hw4 - masses? 3. Consider the Higgs potential in the...

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Homework Set No. 4, Physics 880.02 Deadline – Thursday, February 26, 2009 1. (10 pts) Consider a non-Abelian gauge theory with the gauge feld A a μ and the La- grangian L = - 1 4 F a μν F a μν . Here F a μν = μ A a ν - ν A a μ + g f abc A b μ A c ν with f abc the structure constants oF the gauge group SU ( N ). Write the equations oF motion For this theory. IF we defne J a μ by ν F a νμ = J a μ what is J a μ For the above Lagrangian? 2. (10 pts) Consider the Lagrangian For a complex scalar feld φ coupled to an Abelian gauge feld A μ : L = ( μ + i g A μ ) φ * ( μ - i g A μ ) φ + μ 2 φ * φ - λ ( φ * φ ) 2 - 1 4 F μν F μν with μ 2 , Λ > 0 positive constants. Like in problem 2 oF HW3 write φ = χ 1 + i η 2 with χ 1 and η real felds and with χ 1 = v + χ , where v is a real number and χ is a real feld. What is the value oF v For which the linear terms in χ disappear From the Lagrangian? Write the Lagrangian in terms oF A μ , χ and η . What are the physical particles and what are their
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Unformatted text preview: masses? 3. Consider the Higgs potential in the Standard Model: V ( ) =- 2 + ( ) 2 (1) with the Higgs feld = + = 1 2 . 1 The potential is SU (2) U (1) invariant. a. (5 pts) In principle one can write SU (2) U (1) quadric invariant of the form V 1 ( ) = 1 ( ~ ) ( ~ ) (2) with ~ the Pauli matrices and 1 a constant. Show that this quadric term can be reduced to that in V ( ) using the Fierz identity 3 X a =1 a 2 ij a 2 kl = 1 2 jk il-1 4 ij kl . b. (5 pts) The same as in part a for V 2 ( ) = 1 3 X a,b =1 ( a b ) ( a b ) . (3) 2...
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hw4 - masses? 3. Consider the Higgs potential in the...

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