Analysis of radial scaling in single-particle inclusive reactions

Abstract

An analysis of an extensive sample of the world's data has been performed to test the hypothesis of radial scaling. We have studied the inclusive reactions p + p --+ (~* %~ or K * or p or 9) +anything to determine the behavior of the invariant cross section as a function of p,, x, = E*/E*,,,, the radial scaling variable, and s.The data cover a range in p, from 0.25 to -6.0 GeV/c and a range in d s from 3.0 to 63 GeV. For small x, and all available p, the single-particle inclusive cross sections for the reactions studied scale to a good approximation for all v ' s , even down to the kinematic threshold. For large x,, the single-particle inclusive cross sections for increasing d s show a rapid approach to the scaling limit from above. In these cases the scaling limit is always approached by v ' s z 10 GeV. Thus, data for all particles to a good approximation exhibit radial scaling at all available p, and x, over the CERN ISR energy range. A comparison of radial scaling with Feynman scaling is given. It is shown that in the Feynman case the cross sections for small xll ( x , , = p*,, /p*,.,) approach their scaling limit from below, and that the approach to the scaling limit is slower than is exhibited for the case of small x,. The systematic differences among the inclusive cross sections of various particles are discussed in the range of v ' s where radial scaling has been shown to be valid. In particular, the p, and x, distributions of E d v / d p 3 are examined.

On particle production for high energy neutrino beams

“…A reasonable fit to data has been obtained by following the empirical observation that the longitudinal momentum distributions of the leading nucleon in pp collisions is flat [24], which translates into a linear rise of the inclusive invariant cross section as a function of x R . The transverse momentum distribution is also affected by the leading particle effect, resulting in an enhanced leading particle production in the forward direction (see for example [19,23]). In the proton fit, this is empirically accounted for by the term (…”

“…An alternative scaling variable x R = E * /E * max , defined as the ratio of the energy of the detected particle in the centre-of-momentum frame and the maximum energy kinematically available to the detected particle, was suggested by Yen [18] and Taylor et al [19] and shown to greatly extend the range of validity of scaling at subasymptotic energies. As discussed in section 4.2, in the comparison of NA56/SPY and NA20 data to pBe data collected at lower energies, we have not observed an improved scale invariance when x R is used.…”

“…For cylindrical symmetry this allows the analytical integration of equations (18) and (19). In case of continuous targets of radius R, the integration yields:…”

Observation of Disappearance of Muon Neutrinos in the NuMI Beam

“…The x R = E ′ /E ′ max , defined as the ratio of the energy of the detected particle in the center of mass frame and the maximum energy kinematically available extends the range of validity of scaling [69,70]. For our purposes even the variable x lab = p L /p 0 where p L is the longitudinal component of the momentum of the secondary and p 0 is the momentum of the primary beam is sufficiently good scaling variable.…”

Measurement of the Muon Neutrino Inclusive Charged Current Cross Section on Iron using the MINOS Detector

“…It allows particle production to be quantitatively described in the most simple way possible and allows data collected with different initial proton beam momenta to be compared. Under two conditions, the produced particle is ultrarelativisic, E ≫ m, and it's divergence is very small, p z ≫ p T , the Feynman scaling variable can be expressed as x F ≃ p z /p 0 , where p 0 is the momentum of the incident particle [61,62].…”