On the Flavor Problem in Strongly Coupled Theories - THEP Mainz
On the Flavor Problem in Strongly Coupled Theories - THEP Mainz
On the Flavor Problem in Strongly Coupled Theories - THEP Mainz
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80 Chapter 2. The Randall Sundrum Model and its Holographic Interpretation<br />
<strong>the</strong> unitary diagonalization matrices U q q<br />
R and UL [70], which read<br />
(U q<br />
L )ij = (uq)ij<br />
(U q<br />
R )ij = (wq)ij e iφj<br />
⎧<br />
F (cQi<br />
⎪⎨<br />
)<br />
, i ≤ j ,<br />
F (cQj )<br />
⎪⎩<br />
F (cQj )<br />
, i > j ,<br />
F (cQi )<br />
⎧<br />
F (cqi<br />
⎪⎨<br />
)<br />
, i ≤ j ,<br />
F (cqj )<br />
⎪⎩<br />
F (cqj )<br />
, i > j .<br />
F (cqi )<br />
⎛<br />
⎜<br />
uq = ⎜<br />
−<br />
⎜<br />
⎝<br />
1<br />
∗<br />
(Mq) 21<br />
(Mq) ∗<br />
11<br />
(Mq) ∗<br />
31<br />
(Mq) ∗<br />
11<br />
(Mq) 21<br />
(Mq) 11<br />
−<br />
⎛<br />
⎜<br />
1<br />
⎜<br />
wq = ⎜<br />
⎜−<br />
⎜<br />
⎝<br />
(Mq) 12<br />
(Mq) 11<br />
(Mq) 13<br />
(Mq) 11<br />
1<br />
∗<br />
(Yq) 23<br />
(Yq) ∗<br />
33<br />
(Mq) ∗<br />
12<br />
(Mq) ∗<br />
11<br />
1<br />
− (Yq) 32<br />
(Yq) 33<br />
⎞<br />
(Yq) 13<br />
(Yq) ⎟<br />
33 ⎟<br />
(Yq) ⎟<br />
23 ⎟<br />
(Yq) ⎟ ,<br />
33 ⎟<br />
⎠<br />
1<br />
(2.159)<br />
(Yq) ∗<br />
31<br />
(Yq) ∗<br />
33<br />
(Yq) ∗<br />
32<br />
(Yq) ∗<br />
⎞<br />
⎟ ,<br />
33 ⎟<br />
⎠<br />
1<br />
(2.160)<br />
<strong>in</strong> which <strong>the</strong> phase factor<br />
e i φj<br />
�<br />
= sgn F (cQj )F (cfj )�e −i θj ,<br />
⎛<br />
⎞<br />
arg(det Y q) − arg((Mq)11)<br />
θ = ⎝ arg((Mq)11) − arg((Yq)33) ⎠ ,<br />
arg((Yq)33)<br />
(2.161)<br />
appears as a consequence of <strong>the</strong> convention to choose <strong>the</strong> diagonal entries of U q<br />
L real.<br />
The CKM matrix VCKM = (U u L )† U d L can <strong>the</strong>n be expressed <strong>in</strong> terms of <strong>the</strong> zero mode<br />
profiles and <strong>the</strong> 5D Yukawa coupl<strong>in</strong>gs alone. The Wolfenste<strong>in</strong> parameters read<br />
|Vus|<br />
λ = �<br />
|Vud| 2 + |Vus| 2<br />
, A = 1<br />
� �<br />
� Vcb �<br />
� �<br />
λ �Vus<br />
� , ¯ρ − i¯η = −V ∗ udVub V ∗<br />
cdVcb , (2.162)<br />
so that we obta<strong>in</strong> from (2.159)<br />
|F (cQ1 )|<br />
λ =<br />
|F (cQ2 )|<br />
�<br />
�<br />
�<br />
(Md) 21<br />
� −<br />
(Md) 11<br />
(Mu)<br />
�<br />
�<br />
21 �<br />
|F (cQ2<br />
(Mu) � , A =<br />
11<br />
)|3<br />
|F (cQ1 )|2 |F (cQ3 )|<br />
�<br />
�<br />
� (Yd) 23<br />
�<br />
� (Yd) 33<br />
��<br />
�<br />
� (Md) 21<br />
� (Md) 11<br />
¯ρ − i¯η =<br />
(Yd) 33 (Mu) 31 − (Yd) 23 (Mu) 21 + (Yd) 13 (Mu) 11<br />
(Yd) 33 (Mu) 11<br />
� (Yd) 23<br />
(Yd) 33<br />
− (Yu) 23<br />
(Yu) 33<br />
� � (Md) 21<br />
(Md) 11<br />
− (Mu) 21<br />
(Mu) 11<br />
− (Yu) 23<br />
(Yu) 33<br />
− (Mu) 21<br />
(Mu) 11<br />
� .<br />
� 2<br />
�<br />
�<br />
�<br />
�<br />
�<br />
� ,<br />
�<br />
�<br />
�<br />
(2.163)<br />
Note, that ρ and η are <strong>in</strong> lead<strong>in</strong>g order <strong>in</strong>dependent of <strong>the</strong> zero mode profiles and thus<br />
of <strong>the</strong> quark localization. S<strong>in</strong>ce <strong>the</strong> Yukawa coupl<strong>in</strong>gs are not supposed to add any
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