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January 6, 2012 1:16 WSPC/INSTRUCTION FILE art10 International JournalofModernPhysicsE (cid:13)c WorldScientificPublishingCompany 2 1 0 MAJORANA NEUTRINO MAGNETIC MOMENTS 2 n a J MarekGó´zd´zandWiesl awA.Kamin´ski 5 Department of Theoretical Physics, Maria Curie-Sk lodowska University, Lublin, Poland [email protected] ] [email protected] h p FedorSˇimkovic - p Department of Nuclear Physics, Comenius University,Bratislava, Slovakia e [email protected] h [ Received(xxOctober 2005) 1 v ThepresenceoftrilinearR-parityviolatinginteractionsintheMSSMlagrangianleadsto 7 existenceofquark–squarkandlepton–sleptonloopswhichgeneratemassoftheneutrino. 3 Byintroducing interaction with an external photon the magnetic moment is obtained. 2 Wederiveboundsonthatquantitybeingaroundoneorderofmagnitudestrongerthan 1 thosepresentintheliterature. . 1 Thanks to more than 40 years of intensive work oscillations of neutrinos are 0 treated as a well established experimental fact. Up to our best knowledge this 2 1 situation implies that neutrinos are massive particles and therefore the Standard : Model (SM) of elementary particles and interactions should be extended. All such v i attempts to go beyond SM are called non-standard physics and include supersym- X metry (SUSY), theories of grand unification (GUT), extra dimensions and others. r a The problem of generation of very small neutrino masses has recently received a great deal of attention. There is, among others, a mechanism that relies on the R-parity violation in supersymmetric Standard Model. The R-parity is defined as R = +1 for ordinary particles and R = −1 for their supersymmetric partners. R-parity conservation makes life easier because SUSY particles cannot decay into ordinary particles and vice-versa. Theoretically, however, nothing motivates such behavior, therefore many models consider violation of R-parity. This implies, in turn, non-conservation of lepton and baryon numbers, which opens the possibility for exotic nuclear processes to appear. We will use the minimal supersymmetric standard model (MSSM), with su- 1 persymmetry broken by supergravity (SUGRA). The model is fully described by the following superpotential and lagrangian. The R-parity conserving part of the superpotential has the form WMSSM =ǫ [(Y ) LaHbE¯ +(Y ) QaxHbD¯ ab E ij i 1 j D ij i 1 jx +(Y ) QaxHbU¯ +µHaHb], (1) U ij i 2 jx 1 2 1 January 6, 2012 1:16 WSPC/INSTRUCTION FILE art10 2 M. Gó´zd´z, F. Sˇimkovic, and W. A. Kamin´ski while its R-parity violating part reads 1 WRpV =ǫ λ LaLbE¯ +λ′ LaQxbD¯ ab(cid:20)2 ijk i j k ijk i j kx(cid:21) 1 + ǫ λ′′ U¯xD¯yD¯z +ǫ κiLaHb. (2) 2 xyz ijk i j k ab i 2 HereY’sare3×3Yukawamatrices,LandQstandforleptonandquarkleft-handed SU(2) doublet superfields while E¯, U¯ and D¯ denote the right-handed lepton, up- quark and down-quark SU(2) singlet superfields, respectively. H1 and H2 mean two Higgs doublet superfields. We have introduced color indices x,y,z = 1,2,3, generation indices i,j,k=1,2,3 and the SU(2) spinor indices a,b,c=1,2. TheintroductionofR-parityviolationimplies theexistence ofleptonorbaryon numberviolatingprocesses,liketheunobservedprotondecayandneutrinolessdou- ble beta decay (0ν2β). Fortunatelly one may keep only one type of terms and it is not necessary to have both non-zero at one time. In order to get rid of too rapid proton decay and to allow for lepton number violating processesit is customary to set λ′′ =0. We present here for completeness the mass term for scalar particles: Lmass = m2H1h†1h1+m2H2h†2h2+q†m2Qq+l†m2Ll + um2u†+dm2 d†+em2e†, (3) U D E soft gauginos mass term (α=1,...,8 for gluinos): Lgaug. = 1 M1B˜†B˜+M2W˜i†W˜i+M3g˜α†g˜α+h.c. , (4) 2 (cid:16) (cid:17) aswellastheSUGRAmechanism,byintroducingthesoftsupersymmetrybreaking Lagrangian Lsoft =ǫ [(A ) lahbe¯ +(A ) qaxhbd¯ ab E ij i 1 j D ij i 1 jx +(AU)ijqiaxhb2u¯jx+Bµha1hb2+B2ǫiliahb2], (5) wherelowercaselettersstandforscalarcomponentsofrespectivechiralsuperfields, and3×3matricesAaswellasBµandB2 arethesoftbreakingcouplingconstants. d d d d jL jR kR kL νiL νi′L νiL νi′L λ′ λ′ λ′ λ′ ijk i′kj ijk i′kj d˜kR d˜kL d˜jL d˜jR Fig. 1. Feynman diagrams leading to Majorana neutrino masses. By attaching photon to the internallineofthelooponegets theneutrinomagneticmoment. January 6, 2012 1:16 WSPC/INSTRUCTION FILE art10 Majorana neutrino magnetic moment 3 The loops showed in Fig. 1 lead to Majorana neutrino mass term.2,3 Detailed calculation gives for the quark–squark loop the following contribution: 3 sin(2θk) Miqi′ = 16π2 λ′ijkλ′i′kj(cid:20)mqj 2 f(xj2k,xj1k)+(j ↔k)(cid:21), (6) Xjk where f(a,b)=log(a)/(1−a)−log(b)/(1−b), m is j-th generation down quark dj mass,θk isthesquarkmixinganglebetweenthek-thsquarkmasseigenstatesM , q˜1k,2 and xjk =m2 /M2 . 1,2 dj q˜1k,2 A similar contribution comes from loops containing lepton–slepton pairs. It reads: 1 sin(2φk) Miℓi′ = 16π2 λijkλi′kj(cid:20)mej 2 f(y2jk,y1jk)+(j ↔k)(cid:21), (7) Xjk where allthe quantities are defined in complete analogywith the previous case,by replacingsquarkswith sfermionsandquarkswithfermions.The factor3 inEq.(6) comes fromsummation over three colorsof quarks and therefore it is absentin the case of fermions. Theleft-handsidesofEqs.(6)and(7)maybeobtainedbyconsideringneutrino oscillations phenomenology or by using data from neutrinoless double beta decay experiments. The half-life of this exotic process depends on the effective neutrino mass hm i = 3 |U |2m , where U is the neutrino mixing matrix and m are ν i=1 ei i i neutrinomassPeigenstates.Thereisnofirmexperimentalevidenceforobservationof the0ν2β process.TheconservativeconstraintcomingfromtheHeidelberg–Moscow experimentisfornowT0ν ≥1.9×1025years.Byassumingnuclearmatrixelements 1/2 4 of Ref. this value translates into hm i ≤ 0.55 eV, which in turn implies the ν neutrino mass matrix in the form 0.550.780.66 |MHM|≤0.780.560.69eV. (8) 0.660.690.81   The entries of MHM may be used as the left-hand side of Eqs. (6) and (7), con- straining non-standard physics parameters. By attaching a photon to the internal line of the loop one has an effective neutrino–neutrino–photon vertex, which allows to calculate the neutrino magnetic moment. In the case of Majorana neutrino, the CPT theorem allows only for the transitional magnetic moment between two neutrinos of different flavors. It is described by the effective hamiltonian Heff = µii′ν¯iL(σab/2)νic′RFba, F being the electromagnetic tensor. After evaluating the loop amplitude one gets the following expression for the magnetic moment: 3 2sin(2θk) µqii′ =(1−δii′)16π2me1Qd λ′ijkλ′i′kj(cid:12) m g(xj2k,xj1k)−(j ↔k)(cid:12)µB, (9) Xjk (cid:12) qj (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) January 6, 2012 1:16 WSPC/INSTRUCTION FILE art10 4 M. Gó´zd´z, F. Sˇimkovic, and W. A. Kamin´ski me1 being the electron mass, g(a,b)=(alog(a)−a+1)/(1−a)2−(blog(b)−b+ 1)/(1−b)2,Q =1/3isthedown-quarkelectriccharge,andµ theBohrmagneton. d B A similar expression may be obtained for the lepton–slepton contribution. We have calculated the low energy MSSM spectrum by starting from certain unification conditions at the GUT scale ∼ 1016 GeV and evolving the running masses and coupling constants down to the m scale using renormalization group Z equations (RGE). The so-obtained spectrum is tested against various constraints on SUSY masses, proper electroweak symmetry breaking, FCNC phenomenology 3 and others. The procedure has been described in detail elsewhere. We use the so-called conservative approach, assuming that only one mechanism dominates at atime.Itmeansthatweperformcalculationsforsomegivenj,k withoutdoingthe summation. We pick up the highest obtained value, which constitutes the “most optimistic” result: µq ≤4.0×10−17µ , µℓ ≤1.6×10−16µ , eµ B eµ B µq ≤3.4×10−17µ , µℓ ≤1.4×10−16µ , (10) eτ B eτ B µq ≤3.6×10−17µ , µℓ ≤1.4×10−16µ . µτ B µτ B These values were obtained for the following GUT conditions: A0 = 100 GeV, m0 = m1/2 = 150 GeV, tan(β) = 19, µ > 0, where A0 denotes the common value of soft breaking couplings, m0 and m1/2 the common scalar and fermion masses, andtan(β)istheratioofHiggsvacuumexpectationvalues.Bytaking,forexample, A0 ∼ 1000 GeV, the results become smaller by at least one order of magnitude. We would like to mention at this point that our bounds are around one order 2 of magnitude stronger than those published earlier. It is due to the more exact procedure (GUT and RGE) and partially due to inclusion of sparticle mixing. It is also worth mentioning that the present experimental limits are µii′ ≤10−10µB. In further discussion one should analyze the influence of quark mixing on µ , ν whichmaygiveimportantcorrections.Thisdiscussionis,however,beyondthescope of the present work and will be postponed to a separate regular paper. Acknowledgements This work was supported by the VEGA Grant agency of the Slovak Republic under contract No. 1/0249/03, by the EU ILIAS project undercontractRII3-CT-2004-506222,andthePolishStateCommitteeforScientific Research under grants no. 2P03B 071 25 and 1P03B 098 27. References 1. see eg. H.E. Haber and G. L.Kane, Phys. Rep. 17, 75 (1985). 2. O.Haug,J.D.Vergados,A.Faessler,S.Kovalenko,Nucl.Phys.B 565,38(2000);G. Bhattacharyya, H. V. Klapdor-Kleingrothaus, H.Pa¨s, Phys. Lett. B 463, 77 (1999); A. Abada, M. Losada, Phys. Lett. B 492, 310 (2000); A. Abada, M. Losada, Nucl. Phys. B 585, 45 (2000); 3. M.Gó´zd´z,W.A.Kamin´ski,F.Sˇimkovic,Phys.Rev.D 70,095005(2004);M.Gó´zd´z, W. A.Kamin´ski, F. Sˇimkovic,A. Faessler, to appear in Phys. Rev. D. 4. V. A.Rodin, A.Faessler, F. Sˇimkovic,P. Vogel, Phys. Rev. C 68, 044302 (2003).