I have come across the term "normalized hypercharge" in context with group theory in particle physics. I want to know what is the meaning of "normalization" of hypercharge and how to compute this.
Basabendu,
Yes, an existing normalization convention in standard model is the following. Consider fermion multiplets, with electric charge Q= T3L + Y
(3,2,1/6) , (3bar,1, -2/3), (3bar 1, 1/3), (1,2,1/2), (1,1,1)
QCD multiplets are normalized to (1/2 x 2 + 1/2 + 1/2)=2
SU(2)L multiplets are also normalized to ( 1/2 x 3 + 1/2)=2
But U(1) is normalized to 6 x (1/6)2 + 3 x (2/3)2 + 3 x (1/3)2 + 2 x (1/2)2 + 1=10/3
That means U(1) charges are to be multiplied by √(3/5) so that the total becomes 2 in the place of 10/3.
Sometime this is called an SU(5) normalization. This is because to embed matter multiplets into unified large multiplets of a GUT group it is necessary to normalize charges of each factor group of low energy theory identically.
Dear Basabendu,
there is no unique normalisation of hypercharge, like for any U(1) charge. The same is true for electric charge: for example, we usually define the charge of the electron to be (-1), however, we could euqally well define it as (+7/8), as long as we change the definition of all other charges accordingly. Of course it is convenient to define the unit of electric charge in terms of the charge of a known particle, but our formalism would not force us to do so.
Now, for hypercharge, we do not even have such an intuitive example system as the electron. This is why you can in fact find two different conventions for hypercharged normalisation in the literature on the Standard Model, whose difference you can e.g. see in the Gell-Mann/Nishijma relation: Q=T_3+Y or Q=T_3+Y/2, where Q is the electric charge and T_3 is the weak isospin (e.g. +1/2 for a neutrino or -1/2 for an electron).
Best regards,
Alexander
Dear Alexander,
Thank you so much for the explanation. I have one doubt though. Is there any convention of choosing the so called "normalization" factor for a particular group, say SU(n) ?
Dear Basabendu,
for other groups you don't have the same freedom. For example, for SU(2), you can show that only integer and half-integer values can exist for the angular momentum quantum number j, because you have just a countable set of different representation.
Best regards,
Alexander
PS: A good reference is the group theory book by Pierre Ramond.
Basabendu,
Yes, an existing normalization convention in standard model is the following. Consider fermion multiplets, with electric charge Q= T3L + Y
(3,2,1/6) , (3bar,1, -2/3), (3bar 1, 1/3), (1,2,1/2), (1,1,1)
QCD multiplets are normalized to (1/2 x 2 + 1/2 + 1/2)=2
SU(2)L multiplets are also normalized to ( 1/2 x 3 + 1/2)=2
But U(1) is normalized to 6 x (1/6)2 + 3 x (2/3)2 + 3 x (1/3)2 + 2 x (1/2)2 + 1=10/3
That means U(1) charges are to be multiplied by √(3/5) so that the total becomes 2 in the place of 10/3.
Sometime this is called an SU(5) normalization. This is because to embed matter multiplets into unified large multiplets of a GUT group it is necessary to normalize charges of each factor group of low energy theory identically.
Thank you so much Sir. I was looking for an example somewhat like this.
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