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Charming New Physics in Beautiful Processes?

Part of the book series: Springer Theses ((Springer Theses))

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Abstract

In this chapter we explore some of the concepts, tools and methods which will be used in the rest of the thesis. The idea of effective field theories (EFTs) is one of the most powerful in physics, and we will explain them in Sect. 2.1, along with a specific example of an EFT in Sect. 2.3. Another omnipresent tool is the Heavy Quark Expansion, which we see in Sect. 2.2.

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Notes

  1. 1.

    In the case of the total lifetime and \(\Gamma _{12}\), you also find that an overall factor of \(\sqrt{1 - M_f^2/m_b^2}\), where \(M_f\) is the total mass of the final state quarks, appears in the calculations.

  2. 2.

    In the language of amplitudes, the optical theorem can be written in the form

    $$i \mathcal {M} (i \rightarrow f) + \left[ i \mathcal {M} (f \rightarrow i) \right] ^* = - (2\pi )^4 \sum _X \int _\text {p.s.} \left[ i \mathcal {M} (f \rightarrow X) \right] ^* i \mathcal {M} (i \rightarrow X) \,.$$
  3. 3.

    Note that the explicit \(\mu \) dependence in this result cancels with that of \(\alpha _s\) (from Eq. 2.4.8) to give a \(\mu \) independent result for \(C_1\) and \(C_2\), as expected since these are bare coefficients.

  4. 4.

    I.e. not at all.

  5. 5.

    Note that while this hierarchy is true for and mesons, they do not hold for mixing.

  6. 6.

    A full account of this calculation can be found in Chap. 1.2.1 of [29].

  7. 7.

    See for example [33] for a set of Feynman rules which make this sign clear.

  8. 8.

    For this reason, \(a_\text {sl}\) is often referred to as the flavour specific asymmetry, or \(a_\text {fs}\).

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  1. Dipartimento di Fisica, La Sapienza, University of Rome, Rome, Italy

    Matthew John Kirk

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Kirk, M.J. (2019). Theoretical Tools. In: Charming New Physics in Beautiful Processes?. Springer Theses. Springer, Cham. https://doi.org/10.1007/978-3-030-19197-9_2

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