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Constraints on Hadronic Form Factors

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Functional Analysis and Optimization Methods in Hadron Physics

Part of the book series: SpringerBriefs in Physics ((SpringerBriefs in Physics))

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Abstract

In this chapter, we discuss several applications which illustrate the usefulness of the functional analysis and optimization methods for improving the knowledge of the weak and electromagnetic hadronic form factors. We first present the method of “unitarity bounds”, proposed in the early 1970s by Meiman and Okubo for deriving model-independent bounds on the semileptonic form factors. The development of the method in the frame of the Standard Model is then reviewed, emphasizing the increased strength of the formalism when it is combined with additional theoretical information provided by heavy-quark symmetry, chiral perturbation theory or lattice QCD. Finally, we show how this approach leads to precise predictions for the pion electromagnetic form factor, in particular for the charge radius of the pion. We briefly describe also the way in which the rigorous bounds can be merged with statistical simulations.

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Notes

  1. 1.

    Other choices of invariant amplitudes which satisfy unsubtracted dispersion relations have been also investigated in the literature.

  2. 2.

    As shown in [37], the world average of R(D) and \(R(D^*)\) measured by BABAR, Belle and LHCb is in tension with the SM expectation at the \(4\sigma \) level.

  3. 3.

    For the form factors of interest, the thresholds occur at the minimum values of \(\sqrt{t}\) at which the relevant \(B^{(*)} D^{(*)}\) pairs can be produced, i.e. at \((m_B+m_D)\approx 7.15\,\text{ GeV }\), \((m_B+m_{D^*})\approx 7.29\,\text{ GeV }\), \((m_{B^*}+m_D)\approx 7.19\,\text{ GeV }\), or \((m_{B^*}+m_{D^*})\approx 7.33\,\text{ GeV }\).

  4. 4.

    A similar quantity, not observable but of theoretical interest for \(\chi \)PT, is the pion scalar form factor, defined from the matrix element between pion states of a suitable scalar operator. The pion scalar form factor was investigated in the Meiman-Okubo approach in [38, 39].

  5. 5.

    As discussed below (4.62), the minors of the determinant should be also nonnegative. Some of these conditions involve only input quantities, and are violated if the input values are not consistent.

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Authors and Affiliations

  1. Department of Theoretical Physics, National Institute of Physics and Nuclear Engineering, Bucharest-Magurele, Romania

    Irinel Caprini

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  1. Irinel Caprini
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Correspondence to Irinel Caprini .

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Caprini, I. (2019). Constraints on Hadronic Form Factors. In: Functional Analysis and Optimization Methods in Hadron Physics. SpringerBriefs in Physics. Springer, Cham. https://doi.org/10.1007/978-3-030-18948-8_5

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