under rotation of the applied field. Contrastingly, f transforms like a pseudo-vector since it has acquired f0 an angular momentum. This invariance of the singlet al- ready points to its correspondence to the sterile neutrino, which is expected to not carry a flavor. fy fx

B. Quasi-particle symmetry relations and FIG. 3. A summary of the first order transitions associated Majorana fermions with the cascade effect [81] discussion in the text for a Bloch domain wall (two components). Triplet components may oscillate into the singlet component along the arrows if a field The time reversal symmetry relations of the compo- (black lines) are present in that direction. Broken lines indi- nents in the condensed matter system are [78] cate the field never points in that direction. ˜ f0(ωn) = f0(−ωn) and ˜f(ωn) = −f(−ωn) (21) 7 at each position where ωn is the Matsubara frequency gether is an addition to the Standard Model, LSM, that [68]. This reflects the nature of the triplet pairing to appears as [53] be odd in frequency but even in momentum (the limit 1 p → 0 for the diffusive limit is shown) under time rever- L = L − ν¯ m νc + h.c. (22) sal denoted by a tilde. Note that these symmetries can SM 2 L ν L be derived from the transport equations by examining the symmetry of the pairing potential, ∆(r) [78]. The where νL is a left-handed Majorana field with mass mν symmetry relations do not depend on the energy scale or for each neutrino and c is the charge conjugation oper- the spin, so the reduction of Eq. (23) to Eq. (28) should ator. When this term is written for energies above the not cause any issues in using Eq. (21). electro-weak symmetry breaking, Eq. (22)–requiring two The time reversal symmetry operator in the quantum Higgs bosons–is of mass dimension 5 [15, 52] which re- field theory, based on the symmetries of the Dirac equa- quires a coupling constant of negative mass dimension. tion, does not mix particle and anti-particle components Such a term is therefore non-renormalizable and indi- of the 4-vector [15]. This implies that Eq. (21), for the cates that some degree of freedom has been integrated out [91]. singlet f0 cannot be a Dirac particle since it relates the particle (pairs with ω > 0 as assigned in the PBKY A candidate for the integrated out quantity is a right model) with the antiparticle (ω < 0). The neutrino is handed sterile neutrino. The Standard Model can be also not likely to be a scalar particle since it is expected to extended to include a right handed field for the sterile have non-zero spin. The remaining possibility is that the neutrino, ν0, [53] neutrino is a Majorana particle [85]. The Majorana par- L = L + iν¯ ∂ν/ − E¯ F ν Φ˜ − ν¯ F †E Φ˜ † (23) ticle has no distinction between particle and anti-particle SM 0 0 L 0 0 L as they are equivalent, so the Majorana particle satisfies 1 c c − (¯ν0MM ν0 +ν ¯0MM ν0) the symmetry relations given. The field corresponding to 2 f can be a Dirac field [52]. where F is a tensor of Yukawa couplings, Φ is the For the condensed matter case, note that the Cooper Higgs boson with Φ˜ = (Φ)† where is the SU(2) anti- pair is a Majorana particle when the weights of the symmetric tensor, and EL is a vector of doublets for each Bogolyubov-Valatin transformation [86, 87] are equal generation in LSM. The possibility of several sterile neu- which is physically realized in p+ip superconductors [85]. trinos are allowed with the Majorana mass matrix, MM . The formalism used, that of Usadel, is still valid for these Equation (23) contains both the Majorana and Dirac systems [88]. masses. Only one sterile neutrino is required based on Another type of Majorana mode may be found in mag- arguments in Sec. III A but more sterile neutrinos may netic systems with a superconducting gap [89] but this be desirable [53, 57]. The following arguments are with- is a distinct type of Majorana mode from the fermion out a loss of generality to more sterile neutrinos and MM of present interest. The individual Majorana particle is is reduced to scalar m0 in the following. different from the topological state of matter with re- Taking Eq. (23) at low energies and in the case of free gards to the properties of the sterile, f0 neutrino. The neutrinos, the equations of motion with respect toν ¯ρ and superconducting gap is realized in the neutrino gas, so ν¯0, respectively, can be written as the topological Majorana mode may be realized in the neutrino gas’s case. In any case, this is distinct from the i∂ν/ ρ = mρν0 (24) Majorana fermion. m0 X i∂ν/ = νc + m ν (25) 0 2 0 ρ ρ ρ

C. Minimally extended Standard Model where an index ρ indexes the neutrino flavors, ρ ∈ {e, τ, µ}. Adding Eqs. (24–25), Given that BCS theory appeared for neutrino oscilla- X m0 X tions in the PBKY model, the Lagrangian formulation i∂ν/ + i∂ν/ = νc + m ν + m ν . (26) 0 ρ 2 0 ρ 0 ρ ρ is provided by GL theory. This sections shows that a ρ ρ reduction of the MESM reduces to the GL Lagrangian, although it is possible that other model extensions also The constants multiplying each field are of the same order reduce to the desired result. This is the simplest exten- of magnitude, and so an approximation that all magni- sion that does not change the symmetry group of the tudes, m, are equal is made for simplicity. model. The appearance of the sterile neutrino is justi- Converting all fields to a scalar field (as is done com- fied by its appearance in the superconducting case (see monly in scalar quantum electrodynamics calculations Sections IV A and IV B). [91] since the neutrino’s spin should not affect flavor os- The simplest term one can add to the Standard Model cillations), the equations of motion can be rewritten in to obtain Eq. (1) is the Pontecorvo-Maki-Nakagawa- terms of one master scalar field, Sakata (PMNS) matrix [11, 90]. Incorporating a Ma- jorana mass term that couples left-handed particles to- χ = χ0 + ˆw · χ˜ (27) 8 with ~χ = hχe, χτ , χµi and ˆw represents the flavor basis of rotating fields (J). Thus, the description of the neu- unit vector. This expansion is reminiscent of the symme- trino oscillations will only involve one basis: the mass- try breaking expansion used in the linear sigma model flavor basis with a rotating preferred direction provided (but is used in the same way as the condensed matter by the sources of the field. system in Sec. IV A). The two-flavor case corresponds to a rotating Bloch The Lagrangian in the scalar field χ is domain wall as in the superconductor-exchange spring system of Ref. 82 which contains a graphical depiction 2 1 m 2 of the system. Only two of the three cartesian directions L(∂χ, χ) = χχ − χ + Bχ (28) 2 2 have a non-zero magnetization in the Bloch domain wall, P giving non-zero amplitude to two of the components of where Bχ(=Bχ0 + B ρ χρ from Eq. (27)) is introduced in a purely mathematical sense for the purpose of calcu- f, and this makes the Bloch domain wall an ideal system lating correlation functions. The physical origin of this to study the two-flavor physics of the PBKY model. source field may come from several places. Terms that In the gapless case (free neutrinos), the Cooper pair were neglected in the analysis so far (i.e., coupling of fla- will eventually break at some distance from the super- vored neutrinos to leptons via gauge bosons–in particular conductor, and the physical effects of interest, namely the W boson) might account for the external field cou- the FFLO effect, appears close to the superconductor, P typically within a few nanometers, though that length pling to the flavored states (B χρ). The coupling of ρ scale may be much different in the neutrino case as the the field to the sterile state (Bχ ) may come from many 0 diffusion coefficient is defined differently. The gapless sources [92–94] or is an artifact of writing the neutrino case is the vacuum limit for the neutrinos and this pair fields as one field. The physical origin of this field is not breaking can be regarded as a loss of quantum coherence of particular interest here and it is sufficient to introduce from the source if the neutrino is a fundamental particle, it mathematically. though it is not explicitly ruled out here that it may be Note that one term is missing from Eq. (28) that ap- composed of other particles. pears in GL, a density-density term proportional to χ4 must be added for the neutrino gas (this term is equiva- The complete computational details can be found in lent to Eq. (11)). Ref. 62 where a parameterization by Ivanov-Fominov The field χ represents a neutrino particle whose oscil- [96, 97] which parameterizes the Green and Gor’kov func- lations are different representations of χ in the expansion tions in terms of trigonometric functions [98]. This pa- of Eq. (27). There is a question of momentum and en- rameterization allows for the exact identification of sin- ergy conservation for such a form: since the mass of the glet and triplet pairing. It is useful to examine the sys- three neutrino flavors are different, a free neutrino can tem in one dimension where the physics of the full three not conserve its momentum and energy while changing dimensional system can be exposed with a reduced com- its mass [95]. This point will be revisited in Sec. IV F putational cost with reliable results in comparison with when it is shown in the proximity system that turning experiment. off the external field freezes the oscillations between the Having identified all the components of F in the con- components. densed matter case with their corresponding particle physics representation in the previous section, now the results of previous works [62, 81, 82] are used to implement D. Flavor and mass basis: Connection to a solution of Usadel’s equations for a superconductor- superconductor-exchange spring proximity systems exchange spring system and identify behaviors of the neutrino oscillations. Neutrinos are typically written as in Eq. (3) where it is recognized that the particle has two different representations: the neutrinos can either be written in terms of E. Role of f0, χ0 in proximity systems and flavor the mass basis or the flavor basis. The classic description oscillations of mass-flavor oscillations is that a propagating neutrino will oscillate flavors by first oscillating into one of the Shown are four panels in Fig. 4 corresponding to the mass states and then back to a flavor state which may domain walls of the twisted exchange spring correspond- not be the same as the original flavor [34]. This rota- ing to the rotation of the domain wall (top panel), the tion in mass-flavor space is connected rigorously in the singlet pairing (second panel), triplet pairing in the ˆy PBKY model by a unitary transformation between the direction (third panel), and the triplet pairing in the fields describing the mass and flavor states (i.e. rotating ˆz direction (last panel). One may think of the super- the fields in Eqs. (7–8)). conductor (not pictured; left of amplitudes) as an arbi- However, the superconducting-magnetic system does trary bath for neutrinos corresponding to the f0 com- not use this two field (mass and flavor) construction. ponent (sterile neutrino) in the system characterized by p 2 2 Instead, one can observe that a unitary rotation of f0(ωn) = ∆/ ωn + |∆| for some pairing potential, ∆, the Hamiltonian Eq. (11) and (10) can equivalently be and f = 0. All components are set to zero on the op- phrased in terms of a rotating magnetic field (B) instead posite side [62]. Singlet and triplet amplitudes oscillate 9

π when a magnetic ﬁeld is oriented in a constant direction (shown by oscillations on the ﬁgures) and corre-

�π spond to s = 0. Meanwhile, the symmetry protected � states (s = ±1) decay exponentially with a longer coherence length–a behavior characteristic of normal metals π (no magnetic field). The long range decays (perpendicu- � ϕ ( x ) lar components of the Gor’kov function to the field) can be assigned to mass eigenstates and oscillating curves π � (parallel components) to flavor eigenstates [62], following Eq. (15). As the magnetization twists more and more in the top panel of Fig. 4 (descending curves), more of the f0 states ��-� are deposited into the system as in Fig. 3. The singlet/sterile states appear to decay exponentially without oscillation in the interval x > −2ξc. These ‘short-ranged’ -� �� components appear more as a ‘long-ranged’ triplet com- 0

f ponent with an exponential decay, albeit a orders of magnitude lower than the triplet states. In truth, these ex- ��-� ponentially decaying singlets were present even close to the superconductor but only reveal themselves with the ��-� oscillating singlets from the superconductor (x < −2ξc) decay enough in their amplitude to reveal the exponential behavior [62, 83, 99, 100]. ��-� Reference 101 suggests that sterile neutrinos can be found close to sources. This is wholly consistent with

-� this analogy. In that situation, the flavored neutrinos �� would be produced and oscillate into the sterile state so y f long as an external field is present to change the fla- ��-� vor. That the sterile neutrinos are observable near the source merely implies that there are abrupt flavor oscillations near this region. Away from this region, the flavor ��-� oscillations do not interact with the field and produce less sterile components. Note also that f0 need not be large even in a slow but constant rotation of the field. Contrastingly, abrupt rotations of the domain wall cause ��-� large populations of singlets to appear, for example, in discrete rotation of the domain wall [62, 81, 83, 100]. The

z -� f ��

��-�

��-� -� -� -� � � � � x/ξc

FIG. 4. The domain walls, singlet, and triplet Gor’kov functions. See text for a full discussion. Parameters for the ferromagnetic system are chosen to be most closely aligned with FIG. 5. A proximity system with two homogeneous ferromag- a Cobalt/Permalloy system with parameters K /K = 625, 1 2 nets oriented perpendicularly to each other [62] corresponds A /A = 1, domain walls found at h = 14πT for Co and 1 2 c to the case where the sterile neutrino, ν , encounters a back- 8πT for Py, with an interface placed at x = 0.625ξ , a total 0 c c ground of one type of lepton and then another. In this depic- distance of d = 13.75ξ , T = 0.2T , and ∆ = 0. F c c tion, it encounters a background of electron lepton flavor and then the µ lepton flavor. This is a rotation in isospin space of the external field (wavy line) is analogous to the rotating the magnetic field in cartesian coordinates in a magnetic system consisting of two perpendicular ferromagnets. 10

analogous situation for neutrino oscillations is where the π conserved background current is abruptly changed as is

depicted in Fig. 5. �π The bottom two panels in Fig. 4 correspond to com- � ponents of f. The component fx is zero everywhere since the magnetization never points in this direction. Again, π � oscillations corresponding to s = 0 components are seen, ϕ ( x ) and so are s = ±1 components with exponential decays π propagating much further into the system. Note that ex- � pressing these components as parallel and perpendicular to the ﬁeld shows clear ‘short’ and ‘long’ ranged behavior and is discussed in Ref. 62. Adding back in the pairing amplitude of the supercon- ��-� ductor corresponds directly to the neutrino gas. This allows for neutrino-neutrino scattering in Eq. (11) and is represented by the BCS order parameter. Fig. 6 shows ��-�

the superconductor-exchange spring with a gap for refer- 0 f

ence. The oscillations are evident near the bath of f0 but -� neutrino amplitudes saturate far from the source to a con- �� stant value. The domain walls of the exchange spring are chosen to match Fig. 4 even though the distance has been ��-� increased by a factor of two. This is to avoid a finite size effect from the extremely long correlation length of the pairing potential, ∆. Saturation to two different values ��-� on either side of the interface (dF = 0.625ξc ungapped; dF = 1.25ξc gapped) reflects the changing magnetization strength. There is a jump in the value at the interface ��-�

also for these reasons. As one twists the domain wall, y f the singlet is relatively unchanged due to its rotational -� invariance. �� The singlet in Fig. 6 is at least an order of magnitude lower than the triplet amplitudes which also saturate. ��-� Oscillations can be seen to the right of the interface in the fz component as the domain walls increase their twist and more singlets are generated on that half of the mag-

netic system. -� The summary statement is that the sterile neutrino �� controls the oscillations of the ﬂavored neutrino states.

The amplitude of the sterile neutrino is much less than z -� f �� the ﬂavored states and is directly controlled by the magnitude and rotation of the applied ﬁeld. ��-�

F. Weak process states and symmetry protected ��-� triplet components -�� -� � � �� x/ξc Triplet states with s = ±1 have the same character as

a weak process state in the neutrino case [95, 102, 103]. FIG. 6. The same as Fig. 4 but with ∆ = 0.3πTc and Flavored neutrino states require modification so that dF = 27.5ξc. The same angles at the edges of the ferromag- decays involving the weak force guarantee the creation net were used as in Fig. 4 for comparison purposes only and of the correct flavor (i.e., beta decays must produce not recalculated for the longer layer; a longer layer is called electron-type anti-neutrinos where the guarantee of the for to avoid a finite size effect for this gapped case; this can be thought of as changing the domain wall length or reduc- electron type is desired). p Reference 95 uses this idea to construct a theory where ing the coherence length ξc = D/(2πT c) for some diffusion coefficient, D, and critical temperature Tc. Effectively, the beta decays are guaranteed to give electrons and electron- domain wall from Fig. 6 is re-scaled for this longer system. type neutrinos only but also that entangle leptons and their partner neutrinos. After some distance, Ref. 95 presumes that these entangled states decay and the pro- 11 jection of the neutrino’s flavor onto the other isospin di- type neutrinos and the same effect occurs if the exter- rections gives the oscillation. According to the PBKY nal field either has a larger magnitude or points in one model and the discussion above, these protected states direction in a region of space. In the condensed matter do exist in analog with the s 6= 0 triplets components system, if the magnetic field is oriented in the ˆz direc- when there is no external field of like flavor present, tion, more triplets in the ˆz direction appear. This is the but upon encountering another region where the external essence of the MSW effect. field points in the same direction as the triplet’s angu- That the MSW effect appears so similar to a particular lar momentum, the symmetry protected state reverts to orientation of the domain wall implies the external field a s = 0 triplet and may undergo oscillations. In other is connected to leptons, but this can be no more than a words, if the isospin of the symmetry protected state does conjecture based on the analysis here. not match the isospin carried by the external field, then there is no reversion to the s = 0 triplet states and the state remains symmetry protected. Reference 95 might I. FFLO phases and CP violation be regarded as an effective theory of only the symmetry protected states but not of the oscillations themselves. Taking the suggestion that an external field splits the With regards to the energy and momentum conserva- † momentum the Cooper pair, written ukck,↑ + vkc−k,↓ tion discussed after Eq. (28), the situation can be resolved with coefficients u, v [86, 87], the same relation can be if the neutrino does not oscillate (mass does not change). applied to a Majorana spinor in its quantized form as According to the condensed matter system, this occurs [15] when there is no external field or a field that is perpen- ∗ † 2 2 dicular to the flavor direction of the neutrino. This was χ(p) = ξ−paˆ−p + (iσy)ξpaˆp δ(p − m ) (29) the physics derived in Eq. (15) in the PBKY model where there was no field found for the mass basis. for a spinor ξ. Under the FFLO effect, having a momentum q being transferred by the external field, this becomes

G. Angular momentum and lepton number 2 2 χ(p, q) = ξ−p+qaˆ−p+qδ((−p + q) − m ) (30) ∗ † 2 2 The clearest meaning of ` available from the PBKY +(iσy)ξp+qaˆp+qδ((p + q) − m ). connection is replacing the moment of inertia by a quan- This breaks the symmetry between the parts of the Ma- tity in flavor space to give a flavor angular momentum. jorana field that are particles and those components that It is tempting to call the angular momentum quantum are anti-particles in a way that is similar to the split- number, `, for the neutrinos as a lepton number. This ting of paired electrons. This would add an extra phase is since the sterile state has ` = 0 and the flavor com- factor to calculated scattering amplitudes. A weak field ponents that have ` = 1. However, it is not clear if the produces a weak splitting. quantum number appears similarly for leptons in the the- In the CP violation problem, a phase factor can ac- ory. Note that if components of f leave the region where count for particle-anti-particle a-symmetry seen in the the external field may act on it, it remains in the sym- universe [110, 111]. The entire discussion of Majorana metry protected state (symmetry protected states from a particles that are split by a complex phase as in Eq. (30), ferromagnet survive in a normal metal [104]). So, ` will which in effect adds a factor of eiq·x to the Majorana, is not change unless in response to an external field. very reminiscent of this issue. The tangible consquence is that when calculating a given scattering amplitude with the neutrino in the presence of an external field, the ex- H. FFLO and MSW effects tra phase appears. This phase produces complex phase factors in the PMNS matrix and Cabibbo-Kobayashi- Taking into account the effects of the external field Maskawa (CKM) [112, 113]. from Sec. IV D, the behavior of the components of f be- Experimentally, the violation can be measured in kaon have similarly to neutrinos in the presence of matter. oscillations of color singlets (consisting of, for example, The MSW effect is a modification of neutrino couplings a strange and down quark) as moderated by the ex- or masses in the presence of matter [64, 105–107]. This ternal fields [15]. It is conjectured that this also oc- can cause mixing angles [108] to alter. These effects are curs for leptons. The other place where it is expected observed, for example, inside of the sun where the high that the complex phase to enter is in the derived mass density of leptons alters these couplings. There is also term above. The neutrinos acquire this phase in the ik·x experimental evidence that neutrino fluxes are greater mν¯0νρ → e mν¯0νρ where k contains the momentum of when viewing them coming from the earth as opposed to the sterile and flavored neutrino. Note that the phase fac- from space [109]. Both of these effects have a similarity tor considered here is different from the Majorana phase with the proximity effects in that they are polarizing the (set to one throughout this paper) [64, 107]. flux of particles moving through the system. If only elec- For other particles that do not undergo flavor oscilla- trons are around, then it is expected that more electron tions, there is no equivalent external field in the PBKY 12 model (i.e., Eq. (10) does not appear since Eq. (1) does Neutrino oscillations can either be symmetry protected not mix 1 and 2). This is one reason why leptons may (analogous to mass eigenstates) or oscillate into another not undergo this effect, though it is not ruled out. Two neutrino type (flavor eigenstates). This allows for the Dirac particles that are paired together, however, as in conservation of mass and energy in oscillations by only the kaon’s or Cooper pair’s case, form an effective par- allowing them when an external source field is present. ticle that can receive a phase by the FFLO effect, for A cascade between all neutrino types available in a given example. system occurs when a changing external field is present. Many other explanations may account for CP violation The symmetry relations of the quasi-particle states in [114, 115]. the condensed matter case show that the sterile neutrino is a Majorana fermion. The sterile neutrinos are the intermediate states between other flavor states and appear J. Further consequences and exotic possibilities wherever neutrinos are found–especially when the external field is rotated abruptly through the direction of the neutrino flavor. If the sterile neutrino is set to zero, no Some exotic possibilities may also be realized. The flavor oscillations occur. literal analog of the connection between Josephson junc- A quantum number analogous to the angular momen- tions and neutrinos is the appearance of a current flow- tum quantity of the proximity system is defined for a ing between two superconductors in the former and the neutrino; the sterile state has a quantum number of zero transfer of particles between two interstellar bodies in while the flavored neutrinos have a value of one. Weak the latter, but the same mechanism may appear in both process states bear a close similarity with symmetry pro- systems (though the stellar phenomenon might be more tected s = ±1 triplet states of the superconductor. of an instantaneous transfer akin to a lightning bolt as Other effects such as the MSW effect are contained in a local phase may be responsible in the particle physics the physics of the proximity effect since both polarize the case). field. Phase factors acquired from the FFLO effect might Many non-intuitive effects exist in the proximity ef- account for the CP violation seen in kaon oscillations by fect literature. It may be possible to realize these in the adding the appropriate phase factors and also allow for context of the particle or astrophysics sense such as su- their effects in neutrinos. perharmonicity in Refs. 116 and 117. The single angle approximation from Ref. 34 restricts the equivalent superconductor to that of an s-wave [118]. VI. ACKNOWLEDGEMENTS If there is a situation where this approximation breaks down, other types of pairing symmetry may be realized Support is gratefully provided by the Pat Beckman in the neutrino gas, such as p-wave. Memorial Scholarship from the Orange County Chapter The general discussion of FFLO effects in this arti- of the Achievement Rewards for College Scientists Foun- cle applies to any flavor mixing process corresponding to dation. Eq. (1). For example, FFLO effects on bound quark pairs describe glitches in the radio frequency pulses of a neu- tron star [119]. The singlet and triplet states used in this Appendix A: Overview of Anderson’s reformulation article are most likely not significant for bound quarks of BCS superconductivity (such as in color-superconductivity) in the same way as for neutrinos. For the quark case, the paired quark states In the rewritten theory of Ref. 61, the vectors Jp (or correspond to a single Cooper pair. The CKM matrix, J~p) appear as spin vectors. In a slight departure from which is the analog of the PMNS matrix, is therefore the PBKY model, the Hamiltonian appears as [61] describing two particles bound together which is differ- X ent from the analysis in this paper (i.e., a mixing matrix H = Hk · Jk (A1) describing one particle with the FFLO effect). k P where H = 2(k −F )ˆz+ k0 Vkk0 J⊥k0 with ˆz picking the z component of Jp and “⊥” selecting the perpendicular V. CONCLUSION components x and y. Note that a Fermi-energy, F , was defined and is equal to the chemical potential at zero This paper connects the physics of neutrino flavor os- temperature. cillations and superconducting proximity effects. The The physics of this model can be understood in terms FFLO effect causes singlet-triplet oscillations of the su- of a spin chain (see Fig. 7). Below (above) the Fermi perconducting state that physically resemble flavor os- surface, the vectors point down (up) if there is no pairing cillations in the neutrino case. This demonstrates the potential. When the pairing potential is turned on, a 1 physical need for sterile neutrinos since they correspond domain wall appears for the spin- 2 chain. Spin up states to singlet states in the superconductor. Both are neces- (occupied) correspond to one type of neutrino (ν1) while sary for triplet/flavored states to oscillate. spin down states correspond to the other (ν2). 13

Note that the magnetic field H is not physically applied |H| to the electrons in the condensed matter system, it is a mathematical construction. Adding a magnetic field B into the model introduces a spin-dependent which F (ϵk-μ) shifts the vertical dashed lines in Fig. 7 for each spin. This is understood in the neutrino case as an increase |H| in one neutrino type and a decrease in another which is consistent with the CP violation induced by the FFLO effect (see Sec. IV I). (ϵk-μ)

FIG. 7. The arrangement of spin vectors (arrows) Jp in An- derson’s reformulation of BCS theory from Ref. 61. The upper ﬁgure is the gapless case ∆ = 0 with single particle energy shown by the positively sloped line. The lower ﬁgure shows the gapped case (∆ 6= 0).

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