Feynman–Hellmann theorem for resonances and the quest for QCD exotica

The European Physical Journal C, Oct 2017

The generalization of the Feynman–Hellmann theorem for resonance states in quantum field theory is derived. On the basis of this theorem, a criterion is proposed to study the possible exotic nature of certain hadronic states emerging in QCD. It is shown that this proposal is supported by explicit calculations in chiral perturbation theory and by large-\(N_c\) arguments. Analyzing recent lattice data on the quark mass dependence in the pseudoscalar, vector meson, baryon octet and baryon decuplet sectors, we conclude that, as expected, these are predominately quark-model states, albeit the corrections are non-negligible.

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Feynman–Hellmann theorem for resonances and the quest for QCD exotica

The European Physical Journal C October 2017, 77:659 | Cite as Feynman–Hellmann theorem for resonances and the quest for QCD exotica AuthorsAuthors and affiliations J. Ruiz de ElviraU.-G. MeißnerA. RusetskyG. Schierholz Open Access Regular Article - Theoretical Physics First Online: 04 October 2017 5 Shares 451 Downloads 2 Citations Abstract The generalization of the Feynman–Hellmann theorem for resonance states in quantum field theory is derived. On the basis of this theorem, a criterion is proposed to study the possible exotic nature of certain hadronic states emerging in QCD. It is shown that this proposal is supported by explicit calculations in chiral perturbation theory and by large-\(N_c\) arguments. Analyzing recent lattice data on the quark mass dependence in the pseudoscalar, vector meson, baryon octet and baryon decuplet sectors, we conclude that, as expected, these are predominately quark-model states, albeit the corrections are non-negligible. 1 Introduction The celebrated Feynman–Hellmann theorem [1, 2] addresses the situation when a quantum-mechanical Hamiltonian \(H(\lambda )\) of a given system depends on a some external parameter \(\lambda \). The energy spectrum \(E_n(\lambda )\) and the wave functions \(|\Psi _n(\lambda )\rangle \) will then depend on this parameter as well. The theorem relates the \(\lambda \)-dependence of the energy spectrum to the matrix element of the operator \(\mathrm{d}H(\lambda )/\mathrm{d}\lambda \): $$\begin{aligned} \frac{\mathrm{d}E_n(\lambda )}{\mathrm{d}\lambda } =\biggl \langle \Psi _n(\lambda )\biggl |\frac{\mathrm{d}H(\lambda )}{\mathrm{d}\lambda }\biggr |\Psi _n(\lambda )\biggr \rangle . \end{aligned}$$ (1) In the context of QCD, one often identifies the abstract parameter \(\lambda \) with the quark masses \(m_q\) and studies the dependence of the hadron spectrum on the quark masses. Since the quark-mass-dependent part of the QCD Hamiltonian takes the form \(H_m=\sum _qm_q{\bar{q}}q\), the dependence of, say, the nucleon mass on the quark masses is given by $$\begin{aligned} \frac{\mathrm{d}m_N}{\mathrm{d}m_q}=\frac{1}{2m_N}\,\langle N|{\bar{q}}q|N\rangle . \end{aligned}$$ (2) Here, the factor \(1/(2m_N)\) emerges from the relativistic normalization of the one-particle states, \(\langle N'|N\rangle =(2\pi )^32E_N\delta ^3(\mathbf{p}'_N-\mathbf{p}^{}_N)\). Below, we shall explicitly consider only the three light quark flavors \(q=u,d,s\) and, for simplicity, assume that isospin is conserved: \(m_u=m_d={\hat{m}}\). The non-strange and strange \(\sigma \)-terms of the nucleon are defined, respectively, as $$\begin{aligned} \sigma _N=\frac{{\hat{m}}}{2m_N}\langle N|{\bar{u}}u+{\bar{d}}d|N\rangle ,\quad \sigma ^s_N=\frac{m_s}{2m_N}\langle N|{\bar{s}}s|N\rangle , \end{aligned}$$ (3) and the strangeness content of the nucleon is given by $$\begin{aligned} y=\frac{2\langle N|{\bar{s}}s|N\rangle }{\langle N|{\bar{u}}u+{\bar{d}}d|N\rangle }. \end{aligned}$$ (4) These \(\sigma \)-terms contain important information as regards the effect of the explicit chiral symmetry breaking (\(m_q\ne 0\)) on the hadronic observables. In addition, the nucleon \(\sigma \)-term is an important input for the estimates of WIMP cross sections in dark matter direct detection experiments see, e.g., [3, 4, 5, 6, 7, 8, 9], as well as in searches for the lepton flavor violation [10, 11] and electric dipole moments [12, 13, 14, 15, 16, 17]. The strange \(\sigma \)-term of the nucleon is relevant for the kaon condensation and the formation of the neutron stars, as well as the study of the heavy ion collisions [18, 19, 20], etc. The extraction of the \(\sigma \)-terms from the experimental data is a very delicate issue since, in particular, it implies an analytic continuation of the amplitudes below threshold (to the Cheng-dashed point). In this respect, we note that, from a thorough theoretical analysis of the problem on the basis of dispersion relations and using the input from chiral perturbation theory (ChPT), in Ref. [21] the value \(\sigma _N\simeq 45~\text{ MeV }\) was obtained for the non-strange \(\sigma \)-term. The most recent and comprehensive analysis, carried out in Ref. [22], is based on the Roy–Steiner equations for \(\pi N\) scattering and yields a larger value \(\sigma _N=(59.1\pm 3.5)~\text{ MeV }\). As explained in detail in Ref. [23], most of the difference can be traced back to the new and improved values of the pion–nucleon scattering lengths deduced from pionic hydrogen and deuterium. This conclusion has been strengthened by a recent reanalysis of the low-energy pion–nucleon scattering data [24]. In recent years, the \(\sigma \)-terms have also been measured on the lattice [25, 26, 27, 28, 29, 30, 31]. In general, two methods are employed in these measurements: a direct measurement of the matrix element and extracting the \(\sigma \)-terms from the quark mass dependence of the hadron masses with the use of the Fey (...truncated)


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J. Ruiz de Elvira, U.-G. Meißner, A. Rusetsky, G. Schierholz. Feynman–Hellmann theorem for resonances and the quest for QCD exotica, The European Physical Journal C, 2017, pp. 659, Volume 77, Issue 10, DOI: 10.1140/epjc/s10052-017-5237-3