Quantitative geometrical theory of diffraction-dissociation $d\sigma/dt$ beyond the zeroth-order source $\Omega(b)e^{-\Omega(b)}$

The open question left by T. T. Chou and C. N. Yang's 1980 geometrical treatment of dip and kink structures in hadron-nucleus and hadron-hadron diffraction dissociation (Phys. Rev. D 22, 610) is to construct a quantitative theory of the diffractive differential cross section beyond their zeroth-order source. Using the zeroth-order source $\Omega(b)e^{-\Omega(b)}$ with the opacity $\Omega(b)$ fixed independently by electromagnetic form factors, total cross sections, and nuclear densities, the 1980 work predicted diffractive dip positions with no adjustable parameters, in agreement with observed kink positions in $pp$ and pion-nucleus data — but only dip positions, not shapes or normalizations. The open task beyond this zeroth order is a quantitative multi-channel theory: allow for the nonvanishing real part of the amplitude (complex opacity), spin-dependent amplitudes, differing absorptions of the incoming hadron and the outgoing excited system, coherent channel coupling among internal states, and a source function derived from dynamics, so that measured $d\sigma/dt$ shapes and normalizations are reproduced and the range of validity in $|t|$ is delimited. The question remains open: the only follow-ups within the geometrical dissociation program are asymptotic extensions of the dip-position program (Chou, Wu, and Yang, 1982) and partial re-examinations on nuclei (Hoang, 1985).

Background

High-energy collisions of hadrons at small squared four-momentum transfer $t$ are dominated by diffraction: the scattering is elastic or nearly elastic, the amplitudes are peaked at $t=0$, and the differential cross section $d\sigma/dt$ (cross section per unit $t$) exhibits characteristic interference structures called dips (minima) and kinks (abrupt slope changes) as a function of $t$. In diffraction dissociation (DD), the incoming hadron traverses the target and emerges not in its ground state but as a resonant or continuum hadronic system with the same internal quantum numbers, while the target stays intact, leaving a large rapidity gap (an interval of emission angles around the beam direction in which no particles are produced) in the final state. Good and Walker showed in 1960 that such diffractive excitation occurs when the scattering operator fluctuates over the internal quantum states of the projectile, so that components which scatter differently acquire relative phases and the outgoing state becomes a coherent superposition — dissociation without exchange of internal quantum numbers.

The geometrical model of Chou and Yang (1968) describes high-energy elastic scattering in optical terms: the hadron is an extended distribution of strongly interacting matter, and absorption at impact parameter $b$ (the transverse separation of the colliding hadrons' centers) is encoded in a real opacity or 'blackness' profile $\Omega(b)$, with the elastic profile function $1-e^{-\Omega(b)}$ and

$$\left(\frac{d\sigma}{dt}\right)_{\rm el}=\pi\left|\mathcal{F}_{2D}\{1-e^{-\Omega}\}\right|^2,$$

where $\mathcal{F}_{2D}$ is the two-dimensional Fourier transform from $b$ to transverse momentum ($-t=k^2$). Crucially, $\Omega(b)$ is not fitted to the scattering data being explained but fixed by independent inputs: electromagnetic form factors measured in electron scattering and total cross sections via the optical theorem (the exact relation fixing the total cross section from the imaginary part of the forward elastic amplitude). Zeros of the Fourier-Bessel transform then predict diffraction minima; for a black disc of radius $R$ the elastic amplitude is governed by the Bessel function $J_1$ with its first zero at $\sqrt{-t}\,R\approx3.8$, accounting for observed elastic dips such as the proton-proton dip near $|t|\approx1.3\ (\mathrm{GeV}/c)^2$ at CERN Intersecting Storage Rings energies.

In 1980, Chou and Yang extended this picture to diffraction dissociation. They proposed that the 'source' distribution of the outgoing dissociated system is, to zeroth order, $\Omega(b)\,e^{-\Omega(b)}$: a production probability proportional to the local thickness of matter traversed, $\Omega(b)$, attenuated by absorption of both the incoming and outgoing amplitudes, $e^{-\Omega(b)}$. For a diffractive channel $A+B\to A+B^*$,

$$\left(\frac{d\sigma}{dt}\right)_{\rm DD}\propto\left|\mathcal{F}_{2D}\{\Omega(b)\,e^{-\Omega(b)}\}\right|^2.$$

For pion-nucleus systems they built $\Omega(b)$ from the measured $\pi N$ blackness integral (14.34 mb, converted to an opacity of 0.244 per fermi in nuclear matter) folded with a Woods-Saxon nuclear density (a Fermi-function profile with radius $r_0=1.12\,A^{1/3}$ fm and surface thickness $a=0.5$ fm); for proton-proton they used the dipole form factor $(1+|t|/0.71)^{-2}$ and $\sigma_T=42.1$ mb. The computed first dips — $|t|\approx0.0086\ (\pi\mathrm{Pb})$, $0.020\ (\pi\mathrm{Cu})$, and $0.50\ (pp)$ in $(\mathrm{GeV}/c)^2$ — agreed with the observed kink positions ($\approx0.009$, $\approx0.019$, and $0.4$-$0.5$) with no adjustable parameters. In the black-disc limit the diffractive source transforms as $|J_0(\sqrt{-t}R)|^2$ with its first zero at $\sqrt{-t}\,R\approx2.4$, so the diffractive dip is expected at roughly $1/2.5$ of the elastic dip's $|t|$ — a parameter-free geometrical explanation of why diffractive dips sit at markedly smaller momentum transfer than elastic ones.

The 1980 treatment is zeroth order in several structural respects. The opacity was taken real, so the diffractive amplitude is purely imaginary and any nonvanishing real part of the amplitude is ignored (admixing an imaginary part into $\Omega$, which generates such a real amplitude component, had been reported to improve fits); spin-dependent amplitudes are neglected; the mean free paths of the incoming hadron and the outgoing excited system in matter are assumed equal (a ratio parameter $\gamma$ moves the predicted $\pi$Cu dip between $0.014$ and $0.025\ (\mathrm{GeV}/c)^2$); coherent switching among the many internal states of the traversing hadron — the multi-channel Good-Walker problem — is absent from the one-state source $\Omega e^{-\Omega}$; and the source function itself is postulated rather than derived from dynamics. The description was also expected to fail at larger $|t|$, above a few $(\mathrm{GeV}/c)^2$, where hard, short-distance mechanisms become important. Consequently the 1980 work explained dip positions qualitatively and parameter-free, but not the measured point-by-point shapes or normalizations of diffractive $d\sigma/dt$. In the modern era, single diffractive dissociation at the LHC is measured only at small $|t|$, where $d\sigma/dt$ is a smooth exponential interpreted in triple-Regge phenomenology (the description of diffraction through exchanges of Regge trajectories such as the pomeron), so the dip-and-kink structures remain a fixed-target/ISR-era data feature whose quantitative understanding has never been completed.

Problem Statement

The problem left open by Chou and Yang (1980) is to turn the zeroth-order geometrical description of diffraction dissociation into a quantitative theory of the diffractive differential cross section. Construct, within the eikonal/geometrical framework (the eikonal approximation: the high-energy limit in which the incoming wave acquires absorption and phase while traversing the target at a fixed impact parameter), or within a dynamical framework that reduces to it, a treatment of hadron-hadron and hadron-nucleus diffraction dissociation that: (i) allows a complex opacity, so the diffractive amplitude acquires a nonvanishing real part fixed by independent data; (ii) includes spin-dependent profile functions; (iii) allows the mean free paths of the incoming hadron and the outgoing excited system to differ; (iv) treats coherent channel coupling among the hadron's internal excited states — the multi-channel Good-Walker eigenstate problem — instead of a single effective state; and (v) derives the dissociation source function from underlying dynamics. The theory must keep the opacity fixed by inputs independent of the dissociation data — electromagnetic form factors, total cross sections, nuclear densities — with no parameters tuned to the diffractive data being explained; the further structures required by (ii)–(iv) (spin-dependent profile functions, the incoming/outgoing mean-free-path ratio $\gamma$, and the Good-Walker eigenstate distribution) must likewise be determined either from first principles or from data independent of the diffractive dissociation cross sections being reproduced. The theory must reproduce the measured $d\sigma/dt$ of diffraction dissociation in shape and normalization (positions, depths, and filling of the dips, and absolute cross sections) for the systems in which diffractive dip and kink structures were measured — $pp$ and pion-nucleus ($\pi$Pb, $\pi$Cu) — up to $|t|$ of a few $(\mathrm{GeV}/c)^2$, delimiting the momentum-transfer region beyond which hard, large-$|t|$ mechanisms must be included.

References

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