On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

32 Chapter 1. Introduction: Problems beyond the Standard Model
orders in λ)
(Cd 11 + Cd 22 )λ2
Λ2 (
Fl
¯ Q2 d c 1)( ¯ Q2 d c 1) , (1.69)
and there is no reason why, considering an abelian symmetry, the diagonal coefficients
Cd ii should be related. In Nelson Strassler type models (or in warped extra dimensions)
there is also no reason to assume this, but the coefficients are individually small,
because they are proportional to the localizations of the involved light flavors or the
corresponding mixing angles (compare the discussion at the end of 1.1). Further,
models with Yukawa hierarchies induced from strong renormalization group running
do not actually have an underlying flavor symmetry. At the flavor scale, the Yukawa
coupling is an anarchic order one matrix, which does not increase the global symmetry
group of the Lagrangian compared to the low energy theory. This is an advantage,
because in the Froggatt Nielsen model the Yukawa couplings emerge as effective operators
from a fundamental flavor symmetric theory. The breaking of this continuous
symmetry by the vev of φ results in massless Goldstone bosons. As a consequence, it
can not be realized exactly and even then, light Would-be Nambu Goldstone bosons
are bound to appear. This is the reason why, after it became known that a rich flavor
sector is realized in nature (basically after the discovery of the charm quark), most of
the proposals for a underlying symmetry were based on discrete groups, most of them
non-abelian ones (see [73, Ref.5]).
Non Abelian Flavor Symmetry
Another way to avoid the FCNC problem of ablian flavor theories is to assume nonabelian
flavor symmetries, which can ensure the coefficient matrix of (1.68) to be
universal. In these approaches, the maximal U(3) 3 = U(3)Q × U(3)u × U(3)d quark
flavor symmetry, or a non-abelian subgroup, is considered to be broken by flavon
fields which transform in a χ ∼ (¯3, 3) under the respective U(3)s (or a corresponding
equivalent for smaller flavor groups) in order to make the Yukawa couplings
1
ΛFl
χu ¯ Q ˜ Hu c + 1
ΛFl
χd ¯ QHd c
(1.70)
invariant. The problems regarding continuous groups mentioned at the end of the last
section have lost nothing of their validity and it is therefore preferable to consider
a discrete flavor symmetry, in this case a discrete subgroup of U(3) 3 . In principle,
another solution would be to gauge the flavor group in order to get rid of the massless
Goldstone modes. However, the small breaking in the light flavor sector will generate
very light flavor gauge bosons, as was already pointed out in the very first papers
considering gauged flavor groups [73]. But also if a discrete non-abelian subgroup is
realized, it should emerge from a gauged group via spontaneous breaking, because
fundamental global symmetries are not respected by quantum gravity (e.g. a black
hole may eat a proton and Hawking radiate a positron). Assuming such an explanation
is found, renormalizable terms in the flavon potential might still be invariant under
an accidental continuous symmetry, leading to Goldstone bosons. Higher dimensional