Suppose the hypercharges of one generation were not known

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Particle Physics & Quantum Fields · Beyond the Standard Model

Symmetry and Flavour, Ch Grand unification: running couplings, SU(5), proton decay and the hierarchy problem

Problem

Suppose the hypercharges of one generation were not known. Keep the fields and their

SU(3)\times SU(2)

content,

Q\,(3,2)

u^c\,(\bar3,1)

d^c\,(\bar3,1)

L\,(1,2)

and

e^c\,(1,1)

, but give them unknown hypercharges

y_Q

y_u

y_d

y_L

and

y_e

, with

y_Q\neq0

. Require the four anomaly sums of this chapter to vanish: the mixed sum over the colour triplets and antitriplets, the mixed sum over the weak doublets,

\sum Y=0

and

\sum Y^3=0

, each state counted with its multiplicity. ((a)) Write the two mixed sums and show that they give

y_u+y_d=-2y_Q

and

y_L=-3y_Q

. ((b)) Use

\sum Y=0

to find

y_e

in terms of

y_Q

. ((c)) Use

\sum Y^3=0

to show that

y_u

and

y_d

are

-4y_Q

and

2y_Q

in some order. Calling

u^c

the field with hypercharge

-4y_Q

and taking

y_Q=\tfrac16

, find

y_L

y_e

y_u

and

y_d

. ((d)) Show that, for any

y_Q

, the hypercharges of

d^c

and

L

satisfy

3y_d+2y_L=0

, the tracelessness of hypercharge on the

\bar5

that

SU(5)

requires.

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