Merging with massive matrix elements

The matching to first-order matrix-elements is well-defined for massless quarks, and was originally used unchanged for massive ones. A first attempt to include massive matrix elements did not compensate for mass effects in the shower kinematics, and therefore came to exaggerate the suppression of radiation off heavy quarks [Nor01,Bam00]. Now the shower has been modified to solve this issue, and also improved and extended to cover better a host of different reactions [Nor01].

Table: The processes that have been calculated, also with one extra gluon in the final state. Colour is given with 1 for singlet, 3 for triplet and 8 for octet. See the text for an explanation of the $\gamma_5$ column and further comments.

colour	spin	$\gamma_5$	example	codes
$1 \to 3 + \overline{3}$	--	--	(eikonal)	6 - 9
$1 \to 3 + \overline{3}$	$1 \to \frac{1}{2} + \frac{1}{2}$	$1,\gamma_5,1\pm\gamma_5$	$\mathrm{Z}^0 \to \mathrm{q}\overline{\mathrm{q}}$	11 - 14
$3 \to 3 + 1$	$\frac{1}{2} \to \frac{1}{2} + 1$	$1,\gamma_5,1\pm\gamma_5$	$\t\to \b\mathrm{W}^+$	16 - 19
$1 \to 3 + \overline{3}$	$0 \to \frac{1}{2} + \frac{1}{2}$	$1,\gamma_5,1\pm\gamma_5$	$\mathrm{h}^0 \to \mathrm{q}\overline{\mathrm{q}}$	21 - 24
$3 \to 3 + 1$	$\frac{1}{2} \to \frac{1}{2} + 0$	$1,\gamma_5,1\pm\gamma_5$	$\t\to \b\H ^+$	26 - 29
$1 \to 3 + \overline{3}$	$1 \to 0 + 0$	$1$	$\mathrm{Z}^0 \to \tilde{\mathrm q}\overline{\tilde{\mathrm{q}}}$	31 - 34
$3 \to 3 + 1$	$0 \to 0 + 1$	$1$	$\tilde{\mathrm q}\to \tilde{\mathrm q}'\mathrm{W}^+$	36 - 39
$1 \to 3 + \overline{3}$	$0 \to 0 + 0$	$1$	$\mathrm{h}^0 \to \tilde{\mathrm q}\overline{\tilde{\mathrm{q}}}$	41 - 44
$3 \to 3 + 1$	$0 \to 0 + 0$	$1$	$\tilde{\mathrm q}\to \tilde{\mathrm q}'\H ^+$	46 - 49
$1 \to 3 + \overline{3}$	$\frac{1}{2} \to \frac{1}{2} + 0$	$1,\gamma_5,1\pm\gamma_5$	$\tilde{\chi} \to \mathrm{q}\overline{\tilde{\mathrm{q}}}$	51 - 54
$3 \to 3 + 1$	$0 \to \frac{1}{2} + \frac{1}{2}$	$1,\gamma_5,1\pm\gamma_5$	$\tilde{\mathrm q}\to \mathrm{q}\tilde{\chi}$	56 - 59
$3 \to 3 + 1$	$\frac{1}{2} \to 0 + \frac{1}{2}$	$1,\gamma_5,1\pm\gamma_5$	$\t\to \tilde{\mathrm t}\tilde{\chi}$	61 - 64
$8 \to 3 + \overline{3}$	$\frac{1}{2} \to \frac{1}{2} + 0$	$1,\gamma_5,1\pm\gamma_5$	$\tilde{\mathrm{g}}\to \mathrm{q}\overline{\tilde{\mathrm{q}}}$	66 - 69
$3 \to 3 + 8$	$0 \to \frac{1}{2} + \frac{1}{2}$	$1,\gamma_5,1\pm\gamma_5$	$\tilde{\mathrm q}\to \mathrm{q}\tilde{\mathrm{g}}$	71 - 74
$3 \to 3 + 8$	$\frac{1}{2} \to 0 + \frac{1}{2}$	$1,\gamma_5,1\pm\gamma_5$	$\t\to \tilde{\mathrm t}\tilde{\mathrm{g}}$	76 - 79
$1 \to 8 + 8$	--	--	(eikonal)	81 - 84

The starting point is the calculation of processes $a \to bc$ and $a \to bc\mathrm{g}$, where the ratio

$$W_{\mathrm{ME}}(x_1,x_2) = \frac{1}{\sigma(a \to bc)} \, \frac{\d\sigma(a \to bc\mathrm{g})}{\d x_1 \, \d x_2}$$ (180)

gives the process-dependent differential gluon-emission rate. Here the phase space variables are $x_1 = 2E_b/m_a$ and $x_2 = 2E_c/m_a$, expressed in the rest frame of particle $a$. Using the standard model and the minimal supersymmetric extension thereof as templates, a wide selection of colour and spin structures have been addressed, as shown in Table . When allowed, processes have been calculated for an arbitrary mixture of `parities', i.e. without or with a $\gamma_5$ factor, like in the vector/axial vector structure of $\gamma^* / \mathrm{Z}^0$. Various combinations of 1 and $\gamma_5$ may also arise e.g. from the wave functions of the sfermion partners to the left- and right-handed fermion states. In cases where the correct combination is not provided, an equal mixture of the two is assumed as a reasonable compromize. All the matrix elements are encoded in the new function PYMAEL(NI,X1,X2,R1,R2,ALPHA), where NI distinguishes the matrix elements, ALPHA is related to the $\gamma_5$ admixture and the mass ratios $r_1 = m_b / m_a$ and $r_2 = m_c/m_a$ are free parameters. This routine is called by PYSHOW, but might also have an interest on its own.

In order to match to the singularity structure of the massive matrix elements, the evolution variable $Q^2$ is changed from $m^2$ to $m^2 - m_{\mathrm{on-shell}}^2$, i.e. $1/Q^2$ is the propagator of a massive particle [Nor01]. For the shower history $b \to b\mathrm{g}$ this gives a differential probability

$$W_{\mathrm{PS,1}}(x_1,x_2) = \frac{\alpha_{\mathrm{s}}}{2\p... ...{2\pi} \, C_F \, \frac{2}{x_3 \, (1 + r_2^2 - r_1^2 - x_2)} ~,$$ (181)

where the numerator $1 + z^2$ of the splitting kernel for $\mathrm{q}\to \mathrm{q}\mathrm{g}$ has been replaced by a 2 in the shower algorithm. For a process with only one radiating parton in the final state, such as $\t\to \b\mathrm{W}^+$, the ratio $W_{\mathrm{ME}}/W_{\mathrm{PS,1}}$ gives the acceptance probability for an emission in the shower. The singularity structure exactly agrees between ME and PS, giving a well-behaved ratio always below unity. If both $b$ and $c$ can radiate, there is a second possible shower history that has to be considered. The matrix element is here split in two parts, one arbitrarily associated with $b \to b\mathrm{g}$ branchings and the other with $c \to c\mathrm{g}$ ones. A convenient choice is $W_{\mathrm{ME,1}} = W_{\mathrm{ME}} (1 + r_1^2 - r_2^2 - x_1)/x_3$ and $W_{\mathrm{ME,2}} = W_{\mathrm{ME}} (1 + r_2^2 - r_1^2 - x_2)/x_3$, which again gives matching singularity structures in $W_{\mathrm{ME,}i}/W_{\mathrm{PS,}i}$ and thus a well-behaved Monte Carlo procedure.

Also subsequent emissions of gluons off the primary particles are corrected to $W_{\mathrm{ME}}$. To this end, a reduced-energy system is constructed, which retains the kinematics of the branching under consideration but omits the gluons already emitted, so that an effective three-body shower state can be mapped to an $(x_1, x_2, r_1, r_2)$ set of variables. For light quarks this procedure is almost equivalent with the original one of using the simple universal splitting kernels after the first branching. For heavy quarks it offers an improved modelling of mass effects also in the collinear region.

Some further changes have been introduced, a few minor as default and some more significant ones as non-default options [Nor01]. This includes the description of coherence effects and $\alpha_{\mathrm{s}}$ arguments, in general and more specifically for secondary heavy flavour production by gluon splittings. The problem in the latter area is that data at LEP1 show a larger rate of secondary charm and bottom production than predicted in most shower descriptions [Bam00,Man00], or in analytical studies [Sey95]. This is based on applying the same kind of coherence considerations to $\mathrm{g}\to \mathrm{q}\overline{\mathrm{q}}$ branchings as to $\mathrm{g}\to \mathrm{g}\mathrm{g}$, which is not fully motivated by theory. In the lack of an unambiguous answer, it is therefore helpful to allow options that can explore the range of uncertainty.

Further issues remain to be addressed, e.g. radiation off particles with non-negligible width. In general, however, the new shower should allow an improved description of gluon radiation in many different processes.