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HW9 - determine 5 X s =1 ± μν ~ k,s ± ρσ ~ k,s in...

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PHY5667 Problem Set #9 (due Tue Oct 26) (1) Let’s consider a massive spin-2 particle. Analogous to the polarization vectors μ ~ k,s for a spin-1 field, we have polarization tensors μν ~ k,s for a spin-2 field. (i) In the rest frame ( k = ( m, 0 , 0 , 0)), we should have 0 ν = μ 0 = 0 while ij 6 = 0 corresponding to the five polarizations of the spin-2 particle. Of course, a generic ij would have 9 independent components, so we need to impose some constraints on ij to reduce 9 to 5. Identify these constraints. [Hint: there are two constraints.] (ii) Still staying in the rest frame, find five polarization tensors ( ij s with s = 1 , · · · , 5) normalized as X i,j ij s ij s 0 = δ ss 0 . (iii) Still staying in the rest frame, determine an expression for 5 X s =1 ij ~ k,s k‘ * ~ k,s in terms of the metric δ ij . [Hint: Constrain the possible form of the expression as much as possible using the conditions identified in (i).] (iv) Now promote the expression found in (iii) to the form valid in all frames, that is,
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Unformatted text preview: determine 5 X s =1 ± μν ~ k,s ± ρσ * ~ k,s in terms of the metric η μν , the 4-momentum k μ , and the mass m . (The point of this problem is to find the spin-2 version of the spin-1 formula ∑ s ± μ ~ k,s ± ν * ~ k,s =-η μν + k μ k ν /m 2 .) (2) Consider a real scalar φ with mass m and a real massless vector A μ . Suppose they interact via an interaction L int = λ 4 φF μν F μν , where F μν = ∂ μ A ν-∂ ν A μ . Calculate the decay rate for φ → AA , with the all polarizations of A summed over. (If you don’t see what the Feynman rules should be, do perturbation theory algebraically (that is, using contractions) to obtain the amplitude, then interpret the result diagrammatically.) 1...
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