Comment on " Topological phase in two flavor neutrino oscillations"
aa r X i v : . [ h e p - ph ] O c t Comment on “ Topological phase in two flavorneutrino oscillations”
Rajendra Bhandari
October 17, 2018
Raman Research Institute*,Bangalore 560 080, India.email: [email protected],[email protected]
Abstract
We critically analyze the claims with regard to the relevance of topologicalphases in the physics of neutrino oscillation made in a recent paper [Phys.Rev. D 79, 096013 (2009)] and point out some inappropriate exaggerationsand misleading statements. We find that the π phase described in this paper,while interesting, is an artefact of two major approximations made in thepaper. We point out a more robust and more familiar π phase in the neutrinooscillation formulae which can be interpreted as a pure Pancharatnam phase.We also make some relevant remarks on the distinction between the geometricand the topological phase made in the commented paper.PACS numbers: 03.65.Vf, 14.60.Pq1n a recent paper [1] it has been claimed that “ for the minimal case oftwo flavors and CP conservation, there is a geometric interpretation of theneutrino oscillation formulae for the survival and detection probabilities ofneutrino species”. In this paper we first recall the derivation of the mainresult in [1]with a slightly different notation and then show that (i) there is amore robust and more familiar π phase in neutron oscillation formulae thanthe one discussed in [1] which can be seen as a pure Pancharatnam phase, (ii)the geometric interpretation and the π phase discussed in [1] is an artefact oftwo important approximations made in the paper and thus lacks fundamentalsignificance, (iii) the geometric interpretation belongs only to an aspect ofthe neutron oscillation formulae and not to the formulae themselves, (iv) thedistinction between the topological phase and geometric phase introducedin the paper is inappropriate and (v) discuss certain misleading statementsmade in the paper.As in [1], let | ν α > and | ν β > be the two flavor eigenstates representedby the two antipodal points lying along the z-axis and let | θ, ± > be the twoorthogonal mass eigenstates which lie on the line making an angle θ withrespect to the z-axis. Let the initial state be | ν α > = c + | θ , + > + c − | θ , − >, (1)where the coefficients ν α + and ν α − of [1] have been replaced by c + and c − ,the rest of the being the same. This state evolves in time t to the state | ν α > ′ = e iD + c + | θ , + > + e iD − c − | θ , − >, (2)where D + and D − are the dynamical phases acquired during the adiabaticevolution of the states | θ, ± > from | θ , ± > to | θ , ± > .The survival probability P α and the transition probability P β are givenby, P α = | < ν α | ν α > ′ | = | c + | | < ν α | θ , + > | + | c − | | < ν α | θ , − > | +( c + ∗ c − e i ( − D − + D + ) < ν α | θ , + > ∗ < ν α | θ , − > + c.c. ) , (3) P β = | < ν β | ν α > ′ | = | c + | | < ν β | θ , + > | + | c − | | < ν β | θ , − > | +( c + ∗ c − e i ( − D − + D + ) < ν β | θ , + > ∗ < ν β | θ , − > + c.c. ) . (4)2e first note a rigorous result within the two-flavor model which does notdepend on any approximations. Since it has been assumed that there is nodecay, P α and P β must add to 1. An inspection of eqns. (3) and (4) makesit obvious that this is possible only if the cross terms in the two equationsadd to zero. This implies that the complex numbers A α = < ν α | θ , + > ∗ <ν α | θ , − > and A β = < ν β | θ , + > ∗ < ν β | θ , − > should be equal in magni-tude and differ in phase by π , i.e. the phase of the number A β ∗ A α must beequal to ± π . Now A β ∗ A α = < ν β | θ , + >< ν β | θ , − > ∗ < ν α | θ , + > ∗ < ν α | θ , − > . (5)Using the fact that < ν β | θ , − > ∗ = < θ , −| ν β > etc. and rearranging theterms we get A β ∗ A α = < ν β | θ , + > < θ , + | ν α >< ν α | θ , − >< θ , −| ν β > . (6)By Pancharatnam’s theorem, the phase of the complex number on the righthand side of Eq.(6) is equal to half the solid angle subtended by the closedgeodesic curve starting at the state | ν β > , passing through thestates | θ , − > , | ν α > , | θ , + > and ending at | ν β > , which is a great circle on the Poincar´esphere. This phase is obviously equal in magnitude to π . This is the wellknown π phase between the oscillations of intensities of the two differentflavours which is seen here as a pure Pancharatnam phase of magnitude π .It follows from the condition of unitarity and is an elegant example of inter-nal consistency of different principles of physics having apparently differentorigins. Within the two-state model, this phase is independent of any ap-proximations. It can be shown easily that this π phase does not dependeven on the adiabatic approximation. Let us also note that (a) this phaseis independent of the phases of the individual states occurring in Eq.(6) and(b) this phase can be looked upon as the phase acquired by the state | ν α > if it evolved along the closed great circle under the action of a constant uni-tary hamiltonian that represents rotation about an axis perpendicular to thegreat circle, i.e. under an SU(2) element that represents a 2 π rotation on thePoincar´e sphere about this axis.If one stares at Eqs.(3) and (4) for a while it becomes obvious that thecontent of the above result can be exactly simulated by the following polar-ization experiment: Let polarization states | θ , + > and | θ , − > be incident3n the two slits of an interferometer and a polarizer that passes the state | ν α > be placed in front of the screen and the position of the fringes noted.Let the polarizer now be replaced by one that passes the state | ν β > and theshift in the fringes measured. The above result says that irrespective of theincident states the measured phase shift must be equal to π in magnitude.The above result has in fact been demonstrated in optical interferenceexperiments [2] using the two-state system of light polarization. The resultsshown in Fig. 3 of [2] show that a rotation of a linear polarizer through 90 ◦ always results in a phase shift of ± π irrespective of the polarization states ofthe interfering beams.We next come to the π phase discussed by Mehta [1]. In the cross termon the right hand side of Eq. (4), if we substitute for c + and c − from Eq.(1)we get, after rearranging the terms, the product < ν α | θ , + > < θ , + | ν β ><ν β | θ , − >< θ , −| ν α > multiplying the exponential term. To make this prod-uct correspond to evolution of a state along a closed great circle one needstwo more terms < θ , −| θ , − > and < θ , + | θ , + > which are missing fromthe product. To compensate for the missing terms, the author first sacrificesthe arbitrariness of the phases of the individual states in the product andthen makes the approximation that the hamiltonian is CP non-violating. Letus note that the approximation of adiabatic evoution of the states | θ , − > and | θ , + > to the states | θ , − > and | θ , + > has already been made.Under these two approximations, the author argues rightly that the phasesof the missing terms are accounted for exactly if the phases of the pairs ofstates | θ , − > , | θ , − > and | θ , + > , | θ , + > in the product are relatedby parallel transport and that the phase of the product is then equivalentto that acquired in a unitary evolution along a great circle under a constanthamiltonian, i.e. equal in magnitude to π . If any of the two approximationsis dropped the result is no more true. In fact the author shows in a separatepaper [3] that if the adiabatic approximation is retained but the hamiltonianis allowed to be CP-violating, the product is no longer equivalent to evolutionalong a closed great circle and the phase of the product is no longer π but isequal to that determined by the solid angle of the distorted curve! The phaseof magnitude π is thus an artefact of the restriction on the hamiltonian.In order to evaluate the significance of the result let us first consider thecase of evolution in free space or in constant density matter where the states | θ > and | θ > are the same. In this case the variations of the flavor in-tensities are pure sinusoidal oscillations. The sinusoidal oscillation has threeattributes: (A) amplitue of the oscillation which is determined by the mod-4lus of the cross terms in Eqs. (3) and (4), (B) frequency of the oscillationwhich is determined by the dynamical phase term containing D + and D − and (C) a constant phase which is equal to 0 in case of the survival proba-bility given by Eq. (3) and equal in magnitude to π in case of the transitionprobability given by Eq. (4). The main results of the paper are concernedwith only the attribute (C), i.e. the constant phase of the oscillation. Some-one who had never heard of the Pancharatnam phase would fix this constantphase trivially by requiring that the survival probability P α and the transi-tion probability P β be equal to 1 and 0 respectively at time t=0. Surely itcan be seen as a Pancharatnam phase, but the claim on this basis, as in theabstract of the paper, that “the neutron oscillation formulae have a geometricinterpretation” is, in our view, a gross exaggeration. The statement “Moreprecisely, the standard result for neutrino oscillations is in fact a realizationof the Pancharatnam topological phase” on page 9 is also an inappropriateexaggeration. It is equivalent to claiming in the first example discussed inthis comment that unitarity is a realization of the Pancharatnam phase !. Inour view the frequency of the oscillation determined by the dynamical phaseand the amplitude of the oscillation determined by the mixing angle are atleast as important parts of the neutron oscillation formulae as the absolutephase of the oscillation.When variable matter density is introduced, the dynamical phase is nomore a linearly varying function of time nor is the amplitude of the cross termconstant in time. Both these effects are just as important to the variation offlavour intensities along the path as the geometric part of the phase and areparts of the neutron oscillation formulae. In fact in the adiabatic limit, sincethe dynamical phase is large compared to the geometric phase by definition,the variation of dynamical phase due to the presence of variable matter den-sity could easily dominate over the geometric term. Moreover, as discussedabove, the value π for the phase found by the author is a consequence ofthe restriction on the hamiltonian and the adiabatic approximation and is ,therefore, not fundamental. It is just a special value obtained under specifiedconstraints.A distinction in terminology between the π phase obtained when CPviolation is absent and the non- π phase obtained when it is present hasbeen made in the paper, the former being called topological and the lattergeometric. We find this distinction unnatural and inappropriate since bothare parts of a single phenomenon and are manifestations of the same basicsingularity associated with the SU(2) group. A unified description of this5ingularity has been described in several papers. In the context of adiabaticevolution, a three-dimensional generalization of the sign change rule has beendescribed in [4]. In the context of nonadiabatic evolution an operationaldescription of this singularity has been given in [5, 6, 7] and experimentaldemonstrations with polarization of light have been reported in [6, 2, 8]. Webriefly recall this work below.Consider the evolution of a spin-1/2 state under the action of a hamilto-nian which is a function of three parameters x , y and z . First let us considerevolution along a closed circuit in a two-dimensional space, i.e. the plane y = 0. Let the hamiltonian be degenerate at an isolated point ( x , , z ) inthis plane. Let the parameters of the hamiltonian be changed so that theclosed circuit moves from a condition where it does not encircle the point( x , , z ) to a condition where it does. Then the considerations in [1] saythat at the transition point where the boundary of the closed circuit crossesthe degeneracy, there is a sudden jump of magnitude π in the phase of thestate, assuming that the dynamical phase has been subtracted out. We pointout that at the point of crossing of the degeneracy the adiabatic approxima-tion must break down. It was pointed out in [4] that if the same circuit werelocated in the plane y = ǫ , where ǫ is small and the same motion of the circuitcarried out so that this time there is no actual crossing of the degeneracy,there is a measurable phase shift of a magnitude approximately equal to, + π whereas if the same operation were carried out in the plane y = − ǫ there is ameasurable phase shift of a magnitude approximately equal to − π (only therelative sign being important) [9]. It was further shown that if the circuitwere taken around a closed loop such that it loops the degeneracy, there isa measurable phase shift equal to ± π . A monopole of strength 1/2 locatedat the degeneracy point gives a good unified description of the two effectswhich have been termed differently in [1].To consider the more general nonadiabatic case, an element of SU(2)which corresponds to a 2 π rotation about any axis on the Poincar´e sphere isrepresented by the matrix − . Any state acted upon by this element thereforeacquires a phase of magnitude π . This is the well-known phenomenon of “4 π spinor symmetry”. States with different polar angles with respect to therotation axis execute small circles of different diameter but the total phaseacquired in one full cycle is always of magnitude π . There is, however, anotherdimension to the problem. A phase shift has a magnitude as well as a sign. Ifthe phase of a state evolving under the above hamiltonian were continuouslymonitored with an interferometer with reference to some reference state | R > ,6he total measured phase shift will be + π for some states and − π for others.At critical values of the parameters, the phase shift jumps suddenly frombeing + π to being − π . This has been verified in polarization experiments[6, 2]. This is due to the presence of phase singularities which can be identifiedas follows. Let | R > stand for the reference state in the interferometer and | ˜ R > the diametrically opposite state on the sphere which is orthogonal to | R > . At points during the evolution where the evolving state is equal to | ˜ R > the interference pattern vanishes and the phase is undetermined. In thevicinity of this point the phase shifts vary sharply. In general, if a state | I > incident on one arm of an interferometer undergoes an SU(2) transformationU which is a function of some variable parameters to yield a final state | F > which interferes with a state | R > in the reference arm then a closed cycle ofthe parameters of U around an isolated point at which | F > = | ˜ R > yields atotal phase shift equal to ± nπ where n is an integer index representing thestrength of the singularity. Some examples of such phase shifts have beendemonstrated in interference experiments [8]. This result is conceptuallysimpler than the adiabatic result as it does not depend on subtraction of alarge dynamical phase from the total phase. Note that in both the abovediscussions the phase shift associated with the singularity is of magnitude2 nπ and not π . .Finally we point out that at the end of page 2 of [1], the statement “Nowthis open loop (noncyclic) Schrodinger evolution of a quantum state over atime τ can be closed by a collapse of the time-evolved quantum state at τ onto the original state at τ = 0 by the shortest geodesic curve joining thetwo states in the ray space [10].” is a misrepresentation of historical facts.A similar statement made by Samuel on page 960 of [11] is also false. Whatwas said in Samuel and Bhandari [10] was just the opposite. It was statedrepeatedly on page 2341 of [10] that the final state can be connected to theinitial state by any geodesic arc. After a careful discussion with the firstauthor of [10] the author of [1] has stated in [12] that the word “any” in “anygeodesic arc” refers to any of the gauge copies of the geodesic in the N - space.Our response to this is that the expression “any geodesic arc” includes allgeodesic arcs in N , i.e. those that project down to the shorter geodesic aswell as those that project down to the longer geodesic in the ray space. Thefootnote from [10] cited in [12] is merely a description of a property of theshorter geodesic and does not constitute a restriction on the definition of thenoncyclic geometric phase. In fact the footnote does not form part of thediscussion of the noncyclic geometric phase on p. 2341 where the expression7any geodesic curve” occurs again. In response to the following commentmade in [12] : “the author has been caught in the unfortunate position ofhaving been scooped by himself”, a statement we do not understand, wereiterate that the first clear statement that the noncyclic geometric phaseshould be defined as half the solid angle of the area obtained by closing theopen curve in the ray space with the shortest geodesic arc connecting thefinal state to the initial state was made in [5], i.e. [R. Bhandari, Phys. Lett. A 157 , 221 (1991)] and not in [10], i.e. [J. Samuel and R. Bhandari, Phys.Rev. Lett. , 2339 (1988)]. This led to the prediction of observable ± π phase jumps in two-state systems which were later verified in interferenceexperiments [6, 2] and to the nonmodular view of the topological phase, thusconstituting a conceptual advance in the subject. The restriction to theshorter geodesic is thus much more than a footnote. It may also be pointedout that the definition of the noncyclic geometric phase proposed in [5] asthe difference of the total Pancharatnam phase of the evolving state and thedynamical phase as defined by Aharonov and Anandan does not depend ona geodesic rule and is thus particularly useful for systems with more thantwo states where the geometry of the ray space can not be easily visualized.An extension of the definition to the case of an arbitrary reference state hasbeen proposed in [7]. To end this discussion we note that the fact that theshortest geodesic rule was not used in [10] was also pointed out in [4].To sum up, the net content of [1] would be precisely summarized if the ab-stract of the paper read: “We show that, under the adiabatic approximation,the phase appearing in the neutrino oscillation formulae has a geometriccontribution which, under the constraint of CP non-violation, is equal inmagnitude to π ”. Considering that the phase in a quantum evolution ingeneral has a geometric part, this is not very significant. Acknowledgements :I thank the author of the criticized paper for clarifying her work in [12].
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