Transcript Slide 1

Quark Mass Matrix
from
Dimensional Deconstruction
Andrea Soddu
National Taiwan University
Academia Sinica
P.Q Hung, A.S., N.-K. Tran
Taipei November 17, 2004
hep-ph/0410179
OUTLOOK
What if the world is not four dimensional ?
Dimensional Deconstruction
Deconstructing 5D QED
Deconstructing 5D SU(2)xU(1)
Fermion zero modes and Quark Mass Matrices
Numerical results
Possible projects
Conclusions
World is apparently four dimensional
The Universe may be better described by a theory with more than
the conventional one time and three space coordinates at energies
not yet probed
Simple scenario: theory of fields living in a space-time with four
extended dimensions, plus one or more additional compact dimensions
At distances large compared to the size of compact dimensions,
the world appears four dimensional: gauge forces fall off like the
square of the distance
At energies corresponding to the inverse compactification size,
Kaluza-Klein excitations appear with a spectrum depending on the
detailed nature of the compact space
At energies much higher than the compactification scale the extra
dimensions become manifest. Physics is insensitive to the
compactification and the theory appears higher dimensional
Higher-dimensional field theories have dimensionful couplings
and therefore require a cutoff
As energy approaches the cutoff physics becomes strongly coupled
and this makes difficult to address what happens at energies above the
cutoff
Dimensional (De)Construction provides a possible scenario for UV
completing higher-dimensional field theories
N. Arkani-Hamed, A.G. Cohen, and H. Georgi
Dimensional Deconstruction provides a manifestly gauge invariant
description of n KK modes for an SU(m) gauge theory in the bulk
C. T. Hill, S. Pokorski, and J. Wang
Dimensional Deconstruction provides at low energy a theory
with a similar spectrum of particles as higher dimensional theories
A natural light axion can be obtained as a WLPNGB
C. T. Hill, and K. Leibovich
U(1) gauge theory in 1+4 dim.’s periodically compactified to 1+3 dim.’s
Latticization of a U(1) gauge theory with periodic 5th dim. gives a
theory in 1+3 dim.s for N copies of the U(1) gauge group
1=N+1 2
n
…
n+1
N-1
N
…
invariant under
has replaced
translational invariance
Good approximation in the large N case for quantities that are
insensitive to short distance (UV) structure of the theory
B.C.
causes each
to develop a common VEV
compactification scale
From the point of view of 1+3 dim.’s each
a non linear model field
is effectively
gauge transformation
is the zero mode of
( SM photon )
tower of doubled KK modes ( massive photons )
N-1
are eaten to become longitudinal modes
is a massless scalar field (WLPNGB)
At low energy the deconstructed model has the same spectrum as
an higher dimensional model
The master gauge group U(1)N is broken to the diagonal subgroup
U(1) by the
N-1 of the link fields
fields (KK modes)
are eaten, giving N-1 massive vector
One massless vector field
are left
When fermions are introduced
The mass of the
and one massless scalar field
becomes massive
field is obtained by expanding about
the minimum of its Coleman-Weinberg potential
Similar result is obtained in a non-Abelian gauge theory
has been proposed as a candidate for an axion
Moose Diagram
Moose not for “chocolate moose”
Moose is a large deer, “ELK”
Deconstructing SU(2)xU(1)
2x2 matrix
Because of the potential
develop a VEV
[SU(2)xU(1)]N is broken down to the diagonal group SU(2)xU(1)
Four zero modes
are left
We are interested in the SM Chiral Fermions
Impose Chiral Boundary Conditions on Fermions
One more LH d.o.f over RH for Q field
One more RH d.o.f over LH for U and D fields
all couplings remain real
is complex
for
for
Zero Mode
Zero Eigenvalue
for
for
Zero Mode
Zero Eigenvalue
for
for
Zero Mode
Zero Eigenvalue
has the same
eigenvalues as
except the zero eigenvalue
unitary (not just orthogonal)
Zero Mode localized at n = 1
C.B.C.
N.C.
Zero Mode localized at n = N
N.C.
C.B.C.
localized in n=1
localized in n=1
localized in n=N
theory space
bigger is a more localized is the zero mode w. f.
The diagonal group SU(2)xU(1) is broken down to U(1)Q by SSB
Fermion zero modes obtain mass through Yukawa interaction
with a Higgs field as in the SM
SU(2)xU(1) is equally broken at each site n
Comment: in Higgless theories [SU(2)xU(1)]N is broken completely
through the deconstruction process. The SM massive gauge bosons
correspond to the lightest KK modes.
R.S. Chivukula, M. Kurachi, and M. Tanabashi
zero modes are the SM particles
Simulated Annealing Method
When a liquid is cooled down sufficiently slowly, the atoms are often
able to line themselves up and form a pure crystal, state of minimum
energy
Assign a probability
from a configuration with energy
to the change
to one with energy
assign
The system always takes a downhill step
transition
no transition
The system sometimes takes an uphill step
The temperature T is reduced with a given schedule
Q, U and D localized at n=1
Q, U and D localized at n=1
Q and U localized at n=1
D localized at n=N
Q and U localized at n=1
D localized at n=N
is localized at
is delocalized
lives in the bulk
new source of CP violation
mediated by KK gluons
One can have CP violation with only two families
A. Delgado, A. Pomarol, and M. Quiros
KK gluons mediate FCNC at tree level
In a split fermion scenario from deconstructed models
one can have a new source of CP violation as well
One can constrain
from FCNC
Conclusions
The world could have more than four dimensions
Deconstructed theories have the same low energy behavior as
higher dimensional theories but gauge invariance is preserved
(De)constructed theories can offer a UV completion of higher
dimensional theories
Zero mode fermions can be localized in the theory space
Realistic quark mass matrices can be constructed starting from
universal Yukawa coupling
Future projects
Study of split fermion scenarios from deconstruction and FCNC
Generation of small Dirac neutrino masses without See-Saw