Sum rules for quasifree scattering of hadrons

The areas $d\sigma /d\mathrm{\Omega }$ of fitted quasifree scattering peaks from bound nucleons for continuum hadron-nucleus spectra measuring $d^2\sigma /d\mathrm{\Omega }d\omega$ are converted to sum rules akin to the Coulomb sums familiar from continuum electron scattering spectra from nuclear charge. Hadronic spectra with or without charge exchange of the beam are considered. These sums are compared to the simple expectations of a nonrelativistic Fermi gas, including a Pauli blocking factor. For scattering without charge exchange, the hadronic sums are below this expectation, as also observed with Coulomb sums. For charge exchange spectra, the sums are near or above the simple expectation, with larger uncertainties. The strong role of hadron-nucleon in-medium total cross sections is noted from use of the Glauber model.

Figure 1

Inclusive pion scattering spectra for carbon [8] are shown, together with fitted lines above an assumed linear background. Short vertical lines indicate the simplest energy losses for quasifree scattering, $q^2/2M+20\mathrm{MeV}$ (for binding, as Ref. [14]), with $M$ the free nucleon mass. The areas of the quasifree peaks are 27(3), 14.3(3), 7.75(1.6), and 4.06(2) mb/sr at $q=350$, 450, 550, and 650 MeV/$c$. These singly-differential cross sections were used to compute the experimental sum rules.

Figure 2

Bottom: Measured areas of hadronic quasifree scattering peaks without charge exchange (NCX) on ${}^6\mathrm{Li}$ from Refs. [8, 24], presented as sum rules as the momentum transfer $q$ to the struck nucleon increases. Top: Single charge exchange (SCX) areas (in red) on natural lithium from Refs. [37, 43, 44], as sum rules. Curves show the expected sums for a Fermi gas, including the Pauli blocking factor in the text, with $k_{\mathrm{F}}=165\mathrm{MeV}/c$.

Figure 3

As Fig. 2, but for a carbon sample, with $k_{\mathrm{F}}=228\mathrm{MeV}/c$. Data sources are presented in the text. Dashed curves show the one-body calculations for charge (bottom) and magnetic (top) electron scattering [26]. The charge electron scattering (EEL) sums among the NCX points and the transverse (EET) sums among the SCX points are from Ref. [26], with no tails added.

Figure 4

As Fig. 2, but for an aluminum sample, with $k_{\mathrm{F}}=236\mathrm{MeV}/c$. Data sources are presented in the text.

Figure 5

As Fig. 2, but for a calcium sample with NCX hadron quasifree sums, with $k_{\mathrm{F}}=241\mathrm{MeV}/c$. Data sources are presented in the text.

Figure 6

As Fig. [2], but for zirconium and niobium samples, with $k_{\mathrm{F}}=245\mathrm{MeV}/c$. Data sources are presented in the text.

Figure 7

As Fig. 2, but for lead and bismuth samples, with $k_{\mathrm{F}}=248\mathrm{MeV}/c$. Data sources are presented in the text.

Figure 8

The (${\pi }^{-},{\pi }^{0}x$) data for carbon of Ref. [37] are shown, as are fits above an assumed quadratic background (as curves). Vertical lines indicate the loci of free scattering ($q^2/2M$) + 12.17 MeV for binding [14], Coulomb, and $Q$-value effects.

Figure 9

As Fig. 2, but for a copper sample only for SCX spectra, with $k_{\mathrm{F}}=241\mathrm{MeV}/c$. The data are from Ref. [37].

Figure 10

Sums extracted from hadron quasifree spectra with momentum transfers $q$ near 500 MeV/$c$ presented as the target mass $A$ increases. The data sources are the same as noted in the preceding figures. Bottom, NCX results, and top, SCX results, incremented by one, in red. The 70% factor for free-space hadron-nucleon total cross sections in the Glauber model is used for this figure. Coulomb sums are included among the NCX results. The simplest expectation for these sums would be unity. The ${}^{3,4}\mathrm{He}$ Coulomb sums are from Ref. [4].