Constraining an R-parity violating supergravity model with the Higgs induced Majorana neutrino magnetic moments

1.2 MB·English

by  Marek Góźdź

Constraining an R-parity violating supergravity model with the Higgs induced
Majorana neutrino magnetic moments
Marek G´o´zd´z∗
Department of Informatics, Maria Curie-Skl odowska University,
pl. Marii Curie–Sk lodowskiej 5, 20-031 Lublin, Poland
It is well known, that R-parity violating supersymmetric models predict a non-zero magnetic
moment for neutrinos. In this work we study the Majorana neutrino transition magnetic moments
within an RpV modified minimal supergravity model. Specifically, we discuss the contributions
coming from the charged Higgs bosons, higgsinos, leptons, sleptons, charged gauge bosons, and
charginos. We use the experimental results from the MUNU collaboration to restrict the model’s
parameterspace. Acomparisonwithtwoothertypesofcontributions(onlytrilinearRpVcouplings
and trilinear plus neutrino–neutralino mixing) is also presented. We have found that the presently
2 discussed processes dominate significantly, exceeding in some cases even theexperimental limits.
1
0
PACSnumbers: 11.30.Pb,12.60.Jv,14.60.Pq
2
Keywords: neutrinomagneticmoment,supersymmetry,R-parity

n
a
J I. INTRODUCTION magnetic moments has been presented in our previous
4 works [13, 16–18]. The SUGRA model has been used
to calculate masses and magnetic moments in Ref. [16].
Neutrinos are one of the most mysterious particles
] The contribution coming from neutrino–neutralino mix-
h known. They interact very weakly with matter, taking
inghasbeendiscussedinRef.[17],whilethecontribution
p partonlyinthe weak(andpresumablygravitational)in-
from generic diagrams with two mass insertions on the
-
teractions. Due to mixing [1] their mass eigenstates and
p internal lines was presented in [13]. Calculations within
e interaction eigenstates differ substantially, which gives
the gauge-mediated model were performed in [18]. We
h risetothegenerationleptonnumberviolatingoscillations
have studied the Majorana neutrino masses and transi-
[ [2]. Theyarealsothelightestfermionseverobserved–in
tionmagneticmomentsusingexactanalyticformulasand
1 factthesmallnessofneutrinomassescannotbeexplained up-to-date experimental data concerning neutrino oscil-
v using the standard Higgs mechanism [3]. All this indi-
lationsandthenon-observabilityoftheneutrinolessdou-
3 cates, that neutrinos are a window to some new physics
ble beta decay (0ν2β). Our numerical procedure rested
7 beyond the standard model.
upon the assumption that most of the free parameters
8
0 The possible electromagnetic properties of neutrinos of the supersymmetric model unify in certainway at the
1. in the form of an induced magnetic moment are partic- GUT scale mGUT ∼ 1.2×1016 GeV. This allows to re-
ularly interesting, since the electromagnetic interactions ducethenumberoffreeparameterstofew,andcalculate
0
2 are much easier to control in any precise measurement. the rest by performing renormalization group evolution.
1 This problem has been studied in the standard model Such approach gives a much more convincing picture of
: [4, 5] and its extensions. In Ref. [6] the magnetic mo- the low energy spectrum than just assessing the running
v
ments have been reviewed in models with spontaneously massesofthesuperparticles. Wehavealsodemonstrated
i
X broken lepton number. A lot of work has been devoted theinfluenceofthemixingamongsquarksandquarkson
r to discuss the neutrino masses in the R-parity violat- the results. The evident weakpoint of our approachwas
a ing (RpV) supersymmetry [7–14]. In these models, the the lack ofknowledgeofthe trilinearcoupling constants,
neutrino mass is generated mostly at the one-loop level which forced us to use additional source of constraints
through an RpV loop containing in the simplest ver- (neutrino oscillations, 0ν2β decay).
sion a particle–sparticle pair (lepton–slepton or quark– In this paper we study the Majorana neutrino transi-
squark). The approach usually used in the literature tionmagneticmomentsgeneratedinthe R-parityviolat-
assumed no mixing among particles, and some average ing modified minimal supergravity model. This model,
commonvalueforthemassesofthesuperparticlesatlow basedonthe Ref. [19], takesinto accountthe full depen-
energies. In Ref. [15] a basis independent classification dence of the renormalization group equations (RGE) on
ofvariousdiagramsleadingtoMajorananeutrinomasses the RpV couplings and their soft SUSY breaking ver-
hasbeenpresented. Thestudyofneutrinomassescanbe sions. All these couplings have been incorporated in
directlyextendedtothe magneticmomentcasebyintro- the unification scheme, so, within this assumption, the
ducing an interaction with an external photon. whole model canbe calculatednumerically. No other in-
A more systematic approachto the calculations of the put, in particular experimentaldata, is needed. We con-
centrate on the contributions coming from the charged
Higgs bosons, charged higgsinos, charged gauge bosons,
charginos, leptons, and sleptons. The neutral particles,
∗Electronicaddress: [email protected] like the neutral components of the Higgs bosons, neu-
2
tralinos and sneutrinos will contribute to the neutrino which might suggest that it is only an accidental (not
mass, but will not contribute to the magnetic moment, exact) symmetry present in the low-energy regime. It
and as such will not be discussed here. is generally expected, that the higher-energy extensions
The paper is organized as follows. After the intro- of the SM will not preserve at least the lepton number
duction, in Sect. II the minimal supersymmetric stan- (the protonstabilitymuststillhold). Suchmodelsbased
dard model with R-parity violation and supersymme- on supersymmetry introduce new interactions, or in fact
try brokenby the gravitationalinteractions is presented. do not rule out certain terms, that should be present in
Sect. III is devoted to the technical problems of han- the superpotential,whichviolateR-parity[23–27]. They
dling the free parameters and calculating the coupling have a much richer phenomenology and predict lots of
constantsintheHiggssectorduringtheelectroweaksym- newphenomena,liketheleptonnumberviolatingneutri-
metry breaking. The Majorananeutrino transitionmag- noless double beta decay, RpV loop-corrected neutrino
netic moments, and the interesting for us contributions mass and magnetic moment, and others.
to this quantity are discussed in Sect. IV. At the end We work within the slightly modified model described
numerical results are presented. in details in Ref. [19]. It is defined by the following R-
parity conserving and R-parity violating parts of the su-
perpotential
II. R-PARITY VIOLATING SUGRA
W =W +W , (1)
RpC RpV
Theminimalsupersymmetricstandardmodel(MSSM)
[21, 22] is the minimal extension of the standard model
where
(SM) of particles and interactions, which introduces su-
persymmetry (SUSY). In short, for each particle there
is a new superpartner (fermionic for bosons and bosonic WRpC = ǫab (YE)ijLaiHdbE¯j +(YD)ijQaixHdbD¯jx
forfermions)sothatthenumberoffermionicandbosonic h
degrees of freedom in the MSSM are equal. The one ad- + (YU)ijQaixHubU¯jx−µHdaHubi, (2)
ditional multiplet, not present in the SM, is a second
1
Higgs doublet needed to cancel the gauge anomalies and WRpV = ǫab(cid:20)2(ΛEk)ijLaiLbjE¯k+(ΛDk)ijLaiQxjbD¯kx(cid:21)
to give masses to both the up and down components of
1
tthheesSoU-c(a2l)leddoRub-pleatrsi.tyT,whihsicvherissioanmoufltthipelimcaotdiveel pqrueasnetruvmes + 2ǫxyz(ΛUi)jkU¯ixD¯jyD¯kz −ǫabκiLaiHub. (3)
number defined as R=( 1)2S+3(B−L), S being the spin
of the particle, B the bar−yon, and L the lepton number. Here Y’s are the 3 3 trilinear Yukawa-like couplings,
Oneeasilysees,thatR=+1forordinarystandardmodel µ the bilinear Higgs×coupling, and Λ and κi represent
particles,whileR= 1forthesupersymmetricpartners. the magnitudes of the R-parity violating trilinear and
PreservingR not onl−y gives the baryonandlepton num- bilinear terms. L and Q denote the SU(2) left-handed
ber conservationsbut alsoforbids decays ofSUSY parti- doublets,while E¯, U¯ andD¯ arethe right-handedlepton,
clesinto SMparticles,leavingthe lightestSUSYparticle up-quark and down-quark SU(2) singlets, respectively.
stable. Hd and Hu mean two Higgs doublets. We have intro-
The problem of the lepton and baryon number con- duced color indices x,y,z = 1,2,3, generation indices
servation is still disputable. On one hand, our observa- i,j,k = 1,2,3 = e,µ,τ and the SU(2) gauge indices
tionssuggest,thatbothB andLareconservedquantum a,b = 1,2. Finally, ǫab and ǫxyz with ǫ12 = ǫ123 = 1
numbers. Especially strong limits on B violation come are the totally antisymmetric tensor densities.
from the non-observation of the proton decay. On the The supersymmetry is not observed in the regime of
other hand, in the SM not only the total lepton num- energiesaccessibletoourexperiments. Thereforeitmust
ber but also generation lepton numbers L , L , and L be broken at some point. A convenient method to take
e µ τ
areconserved,andthis rule has been provenwrongafter thisfactintoaccountistointroduceexplicitterms,which
the discovery of neutrino oscillations. From the theo- breaksupersymmetryinasoftway,ie.,theydonotsuffer
retical point of view, there is no underlying symmetry from ultraviolet divergencies. We add them in the form
or mechanism, which supports conservation of B and L, of a scalar Lagrangian[19],
= m2 h†h +m2 h†h +l†(m2)l+l †(m2 )h +h†(m2 )l +q†(m2)q+e(m2)e†+d(m2 )d†+u(m2)u†
−L Hd d d Hu u u L i LiHd d d HdLi i Q E D U
+ 1 M B˜†B˜+M W˜ †W˜i+M g˜†g˜α+h.c. (4)
1 2 i 3 α
2
(cid:16) (cid:17)
+ [(A ) l h e +(A ) q h d +(A ) q h u Bh h
E ij i d j D ij i d j U ij i u j d u
−
+ (AEk)ijliljek+(ADk)ijliqjdk+(AUi)jkuidjdk−Dilihu+h.c.],
3
where the lowercaseletter denotes the scalarpartofthe squark mass eigenstates as
respective superfield. M are the gauginomasses, and A
i
(B, Di)arethesoftsupersymmetrybreakingequivalents qmin =√mt˜1mt˜2. (6)
of the trilinear (bilinear) couplings fromthe superpoten-
tial. At this scale the radiative corrections to the Higgs po-
tential coming from the squarks are minimal. We start
by fixing the Yukawa couplings at m using the quark
Z
III. FREE PARAMETERS AND THE and lepton mass matrices M
U,D,E
LOW-ENERGY SPECTRUM
M = v S YTS† ,
The main problem of most supersymmetric models is U u UR U UL
their large number of free parameters. Therefore addi- MD = vdSDRYDTS†DL, (7)
tional constraints are usually imposed, which introduce M = v S YTS† ,
certain relations among them, effectively reducing their E d ER E EL
number. Itis awellknownfact,thatthe RGEequations
where S matrices perform diagonalization so that one
forthegaugecouplingconstantsleadintheMSSMmodel
obtains eigenstates in the mass representation. The
to their unification at energy m 1.2 1016 GeV.
This feature is absent in the SMG, UalTth≈ough s×uggested by Yukawa-like RpV couplings at mZ are all set to
the extrapolation of the LEP1 data. It may therefore
Λ=Λ 1, (8)
seem natural to assume also certain type of unification 0
of other parameters. Let us introduce at m :
GUT 1 being a 3 3 unit matrix. Next, we evolve the gauge
×
the common mass of all the scalars m , and Yukawa couplings from m to the unification scale.
• 0 Z
During this running we turn off the Higgs sector, µ =
the common mass of all the fermions m ,
1/2 κ = B = D = 0, and set the sneutrino vev’s v = 0.
• i i i
These parameters will be calculated later.
the common proportionality factor A for the soft
0
• SUSY breaking couplings At mGUT we impose the GUT conditions as described
above, ie., we set the massess of all the scalars to m , of
0
AU,D,E =A0YU,D,E, allthe fermions to m1/2, andthe softbreakingcouplings
AUi,Di,Ei =A0YU,D,E (i=1,2,3). (5) to A0Y. The Yukawa couplings Y’s are left unchanged
inordertoreproducecorrectlythemassesofleptonsand
quarkswhenevolveddownto m . So arethe Λ’s,which
Z
ThisschemeofunificationfollowsthemSUGRAassump-
are free parameters here. We evolve all the running pa-
tionswith theexceptionthatwedonotassumethe total
rameters,couplingconstantsandmasses,down,andfind
universality of the A couplings, but vary them keeping
the bestscale for minimizing the scalarpotentialEq.(6)
them proportional to the Yukawa Y coupling constants
and atthis scale calculate µ=sgn(µ) µ2 andB using
at the unification scale. As a free parameter remains | |
[19] p
also the ratio of the Higgs vacuum expectation values,
tanβ = v /v , and the sign of the µ coupling constant,
u d m2 m2 tan2β M2
sgn(µ). Thesixthfreeparameterinourconsiderationsis µ2 = Hd − Hu Z, (9)
theinitialvalueoftheΛ’satm ,Λ . Weleavethisvalue | | tan2β 1 − 2
Z 0 −
freeandinvestigateitsinfluence onthe results. The idea sin2β
is that we want to have Λ’s non-zero and contributing B = 2 (m2Hd −m2Hu +2|µ|2). (10)
from the beginning to the RGE running of other param-
eters. In this way, Λ controls the amount of R–parity Runningallthe parametersdownq m non-zeroµ
0 min Z
→
violation in the low energy regime. Notice, that Λ’s will andBgeneratenon-zeroκ andD . Theseareusedasthe
i i
not unify at m , although their RGE running is al- startingvaluestothe secondlongrun. Next, weperform
GUT
most flat (cf. Fig. 1 in Ref. [20]). once again the RGE running up to the m scale, but
GUT
Theprocedureofobtainingthelowenergyspectrumof now with all the Higgs parameters contributing to the
the model involves few renormalization group runnings running. WeimposeGUTconditions,godown,calculate
between the low (m ) and high energy scale (m ), the new q scaleand onceagainevaluate the corrected
Z GUT min
and between m and the scale at which the electroweak Higgs parameters, this time together with the sneutrino
Z
symmetry breaking occurs, which is defined by the top vev’s, according to
1 v v 1 v M2
µ2 = m2 +m2 i +κ∗µ i m2 + κ 2 (g2+g2)v2 D i tan2β Z, (11)
| | tan2β 1(cid:26)(cid:20) Hd LiHdv i v (cid:21)−(cid:20) Hu | i| − 2 2 i − i v (cid:21) (cid:27)− 2
d d u
−
4
sin2β v v
B = (m2 m2 +2µ2+ κ 2)+(m2 +κ∗µ) i D i , (12)
2 (cid:20) Hd − Hu | | | i| LiHd i v − i v (cid:21)
d u
v [(m2) + κ 2+D′]+v [(m2) +κ κ∗]+v [(m2) +κ κ∗]= [m2 +µ∗κ ]v +D v ,
1 L 11 | 1| 2 L 21 1 2 3 L 31 1 3 − HdL1 1 d 1 u
v [(m2) +κ κ∗]+v [(m2) + κ 2+D′]+v [(m2) +κ κ∗]= [m2 +µ∗κ ]v +D v , (13)
1 L 12 2 1 2 L 22 | 2| 3 L 32 2 3 − HdL2 2 d 2 u
v [(m2) +κ κ∗]+v [(m2) +κ κ∗]+v [(m2) + κ 2+D′]= [m2 +µ∗κ ]v +D v ,
1 L 13 3 1 2 L 23 3 2 3 L 33 | 3| − HdL3 3 d 3 u
where D′ =M2cos2β +(g2+g2)sin2β(v2 v2 v2). The tadpole equations (13) can be easily solved and we get for
Z 2 2 2 − u− d
the sneutrino vev’s
detW
i
v = , i=1,2,3, (14)
i
detW
where
(m2) + κ 2+D′ (m2) +κ κ∗ (m2) +κ κ∗
W =((mmL2L2))1112++|κκ21κκ|∗1∗ ((mmL2L2))2212++|κκ12κ|22∗+D′ ((mmL2L2))3312++κκ12κ23∗3+D′ , (15)
 L 13 3 1 L 23 3 2 L 33 | 3| 
and W can be obtained from W by replacing the i-th column with
i
[m2 +µ∗κ ]v +D v
−−[[mmH2H2ddLL12 ++µµ∗∗κκ12]]vvdd++DD12vvuu . (16)
− HdL3 3 d 3 u 
Thenewlyobtainedv ’sareincorporatedinthescheme or (Λ )2. Detailed discussions can be found, e.g., in
i E
in the following way: Refs. [7–12, 15].
Neutrinos are neutral particles and as such cannot in-
v = sinβ v2 (v2+v2+v3), (17) teract with photons. However, this interaction is possi-
u q − 1 2 3 ble through the one-loop mechanism, in which charged
v = cosβ v2 (v2+v2+v3), (18) particles appear in the loop. Therefore,aneffective neu-
d q − 1 2 3 trino magnetic moment may be generated. Due to the
CPTtheorem,Majorananeutrinoscannothavediagonal
v2 = (246 GeV)2, which preserves the definition of the
magnetic moments,whichactbetweenthe sameflavours
tanβ andv2 =v2+v2+v2+v2+v2. Notice,thatinthis
u d 1 2 3 of neutrinos, but they may have the off-diagonal tran-
model sneutrino vev’s contribute to the mass of the Z
sition magnetic moments [4]. To be more precise, the
and W bosons. The equations (11)–(13) are solved sub-
transition Majorana magnetic moment acts between ν
sequently until convergence and self-consistency of the iL
and νc chiral components of Majorana neutrinos, as-
results is obtained. In practice three iterations turned jL
suminggaugetheorywithonlyleft-handedneutrinos. As
out to be enough. After this procedure we add also the
aconsequence itviolatesthe totalleptonnumber by two
dominant radiative corrections [28], and get back to the
units (∆L = 2), and can be discussed provided that R-
m scale to obtain the mass spectrum of the model.
Z parity violation occurs. The effective Hamiltonian takes
the form
IV. NEUTRINO MAGNETIC MOMENTS 1
HM = µ ν¯ σαβνc F +h.c. (19)
eff 2 ij iL jL βα
ItiswellknownthatintheRpVsupersymmetricmod-
elsonlyoneneutrinogainsmassafterthediagonalization The contributing Feynman diagrams are presented on
of the neutrino–neutralino mass matrix. The remaining Fig. 1. We will evaluate the amplitudes of these pro-
contributions come from the sneutrino vev’s at the tree cesses in the interaction basis, in which the vertex cou-
levelandfromtheone-loopprocesses. Themainone-loop pling constants are known directly from the mixing ma-
mechanism involves decomposing a Majorana neutrino– trices. However, one has to take into account, that the
neutrinovertexintoaparticle–sparticleloop,whichbasi- propagators of particles travelling inside the loop must
callyiseitherthequark–squark,orthe(charged)lepton– be written for their mass eigenstates. This can be done
sleptonloop. Thesecontributionswillbeproportionalto using the mixed-propagator formalism [29]. Assume the
some functions of the masses of the particles and (Λ )2 following relation between the mass (j) and interaction
D
5
(a) (b) (c) (d) (e)
FIG.1: FeynmandiagramswithtwomassinsertionsinsidetheloopcontributingtotheMajorananeutrinotransitionmagnetic
moments.
(α) eigenstates of some particle, integral, so it may be written in a general form as
Φ = V Φ . (20)
α αj j
Xj µ = N C C X X Q 2m µ , (23)
ν c 1 2 1 2 e B
hX Ii
Then, the amplitude in the weak basis can be written in
terms of the Green function and expanded in the mass
basis where Nc is the color index (=3 for quarks and squarks,
=1 otherwise), C are the trilinear coupling constants,
1,2
G = V G V† , (21) X are the mass insertions residing inside the loop, Q
Aα ∼ Φα αj j jα 1,2
Xj is the charge of the particle the photon is attached to
(in units of the elementary charge), m is the electron
e
so the flavour changing amplitude may be written as mass (2m 10−3 GeV), and µ the Bohr magneton.
e B
≈
Thesumrunsoverallmasseigenstateswhichformevery
A(α→β)= VβjAjVj†α. (22) mixed particle present in the loop.
Xj
The masses and couplings in our approach are calcu-
A more detailed derivation and discussion of this result latedfromtheboundaryconditionsatthem andm
Z GUT
can be found in Sect. 4.2 of Ref. [29]. scales (see Sect. III). The loop integrals can be de-
I
The matrix V is obtained during the diagonalization rived analytically using standard techniques of integra-
procedureofthemixingmatricesanditscolumnsarethe tion in the Minkowski space. Denoting by f ,f ,... and
1 2
eigenvectorsofthemixingmatrix. Themagneticmoment b ,b ,... the fermions and bosons which appear in the
1 2
will consist of the product of numerical constants, cou- loop, and by m their respective masses, the loop inte-
f,b
pling constants, and the function describing the loop grals for diagrams (a)–(e) depicted on Fig. 1 read
I
(a) = m m (f ,f ,b ,b )+ (f ,f ,b ,b )+m m (f ,f ,b ,b )+ (f ,f ,b ,b ), (24)
I f1 f2 F3 1 2 1 2 F4 1 2 1 2 f1 f2 F3 2 1 1 2 F4 2 1 1 2
(b) = m (f ,b ,b ,b ), (25)
I f1 F3 1 1 2 3
(c) = 4 (b ,f ,f ,f ), (26)
4 1 1 2 3
I F
(d) = m m (m +m ) (f ,f ,f ,b )+(2m +3m +m ) (f ,f ,f ,b ), (27)
I f1 f2 f2 f3 F3 2 1 3 1 f1 f2 f3 F4 2 1 3 1
(e) = m (m m +m m +m m ) (f ,f ,f ,b )+(5m +2m +2m ) (f ,f ,f ,b )
I f1 f1 f2 f2 f3 f3 f1 F3 1 2 3 1 f1 f2 f3 F4 1 2 3 1
+ m (m m +m m +m m ) (f ,f ,f ,b )+(5m +2m +2m ) (f ,f ,f ,b ), (28)
f3 f1 f2 f2 f3 f3 f1 F3 3 1 2 1 f3 f2 f1 F4 3 1 2 1
where the functions are given by
3,4
F
∞ d4k 1
(16π2) (a,b,c,d) = (16π2)
F3 Z (2π)4(k2 m2)2(k2 m2)(k2 m2)(k2 m2)
−∞ − a − b − c − d
= m2alog(m2a)(cid:16)m2a−1m2b + m2a−1m2c + m2a−1m2d −1(cid:17)−1 m2blog(m2b)
(m2 m2)(m2 m2)(m2 m2) − (m2 m2)(m2 m2)(m2 m2)
a− b a− c a− d b − a b − c b − d
m2log(m2) m2log(m2)
c c d d , (29)
− (m2 m2)(m2 m2)(m2 m2) − (m2 m2)(m2 m2)(m2 m2)
c − a c − b c − d d− a d− b d− c
∞ d4k k2
(16π2) (a,b,c,d) = (16π2)
F4 Z (2π)4(k2 m2)2(k2 m2)(k2 m2)(k2 m2)
−∞ − a − b − c − d
6
= m4alog(m2a)(cid:16)m2a−1m2b + m2a−1m2c + m2a−1m2d −2(cid:17)−1 m4blog(m2b)
(m2 m2)(m2 m2)(m2 m2) − (m2 m2)(m2 m2)(m2 m2)
a− b a− c a− d b − a b − c b − d
m4log(m2) m4log(m2)
c c d d . (30)
− (m2 m2)(m2 m2)(m2 m2) − (m2 m2)(m2 m2)(m2 m2)
c − a c − b c − d d− a d− b d− c
WehavekeptdenotationsfromRef.[13]. Pleasenotice A. Constraining the model’s parameter space with
also that a typo has been correctedin Eq.(25), andnew the magnetic moments
terms appeared in Eqs. (24) and (28).
Therearedifferentassessmentsandconstraintsputon
the neutrino magnetic moment. Analyzing the impact
V. RESULTS of the solar neutrino data on the neutrino spin-flavour-
precessionmechanism,theauthorsofRef.[31]havefound
In order to contribute to the neutrino magnetic mo- amodel-dependentuppervalueforthemagneticmoment
ment, the loop must contain charged particles. The to be few 10−12µB. On the other hand, direct searches
×
quarks–squarks and leptons–sleptons have already been in the MUNU experiment resulted in the upper limit to
discussed elsewhere (see Sect. I). The discussion on be9 10−11µB [32]. Inthispaperweadoptaconservative
the bilinear insertions on the neutrino lines, possible limit×andrejectpoints,forwhichµν 10−10µB. Amore
≥
due to the neutrino–neutralino mixing, can be found strict approach would result in a substantially narrower
in Refs. [14, 17]. In this work we have focused on the parameter space of the model.
charged Higgs–sleptons and the leptons–charginos inter- TheresultsarepresentedonFigs.2–9,forµ>0,Λ0 =
action eigenstates. 10−4, A0 = 200 1000 GeV in steps of 100, m0,1/2 =
−
Altogetherwehaveidentified45diagramsoftheforms 200 1000 GeV in steps of 20, and tanβ = 5 40 in
− −
given on Fig. 1 containing the particles in question. The steps of 5. Each point represents a valid set of input
way of obtaining them is a straightforward but rather parameters, which results in a physically accepted low
tedious task, and we have used a computer program to energy spectrum. Each figure represents a fixed tanβ,
matchallpossibleverticesandinternallinestothe given and panels read in rows from left to right correspond to
patterns. The trilinear vertices can be constructed di- A0 taking values 200 GeV, 300 GeV, ..., 1000 GeV.
rectly from the superpotential (1), and the possible bi- Ingeneral,thereisnoglobalregularityintheseresults.
linearmassinsertionsaredefinedbythemixingmatrices, Certainregionsareexcludedduetounacceptedvaluesof
as given in Ref. [19]. the masses. The excluded stripes, visible in several of
A few conditions must have been met for the result the panels, come from too high values of µν. Starting
to be accepted. First of all, during the RGE evolution from tanβ = 15, a parabolic-like shape, whose position
some of the particles may get negative mass parame- depends on the A0 parameter, appears. It starts to be
ters squared. Such tachyonic behaviour resulted in im- visible for A0 = 400 GeV and crosses the m0 m1/2
−
mediate rejecting of the given input parameters. Also, plane with increasing A0. [We are not sure about the
finite values for the Yukawa couplings, which tend to origin of this ‘stability line’. As a side remark we add,
‘explode’ for too low or too high values of the tanβ, that it appeared also in our analysis of the masses of
were required. When analyzing the results, we have re- the lightest Higgs bosons h0 in the context of the CDF-
jected points, for which the masses of the superparti- D0 discovery [33]. Constructing similar plots, but using
cles where too low. The mass limits we have used where the condition, that 120 GeV<mh0 <140 GeV, we have
takenfromtheParticleDataGroupreport[30],andthey observedverysimilarparabolas.] Wenoticealso,thatfor
rmeaχ˜d04:>m1χ˜1016>Ge4V6,GmeχV˜±1, >mχ˜9024>Ge6V2,GmeχV˜±2, m>χ˜9034>Ge1V00, mGee˜V>, ttvahanalutβetsh=oef4mt0hotedheYelubpkroaeiawnktasscdlooouowpknlitniongtsat.hlliysrreagnidoonmd,uwehtoicthooshhoiwghs
107 GeV, m >94 GeV, m >82 GeV, m >379 GeV,
µ˜ τ˜ q˜
m > 308 GeV. This is by far the conservative choice
g˜
of constraints. The new results from the LHC projects
push the quoted above limits to higher values, closer to B. Λ0 dependence
the 1 TeV range. We do not use them here for various
reasons. Firstly, most of the results are still marked as Another interesting problem is how the initial value
preliminary,and more careful data analysis is being per- of the RpV couplings at m , represented by a common
Z
formed right now. Also, the LHC data are interpreted parameter Λ , influences the results. We have chosen to
0
within the simplest SUSY models, with many additional check this relation for a few specific sets of parameters.
assumptions, like the absence of R–parity violation, and Forthis,theSnowmassSUSYbenchmarkpoints[34]were
equality of the gluino and squark masses. Theses as- used. Thedependenceoftheresultingcontributiontothe
sumptions are not true in the model used here. transition magnetic moments have been calculated for
7
1000
800 tanβ=5 tanβ=5 tanβ=5
A =200 GeV A =300 GeV A =400 GeV
0 0 0
600
400
200
1000
V] 800 tanβ=5 tanβ=5 tanβ=5
e A =500 GeV A =600 GeV A =700 GeV
G 0 0 0
600
[2
1/
m 400
200
1000
800 tanβ=5 tanβ=5 tanβ=5
A =800 GeV A =900 GeV A =1000 GeV
0 0 0
600
400
200
200 400 600 800 200 400 600 800 200 400 600 800 1000
m [GeV]
0
FIG.2: Allowedparameterspaceofthemodelfortanβ =5andµ>0. TheparameterA0 changesasindicatedonthepanels.
1000
800
600
400
200
1000
V] 800
e
G
600
[2
1/
m 400
200
1000
800
600
400
200
200 400 600 800 200 400 600 800 200 400 600 800 1000
m [GeV]
0
FIG. 3: LikeFig. 2 but for tanβ=10.
8
1000
800
600
400
200
1000
V] 800
e
G
600
[2
1/
m 400
200
1000
800
600
400
200
200 400 600 800 200 400 600 800 200 400 600 800 1000
m [GeV]
0
FIG. 4: LikeFig. 2 but for tanβ=15.
1000
800
600
400
200
1000
V] 800
e
G
600
[2
1/
m 400
200
1000
800
600
400
200
200 400 600 800 200 400 600 800 200 400 600 800 1000
m [GeV]
0
FIG. 5: LikeFig. 2 but for tanβ=20.
9
1000
800
600
400
200
1000
V] 800
e
G
600
[2
1/
m 400
200
1000
800
600
400
200
200 400 600 800 200 400 600 800 200 400 600 800 1000
m [GeV]
0
FIG. 6: LikeFig. 2 but for tanβ=25.
1000
800
600
400
200
1000
V] 800
e
G
600
[2
1/
m 400
200
1000
800
600
400
200
200 400 600 800 200 400 600 800 200 400 600 800 1000
m [GeV]
0
FIG. 7: LikeFig. 2 but for tanβ=30.
10
1000
800
600
400
200
1000
V] 800
e
G
600
[2
1/
m 400
200
1000
800
600
400
200
200 400 600 800 200 400 600 800 200 400 600 800 1000
m [GeV]
0
FIG. 8: LikeFig. 2 but for tanβ=35.
theSUGRAbenchmarkpointsSPS1a,SPS1b,SPS2,and SPS3).
SPS3, are shown on Figs. 10–13. In this calculation we Letusalsocomparethenewlycomputedcontributions
have not discriminated the results which were exceeding with similar contributions coming from simplest loops
the upper limit for µν. with no mass insertions (trilinear RpV couplings only)
The existing limits on Λ’s, obtained withtin SUSY and loops with bilinear insertions on the external neu-
models, oscillate around 10−2 10−5, depending on the trino lines (neutrino–neutralino mixing). We do it for
−
method used [11, 35]. We check here the range between two points, for which earlier calculations have been pre-
few 10−2 and 10−10. We notice that the RGE running sented in Refs. [14, 16] – the unification with low val-
decr×eases the values of Λ’s for higher energies [20], thus ues: A =100 GeV, m =m =150,andhigh values:
0 0 1/2
the parameter Λ sets the upper limit on them. A = 500 GeV, m = m = 1000. In both of these
0 0 0 1/2
Wefirstnotice,thatbelowroughlyΛ =10−5(insome cases tanβ =19 and µ>0.
0
cases even more) the results stabilize and do not change The results are shown in Tab. I for the low point
with decreasing Λ , which makes them almost indistin- and Tab. II for the high unification point. Below the
0
guishable with the situation when all RpV couplings are numericalvalues, results for pure trilinear loops [16] and
set to zero. On the other hand, the calculations broke bilinear contributions [14] are given as ranges, obtained
down for Λ > few 10−2. This leaves a rather nar- fordifferentcases(assumptionsofthenormalorinverted
0
×
row region of Λ , between 1 and 3 orders of magnitude hierarchy of neutrino masses, data from the neutrinoless
0
and only modestly exceeding 10−2, for which the RpV double beta decay searches).
couplings play a role. Forthelowpointthesmallestcontributionsdiscussed
We see also that for small Λ the general alignment in this paper are at least 5 orders of magnitude greater
0
ofthe µ’sresembleshierarchicalstructure,while forhigh than any other calculated so far. This clearly shows,
Λ this hierarchy vanishes, and the results show no reg- that the loops with heaviest particles (like higgsinosand
0
ular pattern. Too high values of Λ can boost them charginos) tend to dominate the overall contribution to
0
to values close to 10−6µ , which are excluded by ex- the magnetic moments. This result is not totally un-
B
periments like MUNU [32] (recall upper limit from the expected, but it was not fully clear, if the higher order
MUNU 10−10µ ). Curiously, for small (or zero) val- process (amplitude proportional to four instead of two
B
∼
uesofΛ ,allthetransitionmagneticmomentstendtobe coupling constants) will not suppress the effect coming
0
ofthe orderof10−10µ (SPS1a,SPS1b)orlower(SPS2, directly from the masses of the heavy particles. That
B