One-loop flavor change in Little Higgs models

0.35 MB·English

by  Jose I. Illana

1 One-loop flavor change in Little Higgs models
∗
1
0 Jos´e I. Illanaa and Mark D. Jenkinsa
2
aCAFPE and Departamento de F´ısica Teo´rica y del Cosmos,
n
a Universidad de Granada, E-18071Granada, Spain
J

0 TheLittleHiggs(LH)ideaattemptstocurethelittleamountoffine-tuningnecessarytobridgethegapbetween
1 theHiggsmass(electroweakscale)andthenewphysicsscalesuggestedbyelectroweakprecisiontests(∼10TeV).

However, we show that LH models do not survive the confrontation with experimental limits on lepton flavor
]
h mixing,assumingthesamenaturalnessargumentsthatmotivatetheirintroduction. TwodifferentLHmodelsare
p analyzed and several aspects of their one-loop predictions for lepton flavor-changing processes are discussed.
-
p
e
h
[ 1. The hierarchy and the flavor problems sigma model. For physics above Λ, ultraviolet

completion of the theory would be required, that
2 The Standard Model (SM) suffers a hierarchy
is both unknown and beyond the current experi-
v
problem: the Higgs mass must be of the or-
3 mental reach. The global symmetry is explicitly
der of the electroweak scale in spite of receiv-
7 brokenby gaugeandYukawa interactions,giving
5 ing quadratic loop corrections of the order of the the Higgs a mass and non-derivative interactions
3 Planck scale (theory cutoff), when gravity sets
while preserving the cancellation of just the one-
7. in. Naturalnessargumentsthenrequirethatsome loop quadratic corrections, thanks to the collec-
0 new physics must show up earlier, around the
tive symmetry breaking [3]. The sensitivity to a
0 TeV, just the region to be explored by the LHC.
10 TeV cutoff at two loops is not considered un-
1
Supersymmetry is one of the favourite sce-
: natural.
v narios for physics beyond the SM. In su-
Ontheotherhand,neithersupersymmetricnor
i persymmetric models extra loop contributions
X LH models are taylored to explain the observed
from superpartners approximately cancel the
r suppression of flavor-changing neutral currents
a quadratic Higgs mass corrections, provided that
(FCNC), that can be naturally accommodated
thesupersymmetry-breakingtermsareatmostof
in the SM by the GIM mechanism but demands
the order of the TeV. However, a little hierarchy
alignmentbetweenthestandardYukawamatrices
problem persists because in most of the param-
and the new physics interactions. It can be ar-
eter space of these theories the Higgs mass lies
guedthatsupergravityprovidesthe initialcondi-
below the current experimentallower bound, im-
tionsinthesupersymmetryframeworktoexplain
plying a fine-tuning at the few percent level [1].
the needed fine tuning, but in LH models the
The Little Higgs (LH) models [2] were intro-
introduction of new gauge bosons and fermions
duced to offer an explanation to the little hierar-
with arbitrary flavor structure challenges the fla-
chy between the electroweak scale v = 246 GeV
vor problem.
and the expected new physics scale f 1 TeV.
∼ The purpose of this talk is to report on recent
In these models the Higgs is a pseudo-Goldstone
works[4,5,6]thathaveexploredthenaturalnessof
boson of an approximate global symmetry bro-
two different LH models as they confront the ex-
ken at f. The physics below a cutoff Λ 4πf
∼ ∼ perimental limits on lepton flavor-changing pro-
10 TeV, is effectively described by a non-linear
cesses. Previouscomplementarystudies onflavor
violation focused on a single model, the Littlest
∗TalkgivenbyJ.I.I.at“LoopsandLegsinQuantumField Higgswithout[7]andwithT-parity,forthequark
Theory”, 10th DESY Workshop on Elementary Particle
[8,9] and the lepton [10,11] sectors.
Theory,25-30April2010,Wo¨rlitz,Germany.
1
2
2. Lepton flavor mixing in Little Higgs heavymassesoforderf totheextrafermiondou-
blet, respecting gauge an T symmetries. Then
There are essentially two types of LH models:
the gauge interactions of the light left-handed
theProductgroupmodels,wheretheSMSU(2)
L fermions to the heavy ones are fixed and involve
groupcomesfromthediagonalbreakingoftwoor
the Goldstone fields [17], introducing corrections
moregaugegroups,andtheSimplegroupmodels,
to the vertices coupling two heavy right-handed
where the SU(2) comes from the breaking of a
L fermions to a SM gauge boson [4,9] that are cru-
larger group. We analyze below an example of
cial to keep the one-loop amplitudes ultraviolet
each type.
finite. Another subtlety concerns the SM right-
handed singlets, whose hypercharges can only be
2.1. Littlest Higgs with T-parity
accommodatedbyenlargingSU(5)withtwoextra
The Littlest Higgs with T-parity (LHT) [12,
U(1) factors [9].
13,14] is a non-linear sigma model based on the
In the case of three generations, the interac-
breaking at a scale f of a global SU(5) SO(5)
→ tion of a heavy gauge boson with a light and a
by the VEV of a 5 5 symmetric tensor. The
× heavy left-handed fermion is proportional to the
broken generators expand 14 Goldstone fields. A
misalignment of the respective Yukawa matrices:
subgroup [SU(2) U(1)] [SU(2) U(1)]
1 2
SU(5) is gauged×and gets×broken ×to the SM⊂ quarks: V =Vq†V , V =Vq†V , (1)
Hu H u Hd H d
StoUt(h2)eL4×unUb(r1o)kYe.n gTehneergaatourgse(bWos±o,nZs,γas)sowciilaltreed- leptons: VHν =VHl†Vν , VHℓ =VHl†Vℓ , (2)
main massless and the other 4 (WH±,ZH,AH) where VCKM = Vu†Vd and VHq is the rotation
get masses of the order of f after eating the performed to heavy left-handed quarks to diag-
corresponding would-be-Goldstone bosons. In a onalize the Yukawa matrix of the heavy fermions
second step, the electroweak symmetry breaking (analogously, VPMNS = VνVℓ† in the lepton sec-
(EWSB)SU(2) U(1) U(1) takesplace tor). One has in particular lepton-flavor chang-
when a compleLx×SU(2)Ydo→ublet (Q4EDof the 10 re- ing interactions of charged and neutral currents
maining Goldstone bosons) is identified with the proportionaltoVHiαℓν¯Hi L/WH†ℓαL,VHiαℓℓ¯iHL/A†HℓαLand
SM Higgs (3 would-be-Goldstone bosons, to be Viαℓ¯i /Z† ℓα.
Hℓ HL H L
eaten by W±,Z, plus the physical h). In addi-
2.2. Simplest Little Higgs
tion,adiscretesymmetry(T-parity)isintroduced
The Simplest Little Higgs (SLH) [18] is based
underwhichtheSMfieldsareevenand(mostof)
on the breaking of the global symmetry G
the new ones are odd, in order to be consistent ≡
[SU(3) U(1)] [SU(3) U(1)] [SU(2)
with electroweak precision data [15,16]. × 1 × × 2 → ×
U(1)] [SU(2) U(1)] when two scalartriplets
The gauge and T-parity invariant Lagrangian 1× × 2
Φ [(3,1)] and Φ [(1,3)] G acquire VEVs
of the gauge and scalar sectors is easy to ob- 1 2 ⊂
fcosβ and fsinβ, respectively. The brokengen-
tain in terms of covariant derivatives [8]. The
erators expand 10 Goldstone fields. A subgroup
fermion sector is much more complicated. For
SU(3) U(1) is gauged and gets broken to the
eachgeneration,onehasaSM(extra)left-handed × χ
SM group. Then 4 gauge bosons remain mass-
fermion doublet from a T-even (T-odd) combi-
less (W±,Z,γ) and 5 get heavy masses of or-
nation of two incomplete representations under
SU(5), Ψ [¯5] and Ψ [5]. In addition, an extra der f (X±,Y0,Y¯0,Z′) after eating 5 would-be-
1 2
Goldstone bosons. The other 5 Goldstones in-
right-handeddoubletisintroducedinanon-linear
five-dimensionalrepresentationΨ ofSO(5).1 Ψ cludetheSMHiggsdoublet,thatprovidesmasses
R 1
to the weak bosons after the EWSB at a lower
and Ψ are connected by a T-transformation
2
scale.
whereas Ψ is a T-eigenstate. A Yukawa La-
R
Again it is straightforward to express the La-
grangianthat couples Ψ and Ψ to Ψ provides
1 2 R
grangianforthescalarandgaugesectorsinterms
1Actually,ξΨR isa5underSU(5),whereξ=exp{iΠ/f} ofcovariantderivatives. Thewould-be-Goldstone
istheexponential oftheGoldstonefields. fields in the ’t Hooft-Feynman gaugeare also ob-
3
tained after a good amount of algebra [6]. To study all lepton flavor-changing processes avail-
build the fermion sector, the SM fermions are able: µ eγ, µ eee and µN eN (µ to e
→ → →
included into SU(3) multiplets together with a conversion in nuclei). The contributions to all
newheavyfermion. Eachleptonfamily iconsists them vanish at tree level and those at one-loop
ofanSU(3)left-handedtriplet[3],(ν ,ℓ ,iN ) , are organized in form factors written in terms of
L L L i
and two right-handed singlets, ℓ and N . For genericcouplingsforthemostgeneralverticesin-
Ri Ri
quarks there are two options. In the universal volving scalar, fermion and vector fields. The in-
embedding the quarks are introduced similarly to variantamplitudesareexpressedascombinations
the leptons: (u ,d ,iU ) , u , d and U . ofstandardloopintegrals[21],includingbubbles,
L L L i Ri Ri Ri
Howeverthis embedding is notanomalyfree. Al- trianglesandboxes,thathavebeencomputedal-
though this is not necessarily a problem in an ef- gebraically and reduced to a simple form.
fective theory, because additional fermions could The leading order of the amplitudes is v2/f2.
be addedatthe cutoff scaleto cancelthe anoma- InthecaseoftheLHTmodel,theT-parityforces
lies, one can construct an anomaly-free embed- only heavy particles (T-odd) in the loop. As a
ding with no additional degrees of freedom [20]: consequence, the loop integrals have a well de-
thethirdfamilyisanalogoustotheleptonsector, finedorder. Inconstrast,inthe SLHmodelthere
(t ,b ,iT ), t , b , T , but the first two fami- are heavy and light neutrinos or gauge bosons in
L L L R R R
liescontainconjugatetriplets[¯3],(d , u ,iD ), thesameloop,thatdemandsacarefulv/f expan-
L L L
−
d , u , D and (s , c ,iS ), s , c , S , re- sion of the loop functions for consistency, since
R R R L L L R R R
−
spectively. the coupling constants in the Feynman rules are
The flavor mixing in the quark sector is ex- obtained to a given order [6].
tremelyrich. Ingeneral,allquarksmixwithother Asanimportantremark,allformfactorsareul-
heavy and light quarks of every family. However travioletfiniteinbothmodels. Boxcontributions
we will be interested just in the effects that are are finite by simple power counting. The dipole
consequence of the mixing in the lepton sector, partofthepenguinsisfiniteevenforgenericcou-
soquarkintergenerationmixingwillbeneglected plingsanditissubleadingexceptforphotonpen-
[19]. Concerning leptons, after the EWSB the guins. The chirality-conserving(non-dipole) part
light and the heavy neutrino of the same family of the penguins is left-handed and receives con-
mix at order v/f and there is family mixing as tributionsfromtriangleandflavor-changingbub-
longastheYukawamatrixofheavyneutrinosand ble diagrams, with divergences that cancel ex-
that of charged leptons are not aligned. Thus, in actly thanks to the unitarity of the flavormixing
thebasiswheretheheavyneutrinosarediagonal: matrices. Incidentally, in the LHT model the Z-
penguinreceivescontributionsonlyfromtheW ,
ℓ0 = [V ℓ ] , (3) H
Li ℓ L i the heavy partner of the W (those from the A
H
νL0i = 1 −δν [VℓνL]i + (δ2),(4) and ZH vanish), and its form factor has exactly
(cid:20) NL0i (cid:21) (cid:20) δν 1 (cid:21)(cid:20) NLi (cid:21) O ν thesameappearanceasthatoftheSMwithmas-
sive neutrinos [22].
with δ = v/(√2ftanβ). This induces
ν
− Inordertosimplifythephenomenologicalanal-
lepton-flavor changing interactions of charged
ysis,weassumejusttwoleptongenerations. Then
currents proportional to ViαN /X†ℓ and
ℓ Li Lα theheavyleptonsectorconsistsofν andℓ for
δ ViαN /W†ℓ . Although the latter is sup- Hi Hi
ν ℓ Li Lα the LHTmodel2 andN forthe SLHmodel, with
pressedby v/f,both types contribute to the am- i
i=1,2. The mixing matrix is parameterized by
plitudes at the leading order.
V1e V1µ cosθ sinθ
V = = . (5)
3. Lepton flavor-changing processes (cid:20) V2e V2µ (cid:21) (cid:20) sinθ cosθ (cid:21)
−
After expandingthe Lagrangiansofbothmod- The gauge boson masses are determined by the
els to the required order in v/f one obtains the 2Adifferenceoforderv2/f2 inthemassesofνHiandℓHi
complete set of Feynman rules needed [4,6]. We isneglected.
4
LHT SLH
102 102
101 101
100 100
10−1 10−1
y=4 y=1
.10−2 . .10−2 .
0 e 5 10 15 20 0 e 5 10 15 20
f [TeV] f [TeV]
102 102
101 101
100 100
10−1 10−1
y=4 y=1
.10−2 . .10−2 .
-0.2 e -0.1 0 0.1 0.2 -0.2 e -0.1 0 0.1 0.2
sin2θ sin2θ
105 105
104 104
103 103
102 102
101 101
100 100
10−1 10−1
.10−2 . .10−2 .
10−3 10−2 10−1 100 101 102 103 10−3 10−2 10−1 100 101 102 103
y˜ y˜
Figure 1. Ratios of expectations to current limits for µ eγ (solid), µ eee (dashed), µ Ti e Ti
→ → →
(dotted) and µ Au e Au (dot-dashed) in the LHT (left) and the SLH (right). The µN eN curves
→ →
in the SLH model are doubled: the lower (upper) ones assume anomaly-free (universal) embedding.
5
breaking scale f and the heavy fermions masses reference value y =4, still of order one, has been
can be defined as m2 y M2 (LHT) or chosen for the LHT model. All these plots are
m2 y M2 (SLH). WHei a≡re lefit wWitHh just 4 free for degenerate qeuark masses. We have checked
Hi ≡ i X
parameters. Our choice is that in the LHT model, as long as we are far
from a cancellation point, the quark mixing of
m2 m2
f , θ , δ = H2− H1 , y =√y1y2 . (6) non-degenerate quarks does not change the re-
m m
H1 H2 sults much [5]. As a final remark, µN eN de-
The amplitudes scale as v2e/f2, are roughly pro- pends on β in the SLH model [6]. A s→afe refer-
portionaltosin2θandδandvarywithy. Namely, encevalueoftanβ =1wastakenalongthestudy.
the flavor violation effects are suppressed by the Cancellations for other values are possible.
Little Higgs scale and vanish unlessethe heavy
fermions mix and are non degenerate. Natural-
4. Conclusions
ness suggests that these parameters take the fol-
lowing reference values: Tables 1 and 2 summarize the present and fu-
ture experimentalconstraints onthe relevantpa-
f 1 TeV , sin2θ 1 , δ 1 , y 1 . (7)
∼ ∼ ∼ ∼ rameters of the LHT and SLH models, based
For the process µN eN the heeavy quarks also on naturalness arguments and avoiding acciden-
→
play a role. By default we will assume that they tal cancellations. Presently, µ e conversion
→
are degenerate and with masses of the order of in Au is the most stringent process. LH models
M (M ) in the LHT (SLH) model. solve the little hierarchy problem if f 1 TeV.
WH X ∼
Figure1showsourpredictionsfor the ratiosof However,itisremarkablethatcurrentlimitsfrom
branching fractions of µ eγ, µ eee and the lepton flavor-changing processes indicate a much
→ →
conversion rates of µN eN (in gold and tita- heavier scale f 10 TeV or else indicate a large
→ ∼
nium) to the present experimental limits [23] in amount of fine-tuning in the mixing with (or the
thetwoLittleHiggsmodelsunderstudy. Onepa- mass splitting of) the heavy fermion sector.
rameter is varied while the others are set to the Acknowledgments. The authors are grateful to
reference values above. Ratios above unity are PacodelA´guilaforafruitfulcollaboration. Work
excluded. The curves as a function of δ are not supported by the Spanish MICINN (FPA2006-
displayed but are almost indistinguishable from 05294), Junta de Andaluc´ıa (FQM 101, FQM
those of sin2θ, which provides evidence for the 03048) and European Union (MRTN-CT-2006-
behaviour of the amplitudes pointed out above. 035505: “HEPTOOLS”).
Also the scaling with f−4 of the ratios is obvi-
ous. The dependence on y is less evident and
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Table 1
Bounds onLHTparametersfrompresent[23]andfuture∗ [24]limits onleptonflavor-changingprocesses.
LHT µ eγ µ eee µAu eAu µTi eTi
→ → → →
Limit 1.2 10−11 10−13∗ 10−12 10−14∗ 7 10−13 4.3 10−12 10−18∗
× × ×
f/TeV> 3.00 9.93 3.98 12.6 9.11 5.13 234
sin2θ < 0.111 0.010 0.055 0.006 0.012 0.038 <10−4
δ < 0.106 0.010 0.062 0.006 0.012 0.039 <10−4
| |
Table 2
Bounds on SLH parametersfrompresent[23] andfuture∗ [24]limits onlepton flavor-changingprocesses.
For µN eN the numbers correspond to the anomaly-free (universal) embedding.
→
SLH µ eγ µ eee µAu eAu µTi eTi
→ → → →
Limit 1.2 10−11 10−13∗ 10−12 10−14∗ 7 10−13 4.3 10−12 10−18∗
× × ×
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| |
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