Suppose a fourth generation of quarks existed

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Particle Physics & Quantum Fields · Linear algebra and vector spaces

Symmetry and Flavour, Ch Quark mixing: the Cabibbo angle, the GIM mechanism and the CKM matrix

Problem

Suppose a fourth generation of quarks existed, with a down-type quark

b'

, so that the up quark could also turn into

b'

and the first row of the mixing matrix gained a fourth entry

V_{ub'}

. The measured first row gives

|V_{ud}|^2 + |V_{us}|^2 + |V_{ub}|^2 = 0.9985 \pm 0.0007

; take

\lambda = 0.225

and

|V_{ub}| = 0.0037

. ((a)) How many mixing angles and how many physical phases would the

4\times4

mixing matrix carry? ((b)) Find the

|V_{ub'}|

that makes the four-entry first row exactly unitary, and the range of

|V_{ub'}|

allowed if the measured sum lies anywhere within one standard deviation of its central value. ((c)) Express the central

|V_{ub'}|

as a multiple of

\lambda^2

and of

|V_{ub}|

, and say whether the data require a fourth generation.

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