First-order QCD matrix elements

The Born process $\mathrm{e}^+\mathrm{e}^-\to \mathrm{q}\overline{\mathrm{q}}$ is modified in first-order QCD by the probability for the $\mathrm{q}$ or $\overline{\mathrm{q}}$ to radiate a gluon, i.e. by the process $\mathrm{e}^+\mathrm{e}^-\to \mathrm{q}\overline{\mathrm{q}}\mathrm{g}$. The matrix element is conveniently given in terms of scaled energy variables in the c.m. frame of the event, $x_1 = 2E_{\mathrm{q}}/E_{\mathrm{cm}}$, $x_2 = 2E_{\overline{\mathrm{q}}}/E_{\mathrm{cm}}$, and $x_3 = 2E_{\mathrm{g}}/E_{\mathrm{cm}}$, i.e. $x_1 + x_2 + x_3 = 2$. For massless quarks the matrix element reads [Ell76]

$$\frac{1}{\sigma_0} \, \frac{\d\sigma}{\d x_1 \, \d x_2} = ... ...{s}}}{2\pi} \, C_F \, \frac{x_1^2 + x_2^2}{(1-x_1)(1-x_2)} ~,$$ (24)

where $\sigma_0$ is the lowest-order cross section, $C_F = 4/3$ is the appropriate colour factor, and the kinematically allowed region is $0 \leq x_i \leq 1, i = 1, 2, 3$. By kinematics, the $x_k$ variable for parton $k$ is related to the invariant mass $m_{ij}$ of the other two partons $i$ and $j$ by $y_{ij} = m_{ij}^2/E_{\mathrm{cm}}^2 = 1 - x_k$.

The strong coupling constant $\alpha_{\mathrm{s}}$ is in first order given by

$$\alpha_{\mathrm{s}}(Q^2) = \frac{12\pi}{(33-2n_f) \, \ln(Q^2/\Lambda^2)} ~.$$ (25)

Conventionally $Q^2 = s = E_{\mathrm{cm}}^2$; we will return to this issue below. The number of flavours $n_f$ is 5 for LEP applications, and so the $\Lambda$ value determined is $\Lambda_5$ (while e.g. most Deeply Inelastic Scattering studies refer to $\Lambda_4$, the energies for these experiments being below the bottom threshold). The $\alpha_{\mathrm{s}}$ values are matched at flavour thresholds, i.e. as $n_f$ is changed the $\Lambda$ value is also changed. It is therefore the derivative of $\alpha_{\mathrm{s}}$ that changes at a threshold, not $\alpha_{\mathrm{s}}$ itself.

In order to separate 2-jets from 3-jets, it is useful to introduce jet-resolution parameters. This can be done in several different ways. Most famous are the $y$ and $(\epsilon, \delta)$ procedures. We will only refer to the $y$ cut, which is the one used in the program. Here a 3-parton configuration is called a 2-jet event if

$$\min_{i,j} (y_{ij}) = \min_{i,j} \left( \frac{m_{ij}^2}{E_{\mathrm{cm}}^2} \right) < y ~.$$ (26)

The cross section in eq. () diverges for $x_1 \rightarrow 1$ or $x_2 \rightarrow 1$ but, when first-order propagator and vertex corrections are included, a corresponding singularity with opposite sign appears in the $\mathrm{q}\overline{\mathrm{q}}$ cross section, so that the total cross section is finite. In analytical calculations, the average value of any well-behaved quantity ${\cal Q}$ can therefore be calculated as

$$\left\langle {\cal Q} \right\rangle = \frac{1}{\sigma_{\ma... ...{3parton}}}{\d x_1 \, \d x_2} \, \d x_1 \, \d x_2 \right) ~,$$ (27)

where any explicit $y$ dependence disappears in the limit $y \rightarrow 0$.

In a Monte Carlo program, it is not possible to work with a negative total 2-jet rate, and thus it is necessary to introduce a fixed non-vanishing $y$ cut in the 3-jet phase space. Experimentally, there is evidence for the need of a low $y$ cut, i.e. a large 3-jet rate. For LEP applications, the recommended value is $y = 0.01$, which is about as far down as one can go and still retain a positive 2-jet rate. With $\alpha_{\mathrm{s}}= 0.12$, in full second-order QCD (see below), the $2:3:4$ jet composition is then approximately $11 \% : 77 \% : 12 \%$.

Note, however, that initial-state QED radiation may occasionally lower the c.m. energy significantly, i.e. increase $\alpha_{\mathrm{s}}$, and thereby bring the 3-jet fraction above unity if $y$ is kept fixed at 0.01 also in those events. Therefore, at PETRA/PEP energies, $y$ values slightly above 0.01 are needed. In addition to the $y$ cut, the program contains a cut on the invariant mass $m_{ij}$ between any two partons, which is typically required to be larger than 2 GeV. This cut corresponds to the actual merging of two nearby parton jets, i.e. where a treatment with two separate partons rather than one would be superfluous in view of the smearing arising from the subsequent fragmentation. Since the cut-off mass scale $\sqrt{y} E_{\mathrm{cm}}$ normally is much larger, this additional cut only enters for events at low energies.

For massive quarks, the amount of QCD radiation is slightly reduced [Iof78]:

$$\frac{1}{\sigma_0} \, \frac{\d\sigma}{\d x_1 \, \d x_2} = \frac{\alpha_{\mathrm{s}}}{2\pi} \, C_F \, \left\{ \frac{x_1^2 + ... ...4 m_{\mathrm{q}}^2}{s} \left( \frac{1}{1-x_1} + \frac{1}{1-x_2} \right) \right.$$

$$- \left. \frac{2 m_{\mathrm{q}}^2}{s} \left( \frac{1}{(1-x_1)^2} ... ...thrm{q}}^4}{s^2} \left( \frac{1}{1-x_1} + \frac{1}{1-x_2} \right)^2 \right\} ~.$$ (28)

Properly, the above expression is only valid for the vector part of the cross section, with a slightly different expression for the axial part, but here the one above is used for it all. In addition, the phase space for emission is reduced by the requirement

$$\frac{(1-x_1)(1-x_2)(1-x_3)}{x_3^2} \geq \frac{m_{\mathrm{q}}^2}{s} ~.$$ (29)

For $\b$ quarks at LEP energies, these corrections are fairly small.