The Underlying Event and the Total Cross Section from Tevatron to the LHC
Abstract:
Multiple partonic interactions are widely used to simulate the hadronic final state in high energy hadronic collisions, and successfully describe many features of the data. It is important to make maximum use of the available physical constraints on such models, particularly given the large extrapolation from current high energy data to LHC energies. In eikonal models, the rate of multiparton interactions is coupled to the energy dependence of the total cross section. Using a Monte Carlo implementation of such a model, we study the connection between the total cross section, the jet cross section, and the underlying event. By imposing internal consistency on the model and comparing to current data we constrain the allowed range of its parameters. We show that measurements of the total protonproton crosssection at the LHC are likely to break this internal consistency, and thus to require an extension of the model. Likely such extensions are that hard scatters probe a denser matter distribution inside the proton in impact parameter space than soft scatters, a conclusion also supported by Tevatron data on doubleparton scattering, and/or that the basic parameters of the model are energy dependent.
KATP132008
MCnet/08/03
1 Introduction
Hadronhadron collision events at high energies often contain high transverse energy jets, which in QCD arise from gluon or quark (generically, parton) scattering followed by QCD radiation and hadronization. This model is generally taken to be realistic above some minimum transverse momentum scale, . The contribution of these events to the total cross section rises with hadronhadron centreofmass energy, , since the minimum value of the probed is given by , and the parton densities rise strongly for or so [1, 2].
One reason that this rising contribution to the cross section is of interest is that while perturbative QCD cannot predict total hadronic cross sections (since in many events no hard perturbative scale is present), rising hadronic cross sections are a feature also seen in successful nonperturbative approaches [3, 4], the behaviour of which must presumably emerge from the QCD Lagrangian in some manner. Thus by comparing the behaviour of the hard contribution to the cross section to the behaviour expected from fits to the total cross section, it may be possible to learn something about the connection between these approaches and about hadronic cross sections in general.
The connection between the hard partonic cross section and the total cross section is not onetoone, however. There are certainly hadronic scatters in which no hard jets are produced, and some nonperturbative scattering process must be added to the perturbative jet contribution to model the total cross section. In addition, at the high parton densities probed at recent, current and future colliders, simple assumptions lead to the conclusion that the probability of multiple partonic scatters in a single hadronhadron collision is significant. In fact, Fig. 1 shows that for values below about 5 GeV, the total “hard” cross section calculated assuming one partonparton scatter per protonproton collision exceeds the total cross section as extrapolated using the nonperturbative fits, at LHC energies. This strongly implies that the average number of partonic scatters in an inelastic collision must be greater than one.
Introducing the possibility of such multiparton interactions also seems to be required in order to describe the hadronic final state[5, 6, 7]. In general, softer additional scatters occurring in a highevent manifest themselves as additional particles and energyflow, the socalled “underlying event”.
In this paper we examine the predictions of the model that was discussed, for example in [8], and implemented in [9, 10, 11, 12] includingthe possibility of soft scatters. We explore the consistency constraints that would be imposed by comparing a given value of the total cross section to the predicted jet cross section, and attempt to identify allowed regions of parameter space within which the model must lie if it is to be consistent with the measured cross section at the LHC. We also discuss ways in which energy dependencies in the parameters could arise, and their impact upon these constraints. The studies are all carried out using the new implementation in Herwig++ [11, 12]; however, they are also relevant to the fortran implementation Jimmy [9], if the same hard cross section is used.
2 Total and elastic cross section parameterizations
Throughout this paper we will exploit the connection that can be established between the eikonal model of Refs. [9, 10, 11] and the total cross section. To give a reasonable range of expectations for the latter, we use the successful parameterization of Donnachie and Landshoff [3, 4]. We will use three different variations;

The standard parameterization from [3] with the following behaviour at high energies:
(1) 
Using the same energy dependence but normalizing it to the measurement [13] by CDF:
(2) 
Using the most recent fit [4], which takes the contributions from both hard and soft Pomerons into account:
(3)
Other parameterizations and models for the total cross section exist [14, 15], but their predictions for the total cross section at 14 TeV generally lie within the range covered by these three^{1}^{1}1The most recent models of [16, 17] predict mb, which is 10 % below the smallest expectation we use. Since the difference this introduces is similar to the one between our first and second parameterization it can easily be estimated by the reader.. As will be seen, the range is wide, and early measurements of the total cross section at the LHC can be expected to have a big impact [18].
We will also find it useful to compare our model with the elastic slope parameter, , defined in terms of the differential elastic scattering cross section, , as
(4) 
In the DonnachieLandshoff parameterization, this is given by:
(5) 
with . Together with the CDF data[19], this implies
(6) 
The most recent fit[4] has the same value for and hence , while those of [14, 15] are a little higher: 20–22. We therefore use the CDF value for the Tevatron energy and the range 19–22 to represent the range of possible measurements from the LHC.
3 Eikonal model
The scattering amplitude can be expressed as the Fourier transform of the elastic scattering amplitude in impact parameter space as
(7) 
where is the transverse momentum transfer vector, with, in the high energy limit, . In this limit, can be assumed to be purely imaginary and therefore be expressed in terms of a real eikonal function , as
(8) 
Using (7) and (8) the total cross section for can be expressed as
(9) 
The elastic cross section is then
(10) 
The inelastic cross section thereby reads
(11) 
The elastic slope parameter at zero momentum transfer is also calculable within this framework and yields [20]
(12) 
3.1 Multiparton scattering model
The preceding expressions are completely general and modelindependent, but we now introduce an explicit model[8, 9, 10, 11] to predict the form of the eikonal function, . First we assume that it can be decomposed into the sum of independent soft and hard parts,
(13) 
and start by considering the hard part. We consider a model in which partons are distributed across the face of the colliding hadrons with a spatial distribution that is independent of their longitudinal momentum. We assume that pairs of partons in the colliding hadrons scatter with independent probabilities, leading to the distribution of number of scatters at fixed impact parameter obeying Poisson statistics. We further assume that any hadronhadron collision in which there is an elastic partonparton collision above some cutoff will lead to an inelastic hadronic final state. Finally, we require that the inclusive cross section for hadronhadron collisions to produce partons above be equal to the inclusive partonic cross section folded with standard inclusive parton distribution functions, as given by the factorization theorem. The result of this model is that the inelastic cross section is given by an expression identical to Eq. 11, but with replaced by
(14) 
where describes the overlap distribution of the partons in impact parameter space and denotes the inclusive cross section above a transverse momentum cutoff , given by the perturbative result
(15) 
where denotes the inclusive cross section for partons of types and and invariantmasssquared to produce jets of a given .
We assume that the soft eikonal function has the same form,
(16) 
where is the purely nonperturbative cross section below , which is a free parameter of the model. That is, we assume that soft scatters are the result of partonic interactions that are local in impact parameter.
The elastic slope parameter discussed above relates to bulk interactions of the proton. Thus it can be taken as directly constraining the matter distribution “seen” by soft scatters. Higher scatters might be expected to see a different matter distribution, for example they might probe denser “hot spots” within the proton. However, at present we take the simplest assumption for the perturbative part of the eikonal function, i.e. that the semihard scatters “see” the same matter distribution as the soft ones and therefore take . This is clearly a simplifying assumption, but a reasonable one until proven otherwise.
It is worth mentioning that according to the definition in Eq. 11, the inelastic cross section contains all inelastic processes, including diffractive dissociation. This is consistent with the calculation of the inclusive hard cross section, which is calculated from the conventional parton distribution functions, which describe the inclusive distribution of partons in a hadron whatever their source, i.e. whether the proton remains intact or not.
3.2 Overlap parameterization
The only remaining freedom in the eikonal model is the functional form of the overlap function . is the convolution of the individual spatial parton distributions of the colliding hadrons,
(17) 
In Refs. [9, 10, 11], is taken to be proportional to the electromagnetic form factor,
(18) 
is the only parameter and has the dimensions of an inverse radius. In scattering its value was measured to be . This is a loose constraint, since the distribution of partons may not necessarily coincide with the distribution of electromagnetic charge. Actually, using the results from the previous section, the CDF data on the total cross section ( mb [13]) and the elastic slope ( [19]) one can solve for the total inclusive cross section and for , yielding .
In order to investigate the dependence on the assumed shape of the
matter distribution, we have compared our default results with those
obtained with a doubleGaussian distribution, as chosen in
Refs. [21, 22, 23],
(19) 
Here and are all free parameters, but we choose to fix and at values that are reasonably generic, but also close to the tuned values used in [21, 22, 23], with the relative strengths given by and the relative widths by , and view as the only free parameter. The parameters and in the two models are arbitrary and should ultimately be fit to data. However, in order to have a likeforlike comparison, we choose to relate them in such a way that the rms value of is identical. That is, we describe the doubleGaussian also as being a function of , with set via . We illustrate the shapes of the two resulting overlap functions for two different values of in Fig. 2.
We find that for small values of () the results of the two models are extremely similar, differing at most by %. For large values () they differ more, the double Gaussian distribution giving a larger deeikonalized cross section (see next section) by between 30 % with the standard Donnachie–Landshoff total cross section prediction at the LHC and 150 % with the soft+hard Pomeron prediction. While these lead to somewhat different predictions, in our final results they effectively correspond to a distortion of the axis. The effect on our final plots, Figure 6, is small, since our consistency requirement is mainly active at small .
3.3 Connection to the total cross section
For a given point in the parameter space () of our model, we are able to calculate using Eq. 14. The remaining uncertainty is in , which depends on the PDF choice, the treatment of etc. If we now choose a value for the nonperturbative cross section below , , we have the full eikonal function at hand and can calculate the total cross section from Eq. 9.
We will, however, turn this argument around and use the value of the total cross section as input to fix the additional parameter in our nonperturbative part of the eikonal function (). For energies at which there are not yet measurements, we use the parameterizations of [3, 4] to give an indication of the likely range of the total cross section. We will extract the sum from Eq. 9 and call this cross section the deeikonalized cross section. That is, the deeikonalized cross section is given by the solution to
(20) 
for a given value of and a given value of the total cross section, . Clearly, is a function only of these two quantities. Since is dependent, this implies that the value of we extract by this procedure is also dependent ( is a matching scale between the two subprocess cross sections and the sum of the two is independent of it).
In Fig. 3 we plot the deeikonalized cross sections for the three different total cross section extrapolations as a function of the centreofmass energy. Furthermore we show the value of using different cutoffs. is now given by the difference of these curves. If we interpret as a physical cross section (the inclusive cross section for two partons to undergo a nonperturbative soft scattering), it cannot be negative. Thus the implication is that whenever the inclusive hard cross section is larger than the deeikonalized one, the model is inconsistent. We will investigate this behaviour in more detail in Sect. 4.1.
From the plots in Fig. 3 the values for can in principle be read off. However, due to the logarithmic scale it is not easy to see what is implied for the energy dependence of the soft cross section. Therefore, for selected points in parameter space, is shown separately in Fig. 4. Note that where the inclusive hard cross section line for GeV crosses and recrosses the total crosssection extrapolation in the top left plot of Fig. 3, the soft cross section in the top left plot of Fig. 4 first becomes negative and then positive again. The dependence of on the centreofmass energy reveals two main points: First, it is noticeable that one observes a moreorless constant behaviour with increasing energy only in a limited range of our parameter space. This behaviour is mainly present for lower values of . Second, for the most extreme total cross section prediction, is never sufficient to explain the strong rise with energy. There, essentially all parameter choices require a strongly rising soft cross section, in addition to the expected strong rise in the hard cross section. This is, at the very least, counterintuitive, and one might conclude that, within our model, the rise of the cross section in parametrization (2.3) is too extreme.
3.4 Parton saturation physics
The main motivation for allowing multiparton scatterings is the rise of the inclusive cross section, for fixed , with increasing centreofmass energy. Multiparton scattering provides a mechanism to allow this quantity to exceed the total cross section. Eikonal models that incorporate this fact unitarize the inclusive cross section. There is however a second source of unitarization, the physics of parton saturation, which is a competing effect. To estimate the influence on our studies, we have used a simple modification of the PDFs recently introduced [24] into Herwig++ to mimic parton saturation effects for any PDF. The modification replaces below by
(21) 
where X0 and Exp are changeable parameters. HERA data indicate that saturation is unlikely to be a strong effect above . Therefore, the strongest reasonable influence from this effect is obtained by setting . The results are shown in Fig. 3, where the effect is visible, but small, at LHC energies.
4 Parameter space constraints from data
In discussing the deeikonalized cross section, we noted that for some parameter values the hard partonic cross section exceeds it. This implies in our model that the soft cross section should be negative. We take this as an inconsistency that would, for a given measured at the LHC, rule out such parameter space points. In this section we discuss the extent to which the space of parameter values can be limited by this and other constraints.
4.1 Consistency
The horizontal band shows the range of values allowed for a given value of the elastic slope in conjunction with the indicated . For Tevatron energies, both and are chosen according to the CDF measurement from Refs. [13] and [19] respectively.
Our expectations on the value of the elastic slope at LHC energies simply reflect the range of predictions that the models of [3, 4, 14, 15] give, as discussed in Sect. 2. For the value of at the LHC we show a range of possible values motivated by these parameterizations.
For a particular value of (or for a given range of possible values at the LHC), we are able to extract constraints on the allowed parameters, by simply requiring a sensible performance of the eikonal model. The most basic requirement, which was just mentioned, is that the nonperturbative cross section that is needed to match the total crosssection prediction is positive. A negative value means that the model cannot be applied and therefore this requirement puts a stringent limit on the allowed values of and . This limit will depend on the value of , which is not a stable prediction itself. We therefore calculate this limit with several variations. We use three different PDF sets [25, 26, 27], vary the running of from 1loop, which is the default in Herwig++ to 2loop and finally apply the modifications to the PDF’s described in Sect. 3.4. The solid lines in Figs. 5 and 6 show these limits, where the entire range below the curves is excluded. The limits impose a minimal for any given value of . Points on that line are parameter sets where mb.
Another, weaker, consistency constraint we apply is related to the simulation of the final state of these collisions. We observe that when we embed them into the full simulation of Herwig++, including backward evolution of the initial state, each collision consumes, on average, about a tenth of the available total energy, so that the approximation that individual hard scatters are independent must break down, at least due to energy conservation, when there are more than about ten of them. We therefore indicate on Fig. 6 the points in parameter space where the average multiplicity of scatters above reaches 10. This is certainly not a stringent limit but a sensible parameter choice most likely avoids this region.
The classic CDF analysis of the distribution in azimuth of the mean charged multiplicity and scalar sum as a function of the transverse momentum of the leading jet[6] also provides constraints on the model as embedded in Herwig++. Reference [11] described the implementation of multiparton scattering into Herwig++ (i.e. the simulation of the final state corresponding to ) and made a twoparameter fit ( and ) to these data. Since Herwig++ does not yet include a simulation of the final state corresponding to , we do not take the results of this fit as a strong constraint on the parameter space, but an indication of the effect such a tuning could have once a complete description is available. The result is that, although one obtains a best fit with the values , , the bestfit values of the parameters are strongly correlated, with the function having a long, thin, rather flat valley running from to , and beyond. For any given value of in this range one can find a value that gives a good description of these data.
Combining these constraints at the Tevatron, a small allowed region remains around and .
At the LHC, this region would be ruled out for all the values of we have considered. Note that if the LHC measurement were as high a 164 mb, this would on its own imply an energydependent , in contradiction with our initial assumptions.
In the next section we discuss different ways in which the assumptions of the model might be modified to account for this potential inconsistency.
4.2 Extensions to the model
Some authors have suggested, within multiparton scattering models, that the parameters of the model, analogous to our and , should be energy dependent. In this section we briefly discuss the arguments for these models.
In [28] a simple model of the spatial/momentum structure of a hadron was constructed from which it was argued that the colour screening length decreases slowly with increasing energy. This translates into a that increases slowly with energy, for which they estimated with in the range 0.025 to 0.08. The actual value used in Refs. [21, 22, 23] is 0.08, leading to a 35% increase in from the Tevatron to the LHC.
In [29, 30] a multiparton model was constructed that is very similar to ours at low energy, with an impact parameter distribution of partons given by the electromagnetic form factor. However, soft gluon effects were estimated and summed to all orders, to give a mean partonparton separation, , that falls with energy, quickly at first, but then saturating: the value at 1 TeV is about a factor of two smaller than at low energy, while the value at 14 TeV is only about 10% smaller still. In terms of our simple model in which the matter distribution always has the form factor form and is parameterized by , and this corresponds to at the Tevatron and at the LHC. Not only would this introduce an energy dependence in , but the values imply a different for hard partonic interactions than that derived from the measured elastic slope parameter, a point that we will return to below.
Note that both these sources of energy dependence would act in the right direction to evade the potential consistency constraints at the LHC. Allowing and/or to increase with energy would move the model towards the allowed region in Fig. 6.
The CDF collaboration have published measurements of the doubleparton scattering cross section[31, 32]. As pointed out in Ref. [33] the quantity called there is not the effective cross section as it is usually defined,
(22) 
but is related to the latter by a small correction. Using the value of this correction estimated in Ref. [34], we obtain mb. In our form factor model, this corresponds to . It is interesting to note that this value is close to the one predicted by the analysis of Refs. [29, 30] mentioned earlier. Again, this value is inconsistent with the assumption that the hard scatters “see” a form factor matter distribution derived from the elastic slope parameter. Recall from our earlier discussion that we do not expect significant qualitative differences for other models of the matter distribution in the proton, merely some distortions of the parameterspace plane.
Improved analyses of these and other observables are under way and, once completed, in particular with a simulation of the final state of , will provide strong constraints on the values of the parameters and in our model.
5 Conclusions
The connections between our underlying event model and the total protonproton crosssection have been discussed. Requiring consistency of the model up to LHC energies imposes constraints on the allowed parameter values, for a given range of possible measurements of at the LHC. Our main result is summarized in Fig. 6, which shows these constraints for various values of the total cross section at the LHC and various inputs to the perturbative cross section calculation. Taking the Tevatron data together with the wide range of possible values of considered at LHC, no allowed set of parameters (, ) remains for our simple model.
This would imply that soft and hard scatters see different matter distributions as a function of impact parameter and/or that the parameters of our model are energy dependent. The measurement of doubleparton scattering at the Tevatron supports the idea that hard scatters see a more dense matter distribution than is implied by the slope of the elastic cross section. Various phenomenological models also predict such effects.
With steadily improving data from the Tevatron, more detailed phenomenological analyses being performed and the prospect of data from the LHC, there is a real hope that the various simplifying assumptions that go into our model will be tested to the limit and we will discover where, if anywhere, more detailed understanding of the dynamics of underlying event physics is needed. The biggest unknown in our analysis is the total cross section at the LHC. With even a first imprecise measurement of this cross section, we could strengthen our parameter space analysis considerably.
Acknowledgements
We are grateful for discussions with Stefan Gieseke and Jeff Forshaw. This work was supported in part by the European Union Marie Curie Research Training Network MCnet under contract MRTNCT2006035606. MB was supported by the Landesgraduiertenförderung BadenWürttemberg.
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