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samalkhaiat

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[tex]

J^{ \mu }_{ \mbox{ em } } ( x ) = J^{ \mu }_{ \mbox{ em } , I = 0 } ( x ) + J^{ \mu }_{ \mbox{ em } , I = 1 , I_{ 3 } = 0 } ( x )

[/tex]

or

[tex]

J^{ \mu }_{ \mbox{ em } } ( x ) = S^{ \mu } ( x ) + V^{ \mu }_{ 3 } ( x )

[/tex]

The two pieces are separately conserved, reflecting conservation of Baryon number and 3rd component of iso-spin. Indeed, it is easy to see that

[tex]

I^{ 2 } V^{ \mu }_{ 3 } ( 0 ) | 0 \rangle = I_{ i } [ I_{ i } , V^{ \mu } ( 0 ) ] | 0 \rangle = \epsilon_{ i k 3 } \epsilon_{ i k l } V^{ \mu }_{ l } ( 0 ) | 0 \rangle = 2 V^{ \mu }_{ 3 } ( 0 ) | 0 \rangle .

[/tex]

This means that [itex]V^{ \mu }_{ 3 } ( 0 ) | 0 \rangle[/itex] is an eigenstate of [itex]I^{ 2 }[/itex] with eigenvalue [itex]1 ( 1 + 1 ) = 2[/itex], i.e. the iso-vector part of the hadronic em-current connects the vacuum to states with [itex]I = 1 , I_{ 3 } = 0[/itex].

I leave you to conclude that

[tex]

I ( I + 1 ) \langle 0 | S^{ \mu } ( 0 ) | I , I_{ 3 } = 0 \rangle = 0 ,

[/tex]

which means that the iso-scalar part of the em-current, only connects the vacuum to states of total iso-spin zero.

Good luck

Sam

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Thanks for reply.

Why I3 is zero? What type of transitions should we understand? Why is there Delta?

Why I3 is zero? What type of transitions should we understand? Why is there Delta?

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samalkhaiat

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I don’t understand what you mean by “what type of transitions”. I also don’t know how much you know about the subject. However, I can tell you what you need to know.Thanks for reply.

Why I3 is zero? What type of transitions should we understand? Why is there Delta?

The Lagrangian of the quarks iso-doublet has three symmetries:

i) The invariance under a simultaneous phase change of the up and down quark fields, [itex]U_{ B } (1)[/itex], given by

[tex]\delta \Psi = - i \alpha \Psi ,[/tex]

leads to the conserved current

[tex]S^{ \mu } ( x ) = \bar{ \Psi } ( x ) \gamma^{ \mu } \Psi ( x ) ,[/tex]

and to the constant charge

[tex]

B = \int d^{ 3 } x \ S^{ 0 } ( x ) = \int d^{ 3 } x \ \left( u^{ \dagger } (x) u ( x ) + d^{ \dagger } ( x ) d ( x ) \right) ,

[/tex]

which we call the Baryon Number.

ii) The invariance of the theory under iso-spin transformations, [itex]SU( 2 )[/itex],

[tex]\delta \Psi = - i \vec{ \beta } \cdot \frac{ \vec{ \tau } }{ 2 } \Psi ,[/tex]

leads to the conserved isotopic spin current

[tex]\vec{ J }_{ \mu } ( x ) = \bar{ \Psi } ( x ) \gamma_{ \mu } \frac{ \vec{ \tau } }{ 2 } \Psi ( x ) .[/tex]

Its associated constant charges (iso-vector) are given by

[tex]

T^{ a } = \frac{ 1 }{ 2 } \int d^{ 3 } x \Psi^{ \dagger } ( x ) \tau^{ a } \Psi ( x ) .

[/tex]

iii) The invariance under [itex]U_{ em } (1)[/itex] transformation, given by,

[tex]

\delta \Psi = - i \epsilon ( \frac{ I }{ 6 } + \frac{ \tau^{ 3 } }{ 2 } ) \Psi ,

[/tex]

gives us the conserved electromagnetic current

[tex]J_{ em }^{ \mu } = \bar{ \Psi } \gamma^{ \mu } \left( \frac{ I }{ 6 } + \frac{ \tau^{ 3 } }{ 2 } \right) \Psi .[/tex]

This can be written as

[tex]J_{ em }^{ \mu } ( x ) = \frac{ 1 }{ 6 } S^{ \mu } ( x ) + J^{ \mu }_{ 3 } ( x )[/tex]

Integrating the time component of this current leads to the well-known relation

[tex]Q_{ em } = \frac{ B }{ 6 } + T_{ 3 } .[/tex]

In order to study the collisions of hadrons, we need to know the properties of all possible intermediate states. Therefore, we are led to consider the matrix elements [itex]\langle 0 | J_{ em }^{ \mu } ( 0 ) | n \rangle[/itex], where [itex]| n \rangle[/itex] is a set of ( small mass) intermediate states. From the properties of the vacuum, [itex]| 0 \rangle[/itex], and [itex]J_{em}^{ \mu }[/itex] (under the above symmetries), we can prove the following properties of [itex]| n \rangle[/itex]:

1) [itex]| n \rangle[/itex] has zero electric charge and zero Baryon number.

2) [itex]| n \rangle[/itex] is an eigenstate of the charge conjugation operator [itex]C[/itex], with eigenvalue [itex]C = - 1[/itex].

3) The total angular momentum and parity ([itex]J^{ P }[/itex]) of [itex]| n \rangle[/itex] is [itex]1^{ - 1 }[/itex].

4) The total isotopic spin of [itex]| n \rangle[/itex], [itex]T[/itex], is either [itex]0[/itex] or [itex]1[/itex].

Try to prove these properties yourself. If you got stuck on any one, you can ask me for help.

Regards

Sam

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thanks.

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