Theoretical Studies of Flavor‐
Changing Neutral Currents in 1980’s
C.S. Lim (Chong‐Sa Lim, 林 青司)
Kobe Univ., Japan@ KAON09, June 9‐12 (‘09), Tsukuba

Ｉ． A little (biased) history of flavor physics mainly in 1980’s
If there is a perfect symmetry among generations (or flavors), generation
number or flavor is exactly conserved ( ← Noether). Thus Flavor‐Changing
Neutral Current (FCNC) stems from the violation of flavor symmetry .
The Weinberg‐Salam model with only 3 favors breaks the symmetry, and toremedy the problem of too large FCNC c quark was introduces (GIM).
FCNC is sensitive to the breaking of the flavor symmetry due to the mass‐squared differences among quarks.
M. Gaillard and B.W. Lee (‘74)
They could predict
before the discovery of:ideal work in the field of phenomenology.
(Their work greatly influenced ours.)

How about t quark introduced by Kobayashi and Maskawa ?
In 80’s t quark was regarded to be not so heavy,
The formula by Gaillard‐Lee , using the approximation
to give reasonable results for FCNC.

.
, still seems

However, Takeo Inami (Chuo Univ.) and C.S. L. published a paper(“Effects of Superheavy Quarks and Leptons in Low‐Energy Weak Processes
and” , Prog. Theor. Phys. 65(‘81) 297).
In contrast to the common sense, we assumed the presence of “arbitrarily” heavy(but not so heavy as to spoil perturbation) intermediate quarks and leptons(t, t’, L). We derived several functions describing the quantum effects of theseheavy fermions to the FCNC processes, without invoking to theapproximation.

A strong theoretical motivation
We were interested in the possible non‐decoupling effects of heavy fermions,anticipated from the large Yukawa couplings of heavy fermions in the theories
with spontaneous symmetry breaking (SSB) .
(Purpose) : to make clear whether the quantum effects due to the heavyfermions really give some non‐decoupling effects or not.
Eventually, it turned out→ We were lucky.

Rξ gauge (K. Fujikawa, B.W. Lee and A.I. Sanda, Phys. Rev. D6(‘72)2923)
Each diagram has ξ ‐ dependence, but the final result should be ξ‐ independent.
A good check of calculation.
In this gauge, the (would‐be) Nambu‐Goldstone boson has Yukawa couplings,
which are proportional to masses of intermediate heavy fermions → thepossibility of non‐decoupling effects of heavy fermionsA very useful suggestion by K. Fujikawa
The ξ ‐dependence appearing in the FCNC Z‐vertex gave ξ‐ dependent 4
fermi effective lagrangian with vector‐like coupling of charged lepton:ξ ‐dependence

The problem seemed to be solved by adding “QED‐penguin” diagram, though the penguin diagram has nocontribution to the decay processes .
I asked Prof. Fujikawa his advise and he recommended
me to calculate thevertex as well.
The calculated
vertex was later used in the analysis of FCNC radiativedecays, e.g.
We greatly appreciate the useful advice.

Attempts to estimate mt
Is it possible to estimate mt by comparing with the data on the FCNC
processes ?
A.J. Buras attempted to estimate mt(“An Upper Bound on the Top Quark Mass from Rare Processes”,
Phys. Rev. Lett. 46(‘81) 1354)
Because of large uncertainty of mixing angles, CP phase at that time made theestimation difficult.
We thank Prof. Buras very much for bringing public attention to our work.

2. Non‐decoupling effects due to heavy fermions
There is a ``decoupling theorem” (T. Appelquist and J. Carazzone (‘75)), sayingthat the effects of heavy (unknown) particles are suppressed by the inverse
powers of their large masses.
@ tree level
An example: see‐saw mechanism” (Gell‐Mann, Ramond & Slansky,T. Yanagida)：
The Majorana mass term for the left‐handed neutrino does not exist as arenormalizable operator in the theory and is described in terms of gaugeinvariant operator with higher mass dimension (> 4),: reflects the scale of unknown new physics (NP)→ decoupling@ quantum level
The decoupling is not so trivial , as the energy of intermediate states can be large.
In fact, in QED assumed heavy lepton with mass M does contribute to therenormalization factor, as
After the renormalization, all remaining observables (described by operatorswith higher mass dimensions) are suppressed by the powers of 1/M
In gauge theories with spontaneous symmetry breaking (SSB), however, thesituation can be quite different.
The large masses of heavy particles are provided not by some New Physics (NP)mass scale M, but from the weak scale MW itself, or the VEV of Higgs.
： no suppression like 1/M
Since every masses come form the VEV, heavy means “strong” Yukawa
coupling. → We can expect some non‐decoupling effects
In fact, in our work (‘81), we found non‐decoupling effects of heavy fermion,say t quark, in FCNC processes, growing asFor instance, in the box diagram, the exchange of N‐G bosonquark

Qualitative difference between Z andand tvertices

Z vertex:
Vertex:CVC

To some extent, the vertex mimics QED, where decoupling theorem holds.
→ The contribution of 4th generation, if it ever exists, may be remarkable inthe Z vertex (weak‐penguin diagram).

3. CP violation in the Kobayashi‐Maskawa model and the non‐decoupling effects
It was great news that the Nobel prize in physics of 2008 was awarded toProfessors Nambu, Kobayashi and Maskawa.
In the Kobayashi‐Maskawa (KM) model, CP violation is accompanied by thebreaking of flavor symmetry. Namely, if there is mass degeneracy between
different generations, CP violation is known to disappear even for non‐vanishing
CP phase. Intuitively, in this case the number of generations is effectively
reduced from 3 to 2.
→ CP violating observables are induced through FCNC processes
To get CP violation in the KM model, all of three generations have to participate.
In fact, in the neutral kaon system CP violating observable
is dominated by
the non‐decoupling effect due to t quark :As far as all three generations participate, CP violation should happeneverywhere, in particular in the B meson system.
In addition, since the non‐decoupling effect due to the top quark is thedominant source, such CP violating effect is expected to be large in the B
system , which couples to the t quark more directly (A.I. Sanda et al.)→B‐factory experiment

4. The ideas of NP and the non‐decoupling effects“Puzzles “(1) Rather large rate of

mixing, while the short distance contribution of the Standard Model is estimated to bevery small due to small mass‐squared differences among d, s, b quarks,(2)Possible deviation from the Standard Model predictions for CP asymmetries
in B decays ( e.g., A. Soni et al., arXiv: 0807.1971[hep‐ph] )・

predicted by use ofandis a little larger than directly
measured value by
・ Rather large difference in the direct CP asymmetry,・ Indication of large non‐standard CP phase in the Bs decay,

Among various possibilities of NP, the presence of fourth family may be anatural cause of these puzzles.
Since the heavy fermions of fourth generation should get their masses
through SSB, they yield non‐decoupling effects enhanced as,compared with those by third generation, except for the factors ofgeneration mixings.
The enhancements by the powers ofwill be realized in the box
diagram and FCNC Z‐vertex (weak penguin), thus making the characteristic
predictions of this scenario possible.

5. Summary(1) The question of whether heavy fermions decouple from low energy
processes or not was an interesting theoretical issue in early 80’s.
(2) The top quark turned out to be unexpectedly heavy, and was found to haveimportant non‐decoupling contributions to FCNC processes, including CP
violating processes, as the reflection of SSB.
(3) The extent of the non‐decoupling effects is different , e.g. between QED‐ andweak‐penguin diagrams.
(4) In the Kobayashi‐Maskawa model, CP violating observables are inevitablyinduced by FCNC processes.
(5) The non‐decoupling effects of the top quark play crucial roles in the CP
violating observables both in K and B meson systems.
(6) Recent claims on the possible deviations from the Standard Model predictions
concerningand CP asymmetries in B decay processes, may
signal some New Physics (NP).
(7) If we take the possibility of the 4th generation as NP, the unknown heavy
fermions are expected to have enhancement factor, such as.
(8) The fact that the enhancement is remarkable in the box and weak‐penguin
diagrams, make the characteristic predictions of the 4th generation scenario
possible.