Perturbative Calculations

Perturbative QCD calculations can be performed in the formalism of collinear factorization in which the DGLAP evolution equations [59] are used to describe the radiation of partons from the initial parton distribution in the proton and in the photon. In the collinear factorization ansatz the parton distributions in the proton (and the photon) are assumed to depend only on the scaling variable $x$ and an energy scale $\mu$, usually the photon virtuality $Q^2$. In particular, the initial partons in the proton are assumed to carry no transverse momentum. In the evolution, the partons are treated as massless on-shell particles. Factorization and renormalization scale parameters are used to absorb divergent parts of the perturbation series into parton distributions and $\alpha_S$. In the DGLAP scheme, calculations up to next-to-next-to-leading order (NNLO) have become available recently, e.g.for inclusive cross sections [60].

In other approaches, such as the BFKL evolution equation [61,62,63,64], and later the CCFM evolution equation [65], the so-called $k_t$ factorization formalism is used which is described in section 2.5.

Heavy Quark production poses a particular theoretical challenge as the presence of the heavy quark mass $m_c$ or $m_b$ introduces a new scale into the perturbative calculations. Quantitative calculations for heavy quark production at HERA have been performed by a number of authors, providing detailed results up to next-to-leading order in perturbation theory. Predictions for inclusive heavy quark distributions, as derived from fits to inclusive data from HERA and fixed target experiments [66,67,68,69], are available as a function of $Q^2$ and $x$. In global fits also other data, such as dijet data have been used [70,71,72,73,74,75,76]. Predictions for exclusive processes in which the topologies of the two quarks are explicitly taken into account, are available as a function of a number of variables, such as the transverse momenta $p_t$ and/or pseudo-rapidity $\eta$ for one or both of the heavy quarks and/or jets (see section 2.1).

Different schemes to calculate heavy quark production processes have been developed in the framework of collinear factorization which are expected to be valid in different kinematic regions:

• In calculations for processes with light quarks, the mass of the light quarks is assumed to be zero. The quarks are treated as active partons in the proton, i.e.a density distribution for the quarks in the proton is used to describe the non-perturbative part of the calculation. The perturbative series is expanded using a scale-parameter $\mu$ as given by the photon virtuality $Q^2$ or the jet momentum $p_t$. Perturbative calculations are expected to converge for $\mu \gtrsim \Lambda_{QCD}$. Due to the heaviness of the quark mass $m_q$, this approach does not work for heavy quarks except in the extreme limit $\mu \gg m_q$, in which the heavy quarks can be treated as massless. In this `massless' scheme, at leading order (LO), the quark parton model (QPM) process ( $\gamma q \rightarrow q$, fig. 2a) is the dominant contribution. At next-to-leading order (NLO), virtual corrections are included (fig. 2b) and the QCD Compton ( $\gamma q \rightarrow qg$, fig. 2c) and photon gluon fusion ( $\gamma g \rightarrow q\bar{q}$, fig. 2d) processes also contribute. The massless approach is often referred to as the zero mass variable flavor number scheme (ZM-VFNS) [70,71]. In this approach the heavy quarks are treated as infinitely massive below some scale $\mu \sim m_q$ and massless above this threshold.

  Figure 2: Leading diagrams for heavy quark production in the massless scheme at leading order (a) and next-to-leading order (b-d).

  
• At values of $\mu^2 \sim M^2$, the `massive' scheme [77,78,79], in which the heavy flavor partons are treated as massive quarks is more appropriate. In the massive scheme the dominant LO process is photon gluon fusion (PGF, fig. 3a) and the NLO diagrams are of order $\alpha_s^2$ (figs. 3b-c). The scheme is often referred to as the fixed flavor number scheme (FFNS). As $\mu^2$ becomes large compared to $M^2$, the FFNS approach is unreliable due to large logarithms in $\ln (\mu^2/M^2)$ in the perturbative series. Generator programs in this scheme which are applicable to HERA physics are available to next-to-leading order (FMNR [80], HVQDIS [81]). The fixed order massive scheme is also used in various Monte Carlo event generator programs which implement leading order matrix elements and parton showers to simulate higher order effects. A description of these programs is given in section 3.

  Figure 3: Leading diagrams for heavy quark production in the massive scheme at leading order (a) and next-to-leading order (b-d).

  

• In order to provide reliable pQCD predictions for the description of heavy flavor production over the whole range in $\mu^2$, composite schemes which provide a smooth transition from the massive description at $\mu^2 \sim M^2$ to the massless behavior at $\mu^2 \gg M^2$ have been developed. These composite schemes are commonly referred to as variable flavor number schemes (VFNS). The VFNS approach has been incorporated in various different forms to order $\alpha_s$ [82,83,84,85,75,86,87,88] and to order $\alpha_s^2$ [89,90].

  In resummed or `matched' next-to-leading-order QCD predictions the divergent logarithms are controlled by resummation techniques. Here, matched means that measures are taken to avoid double-counting of contributions which are contained in both the perturbative and the resummed part of the calculation, i.e.

  $${\rm FONLL=FO + (RS - FOM0)} \times G(m,p_t),$$ (2)

  
  
  where FONLL stands for fixed-order plus next to leading logarithms, and FO is the fixed order, ${\cal{O}}(\alpha_{em}\alpha^2_s)$ result. RS is the resummed result, which includes all terms of the form $\alpha_{em}\alpha_s(\alpha_s \log p_t/m)^i$ and $\alpha_{em}\alpha_s^2(\alpha_s \log p_t/m)^i$ and neglects all terms suppressed by powers of the heavy quark mass $m$. FOM0 is the massless limit of FO, in the sense that all terms suppressed by powers of $m$ are dropped, while logarithms of the mass are retained; thus FOM0 is the truncation of RS to order $\alpha_{em}\alpha_s^2$. $G(m, p_t)$ is an arbitrary dumping function, that must be regular in $p_t$, and at large $p_t$, it must approach unity up to terms suppressed by powers of $m/p_t$. In the matched calculation FONLL [87,88,91,92,93] differential cross sections are calculated to next-to-leading-log precision using perturbative fragmentation functions [94]. A similar approach is pursued in [95,96,97,98,99,100]. Here, the ZM-VFNS is used as a starting point, adjusting the factorization scales such that the massive calculation is reached for $m \rightarrow 0$.