Transverse evolution

We have above seen that two parton lines may be defined, stretching back from the hard scattering to the initial incoming hadron wavefunctions at small $Q^2$. Specifically, all parton flavours $i$, virtualities $Q^2$ and energy fractions $x$ may be found. The exact kinematical interpretation of the $x$ variable is not unique, however. For partons with small virtualities and transverse momenta, essentially all definitions agree, but differences may appear for branchings close to the hard scattering.

In first-order QED [Ber85] and in some simple QCD toy models [Got86], one may show that the `correct' choice is the `$\hat{s}$ approach'. Here one requires that $\hat{s} = x_1 x_2 s$, both at the hard-scattering scale and at any lower scale, i.e. $\hat{s}(Q^2) = x_1(Q^2) \, x_2(Q^2) \, s$, where $x_1$ and $x_2$ are the $x$ values of the two resolved partons (one from each incoming beam particle) at the given $Q^2$ scale. In practice this means that, at a branching with the splitting variable $z$, the total $\hat{s}$ has to be increased by a factor $1/z$ in the backwards evolution. It also means that branchings on the two incoming legs have to be interleaved in a single monotonic sequence of $Q^2$ values of branchings. A problem with this $x$ interpretation is that it is not quite equivalent with an $\overline{\mbox{\textsc{ms}}}$ definition of parton densities [Col00], or any other standard definition. In practice, effects should not be large from this mismatch.

For a reconstruction of the complete kinematics in this approach, one should start with the hard scattering, for which $\hat{s}$ has been chosen according to the hard-scattering matrix element. By backwards evolution, the virtualities $Q_1^2 = -m_1^2$ and $Q_2^2 = -m_2^2$ of the two interacting partons are reconstructed. Initially the two partons are considered in their common c.m. frame, coming in along the $\pm z$ directions. Then the four-momentum vectors have the non-vanishing components

$$E_{1,2} = \frac{ \hat{s} \pm (Q_2^2 - Q_1^2)}{2 \sqrt{\hat{s}}} ~,$$

$$p_{z1} = - p_{z2} = \sqrt{ \frac{ (\hat{s} + Q_1^2 + Q_2^2 )^2 - 4 Q_1^2 Q_2^2 }{ 4 \hat{s} } } ~,$$ (190)

with $(p_1 + p_2)^2 = \hat{s}$.

If, say, $Q_1^2 > Q_2^2$, then the branching $3 \to 1 + 4$, which produced parton 1, is the one that took place closest to the hard scattering, and the one to be reconstructed first. With the four-momentum $p_3$ known, $p_4 = p_3 - p_1$ is automatically known, so there are four degrees of freedom. One corresponds to a trivial azimuthal angle around the $z$ axis. The $z$ splitting variable for the $3 \to 1 + 4$ vertex is found as the same time as $Q_1^2$, and provides the constraint $(p_3 + p_2)^2 = \hat{s}/z$. The virtuality $Q_3^2$ is given by backwards evolution of parton 3.

One degree of freedom remains to be specified, and this is related to the possibility that parton 4 initiates a time-like parton shower, i.e. may have a non-zero mass. The maximum allowed squared mass $m_{\mathrm{max},4}^2$ is found for a collinear branching $3 \to 1 + 4$. In terms of the combinations

$$s_1 = \hat{s} + Q_2^2 + Q_1^2 ~,$$

$$s_3 = \frac{\hat{s}}{z} + Q_2^2 + Q_3^2 ~,$$

$$r_1 = \sqrt{s_1^2 - 4 Q_2^2 Q_1^2} ~,$$

$$r_3 = \sqrt{s_3^2 - 4 Q_2^2 Q_3^2} ~,$$ (191)

one obtains

$$m_{\mathrm{max},4}^2 = \frac{s_1 s_3 - r_1 r_3}{2 Q_2^2} - Q_1^2 - Q_3^2 ~,$$ (192)

which, for the special case of $Q_2^2 = 0$, reduces to

$$m_{\mathrm{max},4}^2 = \left\{ \frac{Q_1^2}{z} - Q_3^2 \righ... ...at{s} + Q_1^2} - \frac{\hat{s}}{\hat{s}/z + Q_3^2} \right\} ~.$$ (193)

These constraints on $m_4$ are only the kinematical ones, in addition coherence phenomena could constrain the $m_{\mathrm{max},4}$ values further. Some options of this kind are available; the default one is to require additionally that $m_4^2 \leq Q_1^2$, i.e. lesser than the space-like virtuality of the sister parton.

With the maximum virtuality given, the final-state showering machinery may be used to give the development of the subsequent cascade, including the actual mass $m_4^2$, with $0 \leq m_4^2 \leq m_{\mathrm{max},4}^2$. The evolution is performed in the c.m. frame of the two `resolved' partons, i.e. that of partons 1 and 2 for the branching $3 \to 1 + 4$, and parton 4 is assumed to have a nominal energy $E_{\mathrm{nom},4} = (1/z - 1) \sqrt{\hat{s}}/2$. (Slight modifications appear if parton 4 has a non-vanishing mass $m_{\mathrm{q}}$ or $m_{\ell}$.)

Using the relation $m_4^2 = (p_3 - p_1)^2$, the momentum of parton 3 may now be found as

$$E_3 = \frac{1}{2 \sqrt{\hat{s}} } \left\{ \frac{\hat{s}}{z} + Q_2^2 - Q_1^2 - m_4^2 \right\} ~,$$

$$p_{z3} = \frac{1}{2 p_{z1}} \left\{ s_3 - 2 E_2 E_3 \right\} ~,$$

$$p_{\perp ,3}^2 = \left\{ m_{\mathrm{max},4}^2 - m_4^2 \right\} \, \frac{ (s_1 s_3 + r_1 r_3)/2 - Q_2^2 (Q_1^2 + Q_3^2 + m_4^2)}{r_1^2} ~.$$ (194)

The requirement that $m_4^2 \geq 0$ (or $\geq m_f^2$ for heavy flavours) imposes a constraint on allowed $z$ values. This constraint cannot be included in the choice of $Q_1^2$, where it logically belongs, since it also depends on $Q_2^2$ and $Q_3^2$, which are unknown at this point. It is fairly rare (in the order of 10% of all events) that a disallowed $z$ value is generated, and when it happens it is almost always for one of the two branchings closest to the hard interaction: for $Q_2^2 = 0$ eq. () may be solved to yield $z \leq \hat{s}/(\hat{s} + Q_1^2 - Q_3^2)$, which is a more severe cut for $\hat{s}$ small and $Q_1^2$ large. Therefore an essentially bias-free way of coping is to redo completely any initial-state cascade for which this problem appears.

This completes the reconstruction of the $3 \to 1 + 4$ vertex. The subsystem made out of partons 3 and 2 may now be boosted to its rest frame and rotated to bring partons 3 and 2 along the $\pm z$ directions. The partons 1 and 4 now have opposite and compensating transverse momenta with respect to the event axis. When the next vertex is considered, either the one that produces parton 3 or the one that produces parton 2, the 3-2 subsystem will fill the function the 1-2 system did above, e.g. the rôle of $\hat{s} = \hat{s}_{12}$ in the formulae above is now played by $\hat{s}_{32} = \hat{s}_{12}/z$. The internal structure of the 3-2 system, i.e. the branching $3 \to 1 + 4$, appears nowhere in the continued description, but has become `unresolved'. It is only reflected in the successive rotations and boosts performed to bring back the new endpoints to their common rest frame. Thereby the hard-scattering subsystem 1-2 builds up a net transverse momentum and also an overall rotation of the hard-scattering subsystem.

After a number of steps, the two outermost partons have virtualities $Q^2 < Q_0^2$ and then the shower is terminated and the endpoints assigned $Q^2 = 0$. Up to small corrections from primordial $k_{\perp}$, discussed in section , a final boost will bring the partons from their c.m. frame to the overall c.m. frame, where the $x$ values of the outermost partons agree also with the light-cone definition. The combination of several rotations and boosts implies that the two colliding partons have a nontrivial orientation: when boosted back to their rest frame, they will not be oriented along the $z$ axis. This new orientation is then inherited by the final state of the collision, including resonance decay products.