CP-Symmetry in Scattering of Neutrinos from Nuclei
aa r X i v : . [ h e p - ph ] J a n CP-Symmetry in Scattering of Neutrinos from Nuclei
R.B. Begzhanov and R.S. SharafiddinovInstitute of Nuclear Physics, Uzbekistan Academy of Sciences,Ulugbek, Tashkent 100214, UzbekistanAbstract
The elastic scattering of longitudinal and transversal neutrinos on a spinless nucleus havebeen discussed taking into account the charge, magnetic, anapole and electric dipole momentsof fermions and their weak neutral currents. Compound structure of the neutrino interactioncross section with nuclei have been defined. Invariance of the considered process concerningthe C - and P-operations have been investigated in the polarization type dependence.
1. Introduction
It has been established that the behavior of massive neutrinos plays an important part inunderstanding the physical nature of elementary particles. One of the modes of doing this isto study the possible neutrino-nucleus interaction [1,2].The neutrino interaction with field of emission may be expressed in the form [3,4] of elec-tromagnetic current j emµ = u ( p ′ , s ′ )[ γ µ F ν ( q ) − iσ µλ q λ F ν ( q )++ γ γ µ G ν ( q ) − iγ σ µλ q λ G ν ( q )] u ( p, s ) , (1)where σ µλ = [ γ µ , γ λ ] / , q = p − p ′ is the momentum transfer, p ( s ) and p ′ ( s ′ ) imply the four-momentum (helicities) of initial and final neutrinos, F iν ( q ) and G iν ( q ) are the interactionvector and axial-vector parts respectively. The functions F ν (0) , F ν (0) and G ν (0) give thestatic estimates of the neutrino charge [5], magnetic [6] and electric dipole [7] moments, onwhich there exist experimental and cosmological bounds [8]. Insofar as G ν (0) is concerned,it defines the size of the anapole moment [9], but its value has not yet been measured in thelaboratory [10].It is known that F iν ( q ) are invariant with respect to C - and P-operations because theinteraction of F iν ( q ) with field of emission must be CP-symmetrical. The term G ν ( q ) isCP-even but P-odd [9]. In contrast to this, the term G ν ( q ) must be C-invariant but CP-antisymmetrical [11]. Therefore, the form factors G iν ( q ) may be different from zero only inthe case where P-symmetry is absent.The violation of P-parity leads to the appearance of right-left asymmetry, for example, at thepolarized neutrinos scattering on nuclei. In many works [2,12] the spin phenomena was studiedwith longitudinal neutrinos. Such an investigation is important not only for elucidation ofcompound structure of the interaction between leptons and hadrons but also for observationand refinement of the most diverse symmetries of elementary particles. However, the massiveneutrino must have the longitudinal as well as the transversal polarization. The account of thelatter gives the possibility to directly look at the nature of the discussed processes. n the present work, we investigate the phenomena of symmetricality in the massive neutrinosinteractions with an electroweak field of emission. Section 2 is dedicated to the elastic scatteringof longitudinal polarized neutrinos on the nucleus electric ( Z ) and weak ( Z W ) charges ν ( ν ) + A ( Z, Z W ) γ,Z → ν ′ ( ν ′ ) + A ( Z, Z W ) , (2)going at the expense of neutral and electromagnetic currents. In Sec. 3 the studied processeshave been reanalysed for the transversal case of the neutrino polarization. In Sec. 4 we makesome concluding remarks.
2. Longitudinal Polarized Neutrinos Scattering on a Nucleus
In the framework of the standard theory of electroweak interaction [13], the Hamiltonian ofthe neutrino interaction with field of a nucleus has the form H = 4 παq u ( p ′ , s ′ )[ γ µ F ν ( q ) − iσ µλ q λ F ν ( q )++ γ γ µ G ν ( q ) − iγ σ µλ q λ G ν ( q )] u ( p, s ) J γµ ( q )++ G F √ u ( p ′ , s ′ ) γ µ ( g V ν + γ g A ν ) u ( p, s ) J Z µ ( q ) . (3)Here J xµ ( q ) are the nucleus electromagnetic ( x = γ ) and weak neutral ( x = Z ) currents [14], g V ν and g A ν are the corresponding constants of the neutrino interaction vector ( V ) and axial( A ) parts.In the case of the neutrino longitudinl polarization and of a zero-spin nucleus, the cross-section of the process (2) on the basis of (3) can be presented after the summing of s ′ asfollows: dσ ew ( θ, s ) = dσ em ( θ, s ) + dσ int ( θ, s ) + dσ we ( θ, s ) , (4)where to purely electromagnetic interaction answers the expression dσ em ( θ, s ) d Ω = σ νo (1 − η ν ) − { [ F ν + 2 λ c s p − η ν G ν ] F ν ++ η ν [ F ν + 4 m ν (1 − η − ν ) F ν ] tg θ − sE ν (1 − η ν ) / F ν G ν tg θ − η ν )[ G ν + 4 E ν G ν tg θ } F E ( q ) . (5)The contribution explained by the interference of electroweak interaction is written in theform dσ int ( θ, s ) d Ω = ρσ νo (1 − η ν ) − g V ν { [1 −− λ c s g A ν g V ν p − η ν ][ F ν + λ c s p − η ν G ν ] + η ν F ν tg θ } F EV ( q ) . (6)In the same way one can present the cross-section of purely weak interaction with neutralcurrents dσ we ( θ, s ) d Ω = E ν G F π { g V ν (1 + η ν tg θ g A ν (1 − η ν ) − λ c sg V ν g A ν p − η ν } F W ( q ) cos θ . (7)Here we have also used the size σ νo = α cos θ E ν (1 − η ν ) sin θ , η ν = m ν E ν , ρ = G F q π √ α ,F E ( q ) = ZF c ( q ) , F EV ( q ) = ZZ W F c ( q ) , F W ( q ) = Z W F c ( q ) ,Z W = 12 { β (0) V ( Z + N ) + β (1) V ( Z − N ) } , A = Z + N, M T = 12 ( Z − N ) , where θ is the scattering angle, E ν and m ν are the neutrino mass and energy, F c ( q ) is thecharge ( F c (0) = 1) form factor of a nucleus with isospin T and its projection M T , β (0) V and β (1) V are constants of isoscalar and isovector components of vector neutral hadronic current.The presence of the multiplier s in Eqs. (5)-(7) implies their antisymmetricality concerningthe substitution of the left-handed ( s = −
1) particle with the right-handed ( s = +1) and viceversa. We see in addition that Eqs. (5)-(7) for the neutrino ( λ c = +1) and the antineutrino( λ c = −
1) are different.Taking into account Eqs. (5)-(7), the size of charge asymmetry A ewch = A emch + A intch + A wech = dσ νew − dσ νew dσ νew + dσ νew (8)is defined by the corresponding contributions A emch ( θ ) = 2 s p − η ν F ν G ν { (1 + η ν tg θ F ν ++ (1 − η ν ) G ν + 4 E ν [ s p − η ν G ν − (1 − η ν ) F ν ] tg θ } − , (9) A intch ( θ ) = s p − η ν [ G ν − g A ν g V ν F ν ] { (1 + η ν tg θ F ν −− g A ν g V ν (1 − η ν ) G ν } − , (10) A wech ( θ ) = − s g A ν g V ν p − η ν { (1 + η ν tg θ g A ν g V ν (1 − η ν ) } − . (11)These formulas show clearly that C-invariance of the considered process can be violated onlyin the case when the mirror symmetry is absent. Indeed, taking s = 0 , we find A emch ( θ ) = 0 , A intch ( θ ) = 0 , A wech ( θ ) = 0 , (12)which are true at the conservation of P-parity.Many authors state that one must use the electromagnetic current (1) in the form [15] inwhich an i is absent. If we start with such a procedure, assuming that the interaction magneticand electric dipole terms must not be Hermitian even with q < , we would establish the otherexpressions for the processes cross-sections instead of (5) and (6). They lead to the implication[16] that C-invariance of elastic scattering is basically violated at the expense of the neutrinononzero rest mass. One can also make a conclusion that this influence does not relate to thebehavior of P-symmetry. aking into account that nonconservation of P-parity at the neutrino interaction convenientlycharacterize by the right-left asymmetry, we have A ewRL = A emRL + A intRL + A weRL = dσ Rew − dσ Lew dσ Rew + dσ Lew , (13)from Eqs. (5)-(7), we get A emRL ( θ ) = 2 p − η ν [ λ c F ν G ν −− E ν (1 − η ν ) F ν G ν tg θ { (1 + η ν tg θ F ν ++ (1 − η ν )[ G ν ctg θ E ν ( G ν + (1 − η ν ) F ν )] tg θ } − , (14) A intRL ( θ ) = λ c p − η ν [ G ν − g A ν g V ν F ν ] { (1 + η ν tg θ F ν −− g A ν g V ν (1 − η ν ) G ν } − , (15) A weRL ( θ ) = − λ c g A ν g V ν p − η ν { (1 + η ν tg θ g A ν g V ν (1 − η ν ) } − . (16)The availability of the multiplier λ c in these formulas implies the influence of the interactionC-antisymmetrical structure on the conservation of P-symmetry. Indeed, the average cross-sections, Eqs. (5)-(7), over the two values of λ c would leads us to the equalities A emRL ( θ ) = − E ν (1 − η ν ) / F ν G ν { (1 + η ν tg θ F ν ++ (1 − η ν )[ G ν + 4 E ν ( G ν + (1 − η ν ) F ν ) tg θ } − tg θ , (17) A intRL ( θ ) = 0 , A weRL ( θ ) = 0 , (18)which take place at C-invariance.Thus, it follows that regardless of the behavior of charge symmetry, the right-left asymmetryof the process (2) can be explained by the interference of the interaction axial-vector terms withits vector terms, if neutrinos do not possess any new properties.
3. Interaction of Transversal Polarized Neutrinoswith an Electroweak Field of a Nucleus
Starting from (3) and assuming that the neutrinos are strictly transversal, for the elasticscattering cross-section we find an explicit expression which can be reduced after the summingof s ′ to the form dσ ew ( θ, ϕ, s ) = dσ em ( θ, ϕ, s ) + dσ int ( θ, ϕ, s ) + dσ we ( θ, ϕ, s ) . (19)As well as in (4), each term here corresponds to the most diverse process and has the differentstructure: dσ em ( θ, ϕ, s ) d Ω = σ νo (1 − η ν ) − { F ν + η ν [ F ν + 4 m ν (1 − η − ν ) F ν ] tg θ λ c sη ν p − η ν F ν G ν tg θ cos ϕ ++ (1 − η ν )[ G ν + 4 E ν G ν tg θ } F E ( q ) , (20) dσ int ( θ, ϕ, s ) d Ω = ρσ νo (1 − η ν ) − g V ν { F ν + η ν [1++ λ c s g A ν g V ν η − ν p − η ν ctg θ cos ϕ ] F ν tg θ −− λ c sη ν p − η ν [ tg θ cos ϕ + λ c s g A ν g V ν η − ν p − η ν ] G ν } F EV ( q ) , (21) dσ we ( θ, ϕ, s ) d Ω = E ν G F π { g V ν (1 + η ν tg θ g A ν (1 − η ν ) −− λ c sg V ν g A ν η ν p − η ν tg θ cos ϕ } F W ( q ) cos θ , (22)where ϕ is the azimuthal angle.Using (20)-(22) and taking (8), for the C-odd asymmetry in the case of the neutrino transver-sal polarization we get A emch ( θ, ϕ ) = 2 sη ν p − η ν F ν G ν { (1 + η ν tg θ F ν ++ (1 − η ν )[ G ν + 4 E ν ( G ν + (1 − η ν ) F ν ) tg θ } − tg θ cos ϕ, (23) A intch ( θ, ϕ ) = − sη ν p − η ν [ G ν − g A ν g V ν F ν ] { (1 + η ν tg θ F ν −− g A ν g V ν (1 − η ν ) G ν } − tg θ cos ϕ, (24) A wech ( θ, ϕ ) = − sη ν g A ν g V ν p − η ν { (1 + η ν tg θ g A ν g V ν (1 − η ν ) } − tg θ cos ϕ. (25)The solutions (23)-(25) at s = 0 coincide with the corresponding size from (12) and that,consequently, the behavior of C-invariance in the P-symmetrical interactions does not dependon the type of polarization.In the same way one can see that the P-odd characteristics of elastic scattering, accordingto (13), (20)-(22), has the form A emRL ( θ, ϕ ) = 2 λ c η ν p − η ν F ν G ν { (1 + η ν tg θ F ν ++ (1 − η ν )[ G ν ctg θ E ν ( G ν + (1 − η ν ) F ν )] tg θ } − tg θ cos ϕ, (26) A intRL ( θ, ϕ ) = − λ c η ν p − η ν [ G ν − g A ν g V ν F ν ] { (1 + η ν tg θ F ν − g A ν g V ν (1 − η ν ) G ν } − tg θ cos ϕ, (27) A weRL ( θ, ϕ ) = − λ c η ν g A ν g V ν p − η ν { (1 + η ν tg θ g A ν g V ν (1 − η ν ) } − tg θ cos ϕ. (28)However, due to C-parity, it follows from (26)-(28) that A emRL ( θ, ϕ ) = 0 , A intRL ( θ, ϕ ) = 0 , A weRL ( θ, ϕ ) = 0 . (29)Comparing (17) and (18) with (29), it is easy to observe the differences which may serve as anindication to the type of polarization dependence of C-invariant processes right-left asymmetry.
4. Conclusion
We have established an explicit form of the differential cross sections describing the elasticelectroweak scattering of longitudinal and transversal polarized neutrinos (antineutrinos) onspinless nuclei as a consequence of the availability of rest mass, charge, magnetic, anapole andelectric dipole moments of elementary particles and their weak neutral currents. With the useof these formulas a proof has been obtained regardless of the nature of C, nonconservation ofP can be explained by the interference of the interaction vector and axial-vector parts.One of the new features of our results is the connection between the P-odd phenomena andpossible polarization types. Unlike the behavior of C-parity in the P-symmetrical scattering,coefficients of right-left asymmetries A ewRL ( θ ) and A ewRL ( θ, ϕ ) in the C-invariant processes withlongitudinal and transversal neutrinos are different.Furthermore, if neutrinos are of high energies ( E ν ≫ m ν ) then A ewRL ( θ, ϕ ) = 0 , and the sizeof A ewRL ( θ ) is reduced to the form A ewRL ( θ ) = − F ν G ν F ν + G ν . (30)It is expected that measurement of right-left asymmetry A ewRL ( θ ) for any two values of largeenergies will testify in favor of the equality of the neutrino magnetic and electric dipole moments. References
1. P.H. Frampton and P. Vogel,
Phys. Rep. , 339 (1982); F. Boehm and P. Vogel, Ann.Rev. Nucl. Part. Sci. , 125 (1984); P. Vogel and J. Engel, Phys. Rev.
D39 , 3378(1989).2. B.K. Kerimov, T.R. Aruri and M.Ya. Safin,
Izv. Acad. Nauk SSSR. Ser. Fiz. , 1768(1973); B.K. Kerimov and M.Ya. Safin, Izv. Russ. Acad. Nauk Ser. Fiz. , 657 (1997).3. M.A.B. Beg, W.J. Marciano and M. Ruderman, Phys. Rev.
D17 , 1395 (1978).4. W. Bernreuther and M. Suzuki,
Rev. Mod. Phys. , 313 (1991).5. R.S. Sharafiddinov, Dokl. Akad. Nauk Ruz. Ser. Math. Tehn. Estest. , 25 (1998). . K. Fujikawa and R.E. Shrock, Phys. Rev. Lett. , 963 (1980).7. R.B. Begzhanov and R.S. Sharafiddinov, in Proc. Int. Conf. on Nuclear Physics , Moscow,June 16-19, 1998, St. Petersburg, 1998, p. 354.8. S. Davidson, B. Campbell and K.D. Bailey,
Phys. Rev.
D43 , 2314 (1991); J.A. Morganand D.B. Farrant,
Phys. Lett.
B128 , 431 (1983).9. Ya. B. Zel’dovich,
Zh. Eksp. Teor. Fiz. , 1531 (1957); Ya. B. Zel’dovich and A. M.Perelomov, ibid. , 1115 (1960).10. M.J. Musolf and B.R. Holstein, Phys. Rev.
D43 ,1956 (1991).11. L.D. Landau,
Zh. Eksp. Teor. Fiz. ,405 (1957); Nucl. Phys. , 127 (1957).12. B.K. Kerimov and M.Ya. Safin, Izv. Russ. Acad. Nauk Ser. Fiz. , 93 (1993).13. S.L. Glashow, Nucl. Phys. , 579 (1961); A. Salam and J.C. Ward, Phys. Lett. , 168(1964); S. Weinberg, Phys. Rev. Lett. , 1264 (1967).14. T.W. Donnelly and R.D. Peccei, Phys. Rep. , 3 (1979).15. J.Bernstein, M. Ruderman and G. Feinberg, Phys. Rev.
Proc. Int. Conf. on Nuclear Physics , Dubna,April 21-24, 1999, St. Petersburg, 1999, p. 408., Dubna,April 21-24, 1999, St. Petersburg, 1999, p. 408.