Topological phase in two flavor neutrino oscillations
aa r X i v : . [ h e p - ph ] M a y Topological phase in two flavor neutrino oscillations
Poonam Mehta ∗ Raman Research Institute, C. V. Raman Avenue, Bangalore 560 080, India (Dated: October 24, 2018)We show that the phase appearing in neutrino flavor oscillation formulae has a geometric andtopological contribution. We identify a topological phase appearing in the two flavor neutrinooscillation formula using Pancharatnam’s prescription of quantum collapses between nonorthogonalstates. Such quantum collapses appear naturally in the expression for appearance and survivalprobabilities of neutrinos. Our analysis applies to neutrinos propagating in vacuum or throughmatter. For the minimal case of two flavors with CP conservation, our study shows for the firsttime that there is a geometric interpretation of the neutrino oscillation formulae for the detectionprobability of neutrino species. PACS numbers: 03.65.Vf,14.60.Pq
I. INTRODUCTION
The phenomenon of neutrino flavor oscillation resultsfrom the phase difference acquired by the mass eigen-states due to their time evolution while propagating invacuum or in matter. The observation of neutrino fla-vor oscillations in solar, atmospheric, reactor, and accel-erator experiments reveal the remarkable fact that theneutrinos exhibit sustained quantum coherence even overastrophysical length scales [1, 2]. It is then natural toask what we can learn about neutrinos from these co-herent phases. Here, we address the issue of geometricand topological phases involved in the physics of neutrinooscillations.On the theoretical front, it is well known that thephenomenon of neutrino oscillations cannot be accom-modated within the standard model (SM) of particlephysics. Therefore, the experimental observation of neu-trino oscillations provides a concrete evidence for the re-quirement of physics beyond the SM and neutrinos havebeen an intensive area of research in the past severalyears.The study of geometric phases in the context of neu-trino oscillations has been carried out in the past by sev-eral authors [3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16,17, 18], but none of the papers seem to provide a uni-fied perspective on the problem taking into account thedifferent avatars of geometric phase. It is worthwhile tostress here that one needs to be cautious while interpret-ing claims in the literature as they crucially depend onwhich version of the geometric phase one is dealing with.We will first summarize the related literature and thenfocus on the specific question that we address in thispaper. We mostly restrict our attention to the case oftwo neutrino flavors and the CP ( CP stands for charge-conjugation and parity) conserving situation, which is theminimal scenario for studying the physics of oscillations.We find, in contrast to earlier studies of this problem ∗ Electronic address: [email protected] that the geometric phase appears even in this minimalcontext.Let us first review the papers that are connected toBerry’s [19] cyclic adiabatic phase. Berry studied phasesthat appear when the Hamiltonian of a quantum sys-tem depends on parameters that are varied slowly andcyclically. Nakagawa [3] followed this work by an ele-gant paper in which he pointed out that the geometricphase could also arise in systems where adiabatic theo-rem did not hold. The key point made by Nakagawa wasthat while for existence of geometric phases, adiabaticcondition was not necessary (this was also independentlypointed out by Aharonov and Anandan [20]), the adia-batic theorem itself could be most easily understood interms of geometric arguments. As an application of hisgeneral formalism, Nakagawa considered two flavor neu-trino oscillations in matter. He concluded that the Berryphase played no role in this situation. The topologicalphase in the two flavor neutrino case, which is the cen-tral result of the present paper was missed in his workbecause he restricted himself to a limited region in the pa-rameter (ray) space and did not consider generalizationsof the geometric phase that allow for quantum collapse.Subsequent work on Berry’s geometric phase and neu-trinos exploited the spin degree of freedom of neutrinosand its interaction with the transverse magnetic fieldleading to geometric effects and spin flip. Since at thattime, spin precession was a plausible solution to the solar-neutrino problem, there is a body of work by several au-thors on the subject of geometric phase effects in thiscontext, both in the absence and presence of matter andmass-splitting terms [4, 5, 6, 7, 8, 9, 10, 11]. However, inthe present scenario, spin flavor precession is disfavoredas the leading solution to the solar-neutrino problem at99.86 % C.L. [21], which makes it phenomenologicallyuninteresting. Also, we would like to mention that in thepresent study, spin plays only a passive role, and we shallnot discuss this particular aspect any further.Naumov [12, 13, 14, 15] studied geometric phases fortwo and three flavor neutrino oscillations taking into ac-count the optic potentials [22] induced by coherent for-ward scattering of neutrinos against the background mat-ter via SM interactions. The slowly changing parametersin the Hamiltonian were identified as a set of optic poten-tials q ( t ), which were connected to the refractive indicesof neutrinos in a medium. For the naturally existingcyclic cases like spherically symmetric or sandwich-likedensity profiles, he found that the geometric (or topolog-ical) phase was zero for both two and three flavors dueto only one of the optic potentials appearing in an essen-tial manner in the Hamiltonian. Note that the two terms“topological” and “geometric” were used interchangeablyin Naumov’s works. Here we will make a distinction be-tween the two terms. The topological phase refers tophase factors that are insensitive to small changes inthe circuit, while geometric phases are sensitive to suchchanges.In a more recent paper, He et. al. [16] carried out adetailed study of the Berry phase in neutrino oscillationsfor both two and three flavors, active and sterile mixing,and with inclusion of nonstandard interactions. For theparticular case of two flavor oscillations in matter, theyclaimed that the Berry phase can only appear if non-standard (R-parity violating supersymmetry) neutrino-matter interactions are taken into account.All the above papers [3, 12, 13, 14, 15, 16] claimthat the geometric phases do not appear in the oscil-lation probabilities for the case of two flavor neutrinoswith CP conservation in vacuum or in matter as long asneutrino-matter interactions are standard i.e. coherentforward scattering is induced by charged current inter-action of electron neutrino ( ν e ) with electrons in matter.The above claims can be understood as the necessity ofhaving at least two essential parameters in the Hamil-tonian to detect curvature. Because of the absence offlavor changing neutral currents in the SM, it turns outthat for the case of ordinary electrically neutral matter,even though one has two varying parameters - electronnumber density ( n e ) and neutron number density ( n n ),only one of these will appear in an essential way in theHamiltonian and hence the Berry’s geometric phase isexpected to be zero. The other parameter n n just addsa global phase to the time-evolved neutrino flavor stateand hence does not affect oscillation. But also it is worthstressing that if both the conditions of having a nontriv-ial multidimensional parameter space as well as cyclicevolution of the states in parameter space were satisfied,the net geometric phase (resulting from the difference be-tween the geometric phases picked up by the individualmass eigenstates) would have appeared in the formulaefor detection probability and hence been observable.Next, we will briefly review and summarize papersdealing with geometric phases that are generalizationsof the Berry phase [20, 23, 24] in the context of neutri-nos [17, 18]. Such geometric effects can appear underless restrictive conditions than those required for Berry’sversion of the geometric phase. Infact such phases canappear even in situations where there are no parametersvarying in the Hamiltonian and the evolution is not nec-essarily cyclic or unitary. Note, however, that in generalthe geometric phases appearing in transition amplitudes are global phases that do not have any observable conse-quences. To observe such a phase one needs a split-beaminterference experiment in which a beam is spatially sep-arated into two parts that suffer different histories. Suchan experiment is hard to design for neutrinos becausethey interact so weakly and are nearly impossible to de-flect or confine. This renders such phases uninteresting asthey are not observable as far as neutrinos are concerned.Our aim here is to explore whether there are geometriceffects that survive at the level of detection probabilitiesthat are directly measurable quantities.Blasone et. al. [17] claimed that Berry’s phase waspresent in the physics of neutrino oscillation in vacuumeven for the two flavor CP conserving case. Their argu-ment is based on the fact that under Schr¨odinger evolu-tion, the pure flavor states come back to themselves afterone period ( T ) of oscillation having acquired an overallphase. This overall phase was shown to be a sum of apure dynamical phase and a part that depended on themixing angle only and independent of energy and massesof the two mass states (hence, geometric). They calledthis extra phase the Berry phase. Note that this phasepicked up by a neutrino flavor state arises purely due toSchr¨odinger evolution of the system giving a closed loopin the Hilbert space but not due to any slowly varyingparameters leading to adiabatic evolution of the Hamil-tonian itself. Hence strictly speaking it is the Aharonov-Anandan cyclic phase [20] that generalized Berry’s adi-abatic phase to situations where the adiabaticity con-straint did not apply and only the cyclic condition ismet. Also, we should note that since the phase obtainedwas a global phase at the amplitude level, it does not ap-pear in measurable quantities like neutrino appearanceor survival probabilities as mentioned above.After Berry’s [19] seminal paper on this subject, Ra-maseshan and Nityananda [25] pointed out that Berry’sphase had a connection with the phase obtained by Pan-charatnam [23] in the fifties in his study on interferenceof polarized light. These insights were carried over to theray space of quantum mechanics by Samuel and Bhan-dari [24]. They showed that the two seemingly differ-ent geometric phases obtained by Berry and Pancharat-nam (appearing under different sets of conditions) couldbe described in a unified framework. They also pointedout that geometric phases are not restricted to unitary,cyclic and adiabatic evolution [19] of a quantum systemand can appear in an even more general context that al-lows for quantum collapses, which occur during measure-ments. Following this line of thought, Wang et. al. [18]extended the study of Blasone et. al. [17] to obtain non-cyclic geometric phases for two and three flavor neutrinosin vacuum. Their claim can be understood as follows.Consider the Schr¨odinger evolution of a quantum stateover an arbitrary time period from τ = 0 to τ . Now thisopen loop (noncyclic) Schr¨odinger evolution of a quan-tum state over a time τ can be closed by a collapse ofthe time-evolved quantum state at τ onto the originalstate at τ = 0 by the shorter geodesic curve joining thetwo states in the ray space [24]. The phase associatedwith the complex number ( re i ß ) representing the innerproduct of the original state vector and the time-evolvedstate vector (with the dynamical phase removed) has apure geometric origin. This noncyclic geometric phasewas evaluated by Wang et. al. [18] for both the twoand three flavor cases. But, again note that this phasewill be unobservable as it only appears at the level ofamplitude.The main purpose of the present work is to estab-lish that Pancharatnam’s phase does appear in detec-tion probabilities and hence is directly observable. Forthe simplest case of two flavors in vacuum or in con-stant density matter (restricting to SM interactions) with CP conservation, we obtain a Pancharatnam phase of π ,and this leads to an elegant geometric interpretation ofthe neutrino oscillation formulae. We also make a directconnection of this phase with the Herzberg and Longuet-Higgins topological phase [26] in molecular physics. Weshow that the Pancharatnam phase of π remains even inthe presence of slowly varying matter density and thiscan be ascribed to the topological nature of this phase.Inclusion of CP violation can change the topological na-ture of the phase and make it a path-dependent geometricphase.Although one should do a full three flavor analysis fora complete treatment, we work in an effective two fla-vor approximation that is fairly justified [27, 28] due tothe smallness of Θ , and hierarchy of mass splittings( | δm /δm | <<
1) and in addition on matter interac-tions being standard [50]. In many physical situations,observations depend on mainly one mixing and one masssquared splitting. Conventionally, Θ and δm describeoscillations of solar neutrinos, while Θ and δm areused to describe atmospheric neutrinos. The mixing an-gle Θ gives small effects on both solar and atmosphericneutrinos. Working with only two flavors is of courseadvantageous as the results obtained are physically moretransparent and can be visualized in analogous situationsin optics and the Poincar´e sphere can be used as a cal-culational tool to study the system.For the ease of visualization of the phenomena of os-cillations, in the past several authors have discussed sim-ple pictorial depiction of neutrino oscillations in terms ofprecession of a (pseudo) spin vector in three-dimensionalspace in a variety of contexts for the case of two neu-trino flavors [29, 30, 31, 32, 33, 34, 35, 36]. Below wegive a brief account of the papers dealing with geomet-ric representation of neutrino flavor oscillations. Harrisand Stodolsky [29] addressed the question of a unifiedtreatment of generic two-state systems (including parti-cle mixing involving two neutrino types) in media usingdensity matrices. It was shown that the equation of mo-tion for the polarization vector represented the precessionof polarization vector about a vector representing an ef-fective magnetic field (which could result from the massterms in vacuum or matter terms). Kim et. al. [30] dis-cussed the analogy of solar-neutrino oscillations with thatof precession of electron spin in a time-dependent mag-netic field. They applied this picture in the limit of adia- batic approximation. Stodolsky [31] described the evolu-tion of a statistical ensemble (neutrinos from supernovaeor in the early Universe) applying the density matrix ap-proach [29] and showed that oscillations in presence ofmixing and matter interactions in a thermal environmentcould be viewed in terms of precession. Kim et. al. [32]derived the geometric picture for two and three flavorneutrinos and applied it to nonadiabatic as well as adia-batic cases. Thomson and McKellar [33] treated the caseof neutrino background giving rise to nonlinear feedbackterms in the equation of motion for polarization vectorsand gave a pictorial representation for the same. En-qvist et. al. [34] describe visualization of oscillations ofa thermal neutrino ensemble of the early Universe. Thegeometrical representation in wave packet treatment ofoscillations was discussed by Giunti et. al. [35]. As inoptics, the Poincar´e sphere is a convenient tool for visu-alizations and calculations pertaining to neutrino oscilla-tions, particularly in looking for geometric effects.This paper is organized as follows. In Sec. II, we de-velop an analogy between the neutrino flavor states andpolarized states in optics since such a mapping allowsfor a convenient visualization of geometric effects. Wethen go on to show in Sec. III that the Pancharatnamphase does appear in the detection probabilities of neu-trino species in the two flavor neutrino system in vacuumand also in matter. We conclude with a discussion of ourkey result and future directions in Sec. IV. Throughoutwe set ~ = c = 1. II. CORRESPONDENCE BETWEEN TWOFLAVOR NEUTRINOS AND POLARIZATIONSTATES IN OPTICS
Since the concept of Pancharatnam’s phase was devel-oped in the context of optics, it is worthwhile to firstdevelop a correspondence between the mathematics oftwo flavor neutrino states and polarization states in op-tics. Let us first recall the conditions under which thetwo flavor neutrinos and polarization states in optics canbe analyzed within an unified framework.
A. Two flavor neutrinos
In the ultra-relativistic limit, the Dirac equation fortwo flavor neutrinos (antineutrinos) can be reduced toa Schr¨odinger form [22, 37] written in terms of a two-component vector of positive (negative) energy probabil-ity amplitude. This is analogous to Maxwell’s equationsreducing to the linear Schr¨odinger form for the polariza-tion states in optics in the paraxial limit [38].The two neutrino flavor states can be mapped to atwo-level quantum system with distinct energy eigen-values, E i ≃ p + m i / p in the ultrarelativistic limitalong with the assumption of equal fixed momenta (orenergy) [36, 39]. In the presence of matter, the relativis-tic dispersion relation E i = f ( p, m i ) gets modified due tothe neutrino-matter interactions (in an electrically neu-tral homogeneous medium) leading to E i = ∓ = (cid:18) p + m + m p + V C V N (cid:19) ∓ q ( ω sin 2Θ) + ( V C − ω cos 2Θ) , (1)where ω = δm / p with mass splitting δm = m − m and p ≃ E being the fixed momentum (energy) of theneutrino. Θ is the mixing angle in vacuum. V C = √ G F n e = 7 . × − Y e ρ eV and V N = −√ G F n n / − . × − Y n ρ eV are the respective effective poten-tials due to coherent forward scattering of neutrinos withelectrons (via charged current interactions) and neutrons(via neutral current interactions). G F = 1 . × − GeV − parameterizes the weak interaction strength(Fermi constant). V C and V N depend on the electron( n e ) and neutron ( n n ) number densities (in units ofcm − ). n e/n = ρY e/n N Avo , where ρ is the mass den-sity in g cm − , Y e/n is the relative electron (neutron)number density and its value is roughly ∼ . N Avo is the Avogadro’s number. Setting V C = V N = 0, we recover the vacuum case.Note the fact that although there are two densities n e and n n appearing in the eigenvalues, it is only n e thatappears in a nontrivial way (through V C ) in the flavorHamiltonian, H ν = (cid:18) p + m + m p + V C V N (cid:19) I + 12 (cid:18) V C − ω cos 2Θ ω sin 2Θ ω sin 2Θ − ( V C − ω cos 2Θ) (cid:19) . (2)The above Hamiltonian (Eq. 2) also describes an inhomo-geneous medium provided the scale of variation of mat-ter induced potential V C is slow compared to the scaleof the order of ~ / ( E + − E − )), hence ensuring no tran-sitions between the mass eigenstates. This defines theadiabaticity condition [36, 39]. As neutrinos traverse adensity gradient, at a particular value of n e the diagonalelements of H ν can vanish causing an interchange of fla-vors irrespective of the value of the vacuum mixing angleΘ. This phenomenon of resonant conversion in matteris known as the Mikheyev-Smirnov-Wolfenstein (MSW)effect [40, 41].The off-diagonal form of the Hamiltonian in flavor ba-sis (both in vacuum and matter) leads to flavor oscil-lations of neutrinos, which is the only mechanism thatmixes the neutrinos of different generations or flavorswhile preserving the lepton number (note that the ab-sence of flavor changing neutral currents prevents anyflavor change within the SM). Also note that the matterterm appears in diagonal elements only so in the absenceof vacuum mixing, neutrinos of different flavors cannotmix. The term proportional to the identity gives an over-all phase to each of the mass eigenstates and hence doesnot affect oscillations. This corresponds to the gaugefreedom of any state of a two-level quantum system [3]. In the next subsection, we describe the polarized statesin optics in the language of quantum mechanics. B. Polarized states in optics
Polarization optics is mathematically identical to theevolution of a two state quantum system. In a helicitybasis for polarized light, we can write | R i and | L i rep-resenting right and left circular polarizations. A generalpolarized light beam | Ψ i can then be expanded in thisbasis as | Ψ i = α | R i + β | L i where | α | + | β | = N , theintensity of the beam of polarized light. We can param-eterize an arbitrary state of polarized light by | Ψ i = √ N exp { iη } (cid:18) cos( θ/
2) exp( − iφ/ θ/
2) exp( iφ/ (cid:19) , (3)where N is the total intensity, which is normalized tounity, and the angles θ and φ (where 0 ≤ θ ≤ π and0 ≤ φ ≤ π ) describe the state of polarization of thebeam, represented on the two-dimensional unit sphere( S ) called the Poincar´e sphere. Orthogonal polarizationstates are antipodal points of the sphere. η is the overallphase of the beam. The states on the sphere are de-fined modulo this overall phase of η and represent theray space [42]. The north pole ( θ = 0) represents rightcircular light and the south pole ( θ = π ) represents leftcircular light. States on the equator ( θ = π/
2) repre-sent linear polarizations. Any other point on the sur-face of the sphere represents elliptic polarization. ThePoincar´e sphere is a useful device to visualize the changesin the state of polarization of a light beam traversingthrough a medium.The mapping between the polarized states and a two-level quantum system originates from the following fact.Neglecting absorption effects [51], the effect of differentmedia can be encoded in terms of 2 × H = A σ x + B σ y + C σ z + D I , (4)where, the coefficients of the three traceless Pauli ma-trices, A , B and C are responsible for generating rota-tions of incident optical states about x, y, z axes on thePoincar´e sphere. D just adds an overall phase that canbe absorbed in a redefinition of the state. Hence givenan arbitrary medium, it can be represented by a Hamil-tonian as mentioned above, and the eigenstates of theHamiltonian represent those optical states that do notsuffer any change (when incident on such a medium) intheir state of polarization except for picking up an overallphase shift. The polarization of any other state (otherthan the eigenstates) incident on this medium will un-dergo a periodic change. On the Poincar´e sphere this canbe visualized as a rotation of the incident state vector xy z | ϑ, + i| ν α i| ϑ, − i ϑ | ν α i| ν β i FIG. 1: Neutrino states on the Poincar´e sphere. The flavorstates | ν α i and | ν β i are the two antipodal points on the z axis while | ϑ, ± i correspond to the mass (energy) eigenstateslying on an axis making an angle ϑ with respect to the z axis. about the axis defined by a line joining the two eigen-states of the Hamiltonian. Mathematically, these uni-tary rotations on the Poincar´e sphere are generated by e − i H t . This is identical to unitary time evolution gen-erated by the Hamiltonian of the quantum states in theHilbert space. The quantum-mechanical analogue of thePoincar´e sphere is the Bl¨och sphere, which geometricallyrepresents the space of pure states of a two-level quantumsystem.Nonvanishing values of A , B , C simultaneously param-eterize the effect of an elliptically birefringent medium.Circular (linear) birefringence are special cases wherethe conditions A , B = 0 and C , D 6 = 0 ( B , C = 0 and A , D 6 = 0) are satisfied.
C. Neutrinos and optics analogy
We can now describe the isomorphism between neu-trino states and polarized states in optics. The completeset of states for two flavor neutrino system can be repre-sented on the Poincar´e sphere just like the optical statesas depicted in Fig 1. For convenience we define a newcoordinate ϑ , which goes from 0 → π as we traversethe unit great circle in the x − z plane. In terms of theold coordinates, the points θ, φ = 0 are now labeled by ϑ = θ and the points θ, φ = π are labeled by ϑ = 2 π − θ .If we assume that the flavor states are the north andsouth poles of the Poincar´e sphere, then the mass eigen-states are represented by the two antipodal points ly-ing on an axis making an angle 2Θ = ϑ with respect tothe polar axis. States on the equator coincide with themass eigenstates for the special case of maximal mixing(Θ = ϑ/ π/
4) which corresponds to complete flavorconversion (MSW effect). Geometrically, the MSW ef-fect can be viewed as rotation about an equatorial axis, rotating the north pole into the south pole.Ignoring the term proportional to the Identity, the neu-trino Hamiltonian (Eq. 2) both in vacuum or matter canbe recast in exactly the same form given by (see Eq. 4) H ν = ω ϑ ) σ x − (cos ϑ ) σ z ] , (5)where ω = δm / p and the mixing angle Θ is replacedby ϑ/ e − i H ν t on the Poincar´e sphere. In the languageof neutrino optics, both vacuum and matter exhibit el-liptic birefringence property with different elliptic axes.The absence of flavor changing neutral currents in theSM gives rise to a real form of the Hamiltonian ( B = 0),and it corresponds to a CP -conserving situation. Theeigenvectors (also called mass eigenstates) of Eq. 5 aregiven by | ϑ, + i = (cid:18) cos( ϑ/ ϑ/ (cid:19) and | ϑ, − i = (cid:18) − sin( ϑ/ ϑ/ (cid:19) . (6)Note that states | ϑ, + i and | ϑ, − i are orthogonal an-tipodal points on the Poincar´e sphere which always lieon the great circle formed by the intersection of the x − z plane with the Poincar´e sphere. Mass eigenstates lyingoutside the x − z plane imply CP violation. This fact hasvery interesting consequences for the physics of geometricphases in CP nonconserving situations [43]. III. PANCHARATNAM’S PHASE IN THE TWOFLAVOR NEUTRINO SYSTEM
The Pancharatnam phase :-
We give a brief introduc-tion to the idea of Pancharatnam’s phase in quantum-mechanical language along the lines of Ref. [24, 42, 44].Given any two nonorthogonal states | A i and | B i in theHilbert space describing a system, a notion of geometricparallelism between the two states can be drawn fromthe inner product h A | B i . The two states are said to beparallel (in phase) if h A | B i is real and positive, whichdefines the Pancharatnam connection (or rule) . Geomet-rically, it implies that the norm of the vector sum ofthe two states || ( | A i + | B i ) || = h A | A i + h B | B i +2 |h A | B i| cos(ph h A | B i ) is maximum. Physically, it im-plies that if we let the two states interfere with each otherthe resulting state will have maximum probability (inten-sity). Note that if | A i is in phase with | B i , and | B i isin phase with | C i , then | C i is not necessarily in phasewith state | A i . The phase difference between the states | C i and | A i is the Pancharatnam phase , and it is equalto half the solid angle Ω subtended by the geodesic trian-gle A , B , C on the Poincar´e sphere for a two-level systemat its center. In general, for an n -level system, the spaceof states is given by CP n − ( CP stands for complex pro-jective) which reduces to the Poincar´e sphere ( S ) for atwo-level system ( n = 2). Nonintegrability of Pancharat-nam’s connection follows from the nontransitivity of therule.Pancharatnam’s phase reflects the curvature of pro-jective Hilbert space (ray space) and is independent ofany parameterization or slow variation. Thus it canalso appear in situations where the Hamiltonian is con-stant in time. All one needs is that the state hasa nontrivial trajectory on the Poincar´e sphere. Thiscondition is met naturally for neutrinos since theyare produced and detected as flavor states (which arenot the stationary mass eigenstates) and hence theyautomatically explore the curvature of the ray space(Poincar´e sphere) under the Schr¨odinger time evolution.Furthermore, note the fact that Schr¨odinger evolution(possibly) interrupted by measurements can lead to Pan-charatnam’s phase. If we take any state and subjectit to multiple quantum collapses (such that consecutivecollapses are between nonorthogonal states) and bringit back to itself, then the resulting state is given by | A ih A | C ih C | B ih B | A i , where the phase of the com-plex number h A | C ih C | B ih B | A i is given by Ω / The Herzberg and Longuet-Higgins phase and CP -conserving neutrino Hamiltonian :- Let us reexaminethe form of the neutrino Hamiltonian given by Eq. 5 andthe eigenvectors given by Eq. 6. Note that the eigenvec-tors depend only on a single parameter ϑ and satisfy | ϑ, ± i = ∓| ϑ + π, ∓ i = −| ϑ + 2 π, ± i = ±| ϑ + 3 π, ∓ i = | ϑ + 4 π, ± i . (7)The minus sign picked up by both the mass eigenstatesas we change ϑ from 0 → π is precisely the Herzbergand Longuet-Higgins phase [26, 45] of π , which was firstobtained in the context of molecular physics in 1963. So,we note that just by looking at the form of the Hamilto-nian for neutrino system, we should expect the Herzbergand Longuet-Higgins phase to appear. Also, note thatthe space of rays for the real neutrino Hamiltonian isthe great circle ( S ) lying on the x − z plane of thePoincar´e sphere (Fig. 1) and global structure of the eigen-vectors is a M¨obius band . The variation of ϑ results inparallel transport of the mass eigenstates (with dynam-ical phase removed) following the parallel transport rulealong ϑ , ℑ m h ϑ ∓ | ddϑ | ϑ ∓ i = 0 . (8)This parallel transport rule (formally referred to as nat-ural connection) has an anholonomy defined on theM¨obius band and this leads to the topological phase of π . The topological phase factor ß depends on the vectorpotential A ϑ given byß = I A ϑ dϑ = I ℑ m h ϑ ∓ | ddϑ | ϑ ∓ i dϑ . (9) This vector potential A ϑ is nonintegrable, and this is theanholonomy of the connection. Physically, this corre-sponds to half a unit of magnetic flux piercing the originof the x − z plane, encircling which leads to this topolog-ical phase. And, the origin of the circle is connected tothe null Hamiltonian ( i.e. all elements are zero), whichcorresponds to the degeneracy point.Naively speaking, one would think that this phase willbe impossible to access for neutrinos because we do nothave a handle on the mixing angle ϑ/ ϑ = 0 → π . The key point tounderstand here is the fact that as long as we carry outa quantum evolution of a state in a closed loop enclos-ing the point of singularity (degeneracy point, origin ofthe Poincar´e sphere), which can be achieved either viaadiabatic variation of ϑ or via Schr¨odinger evolution in-terrupted by collapses, one will always get this phase.However, note that in the former case, the amplitude ofthe initial state undergoing evolution does not changebut in the latter case, it diminishes. In what follows,we will show that the transition probability for neutri-nos actually does carry imprints of such a topologicalphase, which can be explicitly derived using Pancharat-nam’s prescription. We then show that the phase of π actually appears there and is in fact observed by all theexperiments carried out so far. The topological phase in two flavor neutrino oscilla-tions (invoking collapses and adiabatic evolution) :-
Inwhat follows, we consider the most general situation, i.e.neutrinos are traversing through matter with slowly vary-ing density (i.e. ϑ is a slowly varying parameter changingfrom ϑ to ϑ ). Vacuum or constant density matter willbe special cases where ϑ is a constant.In order to see the effect of geometric phases, usuallyone performs a split-beam experiment. In the case of op-tics, one separates a beam into two parts in space andeach part traverses a different path. Finally the beamsare recombined to observe the relative phase shift as theyinterfere. In optics, the reflective and refractive prop-erty of the medium is exploited to make devices like mir-rors and lenses, which facilitates designing of such exper-iments in the laboratory. In the case of neutrinos, sucha procedure is not possible owing to the fact that the re-fractive index is extremely small ( n refr − ≃ − forneutrinos of energy 1 MeV in ordinary matter). Treat-ing the Sun (with density ρ = 150 g cm − in the core)as a spherical lens for a neutrino beam of energy 10 MeVpassing through it, one gets the focal length to be around10 R ⊙ [22], which is about 10 times the size of ourgalaxy. Spatially split-beam interference experimentswith neutrinos are clearly impossible. However, the factthat neutrinos are produced and detected as flavor statesallows us to think of the time evolution of neutrinos asa split-beam experiment in energy space as depicted inFig. 2.Let us consider a neutrino created as a flavor state | ν α i (for example, neutrinos produced inside the Sunare predominantly in the electron neutrino flavor state, | ν e i ) and detected as another flavor state, | ν β i ( | ν β i | ν α i | ν β i| ϑ , + i | ϑ , + i| ϑ , − i | ϑ , − i FIG. 2: Schematic of a split-beam experiment for neutrinos inenergy space. | ν α i and | ν β i are the two flavor states, while | ϑ , ± i and also | ϑ , ± i correspond to two sets of mass (en-ergy) eigenstates. | ϑ , ± i are adiabatically evolved to states | ϑ , ± i , respectively (upon removing the dynamical phase). can either be a | ν e i , i.e. survival of the same electronneutrino flavor or a | ν µ i , i.e. appearance of muon neu-trino flavor), then | ν α i = ν α + | ϑ , + i + ν α − | ϑ , − i , (10)where | ϑ , ± i are the eigenstates of H ν ( ϑ ). Now weconsider an adiabatic evolution of the mass eigenstatesfrom | ϑ , ± i to | ϑ , ± i due to a slow enough variation ofbackground density such that no mixing between the twoeigenstates is ensured under time evolution, and | ϑ , ± i evolves to | ϑ , ± i → e − i D ± | ϑ , ± i with D ± = ± Z t q ( ω sin ϑ ) + ( V C − ω cos ϑ ) dt ′ + Z t (cid:18) p + m + m p + V C V N (cid:19) dt ′ , (11)as the dynamical phases, relevant both for the vacuumcase ( V C = V N = 0) and in the presence of varying mat-ter density profile and t is the time of flight of the neu-trino. The quantities that depend on time (or distance)are V C and V N defined earlier (see Eq. 1). Note thatthe states | ϑ , ± i are | ϑ , ± i are connected via paralleltransport rule (Eq. 8) on the Poincar´e sphere. The twotime-evolved states e − i D ± | ϑ , ± i are finally recombinedto form a flavor state at the detector.In order to see this explicitly, let us proceed as follows:The amplitude for the transition between states ν α → ν β is given by A ( ν α → ν β ) = h ν β | U | ν α i , (12)where U is the unitary evolution operator given by U = e − i D + | ϑ , + ih ϑ , + | + e − i D − | ϑ , − ih ϑ , − | . (13) Inserting two complete sets of states in the amplitude, A ( ν α → ν β ) = + X i,j = − h ν β | ϑ , i ih ϑ , i | U | ϑ , j ih ϑ , j | ν α i = h ν β | ϑ , + ih ϑ , + | U | ϑ , + ih ϑ , + | ν α i + h ν β | ϑ , − ih ϑ , − | U | ϑ , − ih ϑ , − | ν α i . (14)Note that the cross terms do not contribute in the adia-batic limit. Upon substituting Eq. 13 in Eq. 14, we get A ( ν α → ν β ) = e − i D + h ν β | ϑ , + ih ϑ , + | ν α i + e − i D − h ν β | ϑ , − ih ϑ , − | ν α i . (15)Then the probability for flavor transition ν α → ν β isgiven by P ( ν α → ν β ) = |A ( ν α → ν β ) | = h ν α | ϑ , + ih ϑ , + | ν β ih ν β | ϑ , + ih ϑ , + | ν α i + h ν α | ϑ , − ih ϑ , − | ν β ih ν β | ϑ , − ih ϑ , − | ν α i + [ h ν α | ϑ , − i e i D − h ϑ , − | ν β ih ν β | ϑ , + i e − i D + h ϑ , + | ν α i + c . c . ] . (16)The cross term term in Eq. 16 is related to the inter-ference term resulting from the two path interferometerdepicted in Fig. 2. Upon dropping the dynamical phase,we have h ν α | ϑ , − ih ϑ , − | ν β ih ν β | ϑ , + ih ϑ , + | ν α i which can be viewed as a series of closed loop quantumcollapses with intermediate adiabatic evolutions givenby | ν α i → | ϑ , + i → | ϑ , + i → | ν β i → | ϑ , − i →| ϑ , − i → | ν α i that essentially covers a great circle inthe x − z plane as is shown in Fig. 3(a). This closedtrajectory subtends a solid angle of Ω = 2 π at the centerof the great circle. Hence without any further calcula-tion, we can immediately predict that the phase of theinterference term will be π (half the solid angle) due toPancharatnam’s prescription. On the circle, each of theindividual collapse processes which essentially projects astate with given angle ϑ to another state with differentangle ϑ ′ can be thought of as an infinite series of infinites-imally close collapses between states defined as | ϑ i and | ϑ + δϑ i as far as geometric phases are concerned. Theentire closed loop of collapses with intermediate adiabaticevolutions mentioned above can be viewed as a smoothvariation of ϑ from 0 → π in the limit δϑ → δϑ → α = β , i.e. survival probability,it is easy to see that the collapses do not form a closedloop enclosing the origin and therefore the interferenceterm will not pick up any phase. This case is depicted inFig. 3(b).In a simpler situation when ϑ does not change, i.e. thecase of vacuum or constant density matter, the number z x z x | ν β i | ϑ , + i| ϑ , + i| ϑ , − i| ϑ , − i | ν α i | ν α i | ϑ , + i| ϑ , + i| ϑ , − i| ϑ , − i ϑ ϑ ϑ ϑ ( b )( a ) FIG. 3: Two representative cases depicting the collapse processes (dashed red lines) with intermediate adiabatic evo-lutions upon removing the dynamical phase (dotted blue lines) on the great circle ( S ) arising due to the cross term h ν α | ϑ , − ih ϑ , − | ν β ih ν β | ϑ , + ih ϑ , + | ν α i in the probability. The initial flavor state | ν α i is on the positive z axis, whilethe final flavor state | ν β i is not necessarily its antipodal point. The two sets of mass eigenstates are antipodal points ontwo axes making angles ϑ and ϑ respectively with respect to the z axis. Case (a) corresponds to appearance probability[ P ( ν α → ν β )] for which we get a cyclic loop in ϑ space. (b) The collapse processes for survival probability [ P ( ν α → ν α )] do notenclose any loop. of states will be fewer (in the absence of variation of den-sity, | ϑ , ± i is the same as | ϑ , ± i ) and the collapses aregiven by | ν α i → | ϑ , + i → | ν β i → | ϑ , − i → | ν α i . Aslong as the collapses lead to closed loop encircling the ori-gin, we will obtain this topological phase. So this phaseof π appears whether we consider vacuum and/or ordi-nary matter with constant density or with slowly chang-ing (but noncyclic) electron number density. This is dueto the topological character of this phase, which will bepreserved as long as we have CP -conserving (real) Hamil-tonian and states are always lying on a great circle in the x − z plane in the Poincar´e sphere.Next we write down an explicit expression for the ob-servable quantities, i.e. appearance and survival prob-abilities for two neutrino flavors. Using the general ex-pression obtained in Eq. 16, the appearance probabilityfor transition ν e → ν µ is given by [53] P ( ν e → ν µ ) = U ⋆e + (Θ ) U µ + (Θ ) U ⋆µ + (Θ ) U e + (Θ )+ U ⋆e − (Θ ) U µ − (Θ ) U ⋆µ − (Θ ) U e − (Θ )+ [ U ⋆e − (Θ ) e i D − U µ − (Θ ) U ⋆µ + (Θ ) e − i D + U e + (Θ ) + c . c . ] . (17)Note that the matrix U (Θ) is the lepton mixing matrix(defined in a basis where the charged lepton mass matrixis diagonal). It is also referred to as the Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix [46, 47] and connectsthe flavor states to the mass eigenstates. For the 2 × U (Θ) = (cid:18) cos Θ sin Θ − sin Θ cos Θ (cid:19) . (18)Substituting the elements of U (Θ) we get P ( ν e → ν µ ) = cos Θ sin Θ + sin Θ cos Θ + [2 cos( D + − D − )]( − sin Θ ) cos Θ sin Θ cos Θ . (19) We note that there are four inner products appearing inthe interference term in the final expression for the prob-ability out of which the first three inner products, viz., h ϑ , + | ν e i = U e + (Θ ) = cos Θ > h ν µ | ϑ , + i = U ⋆µ + (Θ ) = sin Θ > h ϑ , − | ν µ i = U µ − (Θ ) =cos Θ > h ν e | ϑ , − i = U ⋆e − (Θ ) = − sin Θ < | ν e i antiparallel to | ϑ , − i , since thephysically allowed values for the mixing angles Θ andΘ are within the interval [0 , π/
2] for δm > ϑ and ϑ cantake values between [0 , π ]). The minus sign appearing inthe interference term is thus the Pancharatnam’s phaseof π appearing in the neutrino oscillation formula (seeFig. 3(a)).If in a hypothetical situation, for some range of param-eters Θ and Θ , the first three of the inner products arereal and negative ( i.e. the states are aligned antiparal-lel to each other or completely out of phase), while thefourth inner product is real and positive (the states arein phase) then also we will have this minus sign. Thenontransitivity also holds here leading to the non-trivialtopological phase of π . This situation where the innerproduct becomes real and negative defines an “antipar-allel” rule (in the same spirit in which Pancharatnamdefined his rule of two states being “in phase or paral-lel”) would correspond to the norm of the vector sum ofthe two states being at its minimum value. Physically,this implies the interference of the two given states wouldbe destructive and the resulting state will have minimumintensity or a dark fringe in optics.The existence of Pancharatnam’s phase of π can besimply connected to the fact that the mixing matrix U (Θ)matrix for two flavors is an orthogonal rotation matrixparameterized by the mixing angle Θ of which one ele-ment has a negative sign. Thus, this phase is built into the structure of U (Θ) matrix.The survival probability is given by P ( ν e → ν e ) = cos Θ cos Θ + sin Θ sin Θ + [2 cos( D + − D − )] sin Θ cos Θ sin Θ cos Θ . (20)Note that in the case of survival probability, the crossterm does not pick up any nonzero topological phase, andgeometrically this is exactly what we had expected fromFig 3(b). The loop in ϑ -space is open in this case, andthis is what leads to this result. The topological phase ofthe interference term in survival probability is zero, whileit is π in the case of the appearance probability, and thisfact is in accord with unitarity.The above expressions (Eqs. 19 and 20) reduce to thestandard results [27, 36, 39, 48] for vacuum if we substi-tute Θ = Θ = Θ, P ( ν e → ν µ ) = sin
2Θ sin δm l E and P ( ν e → ν e ) = 1 − sin
2Θ sin δm l E , (21)where in the ultrarelativistic limit, we can use t ≃ l and p ≃ E leading to D ± = ± δm l/ E (see Eq. 11) for thevacuum case ( V C = V N = 0). In constant density mat-ter, the quantities Θ and δm in Eq. 21 are replaced bytheir respective renormalized values in matter, Θ m and( δm ) m but the form of the expression will remain thesame. Hence our result is consistent with the standardneutrino oscillation formulation, and it provides a cleargeometric interpretation of the phenomenon of neutrinooscillations. More precisely, the standard result for neu-trino oscillations is in fact a realization of the Pancharat-nam topological phase. IV. DISCUSSION
As mentioned in the introduction, the existing workon the subject of geometric phases in neutrino oscilla-tions led to the widespread belief that the two flavorneutrino oscillation formulae in CP conserving situationswere devoid of any geometric or topological phase com-ponent. Appearance of the cyclic Berry phase was dis-missed on the grounds of not having any time-varyingparameter in vacuum and having only one essential pa-rameter (thereby enclosing no area) in the case of normalmatter [3, 12, 13, 14, 15, 16]. Concerning the appearanceof the general geometric phase in the two flavor neutrinocase for propagation in vacuum, there are claims report-ing its appearance [17, 18]. But, it should be noted thatsuch terms appeared only at amplitude level and as ar-gued earlier, a phase appearing in the amplitude can beobserved only via a split-beam experiment, which is notfeasible to design in the case of neutrinos. In this paper, we have examined the minimal caseof two flavor neutrino oscillations and CP conservation.Contrary to all existing claims in the literature concern-ing the geometric or topological phase in two flavor neu-trino oscillation probabilities, our study provides the firstclear prediction that a topological phase of π exists at theprobability level even in the minimal case of CP con-servation. We show that it is inherently present in thephysics of neutrino oscillations via the structure of thePMNS neutrino mixing matrix. This existence of thistopological phase is linked to the presence of a flux lineof strength π at the origin of ray space, which is con-nected to the degeneracy point associated with the nullHamiltonian.Pancharatnam’s idea is quite useful in terms of pre-dictive power as it allows for a clear visualization of theappearance of such a phase due to geometric effects with-out doing any algebra. Our prescription is general as itcontains effects due to collapses and also due to adiabaticevolution. In the absence of either of these, one wouldget the same phase. So no matter what the details are,as long as the singular (degeneracy) point is enclosed bya cyclic loop (in the space of rays) as ϑ is varied from0 → π , we will get this phase, and this is due to itstopological robustness. The adiabatic and collapse pro-cesses both conspire in such a fashion that the net phasewould always be π . This does not happen for geometricphases.The topological phase obtained in this paper is a con-sequence of anholonomy, which can arise in situationseven when there is no curvature. The most striking ex-ample of this is the Aharonov-Bohm effect [45]. To expe-rience the effect of anholonomy, the main requirement isto encircle the singular point, this fact was exploited byHerzberg and Longuet-Higgins in pointing out the topo-logical phase in molecular physics. On the other hand, forBerry’s phase to appear, a net curvature is a must whichis fulfilled by having at least two essential parameters inthe Hamiltonian varying cyclically. This is an importantdistinction between the geometric phases as obtained byHerzberg and Longuet-Higgins and by Berry.If we consider mixed flavor states [54] instead of thepure flavor states, there will be a greater number of phys-ical situations (or, possible diagrams for the interferenceterm like the ones shown in Fig. 3 for pure flavor states)that can be explored to see if one encircles the singularpoint or not. A mixed state corresponds to a generalpoint on the surface of the Poincar´e sphere like an el-liptically polarized state in optics. If the mixed statesare such that they lie on the x − z plane, it will al-ways lead to the same quantized topological phase of π .But, for a general mixed state lying anywhere else on thePoincar´e sphere, the phase will be geometric in nature.It might be a nontrivial task to extend our geometri-cal interpretation to the case of three neutrinos flavorsbecause it will involve a higher dimensional sphere (theray space is CP for the three level quantum system).It is natural to ask what happens when we invoke CP violation. In vacuum, CP violation cannot be induced0in the two flavor case as a consequence of CP T invari-ance and unitarity [28]. However, matter with constantor varying density can induce CP violation via the co-herent forward scattering of neutrinos with backgroundmatter. If we introduce CP violation induced by back-ground matter with constant density [28], we still expectto get the same phase of π as we have two pairs of or-thogonal states that will always lie on a great circle. Ifthe density is varying slowly (adiabatic condition holds),then the intermediate states (connected by adiabatic evo-lution) will be lifted from the great circle, hence resultingin a path-dependent solid angle, and the phase will be ge- ometric [43]. Acknowledgments
The author is deeply indebted to Joseph Samuel andSupurna Sinha for numerous useful discussions lead-ing to the present work and critical comments on themanuscript. Support from the Weizmann Institute ofScience, Israel during the initial stages of this project isgratefully acknowledged. [1] M. C. Gonzalez-Garcia and M. Maltoni, Phys. Rept. ,1 (2008), 0704.1800.[2] R. Z. Funchal, talk on “Global overview of neutrino mix-ing and masses” given at Neutrino-2008, New Zealand.[3] N. Nakagawa, Ann. Phys. , 145 (1987).[4] J. Vidal and J. Wudka, Phys. Lett.
B249 , 473 (1990).[5] C. Aneziris and J. Schechter, Int. J. Mod. Phys. A6 , 2375(1991).[6] C. Aneziris and J. Schechter, Phys. Rev. D45 , 1053(1992).[7] A. Y. Smirnov, Pis’ma Zh. Eksp. Teor. Fiz. , 280(1991).[8] A. Y. Smirnov, Phys. Lett. B260 , 161 (1991).[9] E. K. Akhmedov, A. Y. Smirnov, and P. I. Krastev, Z.Phys.
C52 , 701 (1991).[10] M. M. Guzzo and J. Bellandi, Phys. Lett.
B294 , 243(1992).[11] V. M. Aquino, J. Bellandi, and M. M. Guzzo, Phys. Scr. , 328 (1996).[12] V. A. Naumov, JETP Lett. , 185 (1991).[13] V. A. Naumov, Sov. Phys. JETP , 1 (1992).[14] V. A. Naumov, Int. J. Mod. Phys. D1 , 379 (1992).[15] V. A. Naumov, Phys. Lett. B323 , 351 (1994).[16] X.-G. He, X.-Q. Li, B. H. J. McKellar, and Y. Zhang,Phys. Rev.
D72 , 053012 (2005), hep-ph/0412374.[17] M. Blasone, P. A. Henning, and G. Vitiello, Phys. Lett.
B466 , 262 (1999), hep-th/9902124.[18] X.-B. Wang, L. C. Kwek, Y. Liu, and C. H. Oh, Phys.Rev.
D 63 , 053003 (2001).[19] M. V. Berry, Proc. Roy. Soc. Lond.
A392 , 45 (1984).[20] Y. Aharonov and J. Anandan, Phys. Rev. Lett. , 1593(1987).[21] O. G. Miranda, J. Phys. , 121 (2006).[22] G. G. Raffelt, Stars as Laboratories for Fundamen-tal Physics: The Astrophysics of Neutrinos, Axions,and Other Weakly Interacting Particles (University ofChicago Press, Chicago, 1996).[23] S. Pancharatnam, Proc. Ind. Acad. Sci.
A44 , 247 (1956).[24] J. Samuel and R. Bhandari, Phys. Rev. Lett. , 2339(1988).[25] S. Ramaseshan and R. Nityananda, Curr. Sci. , 1225(1986).[26] G. Herzberg and H. C. Longuet-Higgins, Disc. FaradaySoc. , 77 (1963).[27] T. K. Kuo and J. Pantaleone, Rev. Mod. Phys. , 937(1989).[28] E. K. Akhmedov, Phys. Scr. T121 , 65 (2005), hep- ph/0412029.[29] R. A. Harris and L. Stodolsky, Phys. Lett.
B116 , 464(1982).[30] C. W. Kim, W. K. Sze, and S. Nussinov, Phys. Rev.
D35 ,4014 (1987).[31] L. Stodolsky, Phys. Rev.
D36 , 2273 (1987).[32] C. W. Kim, J. Kim, and W. K. Sze, Phys. Rev.
D37 ,1072 (1988).[33] M. J. Thompson and B. H. J. McKellar, Phys. Lett.
B259 , 113 (1991).[34] K. Enqvist, K. Kainulainen, and J. Maalampi, Nucl.Phys. B , 754 (1991).[35] C. Giunti, C. W. Kim, and U. W. Lee, Phys. Lett.
B274 ,87 (1992).[36] C. W. Kim and A. Pevsner,
Neutrinos in physicsand astrophysics (Harwood Academic Publishers, Chur,Switzerland, 1993).[37] A. Halprin, Phys. Rev.
D34 , 3462 (1986).[38] N. Mukunda, R. Simon, and E. C. G. Sudarshan, J. Opt.Soc. Am. A2 , 1291 (1985).[39] M. C. Gonzalez-Garcia and Y. Nir, Rev. Mod. Phys. ,345 (2003), hep-ph/0202058.[40] L. Wolfenstein, Phys. Rev. D17 , 2369 (1978).[41] S. P. Mikheyev and A. Y. Smirnov, Sov. J. Nucl. Phys. , 913 (1985).[42] J. Samuel, Pramana , 959 (1997), quant-ph/9705019.[43] P. Mehta, work in progress.[44] M. V. Berry, Jour. Mod. Optics , 1401 (1987).[45] A. Shapere and F. Wilczek, Geometric Phases in Physics (World Scientific, Singapore, 1989).[46] B. Pontecorvo, Sov. Phys. JETP , 172 (1958).[47] Z. Maki, M. Nakagawa, and S. Sakata, Progress of Theo.Phys. , 870 (1962).[48] J. D. Walecka, Introduction To Modern Physics: Theo-retical Foundations (World Scientific, Singapore, 2008).[49] A. Bandyopadhyay et al. (ISS Physics Working Group)(2007), 0710.4947.[50] It turns out that in the presence of non-standard interac-tions during propagation, it is possible to do the analysiswith only two flavors for the case of solar neutrinos whilea complete three flavor analysis is needed for the case ofthe atmospheric neutrinos [49].[51] The incoherent scattering cross section for neutrinos(10 − cm for 1 MeV neutrinos impinging on target ofmass 1 MeV) is extremely small as compared to photonsin a medium.[52] In defining the Poincar´e sphere, it is useful to work with half angles ϑ/ S as ϑ changesfrom 0 to 4 π .[53] In order to connect with the standard expressions used inneutrino literature, we shall revert to Θ instead of ϑ/ϑ/