Electroweak Symmetry Breaking from SUSY Breaking with Bosonic See-Saw Mechanism

0.12 MB·English

by  Hyung Do Kim

Electroweak Symmetry Breaking from SUSY Breaking
with Bosonic See-Saw Mechanism
Hyung Do Kim ∗
School of Physics, Seoul National University, Seoul 151-747, Korea
We introduce the idea of bosonic see-saw mechanism in analogy with the see-saw mechanism.
Bosonic see-saw is a new symmetry breaking mechanism and we apply it to explain electroweak
symmetry breaking as an inevitable consequence of supersymmetry breaking. The breaking of
electroweak symmetry occurs at tree level once supersymmetry is broken. Absence of color/charge
breaking in this model is related to doublet-triplet splitting in grand unified theory. An extension
of MSSM with a weak triplet shows very interesting results especially when µ=0. It provides the
mostnaturalunderstandingofwhywehaveonlyelectroweaksymmetrybreakingratherthanhaving
5
color/charge breaking. In the limit µ = 0, the model predicts very light chargino mass, 104 GeV
0
while Higgs is heavy,130 GeV.
0
2

n The standard model(SM) has a beautiful structure of the electroweak symmetry breaking is a surprising can-
a explaining all the matters and forces except gravity in cellation of µ2 (500 GeV)2 and the Higgs soft scalar
0 J terms of quark/lepton(s) and gauge interactions. All masssquaredm∼2Hu [6]. Ifµwereslightlylarger,wewould
the quarks and leptons are massless as long as elec- never have the electroweak symmetry breaking. And if
1
troweak symmetry is unbroken, and they can get mass µ were slightly smaller, the Higgs would develop its vac-

1 from Yukawa interactions only after electroweak sym- uum expectation value (VEV) exponentially larger than
v metry breaking. Within the framework of the standard the weak scale. Thus it is desirable to consider models
9 model, Higgs potential can be arbitrary and we choose in which the weak scale electroweak symmetry breaking
5
the sign of the coefficient of quadratic (quartic) term to can be explained for a broad range of parameters.
0
be negative (positive) such that the Higgs potential has In this paper we first introduce a simple idea called
1
0 a desired mexican hat shape. It would not be easy to ’bosonic see-saw mechanism’. We apply ’bosonic see-
5 understand why the quadratic term has a negative sign saw mechanism’ to explain the electroweak symmetry
0 while the quartic term is positive within the standard breaking. When down type Higgs couples to the extra
/
h model. vector-like pair of Higgs, we can soften the little hierar-
p chy problem. Finally we consider a model in which the
- The SM is just regarded as a low energy effective the- electroweaksymmetryiscloselytieduptothesupersym-
p
e ory of some extended one and gauge hierarchy problem metry
h suggests a modification of the standard model at TeV Let us briefly discuss the conventional see-saw mech-
v: scale. Supersymmetry(SUSY) [1, 2] is one of the most anism explaining the lightness of neutrino masses [7, 8].
Xi promising candates for it. In supersymmetric extensions For ν, the left-handed neutrino which is an SU(2)L dou-
of the standard model, we can get a better understand- blet, and N a singlet, the possible interactions are
r ing of the electroweak symmetry breaking. First of all,
a
the Higgs potential is no longer arbitrary and should = MNN +l HLN +h.c., (1)
ν ν
L −
be a sum of supersymmetric F/D terms and soft super-
symmetry breaking terms. The quartic term is calcu- where H = (H+,H0) is the Higgs doublet and L =
lated from gauge couplings and is positive definite. The (ν,l−). After the electroweak symmetry breaking, it be-
quadratic term (soft terms) is a sum of supersymmetric comes
mass term (µ term) and soft supersymmetry breaking
ν 1 0 m ν
terms which are calculable in certain mediation mecha- Mν(cid:18)N (cid:19) = 2(cid:18)mD MD (cid:19)(cid:18)N (cid:19), (2)
nism of supersymmetry breaking. In the minimal super-
symmetricstandardmodel(MSSM)withgaugemediated where m = l H0 . The lightest neutrino mass for
D ν
SUSY breaking [3, 4, 5], a radiative correction by large h i
m M is then
D
top Yukawa coupling gives negative Higgs mass squared. ≪
If µ were zero, the above picture might have been beau- m2
m = D. (3)
tifulandcouldbeconsideredasapossibleexplanationof ν − M
the electroweaksymmetry breaking. However,in reality,
Note the sign of the lighter eigenvalue. As the deter-
minant of the matrix is negative definite (-m2 ) and the
D
heavieroneisnearlyM,thelightereigenvalueisnegative
∗[email protected] definite. TheresultisvalidaslongasmD M. Wecan
≪
2
make the mass term to be positive definite by the field From now on, we focus on the application of bosonic
redefinition of neutrinos. Therefore, this observation is see-saw mechanism to the electroweak symmetry break-
not important for neutrinos (fermions) but it will turn ing. As we have two Higgs fields H and H in MSSM,
u d
out to be very important for later consideration. there are three possibilites. First, H couples to heavy
u
Bosonicsee-sawmechanismworksforbosonsinsteadof Higgs. Second, H couples to it. Finally, both of them
d
fermions(neutrinos). Althoughthemechanismworksfor couple to it. When there is no radiative correction, the
any scalar fields (superfields), here we take Higgs as an first option looks the most natural. However, we know
example for a clear illustration. Supersymmetric exten- thattopYukawagiveslargeradiativecorrectionsandthe
sion of the standard model requires two Higgs chiral su- secondoptionisthebest. Ifbothofthemcoupletoheavy
perfieldsH andH . Supposethereisanadditionalmas- Higgs and X, we can not make them light and the third
u d
sive pair H′ and H′, the electroweak doublets with the option does not work. Thus we consider only the second
u d
oppositehypercharge. LetX = +F θ2beasuperfield possibility in this paper.
X
···
representing supersymmetry breaking F = F = 0. As H′ andH′ couple to X directly, they are the mes-
h Xi 6 u d
For the superpotential sengersof SUSY breakingand M is the messenger scale.
We can calculate soft terms mediated by gauge inter-
W = l1XHu′Hd+MHu′Hd′, (4) actions. We also assume that there is a pair of color
triplet Higgs fields which complete the messenger fields
the scalar mass squared matrix for H ,H′∗ is
d u into SU(5) multiplets. The soft terms from gauge medi-
ation are positive definite [9].
0 l∗F∗
ˆ2 = 1 . (5)
M (cid:18)l1F |M|2 (cid:19) m2 = 2c α 2Λ2. (8)
Φ i 4π
When √F M, the lightest scalar mass squared be- Xi (cid:16) (cid:17)
≪
comes negative definite, where Λ = l2F/M and ci is the quadratic Casimir of
2 i-th gauge g|roup. N|ote that l1 10−2l2 is required to
m2Hd = −(cid:12)(cid:12)lM1F(cid:12)(cid:12) . ohnavlyedΛiff∼ere1n0ceTweVithwthhieleusmu2HaldM∼S∼S1M00isGtehVe.trTeielllenvoewlctohne-
(cid:12) (cid:12)
Whenever F = 0, the mass sq(cid:12)uare(cid:12)d is negative and we tributionfrombosonicsee-sawwhichisnegativedefinite.
6 The most interesting consequence comes with the ad-
endupwithsymmetrybreaking. Therefore,attreelevel,
dtion of the electroweak triplet Σ. Let us explain why
weobtaintheelectroweaksymmetrybreakingasaconse-
we need Σ and how it brings an interesting result.
quence ofsupersymmetry breaking. We cando the same
InMSSM,thenicemechanismofradiativeelectroweak
thing to H instead of H . Note that the sign here is
u d
symmetry breaking is spoiled by large µ term. Large µ
physical as the matrix is for scalar mass squared. It is
term in MSSM is due to the fact that we have not seen
called ’bosonic see-saw mechanism’ as it is opposed to
Higgs yet. In MSSM, the quartic couplings are given
usual see-saw mechanism which works for the fermions.
by gauge couplings and Higgs mass is predicted to be
The bosonic see-saw mechanism shows a similarity to
light. m2 > 114 GeV requires a large radiative correc-
the (fermionic) see-saw mechanism. H
tion and it is possible only with heavy stop. If stop is
There are heavy states. (heavy Higgs v.s. N) heavy, radiative corrections are too large and the elec-
•
troweaksymmetrybreakingbecomeslargeunlesslargeµ
Thereareinteractionsbetweenheavyandmassless term cancels it. The lightness of Higgs mass in MSSM
•
states. (Hu′ and Hd v.s. ν and N) is mainly due to the small quartic terms from gauge in-
teractions and it can be relaxed if there are additional
Off-diagonal elements are generated if fields get
• VEVs. (X F =0 v.s. H H0 =0) quartic couplings in the theory in addition to the usual
X
→h i6 →h i6 D-term. Thus we consider the modification of MSSM to
The crucial difference is the negative sign of bosonic give the addtional quartic terms.
see-saw mechanism which can not be eliminated by The most transparent application of the bosonic see-
rephasing scalar fields. In general, X = 0 and we can saw mechanism comes out if µ = 0. However, the limit
redefinefieldsandcouplingssuchthahtXi˜6=X X does µ=0 in MSSM poses several problems [10].
not have a scalar VEV ( X˜ =0). Then we ob−thainimore
h i Peccei-Quinn(PQ) symmetry and R symmetry
general superpotential, •
As Higgs fields carry PQ and R charge and the
W = l1XHu′Hd+l2XHu′Hd′ +MHu′Hd′. (6) symmetryisexactinthelimitµ=0,oncetheyget
VEVs, there appears a massless Goldstone boson
Now Yukawa couplings are which is in confict with experiments.
W = l H Quc+l H Qdc+l′H′Quc. (7) Electroweak symmetry breaking
u u d d u u •
3
If Hu gets a VEV from negative m2Hu, Hd gets its where α = 4gπ2 is the SU(2) gauge coupling and αf =
VEV through Bµ term. Therefore,if µ=0 (Bµ= f2 are similarly defined Yukawa couplings for f =
0n)o,ttgheetdtohweinr-mtyapseseqsu.arks and charged leptons can l41π,l2,lΣ1,lΣ2,lt. We assume lt′ ≪1 and neglects its con-
tribution. Other Yukawa couplings are also neglected as
Chargino mass they are small. The effects are summarized as follows.
•
If µ = 0, higgsino can get their mass only by the H : Softscalarmasssquaredisnegativeatthetree
d
electroweak symmetry breaking and the lightest • level from the bosonic see-saw mechanism. There
charginomassisalwayslighterthanMW whichcan are threshold corrections from gauge and Yukawa
not be compatible with the current bound on the interactionsandthesignisopposite. IfYukawaand
lightest chargino mass, 104 GeV. gauge couplings are of similar size, the threshold
These problems can be solved if extra fields are intro- correctionsatthe messengerscalecancelwitheach
duced. InNMSSM,anextrasingletreplacesµterm. The other. Therefore,negativemasssquaredatthetree
singlet gets a VEV and it generates µ term effectively. level dominates.
Thentheelectroweaksymmetrybreakingisafinetuning
H : Threshold corrections at the messenger scale
just as in MSSM. An alternative way is to introduce an • u
are positive for both gauge and Yukawa contribu-
extra weak triplet Σ with no hypercharge.
tions. Wehaveslightlylargerm2 comparedtothe
Letusconsiderthemostinterestinglimitµ=0(µ-less Hu
MSSM with gauge mediation. We should also con-
SSM). We can forbid µ term by a discrete symmetry, so
sider negativ one loop correction from messenger
called’Uparity’,whichisaZ2 subgroupofPeccei-Quinn scale to the weak scale 3 m2log M.
symmetry. Under the U parity, −4π2 t˜ mt˜
(H ,uc,Σ) (H ,uc,Σ). Σ : Threshold corrections are positive for gauge
u → − u • andYukawacontributions. Thuswegetm2 heavier
Σ
ThemostgeneralsuperpotentialconsistentwiththeU
than other soft scalar masses which is necessary to
parity is [11]
suppress the VEV of Σ compared to H and H .
u d
W = M2ΣtrΣ2+lΣ1HuΣHd+lΣ2HuΣHd′. (9) • Third generation Q,uc (→ stop)
Threshold correction from Yukawa mediation is
These terms are enough to break PQ and R symmetry.
negative. We get lighter stop mass compared to
Atthe sametime charginomasscanbeheavierthanM
Z
the MSSM which makes the negative contribution
as we have new sources for it.
to m2 smaller than usual.
Soft supersymmetry breaking terms are Hu
Vsoft = m2Hu|Hu|2+m2Hd|Hd|2+m2ΣtrΣ†Σ Tfrohmemcohsatrcghinaollenmgainssgpbhoeunnodmceonmolboigniecdalctoontshtreaipnrteccoismioens
+AlΣ1HuΣHd+BMΣtrΣ2+h.c.. (10) data. The charginomass is obtainedfrom Yukawa inter-
The neutral componentof Σ gets a VEV once Hu and actions (A : Higgsino-Wino-Higgs and B : Higgsino-ψΣ-
H get VEVs [11], Higgs) in the µ-less theory [15]. A is the gauge coupling
d
vΣ = m2Σ+MΣ2 +lΣ1BMΣ2ΣMvΣ2 + 12lΣ12v2. (11) daabnialdel BSTUi(s(T2a)<nsey0wm.6mY)ureektsratywr.iacTtcshoeluΣpb1loi(nulgΣn1dlΣ<o1n0th.t6ha)te[v1pi5roe]l.caiFtseioosrncluΣvsa1tro∼i--
1, we obtain the lightest chargino mass to be 104 GeV
vΣ < 9 GeV is obtained if mΣ is larger than the elec-
whichistheboundfromLEPII.Thecharginomassesare
troweak scale.
Letus go backto the calculationofsoftterms. As our (104,119,252) GeV for lΣ1 = 1,(M2,MΣ) = (120,150)
GeV. Note thatif we allownonzeroµ, we cansatisfy the
messengerfieldshavedirectcouplingswithmatter/Higgs
fields, there are additional contributions. The Yukawa chargino mass bound with a smaller lΣ1. More precise
calculationofT isneededaswedealwithlightspectrum
mediated ones are calculated using the formalism of an-
(charginos are near 100 GeV). For the neutralinos, the
alytic continuation into superspace [12, 13, 14],
lower mass bound 40 GeV is easily satisfied.
∆m2 = 3α2Σ2 + αΣ2αl2 Λ2 The Higgs mass can be calculated if all the parame-
Hu (cid:20) 4π2 4π2 (cid:21) ters are chosen. As we have a new quartic couplings for
∆m2 = αΣ1αΣ2 Λ2 the Higgs from W =l1HuΣHd, the lightest scalar Higgs
Hd h− 2π2 i massis heavierthanthe oneinthe MSSMandisaround
∆m2 = αtαΣ2 Λ2 120 to 130 GeV before considering the one loop correc-
Q,uc h− 8π2 i tion. Therefore, in this model we do not need to tune
∆m2 = 3αtαΣ2 + 3α2Σ2 + αΣ2αl2 5ααΣ2 Λ2, the parameters to raise up the Higgs mass beyond the
Σ (cid:20) 4π2 4π2 4π2 − 4π2 (cid:21) currentbound114GeV.Thebosonicsee-sawmechanism
4
gives m2 <0 at the messenger scale and m2 is driven tons get their masses. The mechanismworksnicely even
Hd Hu
to be negative by RG running to the weak scale. Both if µ = 0 though we need additional weak triplet. The
m2 and m2 are negative at the weak scale and the chargino remains light (near 104 GeV) when µ = 0 and
Hu Hd
minimum is at around tanβ = vu (1). Unlike in it is robust against radiative corrections. Higgs is heavy
vd ∼ O
the MSSM, the potential is not bound from below for (about130GeVbeforeconsideringradiativecorrections)
m2Hu < 0, m2Hd < 0 as the new quartic coupling l1 pre- but the full spectrum of Higgs can be obtained only af-
vents them from running away along D-flat direction. terconsideringtheradiativecorrectionsandweleavethe
In this paper we proposeda new mechanismto under- detailed calculation of it for future work. The bosonic
standtheelectroweaksymmetrybreaking. AsH couples see-saw mechanism can be applied differently in other
d
directlytothemessengerofsupersymmetrybreaking,the problems.
soft scalar mass squared is negative by the bosonic see-
saw mechanism when supersymmetry is broken. The This work is supported by the ABRL Grant No. R14-
soft scalar mass squared of H is driven to be nega- 2003-012-01001-0,the BK21 program of Ministry of Ed-
u
tive and the symmetry breaking minimum is at around ucation, Korea.
tanβ = vu (1). There is a new SU(2) triplet Σ
vd ∼ O L
whichcouplestoH andH . Thelightestcharginomass
u d
is predicted to be light due to the absence of supersym-
metricmassµandliesjustabovethecurrentmassbound
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