Fragmentation Functions

Fragmentation functions represent the probability for a parton to fragment into a particular hadron carrying a certain fraction of the parton's energy and, like structure functions, cannot be calculated in perturbative QCD, but can be evolved from a starting distribution at a defined energy scale. If the fragmentation functions are combined with the cross sections for the inclusive production of each parton type in the given physical process, predictions can be made for the scaled momentum, x_p, spectra of final state hadrons. Small x_p fragmentation is significantly affected by the coherence (destructive interference) of soft gluons [17], whilst scaling violation of the fragmentation function at large x_p allows a measurement of $\alpha_{s}^{}$ [18].

A natural frame in which to study the dynamics of the hadronic final state in DIS is the Breit frame [19]. In this frame the exchanged virtual boson is purely space-like with 3-momentum $\bf q$ = (0, 0, - Q), the incident quark carries momentum Q/2 in the positive Z direction, and the outgoing struck quark carries Q/2 in the negative Z direction. A final state particle has a 4-momentum p^B in this frame, and is assigned to the current region if p^B_Z is negative, and to the target frame if p^B_Z is positive. The advantage of this frame lies in the maximal separation of the outgoing parton from radiation associated with the incoming parton and the proton remnant, thus providing the optimal environment for the study of the fragmentation of the outgoing parton.

In e⁺e^- annihilation the two quarks are produced with equal and opposite momenta, $\pm$$\sqrt{s}$/2. This can be compared with a quark struck from within the proton with outgoing momentum - Q/2 in the Breit frame. In the direction of the struck quark (the current fragmentation region) the particle momentum spectra, x_p = 2p^B/Q, are expected to have a dependence on Q similar to those observed in e⁺e^- annihilation [20,21,22] at energy $\sqrt{s}$ = Q.

In fig 8 the log(1/x_p) distributions for charged particles in the current fragmentation region of the Breit frame are shown as a function of Q². These distributions are approximately Gaussian in shape with mean charged multiplicity given by the integral of the distributions. As Q² increases the multiplicity increases and the the peak of the distributions shifts to larger values of log(1/x_p). Figure 9 shows this peak position, log(1/x_p)_max, as a function of Q for the HERA data and of $\sqrt{s}$ for the e⁺e^- data. Over the range shown the peak moves from $\simeq$ 1.5 to 3.3. The HERA data points are consistent with those from TASSO and TOPAZ and a clear agreement in the rate of growth of the HERA points with the e⁺e^- data at higher Q is observed.

The increase of log(1/x_p)_max can be approximated phenomenologically by the straight line fit $\mbox{$\log(1/x_{p})_{max}$}$ = blog(Q) + c also shown in figure 9. The values obtained from the fit to the ZEUS data are b = 0.69$\pm$0.01(stat)$\pm$0.03(sys) and c = 0.56$\pm$0.02^+0.08_-0.09. The gradient extracted from the OPAL and TASSO data is b = 0.653$\pm$0.012 (with c = 0.653$\pm$0.047) which is consistent with the ZEUS result. This value is consistent with that published by OPAL, b = 0.637$\pm$0.016, where the peak position was extracted using an alternative method [23]. A consistent value of the gradient is therefore determined in DIS and e⁺e^- annihilation experiments.

Also shown is the statistical fit to the data when b = 1 ( c = 0.054$\pm$0.012) which would be the case if the QCD cascade was of an incoherent nature, dominated by cylindrical phase space. The observed gradient is clearly inconsistent with b = 1 and therefore inconsistent with cylindrical phase space.

The inclusive charged particle distribution, 1/$\sigma_{tot}^{}$ d$\sigma$/dx_p, in the current fragmentation region of the Breit frame are shown in bins of x_p and Q² in fig. 10. The increasingly steep fall-off, at fixed Q²,  towards higher values of x_p as Q² increases, shown in figure 10, corresponds to the production of more particles with a smaller fractional momentum, and is indicative of scaling violation in the fragmentation function. For Q² > 80 GeV² the distributions rise with Q² at low x_p and fall-off at high x_p and high Q². In figure 10 the HERA data are compared at Q² = s to e⁺e^- data [25], again divided by two to account for the production of both a q and $\bar{q}$. In the Q² range shown there is good agreement between the current region of the Breit frame in DIS and the e⁺e^- experiments.