In the rest frame of the ¯B, we define: k|B−RF

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Flavour Physics Problem Set 1.

FS 2019 Andreas Crivellin Rafael Silva Coutinho https://www.physik.uzh.ch/en/teaching/PHY568/FS2019.html Given on: 20. February 2019

Exercise 1. Phase space for B¯ →πµ¯ν decays Consider the decay

B(p)¯ →π(k)µ(q₁) ¯ν(q₂). In the rest frame of the ¯B, we define:

k|B−RF¯ = (k⁰,0,0,−|~k|),

q|_B−RF_¯ = (q⁰,0,0,−|~q|), (1)

whereq =q₁+q₂ is the energy transfer to the leptons. In the rest frame of the lepton pair, we have:

q₁|_`−RF= (q⁰₁,|~q₁|sinθ,0,|~q₁|cosθ),

q2|_`−RF= (q⁰₂,−|~q2|sinθ,0,−|~q2|cosθ). (2) In general, you can write the Lorentz-Invariant Phase Space (LIPS) for an n-body decay by decomposing it into aj-body and an (n−j+ 1)-body phase space as following:

dΦn(p;p1, . . . , pn) =dΦj(qj;p1, . . . , pj) dΦn−j+1(p;pj+1, . . . , pn, qj) dq_j²

2π . (3)

(a) Compute the LIPS elementdΦ2(p;k, q) in the rest frame of the ¯B.

(b) Compute the LIPS elementdΦ₂(q;q₁, q₂) in the lepton restframe for massless leptons.

(c) Give a qualitative argument how the result would change for non-vanishing lepton masses.

(d) Compute the full LIPS elementdΦ₃(p;k, q₁, q₂).

Exercise 2. Photon self-energy

To practice some of the methods commonly used in loop calculations, we will compute the fermionic contribution to the photon self-energy, as shown in Fig. 1. Consider the internal fermions to have a mass m. The amplitude can be written as:

iA=ε_µ(q) iΠ^µν(q², m, µ)

ε^∗_ν(q). (4)

(a) The tensor Π^µν can be decomposed according to Π^µν =g^µνΠ_a+q^µq^ν

q² Π_b. (5)

How can Π_a and Π_b in general be determined?

(b) Under the assumption that the photon is transversely polarized, relate Πa and Πb. 1

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µ ν q

Figure 1: The Feynman diagram corresponding to the amplitude to be computed in Problem 1.

(c) Can the two fermions in the loop be of different flavor in the SM, and why/why not? If not, what would need to happen for it to be possible?

(d) Use Feynman rules to write an explicit expression for Π_a and express it in terms of the scalar integrals

A₀(m) =

Z d^dl (2π)^d

1 l²−m², B₀(p;m₁, m₂) =

Z d^dl (2π)^d

1 l²−m²₁

1

(l+p)²−m²₂ .

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Hint: In convergent integrals, you can shift the integration variable l→l+v.

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