Transcript Slide 1

Higgsless Models
Roberto Casalbuoni
Dipartimento di Fisica, Sezione INFN,
Istituto G. Galilei per la Fisica Teorica
Firenze
[email protected]
Padova, 22 Gennaio, 2009
1
Outline of the talk
(Based on papers by
A. Deandrea, J. Bechi, R.C., F. Coradeschi, S. De Curtis, D. Dolce, D.
Dominici, F. Feruglio, M. Grazzini, R. Gatto)
 Motivations for Higgsless models
 Example of breaking the EW symmetry without Higgs (BESS)
 Linear moose: effective description for extra gauge bosons
 Unitarity bounds and EW constraints
 Degenerate BESS model (DBESS)
 The continuum limit
 Direct couplings to fermions
 The four site model, possibility of detection @ the LHC
 Summary and conclusions
2
Problems of the Higgs sector
Consider the Higgs potential
V(φ) = -μ 2 | φ |2 +λ | φ |4
μ2
λ = 2 , m2H = 2λv 2 , < φ >= v
v
The evolution of the coupling (neglecting gauge fields and fermions
contributions) shows up a Landau pole at MLp
1
λ(M) =
1
3
M2
- 2 log 2
λ(m H ) 4π
mH
M Lp = m H e
4π 2 v 2 /3m2H
● Or MLp pushed to infinity, but then l goes to 0, triviality!
● Or there is a physical cutoff at a scale M < MLp. From this we get a bound on
mH from l(M)>0
3
λ(m H ) <
4π 2 1
m <
v
M
3
log
mH
2
1
2
H
2
3
M
log 2
2
4π
mH
If the cutoff is big (M ~ 1016 GeV), l(mH) is small (~ 0.2). The theory is
perturbative, but the Higgs acquires a mass of order
λ
2
δm = 2 M
8π
2
H
with M of the order of MGUT. The naturalness problem follows and to avoid
it the quadratic divergences should cancel (SUSY).
4
If there is new physics at a scale M of order of TeV, then the theory has a
natural cutoff at M. Then naturally mH ~ M and l(mH) is large (~3 - 4). The
theory is nonperturbative.
1) λ << 1  new particles lighter than 1 TeV
2) λ >> 1  new particles around 1 TeV
In the following the second option
will be considered:
new strong physics at the TeV scale
5
Symmetry Breaking
● Since we are considering a strongly interacting theory: an effective
description of the SB
● We need to break SU(2)LxU(1) down to U(1)em. The SB sector should be
G  H,
of the type
G  SU(2) L  U(1), H  U(1)em
● In the SM the SB sector is the Higgs sector with
G = SU(2) L  SU(2) R , H = SU(2)V
● If the SB sector is strongly interacting one can describe it at low
energies making use of a general s model of the type G/H
●
For instance, in the case SU(2)LxSU(2)R/SU(2)V the model can be
described in terms of a field S in SU(2) transforming as
S  g L Σg , g L  SU(2) L , g R  SU(2) R
†
R
6
● The strong dynamics is completely characterized by the transformation
properties of the field S which can be summarized in the following moose
diagram.
● The breaking is produced by
0 | S | 0  1
2
v
μ †
iπ·τ/v
L   μ S S  , S = e
,
4
π =0
● In this way one could describe also an explicit
breaking SU(2)LxSU(2)R to
U(1)em through an explicit SU(2)V breaking term (the r-parameter is the
standard one)
2
2
2
v
v
μ †
† μ
L   μ S S   ( r  1) Tr(T3S  S) 
4
2
7
● The model can be easily gauged to SU(2)xU(1) introducing the gauge
covariant derivatives
Dμ S = μ S + igWμ S - igSBμ
● With W and B the gauge fields of SU(2) and U(1) respectively. Notice
that with respect to the strong dynamics described by the s model, the
interactions with W and B are to be considered as perturbations.
● The s model can be obtained as the formal limit of the SM in the limit
of MH to infinity.
2
1
M 1
μ
†
†
2
L = Tr  Dμ MD M  TrM
M
v

 +
*
4
8v  2


φ
-φ
1
0
-
M=

* 
1
1
φ
φ
2 0 
+ TrFμν (W)Fμν (W) + TrFμν (B)Fμν (B)
2
2
2
H
2
● Through the definition M = SS, with S is singlet field with a
vev fixed to v in the limit of large Higgs mass.
8
The field S describes the Goldstone fields giving mass to W and Z:
they are related to the longitudinal modes. The interesting amplitude is
WLWL to WLWL strictly related to the physics of the GBs.
Within the SM no bad high-energy behaviour for light Higgs:
• The quartic divergence is cancelled by the gauge contributions
• The quadratic part is cancelled by the Higgs boson contribution
in the high energy limit E >> MW
4
2
p
s
g2 4  g2 4
MW
MW
 
L
μ
pμ
MW
Derivative coupling
g
2
2
s
s
s
2
2
-e 2
=
-g
sin
θ 4
4
s MW
MW
Z
2
s
s
-g 2cos2θ

2
4
s - MW MW
Tgauge
s2
s
2
2
2
 g cos θ 4 - g cos θM Z 4
MW
MW
2
s
 g
M 2W
2
2
9
2
2
1
s
s
M
2
2
2
2
H
(g 2 v)2

g
(gv)
+
g
(
g
v)
=
2
4
4
4
s - MH MW
MW
MW
H
2
s
M
= g 2 2 + g 2 2H
MW
MW
T = Tgauge + THiggs
2
M
 g 2 2H
MW
1 2 M 2H
| a 0 |=
g
1
2
16π M W

M 2W
M  16π 2 = 4πv 2  (1 TeV) 2
g
2
H
10
An important result is the Equivalence Theorem (Cornwall, Levin &
Tiktopoulos, 1974; Vayonakis, 1976):
for E>>MW the scattering amplitudes can be evaluated by replacing the
longitudinal vector bosons with the corresponding Goldstone bosons
TGoldstones
2
2
M
M
M
 H
2
H
λ+λ

λ
=

g
= TGauge  THiggs


2
2
s - MH
MW
 v 
2
H
11
In the limit MW<< s << MH2 the WL WL (or Goldstone) amplitude can be
represented by the non-linear s model
TGoldstones
M 2H
s
λ+λ

 2
2
s<<M 2H
s - MH
v
This coincides with the amplitude that can be extracted from the nonlinear lagrangian expanding at the 4th order in the pion fields
v2
L   μ S μ S†  , S = eiπ·τ/v
4
Since the form of the effective lagrangian depends only on the symmetries,
this result is generally valid (this is the content of the Low Energy Theorem,
LET, see Chanowitz, Golden, Georgi, 1987).
Due to unitarity violation, the validity of this description is up to
1 s
| a 0 |=
 1  E  4 πv  1.7 TeV
2
16π v
12
Summarizing, what have we learned so far? If we assume that the SB
sector is strongly interacting, then:
● The low energy effective action of the strongly interacting sector is
completely fixed by the symmetries (these can be postulated but must
describe at least the breaking SU(2)xU(1) to U(1)em)
● This allows
to determine the behaviour at low energy
description valid up to E ~ 1.7 TeV.
Can we say something else about the strong dynamics to go beyond this
low energy approximation?
● We can explore the possibility of further bound states as, for instance,
vector resonances
● The vector resonances help to improve the energy behaviour of the LET
amplitudes. This amounts to postpone the unitarity bound
● --- very important ----we have to satisfy
the stringent experimental bounds
coming from LEP and SLC data
● Try to learn from QCD, though we will not adopt the point of view of TC
theories, that is scaling QCD up to energies of interest here
13
Enlarging the s model
● We will start enlarging the non-linear s model by introducing vector
resonances. One of the virtues of doing this is that unitarity properties
improve (as it is known from QCD). Of course one has to be consistent
with the non-linear realization. This could be done by standard
techniques, but a tool which is very useful is the one of hidden gauge
symmetries (Bando, Kugo et al. 1985).
● Introduce a non-dynamical gauge symmetry together with a set of
new scalar fields.
● The scalar fields can be eliminated by using the local symmetry and
the theory is equivalent to the non linear model.
● Promoting the local symmetry to a dynamical one allows to introduce
in a simple way vector resonances which are the gauge fields of the
new gauge interaction.
● The new vector resonances are massive due to the breaking of the
local symmetry implied by the non-linear realization.
14
To describe a non- linear theory breaking G to H, we do the following:
● Introduce a mapping g(x) from the space-time to the group G:
g(x)  G
● Contruct a lagrangian invariant under
g(x)  g'(x) = g0g(x)h(x), g0  G, h(x)  H, H  G
L(g, μ g) = L(g', μ g')
● L depends only on the fields defined on the coset G/H. In fact, locally
g(x) = ξ(x)h(x), ξ  G / H, h  H
and using the invariance of L:
1
L(g, μ g) = L(ξ, μξ), g(x)  g(x)h (x)
The theory formulated in G with the (non-dynamical) local symmetry H
is equivalent to the non-linear model formulated over G/H
15
The BESS model
The simplest enlargement of the non-linear model based on SU(2)xSU(2)/SU(2)
is the BESS (Breaking Electroweak Symmetry Strongly) model (RC, De Curtis,
Dominici & Gatto, 1985, 1987, + Feruglio 1988) which introduces a local group
G1=SU(2) with two scalar fields L and R transforming as
S1 (x)  g LS1 (x)h (x), S2 (x)  h(x)S2 (x)g , h  G1
†
†
R
Introduce covariant derivatives, with V the gauge field associated to the
local group transforming as S1 and S2
Dμ S1 = μS1  S1Vμ , Dμ S2 = μS2 - VμS2 , V† = -V
and build up the invariant lagrangian (this is not the most general one):
L = f Tr  Dμ Σ D Σ1  + f Tr  Dμ Σ D Σ2 
2
1
†
1
μ
2
2
†
2
μ
16
Also in this case the transformation properties of the fields can be
condensed in a moose diagram
Notice that in L we have inserted only terms corresponding to the
two links. In principle there is another invariant coupling of the type
Tr (Σ1† Dμ Σ1 )(Σ2 Dμ Σ†2 ) 
which we will not consider here.
Since V is not a dynamical field, it can be eliminated with its e.o.m.
2 † μ
2
μ †
f
S
D
Σ
+
f
Σ
D
S2
μ
1
1
1
2
2
V =f12 + f 22
17
Substituting inside L one gets back the usual non-linear s model
after the identifications
Σ = Σ1Σ2  g L Σ1h hΣ g = g L Σg
†
†
2 r
†
R
4
1 1
1
= 2+ 2 2
2
v
f1 f 2 f
On the other hand, we can make V a dynamical field and by construction
the lagrangian will preserve the total symmetry. It is enough to introduce
the kinetic term for V in a gauge invariant way
1
L = f Tr  Dμ Σ D Σ1  + f Tr  Dμ Σ D Σ2  - Tr [Fμν (V)Fμν (V)]
2
2
1
†
1
μ
2
2
†
2
μ
This model describes 6 scalar fields and 3 gauge bosons. After the breaking
of SU(2)LxSU(2)RxSU(2)local to SU(2) we get 3 Goldstone bosons (necessary to
give mass to W and Z after gauging the EW group) and 3 massive vector
bosons with mass
M 2V =  f12 + f 22  g12 , g1 = gauge coupling of V
18
Electroweak corrections
(Burgess et al.; Anichini, RC, De Curtis)
LEP I puts very stringent bounds on models of new physics. These limits,
assuming universality among different generations, are coded in 3
parameters (using GF, mZ and a as input parameters)
1

m
1
πα(m Z )
= +

2
m
4 2G F mZ (1- ΔrW ) 
 2
2
W
2
Z
ΔrW :
2
And from the modifications of the Z couplings to fermions:
Lneutral
e  Δρ 
μ
μ
=1+
ψ
g
γ
+
g
γ
γ5  ψZμ
A

  V
sθc θ 
2 
1
1
g V = (T3 ) L - sθ2Qem , g A = - (T3 ) L
2
2
1
1 πα(m Z )
2
2
2
sθ = sθ 1+ Δk  , cθ = +
2
4 2G F m 2Z
19
It is usual to introduce another set of parameters i, i=1,2,3 (Altarelli, Barbieri,
1991), or S,T,U (Peskin, Takeuchi, 1990), much more convenient on the
theoretical side
2
s
ò1 = Δρ, ò2 = cθ2 Δρ + θ ΔrW - 2sθ2 Δk, ò3 = cθ2 Δρ + c2θ Δk
c2θ
At the lowest order in the EW corrections the parameters 1 and 2 vanish if
the SB sector has a SU(2) custodial symmetry (as it is the case for the BESS
model). At the same order, 3 has a convenient dispersive representation
g2  ds
ò3 = -  2  Im Π VV (s) - Im Π AA (s) , Π VV(AA) =  J V(A)J V(A)  0
4π 0 s
Assuming vector dominance:
ImΠ VV(AA) (s) = -π g2V(A)δ s - M 2V  , 0 | J μV(A) | V( k )  g V(A)òμ (k )
In the BESS model the decay coupling constants of the vector meson are:
gV(A) = (f12  f22 )g1
20
In the BESS model we get (assuming standard fermions not directly
coupled to the vector boson V):
2
g
g g
f f
1 g 
ò3 =
= 


  
f1 =f 2
4 M
4  g1 
 g1  (f + f )
2
2
V
4
V
2 2
1 2
2
2 2
1
2
2
for g1 ~ g - 5 g, we get
g1 = g  ò3 = 0.25, g1 = 5g  ò3 = 10
-2
Experimentally 3 of order 10-3, we need an unnatural value of g1
bigger than 10g-16g (not allowed by unitarity, see later).
21
A possible way out: couple directly the vector bosons V to the fermions
introducing the following fields:
†
L
1 L
R
2 R
Then, we can add two invariant terms
χ =Σ ψ , χ =Σ ψ
i
i




bχ Liγμ  μ - Vμ + (B - L)Yμ  χ L  b'χ R iγμ  μ - Vμ + (B - L)Yμ  χ R
2
2




The b’ coupling is very much constrained by the KL-KS mass difference
and it is generally ignored.
The b coupling gives rise to a contribution to 3 which, if b>0, is of
opposite sign with respect to the gauge one
2
1 g  b
ò3 =   4  g1  2
ò3exp ~ 10-3
If a direct coupling of V to fermions is present, we can satisfy the bound
but at the expenses of some fine tuning: for g1~5 g, b should be fixed at
the level of 2x10-3. Furthermore the new VB’s would be fermiophobic.
22
Breaking the EW Symmetry without
Higgs Fields
● Let us now generalize the moose construction: general structure
given by many copies of the gauge group G intertwined by link
variables S.
● Condition to be satisfied in order to get a Higgsless SM before
gauging the EW group is the presence of 3 GB and all the moose
gauge fields massive.
● Simplest example:
Gi = SU(2). Each Si describes three scalar
fields. Therefore, in a connected moose diagram, any site (3 gauge
fields) may absorb one link (3 GB’s) giving rise to a massive vector
field. We need:
# of links = # of sites +1
23
● Example:
● The model has two global symmetries related to the beginning
and to the end of the moose, that we will denote explicitly by GL and
GR .
†
Σ1  U L Σ1U1
Σi  U i-1Σi U †i
Σ K+1  U K Σ K+1U †R
● The SU(2)L x SU(2)R global symmetry can be gauged to the standard
SU(2)xU(1) leaving us with the usual 3 massive gauge bosons, W and
Z, the massless photon and 3K massive vectors.
● The BESS model can be recast in a 3-site model (K=1), and its
generalization (Casalbuoni, DC, Dominici, Gatto, Feruglio, 1989) can be recast
in a 4-site model (K=2) (see also Foadi,Frandsen,Ryttov,Sannino, 2007)
24
25
Low-energy limit
From this, after a finite renormalization one can evaluate the EW
parameters i.
26
Electro-weak corrections for the linear
moose
Smoose
K 1
 K 1
i
i
2
† 
  d x   2 Tr  F F    f i Tr (DSi )(DSi )  
i 1
 i1 2gi

4
●
If the vector fields are heavy enough one can derive a low-energy
effective theory for the SM fields after gauging
SU(2)L Ä SU(2)R Þ SU(2) Ä U(1)
One has to solve finite difference equations along the link-line in
terms of the SM gauge fields at the two ends (the boundary)
27
●
From vector mesons saturation one gets
g2
ε3 =
4
å
n
K
æg 2 g 2 ö 2
(1- y i )y i
2
2
-2
2
çç nV - nA ÷
= g g1g Kf1 f K+1 (M 2 )1K = g å
2
çè m 4 m 4 ÷
÷
g
i=1
n
n ø
i
i
yi = å
j=1
●
●
Since
Example:
f2
xi , xi = 2 ,
fi
0 £ yi £ 1 Þ
f i = f c , gi = gc
1 K+1 1
=å 2 Þ
2
f
i=1 f i
ε3 ³ 0
K+1
å
xi = 1
i=1
(follows also from
positivity of M2-1)
1 g2 K(K + 2)
 ε3 =
6 gc2 K +1
 Notice that 3 increases with K (more convenient small K)
28
● Possible solution:
Cut a link, with f i  0, M 2 becomes block diagonal and (M -22 )1K  0
ò3 = 0
Add a Wilson line:
U  S1S 2
S KS K1
Groups connected through weak gauging
Particular example: DBESS model
(R.C. De Curtis, Dominici, Feruglio,
Gatto, Grazzini 1995,1996)
The theory has an enhanced symmetry SU(2)  SU(2) 2 29
L
R
ensuring 3  0 Inami , Lim, Yamada, 1992)


Unitarity bounds for the linear moose
(Chivukula, He; Papucci, Muck, Nilse, Pilaftis, Ruckl;
Csaki, Grojean, Murayama, Pilo, Terning)
●
We evaluate the scattering of longitudinal gauge bosons using
the equivalence theorem, that is using the amplitude for the
corresponding GB’s.
● We choose the following parametrization for the Goldstone fields
(we are evaluating WLWL-scattering)
Si  e
r r
if  /2fi2
K+1
1
1
, 2 = 2
f
i=1 f i
● The resulting 4-pion amplitude is given by
30
A    
f 4 K 1 u f 4 K
   6   Lij  (u  t)(s  M 2 )ij1  (u  s)(t  M 2 ) ij1 
4 i1 f i
4 i, j1
1
1  1
1 
Lij  gi g j  2  2   2  2 


 f i f i 1   f j f j1 
 In the low-energy limit, mW << E << mv, we get the LET:
A      
 In the high-energy limit
A      
4 K 1
f
4
1
u

6
i 1 f i
3
f 
1
u
u
u




 
4  i1 f i2 
4f 2
v2
4
K 1
Best unitarity limit
u
fi = fc  A = (K +1)2 v 2
ΛU = (K +1)Λ HSM  1.7(K +1)TeV
31
 By taking into account all the vectors and using the equivalence
theorem, the amplitudes for the Goldstones are given by
S  e
i
r r
ii  / 2 fi

A    
i
i
i
i
u
 2
4f i
 The unitarity limit is determined by the smallest link coupling.
By taking all equal (see also Chivukula, He, 2002)
Unitarity limit
u
fi = fc  A  (K +1)v 2
ΛU = (K +1)1/2 ΛHSM  1.7(K +1)1/2 TeV
M max
< Λ U , but roughly M max
2 K
V
V
gc
MW
g
Hardly compatible with
electro-weak
experimental constraints

2 K
gc
g
M W < 1.7 KTeV  c < 10
g
g
32
Delocalizing fermions
(RC, De Curtis, Dolce, Dominici; Chivukula, Simmons, He, Kurachi)
 Left- and right-handed fermions, yL (R) are coupled to the ends of
the moose, but they can coupled to any site by using a Wilson line
iL  S†i S†i1
S1†y L , iL  U i iL

i


i
bi  g     igi A  g '(B  L)Y  iL
2


i 
L
(We avoid
delocalization of the
right-handed fermions.
Small terms since they
could contribute to
right-handed currents
constrained by the KLKS mass difference)
33
 g2

1  0, 2  0, 3   yi  2 (1  yi )  bi 
i 1
 gi

K
 Possibility of agreement with EW data with some fine tuning
 Two cases:
f i  f c , g i  g c , bi  bc
(95% CL, with
rad. corrs. as
in the SM with
1 TeV Higgs
and mass 178
GeV for the
top)
34
Local cancellation
g2
bi   2 (1  y i ), with g i  g c , f i  f c
gi

g2 
i 
bi   2  1 

gi  K  1 
(95% CL, with rad.
corrs. as in the SM with
1 TeV Higgs and mass
178 GeV for the top)
35
The continuum limit
● Quite clearly the moose picture for large values of K can be
interpreted as the discretization of a continuum theory along a fifth
direction. The continuum limit is defined by
K  , a  0, Ka  R
bi
2
2
2
2
lim agi  g5 , lim af i  f (y), lim  b(y)
a0
a0
a0 a
●
The link couplings and a variable gauge coupling can be simulated
in the continuum by a non-flat 5-dim metrics. More interestingly, in the
continuum limit, the geometrical structure of the moose has an interpretation
in terms of a geometrical Higgs mechanism in a pure 5-dimensional gauge
theory.
36
. Consider an abelian gauge theory in 4+1 dim
1
1
1
AB
μν
L = - 2 FABF = - 2 Fμν F - 2 Fμ5 Fμ5
2g5
2g5
g5
Fμ5 = μ A5 -  5Aμ
through the gauge transformation:
we get
1
AB  AB  (5 ) ( BA5 )
A5 = 0, Fμ5 = 5Aμ , (Aμ (x, x 5 )   einx5 /R Aμn (x))
n
With a compactified 5 dim on a circle S2 of length 2R, the non
zero eigenmodes An acquire a mass:
n
Mn =
R
absorbing the mode A5n
The zero mode remains massless and a GB is present
37
 Massless modes can be eliminated compactifying on an orbifold,
that is
S / Z, Z : x5  x 5
2
 This allows to define fields as eigenstates of parity:
A B (x  , x 5 )   A B (x  , x 5 )
 Various possibilities, for instance by choosing:
AB
odd  no zero mode  only massive gauge bosons
in the spectrum
By making the theory discrete along the fifth dimension one gets back
the moose structure. In this case one speaks of a deconstructed
gauge theory (Hill, Pokorski, Wang, 2001).
38

A gauge field is nothing but a connection: a way of relating
the phases of the fields at nearby points. Once we
discretize the space the connection is naturally substituted
by a link variable realizing the parallel transport between
two lattice sites
A generalized s - model where
the Higgs mechanism is
realized in a standard way in
terms of a S – field (chiral field)
S  1  iaA5  e iaA5
SS†  1
39
†
S i = 1 - iaA5i- 1 , S i ¾ U¾i Î G¾
®
U
S
U
i- 1 i i
i
● More exactly:
DmS i = ¶ mS i - iAmi- 1S i + iS i Ami = - iaFmi-51
Fmi 5 = ¶ mA5i - ¶ 5Ami - i[Ami , A5i ]
a  1
1
i
i
† 
S   d x 2    Tr  F F   2 Tr (DSi )(DSi )  
g5  2 i
a

4
● Sintetically described by a moose diagram (Georgi, 1986 –Arkani-Hamed,
Cohen, Georgi, 2001)
40
● In order to describe completely the moose structure including also the
breaking, one needs also some kinetic terms on the branes plus BC’s. In the
case of a conformally flat metrics along the fifth direction the complete action
for a SU(2)-moose would be
πR
1
1
S = -  d 4 x  dz e-A(z) 2 (Fμνa ) 2 - 2(Fμ5a ) 2  +
4
g5 (z)
0
πR

1 4
1 3 2
-A(z)  1
a 2
-  d x  dz e
(Fμν ) δ(z) + 2 (Fμν ) δ(z - πR) 
2

4
g'
g

0
a
BC's : A1,2
|
=
0,

A
μ z  R
z μ |z 0 = 0
● Introducing the link variable
Smoose
Σi = e
, i = 1,
, K +1
K 1
 K 1

i
i
  d x   2 Tr  F F    f i2Tr (DSi )(DSi )†  
i 1
 i1 2gi

4
ae-Ai / g5i2 = 1 / g i2 , e-Ai / (ag5i2 ) = f i2
A = Wμ τ a / 2, A
1
μ
-iaA5i
a
K+1
μ
μ
= Y τ3 / 2
FLAT CASE :
f i = f c , gi = gc , e-Ai = 1, g5i2 = agc2
41
2
2
2
1 g K(K + 2)
1 g
1g
ε3 =

aK 
πR
2
2
2
6 gc
K
6 agc
6 g5
Let us go back to the unitarity limit. The 5-dim theory as a natural cutoff
proportional to 1/g52, say Lc, which can be related to the lattice spacing:
1
2
2
a=
 K ~ πRΛ c , Ka  πR, agi  g5
Λc
Then, the unitarity cutoff will be
Λ U ~ πRΛ c Λ HSM
If it would be possible to send a to zero (or Lc to infinity), the unitarity
cutoff would go to infinity. In fact, in the continuum limit there is a
complete cancellation of the terms increasing with the energy in the VB
scattering (Csaki, Grojean, Murayama, Pilo, Terning). However the 5dimensional theory is not renormalizable, and therefore the cutoff Lc makes
the longitudinal vector boson scattering amplitudes not unitary (see later)
42
The 5-dimensional cutoff turns out to be
1
1 
 2=
2 
 g5 2πRg4 
2
12π
m
1
W
It follows
Λ

Using
mW <
c
2
g
R
4
3π g
g
gΛ HSM 

Λc >
 Λ HSM  10 2 Λ HSM  mW =

2
2 g4
g4
8 π 

24π 12π
Λc = 2 =
2
g5
Rg4
3
and
Λ U ~ πRΛ c Λ HSM
2
12π3
20
=
Λ HSM 
Λ HSM
g4
g4
The cutoff is the the smaller between the two. On the other hand
2
1 g 
1g
 3 =   πR =  
6  g5 
12  g 4 
2
43
● Summarizing
 g
1
Λ c  10  2  Λ HSM , Λ U > 20 Λ HSM ,
g4
 g4 
1 g
3   
12  g 4 
2
●Therefore we could increase the cutoff of the theory within a
perturbative context (g4 ~ g), but this would be unacceptable from
the point of view of LEP bounds. On the contrary, if we want to
satisfy the LEP bounds we need g4 ~ 10 g, making the cutoff of the
order of LHSM.
● Introducing fermions in the bulk one gets a positive contribution
(Contino, Pomarol; Panico, Serone, Wulzer; Foadi Schmidt) in analogy to the
discrete case. Therefore with the help of some fine tuning one can
44
solve both problems.
● Since not much difference between the continuum and discrete moose
models, for phenomenological reasons it is simpler the analysis made in
the last case. The simplest possibility would be BESS (3-site model), but
in order to respect the EW bounds one has to make the VB’s almost
fermiophobic. However this is not a general feature, for instance, in the 4site model one can satisfy the EW bounds without having fermiophobic
new VB’s.
45
The Higgsless 4-site Linear Moose model
(Accomando, De Curtis, Dominici, Fedeli)
• 2 gauge groups Gi=SU(2) with global symmetry SU(2)LSU(2)R
plus LR symmetry: g2=g1, f3=f1
• 6 extra gauge bosons W`1,2 and Z`1,2 (have definite parity when g=g`=0)
GL
S1
S3
S2
G1
GR (K=2)
G2
• 5 new parameters {f1, f2, b1, b2, g1} related to their masses and
couplings to bosons and fermions (one is fixed to reproduce MZ)
f1,f2  M1,M2
M1 = f1g1
M2 =
charged and neutral gauge
bosons almost degenerate
M1
 M1
z
z=
f1
2
1
2
2
1
f + 2f
2
e
c,n
M1,2
~ M1,2 + O( 2 )
g1
46
The Higgsless 4-site Linear Moose model
Unitarity and EW precision tests
 g2

1  0 2  0, 3   2 (1  z 4 ) 
 2g1

EWPT
b1=b2=0
M1
=
M2
Best unitarity limit
for f1=f2 or z=1/p3
Unitarity and EWPT are
hardly compatible !
UNITARITY
all channels
WLWL
M2 =
O(e2/g12), b1=b2=0
A direct coupling of the
new gauge bosons to
ordinary matter must be
included: b1,2  0
47
The Higgsless 4-site Linear Moose model
EW precision tests
Calculations O(e2/g12), exact in b1, b2
1,2
 g2
b
 O(b ), 3   2 (1  z 4 )  
2
 2g1
2
2
b=
M2= M1/z
500 < M1 < 1000 GeV
b1 + b2 - (b1 - b2 )z
1+ b1 + b2
700 < M1 < 1600 GeV
48
The Higgsless 4-site Linear Moose model, Z`1,2 production
(500,1250)
M2= M1/z
1) b1=-0.05, b2=0.09
2) b1=0.06, b2=0.02
Z = 0.4
(1000,1250)
b1=-0.08, b2=0.03
b1=0.07, b2=0
Z = 0.8
Total # of evts in a 10GeV-bin versus Minv(l+l-) for L=10fb-1. Sum over e,
49
The Higgsless 4-site Linear Moose model, Z`1,2 production
# of evts for the Z`1,2 DY production within |Minv(l+l-)-Mi|< Gi
50
The Higgsless 4-site Linear Moose model, W`1,2 production
(500,1250)
M2= M1/z
b1=-0.05, b2=0.09
b1=0.06, b2=0.02
(1000,1250)
b1=-0.08, b2=0.03
b1=0.07, b2=0
Total # of evts in a 10GeV-bin versus Mt(l) for L=10fb-1. Sum over e,
51
The Higgsless 4-site Linear Moose model, W`1,2 production
# of evts for the W`1,2 DY-production for
>
The statistical significance for the W`s production is ~ a factor 2 bigger
than for the Z`s but it is less clean.
Neutral and charged channel are complementary
Could be both investigated at the LHC start-up with L ~ 100 pb-1
52
Summary and Conclusions
● Higher dimensional gauge theories naturally suggest the possibility
of Higgsless theories.
● Extensions of the nonlinear s model leads to moose theories.
● Simplest case: the linear moose.
● Difficulties in EW corrections similar to TC models.
● EW corrections and unitarity bounds push in different directions.
● Possibility of easing the theory delocalizing the fermions and
using some fine tuning.
53