Resonance Production

The simplest way to produce a resonance is by a $2 \to 1$ process. If the decay of the resonance is not considered, the cross-section formula does not depend on $\hat{t}$, but takes the form

$$\sigma = \int \int \frac{\d\tau}{\tau} \, \d y \, x_1 f_1(x_1, Q^2) \, x_2 f_2(x_2, Q^2) \, \hat{\sigma}(\hat{s}) ~.$$ (84)

Here the physics is contained in the cross section $\hat{\sigma}(\hat{s})$. The $Q^2$ scale is usually taken to be $Q^2 = \hat{s}$.

In published formulae, cross sections are often given in the zero-width approximation, i.e. $\hat{\sigma}(\hat{s}) \propto \delta (\hat{s} - m_R^2)$, where $m_R$ is the mass of the resonance. Introducing the scaled mass $\tau_R = m_R^2/s$, this corresponds to a delta function $\delta (\tau - \tau_R)$, which can be used to eliminate the integral over $\tau$.

However, what we normally want to do is replace the $\delta$ function by the appropriate Breit-Wigner shape. For a resonance width $\Gamma_R$ this is achieved by the replacement

$$\delta (\tau - \tau_R) \to \frac{s}{\pi} \, \frac{m_R \Gamma_R}{(s \tau - m_R^2)^2 + m_R^2 \Gamma_R^2} ~.$$ (85)

In this formula the resonance width $\Gamma_R$ is a constant.

An improved description of resonance shapes is obtained if the width is made $\hat{s}$-dependent (occasionally also referred to as mass-dependent width, since $\hat{s}$ is not always the resonance mass), see e.g. [Ber89]. To first approximation, this means that the expression $m_R \Gamma_R$ is to be replaced by $\hat{s} \Gamma_R / m_R$, both in the numerator and the denominator. An intermediate step is to perform this replacement only in the numerator. This is convenient when not only $s$-channel resonance production is simulated but also non-resonance $t$- or $u$-channel graphs are involved, since mass-dependent widths in the denominator here may give an imperfect cancellation of divergences. (More about this below.)

To be more precise, in the program the quantity $H_R(\hat{s})$ is introduced, and the Breit-Wigner is written as

$$\delta (\tau - \tau_R) \to \frac{s}{\pi} \, \frac{H_R(s \tau)}{(s \tau - m_R^2)^2 + H_R^2(s \tau)} ~.$$ (86)

The $H_R$ factor is evaluated as a sum over all possible final-state channels, $H_R = \sum_f H_R^{(f)}$. Each decay channel may have its own $\hat{s}$ dependence, as follows.

A decay to a fermion pair, $R \to \mathrm{f}\overline{\mathrm{f}}$, gives no contribution below threshold, i.e. for $\hat{s} < 4 m_{\mathrm{f}}^2$. Above threshold, $H_R^{(f)}$ is proportional to $\hat{s}$, multiplied by a threshold factor $\beta (3 - \beta^2)/2$ for the vector part of a spin 1 resonance, by $\beta^3$ for the axial vector part, by $\beta^3$ for a scalar resonance and by $\beta$ for a pseudoscalar one. Here $\beta = \sqrt{1 - 4m_{\mathrm{f}}^2/\hat{s}}$. For the decay into unequal masses, e.g. of the $\mathrm{W}^+$, corresponding but more complicated expressions are used.

For decays into a quark pair, a first-order strong correction factor $1 + \alpha_{\mathrm{s}}(\hat{s}) / \pi$ is included in $H_R^{(f)}$. This is the correct choice for all spin 1 colourless resonances, but is here used for all resonances where no better knowledge is available. Currently the major exception is top decay, where the factor $1 - 2.5 \, \alpha_{\mathrm{s}}(\hat{s}) / \pi$ is used to approximate loop corrections [Jez89]. The second-order corrections are often known, but then are specific to each resonance, and are not included. An option exists for the $\gamma/\mathrm{Z}^0/\mathrm{Z}'^0$ resonances, where threshold effects due to $\mathrm{q}\overline{\mathrm{q}}$ bound-state formation are taken into account in a smeared-out, average sense, see eq. ().

For other decay channels, not into fermion pairs, the $\hat{s}$ dependence is typically more complicated. An example would be the decay $\mathrm{h}^0 \to \mathrm{W}^+ \mathrm{W}^-$, with a nontrivial threshold and a subtle energy dependence above that [Sey95a]. Since a Higgs with $m_{\mathrm{h}} < 2 m_{\mathrm{W}}$ could still decay in this channel, it is in fact necessary to perform a two-dimensional integral over the $W^{\pm}$ Breit-Wigner mass distributions to obtain the correct result (and this has to be done numerically, at least in part). Fortunately, a Higgs particle lighter than $2 m_{\mathrm{W}}$ is sufficiently narrow that the integral only needs to be performed once and for all at initialization (whereas most other partial widths are recalculated whenever needed). Channels that proceed via loops, such as $\mathrm{h}\to \mathrm{g}\mathrm{g}$, also display complicated threshold behaviours.

The coupling structure within the electroweak sector is usually (re)expressed in terms of gauge boson masses, $\alpha_{\mathrm{em}}$ and $\sin^2 \! \theta_W$, i.e. factors of $G_{\mathrm{F}}$ are replaced according to

$$\sqrt{2} G_{\mathrm{F}} = \frac{\pi \, \alpha_{\mathrm{em}}}{\sin^2 \! \theta_W \, m_{\mathrm{W}}^2} ~.$$ (87)

Having done that, $\alpha_{\mathrm{em}}$ is allowed to run [Kle89], and is evaluated at the $\hat{s}$ scale. Thereby the relevant electroweak loop correction factors are recovered at the $m_{\mathrm{W}}/m_{\mathrm{Z}}$ scale. However, the option exists to go the other way and eliminate $\alpha_{\mathrm{em}}$ in favour of $G_{\mathrm{F}}$. Currently $\sin^2 \! \theta_W$ is not allowed to run.

For Higgs particles and technipions, fermion masses enter not only in the kinematics but also as couplings. The latter kind of quark masses (but not the former, at least not in the program) are running with the scale of the process, i.e. normally the resonance mass. The expression used is [Car96]

$$m(Q^2) = m_0 \left( \frac{\ln(k^2 m_0^2/\Lambda^2)}{\ln(Q^2/\Lambda^2)} \right)^{12/(33 - 2 n_{\mathrm{f}})} ~.$$ (88)

Here $m_0$ is the input mass at a reference scale $k m_0$, defined in the $\overline{\mbox{\textsc{ms}}}$ scheme. Typical choices are either $k=1$ or $k=2$; the latter would be relevant if the reference scale is chosen at the $\mathrm{Q}\overline{\mathrm{Q}}$ threshold. Both $\Lambda$ and $n_{\mathrm{f}}$ are as given in $\alpha_{\mathrm{s}}$.

In summary, we see that an $\hat{s}$ dependence may enter several different ways into the $H_R^{(f)}$ expressions from which the total $H_R$ is built up.

When only decays to a specific final state $f$ are considered, the $H_R$ in the denominator remains the sum over all allowed decay channels, but the numerator only contains the $H_R^{(f)}$ term of the final state considered.

If the combined production and decay process $i \to R \to f$ is considered, the same $\hat{s}$ dependence is implicit in the coupling structure of $i \to R$ as one would have had in $R \to i$, i.e. to first approximation there is a symmetry between couplings of a resonance to the initial and to the final state. The cross section $\hat{\sigma}$ is therefore, in the program, written in the form

$$\hat{\sigma}_{i \to R \to f}(\hat{s}) \propto \frac{\pi}{\ha... ... H_R^{(f)}(\hat{s})} {(\hat{s} - m_R^2)^2 + H_R^2(\hat{s})} ~.$$ (89)

As a simple example, the cross section for the process $\mathrm{e}^- \overline{\nu}_{\mathrm{e}} \to \mathrm{W}^- \to \mu^- \overline{\nu}_{\mu}$ can be written as

$$\hat{\sigma}(\hat{s}) = 12 \, \frac{\pi}{\hat{s}} \, \frac{H... ...(\hat{s} - m_{\mathrm{W}}^2)^2 + H_{\mathrm{W}}^2(\hat{s})} ~,$$ (90)

where

$$H_{\mathrm{W}}^{(i)}(\hat{s}) = H_{\mathrm{W}}^{(f)}(\hat{s}... ...mathrm{em}}(\hat{s})}{24 \, \sin^2 \! \theta_W } \, \hat{s} ~.$$ (91)

If the effects of several initial and/or final states are studied, it is straightforward to introduce an appropriate summation in the numerator.

The analogy between the $H_R^{(f)}$ and $H_R^{(i)}$ cannot be pushed too far, however. The two differ in several important aspects. Firstly, colour factors appear reversed: the decay $R \to \mathrm{q}\overline{\mathrm{q}}$ contains a colour factor $N_C = 3$ enhancement, while $\mathrm{q}\overline{\mathrm{q}}\to R$ is instead suppressed by a factor $1/N_C = 1/3$. Secondly, the $1 + \alpha_{\mathrm{s}}(\hat{s}) / \pi$ first-order correction factor for the final state has to be replaced by a more complicated $K$ factor for the initial state. This factor is not known usually, or it is known (to first non-trivial order) but too lengthy to be included in the program. Thirdly, incoming partons as a rule are space-like. All the threshold suppression factors of the final-state expressions are therefore irrelevant when production is considered. In sum, the analogy between $H_R^{(f)}$ and $H_R^{(i)}$ is mainly useful as a consistency cross-check, while the two usually are calculated separately. Exceptions include the rather messy loop structure involved in $\mathrm{g}\mathrm{g}\to \mathrm{h}^0$ and $\mathrm{h}^0 \to \mathrm{g}\mathrm{g}$, which is only coded once.

It is of some interest to consider the observable resonance shape when the effects of parton distributions are included. In a hadron collider, to first approximation, parton distributions tend to have a behaviour roughly like $f(x) \propto 1/x$ for small $x$ -- this is why $f(x)$ is replaced by $xf(x)$ in eq. (). Instead, the basic parton-distribution behaviour is shifted into the factor of $1/\tau$ in the integration phase space $\d\tau/\tau$, cf. eq. (). When convoluted with the Breit-Wigner shape, two effects appear. One is that the overall resonance is tilted: the low-mass tail is enhanced and the high-mass one suppressed. The other is that an extremely long tail develops on the low-mass side of the resonance: when $\tau \to 0$, eq. () with $H_R(\hat{s}) \propto \hat{s}$ gives a $\hat{\sigma}(\hat{s}) \propto \hat{s} \propto \tau$, which exactly cancels the $1/\tau$ factor mentioned above. Naïvely, the integral over $y$, $\int \d y = - \ln \tau$, therefore gives a net logarithmic divergence of the resonance shape when $\tau \to 0$. Clearly, it is then necessary to consider the shape of the parton distributions in more detail. At not-too-small $Q^2$, the evolution equations in fact lead to parton distributions more strongly peaked than $1/x$, typically with $xf(x) \propto x^{-0.3}$, and therefore a divergence like $\tau^{-0.3}$ in the cross-section expression. Eventually this divergence is regularized by a closing of the phase space, i.e. that $H_R(\hat{s})$ vanishes faster than $\hat{s}$, and by a less drastic small-$x$ parton-distribution behaviour when $Q^2 \approx \hat{s} \to 0$.

The secondary peak at small $\tau$ may give a rather high cross section, which can even rival that of the ordinary peak around the nominal mass. This is the case, for instance, with $\mathrm{W}$ production. Such a peak has never been observed experimentally, but this is not surprising, since the background from other processes is overwhelming at low $\hat{s}$. Thus a lepton of one or a few GeV of transverse momentum is far more likely to come from the decay of a charm or bottom hadron than from an extremely off-shell $\mathrm{W}$ of a mass of a few GeV. When resonance production is studied, it is therefore important to set limits on the mass of the resonance, so as to agree with the experimental definition, at least to first approximation. If not, cross-section information given by the program may be very confusing.

Another problem is that often the matrix elements really are valid only in the resonance region. The reason is that one usually includes only the simplest $s$-channel graph in the calculation. It is this `signal' graph that has a peak at the position of the resonance, where it (usually) gives much larger cross sections than the other `background' graphs. Away from the resonance position, `signal' and `background' may be of comparable order, or the `background' may even dominate. There is a quantum mechanical interference when some of the `signal' and `background' graphs have the same initial and final state, and this interference may be destructive or constructive. When the interference is non-negligible, it is no longer meaningful to speak of a `signal' cross section. As an example, consider the scattering of longitudinal $\mathrm{W}$'s, $\mathrm{W}^+_{\mathrm{L}} \mathrm{W}^-_{\mathrm{L}} \to \mathrm{W}^+_{\mathrm{L}} \mathrm{W}^-_{\mathrm{L}}$, where the `signal' process is $s$-channel exchange of a Higgs. This graph by itself is ill-behaved away from the resonance region. Destructive interference with `background' graphs such as $t$-channel exchange of a Higgs and $s$- and $t$-channel exchange of a $\gamma/\mathrm{Z}$ is required to save unitarity at large energies.

In $\mathrm{e}^+\mathrm{e}^-$ colliders, the $f_{\mathrm{e}}^{\mathrm{e}}$ parton distribution is peaked at $x = 1$ rather than at $x = 0$. The situation therefore is the opposite, if one considers e.g. $\mathrm{Z}^0$ production in a machine running at energies above $m_{\mathrm{Z}}$: the tail towards lower masses is suppressed and the one towards higher masses enhanced, with a sharp secondary peak at around the nominal energy of the machine. Also in this case, an appropriate definition of cross sections therefore is necessary -- with additional complications due to the interference between $\gamma^*$ and $\mathrm{Z}^0$. When other processes are considered, problems of interference with background appears also here. Numerically the problems may be less pressing, however, since the secondary peak is occurring in a high-mass region, rather than in a more complicated low-mass one. Further, in $\mathrm{e}^+\mathrm{e}^-$ there is little uncertainty from the shape of the parton distributions.

In $2 \to 2$ processes where a pair of resonances are produced, e.g. $\mathrm{e}^+\mathrm{e}^-\to \mathrm{Z}^0 \mathrm{h}^0$, cross section are almost always given in the zero-width approximation for the resonances. Here two substitutions of the type

$$1 = \int \delta (m^2 - m_R^2) \, dm^2 \to \int \frac{1}{\pi}... ...frac{m_R \Gamma_R}{(m^2 - m_R^2)^2 + m_R^2 \Gamma_R^2} \, dm^2$$ (92)

are used to introduce mass distributions for the two resonance masses, i.e. $m_3^2$ and $m_4^2$. In the formula, $m_R$ is the nominal mass and $m$ the actually selected one. The phase-space integral over $x_1$, $x_1$ and $\hat{t}$ in eq. () is then extended to involve also $m_3^2$ and $m_4^2$. The effects of the mass-dependent width is only partly taken into account, by replacing the nominal masses $m_3^2$ and $m_4^2$ in the $\d\hat{\sigma}/\d\hat{t}$ expression by the actually generated ones (also e.g. in the relation between $\hat{t}$ and $\cos\hat{\theta}$), while the widths are evaluated at the nominal masses. This is the equivalent of a simple replacement of $m_R \Gamma_R$ by $\hat{s} \Gamma_R / m_R$ in the numerator of eq. (), but not in the denominator. In addition, the full threshold dependence of the widths, i.e. the velocity-dependent factors, is not reproduced.

There is no particular reason why the full mass-dependence could not be introduced, except for the extra work and time consumption needed for each process. In fact, the matrix elements for several $\gamma^* / \mathrm{Z}^0$ and $\mathrm{W}^{\pm}$ production processes do contain the full expressions. On the other hand, the matrix elements given in the literature are often valid only when the resonances are almost on the mass shell, since some graphs have been omitted. As an example, the process $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{e}^- \overline{\nu}_{\mathrm{e}} \mu^+ \nu_{\mu}$ is dominated by $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{W}^- \mathrm{W}^+$ when each of the two lepton pairs is close to $m_{\mathrm{W}}$ in mass, but in general also receives contributions e.g. from $\mathrm{q}\overline{\mathrm{q}}\to \mathrm{Z}^0 \to \mathrm{e}^+\mathrm{e}^-$, followed by $\mathrm{e}^+ \to \overline{\nu}_{\mathrm{e}} \mathrm{W}^+$ and $\mathrm{W}^+ \to \mu^+ \nu_{\mu}$. The latter contributions are neglected in cross sections given in the zero-width approximation.

Widths may induce gauge invariance problems, in particular when the $s$-channel graph interferes with $t$- or $u$-channels. Then there may be an imperfect cancellation of contributions at high energies, leading to an incorrect cross section behaviour. The underlying reason is that a Breit-Wigner corresponds to a resummation of terms of different orders in coupling constants, and that therefore effectively the $s$-channel contributions are calculated to higher orders than the $t$- or $u$-channel ones, including interference contributions. A specific example is $\mathrm{e}^+ \mathrm{e}^- \to \mathrm{W}^+ \mathrm{W}^-$, where $s$-channel $\gamma^*/\mathrm{Z}^*$ exchange interferes with $t$-channel $\nu_{\mathrm{e}}$ exchange. In such cases, a fixed width is used in the denominator. One could also introduce procedures whereby the width is made to vanish completely at high energies, and theoretically this is the cleanest, but the fixed-width approach appears good enough in practice.

Another gauge invariance issue is when two particles of the same kind are produced in a pair, e.g. $\mathrm{g}\mathrm{g}\to \t\overline{\mathrm{t}}$. Matrix elements are then often calculated for one common $m_{\t }$ mass, even though in real life the masses $m_3 \neq m_4$. The proper gauge invariant procedure to handle this would be to study the full six-fermion state obtained after the two $\t\to \b\mathrm{W}\to \b\mathrm{f}_i \overline{\mathrm{f}}_j$ decays, but that may be overkill if indeed the $\t$'s are close to mass shell. Even when only equal-mass matrix elements are available, Breit-Wigners are therefore used to select two separate masses $m_3$ and $m_4$. From these two masses, an average mass $\overline{m}$ is constructed so that the $\beta_{34}$ velocity factor of eq. () is retained,

$$\beta_{34}(\hat{s},\overline{m}^2,\overline{m}^2) = \beta_{... ...frac{m_3^2 + m_4^2}{2} - \frac{(m_3^2 - m_4^2)^2}{4 \hat{s}}.$$ (93)

This choice certainly is not unique, but normally should provide a sensible behaviour, also around threshold. Of course, the differential cross section is no longer guaranteed to be gauge invariant when gauge bosons are involved, or positive definite. The program automatically flags the latter situation as unphysical. The approach may well break down when either or both particles are far away from mass shell. Furthermore, the preliminary choice of scattering angle $\hat{\theta}$ is also retained. Instead of the correct $\hat{t}$ and $\hat{u}$ of eq. (), modified

$$\overline{\hat{t}}, \overline{\hat{u}} = - \frac{1}{2} \lef... ...t\} = (\hat{t}, \hat{u}) - \frac{(m_3^2 - m_4^2)^2}{4 \hat{s}}$$ (94)

can then be obtained. The $\overline{m}^2$, $\overline{\hat{t}}$ and $\overline{\hat{u}}$ are now used in the matrix elements to decide whether to retain the event or not.

Processes with one final-state resonance and another ordinary final-state product, e.g. $\mathrm{q}\mathrm{g}\to \mathrm{W}^+ \mathrm{q}'$, are treated in the same spirit as the $2 \to 2$ processes with two resonances, except that only one mass need be selected according to a Breit-Wigner.