On the Flavor Problem in Strongly Coupled Theories - THEP Mainz

10 Chapter 1. Introduction: Problems beyond the Standard Model
Here the subscript at the vertical bar reads “evaluated at the respective scale”. Therefore,
the quark masses are given by
mq ∼ b 〈QQ〉� � ETC
Λ 2 ETC
= b NTCΛ 3 TC
Λ 2 ETC
, (1.13)
where we have assumed that 〈QQ〉 � � = 〈QQ〉 ETC � � = NTC Λ TC 3 TC . Since ΛTC is set by
the electroweak breaking scale, for fixed NTC, the scale ΛETC can be extracted from
(1.13) if the physical quark masses are used as an input. Allowing for a step-wise
breaking of the extended symmetry at different scales Λ1 > Λ2 > Λ3 would make
it even possible to dynamically generate the hierarchy between the three generations
which will therefore obtain masses mq3 > mq2 > mq1 .
However, ΛETC is severely constrained from the third type of contributions to (1.11),
the c-terms, which generate FCNC processes suppressed by the same scale that enters
the effective Yukawas. From Table 1.1 we know that even in more optimistic scenarios
(assuming no additional CP violation), this bound amounts to at least ΛETC > 10 3
TeV, which translates in a maximal quark mass of (NTC < 10, ΛTC ∼ 1TeV)
mq < b × 10 MeV . (1.14)
Especially the large mass of the top quark poses therefore a problem for theories where
the Higgs is described by a bound state.
This problem can be attenuated by considering theories that are not like QCD [20].
In a QCD-like theory, the running of the coupling is fast because asymptotic freedom
sets in quickly above ΛQCD, as illustrated on the upper left panel of Figure 1.2. The
techniquark condensate will therefore stay roughly the same between the ETC scale
and the TC scale and the assumption going into (1.13) is justified. More precisely,
the running is given by
〈QQ〉 � �� ΛETC
� = exp
ETC
ΛTC
dµ
µ γm(α(µ))
�
〈QQ〉 � � , (1.15)
TC
and QCD-like running corresponds to an anomalous dimension γm(α(µ)) ≈ γα(µ)
≈ γ/ ln(µ), which results in a power-logarithmic enhancement factor proportional
to ln(ΛETC) γ / ln(ΛTC) γ , similar to QCD radiative corrections to semileptonic electroweak
processes.
For general strongly coupled theories, such a behaviour is not mandatory. It may
well be, that the coupling evolves slowly for a large range of scales, before asymptotic
freedom sets in, as depicted on the lower left panel of Figure 1.2. In such a case, the
coupling stays close to a constant, so that γm(α(µ)) ≈ γα(µ) ≈ γα ∗ and the radiative
corrections give a power-law enhancement factor (ΛETC/ΛTC) γ . In terms of the beta
function, this behaviour corresponds to the convergence to a conformal fixed point,
but not quite reaching it, as illustrated in the lower right panel of Figure 1.2. Yang-
Mills theories with this behaviour are called walking and the corresponding walking
technicolor (WTC) theories allow for a significant amplification of the a-and b-terms
in (1.11), while the FCNC inducing c-terms, which do not couple to the technicolor
sector, still feel the full ETC scale suppression. In the WTC scenario, we can thus