59
Alex Pomarol (Univ. Autonoma Barcelona) Models for the Fermi Scale (from the top perspective)
Alex Pomarol (Univ. Autonoma Barcelona)
Models for the Fermi Scale
(from the top perspective)
• Models for the Fermi scale, should be motivated by experimental evidences
• Nevertheless, at 2009, after years of collecting direct or indirect data at the EW scale, either at LEP, Tevatron or B-factories, we do not have a single serious experimental hint for models at the electroweak scale beyond the SM (BSM)
• What can guide us?
Overwhelming cosmo/astro indirect hints for BSM
Not a good guide towards EW models
• Baryon-Antibaryon Asymmetry,• Dark Matter Problem,• Neutrino Masses,• Inflation,• Accelerating Universe, ...
whose explanations could be tied to the electroweak scale, but this is not necessary
A suggestive “experimental evidence”
Existence of the Top quark
(since its heaviness seems to indicate that it plays an important role for EWSB)
Who ordered it?The EWSB agents
for new TeV-physics:
• It’s not a simple spectator of EWSB (as other fermions of the SM)
• It’s a main player → Responsible for EWSB
In this talk, I will single out the top quark and assume
notice that this demands a solution to the hierarchy problem i.e. Higgs mass must be determined by IR-physics
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I will use the the top as a discriminator for models at the electroweak scale
Standard Model (only the Higgs at the EW scale)
Standard Model (only the Higgs at the EW scale)
The top: one more quark!
Standard Model (only the Higgs at the EW scale)
The top: one more quark!
Although...
d!
dt=
116"2
(!6Y 4t + · · · )
110 120 130 140 150
155
160
165
170
175
180
Stable
Metasta
bleUnstab
le
Higgs mass
top
mas
s
Top coupling can induce a negative λ at high energies
☛ Unstable Higgs minimum
d!
dt=
116"2
(!6Y 4t + · · · )
110 120 130 140 150
155
160
165
170
175
180
Stable
Metasta
bleUnstab
le
Higgs mass
top
mas
s
Top coupling can induce a negative λ at high energies
☛ Unstable Higgs minimum
Our present minimum will not last forever:
End of our current inflationary period Arkani-Hamed,Dubovsky,Senatore,Villadoro
SM is almost indifferent vs the top!
Technicolor = EWSB à la QCD
Technicolor = EWSB à la QCD
The top: a nuisance!
Too heavy to be explained by extended technicolor:
Lq
R
(a) (b)
q
TR
TL
R
TL TR
T
TL
! T ! T
Figure 1: Graphs for ETC generation of masses for (a) quarks and leptonsand (b) technipions. The dashed line is a massive ETC gauge boson. Higher–order technicolor gluon exchanges are not indicated; from Ref. [15].
F 2T M2
!T! 2
g2ETC
M2ETC
"TLTRTRTL#ETC . (4)
Here, mq(METC) is the quark mass renormalized at METC . It is a hardmass in that it scales like one (i.e., logarithmically) for energies below METC .Above that, it falls o! more rapidly, like "(p). The technipion decay constantFT = F!/
$N in TC models containing N identical electroweak doublets of
color–singlet technifermions. The vacuum expectation values "TLTR#ETC and"TLTRTRTL#ETC are the bilinear and quadrilinear technifermion condensatesrenormalized at METC . The bilinear condensate is related to the one renor-malized at #TC , expected by scaling from QCD to be
"TLTR#TC = 12"T T #TC ! 2!F 3
T , (5)
by the equation
"T T #ETC = "TT #TC exp
!
" METC
!TC
dµ
µ"m(µ)
#
. (6)
The anomalous dimension "m of the operator TT is given in perturbationtheory by
"m(µ) =3C2(R)
2!#TC(µ) + O(#2
TC) , (7)
where C2(R) is the quadratic Casimir of the technifermion SU(NTC)–representationR. For the fundamental representation of SU(NTC), it is given by C2(NTC) =
8
mq ! "TLTR#M2
ETC
Topcolor assisted Technicolor
Extra ingredient needed
Even if TC scenarios can give a large top mass, still one must check that Zbb is not corrected:
H†DµH qL!µqLHqLtR
Main obstacle:
whatever generates
must not generate
Difficult since tL is with bL in the same weak doublet
How to generate this but not this
tL tR
!
(a)
Z
bL bL
!
(b)
W aLµ! W bL
µ!
!
! !
!
tR
tR
(c)
Figure 4: Diagrams in the 4D holographic theory that generate the top Yukawa coupling (a), acorrection to Z ! bLbL (b), and the T parameter (c). A grey blob represents the 4D CFT dynamicsor the 5D bulk. Another possible diagram contributing to Z ! bLbL, similar to (b) but with two !fields attached, is not shown.
Denoting with !gLb the shift in the coupling of bL to Z, NDA leads to the estimates 6
!gLb
gLb" "2
qN
16#2$2 "
!1
2# cq
"$2 , (45)
%T = "& " "4u
N
(16#2)2'4$2 "
!1
2+ cu
"2
'4 $2
N, (46)
where we have used eqs. (3) and (35). We have also included a new parameter ' to take into account
a possible deviation from NDA in the coupling of the composite fermions to the Higgs (a chirality
flip factor). 7 From eq. (44) we see that the (liberal) bound !gLb/gLb ! 1% from LEP is satisfied
for values of cq close to 1/2. For example, $ $ 0.4 implies cq $ 0.4. From eq. (45), on the other
hand, we see that the 99% CL bound on the T parameter, T ! 0.3 [9] 8, can also be satisfied for
reasonable values of the parameters. For example, setting ' " 1, then $ $ 0.4 implies cu $ #0.1 for
N $ 10. Thus, both estimates give !gLb and T close to the experimental limit for values of the 5D
parameters used in the analysis of section 3.2. This is an indication that our model can succeed in
passing all EWPT, although eqs. (44), (45) should not be taken too seriously, being only estimates
and not exact results. One can take into account the correlation among T , !gLb and the top mass
by making use of the NDA estimate for mt
mt " "q"uN
16#2m" $ ' "
#!1
2# cq
"!1
2+ cu
"4#%N
v ' , (47)
6If fig. 4 is drawn using resonances, one can show that there are two kind of diagrams contributing to !gLb and!" . Either the Higgs couples to a vector resonance, or to a fermionic resonance through a chirality flip. One canshow that the dominant contribution to !gLb and !" are respectively that with zero and four chirality flips.
7This corresponds in the 5D theory to the mass mixing parameters eq. (28).8It corresponds to an extra contribution to the #1 parameter [15] !#1 ! 2.5 · 10!3 relative to the SM value with
mHiggs=115 GeV.
18
tL tR
!
(a)
Z
bL bL
!
(b)
W aLµ! W bL
µ!
!
! !
!
tR
tR
(c)
Figure 4: Diagrams in the 4D holographic theory that generate the top Yukawa coupling (a), acorrection to Z ! bLbL (b), and the T parameter (c). A grey blob represents the 4D CFT dynamicsor the 5D bulk. Another possible diagram contributing to Z ! bLbL, similar to (b) but with two !fields attached, is not shown.
Denoting with !gLb the shift in the coupling of bL to Z, NDA leads to the estimates 6
!gLb
gLb" "2
qN
16#2$2 "
!1
2# cq
"$2 , (45)
%T = "& " "4u
N
(16#2)2'4$2 "
!1
2+ cu
"2
'4 $2
N, (46)
where we have used eqs. (3) and (35). We have also included a new parameter ' to take into account
a possible deviation from NDA in the coupling of the composite fermions to the Higgs (a chirality
flip factor). 7 From eq. (44) we see that the (liberal) bound !gLb/gLb ! 1% from LEP is satisfied
for values of cq close to 1/2. For example, $ $ 0.4 implies cq $ 0.4. From eq. (45), on the other
hand, we see that the 99% CL bound on the T parameter, T ! 0.3 [9] 8, can also be satisfied for
reasonable values of the parameters. For example, setting ' " 1, then $ $ 0.4 implies cu $ #0.1 for
N $ 10. Thus, both estimates give !gLb and T close to the experimental limit for values of the 5D
parameters used in the analysis of section 3.2. This is an indication that our model can succeed in
passing all EWPT, although eqs. (44), (45) should not be taken too seriously, being only estimates
and not exact results. One can take into account the correlation among T , !gLb and the top mass
by making use of the NDA estimate for mt
mt " "q"uN
16#2m" $ ' "
#!1
2# cq
"!1
2+ cu
"4#%N
v ' , (47)
6If fig. 4 is drawn using resonances, one can show that there are two kind of diagrams contributing to !gLb and!" . Either the Higgs couples to a vector resonance, or to a fermionic resonance through a chirality flip. One canshow that the dominant contribution to !gLb and !" are respectively that with zero and four chirality flips.
7This corresponds in the 5D theory to the mass mixing parameters eq. (28).8It corresponds to an extra contribution to the #1 parameter [15] !#1 ! 2.5 · 10!3 relative to the SM value with
mHiggs=115 GeV.
18
?
Even if TC scenarios can give a large top mass, still one must check that Zbb is not corrected:
Main obstacle:
whatever generates
must not generate
Difficult since tL is with bL in the same weak doublet
Too large!
!gb
gb!
yL
yR
mt
m!! 0.07
yL ! yR
}tL tR
!
(a)
Z
bL bL
!
(b)
W aLµ! W bL
µ!
!
! !
!
tR
tR
(c)
Figure 4: Diagrams in the 4D holographic theory that generate the top Yukawa coupling (a), acorrection to Z ! bLbL (b), and the T parameter (c). A grey blob represents the 4D CFT dynamicsor the 5D bulk. Another possible diagram contributing to Z ! bLbL, similar to (b) but with two !fields attached, is not shown.
Denoting with !gLb the shift in the coupling of bL to Z, NDA leads to the estimates 6
!gLb
gLb" "2
qN
16#2$2 "
!1
2# cq
"$2 , (45)
%T = "& " "4u
N
(16#2)2'4$2 "
!1
2+ cu
"2
'4 $2
N, (46)
where we have used eqs. (3) and (35). We have also included a new parameter ' to take into account
a possible deviation from NDA in the coupling of the composite fermions to the Higgs (a chirality
flip factor). 7 From eq. (44) we see that the (liberal) bound !gLb/gLb ! 1% from LEP is satisfied
for values of cq close to 1/2. For example, $ $ 0.4 implies cq $ 0.4. From eq. (45), on the other
hand, we see that the 99% CL bound on the T parameter, T ! 0.3 [9] 8, can also be satisfied for
reasonable values of the parameters. For example, setting ' " 1, then $ $ 0.4 implies cu $ #0.1 for
N $ 10. Thus, both estimates give !gLb and T close to the experimental limit for values of the 5D
parameters used in the analysis of section 3.2. This is an indication that our model can succeed in
passing all EWPT, although eqs. (44), (45) should not be taken too seriously, being only estimates
and not exact results. One can take into account the correlation among T , !gLb and the top mass
by making use of the NDA estimate for mt
mt " "q"uN
16#2m" $ ' "
#!1
2# cq
"!1
2+ cu
"4#%N
v ' , (47)
6If fig. 4 is drawn using resonances, one can show that there are two kind of diagrams contributing to !gLb and!" . Either the Higgs couples to a vector resonance, or to a fermionic resonance through a chirality flip. One canshow that the dominant contribution to !gLb and !" are respectively that with zero and four chirality flips.
7This corresponds in the 5D theory to the mass mixing parameters eq. (28).8It corresponds to an extra contribution to the #1 parameter [15] !#1 ! 2.5 · 10!3 relative to the SM value with
mHiggs=115 GeV.
18
tL tR
!
(a)
Z
bL bL
!
(b)
W aLµ! W bL
µ!
!
! !
!
tR
tR
(c)
Figure 4: Diagrams in the 4D holographic theory that generate the top Yukawa coupling (a), acorrection to Z ! bLbL (b), and the T parameter (c). A grey blob represents the 4D CFT dynamicsor the 5D bulk. Another possible diagram contributing to Z ! bLbL, similar to (b) but with two !fields attached, is not shown.
Denoting with !gLb the shift in the coupling of bL to Z, NDA leads to the estimates 6
!gLb
gLb" "2
qN
16#2$2 "
!1
2# cq
"$2 , (45)
%T = "& " "4u
N
(16#2)2'4$2 "
!1
2+ cu
"2
'4 $2
N, (46)
where we have used eqs. (3) and (35). We have also included a new parameter ' to take into account
a possible deviation from NDA in the coupling of the composite fermions to the Higgs (a chirality
flip factor). 7 From eq. (44) we see that the (liberal) bound !gLb/gLb ! 1% from LEP is satisfied
for values of cq close to 1/2. For example, $ $ 0.4 implies cq $ 0.4. From eq. (45), on the other
hand, we see that the 99% CL bound on the T parameter, T ! 0.3 [9] 8, can also be satisfied for
reasonable values of the parameters. For example, setting ' " 1, then $ $ 0.4 implies cu $ #0.1 for
N $ 10. Thus, both estimates give !gLb and T close to the experimental limit for values of the 5D
parameters used in the analysis of section 3.2. This is an indication that our model can succeed in
passing all EWPT, although eqs. (44), (45) should not be taken too seriously, being only estimates
and not exact results. One can take into account the correlation among T , !gLb and the top mass
by making use of the NDA estimate for mt
mt " "q"uN
16#2m" $ ' "
#!1
2# cq
"!1
2+ cu
"4#%N
v ' , (47)
6If fig. 4 is drawn using resonances, one can show that there are two kind of diagrams contributing to !gLb and!" . Either the Higgs couples to a vector resonance, or to a fermionic resonance through a chirality flip. One canshow that the dominant contribution to !gLb and !" are respectively that with zero and four chirality flips.
7This corresponds in the 5D theory to the mass mixing parameters eq. (28).8It corresponds to an extra contribution to the #1 parameter [15] !#1 ! 2.5 · 10!3 relative to the SM value with
mHiggs=115 GeV.
18
yL yR
yL yL
H†DµH qL!µqLHqLtRHow to generate this but not this ?
If , possible large loop contributions to T-parameter
yR ! yL
!T !
y4R
16!2
m2!
v2! y4
R
Technicolor hates the top!
Supersymmetry = MSSM
Supersymmetry = MSSM
Top quark plays an important role in EWSBif supersymmetry breaking mediated by gauge
interactions (GMSB)
Supersymmetrized SM (fermion↔boson)
+ Susy broken in a extra sector charged under the SM group
But this is not the full story...
!
!
!
!
!
!
! !
"
" "
"
f~
f~
f~
f~
f~
f~
! !
!
!
f~
f~
f~
f~
f~
f~
f~
f~
f~
f~
Figure 1: Feynman diagrams contributing to supersymmetry-breaking gaugino (!) and sfermion(f) masses. The scalar and fermionic components of the messenger fields ! are denoted bydashed and solid lines, respectively; ordinary gauge bosons are denoted by wavy lines.
11
Flavor universal (positive) scalar masses
Higgs mass negative due to the top/stop loops
Theory easy to define:
EWSB thanks to the top:
MSSM+GMSB
Main problem: μ-parameter (Higgsino mass)
116!2
!d4"
X
MH1H2
!X" = F!2
µ ! F
16!2M
Dvali,Giudice,APNeeds extra ingredients to generate
Main problem: μ-parameter (Higgsino mass)
116!2
!d4"
X
MH1H2
!X" = F!2
µ ! F
16!2M
Bµ ! 16!2µ2
A loop factor larger!
Dvali,Giudice,AP
also scalar masses are generated that can be negative → top not needed any more
Needs extra ingredients to generate
but, in general, this is also generated
116!2
!d4"
XX†
M2H1H2
Of course, solutions exist, and many new proposed recentlyGiudice,Kim,Rattazzi;
Murayama, Nomura, PolandPerez, Roy, Schmaltz;
Csaki, Falkowsky, Nomura, Volansky
If Susy is broken in a strongly interacting conformal hidden sector with large anomalous
dimension for the |X|^2 operator
e.g.
becomes irrelevant at low-energies
one finds interesting new conditions (massless scalars!):
m2soft + µ2 = 0Bµ = 0
{
Susy + Conformal dynamics: full of surprises!
116!2
!d4"
XX†
M2H1H2
m2h = m2
Z +3m4
t
2!2v2ln(mstop/mt) + · · ·
m2Z =
m2t
2!2m2
stop ln(MSUSY/mstop) + · · ·
After LEP, a heavy top is essential to keep the MSSM alive
Needed to get above the exp. lower bound on the Higgs-mass
but also a heavy stop is needed, leading to a tension with the EW scale
A heavier top will diminish this tension!
The top, the survivor of the MSSMThe heavier, the better!
The composite Pseudo-Goldstone Boson Higgs
The composite Pseudo-Goldstone Boson Higgs
The top: “raison d’être” of this scenario
Spectrum:
Composite PGB-Higgs scenario is inspired by QCD where one observe that the (pseudo) scalar are the lightest states
!
!
Mass protected by the global QCD symmetry!
Are Pseudo-Goldstone bosons (PGB)
! ! ! + "
!
Spectrum:
Composite PGB-Higgs scenario is inspired by QCD where one observe that the (pseudo) scalar are the lightest states
!
!
Mass protected by the global QCD symmetry!
Are Pseudo-Goldstone bosons (PGB)
! ! ! + "
!
Potential tilted due to the gauging of EM and quark masses → Pseudo-GBs
Spectrum:
!
!
Are Pseudo-Goldstone bosons (PGB)
Mass protected by the global QCD symmetry!
Can the light Higgs be a kind of a pion from a new strong QCD-like sector? Georgi, Kaplan
Composite PGB-Higgs scenario is inspired by QCD where one observe that the (pseudo) scalar are the lightest states
Spectrum of the new QCD-like sector:
!
Pseudo-Goldstone bosons (PGB)
Mass protected by a global symmetry G
h100 GeV
TeV
h! h + !h
Spectrum of the new QCD-like sector:
!
Pseudo-Goldstone bosons (PGB)
Mass protected by a global symmetry G
h100 GeV
TeV{Mass gap: Would explain the
absence of new states at colliders before the LHC
Origin of EWSB
Higgs potential induced by gauge loops + top loops
SM interactions break the global symmetry G
A heavy top essential to break EWSB!
V (h) ! "Y 2
t m2!
16!2h2 + · · ·
Nice idea, but, how to put it in practice?
Not easy, since, as we from QCD, difficult to get predictions within strong interacting theories
One approach: Although the dynamics of the strong sector can be unknown, the low-energy effective lagrangian for a PGB
Higgs can be determined by symmetries (as chiral lagrangian for pions physics)
→ Physics of a PGB-Higgs can be studied see Gripaios’s talk
Another approach: AdS/CFT correspondence
Recent new tool to calculate within strongly coupled theories:
AdS/CFT correspondence
Maldacena 97
Strongly coupled 4D theoriesin the large-N limit
duality Weakly coupled string theories in 10D
Yes, the Sakai-Sugimoto model
Can we find dual examples of strong theories for EWSB?
Geometrical approach to EWSB
z
Chiral symmetry breaking in the Sakai-Sugimoto model:
SU(NF )L ! SU(NF )R SU(NF )V
large number of D4-braneswarped the geometry
D4-D8 system:
At low-energy, this theory is equivalent to a gauge theory in 5Dwith chiral breaking on the boundary
.
Minimal 5D composite Higgs model
AdS5
SO(5)! U(1)
Fermions " 5 of SO(5)
UV-bound.
SU(2)L! U(1)Y
IR-bound.
SO(4)! U(1)
Parameters: g5D, L and 5D fermion masses Agashe, A.P.,Contino
warped extra dim: z
SU(NF )L ! SU(NF )R
z = !z = 0
SU(NF )V
AµL ! A
µR = 0
boundy conditions:
Dirichlet:
Neumann:Fµ5
L + Fµ5
R = 0Extra dim version of TC
Holographic composite PGB Higgs
Agashe,Contino,A.P.
.
Minimal 5D composite Higgs model
AdS5
SO(5)! U(1)
Fermions " 5 of SO(5)
UV-bound.
SU(2)L! U(1)Y
IR-bound.
SO(4)! U(1)
Parameters: g5D, L and 5D fermion masses Agashe, A.P.,Contino
extra dim
Strongly coupled theories with large-N and large ‘tHooft coupling can be described by 5D dimensional models
Lesson from the AdS/CFT correspondence:
Theory simple to define:
Why this symmetry breaking pattern?
We are in 5D: AM = (Aµ,A5)
Massless boson spectrum:
• Aµ of SU(2)L!U(1)Y = SM Gauge bosons
• A5 of SO(5)/SO(4) = 2 of SU(2)L = SM Higgs
!" Higgs-gauge unification
Higgs mass protected by 5D gauge invariance!
Hosotani mechanism
A5 ! A5 + !5"
shifts as a PGB
Spectrum
110-180 GeV
500-1500 GeV
2.5 TeV
Higgs
12/3
gauge KK
color fermionic KK}21/6
27/6
4.2 TeV graviton KK
the higher the spin, the higher the mass
The heavier the top mass,the lighter the KK associated to the top
cuto! scale to be of the order of the mass of the lowest fermionic resonance: 7
mq! ! " ! 900 GeV! mh
150 GeV
" #0.5
!
$, (17)
where in the last equality Eq. (15) has been used. Eq. (17) shows that in composite Higgs
models with a light Higgs and no tuning (! " 1) colored resonances are expected to be not
heavier than " 1 TeV. In our model, the relation between the Higgs mass and the mass of
the lowest fermionic KK turns out to be more complicated than that of Eq. (17). We find
that the points of Fig. 2 are better reproduced by a relation of the form
"2 = a1 m2q! + a2 mq!M + a3 M2 , (18)
where ai are numerical coe#cients, M # m! parametrizes the mass of the heavier resonances
and by mq! we denote the mass of the KK weak doublet with hypercharge Y = 7/6 (the
lightest among the fermionic KKs in Fig. 2). This means that in our model the integral% !
0 dp p [F (p)/F (0)] is not completely cut o! at p " mq! , and that other (heavier) reso-
nances also play a role. A fit to the points of Fig. 2 gives: ai=1,2,3 = ($0.10, 0.35, 0.007) for
the MCHM5 (upper plot) and ai = ($0.14, 0.24, 0.06) for the MCHM10 (lower plot). The
dispersion of the points around the fitted curve (shown in each plot) can be explained as
follows. In Fig. 2 we have fixed N = 8, ! = 0.5 and mpolet = 173 GeV, which leaves two
of the five parameters of Eq. (9) free to vary. If cu is traded for mq! , we are left with one
free parameter, for example, cq. The coe#cients ai of Eq. (18) will thus depend on cq, and
since we have scanned over the values 0.2 < cq < 0.38 (upper plot) and 0.36 < cq < 0.45
(lower plot) to generate the points in Fig. 2, this explains their dispersion. In other words,
the fitted ai given above should be considered as average values.
5 Production and detection of the lightest fermionic
resonances at the LHC
The most promising way to unravel these models is by detecting the lowest fermionic KKs at
the LHC. In particular, detecting the custodian with electric charge 5/3, that we will denote
7Using Eq. (8) we can rewrite Eq. (15) as
m2h !
Nc
!2
m2t
2v2!2 $
4c2h"
f2!
.
The first term is the formula for the Higgs mass one obtains in the SM by defining !2/2 #%dp p in the top
loop. The degree of cancellation between the first and second term gives a measure of the degree of “tuning”needed in our model. This exactly corresponds to #2.
12
Expected: In the SM the top loop diverges
cut-off by the KK fermions:
! ! mKK
The warped geometry gives automaticallylighter KK, for heavier top
Contino,DaRold,AP
This implies a top-mass range of viability: mt
173 GeV
!T
!S
0.003 0.004 0.005 0.006 0.007 0.008
!3
0.003
0.004
0.005
0.006
0.007
0.008
0.009
0.01
!1
Main Problem: the more composite, the more goes out of the S-T ellipse
Elem.
High degree of compositeness
One must keep v f/3 !
v/f
f = Decay constant of the PGB-Higgs
Exists certain tension!
If this is unaffordable, one can try to engineer a model where
V (h) = !m2h2 + !h4
two-loops one-loop
10 TeV
100 GeV h
!
Two-loop gap
Little Higgs!Arkani-Hamed,Cohen,Georgi
10 TeV
100 GeV h
!
Two-loop gap
Little Higgs!Arkani-Hamed,Cohen,Georgi
Collective breaking: Demand two gauge couplings needed to break the PGB symmetry
How?
T, WH , BH
10 TeV
TeV
100 GeV h
! New fields must be introduced
Two-loop gap
How?
Collective breaking: Demand two gauge couplings needed to break the PGB symmetry
Main difficulty: generate a λ at one-loop level
2
II. COLLECTIVE QUARTICS
How does one construct a Higgs quartic that doesnot radiatively generate a quadratically-divergent Higgsmass? In little Higgs theories, one finds a set of opera-tors that each preserve di!erent shift symmetries actingon the Higgs doublet, but collectively break all the sym-metries that protect the Higgs potential [1, 2, 3, 4, 5].Since the quadratically-divergent diagrams only involveone operator at a time, the shift symmetries are su"cientto protect the Higgs mass parameter.
Concretely, in a non-linear sigma model where theHiggs is a pseudo-Nambu-Goldstone boson (PNGB), onenaıvely expects the shift symmetry
h ! h + ! + · · · (4)
to forbid any potential for the Higgs. But if there areadditional PNGBs " with compensating shifts
" ! " "h! + !h
f+ · · · , (5)
then the two operators
V # #1f2
!
!
!
!
" +h2
f+ · · ·
!
!
!
!
2
+ #2f2
!
!
!
!
" $h2
f+ · · ·
!
!
!
!
2
(6)
each preserve one of the Higgs shift symmetries fromeq. (5). Taken alone, neither #i term would give a phys-ical Higgs quartic since each individual quartic could beremoved by a "± % "±h2/f + · · · field redefinition. Col-lectively, though, the two operators yield a Higgs quarticafter " is integrated out:
V #4#1#2
#1 + #2
h4 + · · · . (7)
This is the form of all little Higgs quartics. A small Higgsmass term is generated radiatively from eq. (6), and theresulting potential allows for a parametric separation be-tween the electroweak vev v and the decay constant f .
At this point, we have not specified the quantum num-bers of the scalar ", which is equivalent to specifying thequantum numbers of h2. The possible SU(2)L represen-tations for h2 are determined by
2 & 2 = 3S & 1A, 2& 2 = 3& 1, (8)
where the S/A subscript refers to the representation be-ing symmetric/antisymmetric under the interchange ofthe two doublets. This classification holds regardless ofthe number of Higgs fields.
In a one-Higgs doublet model, the 1A representationvanishes, and " can be a complex triplet, a real triplet,or a real singlet:
hihj ! "ij (3S), (9)
h†$ah ! "a (3), (10)
h†h ! % (1), (11)
where $a are the Pauli matrices, and we use the notation% to refer to a real singlet that carries no other charges.If " is a real or complex SU(2)L triplet, then eq. (6) givesrise to a tree-level quartic coupling yet protects the Higgsmass. A complex " triplet is used in the SU(5)/SO(5)littlest Higgs [3], and a real " triplet is present in theSO(9)/(SO(5)'SO(4)) construction [9] (though this lat-ter model has a pathology that will be understood in thenext section).
However, if " is a real singlet %, then explicit com-putation shows that eq. (6) generates a quadratically-divergent % tadpole and Higgs mass at one-loop!
#1f#2
16&2
"
% +h†h
f+ · · ·
#
$#2f#2
16&2
"
% $h†h
f+ · · ·
#
(12)
Note the sign di!erence between the two terms, whichmeans that the Higgs mass term cannot be forbidden byT -parity [3, 14, 15] with #1 = #2, and a parity that en-forces #1 = $#2 would imply no Higgs quartic couplingin the first place. Therefore, there is no viable one-Higgsdoublet little Higgs model where a collective quartic in-volves a real singlet %. In particular, this explains whyit is impossible to add a collective quartic coupling tothe simplest little Higgs [6] without extending the Higgssector [16].
The reason for this pathology is that the shift sym-metry alone does not forbid a tadpole for %. If % hadnon-trivial quantum numbers (such as being an SU(2)L
triplet), then these extra symmetries would forbid the %tadpole. Famously, the singlet h†h cannot be chargedunder any symmetry (except a shift symmetry), and thesame holds for the singlet %. To illustrate this pathologyfurther, we construct an explicit singlet % model which re-alizes the full non-linear shift symmetries in appendix A.
In a two-Higgs doublet model, one can have quarticsconstructed not only with SU(2)L triplets but also withsinglets. Choosing conventions where h1 and h2 have thesame hypercharge, " can a priori be a complex singletwith or without hypercharge:
hi1h
j2!ij ! " (1A), (13)
h†1h2 ! " (1). (14)
Note however that the quartic constructed from the hy-percharge carrying singlet |hi
1hj2!ij |2 is unsatisfactory be-
cause it vanishes when the h1 and h2 vevs are aligned topreserve electric charge. A hypercharge neutral complex" is used in the SU(6)/Sp(6) antisymmetric condensatemodel [8].
In addition, " can even be a real singlet as long as ithas an extra Z2 symmetry:
Re[h†1h2] ! " (1). (15)
In this case, the symmetry
" ! $", h1 ! $h1, h2 ! h2 (16)
is su"cient to forbid the " tadpole.
2
II. COLLECTIVE QUARTICS
How does one construct a Higgs quartic that doesnot radiatively generate a quadratically-divergent Higgsmass? In little Higgs theories, one finds a set of opera-tors that each preserve di!erent shift symmetries actingon the Higgs doublet, but collectively break all the sym-metries that protect the Higgs potential [1, 2, 3, 4, 5].Since the quadratically-divergent diagrams only involveone operator at a time, the shift symmetries are su"cientto protect the Higgs mass parameter.
Concretely, in a non-linear sigma model where theHiggs is a pseudo-Nambu-Goldstone boson (PNGB), onenaıvely expects the shift symmetry
h ! h + ! + · · · (4)
to forbid any potential for the Higgs. But if there areadditional PNGBs " with compensating shifts
" ! " "h! + !h
f+ · · · , (5)
then the two operators
V # #1f2
!
!
!
!
" +h2
f+ · · ·
!
!
!
!
2
+ #2f2
!
!
!
!
" $h2
f+ · · ·
!
!
!
!
2
(6)
each preserve one of the Higgs shift symmetries fromeq. (5). Taken alone, neither #i term would give a phys-ical Higgs quartic since each individual quartic could beremoved by a "± % "±h2/f + · · · field redefinition. Col-lectively, though, the two operators yield a Higgs quarticafter " is integrated out:
V #4#1#2
#1 + #2
h4 + · · · . (7)
This is the form of all little Higgs quartics. A small Higgsmass term is generated radiatively from eq. (6), and theresulting potential allows for a parametric separation be-tween the electroweak vev v and the decay constant f .
At this point, we have not specified the quantum num-bers of the scalar ", which is equivalent to specifying thequantum numbers of h2. The possible SU(2)L represen-tations for h2 are determined by
2 & 2 = 3S & 1A, 2& 2 = 3& 1, (8)
where the S/A subscript refers to the representation be-ing symmetric/antisymmetric under the interchange ofthe two doublets. This classification holds regardless ofthe number of Higgs fields.
In a one-Higgs doublet model, the 1A representationvanishes, and " can be a complex triplet, a real triplet,or a real singlet:
hihj ! "ij (3S), (9)
h†$ah ! "a (3), (10)
h†h ! % (1), (11)
where $a are the Pauli matrices, and we use the notation% to refer to a real singlet that carries no other charges.If " is a real or complex SU(2)L triplet, then eq. (6) givesrise to a tree-level quartic coupling yet protects the Higgsmass. A complex " triplet is used in the SU(5)/SO(5)littlest Higgs [3], and a real " triplet is present in theSO(9)/(SO(5)'SO(4)) construction [9] (though this lat-ter model has a pathology that will be understood in thenext section).
However, if " is a real singlet %, then explicit com-putation shows that eq. (6) generates a quadratically-divergent % tadpole and Higgs mass at one-loop!
#1f#2
16&2
"
% +h†h
f+ · · ·
#
$#2f#2
16&2
"
% $h†h
f+ · · ·
#
(12)
Note the sign di!erence between the two terms, whichmeans that the Higgs mass term cannot be forbidden byT -parity [3, 14, 15] with #1 = #2, and a parity that en-forces #1 = $#2 would imply no Higgs quartic couplingin the first place. Therefore, there is no viable one-Higgsdoublet little Higgs model where a collective quartic in-volves a real singlet %. In particular, this explains whyit is impossible to add a collective quartic coupling tothe simplest little Higgs [6] without extending the Higgssector [16].
The reason for this pathology is that the shift sym-metry alone does not forbid a tadpole for %. If % hadnon-trivial quantum numbers (such as being an SU(2)L
triplet), then these extra symmetries would forbid the %tadpole. Famously, the singlet h†h cannot be chargedunder any symmetry (except a shift symmetry), and thesame holds for the singlet %. To illustrate this pathologyfurther, we construct an explicit singlet % model which re-alizes the full non-linear shift symmetries in appendix A.
In a two-Higgs doublet model, one can have quarticsconstructed not only with SU(2)L triplets but also withsinglets. Choosing conventions where h1 and h2 have thesame hypercharge, " can a priori be a complex singletwith or without hypercharge:
hi1h
j2!ij ! " (1A), (13)
h†1h2 ! " (1). (14)
Note however that the quartic constructed from the hy-percharge carrying singlet |hi
1hj2!ij |2 is unsatisfactory be-
cause it vanishes when the h1 and h2 vevs are aligned topreserve electric charge. A hypercharge neutral complex" is used in the SU(6)/Sp(6) antisymmetric condensatemodel [8].
In addition, " can even be a real singlet as long as ithas an extra Z2 symmetry:
Re[h†1h2] ! " (1). (15)
In this case, the symmetry
" ! $", h1 ! $h1, h2 ! h2 (16)
is su"cient to forbid the " tadpole.
2
II. COLLECTIVE QUARTICS
How does one construct a Higgs quartic that doesnot radiatively generate a quadratically-divergent Higgsmass? In little Higgs theories, one finds a set of opera-tors that each preserve di!erent shift symmetries actingon the Higgs doublet, but collectively break all the sym-metries that protect the Higgs potential [1, 2, 3, 4, 5].Since the quadratically-divergent diagrams only involveone operator at a time, the shift symmetries are su"cientto protect the Higgs mass parameter.
Concretely, in a non-linear sigma model where theHiggs is a pseudo-Nambu-Goldstone boson (PNGB), onenaıvely expects the shift symmetry
h ! h + ! + · · · (4)
to forbid any potential for the Higgs. But if there areadditional PNGBs " with compensating shifts
" ! " "h! + !h
f+ · · · , (5)
then the two operators
V # #1f2
!
!
!
!
" +h2
f+ · · ·
!
!
!
!
2
+ #2f2
!
!
!
!
" $h2
f+ · · ·
!
!
!
!
2
(6)
each preserve one of the Higgs shift symmetries fromeq. (5). Taken alone, neither #i term would give a phys-ical Higgs quartic since each individual quartic could beremoved by a "± % "±h2/f + · · · field redefinition. Col-lectively, though, the two operators yield a Higgs quarticafter " is integrated out:
V #4#1#2
#1 + #2
h4 + · · · . (7)
This is the form of all little Higgs quartics. A small Higgsmass term is generated radiatively from eq. (6), and theresulting potential allows for a parametric separation be-tween the electroweak vev v and the decay constant f .
At this point, we have not specified the quantum num-bers of the scalar ", which is equivalent to specifying thequantum numbers of h2. The possible SU(2)L represen-tations for h2 are determined by
2 & 2 = 3S & 1A, 2& 2 = 3& 1, (8)
where the S/A subscript refers to the representation be-ing symmetric/antisymmetric under the interchange ofthe two doublets. This classification holds regardless ofthe number of Higgs fields.
In a one-Higgs doublet model, the 1A representationvanishes, and " can be a complex triplet, a real triplet,or a real singlet:
hihj ! "ij (3S), (9)
h†$ah ! "a (3), (10)
h†h ! % (1), (11)
where $a are the Pauli matrices, and we use the notation% to refer to a real singlet that carries no other charges.If " is a real or complex SU(2)L triplet, then eq. (6) givesrise to a tree-level quartic coupling yet protects the Higgsmass. A complex " triplet is used in the SU(5)/SO(5)littlest Higgs [3], and a real " triplet is present in theSO(9)/(SO(5)'SO(4)) construction [9] (though this lat-ter model has a pathology that will be understood in thenext section).
However, if " is a real singlet %, then explicit com-putation shows that eq. (6) generates a quadratically-divergent % tadpole and Higgs mass at one-loop!
#1f#2
16&2
"
% +h†h
f+ · · ·
#
$#2f#2
16&2
"
% $h†h
f+ · · ·
#
(12)
Note the sign di!erence between the two terms, whichmeans that the Higgs mass term cannot be forbidden byT -parity [3, 14, 15] with #1 = #2, and a parity that en-forces #1 = $#2 would imply no Higgs quartic couplingin the first place. Therefore, there is no viable one-Higgsdoublet little Higgs model where a collective quartic in-volves a real singlet %. In particular, this explains whyit is impossible to add a collective quartic coupling tothe simplest little Higgs [6] without extending the Higgssector [16].
The reason for this pathology is that the shift sym-metry alone does not forbid a tadpole for %. If % hadnon-trivial quantum numbers (such as being an SU(2)L
triplet), then these extra symmetries would forbid the %tadpole. Famously, the singlet h†h cannot be chargedunder any symmetry (except a shift symmetry), and thesame holds for the singlet %. To illustrate this pathologyfurther, we construct an explicit singlet % model which re-alizes the full non-linear shift symmetries in appendix A.
In a two-Higgs doublet model, one can have quarticsconstructed not only with SU(2)L triplets but also withsinglets. Choosing conventions where h1 and h2 have thesame hypercharge, " can a priori be a complex singletwith or without hypercharge:
hi1h
j2!ij ! " (1A), (13)
h†1h2 ! " (1). (14)
Note however that the quartic constructed from the hy-percharge carrying singlet |hi
1hj2!ij |2 is unsatisfactory be-
cause it vanishes when the h1 and h2 vevs are aligned topreserve electric charge. A hypercharge neutral complex" is used in the SU(6)/Sp(6) antisymmetric condensatemodel [8].
In addition, " can even be a real singlet as long as ithas an extra Z2 symmetry:
Re[h†1h2] ! " (1). (15)
In this case, the symmetry
" ! $", h1 ! $h1, h2 ! h2 (16)
is su"cient to forbid the " tadpole.
2
II. COLLECTIVE QUARTICS
How does one construct a Higgs quartic that doesnot radiatively generate a quadratically-divergent Higgsmass? In little Higgs theories, one finds a set of opera-tors that each preserve di!erent shift symmetries actingon the Higgs doublet, but collectively break all the sym-metries that protect the Higgs potential [1, 2, 3, 4, 5].Since the quadratically-divergent diagrams only involveone operator at a time, the shift symmetries are su"cientto protect the Higgs mass parameter.
Concretely, in a non-linear sigma model where theHiggs is a pseudo-Nambu-Goldstone boson (PNGB), onenaıvely expects the shift symmetry
h ! h + ! + · · · (4)
to forbid any potential for the Higgs. But if there areadditional PNGBs " with compensating shifts
" ! " "h! + !h
f+ · · · , (5)
then the two operators
V # #1f2
!
!
!
!
" +h2
f+ · · ·
!
!
!
!
2
+ #2f2
!
!
!
!
" $h2
f+ · · ·
!
!
!
!
2
(6)
each preserve one of the Higgs shift symmetries fromeq. (5). Taken alone, neither #i term would give a phys-ical Higgs quartic since each individual quartic could beremoved by a "± % "±h2/f + · · · field redefinition. Col-lectively, though, the two operators yield a Higgs quarticafter " is integrated out:
V #4#1#2
#1 + #2
h4 + · · · . (7)
This is the form of all little Higgs quartics. A small Higgsmass term is generated radiatively from eq. (6), and theresulting potential allows for a parametric separation be-tween the electroweak vev v and the decay constant f .
At this point, we have not specified the quantum num-bers of the scalar ", which is equivalent to specifying thequantum numbers of h2. The possible SU(2)L represen-tations for h2 are determined by
2 & 2 = 3S & 1A, 2& 2 = 3& 1, (8)
where the S/A subscript refers to the representation be-ing symmetric/antisymmetric under the interchange ofthe two doublets. This classification holds regardless ofthe number of Higgs fields.
In a one-Higgs doublet model, the 1A representationvanishes, and " can be a complex triplet, a real triplet,or a real singlet:
hihj ! "ij (3S), (9)
h†$ah ! "a (3), (10)
h†h ! % (1), (11)
where $a are the Pauli matrices, and we use the notation% to refer to a real singlet that carries no other charges.If " is a real or complex SU(2)L triplet, then eq. (6) givesrise to a tree-level quartic coupling yet protects the Higgsmass. A complex " triplet is used in the SU(5)/SO(5)littlest Higgs [3], and a real " triplet is present in theSO(9)/(SO(5)'SO(4)) construction [9] (though this lat-ter model has a pathology that will be understood in thenext section).
However, if " is a real singlet %, then explicit com-putation shows that eq. (6) generates a quadratically-divergent % tadpole and Higgs mass at one-loop!
#1f#2
16&2
"
% +h†h
f+ · · ·
#
$#2f#2
16&2
"
% $h†h
f+ · · ·
#
(12)
Note the sign di!erence between the two terms, whichmeans that the Higgs mass term cannot be forbidden byT -parity [3, 14, 15] with #1 = #2, and a parity that en-forces #1 = $#2 would imply no Higgs quartic couplingin the first place. Therefore, there is no viable one-Higgsdoublet little Higgs model where a collective quartic in-volves a real singlet %. In particular, this explains whyit is impossible to add a collective quartic coupling tothe simplest little Higgs [6] without extending the Higgssector [16].
The reason for this pathology is that the shift sym-metry alone does not forbid a tadpole for %. If % hadnon-trivial quantum numbers (such as being an SU(2)L
triplet), then these extra symmetries would forbid the %tadpole. Famously, the singlet h†h cannot be chargedunder any symmetry (except a shift symmetry), and thesame holds for the singlet %. To illustrate this pathologyfurther, we construct an explicit singlet % model which re-alizes the full non-linear shift symmetries in appendix A.
In a two-Higgs doublet model, one can have quarticsconstructed not only with SU(2)L triplets but also withsinglets. Choosing conventions where h1 and h2 have thesame hypercharge, " can a priori be a complex singletwith or without hypercharge:
hi1h
j2!ij ! " (1A), (13)
h†1h2 ! " (1). (14)
Note however that the quartic constructed from the hy-percharge carrying singlet |hi
1hj2!ij |2 is unsatisfactory be-
cause it vanishes when the h1 and h2 vevs are aligned topreserve electric charge. A hypercharge neutral complex" is used in the SU(6)/Sp(6) antisymmetric condensatemodel [8].
In addition, " can even be a real singlet as long as ithas an extra Z2 symmetry:
Re[h†1h2] ! " (1). (15)
In this case, the symmetry
" ! $", h1 ! $h1, h2 ! h2 (16)
is su"cient to forbid the " tadpole.
it needs an extra triplet or singlet such as transform under the PGB symmetry:
This allow
that after integrating them out:
But triplet gets a VEV → T-parameter
singlet allows for tadpoles → Higgs mass at one-looprecently emphasized by Schmaltz,Thaler
Impose a parity under which extra fields are odd→ effects on EWPT at one-loop
Workable model?
Solution:
LH models with T-parity
Csaki, Heinonen, Perelstein,Spethmann
AdS5
SU(5) ! SU(2) ! U(1)
SU(2)3 ! U(1)3SO(5)!SU(2) ! U(1)
K1
K2
A1
A2
UV IR
z = R " 1/MP l z = R! " 1/(10 TeV)
Figure 2: Geometric setup, gauge symmetries and matter content of the five-dimensionalmodel.
this paper, this symmetry needs to be extended to SU(5)! SU(2)3 ! U(1)3, with [SU(2) !U(1)]3 gauged, in order to incorporate T-parity in the (chiral) fermion sector. So, the 5Dsetup we start with is an SU(5)!SU(2)3 !U(1)3 bulk gauge group. The action of T-parityon the gauge bosons is again given by eq. (2.10). We assume that the gauge symmetry isbroken by boundary conditions (BC’s) for the gauge fields, as in [29]: on the UV brane,
SU(5) ! SU(2) ! U(1) # [SU(2) ! U(1)]3 (UV) , (5.12)
while on the IR brane
SU(5) ! SU(2) ! U(1) # SO(5) ! SU(2) ! U(1) (IR). (5.13)
In the language of the 4D model, this is equivalent to placing the !1,2 fields on the UV braneand the S field on the IR brane, and integrating out the radial models of these fields afterthey get vevs. (Note that this geometric separation of ! and S automatically guaranteesthe absence of the direct potential couplings between them, as needed in our model.) TheseBC’s result in an unbroken [SU(2) ! U(1)]2 gauge group at low energies and leave T-parityunbroken. The gauge fields in [SU(2)!U(1)]3 which are only broken by BC’s on the IR branewill get a mass of order f " 1 TeV. These fields correspond to the T-odd gauge bosons of theLHT model. As discussed above, the full Kaluza-Klein (KK) tower starts at the somewhathigher scale mKK " 10 TeV.
To reduce the group further (down to just the SM) we will assume that the scalars K1,K2 live on the IR brane, getting vevs of order mKK " 10 TeV. Furthermore, to incorporatefermion masses in an SU(5) invariant way, we also add the scalars A1, A2 on the IR brane,with
19
AdS5
SU(5) ! SU(2) ! U(1)
SU(2)3 ! U(1)3SO(5)!SU(2) ! U(1)
K1
K2
A1
A2
UV IR
z = R " 1/MP l z = R! " 1/(10 TeV)
Figure 2: Geometric setup, gauge symmetries and matter content of the five-dimensionalmodel.
this paper, this symmetry needs to be extended to SU(5)! SU(2)3 ! U(1)3, with [SU(2) !U(1)]3 gauged, in order to incorporate T-parity in the (chiral) fermion sector. So, the 5Dsetup we start with is an SU(5)!SU(2)3 !U(1)3 bulk gauge group. The action of T-parityon the gauge bosons is again given by eq. (2.10). We assume that the gauge symmetry isbroken by boundary conditions (BC’s) for the gauge fields, as in [29]: on the UV brane,
SU(5) ! SU(2) ! U(1) # [SU(2) ! U(1)]3 (UV) , (5.12)
while on the IR brane
SU(5) ! SU(2) ! U(1) # SO(5) ! SU(2) ! U(1) (IR). (5.13)
In the language of the 4D model, this is equivalent to placing the !1,2 fields on the UV braneand the S field on the IR brane, and integrating out the radial models of these fields afterthey get vevs. (Note that this geometric separation of ! and S automatically guaranteesthe absence of the direct potential couplings between them, as needed in our model.) TheseBC’s result in an unbroken [SU(2) ! U(1)]2 gauge group at low energies and leave T-parityunbroken. The gauge fields in [SU(2)!U(1)]3 which are only broken by BC’s on the IR branewill get a mass of order f " 1 TeV. These fields correspond to the T-odd gauge bosons of theLHT model. As discussed above, the full Kaluza-Klein (KK) tower starts at the somewhathigher scale mKK " 10 TeV.
To reduce the group further (down to just the SM) we will assume that the scalars K1,K2 live on the IR brane, getting vevs of order mKK " 10 TeV. Furthermore, to incorporatefermion masses in an SU(5) invariant way, we also add the scalars A1, A2 on the IR brane,with
19
Gauge symmetry breaking:
Global symmetry breaking:
Impose a parity under which extra fields are odd→ effects on EWPT at one-loop
Workable model?
Solution:
LH models with T-parity
Workable model?Csaki, Heinonen, Perelstein,Spethmann
AdS5
SU(5) ! SU(2) ! U(1)
SU(2)3 ! U(1)3SO(5)!SU(2) ! U(1)
K1
K2
A1
A2
UV IR
z = R " 1/MP l z = R! " 1/(10 TeV)
Figure 2: Geometric setup, gauge symmetries and matter content of the five-dimensionalmodel.
this paper, this symmetry needs to be extended to SU(5)! SU(2)3 ! U(1)3, with [SU(2) !U(1)]3 gauged, in order to incorporate T-parity in the (chiral) fermion sector. So, the 5Dsetup we start with is an SU(5)!SU(2)3 !U(1)3 bulk gauge group. The action of T-parityon the gauge bosons is again given by eq. (2.10). We assume that the gauge symmetry isbroken by boundary conditions (BC’s) for the gauge fields, as in [29]: on the UV brane,
SU(5) ! SU(2) ! U(1) # [SU(2) ! U(1)]3 (UV) , (5.12)
while on the IR brane
SU(5) ! SU(2) ! U(1) # SO(5) ! SU(2) ! U(1) (IR). (5.13)
In the language of the 4D model, this is equivalent to placing the !1,2 fields on the UV braneand the S field on the IR brane, and integrating out the radial models of these fields afterthey get vevs. (Note that this geometric separation of ! and S automatically guaranteesthe absence of the direct potential couplings between them, as needed in our model.) TheseBC’s result in an unbroken [SU(2) ! U(1)]2 gauge group at low energies and leave T-parityunbroken. The gauge fields in [SU(2)!U(1)]3 which are only broken by BC’s on the IR branewill get a mass of order f " 1 TeV. These fields correspond to the T-odd gauge bosons of theLHT model. As discussed above, the full Kaluza-Klein (KK) tower starts at the somewhathigher scale mKK " 10 TeV.
To reduce the group further (down to just the SM) we will assume that the scalars K1,K2 live on the IR brane, getting vevs of order mKK " 10 TeV. Furthermore, to incorporatefermion masses in an SU(5) invariant way, we also add the scalars A1, A2 on the IR brane,with
19
AdS5
SU(5) ! SU(2) ! U(1)
SU(2)3 ! U(1)3SO(5)!SU(2) ! U(1)
K1
K2
A1
A2
UV IR
z = R " 1/MP l z = R! " 1/(10 TeV)
Figure 2: Geometric setup, gauge symmetries and matter content of the five-dimensionalmodel.
this paper, this symmetry needs to be extended to SU(5)! SU(2)3 ! U(1)3, with [SU(2) !U(1)]3 gauged, in order to incorporate T-parity in the (chiral) fermion sector. So, the 5Dsetup we start with is an SU(5)!SU(2)3 !U(1)3 bulk gauge group. The action of T-parityon the gauge bosons is again given by eq. (2.10). We assume that the gauge symmetry isbroken by boundary conditions (BC’s) for the gauge fields, as in [29]: on the UV brane,
SU(5) ! SU(2) ! U(1) # [SU(2) ! U(1)]3 (UV) , (5.12)
while on the IR brane
SU(5) ! SU(2) ! U(1) # SO(5) ! SU(2) ! U(1) (IR). (5.13)
In the language of the 4D model, this is equivalent to placing the !1,2 fields on the UV braneand the S field on the IR brane, and integrating out the radial models of these fields afterthey get vevs. (Note that this geometric separation of ! and S automatically guaranteesthe absence of the direct potential couplings between them, as needed in our model.) TheseBC’s result in an unbroken [SU(2) ! U(1)]2 gauge group at low energies and leave T-parityunbroken. The gauge fields in [SU(2)!U(1)]3 which are only broken by BC’s on the IR branewill get a mass of order f " 1 TeV. These fields correspond to the T-odd gauge bosons of theLHT model. As discussed above, the full Kaluza-Klein (KK) tower starts at the somewhathigher scale mKK " 10 TeV.
To reduce the group further (down to just the SM) we will assume that the scalars K1,K2 live on the IR brane, getting vevs of order mKK " 10 TeV. Furthermore, to incorporatefermion masses in an SU(5) invariant way, we also add the scalars A1, A2 on the IR brane,with
19
Gauge symmetry breaking:
Global symmetry breaking:
a) SU(5) SU(2)3 U(1)3Q1 ¯
1+2/3
Q2
1+2/3
q3 1!1/6
q4 1!7/6
q5 1!7/6
UR1 11 !2/3
UR2 11 !2/3
uR 11 !2/3
dR 11
+1/3
b) SU(5) SU(2)3 U(1)3Q !
1 ¯1 !2/3
Q !2
1 !2/3q !3 , q !!
3 1+1/6
q !4 1
+7/6q !5 1
+7/6U !
R1 11
+2/3U !
R2 11
+2/3
c) SU(5) SU(2)3 U(1)3L1 ¯
10
L2
10
!3 1+1/2
!4 1!1/2
!5 1!1/2
ER1 11
0ER2 1
10
eR 11
+1("R 1
10 )
Table 3: The complete fermion sector (single generation) and the gauge charge assignments
for the anomaly-free version of the model.
SU(5) SU(3)c SU(2)3 U(1)3
q !!3 1
+1/6
uR 1¯
1 !2/3
dR 1¯
1+1/3
!5 11
!1/2
eR 11
1+1
Table 4: The chiral matter content for one generation of the anomaly-free version of the
model.
13
Impose a parity under which extra fields are odd→ effects on EWPT at one-loop
Solution:
LH models with T-parity
Workable model?Csaki, Heinonen, Perelstein,Spethmann
AdS5
SU(5) ! SU(2) ! U(1)
SU(2)3 ! U(1)3SO(5)!SU(2) ! U(1)
K1
K2
A1
A2
UV IR
z = R " 1/MP l z = R! " 1/(10 TeV)
Figure 2: Geometric setup, gauge symmetries and matter content of the five-dimensionalmodel.
this paper, this symmetry needs to be extended to SU(5)! SU(2)3 ! U(1)3, with [SU(2) !U(1)]3 gauged, in order to incorporate T-parity in the (chiral) fermion sector. So, the 5Dsetup we start with is an SU(5)!SU(2)3 !U(1)3 bulk gauge group. The action of T-parityon the gauge bosons is again given by eq. (2.10). We assume that the gauge symmetry isbroken by boundary conditions (BC’s) for the gauge fields, as in [29]: on the UV brane,
SU(5) ! SU(2) ! U(1) # [SU(2) ! U(1)]3 (UV) , (5.12)
while on the IR brane
SU(5) ! SU(2) ! U(1) # SO(5) ! SU(2) ! U(1) (IR). (5.13)
In the language of the 4D model, this is equivalent to placing the !1,2 fields on the UV braneand the S field on the IR brane, and integrating out the radial models of these fields afterthey get vevs. (Note that this geometric separation of ! and S automatically guaranteesthe absence of the direct potential couplings between them, as needed in our model.) TheseBC’s result in an unbroken [SU(2) ! U(1)]2 gauge group at low energies and leave T-parityunbroken. The gauge fields in [SU(2)!U(1)]3 which are only broken by BC’s on the IR branewill get a mass of order f " 1 TeV. These fields correspond to the T-odd gauge bosons of theLHT model. As discussed above, the full Kaluza-Klein (KK) tower starts at the somewhathigher scale mKK " 10 TeV.
To reduce the group further (down to just the SM) we will assume that the scalars K1,K2 live on the IR brane, getting vevs of order mKK " 10 TeV. Furthermore, to incorporatefermion masses in an SU(5) invariant way, we also add the scalars A1, A2 on the IR brane,with
19
AdS5
SU(5) ! SU(2) ! U(1)
SU(2)3 ! U(1)3SO(5)!SU(2) ! U(1)
K1
K2
A1
A2
UV IR
z = R " 1/MP l z = R! " 1/(10 TeV)
Figure 2: Geometric setup, gauge symmetries and matter content of the five-dimensionalmodel.
this paper, this symmetry needs to be extended to SU(5)! SU(2)3 ! U(1)3, with [SU(2) !U(1)]3 gauged, in order to incorporate T-parity in the (chiral) fermion sector. So, the 5Dsetup we start with is an SU(5)!SU(2)3 !U(1)3 bulk gauge group. The action of T-parityon the gauge bosons is again given by eq. (2.10). We assume that the gauge symmetry isbroken by boundary conditions (BC’s) for the gauge fields, as in [29]: on the UV brane,
SU(5) ! SU(2) ! U(1) # [SU(2) ! U(1)]3 (UV) , (5.12)
while on the IR brane
SU(5) ! SU(2) ! U(1) # SO(5) ! SU(2) ! U(1) (IR). (5.13)
In the language of the 4D model, this is equivalent to placing the !1,2 fields on the UV braneand the S field on the IR brane, and integrating out the radial models of these fields afterthey get vevs. (Note that this geometric separation of ! and S automatically guaranteesthe absence of the direct potential couplings between them, as needed in our model.) TheseBC’s result in an unbroken [SU(2) ! U(1)]2 gauge group at low energies and leave T-parityunbroken. The gauge fields in [SU(2)!U(1)]3 which are only broken by BC’s on the IR branewill get a mass of order f " 1 TeV. These fields correspond to the T-odd gauge bosons of theLHT model. As discussed above, the full Kaluza-Klein (KK) tower starts at the somewhathigher scale mKK " 10 TeV.
To reduce the group further (down to just the SM) we will assume that the scalars K1,K2 live on the IR brane, getting vevs of order mKK " 10 TeV. Furthermore, to incorporatefermion masses in an SU(5) invariant way, we also add the scalars A1, A2 on the IR brane,with
19
Gauge symmetry breaking:
Global symmetry breaking:
a) SU(5) SU(2)3 U(1)3Q1 ¯
1+2/3
Q2
1+2/3
q3 1!1/6
q4 1!7/6
q5 1!7/6
UR1 11 !2/3
UR2 11 !2/3
uR 11 !2/3
dR 11
+1/3
b) SU(5) SU(2)3 U(1)3Q !
1 ¯1 !2/3
Q !2
1 !2/3q !3 , q !!
3 1+1/6
q !4 1
+7/6q !5 1
+7/6U !
R1 11
+2/3U !
R2 11
+2/3
c) SU(5) SU(2)3 U(1)3L1 ¯
10
L2
10
!3 1+1/2
!4 1!1/2
!5 1!1/2
ER1 11
0ER2 1
10
eR 11
+1("R 1
10 )
Table 3: The complete fermion sector (single generation) and the gauge charge assignments
for the anomaly-free version of the model.
SU(5) SU(3)c SU(2)3 U(1)3
q !!3 1
+1/6
uR 1¯
1 !2/3
dR 1¯
1+1/3
!5 11
!1/2
eR 11
1+1
Table 4: The chiral matter content for one generation of the anomaly-free version of the
model.
13
SU(5) SU(2)3U(1)3
!1,2Adj
10
S1
0
K1
!1/2
K2
!1/2
Table 1: Scalar fields and their gauge charge assignments.
Thus, the full gauge group of our model, at high energies,is
SU(5) " SU(2)3 " U(1)3,
(2.6)
where we labeled the extra SU(2)"U(1) factor with a subscript “3” to distinguish it from the
[SU(2)" U(1)]2 subgroup of the SU(5) that survives below 10 TeV. To break the [SU(2) "
U(1)]3 subgroup to the SM electro
weak gauge group, we also need additional bifundamental
scalarsunder SU(5) " SU(2)3, K1 and K2, which will acquire the appropriate vevs (see
eq. (2.9)).
To reproduce the symmetries of the LHT model at low energies,we introduce a set of
scalarfields, summarized
in Table 1. At the 10 TeV scale, the ! fields get vevs of the form
#!1$ = f!
!
"
"
"
"
#
!3!3
22
2
$
%
%
%
%
&
, #!2$ = f!
!
"
"
"
"
#
22
2!3
!3
$
%
%
%
%
&
(2.7)
where f! % 10 TeV. These vevs break the SU(5) down to [SU(2)"U(1)]2 , the gauge group
of the LHT model, and leave the SU(2)3 " U(1)3 unbroken. If the scalarpotential has the
formV = V (!1, !2) + V (S, K1, K2) ,
(2.8)
so that there are no direct couplings between !’s and other scalars, the model will possess
an SU(5) global symmetry below 10 TeV, broken only by gauge interactions. This is the
idea that was first emplyed in the context of SU(6) GUT models in [22], and also in the
”simplest little Higgs” model in [23]. With this assumption, the full gauge/global symmetry
structure of the LHT is reproduced. Of course, this construction is only natural, if there is
a symmetry reasonfor the absence of direct potential couplings between !’s and the other
scalars. In section
5, we will show that the !-vevs can be stabilized at the 10 TeV scale,
either by supersymmetrizing the model or by embedding it into a five-dimensional model
with warped geometry. In both cases, the couplings between ! and the other scalarscan be
naturally suppressed.
5
Impose a parity under which extra fields are odd→ effects on EWPT at one-loop
Solution:
LH models with T-parity
Workable model?Csaki, Heinonen, Perelstein,Spethmann
AdS5
SU(5) ! SU(2) ! U(1)
SU(2)3 ! U(1)3SO(5)!SU(2) ! U(1)
K1
K2
A1
A2
UV IR
z = R " 1/MP l z = R! " 1/(10 TeV)
Figure 2: Geometric setup, gauge symmetries and matter content of the five-dimensionalmodel.
this paper, this symmetry needs to be extended to SU(5)! SU(2)3 ! U(1)3, with [SU(2) !U(1)]3 gauged, in order to incorporate T-parity in the (chiral) fermion sector. So, the 5Dsetup we start with is an SU(5)!SU(2)3 !U(1)3 bulk gauge group. The action of T-parityon the gauge bosons is again given by eq. (2.10). We assume that the gauge symmetry isbroken by boundary conditions (BC’s) for the gauge fields, as in [29]: on the UV brane,
SU(5) ! SU(2) ! U(1) # [SU(2) ! U(1)]3 (UV) , (5.12)
while on the IR brane
SU(5) ! SU(2) ! U(1) # SO(5) ! SU(2) ! U(1) (IR). (5.13)
In the language of the 4D model, this is equivalent to placing the !1,2 fields on the UV braneand the S field on the IR brane, and integrating out the radial models of these fields afterthey get vevs. (Note that this geometric separation of ! and S automatically guaranteesthe absence of the direct potential couplings between them, as needed in our model.) TheseBC’s result in an unbroken [SU(2) ! U(1)]2 gauge group at low energies and leave T-parityunbroken. The gauge fields in [SU(2)!U(1)]3 which are only broken by BC’s on the IR branewill get a mass of order f " 1 TeV. These fields correspond to the T-odd gauge bosons of theLHT model. As discussed above, the full Kaluza-Klein (KK) tower starts at the somewhathigher scale mKK " 10 TeV.
To reduce the group further (down to just the SM) we will assume that the scalars K1,K2 live on the IR brane, getting vevs of order mKK " 10 TeV. Furthermore, to incorporatefermion masses in an SU(5) invariant way, we also add the scalars A1, A2 on the IR brane,with
19
AdS5
SU(5) ! SU(2) ! U(1)
SU(2)3 ! U(1)3SO(5)!SU(2) ! U(1)
K1
K2
A1
A2
UV IR
z = R " 1/MP l z = R! " 1/(10 TeV)
Figure 2: Geometric setup, gauge symmetries and matter content of the five-dimensionalmodel.
this paper, this symmetry needs to be extended to SU(5)! SU(2)3 ! U(1)3, with [SU(2) !U(1)]3 gauged, in order to incorporate T-parity in the (chiral) fermion sector. So, the 5Dsetup we start with is an SU(5)!SU(2)3 !U(1)3 bulk gauge group. The action of T-parityon the gauge bosons is again given by eq. (2.10). We assume that the gauge symmetry isbroken by boundary conditions (BC’s) for the gauge fields, as in [29]: on the UV brane,
SU(5) ! SU(2) ! U(1) # [SU(2) ! U(1)]3 (UV) , (5.12)
while on the IR brane
SU(5) ! SU(2) ! U(1) # SO(5) ! SU(2) ! U(1) (IR). (5.13)
In the language of the 4D model, this is equivalent to placing the !1,2 fields on the UV braneand the S field on the IR brane, and integrating out the radial models of these fields afterthey get vevs. (Note that this geometric separation of ! and S automatically guaranteesthe absence of the direct potential couplings between them, as needed in our model.) TheseBC’s result in an unbroken [SU(2) ! U(1)]2 gauge group at low energies and leave T-parityunbroken. The gauge fields in [SU(2)!U(1)]3 which are only broken by BC’s on the IR branewill get a mass of order f " 1 TeV. These fields correspond to the T-odd gauge bosons of theLHT model. As discussed above, the full Kaluza-Klein (KK) tower starts at the somewhathigher scale mKK " 10 TeV.
To reduce the group further (down to just the SM) we will assume that the scalars K1,K2 live on the IR brane, getting vevs of order mKK " 10 TeV. Furthermore, to incorporatefermion masses in an SU(5) invariant way, we also add the scalars A1, A2 on the IR brane,with
19
Gauge symmetry breaking:
Global symmetry breaking:
a) SU(5) SU(2)3 U(1)3Q1 ¯
1+2/3
Q2
1+2/3
q3 1!1/6
q4 1!7/6
q5 1!7/6
UR1 11 !2/3
UR2 11 !2/3
uR 11 !2/3
dR 11
+1/3
b) SU(5) SU(2)3 U(1)3Q !
1 ¯1 !2/3
Q !2
1 !2/3q !3 , q !!
3 1+1/6
q !4 1
+7/6q !5 1
+7/6U !
R1 11
+2/3U !
R2 11
+2/3
c) SU(5) SU(2)3 U(1)3L1 ¯
10
L2
10
!3 1+1/2
!4 1!1/2
!5 1!1/2
ER1 11
0ER2 1
10
eR 11
+1("R 1
10 )
Table 3: The complete fermion sector (single generation) and the gauge charge assignments
for the anomaly-free version of the model.
SU(5) SU(3)c SU(2)3 U(1)3
q !!3 1
+1/6
uR 1¯
1 !2/3
dR 1¯
1+1/3
!5 11
!1/2
eR 11
1+1
Table 4: The chiral matter content for one generation of the anomaly-free version of the
model.
13
SU(5) SU(2)3U(1)3
!1,2Adj
10
S1
0
K1
!1/2
K2
!1/2
Table 1: Scalar fields and their gauge charge assignments.
Thus, the full gauge group of our model, at high energies,is
SU(5) " SU(2)3 " U(1)3,
(2.6)
where we labeled the extra SU(2)"U(1) factor with a subscript “3” to distinguish it from the
[SU(2)" U(1)]2 subgroup of the SU(5) that survives below 10 TeV. To break the [SU(2) "
U(1)]3 subgroup to the SM electro
weak gauge group, we also need additional bifundamental
scalarsunder SU(5) " SU(2)3, K1 and K2, which will acquire the appropriate vevs (see
eq. (2.9)).
To reproduce the symmetries of the LHT model at low energies,we introduce a set of
scalarfields, summarized
in Table 1. At the 10 TeV scale, the ! fields get vevs of the form
#!1$ = f!
!
"
"
"
"
#
!3!3
22
2
$
%
%
%
%
&
, #!2$ = f!
!
"
"
"
"
#
22
2!3
!3
$
%
%
%
%
&
(2.7)
where f! % 10 TeV. These vevs break the SU(5) down to [SU(2)"U(1)]2 , the gauge group
of the LHT model, and leave the SU(2)3 " U(1)3 unbroken. If the scalarpotential has the
formV = V (!1, !2) + V (S, K1, K2) ,
(2.8)
so that there are no direct couplings between !’s and other scalars, the model will possess
an SU(5) global symmetry below 10 TeV, broken only by gauge interactions. This is the
idea that was first emplyed in the context of SU(6) GUT models in [22], and also in the
”simplest little Higgs” model in [23]. With this assumption, the full gauge/global symmetry
structure of the LHT is reproduced. Of course, this construction is only natural, if there is
a symmetry reasonfor the absence of direct potential couplings between !’s and the other
scalars. In section
5, we will show that the !-vevs can be stabilized at the 10 TeV scale,
either by supersymmetrizing the model or by embedding it into a five-dimensional model
with warped geometry. In both cases, the couplings between ! and the other scalarscan be
naturally suppressed.
5
From simplicity → Intelligent design
Impose a parity under which extra fields are odd→ effects on EWPT at one-loop
Solution:
LH models with T-parity
ConclusionsWhat to expect at the TeV?
From the top perspective,
• Fine-tuned SM: Indifferent vs the top• Technicolor: Extra problem to accommodate the top• MSSM+GMSB • PGB-Higgs
ConclusionsWhat to expect at the TeV?
From the top perspective,
• Fine-tuned SM: Indifferent vs the top• Technicolor: Extra problem to accommodate the top• MSSM+GMSB • PGB-Higgs
if it is an important EWSB player
In all models, already certain tension with EW data
ConclusionsWhat to expect at the TeV?
ConclusionsWhat to expect at the TeV?My attitude (mostly): we are in a
“tip of the iceberg” situation
ConclusionsWhat to expect at the TeV?My attitude (mostly): we are in a
“tip of the iceberg” situation...although doubts sometimes come:
“This could be the discovery of the century. Depending, of course, on how far down it goes”
ConclusionsWhat to expect at the TeV?My attitude (mostly): we are in a
“tip of the iceberg” situation...although doubts sometimes come:
“This could be the discovery of the century. Depending, of course, on how far down it goes”
LHC will tell us!
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