Transcript Diapositiva 1 - Istituto Nazionale di Fisica Nucleare

Deconstructed Higgsless Models
Roberto Casalbuoni
Dept. of Physics, University of Florence
INFN - Florence
Conversano, June 17-20, 2005
June 17-20, 2005
R. Casalbuoni: Deconstructed Higgsless Theories
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Outline of the talk
(Based on papers by R.C., De Curtis, Dominici, 2004 and R.C., De Curtis, Dolce, Dominici, 2005)
 Introduction to higher-dimensional gauge theories
 Breaking the EW symmetry without Higgs and the linear moose
 EW corrections in the linear moose
 Unitarity bounds
 Delocalizing fermions
 Topological relations
 A planar moose
 The continuum limit
 Summary and conclusions
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Higher Dimensional Gauge Theories
 New interest in higher dimensional theories out of the possibility
of sub-millimeter extra dimensions due to softening of
gravitational interactions in a subspace (Arkani-Hamed, Dimopoulos, Dvali,
1998).
 This leads to a relation between the Planck scale in D-dim, MD,
and the one in 4-dim, MP
M R M
2
P
d
2d
D
1 1
1
1
 2d 2 d , r
2 2
MP r
MD r R
M 2D
M 2P
1, if
R
R
M D1
One can choose MD as low as 1 TeV
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 Other bonus of higher-dim theories is the possibility of a
geometrical Higgs mechanism in a pure gauge theory.
 Example: consider an abelian gauge theory in 4+1 dim
1
1
1
AB

L   2 FABF   2 FF  2 F 5F5
2g 5
2g 5
g5
F5   A 5   5A
through the gauge transformation:
we get
1
AB  AB  (5 ) ( BA5 )
A5  0, F5   5A , ( A (x, x5 )   e inx5 / R An (x))
n
With a compactified 5 dim on a circle S2 of length 2pR, 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
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 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
Very nice geometrical structure through discretization of the extra-dim
(Hill, Pokorski, Wang, 2001). One gets a deconstructed gauge theory
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
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  – field (chiral field)
  1  iaA5  e iaA5
†  1
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†
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i )(Di )  
g5  2 i
a

4
● Sintetically described by a moose diagram (Georgi, 1986 –Arkani-Hamed,
Cohen, Georgi, 2001)
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Breaking the EW Symmetry without
Higgs Fields
● Abstracting from the previous construction one can study more
general moose geometries.
● General structure given by many copies of the gauge group G
intertwined by link variables .
● 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.
Gi = SU(2). Each i 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:
● Simplest example:
# of links = # of sites +1
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● 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 .
 1  U L 1U1†
 K 1  U K  1U†R
● This 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. Prototypes of this
theory are the BESS model, K = 1, (R.C, De Curtis, Dominici, Gatto, 1985) and its
generalizations (R.C, De Curtis, Dominici, Gatto, Feruglio, 1989). Notice that SU(2)L x
SU(2)R is a custodial symmetry.
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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i )(Di )  
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, once we gauge
SU(2)L Ä SU(2)R Þ SU(2) Ä U(1)
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●
At the leading order in (MW/MV)2 we get
2
2
v
M
M 2W =
g 2 , M 2Z = 2W , e = gsq = g ' c q
4
cq
4
1
º 2=
2
v
f
K+ 1
å
i= 1
1
fi2
● Only oblique corrections, captured by the epsilon parameters
(Peskin, Takeuchi, 1990/92, Altarelli, Barbieri 1991, Altarelli, Barbieri, Caravaglios, 1998):
Custodial symmetry Þ
æg 2S ö g 2
÷
 3 = çç
=
çè16p ÷
÷
ø 4p
P
J V( A )
0
VV( AA )
vectors
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ò
¥
ds
Im (P
2
s
1 =  2 = 0
VV
( s) - P
AA
= J V( A ) J V( A )
(s))
Vector contributions
to the vector currents
associated to the
global SU(2)Lx SU(2)R
= f12g1A1m + (- )fK2 + 1g K A mK
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●
From vector mesons saturation one gets immediately
2 ö
K
æg 2
g
(1 - y i )y i
2
2
2
çç nV - nA ÷
= g g1g K (M 2 )1K = g å
4 ÷
2
çè m4
÷
m
g
i= 1
n
n ø
i
g2
 =
å
4 n
f2
yi = å xi , xi = 2 ,
fi
j= 1
i
●
●
Since
0 £ yi £ 1 Þ
Example:
fi  fc , g i  gc
1
=
2
f
K+ 1
å
i= 1
 ³ 0
1
Þ
2
fi
K+ 1
å
xi = 1
i= 1
(follows also from
positivity of M2-1)
1 g 2 K(K  2)
 3 
6 gc2
K
3


10
 (K  1), g c  22.3g
 3 exp
 For gc from 2g to 5g, 3
natural value is of order 10-1-10-2 not
compatible with the data
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●
Possible solutions:
1) Cutting a link
ég12 (f12 + f22 ) - g1g 2f22
ê
2
2
2
2
ê - g g f2
g
(
f
+
f
)
g
g
f
1 2 2
2 2
3
2 3 3
ê
2
2
2
2
ê
g
g
f
g
(
f
+
f
)
2
3
3
3
3
4
ê
M2 = ê
O
ê
ê
ê
ê
êë
ù
ú
ú
ú
ú
ú
ú
ú
ú
g K2 - 1 (fK2 - 1 + fK2 ) - g K- 1g K fK2 ú
ú
- g K- 1g K fK2
g K2 (fK2 + fK2 + 1 )ú
û
with fi  0, M 2 becomes block diagonal and (M-22 )1k  0
But we loose a link
3  0
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Add a Wilson line:
U  1 2
 K K1
Groups connected through weak gauging
The theory has an enhanced symmetry
SU(2) L  SU(2) R 
2
ensuring   0 Inami , Lim, Yamada, 1992)
Particular example:
D-BESS model
(R.C. et al. 1995,1996)
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2) Suppressing a link
For instance assume an exponential behavior (no need of extra
links)
fi  f ec(i1) , gi  gc
2
g 2c
3 
 2e
c
gc
gain of about 10-2
already for c  2
But lowering the link
couplings may give
unitarity problems
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Unitarity bounds for the linear moose
●
The worst high-energy behavior comes from the scattering of
longitudinal vector bosons. At energies higher than the vector mass
the amplitude can be evaluated using the equivalence theorem, that
is using the amplitude for the corresponding GB’s.
● To do that one has to choose the unitary gauge with respect to the
vector bosons
i  e
r r
if p / 2fi2
●
The effective lagrangian for the pion and the heavy vector can be
easily evaluated. The resulting 4-pion amplitude is given by
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A p p p p
f 2 K 1 u f 2 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 p p p p  
 In the high-energy limit
A p p p p
f 2 K 1 1
   6u
4 i1 f i
3
f 
1
u
u
u




 
4  i1 f i2 
4f 2
v2
2
K 1
Best unitarity limit
u
f i  fc  A  
(K  1) 2 v 2
moose  (K  1)HSM  1.2(K  1)TeV
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 By taking into account all the vector bosons the eigenchannel
amplitudes, using the equivalence theorem, are given by
  e
r r
ipi  / 2 fi
i

A p p p p
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
f i  fc  A  
(K  1)v 2
moose  (K  1)1/ 2 HSM  1.2(K  1)1/ 2 TeV
M max
  moose , but roughly M max
 KM W
V
V

KM W  1.2(K  1)1 / 2 TeV  K  14

K  3  5   moose  2.4  3 TeV
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Hardly compatible with
electro-weak
experimental constraints
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Delocalizing fermions
 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 (see also Chivukula, Simmons, He, Kurachi, 2005)
iL  †i †i1
1†y L , iL  U i iL

i

 i
i
bi       igi A  g '(B  L)Y   L
2


i 
L
June 17-20, 2005
(We avoid
delocalization of the
right-handed fermions.
Small terms since they
could contribute to
right-handed currents
constrained by the KLKS mass difference)
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 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)
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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)
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Topological relations
●
A moose diagram looks as a planar Feynman diagram via the
identification links = legs, sites = vertices
E = # external links
I = # of internal links
Vl = # of sites with l links
æ
ö
ç
÷
L = I - çå Vl - 1÷, S = I + E çè l
÷
ø
å
Vl
l
L = # of loops
S = # of not eaten up GB multiplets
L = S - (E - 1)
●
Requiring E = 2 to insure the SU(2)L x SU(2)R global symmetry and S = 1
in order to get 3 GB’s implies no loops
L= 0
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A planar moose
●
The most general SU(2) planar moose compatible with our
requirements
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●
These two moose are equivalent through the substitution:
N
1
1
 2
2
g1
j1 g(1j)
●
In general the planar moose is
equivalent to a linear one with an
analogous substitution
June 17-20, 2005
N
1
1
 2
2
gi
j1 g(ij)
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The continuum limit
●
The continuum limit is defined by
K  , a  0, Ka  pR
b
lim agi2  g52 , lim af i2  f 2 (y), lim i  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. For a flat metrics,
corresponding to equal f’s and g’s, one gets
1 g 2 K(K  2)
1 g2
3 

pR
2
2
6 gc
K
6 g5
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● For a Randall-Sundrum metrics, corresponding to exponential f’s
ky
fi  fe c(i 1) 

f
(y)

fe
c  ka
1 g 2 e 4kpR  4kpRe 2kpR  1
1 g2
3 

2
2kpR 2
kpR 1
4k g 5
(1  e
)
4k g 52
● By choosing kR ~ 10 and g52 = p R g42, we get
2
1 g2
g
3
3 

8

10
40p g 42
g 42
could be consistent
with EW data
● For the planar moose:
1 g2 2
1 g 2 pR '
3 
p RR ' (flat case);  3 
(RS case)
2
2
6 g6
4 g6 k
No suppression from the vertical links
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Summary and Conclusions
● Higher dimensional gauge theories naturally suggest the possibility
of Higgsless theories.
● Study in 4-dim through discrete extra-dim 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.
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