Turk J Phys 33 2009), 225 234. c TÜBİTAK doi:10.3906/fiz-0810-1 Study of second-class currents effects on polarization characteristics in quasi-elastic neutrino anti-neutrino) scattering by nuclei Adamou OUSMANE MANGA and Almoustapha ABOUBACAR Department of Physics, Faculty of Sciences, University Abdou Moumouni of Niamey P.O. Box 10662, Niamey-NIGER e-mail: manga adamou@yahoo.com, bmouthe@yahoo.fr Received 09.10.2008 Abstract The differential cross section of quasi-elastic neutrino scattering by nuclei is computed. The numerical analysis of energy dependence of the spin asymmetry, the angular electron-neutrino eν) correlation and the charge asymmetry coefficients shows that the relative contribution of the second-class current SCC) tensor form factor can reach some tens of percents in the case of particular values of final nuclei alignment and polarization. Key Words: correlation. Second-class currents, nuclei polarization, neutrino scattering, spin asymmetry and eν 1. Introduction The most general expressions for the matrix elements of vector currents V μ A μ between nuclei states p 1 and p 2 are given by the formulae [1] and axial - vector currents p 1 V μ p 2 =ūp 1 )[F 1 γ μ + F 2 σ μν q ν + if S q μ ] up 2 ) p 1 A μ p 2 =ūp 1 )[F A γ μ + F T σ μν q ν + if P q μ ] γ 5 up 2 ), 1) where q μ =p 1 p 2 ) μ is the 4-momentum transfer to the nucleus, up 1 ), up 2 ) are Dirac spinor amplitudes, F 1, F A are vector and axial-vector form factors, F 2,F S,F T and F p are respectively weak magnetic, effective scalar, induced and induced pseudo-scalar form factors. All these form factors depend on the square transfer momentumq 2 : 225
) F x q 2 μ = Fx 0)/ 1+qμ 2/855MeV)2) 2, ) F p q 2 μ =2MFA 0)/ ) qμ 2 + m 2 π where x =1, 2,A,T,S,m π and m π is the pion mass, F 1 0) = 1.0, F A 0) = 1.23,F 1 0) + 2MF 2 0) = μ0) = 4.706, F T 0) = 5 10 3 MeV 1. We assume the conserved-vector-current hypothesis CVC) in which case there are no induced scalar currents, and the form factor F S is zero. According to S. Weinberg classification [2], based on the G parity transformation interchanging particles with their anti-particles and rotating the system in isospin space around the T 2 axis), vector and axial-vector currents can be decomposed into first-class currents FCC) and second-class currents SCC). F 1,F 2,F A and F p are the form factors of the first-class currents and F S and F T are the form factors of the second class current. Issue of second-class currents and their existence are being widely discussed. Some contradictions between different experimental data do not allow a definite solution of this issue. For example, the experimental study of angular distribution of positrons in β decay process of polarized 19 Ne nucleus [3] and the study of angular distribution of electron and positron in β decay from polarized 12 B and 12 N [4, 5], show the existence of SCC whose tensor form factor is of the same order of magnitude as the form factor of weak magnetic interaction. Yet, Morita s data [6] show that the value of tensor form factor is next to zero within the limit of experimental errors, in agreement with the hypothesis of SCC shutting. An important development has arose in recent years, in which the experimental study of neutrino interactions with nuclei offers opportunity to investigate subtle detail in the structure of the standard model as well as that of the nucleon. Study of neutrino interaction can offer a solid understanding of cross sections in neutrino-induced reactions, particularly with nuclei such as 12 C, a component of liquid scintillators, and 16 O, the basic component of water Cerenkov detectors [7]. The present study deals with effects due to the possible existence of SCC in quasi-elastic neutrino and anti-neutrino) scattering by nuclei and the influence of polarization on the final nuclei on these effects. 2. Differential cross section of quasi-elastic neutrino anti-neutrino) scattering by nuclei Second class-currents are difficult to detect without any assumption [8], so we try to extract SCC effects by calculating the differential cross section of neutrino scattering by nuclei. In first order of perturbation theory, the process of neutrino anti-neutrino) scattering by nuclei described here can be written as ν ν)+a, Z) A, Z ± 1) + l l + ) 2) with matrix elements expressed as M fi = G F 2 l μ J μ, 3) where G F = 10 5 M 2 is Fermi coupling constant for weak interaction; M is the mass of the nucleus; l μ =ū 2 γ μ 1 + γ 5 )ū 1 and J μ = f dx exp iqx)ĵ μ x) i are, respectively, leptonic and hadronic currents; 226
u j j =1, 2) are Dirac spinor amplitudes; Ĵ μ x) is local current operator of nucleus; q μ =q, iq 0 ) P l P ν ) μ is the 4-momentum transfer to the nucleus; P l is the 4-momentum of electron for l = e and positron for l = e + ); P ν is the 4-momentum of neutrino anti-neutrino). The initial final) state of nucleus is determined by the parity π i π f ), the spin J i J f ), the isotopic spin T i T f ) and also by their projections M i,m Ti M f,m Tf ). Using multipole decomposition [1, 9, 10] in a cyclical basis, and with respect to the spin orientation of the final nuclei, we can put the components of the hadronic current into the following forms: J λ = J 0 = J 3 = J 1,M f J 0,M f J 0,M f i) J 4π2J +1)) 1/2 J f M f ˆMJ0 J i M i D Jf M f M θ,ϕ ), f i) J 4π2J +1)) 1/2 J f M f ˆ J0 J i M i D Jf M f M θ,ϕ ), 4) f i) J 2π2J +1)) 1/2 J f M f λĵj; λ M + ĴJ; λ E J i M i D Jf M f M θ,ϕ ),λ= ±1. f Here, D Jf M f M f θ,ϕ ) is the Wigner D-matrix [11] and J i J f J J i + J f, ˆMJ0, ˆJ0, Ĵ M J; λ and Ĵ E J; λ are the coulomb, longitudinal, magnetic and electric multipole operators [1, 9]; θ and ϕ are the angles used in determining the spin orientation of final nuclei [9]. The differential cross section of the process described in equation 2) is given by the following, by taking into account final nuclei spin orientation: dσ = 2J f +1 dω l 2J i +1 1 odd f f) E l P l G 2 F 2π 0 even f f) P cos θ ) v 1 W1 + v 2W2 + v 3W3 + v 4W4 + v 5W5 ) + [ P cos θ ) v 1 W 1 + v 2 W 2 + v 3 W 3 + v 4 W 4 + v 5 W 5 ) + P 1 cos θ )cosϕ v 6 W 6 + v 7 W 7 + v 8 W 8 + v 9 W 9 )]+ 2 even f f) [ P 1 cos θ ) cosϕ v 6 W6 + v 7 W7 + v 8 W8 + v 9 W9 )+ ] Pcos 2 θ )cos2ϕ v 10 W10 + 3 odd f f) P 2 cos θ )cos2ϕ v 10 W 10 5) 227
Here, 0 2J f and P m cos θ ) are agrange functions; f f) = M f 1) Jf Mf [] J f J f M f M f 0 ) P M f ) is a Fano tensor [9], where [x] = 2x +1 and P M f ) is the population of magnetic substates; v i i = 1, 2,, 10) are leptonic functions and Wk, W k k =1, 2,, 10) are hadronic functions. eptonic functions are defined by: v 1 = 1 2 l 1l 1 + l 2 l 2) v 2 = i 2 l 1l 2 l 2 l 1) v 3 = l 3 l 3 v 4 = 2Re l 3 l 0), v 5 = l 0 l 0 v 6 =2Re l 1 l 3) 6) v 7 = 2Iml 2 l 3 ) v 8 = 2Re l 1 l 0 ) v 9 =2Iml 2 l 0) v 10 = 1 2 l 1l 1 l 2 l 2), where l i i =0, 1, 2, 3) are the components of leptonic currents. Hadronic functions are given by: W 1 = J J A ) { 1;1 P + J +J FEJ F EJ + F MJ F MJ + FEJF 5 EJ 5 + F ) MJF 5 5 MJ P J +J FMJ F EJ + F EJ F MJ + F 5 EJF 5 MJ + F 5 MJF 5 EJ )}, W 2 = J J A ) { 1;1 P + J +J F 5 MJ F EJ + F EJ FMJ 5 + F MJF 5 EJ + F 5 EJ F ) MJ P J +J F 5 EJ F EJ + F 5 MJ F MJ + F MJF 5 MJ + F EJF 5 EJ )}, W 3 A ) 0,0 P + J +J FJ F J + FJF 5 J 5 ), W 4 A ) 0,0 P + ) J +J FJ F CJ + FJF 5 CJ 5, W 5 A ) 0,0 P + ) J +J FCJ F CJ + FCJF 5 CJ 5, W 6 A ) 1;0 P + J +J FEJ F J + FEJF 5 J 5 + F MJF 5 J + F ) MJF 5 J, W 7 A ) 1;0 P J +J FEJ FJ 5 + F EJF 5 J + FMJF 5 J 5 + F ) MJF J, W 8 A ) 1;0 P + J +J FEJ F CJ + FEJ 5 F CJ 5 + F MJF 5 CJ + F 5 MJ F ) CJ, 7) 228
W 9 A ) 1;0 P J +J FEJ FCJ 5 + F EJF 5 CJ + FMJF 5 CJ 5 + F ) MJF CJ, W 10 = J J A ) { 1;1 P + J +J FEJ F EJ F MJ F MJ + FEJF 5 EJ 5 F ) MJF 5 5 MJ + +P J +J FMJ F EJ FEJF 5 MJ + FMJF 5 EJ 5 F )} EJF MJ, W 3 = A ) 0;0 P J J J +J FJ FJ 5 + F ) JF 5 J, W 4 = A ) 0;0 P J J J +J FJ FCJ 5 + F 5 J F ) CJ, W 5 = A ) 0;0 P J J J +J FCJ FCJ 5 + F ) CJF 5 CJ, W 10 = A ) { 1;1 P + J +J F 5 MJ F EJ + F MJ FEJ 5 F EJ 5 F MJ F EJF 5 ) MJ + J J +P J +J F 5 EJ F EJ + F EJ FEJ 5 F MJF 5 MJ F )} MJF 5 MJ, W 1 = W 2, W 2 = W 1, W 6 = W 7, W 7 = W 6, W 8 = W 9, W 9 = W 8 Here, F CJ,F J,F MJ and F EJ FCJ 5,F5 J,F5 MJ and F EJ 5 ) are matrix elements of the vector axial-vector), coulomb, longitudinal, magnetic and electric multipole operators. The coefficients A ) m m and P ± J +J are defined by: ) 1/2 A ) M )! mm [J][J ][] = 1)Ji+Jf + M )! J J m m M ){ J J J f J f J i }, P + J +J = 1 2 1) 1 2 J J) 1+ 1) 1 2J +J) ),P J +J = 1 2 1) 1 2 J J+1) 1 1) 1 2J +J) ). The cross section of the process of neutrino anti-neutrino) scattering by nuclei 2), taking into account initial nuclei spin orientation, is obtained using formulas 5), 6) and 7) in which W k f) and f are replaced by W ki) and f i) defined as W ki) = 2J i +1 2J f +1 1)J +J+ W k, f i) = M f 1) Ji Mi [] J i J i M i M i 0 ) P M i ). After summing over the spin states of electron positron) and, in the case of zero mass neutrino, the leptonic functions are expressed as v 1 =1 β l C 3, v 2 = ηc 1 β l C 2 ), v 3 =1+2β l C 3 cos θ, v 4 = 2C 1 + β l C 2 ), 229
v 5 =1+β l cos θ v 6 =2Eν 2 βl 2 El 2 ) sin θ q 2, v 7 =2ηβ l E l E ν ) sin θ q 8) v 8 = 2β l E l + E ν ) sin θ q, v 9 =2ηβ l sin θ, v 10 = β le l E ν sin 2 θ q 2. Here, E l and E ν are electron positron) and neutrino anti-neutrino) energies; θ is angle between electron positron) and neutrino anti-neutrino) momenta; β l is electron positron) velocity; q = q is momentum transferred to nuclei ; η is +1 for neutrino scattering, 1 for anti-neutrino scattering; and C 1, C 2 and C 3 are coefficients given by the relations C 1 =β l E l cos θ E ν )/q, C 2 =β l E l E ν cos θ)/q, C 3 = C 1 C 2. 3. The differential scattering quasi-elastic cross section of 12 C Consider the process ν ν)+ 12 C 12 N 12 B)+e e + ), 9) for which J i =0 J f =1. Thequantanumbers, J and J are defined then by 0 2J f 0 2, J i J f J J i + J f J =1, J i J f J J i + J f J =1. The differential cross section of the process described in equation 9), obtained from 5), is given by { dσ dω l = E2 l G2 F 2π R 0 0 + A P 2 cos θ )R 0 2 + P 2 1cos θ )cosϕ R1 2 )} +P2 2 cos θ )cos2ϕ R2) 2 + P P1 cos θ )R 0 1 + P1 1 cos θ )cosϕ R 1 1 10) A and P are alignment and polarization coefficients of the final nuclei [12]. Functions R n m are given by the relations R 0 0 = v 1H 1 + v 2 H 2 + v 3 H 3 + v 4 H 4 + v 5 H 5, R 0 2 = 1 2 [v 1H 1 + v 2 H 2 2v 3 H 3 + v 4 H 4 + v 5 H 5 )] R 1 2 = 1 2 6 v 6H 6 + v 8 H 8 ), R 2 2 = 1 4 v 10H 10, 11) R 0 1 = 3 2 v 1H 2 + v 2 H 1 ), R 1 1 = 3 2 6 v 7H 6 + v 9 H 8 ), where H 1 =F M1 ) 2 +F 5 E1) 2, H 2 =2F M1 F 5 E1, H 3 =F 5 1) 2, H 4 = F 5 1F 5 C1, H 5 =F 5 C1) 2, H 6 = F 5 1F 5 E1 + F M1 ), H 8 = F 5 C1 F 5 E1 + F M1), H 10 =F 5 E1 )2 F M1 ) 2. 230
The matrix elements of the vector magnetic, axial-vector electric, coulomb and longitudinal multipole operators computed in the shell model are F M1 = ψ 3 q π 2M F 1 F 1 +2MF 2 )2 y))e y, F 5 E1 = ψ 3 π F A2 y)e y, [ ] 2ψ FC1 5 = 3 q 3 π 2M 2 F A +W 0 F P +2ηMF T )1 y) e y, 12) 2ψ F1 5 = 3 π F A q2 2M F P )1 y)e y, where y =bq/2) 2, b = 1.77 fm is the oscillator parameter, and ψ = 0.003 [1]. 4. Spin asymmetry coefficient Consider the spin asymmetry coefficients defined as A ν E ν,θ)= dσ dσ SN P ) ν S N P ν ) + dσ dσ SN P ) ν S N P ν ), 13) where S N is the final nuclei spin and P ν is the neutrino momentum. Analysis of angular dependence of spin asymmetry shows, for given values of P and A, that the maximum F T contribution calculated due to ΔA ν is moving opposite of increasing neutrino energy see Figures 1a, b)). This maximum also depends on the angle θ for given values of P and A see Table). ΔAν 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 P=1 A=1 E=500MeV E=600MeV -0.1 0 20 40 60 80 100 120 140 160 180 Angle degree) a) ΔAν 0.6 0.5 0.4 0.3 0.2 0.1 0 E=200MeV E=400MeV P=0.7A=0.3-0.1 0 20 40 60 80 100 120 140 160 180 Angle degree) b) Figure 1. Contribution of SCC to the asymmetry coefficient for different value of the neutrino energy as a function of angle. 231
Table. Contribution of SCC to spin asymmetry coefficients, with ΔA ν = A νf T =5 10 3 MeV ) A νf T =0). P =0.7,A =0.3 P =1,A =1 E ν MeV) 200 200 400 400 500 500 600 600 θdegree) 47 163 54 80 41.5 66 34 62 ΔA ν 0.07 0.53 0.27 0.54 0.26 0.73 0.21 0.72 In Figure 2 we show the energy dependence of the contribution D to the SCC, defined as D = A vf T =0) A v F T =5 10 3 MeV 1 ), 14) A v F T =0) relative the asymmetry coefficient for θ =60. It seems that the SCC relative contribution is less than 16% for neutrino energy below 300 MeV. This relative contribution can take value in the range of 75% to 92% when neutrino energy is more than 400 MeV. Figure 2. Relative contribution of SCC to the asymmetry coefficient for θ =60. 5. Electron-neutrino correlation and charge asymmetry coefficients The eν correlation coefficient is defined by the formula A eν = dσθ 0) dσθ π) dσθ 0) + dσθ π). 15) When the final nucleus is oriented in the direction of neutrino and in the ultra relativistic case β l =1),the coefficient A eν takes the form with A eν = D 1 D 2 D 1 + D 2, 16) D 1 =21 A)1 y 1 ) 2 E ν E l )F T + F A ) 2 exp 2y 1 ), 232
D 2 =1+ 1 2 A + 3 2 ηp)2 y 2) 2 E ν + E l )F 2 ηf A ) 2 exp 2y 2 ), y 1,2 =be ν E l )/2) 2. In the maximum polarization case A = P =1), D 1 =0,A eν = 1 andeν correlation do not depend on SCC form factor F T. However, in the case of partial polarization of the final nucleus, the relative contribution of SCC depends on the value of the alignment coefficient A. For example, when A= 0.1andP = 0.5 Figure 3), this value reaches between 1% to 17% in the 80 120 MeV neutrino energy range. When energies E ν are above 200 MeV the coefficient A eν = +1 and it is no more sensitive to SCC form factor FT variation. The charge asymmetry coefficient is defined by the formula B = dσ ν dσ ν dσ ν + dσ ν, 17) where dσ ν dσ ν ) is the differential cross section for neutrino anti-neutrino) scattering. B is determined with respect to the parameter η, which is equal to +1 for neutrino scattering and -1 for anti-neutrino scattering. The curves see Figure 4) for which the neutrino energy is in the range of 350 to 423 MeV show that the charge asymmetry coefficient is negative and takes values between 2.6% to 2.5% when F T =0. Butit becomes positive with value between 30% and 60% when F T =5 10 3 MeV 1. So, in this neutrino energy area, the coefficient B presents only pure SCC effects. Figure 3. Relative contribution of SCC to the eν correlation coefficient. Figure 4. Charge asymmetry coefficient for P = 0.7, A =0.3andθ =60 as a function of energy. 6. Conclusion Theoretical analysis of different characteristics possessed by processes of quasi-elastic neutrino scattering by nuclei has shown that the relative contribution of SCC to spin asymmetry, eν correlation and charge 233
asymmetry coefficients for F T =5 10 3 MeV 1, can reach some tens of percents for particular values of alignment A and polarization P of final initial) nucleus and that of neutrino antineutrino) energy. Therefore, the experimental study of quasi-elastic neutrino antineutrino) scattering processes can allow more accurate expression of the SCC tensor form factor when the nucleus polarization is taken into account. Acknowledgment We would like to thank Dr N. A. Smirnova from the Department of Subatomic and Radiation Physics, University of Ghent, for useful advice and discussions. References [1] T. W. Donnelly, R. D. Peccei, Phys. Rep., 50, 1975), 1. [2] S. Weinberg, Phys. Rev., 112, 1958), 1375. [3] F. P. Calaprice, S. J. Freedman, W. C. Mead and H. C. Vantine, Phys. Rev. ett., 35, 1975), 1566. [4] K. Sugimoto, I. Tanihata and J. Göring, Phys. Rev. ett., 34, 1975), 1533. [5] T. Minamisono et al. Phys. Rev. ett., 80, 1998), 4132. [6] M. Morita, Hyperfine Interaction, 21, 1985), 143. [7] P. Vogel, arxiv: nucl-th/9901027, v1, 12 Jan 1999 [8] B. R. Holstein, www.physics.sc.edu/neutrino/workshop/holstein.pdf, April 2004 [9] T. W. Donnelly, A.S. Raskin, Ann. Phys., 169, 1986), 247. [10] I. Aïzenberg, V. Graïner, Mehanismi vozbujdenia yadr,vol. 2 Atomizdat, Moskva, 1973), p.44 [in Russian]. [11] A. R. Edmonds, Angular momentum in quantum mechanics, Princeton University Press, Princeton, New Jersey, 1957), p.53 [12] E. D. Commins, P. H. Bucksbaum, Weak interactions of leptons and quarks, Energoatomizdat, Moskva, 1987) p. 189 [in Russian]. 234