Quark Mass Matrix from Dimensional Deconstruction

Academia Sinica

Andrea Soddu

Taipei November 17, 2004

National Taiwan University

P.Q Hung, A.S., N.-K. Tran

hep-ph/0410179

OUTLOOK

Dimensional Deconstruction

What if the world is not four dimensional ?

Deconstructing 5D QED

Deconstructing 5D SU(2)xU(1)

Fermion zero modes and Quark Mass Matrices

Numerical results

Conclusions

Possible projects

World is apparently four dimensional

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

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

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 WLPNGBC. 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+12 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.

From the point of view of 1+3 dim.’s each is effectively a non linear model field

compactification scale

causes each to develop a common VEV

gauge transformation

is the zero mode of

tower of doubled KK modes ( massive photons )

( SM photon )

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 are eaten, giving N-1 massive vector fields (KK modes)

One massless vector field and one massless scalar field are left

When fermions are introduced becomes massive

The mass of the 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 FermionsImpose Chiral Boundary Conditions on Fermions

One more LH d.o.f over RH for Q fieldOne more RH d.o.f over LH for U and D fields

is complex

all couplings remain real

for

for

for

for

for

for

Zero Mode Zero Eigenvalue

Zero Mode Zero Eigenvalue

Zero Mode Zero Eigenvalue

unitary (not just orthogonal)

has the same

eigenvalues as

except the zero eigenvalue

Zero Mode localized at n = 1

Zero Mode localized at n = N

C.B.C.

C.B.C.N.C.

N.C.

bigger is more localized is the zero mode w. f.

localized in n=1

localized in n=1

localized in n=N

theory space

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.

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

R.S. Chivukula, M. Kurachi, and M. Tanabashi

zero modes are the SM particles

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

Simulated Annealing Method

Assign a probability to the change

from a configuration with energy to one with energy

assign

The system always takes a downhill step

transitionno 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

KK gluons mediate FCNC at tree level

A. Delgado, A. Pomarol, and M. Quiros

In a split fermion scenario from deconstructed modelsone 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

Study of split fermion scenarios from deconstruction and FCNC Generation of small Dirac neutrino masses without See-Saw

Future projects