Lepton Flavor Violation in Extra Dimension Models

0.29 MB·English

by  We-Fu Chang

Lepton Flavor Violation in Extra Dimension Models
We-Fu Chang1,2, and John N. Ng2,
∗ †
1Institute of Physics, Academia Sinica, Taipei 115, Taiwan
2TRIUMF Theory Group, 4004 Wesbrook Mall, Vancouver, B.C. V6T 2A3, Canada
(Dated: February 7, 2008)
Models involving large extra spatial dimension(s) have interesting predictions on lepton flavor
violating processes. Weconsider some 5D models which are related to neutrinomass generation or
address thefermion masses hierarchy problem. Westudy thesignatures in low energy experiments
that can discriminate the different models. The focus is on muon-electron conversion in nuclei,
µ→eγ and µ→3e processes and their τ counterparts. Their links with the active neutrino mass
matrix are investigated. We show that in the models we discussed the branching ratio of µ → eγ
likerareprocessismuchsmallerthantheonesofµ→3elikeprocesses. Thisisinsharpcontrastto
most of the traditional wisdom based on four dimensional gauge models. Moreover, some rare tau
5
decaysare more promising than therare muon decays.
0
0
PACSnumbers: 13.35.Bv,13.35.Dx,11.10.kk
2

n
a
J

8 I. INTRODUCTION
1

IntheStandardModel(SM)withfifteenfermionsperfamilyneutrinosarestrictlymasslessandthechargedleptons’
1
weakeigenstatescanbe chosento be their mass eigenstates. Thus,eachgenerationhas a separatelyconservedlepton
v
1 number. If one neglects the tiny effects from nonperturbative processes, there is no lepton flavor violating(LFV)
6 interaction in SM. However, recent neutrino experiments show strong evidence that neutrinos have none zero masses
1 and the three active neutrinos mix[1, 2, 3, 4]. Most physicists take this to be a harbinger of new physics beyond the
1 SM. Moreover, finite neutrino masses alone would imply the existence of LFV in charged lepton sector analogous to
0
the quarks. Ifsoweexpectthe Glashow-Iliopoulos-Maiani(GIM)mechanismtobe operativeinthe leptonsectorthen
5
the rateofinducedLFVprocesseswillbe proportionaltotheneutrinomasssquaredifference,whichisoftheorderof
0
<10 3(eV)2. Hence,theywillbehopelesslysmallforexperimentalverification. Therefore,additionalingredientsare
/ −
h essential for a detectable LFV signature. It is very common in model building to have the new physics that generate
p
neutrino massesalsogivenrise to LFVreactions. This link appearsto be naturalalthoughthere is no guaranteethat
-
p thisiscaseinnature. Withthiscautionarynotewewillfocusattentiontonewphysicsthatlinksthetwophenomena.
e AmongthenumerousbeyondtheSMmodels,LFVsignaturesaremostintenselystudiedinsupersymmetric(SUSY)
h ones. The connection with neutrino masses is established through the seesaw mechanism which is the orthodox way
:
v ofgettingasmallmassfortheactiveneutrinos. Sincethe latterhasanaturalsettingingrandunifiedtheories(GUT)
i the end result are rather bedecked supersymmetric seesaw models; see e.g. [5]. Although the details are different the
X
genericsourceofLFVlies inthe mixing ofvarioussfermions. The right-handedMajorananeutrinosplaya secondary
r role in this class of models. In general it is natural to expect B(µ eγ) B(µ 3e) in SUSY models.
a
→ ≫ →
For non-supersymmetric models neutrino mass generation via the seesaw mechanism would required the right-
handed neutrinos to be of the GUT scale. In this simplest version all LFV are undetectable. Attempts are now
made to lower some right-handed neutrinos mandated by the seesaw mechanism to the TeV scale so that the seesaw
mechanismitselfcanbetestedexperimentally. IfsothenonecanoptimisticallyanticipateLFVsignaturesinthenext
round of experiments [6]. Independent of the details of the models one again expects B(µ eγ) B(µ 3e) to
→ ≫ →
hold true.
RecentlyanewavenuehasopenupintheconstructionofmodelsbeyondtheSMthatexploitsthepossibleexistence
ofextraspatialdimensions. Thesetheoriesareparticularlyinterestingphenomenologicallyinthebraneworldcontext.
It is fascinating that many long standing problems in the usual four dimensional (4D) field theories can be overcome
or take on new perspectives in these higher dimensional constructs. For example the hierarchy problem is solved
by invoking large extra dimensions. In this note, we would like to draw the readers’ attention to the models which
involve one or more flat extra spatial dimensions. Furthermore, we focus on those that address the neutrino mass
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
problem. In some cases, we predict a reversed pattern of B(µ 3e) B(µ eγ) compare to SUSY models. On
→ ≫ →
the experimental side, it shall be interesting to see this.
The current experimental limits on muon LFV have already put very stringent constraints on model building. On
the other hand, the limits from tau LFV are rather loose. We shall constraint the extra dimension models by the
muon rare processes data and place upper limits on the rare decays of the τ. To avoidany hadronic uncertainties we
shall focus on purely leptonic processes. We shall also discuss the possible ways to discriminate different models and
their connections to neutrino masses.
Inthisbriefreview,wegivefewexamplesofextradimensionmodelswhichgivepotentiallytestableLFVsignatures.
These LFV processesarealldirectly orindirectly relatedto the generationof neutrino masses. We comparethe LFV
processesinafivedimension(5D)SU(3) [7]andSU(5)[8]GUTmodelswhereneutrinoMajoranamassesaregenerated
W
radiatively without a right-handed neutrino which is viable but less discussed alternative to the seesaw mechanism.
A brief review of this construction is given in [9]. Also, we discuss the LFV processes in split fermion or multi-brane
scenario.
The following is our plan for the paper. In sec.II, we will first review the general operator analysis for the lepton
flavor violating processes. This will also set the notations for the rest of the discussions. Sec. III examines LFV in
a 5D SU(3) model. New results of the one-loop calculations are given here. For details of the model and neutrino
w
mass generation we refer to [7]. In Sec. IV, the LFV processes induced by 5D SU(5) model will be discussed. The
discussion here has not been presented before. The alternative way of studying the flavor problem using the split
fermion model is examined in Sec.V. Calculations of the LFV processes in this scenario involves many new unknown
parameters. The models lack predictive power even semi-quantitatively. However, very general generic trends for
LFV can be discerned even in this early stages of development. Our conclusions are given in Sec. VI. The necessary
5D gauge fixing details, which is crucial for loop calculations, are presented in an appendix.
II. GENERAL OPERATOR ANALYSIS
Firstofall, we collectthe necessarygeneralformulasfor the study ofLFV processes. The mostimportantonesare
the effective interactions ofL l γ and L l Z where we use the notationL to denote the heavierchargedlepton
− − − −
which usually is either µ or τ and l is the lighter daughter lepton which can be µ or e.
In LFV studies, the most important contribution comes from the effective L l γ vertex. The similar vertex
− −
where a virtual Z replaces the γ is subdominant in the class of models we are considering. For definiteness we will
take L=µand l=e. The mostgeneralµ e γ interactionamplitude allowedby Lorentzand gaugeinvariancecan
− −
be written as:
qµqν iσµνq
= eA (q)u (p q) f (q2)+f (q2)γ5 γ gµν + f (q2)+f (q2)γ5 ν u (p ) (1)
M − ∗µ e µ− E0 M0 ν − q2 M1 E1 m µ µ
(cid:26) (cid:18) (cid:19) µ (cid:27)
(cid:2) (cid:3) (cid:2) (cid:3)
with the convention e = e > 0 used through out this paper and qµ is the photon 4-momentum. For real photon
| |
emission, only f and f contribute. But if a off-shell photon is involved, then all 4 form factors contribute. After
E1 M1
proper renormalization, the amplitude is finite as q2 0, so we must have f (0)= f (0)= 0. It is customary to
E0 M0
→
factor out q2 and rewrite the electric and magnetic form factors as
q2 q2
f (q2)= f˜ (q2), f (q2)= f˜ (q2) (2)
E0 m2 E0 M0 m2 M0
µ µ
and now f˜ (q2) and f˜ (q2) are finite at q2 0.
E0 M0
→
A. L→l1l2¯l3 and L→lγ
Using similar notations of [10], the most general effective lagrangianfor µ 3e and µ eγ can be expressed as:
→ →
√2
L = m A e σµνµ F +m A e σµνµ F +g (e µ )(e e )+g (e µ )(e e )
µ R R L µν µ L L R µν 1 R L R L 2 L R L R
− 4G
F
+ g (e γµµ )(e γ e )+g (e γµµ )(e γ e )
3 R R R µ R 4 L L L µ L
+ g (e γµµ )(e γ e )+g (e γµµ )(e γ e )+h.c. (3)
5 R R L µ L 6 L L R µ R
2
where
√2e √2e
A = [f (0)+f (0)] , A = [f (0) f (0)]. (4)
R −8G m2 E1 M1 L −8G m2 M1 − E1
F µ F µ
Also note the anapole form factors f and f have vector like effective contributions to g :
E0 M0 3 6
−
√2e2
δg =δg = f˜ (0) f˜ (0) (5)
3 5 4G m2 E0 − M0
F µ
h i
√2e2
δg =δg = f˜ (0)+f˜ (0) (6)
4 6 4G m2 E0 M0
F µ
h i
which shall be included in the g . The above effective lagrangianleads to
3,4,5,6
B(µ eγ) = 384π2(A 2+ A 2) (7)
L R
→ | | | |
g 2+ g 2
B(µ 3e) = | 1| | 2| +2(g 2+ g 2)+ g 2+ g 2
3 4 5 6
→ 8 | | | | | | | |
+8eRe[A (2g +g )+A (2g +g )]
R 4∗ 6∗ L 3∗ 5∗
m 11
+64e2 ln µ (A 2+ A 2) (8)
R L
m − 8 | | | |
(cid:26) e (cid:27)
if electron mass is ignored.
To carryoutthe calculation,it’s convenientto define two dimensionlessvariables x =2E /m and x =2E /m .
1 1 µ 2 2 µ
However, it is important to keep m2 terms in the intermediate steps in order to properly extract the finite term in
e
the last line of Eq.(8). Our result agrees with [10, 11].
The expressions of Eq.(8), except the last line of dipole operators, can also apply to τ l+γ and τ l l l
1 1 3
→ →
processes . For τ eeµ¯,µµe¯processes, the part from dipole operators have double loop suppression from two flavor
→
violation vertices and resulting in an insignificant branching ratios thus can be safely ignored. For τ decay, the
branching ratios given above are normalized to B(τ eν¯ν ). This holds for subsequent discussions of τ decays.
e τ
→
Tocompletethestory,wealsogivetheexpressionforprocesseswithnoidenticalparticlesinthefinalstate,namely,
τ µe¯e,eµµ¯ or l =l =l . The above expression for the branching ratio is now modified to:
1 2 3
→ 6
g 2+ g 2
B(τ l l l ) = | 1| | 2| +(g 2+ g 2+ g 2+ g 2)
1 2 3 3 4 5 6
→ 4 | | | | | | | |
+8eRe[AR(g4∗+g6∗)+AL(g3∗+g5∗)]
m 3
+64e2 ln τ (A 2+ A 2) (9)
R L
m − 2 | | | |
(cid:26) 2 (cid:27)
with trivial extension of g s. In arriving the last line of Eq.(9), we have ignored the masses difference between m
i e
and m in phase space integration but keep the crucial mass singularity associated with the virtual photon. Not
µ
surprisingly, the approximation agrees very well with the actual numerical integrations.
IfthephotonicdipoleoperatoristheonlydominateLFVsource,wehavethefollowingmodelindependentprediction
for
B(τ eγ) = B(τ µγ) (10)
→ →
B(τ µee¯) 2α m 3
τ
→ = ln (11)
B(τ eγ) 3π m − 2
→ (cid:26) e (cid:27)
B(τ eµµ¯) 2α m 3
τ
→ = ln (12)
B(τ eγ) 3π m − 2
→ (cid:26) µ (cid:27)
to the accuracy of m2/m2. To our knowledge, the last two relations have not been presented before.
µ τ
B. µ−e conversion in nuclei
We can write the effective LFV Lagrangianfor µ e conversionas:
−
Leff = e¯ s pγ5 µ q¯ s p γ5 q+e¯γα v aγ5 µ q¯γ v a γ5 q
q q α q q
√2G − − − −
F q q
(cid:0) (cid:1) X (cid:0) (cid:1) (cid:0) (cid:1) X (cid:0) (cid:1)
3
1
+ e¯ t +t γ5 σαβµ q¯σ q+H.c. (13)
s p αβ
2
q
(cid:0) (cid:1) X
with self explanatory notations. Here, flavor changing terms in the quark sector are not included since they are not
expected to be important here. The effective couplings are normalized to (√2G )1/2. For example, the SM Z boson
F
has a vector coupling to quarks given by
vq =T 2Qsin2θ.
3
−
To calculate the conversion rate, we need to promote the interaction from quark level to the nucleon level by
computing the matrix elements N q¯Γq N = Gq,NN¯ΓN with N denotes a nucleon and Γ = 1,γ5,γ ,γ γ5,σ .
h | | i Γ { α α αβ}
Since the coherent process is the important one only vector and scalar operators matter:
pq¯γ q p =Gq,pp¯γ p, nq¯γ q n =Gq,nn¯γ n (14)
h | α | i V α h | α | i V α
and
pq¯q p =Gq,pp¯p, nq¯q n =Gq,nn¯n. (15)
h | | i S h | | i S
By conserving of vector current, in the q2 0 limit, one can determine that Gu,p = Gd,n = 2 and Gu,n = Gd,p = 1.
∼ V V V V
However,onehasto relyonthe nucleonmodelto evaluatethe scalaroperator. Forqualitativeestimation,we willuse
the result G G from full non-relativistic quark model but the reader should keep in mind that the uncertainty
S V
∼
of nucleon model could be as large as few tens percent[12]. Following the approximationsused in [13], the conversion
rate, normalized to the normal muon capture rate Γ , can be expressed as[10, 13, 14]:
capt
p E G2F2m3α3Z4
B = e e F p µ eff 4eA Z+(s p)S +(v a)Q 2+ 4eA Z+(s+p)S +(v+a)Q 2 (16)
conv 2π2ZΓ | L − N − N| | R N N|
capt
(cid:8) (cid:9)
by assuming that the proton and neutron density are equal and the muon wave function does not change very much
in the nucleus, and F is a form factor whose definition can be found in [13] and p (E ) is the electron momentum
p e e
(energy),E p m . For48Ti(27Al),F 0.55(0.66),Z 17.61(11.62),andΓ 2.6(0.71) 106s 1[15].
e ∼ e ∼ µ 22 13 p ∼ eff ∼ capture ∼ × −
Where the coherent vector and scalar coupling strength of nuclei N are defined as
S su(2Z+N)+sd(2N +Z), (17)
N
≡
Q vu(2Z+N)+vd(2N +Z). (18)
N
≡
Iftherearemorethanonegaugeorscalarbosonsmediatethis process,the aboveexpressioncanbe triviallyextended
with modified couplings:
M2
(s p)S (si pi)Si Z , (19)
± N ⇒ ± NM2
i Hi
X
M2
(v a)Q (vi ai)Qi Z . (20)
± N ⇒ ± NM2
i Zi
X
Note that the form factors f˜ and f˜ in Eq.(2) have extra contribution to the vector couplings:
E0 M0
2eM 2eM 2eM
δv = Wf˜ , δa= Wf˜ , δv = WQ , (21)
E0 M0 q q
− gm − gm gm
µ µ µ
and if Eq.(2) is the only LFV source, then Eq.(16) reduces to the well-known formula given in [13]
8m F2α5Z4 Z
Bγ = µ p eff f +f 2+ f +f 2 . (22)
conv Γ | M1 E0| | M0 E1|
capt
(cid:8) (cid:9)
Also a model-independent relation between the µ e conversionand the µ eγ
− →
m5G2F2α4Z4 Z f +f 2+ f +f 2
Bγ = µ F p eff | M1 E0| | M0 E1| B(µ e+γ). (23)
conv 12π3Γ f 2+ f 2 →
capt (cid:18) | M1| | E1| (cid:19)
The above brief review is sufficient for the phenomenological analysis we do. Next, we will head for the extra-
dimensional models and discuss their LFV signatures.
4
III. 5D SU(3)W UNIFICATION MODEL
It has been known for a long time that the SM lepton left-handed doublet and the right-handed singlet charged
lepton in each family can beautifully form an SU(3) fundamental representation, i.e. L = (e,ν,ec)T [16]. This is
W L
implemented in an electroweak only unification in which SU(2) U(1) is unified to SU(3) . One of the attractive
W
points of this unification model is the tree level prediction of sin×2θ = 1/4. Renormalization group considerations
W
point to a relativelylow scaleof unificationat few TeV. We shalluse U 2,V to denote the SU(3) /(SU(2)
± ± W
∼ { } ×
U(1))gaugebosonswhichhaveSMquantumnumber(2, 3/2). In4D,theSU(3) GUThasafundamentaldifficulty
W
±
of embedding quarks into SU(3) representations. This problem can be circumvented by promoting the model into
W
five dimensional space time [17] and [7]. We give a brief summary of the model construction here.
The extra spatial dimension, with coordinate denoted by y, is compactified into an S /(Z Z ) orbifold. Where
1 2× 2′
the circle S of radius R, or y = [ πR,πR], is orbifolded by a Z which identifies points y and y. The resulting
1 2
− −
space is further divided by a second Z acting on y =y πR/2 to give the final geometry.
2′ ′ −
We now have tow parity transformations P : y y and P : y y under which the bulk fields can be
′ ′ ′
↔ − ↔ −
assignedeitherofthe eigenvalues+ or-. This freedomisusedto breakthe bulkSU(3) symmetryto SU(2) U(1).
W
×
Explicitly, one assigns the following properties to bulk gauge fields
(y)=P ( y)P 1, (y )=P ( y )P 1
µ µ − µ ′ ′ µ ′ ′−
A A − A A −
(y)= P ( y)P 1, (y )= P ( y )P 1 (24)
5 5 − 5 ′ ′ 5 ′ ′−
A − A − A − A −
where the matrices P=diag +++ and P =diag ++ . Now the (Z ,Z ) parities of the SM gauge bosons and
{ } ′ { −} 2 2′
the U,V gauge bosons are (++) and (+ ) respectively. It is easy to work out the Fourier eigenmodes propagating
−
in the bulk and see that only fields with (++) parity have zero modes. In other words, only SM gauge bosons have
zero modes. Both the U,V gauge bosons and all the y components are heavy KK excitation. Note the second Z is
necessary to avoid the presence of zero modes for both−SM gauge boson and the exotic U 2,V boson at the sa2′me
± ±
time.
The SU(3) symmetry is explicitly broken to SU(2) U(1) at the y = πR/2 fixed point, where the 4D quarks
W L
×
field are forced to live on it. The extra degree of freedom in extra dimensional theories is the key to incorporate SM
quarks into the SU(3) symmetry. On the other hand, the lepton fields can be placed anywhere in the bulk or on
W
either two fixed points. We choose to put the 4D lepton triplets at y = 0 which is a SU(3) symmetric fixed point
W
so that they enjoy the SU(3) symmetry. This also avoids possible proton decay contact interactions.
W
One Higgs triplet 3 plus one Higgs anti-sextet 6¯ , denoted as φ , with parities
6
φ (y) = Pφ ( y) ,φ (y )=Pφ ( y )
3 3 3 ′ ′ 3 ′
− −
φ (y) = Pφ ( y)P 1 ,φ (y )= Pφ ( y )P 1. (25)
6 6 − 6 ′ ′ 6 ′ ′−
− − −
is the minimal scalarsetto giveviable chargedfermionmasses (see [7] ). Another Higgs triplet 3 with parities (+ )
′
−
is introduced to transmit lepton number violation essential for generating Majorana neutrino mass through one-loop
diagrams [7] by a triple Higgs interaction of the type of 3T6¯3. This is a 5D realization of radiative neutrino mass
′
generation first proposed in [18]. The resulting mass matrix is necessarily of the Majorana type.
Now we have all the ingredients to write down explicitly the 5D Lagrangiandensity
1
= Tr[G GMN]+Tr[(D φ ) (DMφ )]
5 MN M 6 † 6
L −2
+ (D φ ) (DMφ )+(D φ ) (DMφ )
M 3 † 3 M ′3 † ′3
f3 f′3
+ δ(y) ǫ ij (La)cLbφc +ǫ ij (La)cLbφ′c
" abc√M∗ i j 3 abc√M∗ i j 3
f6
+ ij (La)cφ Lb+L¯iγµD L
√M∗ i 6{ab} j µ #
m
V (φ ,φ ,φ ) φTφ φ +H.c.
− 0 6 3 ′3 − √M 3 6 ′3
∗
+ +quark sector. (26)
GF
L
where G ,M,N = 0,1,2,3,y is the 5D field strength and D is the 5D covariant derivative. The cutoff scale
MN M
{ }
M is introducedto make the coupling constantsdimensionless. The other notations areself explanatory. The quark
∗
sector is not relevant now and will be left out. The complicated scalar potential is gauge invariant and orbifold
5
symmetric and will not be specified since it is not needed here. To perform loop calculations, we need to specify the
5D gauge fixing term, , which will be exhibited later.
GF
L
The fields and their parities of this model are summarized below:
8µ = (1,0) +(3,0) +(2, 3/2) +(2,+3/2)
++ ++ + +
− − −
Bµ Aµ (U,V)µ
8y = |(1,{0z) }+|(3,{0z) }+|(2, 3/2) +{+z(2,+3/2) +}
−− −− − − −
By Ay (U,V)y
3 = |(2,{z1/}2)+|++{z(1,}1)+| {z }
− −
HW1 HS
3 = (2, 1/2) +(1,1)
′ | {z +} | {z++}
− −
HW′ 1 HS′
¯6 = |(3,+1{)z+ +}(2,| 1{/z2)+}++(1, 2)+
− − − −
HT HW2 HS2
wheretheSMquantumnumbersare(S|U(2{z) ,U}(1)| )an{zdthe}sub|scri{pztsla}beltheparitiesP,P . Thenitisstraightfor-
L Y ′
wardtoobtainthe 4Deffective interactionby integratingovery andthe 4Deffective gaugecouplingcanbe identified
as g = g˜/√2πRM . The orbifold construction is engineered such that there is no tree level LFV in the SM gauge
2 ∗
interactions. Thus, the success of that model remains intact. But the tree level LFV interaction emerge in the U,V
gauge interaction which are heavy KK excitation and in the Yukawa interactions.
The LFV charged current is
LCC =g2 eLiγµPL(Ulep)ijecRjUn−,µ2 +H.c.
n=1
X
+g ν γµP ( ) ec V 1+H.c. (27)
2 Li L Ulep ij Rj n−,µ
n=1
X
where the subscripts L and R are kept for book keeping. The matrices U are used to diagonalize the charged
L,R
lepton mass matrix and Ulep =UL†UR∗ is an extra CKM-like unitary mixing matrix for the lepton sector.
The LFV Yukawa interactions are given by
1
LY = √2πRM κn fS3ijecL,iνL,jHS+,n+fH3ij eR,ieL,jHW0 1,n+eR,jνL,iHW−1,n −(i⇔j)
∗ nX=0 h (cid:16) (cid:17) i
+ √2π1RM κn fS′3ijecL,iνL,jHS′+,n+fH′3ij eR,ieL,jHW′01,n+eR,jνL,iHW′−1,n −(i⇔j)
∗ nX=0 h (cid:16) (cid:17) i
1
+ κ f6 ec e H+2 +(ec ν +νc e )H+ +νc ν H0
√2πRM n Tij L,i L,j T,n L,i L,j L,i L,j T,n L,i L,j T,n
∗ nX=0 h (cid:16) (cid:17)
+ fH6ij (eR,ieL,j+eL,ieR,j)HW0 2,n+(eR,iνL,j +νL,ieR,j)HW−2,n +fS6ijeR,iecR,jHS−22,n +H.c. (28)
(cid:16) (cid:17) i
where κn =(√2)1−δn,0 and
fT6 =ULTf6UL, fS6 =URTf6UR, fS(′)3 =ULTf(′)3UL, fH(′)3 =UR†f(′)3UL, fH6 =UR†f6UL. (29)
Note that in the new basis the symmetry of f and f are not changed.
T S
A. L→l+γ transition
Webeginthediscussionbystudyingaspecialcasethatf f ,suchthatU U alsof6,f6 andf6 areroughly
6 ≫ 3 R ∼ L∗ T H S
diagonal. This hierarchical Yukawa structure is also demanded to yield the observed charged lepton mass hierarchy.
In other words, all the LFV sources are in the Yukawa interaction of φ and φ . Since φ has nothing to do with
3 ′3 ′3
the charged lepton masses, we can further assume its LFV contribution is larger than φ , whose coupling is roughly
3
(m/M )(f /f ), and f′3 f′3.
∼ W 3 6 S ∼ H
6
In general this class of decays proceeds via the one-loop diagrams. The ones involving the gauge boson U 2 and
±
V are suppressed by the GIM mechanism. This leaves the singly charged and neutral scalars as the only possible
±
contributors since they both carry two units of lepton charges in the usual scheme. We thus conclude that these
decays are dominated by the scalar induced M1 and E1 operatorsonly. Therefore, they provide unique probes of the
exotic scalar sector. Later we will show that in contrast L 3l probes the gauge interactions of the model.
→
In this case, the leading contribution loop diagrams are shown in Fig.1. Now, briefly discuss the gauge fixing in
00 0(cid:6) 0(cid:6)
HW1 HW1 (cid:13)n;Zn HS (cid:13)n;Zn
li
(cid:22) e (cid:22) (cid:23)i e (cid:22) (cid:23)i e
(a) (cid:13)n;Zn (b) 0(cid:6) (
) 0(cid:6)
H H
W1 S
li
(cid:23)i (cid:23)i
(cid:22) e
HW001 (cid:22) e (cid:22) e
(d) (e) (f)
Zn Zn Zn
FIG.1: Theleadingcontributionsforµ→eγandµ−econversioninthecasedescribedinSEC.IIIA. Thelabelsn≥0indicate
theKK level.
thismodel. Becausethe orbifoldparityforφ ischosentobe(+ ),itcannotdevelopaVEVanddoesn’tparticipate
′3 −
the electroweak breaking. The Goldstone bosons consist of the y-components of gauge bosons and the proper linear
combinations of φ3 and φ6. And the whole 3′, HW′01,HW′±1 and HS′ are physicalHiggs. So now it is straightforwardto
carry out loop calculation. For further details, see Appendix.
The E1,M1 form factors are calculated to be:
m2 ǫ
fLl = µ f + (9f +7f ) (30)
M1 384π2M2 Li 24 Ri Li
S0 Xi h i
m2 ǫ
fLl = µ f (9f 7f ) (31)
E1 384π2M2 Li− 24 Ri− Li
S0 Xi h i
′
where fLi =fR∗i ≡fl∗ifiL, MS0 is the zero mode mass ofHS±, andǫ=(πMS0R)2 ∼O(0.1). On arrivingatthe above
expression, the contributions of all KK scalar excitation running in the loop have been summed. And if we drop the
ǫ-terms, the resulting branch ratio can be expressed as:
2
96π3α α
B(L l+γ) = fLl 2+ fLl 2 f (32)
→ G2m4 | E1| | M1| ∼ 768πG2M4 (cid:12) Li(cid:12)
F µ F S0 (cid:12)i=e,µ,τ (cid:12)
(cid:0) (cid:1) (cid:12) X (cid:12)
4 (cid:12) (cid:12)
= 2.75×10−6 30M0GeV (fS′3,le)∗fS′3,eL+((cid:12)(cid:12)fS′3,lµ)∗fS′3,µ(cid:12)(cid:12)L+(fS′3,lτ)∗fS′3,τL 2 (33)
(cid:18) S0 (cid:19) (cid:12) (cid:12)
Because the Yukawa couplings of triplet scalars are anti-s(cid:12)ymmetric, the L l+γ processes have foll(cid:12)owing forms:
(cid:12) → (cid:12)
4
B(µ e+γ)=2.75 10 6 300GeV (f′3 ) f′3 2 (34)
→ × − M S,eτ ∗ S,µτ
(cid:18) S0 (cid:19) (cid:12) (cid:12)
B(τ →e+γ)=2.75×10−6 30M0GeV 4(cid:12)(cid:12)(fS′3,eµ)∗fS′3,µτ(cid:12)(cid:12)2 (35)
(cid:18) S0 (cid:19) (cid:12) (cid:12)
B(τ µ+γ)=2.75 10 6 300GeV 4(cid:12)(cid:12)(f′3 ) f′3 (cid:12)(cid:12)2 (36)
→ × − M S,eµ ∗ S,eτ
(cid:18) S0 (cid:19) (cid:12) (cid:12)
WehavetakenM =300GeVasthereferencepoint. IfalloftheYukawa(cid:12) couplingsar(cid:12)erealandnoneofthemvanishes,
3′ (cid:12) (cid:12)
their ratios can be further simplified to:
1 1 1
B(µ e+γ):B(τ e+γ):B(τ µ+γ)= : : . (37)
→ → → f′3 2 f′3 2 f′3 2
| S,eµ| | S,eτ| | S,µτ|
7
AtthispointonecanusethedataB(µ e+γ)<1.2 10 11[19]toobtaintheconstrain (f′3 ) f′3 <2.1 10 3.
→ × − S,eτ ∗ S,µτ × −
This is consistent with the expectation from the study of neutrino mass in this as given(cid:12)in [7]. There(cid:12) it was found
that the Yukawa coupling f′3 has to be <10 2 and the f’s exhibit the pattern f′3 >f′3(cid:12)(cid:12)>f′3. Hence(cid:12)(cid:12)it reasonable
eµ − eµ eτ µτ
µ e+γ to occur at a rate less than tw∼o orders of magnitude below current level. Indeed in the SU(3) model we
W
→
can link the various L lγ transition branch ratios to the light neutrino mass matrix elements. Assuming that the
→
light neutrino mass are mostly coming from the one-loop quantum correctioninvolving the zero modes of φ and φ ,
′3 6
we have the prediction :
m 4
µ
B(µ e+γ):B(τ e+γ):B(τ µ+γ) m m :m m :m m (38)
13 23 12 23 12 13
→ → → ∼ m
(cid:18) τ(cid:19)
where m is the (ij) entry of the light neutrino mass matrix. Interestingly the model naturally accommodates an
ij
active neutrino mass matrix of the inverted hierarchy type as follows:
ǫ 1 1
m 1 ǫ ǫ (39)
∼1 ǫ ǫ2 
 
where ǫ 0.1. From the above equations, we see that µ eγ is suppressed compared to the τ lγ decays. This is
∼ → →
a striking feature of the model.
B. µ−e conversion
The µ e conversion in nuclei will be dominated by the virtual photon exchange. Compared to µ eγ it has
− →
additional contributions from the anapole terms. The corresponding photon E0,M0 form factors can be derived as:
f˜ ( k2) = m2µ f + ǫ (3f +f ) 6ǫ(f +f ) ∞ G(δn,xi) (40)
E0 − 576π2MS20 i=e,µ,τ" Li 24 Ri Li − π2 Ri Li n=1(2n−1)2#
X X
f˜ ( k2) = m2µ f + ǫ (3f f ) 6ǫ(f f ) ∞ G(δn,xi) (41)
M0 − 576π2MS20 i=e,µ,τ" Li 24 Ri− Li − π2 Ri− Li n=1(2n−1)2#
X X
where δ =( k2)/M2 , x =m2/( k2), i=e,µ,τ, and
n − Hn i i −
1 √4x+1 1
G(δ,x)= lnδ lnx+ 4x+(1 2x)√1+4xln − . (42)
− − 3 − − √4x+1+1
′
As expected, the principal contribution is from the Fig.1(c) with the HS± zero mode running in the loop. The
logarithmic enhancements in G(δ,x), is due to the exchange of neutral scalars, H′0 ( see Fig.1(a)). Although they
W1
are suppressed by the KK masses we find them to be compatible to the charged singlet contribution.
In this process k2 m2 and G has the following limits
− ∼ µ
m2 1 m2 m2
G ln µ + , G ln µ 1.515, G ln µ 6.978 (43)
e,n ∼− M2 3 µ,n ∼− M2 − τ,n ∼− M2 −
Hn Hn Hn
The KK sum of these logarithmic enhancements are finite:
∞ 1 ln m2µ = π2 ln(m R)2 0.8362 (44)
(2n 1)2 M2 8 µ −
n=1 − Hn
X
So the desired anapole form factors can be expressed as
m2 ǫ 3ǫ
f˜ (m2) = µ f + (3f +f )+ (f +f )[ln(m R)2+η ] (45)
E0 µ 576π2M2 Li 24 Ri Li 4 Ri Li µ i
S0 i=e,µ,τ(cid:20) (cid:21)
X
m2 ǫ 3ǫ
f˜ (m2) = µ f (3f f )+ (f f )[ln(m R)2+η ] (46)
M0 µ 576π2M2 Li− 24 Ri− Li 4 Ri− Li µ i
S0 i=e,µ,τ(cid:20) (cid:21)
X
8
η ,η ,η = 1.011,0.837,6.300 . Again, since f′3 is anti-symmetric, only f = f can contributes. For
{ e µ τ} {− } S Lτ R∗τ
simplicity we assume there is no new CP violation in the scalar sector; then f = f and the µ e conversion rate
L R
−
in 48Ti can be expressed as:
22
Bγ 0.01B(µ e+γ) (47)
conv ∼ →
if taking 1/R = 2TeV and M = 300GeV as a reference point. It is also possible to have extra contributions from
S
KKphotonandKKZ excitation,Fig.1(a-f). Onewillneedtotakecareofthe KKnumberconservationinthescalar-
scalar-gauge boson vertices and sum over all the possible combinations. But generally speaking, their contributions
are further suppressed by (m R)2 <2 10 9 compared to the photon zero mode and can be safely ignored.
µ −
×
The relation of Eq.(47) is basedon the assumption that f f and φ is the dominate LFV source. However,we
should point out that if f is not so small the neutral scalar6z≫ero m3 odes c′3an make µ eγ and Bγ compatible and
3 → conv
deviates a lot from the pure photonic dipole prediction, Eq.(23).
Again, this demonstrates that L lγ and µ e conversion are very important for us to understand the Yukawa
→ −
structure in the SU(3) model.
W
The question now arises about the photonic dipole and anapole contribution to µ 3e. The answer lies in
→
Eqs.(30,31,40, 41). We estimated that
B(µ 3e)<0.04B(µ eγ). (48)
→ →
This predictionis not very sensitive to what the Yukawa pattern is. Moreover,the decays L 3l have overwhelming
→
contribution from other sources of new physics in the model to which we shall turn our attention to next.
C. L→3l
Acharacteristicofthe modelisthe existenceofdoublechargedgaugebosonswithLFVcouplings. Thiswillinduce
µ 3e like processes for the τ. In addition there are also KK scalars H and H which has LFV Yukawa couplings
T 0
→
which are largely unknown. The Feynman diagrams for the L 3l decays are depicted in Fig.2. Since the Yukawa
→
τ/µ l′ τ/µ l′ τ/µ l
τ/µ l′
U±2 H±2 H0
T
H0
′′
l l
′′ ′′ ′′ ′
l l l l l l
FIG. 2: Feynman diagrams which lead to τ(µ)→3l processes.
couplingaretotallyunknown,wewillpostponethediscussionofthecontributionsfromscalarsandlookatthebranch
ratios, normalized to B(τ eν ν¯ ), mediated by U 2 gauge boson alone first:
τ e ±
→
B(τ 3µ) = 2+ 2 2, (49)
τµ µτ µµ
→ F × |U | |U | |U |
B(τ 3e) = (cid:0) τe 2+ eτ 2(cid:1) ee 2, (50)
→ F × |U | |U | |U |
B(τ µ¯ee) = (cid:0) τµ 2+ µτ 2(cid:1) ee 2, (51)
→ F × |U | |U | |U |
B(τ µµe¯) = (cid:0) τe 2+ eτ 2(cid:1) µµ 2, (52)
→ F × |U | |U | |U |
B(τ µee¯) = F (cid:0) 2+ 2 (cid:1) 2+ 2 , (53)
τe eτ eµ µe
→ 8 |U | |U | × |U | |U |
(cid:0) (cid:1) (cid:0) (cid:1)
B(τ eµµ¯) = F 2+ 2 2+ 2 . (54)
τµ µτ eµ µe
→ 8 |U | |U | × |U | |U |
(cid:0) (cid:1) (cid:0) (cid:1)
where =(M πR)4/16=1.56 10 5(2TeV/1/R)4. From the analysis given in sec.II, we know all scalar operators
W −
F ×
give positive contribution. So even though we know nothing about the Yukawa couplings, we can still derive an
interesting lower bond from the unitarity of
lep
U
B(τ 3e) 2 1 2 (55)
ee ee
→ ≥F ×|U | −|U |
(cid:0) (cid:1)
9
for a given 1/R.
If one wants to keep compactification scale 1/R low, say 1.5 TeV, then we would require to be either close
ee
∼ |U |
to zero orone. Furthermore,ifwe takethe upper bound of 1/R<5 TeV derivedfromunification seriouslywe obtain
B(τ 3e)>8.0 10 7 2 1 2 . (56)
− ee ee
→ × |U | −|U |
On the other hand, if we assume that the bilepton gauge boson ex(cid:0)change is(cid:1)the dominating FCNC source, another
interesting upper bond can be derived:
4
2TeV
B(τ 3e)< F =3.9 10−6 (57)
→ 4 × 1/R
(cid:18) (cid:19)
with = 1/√2 in Eq.(55). Actually, if all the LFV Yukawa couplings are associated with φ as discussed in the
|Uee| ′3
previoustwosubsections,thetree-levelbileptonscalarcontributionstoτ 3evanishdue totheantisymmetryofthe
Yukawa couplings. However, the present experimental limit, 1 3 10 7→[20] will indicate that the compactification
−
radius is closer to the upper limit of 5TeV−1 for this particula−r ca×se.
IV. 5D SU(5) MODEL
TheorbifoldSU(3) modeldiscussedabovehasmanyinterestingandnovelfeatures;however,thefactthatquarks
W
andleptonshavetobetreateddifferentlyisanobstacletowardscompleteunification. It’sanaturalattempttofurther
unify the quarks and leptons in a larger GUT group. The simplest group for that is SU(5). Now all fermions are on
equal footing and can be clustered into 2 SU(5) representations,i.e. Ψ¯5 ={dc,L},Ψ10 ={Q,uc,ec}.
Similar to the SU(3) model, the model is embedded in the background geometry of S /Z Z orbifold. The
W 1 2× 2′
bulk SU(5) gauge symmetry is broken to the SM by orbifold parities , with parity matrices diag +++++ and
{ }
diag ++ for Z and Z transformations respectively. These are generalizations of the SU(3) case.
{−−− } 2 2′ W
Sincenoright-handedneutrinosareadded,neutrinomassescanbegeneratedthroughquantumcorrectionbyusing
either 10 or 15 bulk scalars plus the 5(10/15)¯5 interaction mandated by breaking to the SM gauge group. The
′
orbifold parities of 10 or 15 bulk scalars are determined to be (++) by considerations of proton decay. They split
into following components:
2 1
15 (++)=P 6,1, +T (1,3,1) +C 3,2, ,
s 15 15 ++ 15
−3 6
(cid:18) (cid:19)++ (cid:18) (cid:19)+
−
2 1
10 (++)=P ¯3,1, +S (1,1,1) +C 3,2, .
a 10 10 ++ 10
−3 6
(cid:18) (cid:19)++ (cid:18) (cid:19)+
−
Acarefulanalysisshowsthatbyusing15(10)theresultantneutrinomassmatrixfavorthenormal(inverted)hierarchy
[8]. It was also found that extra fine tuning efforts were needed to obtain phenomenologically acceptable neutrino
mass patterns by using 10 alone; so we will only discuss the case which implements 15.
The P components induce tree-level K0 K¯0 mixing. To satisfy the experimental constraints, it is required that
M >105GeV.Ontheotherhand,thetwo−bulkHiggsin5,5 whichareresponsiblefortheSMelectroweaksymmetry
P ′
breaking share the same (++) parities as 15. The brane Yukawa interaction term is easily constructed to be
f˜15
LY =δ(y)" Mij∗/2ψ¯5{iA}cψ¯5{jB}φ{15AB}+H.c.#, (58)
p
where A,B are the SU(5) symmetry indices. It can be seen that to contain the necessary LFV source to generate
neutrinoMajoranamasses.Theneutrinomassmatrixelementsareproportionalto( )ν m f′5f15 wherei,j,k
M ij ∝ k k ik jk
are the generation indices and m is the mass of k-chargedlepton running in the loop.
k
The extra Higgs doublet in the 5 is good for gauge unification. By adding additional dPecaplet bulk fermion pair
′
with (+ ) parity and mass around 10 120 TeV, the unification is achieved at 3 1016 1015 GeV or equivalently
− − × −
1/R 1014 GeV. The high scale unification or tiny radius of extra dimension makes KK excitation decouple from
∼
most phenomenological studies and basically we only need to consider the zero modes.
Below unification scale or equivalently the low energy 4D effective theory is a two Higgs doublets like model. In
general the two Yukawa patterns are not aligned which that can lead to severe tree level charged neutral flavor
changing(FCNC) interaction. A Z symmetry is usually assumedto forbidsuch tree levelFCNC [21]. Inthis model,
2
there is no such freedom since the Yukawa patterns are determined by the geometrical setup. The Ψ of the first
10
10