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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130 Chapter 3. Solv<strong>in</strong>g <strong>the</strong> <strong>Flavor</strong> <strong>Problem</strong> <strong>in</strong> <strong>Strongly</strong> <strong>Coupled</strong> <strong>Theories</strong><br />
to <strong>in</strong>troduce an additional scalar which is ei<strong>the</strong>r a s<strong>in</strong>glet under <strong>the</strong> color groups or<br />
under <strong>the</strong> SU(2)L, <strong>in</strong> order to break <strong>the</strong> residual U(1) <strong>in</strong> <strong>the</strong> potential.<br />
The vacuum expectation values of <strong>the</strong> three scalars are dictated by <strong>the</strong> symmetries of<br />
<strong>the</strong> vacuum<br />
〈 (Hu) i aα〉 = vu<br />
√ δaαδ<br />
2NC<br />
i1 ,<br />
〈 (Hd) i aα〉 = vd<br />
√ δaαδ<br />
2NC<br />
i2 ,<br />
〈h i 〉 = vℓ<br />
√2δ i2 . (3.80)<br />
Here, small lat<strong>in</strong> letters from <strong>the</strong> middle of <strong>the</strong> alphabet denote SU(2)L <strong>in</strong>dices. The<br />
mass term for <strong>the</strong> electroweak gauge bosons (2.102) and for <strong>the</strong> axigluon (3.77) change<br />
accord<strong>in</strong>gly and one f<strong>in</strong>ds,<br />
MW = g5<br />
�<br />
v2 u + v2 d + v2 ℓ<br />
,<br />
2<br />
MZ = MW<br />
cw<br />
, MA = gs5<br />
�<br />
v2 u + v2 d<br />
√ .<br />
2NCsθ cθ<br />
(3.81)<br />
All three scalars are electroweak doublets and consequentially give mass to <strong>the</strong> electroweak<br />
gauge bosons, while only <strong>the</strong> color-charged fields couple to <strong>the</strong> axigluon. It<br />
follows <strong>the</strong>refore for <strong>the</strong> SM vev v = 246 GeV,<br />
�<br />
v = v2 u + v2 d + v2 ℓ , (3.82)<br />
from which one can already <strong>in</strong>fer for <strong>the</strong> factor <strong>in</strong>troduced <strong>in</strong> (3.64) that ξ < 1.<br />
In order to make this explicit, we will repeat <strong>the</strong> match<strong>in</strong>g of <strong>the</strong> 5D to <strong>the</strong> 4D<br />
Lagrangian, which was performed for <strong>the</strong> electroweak sector <strong>in</strong> (2.3), for <strong>the</strong> extended<br />
color sector. Therefore, <strong>the</strong> rotation (3.54) has to be <strong>in</strong>troduced <strong>in</strong> <strong>the</strong> Lagrangian for<br />
<strong>the</strong> SU(3)D × SU(3)S bulk gauge fields<br />
LSU(3)D×SU(3)S = GKM G LN<br />
�<br />
− 1<br />
4 (GrD)KL(G r D)MN − 1<br />
4 (GrS)KL(G r �<br />
S)MN , (3.83)