Table Of ContentConstraining Bilinear R-Parity Violation from Neutrino Masses
Marek G´o´zd´z∗ and Wiesl aw A. Kamin´ski†
Department of Informatics, Maria Curie-Skl odowska University
pl. Marii Curie–Sk lodowskiej 5, 20-031 Lublin, Poland
WeconfronttheR-parityviolatingMSSMmodelwiththeneutrinooscillationdata. Investigating
the 1–loop particle–sparticle diagrams with additional bilinear insertions on the external neutrino
lines we construct the relevant contributions to the neutrino mass matrix. A comparison of the
so-obtainedmatrices with theexperimentalonesassuming normalorinvertedhierarchyandtaking
into account possible CP violating phases, allows to set constraints on the values of the bilinear
coupling constants. A similar calculation is presented with the input from the Heidelberg–Moscow
neutrinoless double beta decay experiment. We base our analysis on the renormalization group
evolutionoftheMSSMparameterswhichareunifiedattheGUTscale. Usingtheobtainedbounds
2 we calculate thecontributions to theMajorana neutrinotransition magnetic moments.
1
0 PACSnumbers: 12.60.Jv,11.30.Pb,14.60.Pq
2 Keywords: Majorananeutrino mass, supersymmetry, R-parity, bilinear R-parityviolation, neutrino oscilla-
tions
n
a
J
I. SUPERSYMMETRIC MODEL WITH that SUSY particles are not allowed to decay to non-
5 R-PARITY VIOLATION SUSY ones. It follows that the lightest SUSY particle
(usually the lightest neutralino χ˜0) must remain stable,
] 1
h givingagoodnaturalcandidateforcolddarkmatter. All
The recent confirmation of neutrino oscillations [1]
p this makesthe R-parityconservingmodelsverypopular.
gives a clear signal of existence of physics beyond
- In practice, however, the R-parity conservation is
p the standard model of particles and interactions (SM).
e Among many exotic proposals the introduction of su- achievedbyneglectingcertaintheoreticallyallowedterms
h in the superpotential. Casting such hand-waving ap-
persymmetry (SUSY) proved to be both elegant and
[ proach away, one should retain these terms, finishing
effective in solving some of the drawbacks of the SM.
withanR-parityviolating(RpV)model,withricherphe-
1 The minimal supersymmetric standard model (MSSM)
v nomenology and many even more exotic interactions [3–
(a comprehensive review can be found in [2]) populates
1 6]. The RpV models provide mechanisms of generat-
the so-called desert between the electroweak and the
4 ing Majorana neutrino masses and magnetic moments,
PlanckscalewithnewheavySUSYparticles,thusremov-
2 describe neutrino decays, SUSY particles decays, exotic
1 ing the scale problem. What is more, using the MSSM
nuclearprocessesliketheneutrinolessdoublebetadecay,
. renormalization group equations for gauge couplings in-
1 andmanymore. Beingtheoreticallyallowed,RpVSUSY
dicatesthatthereisaunificationofg , g andg around
0 1 2 3 theoriesareinterestingtoolsforstudying thephysicsbe-
m ≈ 1.2 × 1016 GeV which means that MSSM in
2 GUT yondtheStandardModel. Themanynever-observedpro-
1 a somehownaturalwayincludes GrandUnified Theories
cesses allow also to find severe constraints on the non-
: (GUTs). This model is also characterized by a heavier
v standard parameters of these models, giving an insight
Higgs boson, comparing with the Higgs boson predicted
i into physics beyond the SM.
X by SM, which is in better agreementwith the known ex-
The violation of the R-parity may be introduced in
r perimentaldata. NewinteractionspresentinMSSMlead
a a few different ways. In the first one R-parity violation
to many exotic processes which opens a completely new
is introduced as a spontaneous process triggered by a
field of research.
non-zero vacuum expectation value of some scalar field
Building the minimal supersymmetric version of the [3]. Another possibilities include the introduction of ad-
StandardModelone usually assumes the conservationof ditional bi- [4] or trilinear [6] RpV terms in the super-
the R-parity, defined as R = (−1)3B+L+2S, where B is potential, or both. In the following we incorporate the
thebaryonnumber,Ltheleptonnumber,andS thespin explicit RpV breaking scenario.
of the particle. The definition implies that all ordinary The R-parity conserving part of the superpotential of
SM particles have R = +1 and all their superpartners MSSM is usually written as
have R = −1. In theories preserving R-parity the prod-
uct of R of all the interacting particles in a vertex of WMSSM = ǫ [(Y ) LaHbE¯ +(Y ) Qa HbD¯x
ab E ij i u j D ij ix d j
a Feynman diagram must be equal to 1. This implies
+ (Y ) Qa HbU¯x+µHaHb], (1)
that the lepton and baryon numbers are conserved, and U ij ix u j d u
while its RpV part reads
∗Electronicaddress: [email protected] WRpV = ǫ 1λ LaLbE¯ +λ′ LaQb D¯x
†Electronicaddress: [email protected] ab 2 ijk i j k ijk i jx k
(cid:20) (cid:21)
2
1
+ 2ǫxyzλ′i′jkU¯ixD¯jyD¯kz +ǫabκiLaiHub. (2) andMχ˜0 is the standardMSSM neutralino mass matrix:
M 0 −1g′v 1g′v
The Y’s are 3×3 Yukawa matrices. L and Q are the 1 2 1 2 2
0 M 1gv −1gv
SU(2)left-handeddoubletswhileE¯,U¯ andD¯ denotethe Mχ˜0 =−1g′v 1gv2 20 1 −2µ2. (8)
right-handed lepton, up-quark and down-quark SU(2) 2 1 2 1
1g′v −1gv −µ 0
singlets, respectively. Hd and Hu mean two Higgs dou- 2 2 2 2
blets. We have introduced color indices x,y,z = 1,2,3,
The matrix (6) has the seesaw–like structure and con-
generation indices i,j,k =1,2,3=e,µ,τ and the SU(2)
tainsthe sneutrino vacuumexpectationvalues(vevs)ω .
spinor indices a,b=1,2. i
These areingeneralfree parameterswhich contributeto
As far as the (in principle unknown) RpV coupling
the gauge boson masses via the relation
constants are concerned, the most popular approach is
to neglect the bilinear terms and to discuss the effects
2
2M
connected with the trilinear terms only. In such a case, v2+v2+ ω2 =v2 = W ≃(246 GeV)2,
if one is not intereseted in exotic baryon number violat- 1 2 i g
i=e,µ,τ (cid:18) (cid:19)
ing processes, one has to additionally set λ′′ = 0, which X (9)
ensuresthe stability ofthe proton. Inthis paper wecon- where v and v are the usual down-type and up-type
1 2
centrate on the bilinear terms only and set all trilinear Higgsbosonvevs,respectively. By introducing the angle
RpV couplings to zero. β defined by tanβ = v /v we obtain four free param-
2 1
For completeness we write down the scalar mass term eters of the theory: tanβ and ω . Fortunately it turns
i
present in our model, outthatinordertoobtainproperelectroweaksymmetry
breakingthesneutrinovevscannotbearbitrary. Wegive
Lmass = m2 h†h +m2 h†h +q†m2q+l†m2l the details in the next section.
Hd d d Hu u u Q L
+ um2u†+dm2 d†+em2e†, (3)
U D E
III. HANDLING THE FREE PARAMETERS
the soft gauginos mass term (α=1,...,8 for gluinos)
Lgaug. = 1 M B˜†B˜+M W˜ †W˜i+M g˜†g˜α+h.c. , The RpV MSSM model introduces several new free
2 1 2 i 3 α parameters when compared with the usual MSSM. For-
(cid:16) (cid:17)(4) tunately their number can be constrained by imposing
aswellasthesupergravitymechanismofsupersymmetry GUT unification and renormalization group evolution.
breaking, by introducing the Lagrangian In this paper we restrict ourselves to the bilinear RpV
couplings only, setting all trilinear couplings (λ, λ′, λ′′)
Lsoft = ǫ (A ) lahbe¯ +(A ) qaxhbd¯ to zero. This assumption simplifies some of the RGE
ab E ij i d j D ij i d jx
equations, which we list below. Such approach leads at
h
+ (AU)ijqiaxhbuu¯jx+Bµhadhbu+Biκiliahbu ,(5) theendtothefollowingsetoffreeparameters: m0,m1/2,
A , tanβ, sgn(µ), and κGUT (i=1,2,3).
i 0 i
wherelowercaselettersstandforscalarcomponentsofthe
respective chiralsuperfields, and3×3matrices A as well
as Bµ and B are the soft breaking coupling constants. A. Masses and soft breaking couplings
i
The masses of all the supersymmetric scalars are uni-
II. NEUTRINO–NEUTRALINO MIXING fiedatmGUT toacommonvaluem0,andofallthesuper-
symmetric fermions to m . The values of the trilinear
1/2
soft SUSY breaking couplings are set according to the
The inclusion of the bilinear RpV terms imply mix-
following relations [8]
ing between neutrinos and neutralinos. In the ba-
sis (ν ,ν ,ν ,B˜,W˜3,H˜0,H˜0) the full 7 × 7 neutrino–
e µ τ d u A = A Y , (10)
neutralino mixing matrix may be written [5] in the fol- E,D,U 0 E,D,U
lowing form: B =B1,2,3 = A0−1. (11)
The RGE equations for the A couplings can be found
0 m
Mνχ˜0 = m3×T3 Mχ˜0 , (6) elsewhere [9–15]. The B couplings are evolved down to
(cid:18) (cid:19) the low energy regime according to the renormalization
group equations
where
m= −−121gg′′ωωe 121ggωωe 00 −−κκe (7) 16π2ddBt = 6 Tr(AUYU†)+6 Tr(ADYD† )
−212g′ωµτ 212gωµτ 0 −κµτ + 2 Tr(AEYE†)+6g22M2+2g12M1,(12)
3
1.05 1.05 1.05
1 1 1
0.95 0.95 0.95
V] 0.9 V] 0.9 V] 0.9
Me 0.85 Me 0.85 Me 0.85
κ [1 0.8 κ [2 0.8 κ [3 0.8
tanβ = 2 tanβ = 2 tanβ = 2
0.75 tanβ = 3 0.75 tanβ = 3 0.75 tanβ = 3
0.7 ttaannββ == 45 0.7 ttaannββ == 45 0.7 ttaannββ == 45
0.65 0.65 0.65
2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18
log10 E [GeV] log10 E [GeV] log10 E [GeV]
1.04 1.04 1.04
1.02
1.02 1.02
1
1 1
V] V] V] 0.98
Me 0.98 Me 0.98 Me 0.96
κ [1 0.96 tanβ = 10 κ [2 0.96 tanβ = 10 κ [3 0.94 tanβ = 10
tanβ = 20 tanβ = 20 0.92 tanβ = 20
0.94 ttaannββ == 3400 0.94 ttaannββ == 3400 0.9 ttaannββ == 3400
0.92 0.92 0.88
2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18 2 4 6 8 10 12 14 16 18
log10 E [GeV] log10 E [GeV] log10 E [GeV]
FIG.1: AnexampleofRGrunningofthebilinearκ couplings. Theunificationscenariowas: m =200 GeV,m =500 GeV,
i 0 1/2
A =200, sgn(µ)=1. All κGUT were equalto 1 MeV.
0 i
dB
16π2 1,2 = 6 Tr(A Y†)+6g2M +2g2M ,(13) C. Vacuum expectation values
dt U U 2 2 1 1
dB
16π2 3 = 6 Tr(A Y†) Atthebeginningofthenumericalprocedurewesetthe
dt U U down and up Higgs vevs to
+ 2 Tr(A Y†)+6g2M +2g2M .(14)
E E 2 2 1 1
v =vcosβ, v =vsinβ, (16)
1 2
where g2 = 5/3 g′2/(4π2) and g = g2/(4π2), 5/3 being
1 2 while the initial guess for the sneutrinos vevs is
the GUT normalization factor.
ω =0. (17)
i
Theactualvaluesofω arecalculatedfromthecondition
B. Bilinear κ couplings i
i that atthe electroweaksymmetry breaking scalethe lin-
ear potential is minimized. By taking partial derivatives
ThethreeκGUT couplingsatGUTscaleremainfreein of the potential one obtains the so-called tadpole equa-
i
our model. After setting them the couplings are evolved tions [8], which are zero at the minimum.
down to the mZ scale according to the renormalization In our procedure we solve three equations, which can
groupequationswhichinourcasetakethefollowingsim- be written as (i=1,2,3)
ple form:
3
16π2dκi = κ (3 Tr(Y Y†)−3g2−g2) κi(v1µ−v2Bi)= ωj Ωji, (18)
dt i U U 2 1 Xj=1
3
where
+ κ (Y Y†) . (15)
j E E ij
j=1 Ω =κ κ +(m2) +δ D, (19)
X ji j i L ji ji
Anexampleoftherunningofκi ispresentedonFig.1. δji being the Kroneckerdelta, and
One sees that for higher tanβ the couplings vary rather
1
weakly (notice the logarithmic scale on the energy axis) D = (g2+g′2)(v2−2v2). (20)
8 2
forthewholeenergyrangebetweentheelectroweakscale
mZ and mGUT. For small tanβ < 10 the difference Notice that they are linear in ω1,2,3 and therefore this
between the mGUT and mZ values are of the order of set has only one solution. After finding it, we use the
≤ 35%. The value 1 MeV at the GUT scale was chosen trigonometric parameterization [8], which preserves the
arbitrarily; we will show later that this is the typical or- definition of tanβ,
derofmagnitudeforwhichagreementwithexperimental
data on neutrino masses and mixing may be obtained. v = vsinα sinα sinα cosβ, (21)
1 1 2 3
4
II
ν ν ν χ0 χ0 ν
νa I III νb
C C2 C3 C
1 4
(a) (b)
FIG. 2: (a) The basic 1–loop diagram giving rise to the Ma-
jorana neutrino mass in the R-parity violating MSSM. (b) FIG. 3: Diagrams with bilinear neutrino–neutralino interac-
1–loop diagram with RpV neutrino–neutralino couplings in- tions leading to the Majorana neutrinomass.
cluded on theexternal lines.
TABLEI:Ninediagramswithneutrino–neutralinomixingon
v2 = vsinα1sinα2sinα3sinβ, (22) the external lines leading to Majorana neutrino mass. I and
ω = vcosα sinα sinα , (23) III are the neutralinos which mix with the neutrinos on the
1 1 2 3
external lines of the diagram depicted on Fig. 3. II defines
ω = vcosα sinα , (24)
2 2 3 thecontent of theloop (uu˜ standsfor theup-type(s)quarks,
ω3 = vcosα3, (25) dd˜stands for the down-type (s)quarks,qq˜ for all (s)quarks,
and l˜l for (s)leptons).
to calculate new values of v and v . We return back
1 2
to the tadpoles with these new values and continue in I II III C1 C2 C3 C4
this way until self-consistency of the results is reached. 1 H˜u uu˜ H˜u κa √2mu/vu √2mu/vu κb
It turns out that due to the expected smallness ofthe ωi 2 H˜u uu˜ B˜ κa √2mu/vu g′/(3√2) g′ωb
−
vevs, the initial guess Eq. (17) is quite a good approx- 3 H˜ uu˜ W˜3 κ √2m /v g/√2 gω
u a u u b
imation. It usually suffices to repeat the whole proce- 4 B˜ qq˜ B˜ g′ω g′/(3√2) −g′/(3√2) g′ω
a b
dure threetimes toobtainself-consistencyatthe levelof 5 B˜ l˜l B˜ g′ω − g′/√2 − g′/√2 g′ω
O(10−4), which is more than enough for our purposes. a − − b
6a W˜3 uu˜ W˜3 gω g/√2 g/√2 gω
The so-obtained set of vevs is used during the determi- a − − b
6b W˜3 dd˜ W˜3 gω g/√2 g/√2 gω
nation of the mass spectrum of the model. a b
7 W˜3 l˜l W˜3 gω g/√2 g/√2 gω
a b
8a B˜ uu˜ W˜3 g′ω g′/(3√2) g/√2 gω
a b
− −
IV. FEYNMAN DIAGRAMS WITH RPV 8b B˜ dd˜ W˜3 g′ωa g′/(3√2) g/√2 gωb
−
COUPLINGS ON THE EXTERNAL NEUTRINO 9 B˜ l˜l W˜3 g′ω g′/√2 g/√2 gω
a b
LINES
Itiswellknownthat,onceallowingforR-parityviola- nos inside the loop, were classified in e.g. Ref. [19] and
tion,aparticle–sparticle1–loopdiagramsgiveimportant discussed in details elsewhere (see [16–23] among oth-
corrections to the usual tree level neutrino mass term. ers). In the present paper we add the possible neutrino–
These processes have been extensively discussed in the neutralinomixingontheexternallines(Fig.2(b)),which
literature [16–22], mainly in the context of constraining leadstoanothercontributionstothe neutrinomass. Ob-
the tree–level alignment parameters Λ or the trilinear viouslythisadditionalcontributionmustbeinagreement
RpV couplings λ and λ′ [20, 23]. withthepresentexperimentaldata. Wediscusstwomain
In general, the explicit RpV effects may be taken into cases, in which either lepton and slepton or quark and
accountin three different ways. One may include the bi- squark are in the loop (in the case of higgsino H˜ only
u
linear RpV couplings or the trilinear couplings, or both. the up-type quarks count). At the same time the neu-
Of course the most complete one is the third possibility, trino may mix either with the gauginos: bino B˜ or wino
which is at the same time the most complicated. There- W˜3, or with the neutral up-type higgsino H˜ . All the
u
fore it is customary to limit the discussion to either tri-
nine cases together with the relevant bi- and trilinear
or bilinear terms only. In this paper we are interested in
coupling constants have been gathered in Tab. I.
bilinear couplings and set all trilinear couplings to zero.
Thecontributionsfromindividualdiagramshavebeen
Inordertodiscussthepossiblemagnitudeofthebilin-
calculated using the same technique as in Refs. [20, 21].
earRpVcouplingsκ weextendthesimplestdiagramsby
i In Ref. [21] we have discussed the possible influence of
includingtheneutrino–neutralinomixingontheexternal
including the quark mixing in the calculations. Here we
lines.
neglect this effect.
The topology of the basic type of 1–loop diagrams
The neutrino mass matrix resulting from the bilinear
we will consider is presented on Fig. 2(a). These dia-
processes only can be written as the following sum:
grams lead to Majorana neutrino mass term, where the
effective interaction vertex is expanded into the RpV
9
particle–sparticle loop. These diagrams and their more M = Mi , (26)
ab ab
complicated versions with the Higgs bosons and sneutri-
i=1
X
5
where the separate contributions read analogous expression involving the lepton and slepton
masses has been named yab. The function coming from
i
1 C C C C
Mi = 1 2 3 4F . (27) integratingoverloopmomentumisf(x,y)=[log(y)/(y−
ab 16π2 mImIII II 1)−log(x)/(x−1)]. The j-sums run over all squarks in
F , F , and F , and over all sleptons in F . The i-
Themassesoftheneutralinosm andm ,andthecou- uu˜ dd˜ qq˜ l˜l
I III sumscountallquarksinF ,up-typequarksonlyinF ,
qq˜ uu˜
pling constants have to be taken from Tab. I. The func-
down-typequarksinF ,andallleptonsinF . The fac-
tions F represent the contributions from the particle– dd˜ l˜l
tor 3 in F , F , and F accounts for summation over
sparticle loops. They read: uu˜ dd˜ qq˜
quarks’ colors. It is absent in the case of leptons.
sin2θj We do not discuss the Mi contributions separately.
Fuu˜ = 3 2 muif(xi2j,xi1j) , (28) The reason is that for different cases the couplings C2
i,j (cid:20) (cid:21) and C enter with opposite signs causing cancellations
X 3
sin2θj between such terms. Since none of the Mi can show
F = 3 m f(xij,xij) , (29)
dd˜ 2 di 2 1 up without the others, only the full sum Eq. (26) gives
Xi,j (cid:20) (cid:21) a meaningful picture.
sin2θj
F = 3 m f(xij,xij) , (30)
qq˜ 2 qi 2 1
i,j (cid:20) (cid:21)
X
sin2φj
F = m f(yij,yij) , (31) V. PHENOMENOLOGICAL MAJORANA
l˜l 2 li 2 1 NEUTRINO MASS MATRIX
i,j (cid:20) (cid:21)
X
where θj and φj are the j-th squark and slepton mass
The neutrino mass matrix can be constructed from
eigenstates’ mixing angles, respectively. For simplic-
the Pontecorvo–Maki–Nakagawa–Sakata mixing matrix
ity we have defined dimensionless quantities xab =
1,2 UPMNS under certainassumptions. The matrix UPMNS
(mqa/mq˜1b,2)2, which are the a-th quark mass over the isusuallyparameterizedbythreeanglesandthree(inthe
b-th squark first or second mass eigenstate ratios. An case of Majorana neutrinos) phases as follows:
c c s c s e−iδ 1 0 0
12 13 12 13 13
UPMNS = −s12c23−c12s23s13eiδ c12c23−s12s23s13eiδ s23c13 0 eiφ2 0 , (32)
s s −c c s eiδ −c s −s c s eiδ c c 0 0 eiφ3
12 23 12 23 13 12 23 12 23 13 23 13
where s ≡sinθ , c ≡cosθ . Three mixing angles θ (i<j) vary between 0 and π/2. The δ is the CP violating
ij ij ij ij ij
Diracphaseandφ ,φ areCPviolatingMajoranaphases. Theirvaluesvarybetween0and2π. Theexplicitexpression
2 3
for the phenomenological mass matrix Mph in terms of m , θ , δ, φ , φ is given by [21]:
αβ i ij 2 3
Mee = c213c212m1 + c213s212m2e−i2φ2 + s213e2iδm3e−i2φ3,
Meµ = −c12c13 c23s12 + c12s23s13e−iδ m1
+c13s12 (cid:0)c23c12 − s23s12s13e−iδ (cid:1)m2 e−i2φ2 + c13s23s13eiδm3e−i2φ3,
(cid:0) (cid:1)
M = −c c −s s + c c s e−iδ m
eτ 12 13 23 12 23 12 13 1
−c13s12 (cid:0)c12s23 + c23s12s13e−iδ (cid:1)m2e−i2φ2 + c23c13s13eiδm3e−i2φ3,
(cid:0) (cid:1)
M = c2 s2 + 2c c s s s e−iδ +c2 s2 s2 e−2δ m
µµ 23 12 23 12 23 12 13 12 23 13 1
+(cid:0)c223c212 − 2c23c12s23s12s13e−iδ + s223s212s213e−2δ(cid:1) m2e−i2φ2 + c213s223m3e−i2φ3,
(cid:0) (cid:1)
Mµτ = − c23s23s212 − c223c12s12s13e−iδ + c12s223s12s13e−iδ − c23c212s23s213e−2iδ m1
−(cid:0)c23c212s23 + c223c12s12s13e−iδ − c12s223s12s13e−iδ − c23s23s212s213e−2iδ(cid:1) m2e−i2φ2
+(cid:0)c23c213s23m3e−i2φ3, (cid:1)
6
M = s2 s2 − 2c c s s s e−iδ + c2 c2 s2 e−2iδ m
ττ 23 12 23 12 23 12 13 23 12 13 1
+(cid:0)c212s223 + 2c23c12s23s12s13e−iδ+c223s212s213e−2iδ(cid:1)m2e−i2φ2 + c223c213m3e−i2φ3.
(33)
(cid:0) (cid:1)
In order to calculate numerical values of elements of
this matrix one needs some additional relations among
0.00240 0.00269 0.00269
the mass eigenstates m . Experiments in which neu-
1,2,3 |M|(NH) = 0.00269 0.02553 0.01951 eV, (37)
trino oscillations are investigated allow to measure the
0.00269 0.01951 0.02553!
absolute values of differences of neutrino masses squared
and the values of the mixing angles. The best-fit values
of these parameters read [1, 24] 0.045267 0.000249 0.000249
|M|(IH) = 0.000249 0.022801 0.022801 eV.
|m21−m22| = 7.1×10−5 eV2, 0.000249 0.022801 0.022801!
|m2−m2| = 2.1×10−3 eV2, (38)
2 3
sin2(θ ) = 0.2857, (34) Yet another possibility is to construct M using con-
12
straintsfromnon-observabilityoftheneutrinolessdouble
sin2(θ ) = 0.5,
23 beta decay (0ν2β). The study of the 0ν2β decay [25] is
sin2(θ ) = 0.
13 one of the most sensitive ways knownto probe the abso-
lute values of neutrino masses and the type of the spec-
The present experimental outcomes are in agreement
trum. The most stringent lower bound on the half-life
with two scenarios:
of 0ν2β decay were obtained in the Heidelberg-Moscow
76Ge experiment [26] (T0ν−exp ≥ 1.9×1025 yr). By as-
• the normal hierarchy (NH) ofmasses imply the re- 1/2
lation m <m <m , sumingthenuclearmatrixelementofRef.[27]weendup
1 2 3
with|m |=U2 m +U2 m +U2 m ≤0.55 eV,where
ββ e1 1 e2 2 e3 3
• the inverted hierarchy (IH) ofmasses imply the re- U is the neutrino mixing matrix Eq. (32). The element
lation m3 <m1 <m2. |mββ|coincideswiththeeeelementofthe neutrinomass
matrixintheflavorbasisandfixingitallowstoconstruct
Notice that in order to keep the same notation for the
the full maximal matrix, which reads:
differences of masses squared and the mixing angles, the
neutrino mass eigenstates are labeled differently in the 0.55 1.29 1.29
NH and IH cases. |M|(HM) ≤ 1.29 1.35 1.04 eV. (39)
max
At this point we are left with four undetermined pa- 1.29 1.04 1.35!
rameters,whicharethephasesandthemassofthelight-
est neutrino. To obtainmoststringentlimits onthe new Inthenextsectionwepresenttheresultsforeachofthese
physicsparametersthelateristakentobezero. Asfaras five cases.
the phasesareconcernedweconsidertwoseparatecases.
First we take allpossible combinations of phases and for
each entry of the matrix we pick up its highest possible VI. CONSTRAINING κ COUPLINGS FROM
THE NEUTRINO MASS MATRIX
value. In this way we obtain unphysical matrices, which
give however some idea about the upper limits on the
non-standardparameters. The maximalmatricesfor the Our aim is to find constraints on the κi coupling con-
NH and IH scenarios read as follows [21]: stants coming from the neutrino mass matrices. As an
example of the unification conditions we take the follow-
ing input:
0.00452 0.00989 0.00989
|M|(NH) ≤ 0.00989 0.02540 0.02540 eV, (35) A =200, m =200 GeV, m =500 GeV, (40)
max 0 0 1/2
0.00989 0.02540 0.02540!
and additionally
0.0452 0.0312 0.0312 tanβ =10, sgn(µ)=1. (41)
|M|(IH) ≤ 0.0312 0.0240 0.0239 eV. (36)
max
0.0312 0.0239 0.0240! We do not expect great differences in the results if the
GUT conditions were changed. The only exception may
The moreconservativeapproachassumesthatthe CP bethetanβ parameter(definedatm scale)whichdom-
z
symmetryispreservedwhichcanbeachievedbyneglect- inates the running of the κ’s. By looking on Fig. 1 only
ing the phases present in the U matrix. In such verylowvaluesofthisparameterwillinfluencetheresults
PMNS
a case the NH and IH matrices take the following forms: significantly.
7
TABLE II:Some results for theSUSYscenario tanβ=10, A =200, m =200 GeV, m =500 GeV.
0 0 1/2
κGUT κGUT κGUT Resulting mass matrix Compare with Remarks
1 2 3
[MeV] [eV]
0.553927 0.864372 0.861090
9.50 14.80 14.80 0.865038 1.349844 1.344714 (HM) µτ elements to big
max
M
0.859783 1.341638 1.336555
0.548762 0.757606 0.753587
9.46 13.02 13.02 0.757853 1.046272 1.040721 (HM)
max
M
0.752398 1.038740 1.033237
0.004520 0.010728 0.010698
0.85 2.03 2.03 0.010734 0.025474 0.025404 (NH) eµand eτ elements to big
max
M
0.010686 0.025361 0.025292
0.045316 0.032954 0.032976
2.72 1.98 1.98 0.032945 0.023958 0.023974 (IH) eµand eτ elements to big
max
M
0.032951 0.023963 0.023978
0.002402 0.007824 0.007802
0.62 2.03 2.03 0.007821 0.025474 0.025404 (NH) eµ,eτ and µτ elements to big
M
0.007787 0.025361 0.025292
0.002402 0.002691 0.002687
0.62 0.70 0.70 0.002688 0.003011 0.003007 (NH)
M
0.002684 0.003007 0.003003
0.045316 0.032053 0.032118
2.72 1.92 1.93 0.032065 0.022680 0.022726 (IH) eµand eτ elements to big
M
0.032084 0.022694 0.022740
0.000453 0.000320 0.000321
0.27 0.19 0.19 0.000320 0.000226 0.000227 (IH)
M
0.000320 0.000226 0.000227
16 16
2 2
14 14
12 12 1.6 1.6
10 10
MeV] 8 MeV] 8 MeV] 1.2 MeV] 1.2
κ [2 κ [3 κ [2 κ [3
6 6 0.8 0.8
4 4
0.4 0.4
2 2
0 0 0 0
0 2 4 6 8 10 0 2 4 6 8 10 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
κ1 [MeV] κ1 [MeV] κ1 [MeV] κ1 [MeV]
16
2
14
κκ33 [[MMeeVV]] κκ33 [[MMeeVV]]
12 1.6
16
10 12 1.26
κ [MeV]3 8 84 κ [MeV]31.2 100...284
6 0 0.8 0
4 8101214 0.4 0 1.21.62
20 κ1 [2MeV]4 6 8 0246 κ2 [MeV] 0 κ1 [MeV0].4 0.8 00.40.8 κ2 [MeV]
0 2 4 6 8 10 12 14 16 0 0.4 0.8 1.2 1.6 2
κ2 [MeV] κ2 [MeV]
FIG. 4: Allowed parameter space in themaximal HMcase. FIG. 5: Allowed parameter space in themaximal NH case.
Weproceedintwosteps. Firstlywefindsuchvaluesof allowed values. It means that the κ’s will not take their
κGUT which will reproduce the diagonal elements of the maximal values simultaneously.
i
massmatrices. Thiscanbeachievedwithgoodaccuracy, Secondly we go down with the κGUT to lower the off-
i
but it turns out that some of the elements (off-diagonal, diagonalelementstotheacceptablelevel. This,however,
seeremarksinTab.II)oftheresultingmatrixexceedthe can be done in many different ways. Some explicit ex-
8
2 2
2 2
1.5 1.5 1.6 1.6
MeV] 1 MeV] 1 MeV] 1.2 MeV] 1.2
κ [2 κ [3 κ [2 κ [3
0.8 0.8
0.5 0.5
0.4 0.4
0 0 0 0
0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
κ1 [MeV] κ1 [MeV] κ1 [MeV] κ1 [MeV]
2
2
κκ33 [[MMeeVV]] κκ33 [[MMeeVV]]
1.5 1.6
2
1.5 1.6
κ [MeV]3 1 0.15 κ [MeV]31.2 100...284
0 0.8 0
0.5 1.5 1.62
0 0 κ01 .[5MeV1] 1.5 2 2.50 0.5 1 κ2 [MeV] 0.04 0 κ1 [Me0V.2] 0.4 0.600.40.81.2κ2 [MeV]
0 0.5 1 1.5 2 0 0.4 0.8 1.2 1.6 2
κ2 [MeV] κ2 [MeV]
FIG. 6: Allowed parameter space in the maximal IH case. FIG. 8: Allowed parameter space in the IH case with con-
served CP symmetry.
2 2
of some of the plots. This behavior is expected, as the
1.6 1.6
mass matrices to be reproduced contain on each entry
MeV] 1.2 MeV] 1.2 the maximal allowed value for it.
κ [2 κ [3 Muchmoreinterestingshapes areobtainedforthe CP
0.8 0.8 conservedcases(Figs.7and8). Theprojectionsontothe
(κ ,κ ) and (κ ,κ ) planes are nearly identical and con-
0.4 0.4 1 2 1 3
tainnon-linear boundaryparts. The mostconstrainedis
0 0 the inverted hierarchy case with conserved CP (Fig. 8)
0 0.2 0.4 0.6 0.8 0 0.2 0.4 0.6 0.8
κ1 [MeV] κ1 [MeV] due to two orders of magnitude differences between the
2 diagonalµµandττ andoff-diagonaleµandeτ elements.
κκ33 [[MMeeVV]]
1.6
2
1.6
κ [MeV]31.2 100...284 VII. THE TRANSITION MAGNETIC MOMENT
0.8 0
The RpV loop diagrams provide not only an elegant
2
0.4 0 1.21.6 mechanismofgeneratingMajorananeutrinomassterms,
0.2 0.4 0.40.8 κ2 [MeV] but also, after a minor modification, may be the source
0 κ1 [MeV] 0.60
0 0.4 0.8 1.2 1.6 2 of the transition magnetic moment µ . This quantity
κ2 [MeV] ab
represents roughly the strength of the electromagnetic
FIG. 7: Allowed parameter space in the NH case with con- interaction of the neutrino. Since the latter is electri-
served CP symmetry.
amples are listed in Tab. II but to find the full allowed
II
parameter space we have prepared scatter plots which
are presented on Figs. 4–8. Each of the plots consists of νa I III νb
C C
roughly 2000 points chosen randomly from the intervals C 2 3 C
1 4
between zero and 1.1 times the assessed upper limit for
given κGUT.
i
The boundaries of the allowed parameter space for
κi in the case of unphysical neutrino mass matrices FIG.9: Feynmandiagramwithneutrino-neutralinomixingon
|M|(HM,NH,IH) are nearly box-shaped, except for a small theexternallines,leadingtotheMajorananeutrinotransition
max
region of excluded values in the upper right-hand corner magnetic moment.
9
TABLE III: Contribution to the Majorana neutrino transi- TABLE IV: Like in Tab. III but with A = 100, m =
0 0
tion magnetic moments coming from the bilinear neutrino- 150 GeV, m =150 GeV, tanβ =19.
1/2
neutralino mixing, for the GUT scenario: A = 200, m =
0 0
200 GeV, m =500 GeV, tanβ =10. µeµ µeτ µµτ trilinear only
1/2
IH-CP 3.0 10−21 2.9 10−21 2.5 10−19 10−18
µ µ µ trilinear only × × × ≤
eµ eτ µτ IH-max 3.7 10−19 3.6 10−19 2.7 10−19 10−18
IH-CP 7.0 10−22 7.0 10−22 6.0 10−20 10−19 × × × ≤
× × × ≤
IH-max 8.8×10−20 8.5×10−20 6.5×10−20 ≤10−17 NH-CP 3.2 10−20 3.1 10−20 2.2 10−19 10−18
× × × ≤
NH-max 1.2 10−19 1.1 10−19 2.9 10−19 10−18
NH-CP 7.6 10−21 7.6 10−21 5.5 10−20 10−18 × × × ≤
× × × ≤
NH-max 2.8×10−20 2.8×10−20 7.0×10−20 ≤10−17 HM-max 1.0 10−17 9.8 10−18 1.2 10−17 10−16
× × × ≤
HM-max 2.4 10−18 2.3 10−18 2.9 10−18 10−15
× × × ≤
TABLE V: Like in Tab. III but with A = 500, m =
0 0
1000 GeV, m =1000 GeV, tanβ=19.
1/2
callyneutral,the interactionmusttakeplacebetweenan
µ µ µ trilinear only
external photon and a charged particle from inside the eµ eτ µτ
IH-CP 3.7 10−22 3.7 10−22 3.3 10−20 10−20
virtual RpV loop. In practice, only the photon–fermion
× × × ≤
interactions are taken into account, since the photon– IH-max 4.6 10−20 4.6 10−20 3.5 10−20 10−20
× × × ≤
boson (squark or slepton) interaction would be strongly
suppressed by the big mass of the SUSY particle. The NH-CP 4.0 10−21 4.0 10−21 2.9 10−21 10−20
× × × ≤
relevant Feynman diagram is presented on Fig. 9. NH-max 1.4 10−20 1.5 10−20 3.7 10−20 10−20
The contribution to the Majorana neutrino magnetic × × × ≤
momentfromthediscusseddiagramsisgivenby(inBohr HM-max 1.3 10−18 1.2 10−18 1.5 10−18 10−18
magnetons µ ) × × × ≤
B
µ = (1−δ )me1 C C2C3 C anorderofmagnitudeweaker. Thereasonforsuchasitu-
ab ab 4π2 1am m 4b
(cid:18) I III (cid:19) ation is due to the high masses of the neutralinos,which
w(q) w(ℓ) enter the formula Eq. (42) in the denominator. It was
ij ij
× 3m Qqi + m Qℓi µB. (42) possible that they will be compensated by the unknown
i,j " qi ℓi # couplingconstantsproportionaltothesneutrinovacuum
X
expectation values ω, especially that some of them are
Herewehavedenotedtheelectricchargeofaparticle(in
multiplied by negativenumbers, cf. Tab.II.Our explicit
units of e) by Q. The dimensionless loop functions w
calculation showed that it is not the case. Also the ob-
take the forms
serveddifferencesbetweenthevaluesofµ ,reachingnot
ab
sin2θj sin2φj morethanoneorderofmagnitude,aremainlyduetothe
w(q) = g(xij,xij), w(ℓ) = g(yij,yij), changedvalue of the parameter tanβ, and only partially
ij 2 2 1 ij 2 2 1
(43) due to different values of the remaining parameters.
where θ, φ, x , and y are the same as in Eqs. (28)–
1,2 1,2
(31), andg(x,y)=(xlog(x)−x+1)(1−x)−2−(x→y).
The sum over i and j in Eq. (42) accounts for all the VIII. SUMMARY
possible quark-squark and lepton-slepton configurations
forgivenneutralinos. Thefactor3infrontofw(q) counts The R-parity violating MSSM has many free parame-
the three quark colors. terswhichlowerits predictivepower. Ontheotherhand
The results for the alreadydiscussedGUT parameters this fact makes the model very flexible. In this paper
are presented in Tab. III. The last column contains for we have presented a method of constraining the bilinear
comparison upper bounds for the magnetic moment in RpV couplings κ.
the case when only trilinear interactions are taken into We have calculated the contributions to the neutrino
account. One sees that they are at least one order of massmatrixcomingfromtheneutrino–neutralinomixing
magnitudestrongerthanthe discussedbilinearcontribu- inprocessesinwhichtheeffectivevertexisexpandedinto
tions. In Tabs. IV and V we show the results of similar a virtual quark–squark or lepton–slepton loop. These
calculations for two other set of parameters. In Tab. IV contributions have been compared with the phenomeno-
theunificationparametersare‘low’,whileinTab.Vtheir logical mass matrices derived using the best-fit experi-
valuesare‘higher’. Thelastcolumnisgivenaspreviously mental values of the neutrino mixing angles and differ-
forcomparison. Alsoheretheconclusionisclear,thatthe ences of masses squared. We discuss four cases in which
discussed contribution to the main process is at best of normalandinvertedhierarchyisexploredbothwithcon-
thesameorderofmagnitude,inmostcasesbeingatleast served CP symmetry and with maximal values of each
10
matrix element. We also present the fifth case in which stants allows one to discuss many exotic processes, like
the neutrino mass matrix is calculated from the data the neutrino decay and the interaction of neutrino with
published by the Heidelberg–Moscow neutrinoless dou- a photon, to mention only a few. The former may oc-
ble beta decay experiment. cur as a two-step process, first through bilinear mixing
Ingeneralwehavefoundthatsettingtheκcouplingsat with neutralinos, and then the decay of the actual neu-
theunificationscaletovaluesoftheorderof.O(1 MeV) tralino. The later has been presented in the previous
rendersthemasscontributionscorrectlybelowtheexper- section showing by explicit calculation that this contri-
imental upper bound. Another observation is that the bution does not exceed the main 1–loop mechanism.
bilinear RpV mechanism alone is not sufficient to repro- In our calculations we have fixed the GUT unification
ducethewholemassmatrix. Thisis,however,acceptable parameters. Due to technical difficulties in performing
because in the general RpV loop mechanism one has to a full skan over the allowed parameter space we have
sum up the contributions from the tree–level [16], picked only three representatives for which the calcula-
tions were performed. We expect that the results will
Mtiir′ee =ΛiΛi′ g22 not change qualitatively with the changes of the input
M +M tan2θ
× 1 2 W , parameters, which is of course an assumption that may
4(µm2W(M1+M2tan2θW)sin2β−M1M2µ2) be worth checking.
(44)
whereΛ =µω −v κ aretheso-calledalignmentparam-
i i d i
eters, as well as contributions coming from the 1–loop
Acknowledgments
diagrams (see Fig. 2(a)), which are proportional to the
totallyunconstrainedtrilinearcouplingsλandλ′. These
parametersmaybeeasilyfine-tunedtoreproducethefull The first author (MG) is partially supported by the
massmatrix andweshift this discussionto anupcoming PolishStateCommitteeforScientificResearch. Hewould
paper. likealsotoexpresshisgratitudetoprof.A.F¨aßlerforhis
The knowledge of the bounds on the κ coupling con- warm hospitality in Tu¨bingen during the Summer 2006.
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