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/

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 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 5 th 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 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

is complex all couplings remain real

for 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. N.C.

bigger is  more localized is the zero mode w. f. 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 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 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 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 Study of split fermion scenarios from deconstruction and FCNC Generation of small Dirac neutrino masses without See-Saw Future projects