LU TP 99–31\
hep-ph/9910288\
October 1999

[**QCD Interconnection Effects[^1]**]{}\
[Torbjörn Sjöstrand[^2]]{}\
[*Department of Theoretical Physics,*]{}\
[*Lund University, Lund, Sweden*]{}

[**Abstract**]{}\

Heavy objects like the $W$, $Z$ and $t$ are short-lived compared with typical hadronization times. When pairs of such particles are produced, the subsequent hadronic decay systems may therefore become interconnected. We study such potential effects at Linear Collider energies.

This talk mainly reports on work done in collaboration with Valery Khoze [@work].

The widths of the $W$, $Z$ and $t$ are all of the order of 2 GeV. A Standard Model Higgs with a mass above 200 GeV, as well as many supersymmetric and other Beyond the Standard Model particles would also have widths in the range. Not far from threshold, the typical decay times $\tau = 1/\Gamma  \approx 0.1 \, {\mathrm{fm}} \ll  
\tau_{\mathrm{had}} \approx 1 \, \mathrm{fm}$. Thus hadronic decay systems overlap, between pairs of resonances ($W^+W^-$, $Z^0Z^0$, $t\bar{t}$, $Z^0H^0$, …), so that the final state may not be just the sum of two independent decays. Pragmatically, one may distinguish three main eras for such interconnection:

Perturbative: this is suppressed for gluon energies $\omega > \Gamma$ by propagator/timescale effects; thus only soft gluons may contribute appreciably.

Nonperturbative in the hadroformation process: normally modelled by a colour rearrangement between the partons produced in the two resonance decays and in the subsequent parton showers.

Nonperturbative in the purely hadronic phase: best exemplified by Bose–Einstein effects.

The above topics are deeply related to the unsolved problems of strong interactions: confinement dynamics, $1/N^2_{\mathrm{C}}$ effects, quantum mechanical interferences, etc. Thus they offer an opportunity to study the dynamics of unstable particles, and new ways to probe confinement dynamics in space and time [@GPZ; @ourrec], [*but*]{} they also risk to limit or even spoil precision measurements [@ourrec].

So far, studies have mainly been performed in the context of $W$ mass measurements at LEP2. Perturbative effects are not likely to give any significant contribution to the systematic error, $\langle \delta m_W \rangle {\raisebox{-0.8mm}{\hspace{1mm}$\stackrel{<}{\sim}$\hspace{1mm}}}5$ MeV [@ourrec]. Colour rearrangement is not understood from first principles, but many models have been proposed to model effects [@ourrec; @otherrec; @HR], and a conservative estimate gives $\langle \delta m_W \rangle {\raisebox{-0.8mm}{\hspace{1mm}$\stackrel{<}{\sim}$\hspace{1mm}}}40$ MeV. For Bose–Einstein again there is a wide spread in models, and an even wider one in results, with about the same potential systematic error as above [@ourBE; @otherBE; @HR]. The total QCD interconnection error is thus below $m_{\pi}$ in absolute terms and 0.1% in relative ones, a small number that becomes of interest only because we aim for high accuracy.

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More could be said if some experimental evidence existed, but a problem is that also other manifestations of the interconnection phenomena are likely to be small in magnitude. For instance, near threshold it is expected that colour rearrangement will deplete the rate of low-momentum particle production [@lowmom], Fig. \[figlowmom\]. Even with full LEP2 statistics, we are only speaking of a few sigma effects, however. Bose-Einstein appear more promising to diagnose, but so far experimental results are contradictory [@BEstatus].

One area where a linear collider could contribute would be by allowing a much increased statistics in the LEP2 energy region. A 100 fb$^{-1}$ $W^+W^-$ threshold scan would give a $\sim 6$ MeV accuracy on the $W$ mass [@Wilson], with negligible interconnection uncertainty. This would shift the emphasis from $m_W$ to the understanding of the physics of hadronic cross-talk. A high-statistics run, e.g. 50 fb$^{-1}$ at 175 GeV, would give a comfortable signal for the low-momentum depletion mentioned above, and also allow a set of other tests [@othertest; @lowmom]. Above the $Z^0Z^0$ threshold, the single-$Z^0$ data will provide a unique $Z^0Z^0$ no-reconnection reference.

Thus, high-luminosity, LEP2-energy LC (Linear Collider) runs would be excellent to [*establish*]{} a signal. To explore the [*character*]{} of effects, however, a knowledge of the energy dependence could give further leverage.

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In QED, the interconnection rate dampens with increasing energy roughly like $(1 - \beta)^2$, with $\beta$ the velocity of each $W$ in the CM frame [@QED]. By contrast, the nonperturbative QCD models we studied show an interconnection rate dropping more like $(1 - \beta)$ over the LC energy region (with the possibility of a steeper behaviour in the truly asymptotic region), Fig. \[figprob\]. If only the central region of $W$ masses is studied, also the mass shift dampens significantly with energy, Fig. \[figprob\]. However, if also the wings of the mass distribution are included (a difficult experimental proposition, but possible in our toy studies), the average and width of the mass shift distribution do not die out. Thus, with increasing energy, the hadronic cross-talk occurs in fewer events, but the effect in these few is more dramatic.

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The depletion of particle production at low momenta, close to threshold, turns into an enhancement at higher energies [@lowmom]. However, in the inclusive $W^+W^-$ event sample, this and other signals appear too small for reliable detection. One may instead turn to exclusive signals, such as events with many particles at low momenta, or at central rapidities, or at large angles with respect to the event axis, Fig. \[figexclusive\]. Unfortunately, even after such a cut, fluctuations in no-reconnection events as well as ordinary QCD four-jet events (mainly $q\bar{q}gg$ split in $qg + \bar{q}g$ hemispheres, thus with a colour flow between the two) give event rates that overwhelm the expected signal. It could still be possible to observe an excess, but not to identify reconnections on an event-by-event basis. The possibility of some clever combination of several signals still remains open, however.

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Since the $Z^0$ mass and properties are well-known, $Z^0Z^0$ events provide an excellent hunting ground for interconnection. Relative to $W^+W^-$ events, the set of production Feynman graphs and the relative mixture of vector and axial couplings is different, however, and this leads to non-negligible differences in angular distributions, Fig. \[figzzww\]. Furthermore, the higher $Z^0$ mass means that a $Z^0$ is slower than a $W^{\pm}$ at fixed energy, and the larger $Z^0$ width also brings the decay vertices closer. Taken together, at 500 GeV, the reconnection rate in $Z^0Z^0$ hadronic events is likely to be about twice as large as in $W^+W^-$ events, while the cross section is lower by a factor of six. Thus $Z^0Z^0$ events are interesting in their own right, but comparisons with $W^+W^-$ events will be nontrivial.

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As noted above, the Bose–Einstein interplay between the hadronic decay systems of a pair of heavy objects is at least as poorly understood as is colour reconnection, and less well studied for higher energies. In some models [@ourBE], the theoretical mass shift increases with energy, when the separation of the $W$ decay vertices is not included, Fig. \[figboei\]. With this separation taken into account, the theoretical shift levels out at around 200 MeV. How this maps onto experimental observables remains to be studied, but experience from LEP2 energies indicates that the mass shift is significantly reduced, and may even switch sign.

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The $t\bar{t}$ system is different from the $W^+W^-$ and $Z^0Z^0$ ones in that the $t$ and $\bar{t}$ always are colour connected. Thus, even when both tops decay semileptonically, $t \to b W^+ \to b \ell^+ \nu_{\ell}$, the system contains nontrivial interconnection effects. For instance, the total hadronic multiplicity, and especially the multiplicity at low momenta, depends on the opening angle between the $b$ and $\bar{b}$ jets: the smaller the angle, the lower the multiplicity [@topmult], Fig. \[figtop\]. On the perturbative level, this can be understood as arising from a dominance of emission from the $b\bar{b}$ colour dipole at small gluon energies [@dipole], on the nonperturbative one, as a consequence of the string effect [@string].

Uncertainties in the modelling of these phenomena imply a systematic error on the top mass of the order of 30 MeV already in the semileptonic top decays. When hadronic $W$ decays are included, the possibilities of interconnection multiply. This kind of configurations have not yet been studied, but realistically we may expect uncertainties in the range around 100 MeV.

In summary, LEP2 may clarify the Bose–Einstein situation and provide some hadronic cross-talk hints. A high-luminosity LEP2-energy LC run would be the best way to establish colour rearrangement, however. Both colour rearrangement and BE effects (may) remain significant over the full LC energy range: while the fraction of the (appreciably) affected events goes down with energy, the effect per such event comes up. If the objective is to do electroweak precision tests, it appears feasible to reduce the $WW/ZZ$ “interconnection noise” to harmless levels at high energies, by simple proper cuts. It should also be possible, but not easy, to dig out a colour rearrangement signal at high energies, with some suitably optimized cuts that yet remain to be defined. The $Z^0Z^0$ events should display about twice as large interconnection effects as $W^+W^-$ ones, but cross sections are reduced even more. The availability of a single-$Z^0$ calibration still makes $Z^0Z^0$ events of unique interest. While detailed studies remain to be carried out, it appears that the direct reconstruction of the top mass could be uncertain by maybe 100 MeV. Finally, in all of the studies so far, it has turned out to be very difficult to find a clean handle that would help to distinguish between the different models proposed, both in the reconnection and Bose–Einstein areas. Much work thus remains for the future.

[99]{}

V.A. Khoze and T. Sjöstrand, LU TP 99-23, hep-ph/9908408, to appear in the Proceedings of the International Workshop on Linear Colliders, Sitges (Barcelona), Spain, April 28 - May 5, 1999

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[^1]: To appear in the Proceedings of the Workshop on the development of future linear electron-positron colliders for particle physics studies and for research using free electon lasers, Lund, Sweden, 23–26 September 1999

[^2]: torbjorn@thep.lu.se
