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the nobel prize in physics awarded in 2015 to takaaki kajita and arthur mcdonald for studies of the atmospheric and solar neutrinos @xcite does not leave any doubt that neutrinos oscillate and have nonzero masses . the latter fact leads to the well - known possibility for neutrino to have non - trivial electromagnetic properties @xcite , which brought forth the research area that was investigated in details by numerous authors ( see recent review @xcite and references therein ) . in the course of these studies many phenomena that may appear in electromagnetic fields have been recognized and described thoroughly . among them the neutrino spin - flavor oscillations is the one , featuring both the above mentioned basic neutrino aspects  nonzero mass and electromagnetic properties from the one side and mixing from another @xcite . owing to this , in spite of being a longstanding problem , the spin - flavor oscillations can reveal some new aspects of existence of neutrino mass and electromagnetic properties . these concern the problems of neutrino parameters relation in neutrino physical ( mass ) and flavor bases and , generally speaking , of an accurate derivation of the formulas used to describe oscillations . we below give an advanced view on the standard scheme of neutrino spin - flavor oscillations description aiming at the solid determination of parameters involved in the formalism and its rigorous derivations . we consider neutrino mixing and oscillations in presence of an arbitrary constant magnetic field with nonzero transversal @xmath0 and longitudinal @xmath1 components with respect to the direction of neutrino propagation . the electromagnetic interaction of neutrinos is determined by neutrino diagonal and transition magnetic moments that are introduced for the neutrino mass states . explicit expressions for the effective neutrino diagonal and transition magnetic moments for the flavor bases in terms of these values for the mass states are obtained . the effective evolution hamiltonian for the flavor neutrino and the corresponding oscillation probability are derived . the role the longitudinal magnetic field component is examined . in particular , it is shown that : 1 ) @xmath1 coupled to the corresponding magnetic moments shifts the neutrino energy , and 2 ) in case of nonvanishing neutrino transition magnetic moments @xmath1 produces an additional mixing between neutrino states , both in the mass and flovour neutrino bases . consider two dirac neutrino physical states , @xmath2 and @xmath3 , with masses @xmath4 and @xmath5 . also consider the neutrino electromagnetic interaction via magnetic moment matrix @xmath6 ( @xmath7 ) : @xmath8 where @xmath9 is the electromagnetic field tensor , @xmath10 and @xmath11 being the dirac matrices . in a uniform magnetic field the hamiltonian ( [ em_hamilt ] ) becomes @xmath12 where @xmath13 @xmath14 are the pauli matrices .    for considering the neutrino evolution in the ultrarelativistic limit we introduce the 4-component basis @xmath15 of states with definite helicity @xmath16 . with the standard column vector notation , @xmath17 the neutrino evolution equation relevant to electromagnetic interaction has the schr@xmath18dinger - like form , @xmath19 the effective hamiltonian consists of the vacuum and interaction parts @xmath20 where the interaction part @xmath21 is composed of matrix elements of the interaction hamiltonian ( [ b_hamilt ] ) taken over the helicity neutrino states : @xmath22 . let us calculate the effective interaction hamiltonian under the assumption that neutrino moves along the @xmath23-axis . from the magnetic field interaction hamiltonian ( [ b_hamilt ] ) we have : @xmath24 for the spinors representing the free neutrino states we take @xmath25 where @xmath26 is the neutrino @xmath27 momentum . the two - component spinors @xmath28 define neutrino helicity states , and are given by @xmath29 recall that in the ultrarelativistic limit these two states correspond to the right - handed @xmath30 and left - handed @xmath31 chiral neutrinos , respectively . substituting ( [ wave func ] ) into the effective hamiltonian given by ( [ hb ] ) , we get @xmath32 decomposing the magnetic field vector into longitudinal and transversal with respect to neutrino motion components @xmath33 it is possible to show that @xmath34 in the the ultrarelativistic limit @xmath35 one gets : @xmath36 where the quantity @xmath37 , fetched out also in @xcite , we call the transition gamma - factor . similarly , it is also possible to show that @xmath38    introducing an angle @xmath39 between the @xmath40 and @xmath26 vectors and assuming that @xmath41 is aligned along the @xmath42-axis , we further obtain that @xmath43 and , similarly , @xmath44 @xmath45 @xmath46 as it was expected , in neutrino transitions without change of helicity only the @xmath47 component of the magnetic field contributes to the effective potential , whereas in transitions with change of the neutrino helicity the transversal component @xmath48 matters .    performing the remaining simple algebra one can readily write out the @xmath21 matrix . thus , we obtain the evolution equation ( [ schred_eq ] ) in the following form @xmath49 this equation governs all possible oscillations of the four neutrino states with defined masses ( @xmath4 and @xmath5 ) and helicities ( @xmath50 and @xmath51 ) in the presence of a magnetic field . the obtained expression ( [ gen_evol_eq ] ) for the evolution hamiltonian explicitly reveals a particular role of the longitudinal magnetic field component @xmath52 :    \1 ) the longitudinal magnetic field component @xmath52 coupled to the corresponding magnetic moment shifts the neutrino energy ,    \2 ) in case of nonvanishing neutrino transition magnetic moment @xmath53 , the presence of the longitudinal field @xmath52 produces mixing among neutrino species with different masses but with equal helicities .    at the same time , from ( [ gen_evol_eq ] ) it is clearly seen that mixings of different helicity neutrino states are due to the corresponding magnetic moment ( or transition magnetic moment ) interactions with the transversal magnetic field @xmath0 . once having physics in the mass basis in hands , our next step is to bring it to observational terms . this means that we must elaborate a generalization of the mixing matrix for transitions between neutrino state vectors written in two four - component bases @xmath54 and @xmath55 so that @xmath56 this procedure appears to be not quite direct since we should hold the condition that polarization of the fields must be preserved under transformation of the bases elements . that is why , keeping in mind that chiral components are almost the helicity ones , we define @xmath57 then , using eqs . ( [ u_def ] ) and ( [ transformations ] ) , it is easy to obtain that @xmath58    given the transition matrix ( [ u ] ) , derivation of the evolution equation in the flavor basis is straightforward : @xmath59 so that the effective magnetic field interaction hamiltonian @xmath60 has the following structure , @xmath61 here we have introduced the following formal notations intended to manifest an analogy with the standard spin - flavor oscillation formalism ( see below ) : @xmath62 @xmath63    the hamiltonian ( [ h_b_f ] ) has been derived within much consistent procedure than is usually given in literature . it should be noted that eqs . ( [ mu_fl ] ) could be also obtained as results of general consideration @xcite of neutrino mixing and oscillations in an arbitrary constant magnetic field ( the final form like that of eqs . ( [ mu_fl ] ) was not established in @xcite ) . also , an effective neutrino oscillation hamiltonian with terms containing quantities ( [ mu_gammas_fl ] ) , however without account for possibility of spin transitions , was obtained in @xcite . let us now confront the obtained effective magnetic field interaction hamiltonian ( [ h_b_f ] ) with the one typically written straight in the neutrino flavor basis ( see , for instance , @xcite and @xcite ) : @xmath64 where @xmath65 is the common neutrino gamma - factor . first of all , we can ascertain that the structure of the obtained expression ( [ h_b_f ] ) is consistent with the  standard " hamiltonian ( [ h_b_f_old ] ) . at that , the magnitudes ( [ mu_fl ] ) account for the neutrino mixings and represent the straightforward expressions for the effective magnetic moments in the flavor basis in terms of magnetic moments introduced in the mass basis . the neutrino effective  magnetic moments " ( [ mu_gammas_fl ] ) determine neutrino interactions and mixings due to the longitudinal magnetic field @xmath52 .    as in the mass basis , the magnetic field component @xmath52 shifts the neutrino energy also , it contributes to transitions with change of flavor . it is interesting to observe that when the transversal field component is set to zero the hamiltonian structure becomes @xmath66 so that neutrino states with different flavors and the same chirality decouple and form subsystems independently mixed by the field . for example , one would have two neutrino species @xmath67 and @xmath68 mixed in accordance with the equation ( here the standard terms for vacuum oscillations are added ) @xmath69 in this way , neutrino oscillations would be influenced by the longitudinal field @xmath52 ( see also @xcite ) and @xmath52 would produce an additional mixing with respect to the usual one described by the vacuum mixing angle @xmath70 .    the oscillation probability is just strightforward , @xmath71 where @xmath72 is the denominator of the pre - sine factor and the other quantities have usual for the theory of neutrino oscillations meaning . it follows that in case the second term in the denominator is much smaller than the first one then the amplitude of the flavor oscillations @xmath73 gets its maximal value . this can be considered as an effect of neutrinos interaction with the longitudinal magnetic field due to neutrino magnetic moments . one of the authors ( a.s . ) is thankful to nicolao fornengo and carlo giunti for the kind invitation to participate in the 14th international conference on topics in astroparticle and underground physics . the work on this paper was partially supported by the russian basic research foundation grants no . 14 - 22 - 03043-ofi - m , 15 - 52 - 53112-gfen and 16 - 02 - 01023-a . 4 fukuda y _ et al _ ( super - kamiokande collaboration ) 1998 _ phys . lett . _ * 81 * 1562 ahmad q r _ et al _ ( sno collaboration ) 2001 _ phys . lett . _ * 87 * 071301 ahmad q r _ et al _ ( sno collaboration ) 2002 _ phys . lett . _ * 89 * 011301 fujikawa k and shrock r 1980 _ phys . * 45 * 963 giunti c and studenikin a 2015 _ rev . phys . _ * 87 * 531 ( _ preprint _ arxiv:1403.6344 [ hep - ph ] ) cisneros a 1971 _ astrophys . space sci . _ * 10 * 87 schechter j and valle j 1981 _ phys . * d * 24 1883 voloshin m b , vysotsky m i and okun l b 1986 _ sov . . jetp _ * 64 * 446 akhmedov e k 1988 _ sov . j. nucl . phys . _ * 48 * 382 lim c - s and marciano w j 1988 _ phys . _ d * 37 * 1368 likhachev g and studenikin a 1995 _ sov . . jetp _ * 81 * 419 akhmedov e k and khlopov m yu 1988 _ sov . phys . _ * 47 * 689 - 691 akhmedov e k and khlopov m yu 1988 _ mod . lett . _ * a3 * 451 - 457
we consider neutrino mixing and oscillations in presence of an arbitrary constant magnetic field with nonzero transversal @xmath0 and longitudinal @xmath1 components with respect to the direction of neutrino propagation . 
 the electromagnetic interaction of neutrinos is determined by diagonal and transition neutrino magnetic moments that are introduced for the neutrino mass states . 
 explicit expressions for the effective neutrino diagonal and transition magnetic moments for the flavor basis in terms of these values for the mass states are obtained . 
 the effective evolution hamiltonian for the flavor neutrino and the corresponding oscillation probability are derived . 
 the role of the longitudinal magnetic field component is examined . in particular 
 , it is shown that : 1 ) @xmath1 coupled to the corresponding magnetic moments shifts the neutrino energy , and 2 ) in case of nonvanishing neutrino transition magnetic moments @xmath1 produces an additional mixing between neutrino states , both in the mass and flavor neutrino bases .