The process of the neutrino production, oscillation in the vacuum or in matter, and detection in the case of interactions which are beyond the Standard Model is considered. Neutrino states are described by the density
matrix. The final neutrino production rate does not factorize. The known Maki–Nakagawa–Sakata neutrino states and the factorized production rate are recovered in the νSM regime

Syska, Jacek

Zając, S.

Zrałek, Marek

Acta Physica Polonica B

English

Repozytorium Uniwersytetu Śląskiego RE-BUŚ

             Title: Neutrino oscillations in the case of general interaction  Author: Jacek Syska, S. Zając, Marek Zrałek  Citation style: Syska Jacek, Zając S., Zrałek Marek. (2007). Neutrino oscillations in the case of general interaction. "Acta Physica Polonica B" (Vol. 38, no. 11 (2007), s. 3365-3371).   Vol. 38 (2007) ACTA PHYSICA POLONICA B No 11NEUTRINO OSCILLATIONS IN THE CASEOF GENERAL INTERACTION∗J. Syska†, S. Zając‡, M. Zrałek§Department of Field Theory and Particle Physics, Institute of PhysicsUniversity of Silesia, ul. Bankowa 14, 40-007 Katowice, Poland(Received October 24, 2007)The process of the neutrino production, oscillation in the vacuum orin matter, and detection in the case of interactions which are beyond theStandard Model is considered. Neutrino states are described by the densitymatrix. The final neutrino production rate does not factorize. The knownMaki–Nakagawa–Sakata neutrino states and the factorized production rateare recovered in the νSM regime.PACS numbers: 14.60.St, 14.60.Pq, 25.30.Pt1. Introduction: Lagrangian of the extension of the νSMThe aim of this short presentation is to check how neutrino production,oscillation and detection processes are modified if a new physics (NP) de-scribes neutrino interaction. Let us begin with the process of the production(P ) of the massive neutrino νi (i = 1, 2, 3) accompanied by the lepton lα(α = e, µ, τ):lα + P1 → νi + P2 , (1)followed, after traveling along the baseline L, by its detection (D):νi +D1 → lβ +D2 , (2)where P1, P2 and D1, D2 are the accompanied particles.Below we specify the NP extension of the interaction Lagrangian. Itscharged current (CC) part describes the neutrino production and detectionevents, whereas the neutral current (NC) part modifies the neutrino inter-action in a medium.∗ Presented at the XXXI International Conference of Theoretical Physics, “Matter tothe Deepest”, Ustroń, Poland, September 5–11, 2007.†jacek@server.phys.us.edu.pl‡sebkaz@poczta.onet.pl§zralek@us.edu.pl(3365)3366 J. Syska, S. Zając, M. Zrałek1.1. Charged and neutral current Lagrangian for NPThe CC Lagrangian includes the νSM and the new LH and RH currents:LCC = −e2√2 sin θW{∑α,iν¯i [ γµ(1− γ5)εLUL∗αi + γµ(1 + γ5)εRUR∗αi ] lαW+µ+∑α,iν¯i [ (1− γ5)ηLV L∗αi + (1 + γ5)ηRV R∗αi ] lα H++∑u,du¯ [ γµ(1− γ5)ǫqLU∗ud + γµ(1 + γ5)ǫqRU∗ud] d W+µ+∑u,du¯ [ (1− γ5)τLWL∗ud + (1 + γ5)τRWR∗ud ] d H+}+ h.c. , (3)where UL is the A−V νSM unitary neutrino mixing matrix [1]. The UR andV L,R are the NP neutrino mixing matrices in the CC sector, the A+V and(pseudo)scalar (P, S) one, respectively. The coupling constants εL, ǫqL aretaken to be the global factors deviating slightly from their νSM values. Forrelativistic neutrinos the appropriate NC Lagrangian is:LNC = −e4 sin θW cos θW{∑i,jν¯i[γµ(1− γ5)εNνL δij + γµ(1 + γ5)εNνR ΩRij]νj+∑f=e,u,df¯[γµ(1− γ5)εNfL + γµ(1 + γ5)εNfR]f}Zµ . (4)The νSM mixing matrix is diagonal, whereas ΩRij depends on the NP pa-rameters [2]. The couplings εNνL , εNfL and εNfR are the global factors veryclose to their νSM values. The NP factors in LCC and LNC are the smalldeviations from zero.From LCC and LNC we could construct [3] the general low energy four-fermion effective interaction Hamiltonian H which describes the coherentneutrino scattering inside matter. As we consider three massive neutrinosin two possible helicity states (λ = ±1) hence the NP effective HamiltonianHλ,i; η,k in the mass — helicity base |λ, i〉 is the 6× 6 dimensional matrix:H =M+( H−− H−+H+− H++), (5)where the mass term is equal to:M=diag(E01 , E02 , E03 , E01 , E02 , E03 ) with E0i =Eν+m2i2Eν, i=1, 2, 3 . (6)Neutrino Oscillations in the Case of General Interaction 3367Here Eν is the energy for the massless neutrino [4]. The antidiagonal termsin Eq. (5) originate in the tensorial T interaction which appears as a resultof the Fiertz rearrangement applied to the scalar part of LCC. The diagonalterms are connected with the A±V interactions, both the νSM and NP ones.The more detailed description of the νSM and NP decomposition of H couldbe found in [5] and its full analysis is postpone to the forthcoming paper.2. Implementation of the density matrix formalismThe density matrix approach to the neutrino oscillation phenomenonis analyzed e.g. in [6, 7]. Such formalism is here more than merely theelegant form of the analysis. Indeed, for relativistic neutrinos produced anddetected by the left-handed (LH) mechanism of the νSM, the oscillationrates factorize ( [8], see also below) and the analysis is based on the QMevolution of the pure neutrino state |~p, λ, i > of momentum ~p, helicity λ andmass i [8]. Yet, if the NP of the right-handed (RH), scalar, or pseudoscalarinteractions is involved, then the states are mixed and the density matrixformalism becomes the necessity. Below we present the way it works.The neutrino statistical operator of the production process (1) writtenin the mass-helicity basis is equal to [7]:̺αP (~p, L = 0) =∑λ,λ′=±13∑i,i′=1|~p, λ, i〉 ̺αP (~p;λ, i;λ′, i′) 〈~p, λ′, i′| , (7)undergoing in the homogeneous medium the following quantum evolution(with L = T ):ραP (~x=~0, t=0)→ ραP (~x= ~L, t=T )=e−i(HT )ραP (~x=~0, t=0) ei(H T ) , (8)where H is given by Eq. (5). Suppose that in the production process (1) themomentum of the initial lepton lα lies along the z axis. In the case of thelack of the polarization in the process (1), the reduced density matrix forthe neutrino in the production place has the following form:̺αP (λ, i;λ′, i′) =1Nα∑λP2λP1λαAαiλ;λP2λP1 ,λα(~p)(Aαi′λ′;λP2λP1 ,λα(~p))∗, (9)where Aαiλ;λP2λP1 ,λα(~p) are the amplitudes for the production process and topreserve Tr(̺αP ) = 1 the normalization constant Nα has to be equal to:Nα =∑λ=±13∑i=1∑λP2λP1λαAαiλ;λP2λP1 ,λα(~p)(Aαiλ;λP2λP1 ,λα(~p))∗. (10)Here λ and λP2 , λP1 , λα are the helicities for the neutrino and the otherparticles.3368 J. Syska, S. Zając, M. ZrałekAs we are interested in the evolution of the density matrix in the LABframe hence the density matrix ̺αP should be calculated in this frame also,which is more arduous, than in the CM one. However, for the large oscillationbaseline L, only neutrinos which are produced in forward direction in theCM frame will reach the detector. Then for the relativistic limit, it couldbe proved [7] that the Wigner rotation is negligible, that means̺LABP (~pLAB) = ̺CMP (~pCM → ~pLAB) . (11)Therefore we perform the calculation of ̺αP in the CM frame and (for theLagrangian (3) and (4)) from Eq. (9) we obtain the 3×3 density submatrices:̺αP (−1, i; −1, i′) =1Nα[Aε2Lε2LUL∗αi ULαi′ +Aη2Rη2RVR∗αi VRαi′+AεL ηR εLηR (UL∗αi VRαi′ + VR∗αi ULαi′)], (12)̺αP (+1, i; +1, i′) =1Nα[Aε2Rε2RUR∗αi URαi′ +Aη2Lη2LVL∗αi VLαi′+AεRηL εRηL (UR∗αi VLαi′ + VL∗αi URαi′)], (13)and ̺αP (∓1, i; ±1, i′) = 0. The normalization constant Nα is equal toNα = Aε2Rε2R +Aε2Lε2L +Aη2Lη2L +Aη2Rη2R+AεRηLεRηL3∑i=1(UR∗αi VLαi + VL∗αi URαi)+AεL ηR εLηR3∑i=1(UL∗αi VRαi + VR∗αi ULαi). (14)The A-coefficients are the functions of the energies and momenta of the par-ticles in the production process (1) of the neutrino. E.g. Aε2Land Aε2Rarethe amplitudes for the CC processes, A−V in SM and A+V in NP, respec-tively. Similarly, Aη2Land Aη2Rare the amplitudes for the CC processes,S−P and S+P, respectively. Next, AεLηR and AεRηL are the functions ofthe amplitudes that mix different helicities of the neutrino. Finally, fromEqs. (12)–(14) it could be easily noticed that for the νSM interaction thestandard setup for the neutrino, i.e.:̺αP (−1, i; −1, i′) = UL∗αi ULαi′ and ̺αP (+1, i; +1, i′) = 0 , (15)is regained.Neutrino Oscillations in the Case of General Interaction 3369Now, let us discuss the detection process (2) of the neutrino after itpropagates to T = L 6= 0. If the particles D1 are not polarized and the finalparticles polarizations are not measured, the differential cross section for theflavor β neutrino detection in the LAB frame is given by:dσβ αdΩβ=fD∑λ,i;λ′,i′λD1,λD2,λβAβiλ,λD1λβ ,λD2(~pβ) ̺αP (λ, i;λ′, i′;L=T ) (Aβi′λ′,λD1λβ ,λD2(~pβ) )∗ , (16)where ̺αP (λ, i;λ′, i′;L = T 6= 0) is the density matrix of the produced neu-trino after it evolves to the detection point and λ, λD1 , λβ, λD2 are the par-ticles helicities. fD is the kinematical factor:fD =164π2 (2 sD1 + 1) Eν mD1p3β(Eν +mD1) p2β − Eβ (~p · ~pβ). (17)Here mD1 and sD1 are the mass and spin of the detector particle. The neu-trino momentum and energy are noted as ~p and Eν , respectively. Similarly~pβ and Eβ are the momentum and energy of the β lepton. The amplitudesAβiλ=±,λD1λβ ,λD2(~pβ) describe the detection process (2) and are calculated in therest frame of the detector (LAB). The formula (16) could be, after summingover all helicities of the particles, rewritten as follows:dσβ αdΩβ= fD∑i; i′[a−−β;ii′ ̺αP (−1, i; −1, i′;L) + 2 cosϕRe(a+−β;ii′ ̺αP (1, i;−1, i′;L))−2 sinϕ Im(a+−β;ii′ ̺αP (1, i;−1, i′;L)) + a++β;ii′ ̺αP (1, i; 1, i′;L)]. (18)The a-coefficients (e.g. a−−β;ii′) are the functions of the energies and momentaof the particles in the detection process. They depend on the polar angleθ of the momentum of the β lepton but the dependence on the azimuthalangle ϕ is openly factorized out. Further investigations of the shape of thedifferential cross sections enables us to write the a-coefficients (see Eqs.(3)and (4)) as follows:a−−β;ii′ =ALεε|εL|2UL∗βi ULβi′+ALεη(ηLε∗LV LβiUL∗βi′+ h.c.)+ALηη |ηL|2V LβiV L∗βi′ ,a++β;ii′ =ARεε|εR|2UR∗βi URβi′+ARεη(ηRε∗RV RβiUR∗βi′ +h.c.)+ARηη |ηR|2V RβiV R∗βi′ ,a+−β;ii′ =ARLεη (η∗LεRURβiVL∗βi′−ε∗LηRV RβiUL∗βi′ )+ARLεε ε∗LεRURβiUL∗βi′ =(a−+β;i′i)∗ . (19)3370 J. Syska, S. Zając, M. Zrałek3. Density matrix evolution in a medium. The νSM regimeIf the inclusion of the NP parameters is important it should be at firstseen in the cumulative phenomenon of the long propagation in matter. Let(Wi λ; a) ≡ (〈λ, i|a〉) be the unitary transition matrix from the |λ, i〉 helicity-mass eigenbasis to |a〉 energy eigenbasis of the effective Hamiltonian (5), andEa are the eigenvalues of H. As the general NP structure of the effectiveHamiltonian is non-diagonal hence (Wi λ; a) is not diagonal too, and it isclear that the evolution Eq. (8) mixes all entries of the original neutrinodensity matrix (9). Hence the evolution of the density matrix ̺αP :̺αP (λ, i;λ′, i′;L = T 6= 0) =∑nσ∑n′σ′∑a,bWiλ;aW∗nσ;a×̺αP (σ, n;σ′, n′;L = T = 0)ei(Eb−Ea)T Wn′σ′;bW ∗i′λ′;b , (20)is in this respect different in medium and vacuum, where Wi λ; a = δλ,i; a.Clearly in vacuum, Eq. (20) has the form:̺αP (λ, i;λ′, i′;L = T 6= 0) = ̺αP (λ, i;λ′, i′;L = 0)ei(Eλ,i−Eλ′,i′ )L , (21)that does not mix ̺αP entries. Hence for νSM in vacuum, Eq. (18) (after usingEq. (15)) reduces to the νSM, factorized, differential cross section formula:dσβ αdΩβ= fDALεε∑i; i′ULβiUL∗βi′UL∗αi ULαi′eim2i−m2i′2EνL≡ dσβdΩβ(mi = 0)× Pα→β(L) , (22)and we recover the widely used factorization formula for the neutrino pro-duction rate. However, in the general case the full expression (18) for thefinal cross section does not factorize and it should be used in the future,more precise neutrino experiments, anywhere NP effects might be studied.4. Summary and perspectives for the NP correctionsand the νSM medium termsIt has been shown that for the NP Lagrangian (3) and (4), the neutrinoproduction and detection states are in general mixed. We have presentedthe density matrix formalism which enables us to tackle with this sort ofthe NP phenomena. It has been shown that only for relativistic neutrinoproduced and detected by the LH mechanism the oscillation rate factorizes.We indicated that analyzed NP corrections may have the V±A or S±P ori-gin and that the later one may lead to the tensorial spin-flip interactions.Neutrino Oscillations in the Case of General Interaction 3371The phenomenon is well known [3] although its significance has been mainlyignored as it is effective in the presence of the nonzero mean magnetizationof the medium. Yet recent, systematically collected data on the magneticfield of the earth seem to support the view that part of it is connected withthe correlated magnetized structures in the earth’s crust [9]. If so, the spin-flip phenomenon induced by the possible existence of the S−P terms of theproposed NP should not be ignored. Searching on this phenomenon pushedus towards the necessity of the recalculation of the crust magnetization im-pact in the νSM regime also, as it appears in its V±A terms in the CC andNC interactions. The introductory calculations suggest that the effect canbe meaningful [5], depending on the value of the mean magnetization. Theproblem is postpone to the forthcoming paper.We tank for the discussions with F. del Aguila. This work has beensupported by the Polish Ministry of Science under Grant 1 P03 B 049 26,by MEC (FPA2006-05294) and by the European Community’s Marie CurieResearch Training Network under contract MRTN-CT-2006-035505 “Toolsand Precision Calculations for Physics Discoveries at Colliders”.REFERENCES[1] Z. Maki, M. Nakagawa, S. Sakata, Prog. Theor. Phys. 28, 870 (1962); B. Pon-tecorvo, Sov. Phys. JETP 26, 984 (1968).[2] F. del Aguila, J. Syska, M. Zrałek, Phys. Rev. D76, 013007 (2007)[hep-ph/0702182].[3] S. Bergmann, Y. Grossman, E. Nardi, Phys. Rev. D60, 093008 (1999).[4] C. Giunti, C.W. Kim, Found. Phys. Lett. 14, 213 (2001) [hep-ph/0011074].[5] J. Syska, S. Zając, M. Zrałek, Neutrino Oscillations with Density Matrix andMagnetization of the Crust, talk given at the XXXI International Confer-ence of Theoretical Physics, “Matter to the Deepest”, Ustroń, Poland, 2007,http://prac.us.edu.pl/∼us2007/[6] G.G. Raffelt, Stars as Laboratories for Fundamental Physics, The Universityof Chicago Press, 1996, pp. 313–331.[7] M. Zrałek, PHENO 07 conference, (2007); M. Ochman, R. Szafron, M. Zrałek,hep-ph:0707.4089.[8] C. Giunti, C.W. Kim, J.A. Lee, U.W. Lee, Phys. Rev. D48, 4310 (1993);J. Rich, Phys. Rev. D48, 4318 (1993); W. Grimus, P. Stockinger, Phys. Rev.D54, 3414 (1996); R.N. Mohapatra, P.B. Pal, Massive Neutrinos in Physicsand Astrophysics, third ed., World Scientific, Singapore 2004 and citationtherein.[9] J.C. Cain, B. Holter, D. Sandee, J. Geomag. Geoelectr. 42, 973 (1990);C. Voorhies, NASA/TP-1998-208608; M. Purucer, R. Langel, M. 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Neutrino oscillations in the case of general interaction

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Neutrino oscillations in the case of general interaction

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Repozytorium Uniwersytetu Śląskiego RE-BUŚ