Diffractive and non-diffractive wounded nucleons and final states in pA collisions111 Work supported in part by the MCnetITN FP7 Marie Curie Initial Training Network, contract PITN-GA-2012-315877, and the Swedish Research Council (contracts 621-2012-2283 and 621-2013-4287)
We review the state-of-the-art of Glauber-inspired models for estimating the distribution of the number of participating nucleons in and collisions. We argue that there is room for improvement in these model when it comes to the treatment of diffractive excitation processes, and present a new simple Glauber-like model where these processes are better taken into account. We also suggest a new way of using the number of participating, or wounded, nucleons to extrapolate event characteristics from collisions, and hence get an estimate of basic hadronic final-state properties in collisions, which may be used to extract possible nuclear effects. The new method is inspired by the Fritiof model, but based on the full, semi-hard multiparton interaction model of P YTHIA 8.
An important topic in the studies of the strong interaction is the understanding of the features of hot and dense nuclear matter. To correctly interpret signals for collective behaviour in high energy nucleus–nucleus collisions, it is necessary to have a realistic extrapolation of the dynamics in collisions. Here experiments on collisions have been regarded as an important intermediate step. As an example refs. [1, 2] have discussed the possibility to discriminate between the dynamics of the wounded nucleon model and that of the Color Glass Condensate formalism in collisions at the LHC.
An extrapolation of results from to and collisions is generally performed using the Glauber formalism [3, 4]. This model is based on the eikonal approximation, where the interaction is driven by absorption into inelastic channels. Elastic scattering is then the shadow of absorption, and determined by the optical theorem. The projectile nucleon(s) are assumed to travel along straight lines and undergo multiple sub-collisions with nucleons in the target. The Glauber model has been commonly used in experiments at RHIC and LHC, e.g. to estimate the number of participant nucleons, , and the number of binary nucleon–nucleon collisions, , as a function of centrality. A basic assumption is then that one can compare a or an collision, at a certain centrality with, e.g., /2 or times the corresponding result in collisions (for which = 2). A comparison with a fit to collision data, folded by the distribution in /2 or , can then be used to investigate nuclear effects on various observables.
There are several problems related to such analyses, and in this paper we will concentrate on two of them:
Since the actual impact parameter is not a physical observable, the experiments typically select an observable, which is expected to be strongly correlated with the impact parameter (such as a forward energy or particle flow). This implies that the definition of centrality becomes detector dependent, which, among other problems, also implies difficulties when comparing experimental results with each other and with theoretical calculations.
When the interaction is driven by absorption, shadow scattering (meaning diffraction) can contain elastic as well as single and double diffractive excitation. This is important since experiments at high energy colliders show, that diffractive excitation is a significant fraction of the total cross section, and not limited to low masses (see e.g. [5, 6, 7]). Thus the driving force in Glauber’s formalism should be the absorptive, meaning the non-diffractive inelastic cross section, and not the total inelastic cross section.
In the following we will argue that the approximations normally used in this procedure are much too crude, and we will present a number of suggestions for how they can be improved, both in the way and are calculated and the way event characteristics are extrapolated to get reference distributions. In both cases we will show that diffractive processes play an important role.
In Glauber’s original analysis only elastic scattering was taken into account, but it was early pointed out by Gribov , that diffractive excitation of the intermediate nucleons gives a significant contribution. However, problems encountered when taking diffractive excitation into account have implied, that this has frequently been neglected, also in recent applications (see e.g. the review by Miller et al. ). Thus the “black disk” approximation, and other simplifying treatments, are still frequently used in analyses of experimental results.222 The effects of the black disk approximation have also been discussed in ref. .
A way to include diffractive excitation in a Glauber analysis, using the Good–Walker formalism, was formulated by Heiselberg et al. . It was further developed in several papers (see refs. [11, 12, 13, 14] and further references in there) and is often called the “Glauber–Gribov” (GG) model. In the Good–Walker formalism , diffractive excitation is described as the result of fluctuations in the nucleon’s partonic substructure. When used in impact parameter space, it has the advantage that saturation effects can easily be taken into account, which makes it particularly suited for applications in collisions with nuclei.
The “Glauber–Gribov” model has been applied both to data from RHIC and in recent analyses of data from the LHC, e.g. in refs. [16, 17] However, although this formalism implies a significant improvement of the data analyses, also in this formulation the treatment of diffractive excitation is simplified, as the full structure of single excitation of either the projectile or the target, and of double diffraction, is not taken into account. As we will show in this paper, this simplification causes important problems, and we will here present a very simple model which separates the fluctuations in the projectile and the target nucleons.
To guide us in our investigation of conventional Glauber models we use the DIPSY Monte Carlo program [18, 19, 20], which is based on Mueller’s dipole approach to BFKL evolution [21, 22], but also includes important non-leading effects, saturation and confinement. It reproduces fairly well both total, elastic, and diffractive cross sections, and has also recently been applied to collisions . The DIPSY model gives a very detailed picture of correlations and fluctuations in the initial state of a nucleon, and by combining it with a simple geometrical picture of the distribution of nucleons in a nucleus in its ground state, we can build up an equally detailed picture of the initial states in and collisions. This allows us to gain new insights into the pros and cons of the approximations made in conventional Glauber Models.
The DIPSY program is also able to produce fully exclusive hadronic final states in collisions, giving a reasonable description of minimum bias data from e.g. the LHC . It could, in principle also be used to directly model final states in and , but due to some shortcomings, we will in this paper instead only use general features of these final states to motivate a revival of the old Fritiof model [23, 24] with great similarities with the original ”wounded nucleon” model . (For a more recent update of the wounded nucleon model see ref. .)
For energies up to (and including) those at fixed target experiments at CERN, the particle density at mid-rapidity in collisions is almost energy independent. For higher energies the density increases, and the distribution gets a tail to larger values. However, for minimum bias events with lower , the wounded nucleon model still works with the multiplicity scaling with the number of participating (wounded) nucleons, both at RHIC [27, 28] and LHC . For higher the distributions scale, however, better with the number of binary collisions, indicating the effect of hard parton-parton sub-collisions .
We will here argue that, due to the relatively flat distribution in rapidity of high-mass diffractive processes, absorbed and diffractively excited nucleons will contribute to the (and in principle also ) final states in very similar ways, as wounded nucleons. We will also present preliminary results where we use our modified GG model to calculate the number distribution of wounded nucleons in , and from that construct hadronic final states by stacking diffractive excitation events, on top of a primary non-diffractive scattering, using P YTHIA 8 with its semi-hard multi-parton interaction picture of hadronic collisions.
Although this remarkably simple picture gives very promising results, we find that there is a need for differentiating between diffractively and non-diffractively wounded nucleons. We will here be helped by the simple model mentioned above, in which fluctuations in the projectile and the target nucleon are treated separately. The model involves treating both the projectile and target as semi-transparent disks, separately fluctuating between two sizes according to a given probability. The radii, the transparency and the fluctuation probability is then adjusted to fit the non-diffractive nucleon–nucleon cross section, as well as the elastic, single diffractive and double diffractive cross sections. Even though this is a rather crude model, it will allow us to investigate effects of the difference between diffractively and non-diffractively wounded nucleons.
We will begin this article by establishing in section 2 the framework we will use to describe high energy nucleon–nucleon scattering, with special emphasis on the Good–Walker formalism for diffractive excitation. In section 3 we will then use this framework to analyse the Glauber formalism in general and define the concept of a wounded target cross section. In section 4 we dissect the conventional Glauber models and the Glauber–Gribov model together with the DIPSY model and present some comparisons of the resulting number distributions of wounded nucleons in . In section 5 we then go on to present our proposed model for constructing fully exclusive hadronic final states, and compare the procedure to recent results on particle distributions in collisions from the LHC, before we present conclusions and an outlook in section 6.
2 Dynamics of high energy scattering
2.1 Multiple sub-collisions and perturbative parton–parton interaction
As mentioned in the introduction, at energies up to those at fixed target experiments and the ISR at CERN, the cross sections and particle density, are relatively independent of energy. For collisions with nuclei the wounded nucleon model works quite well , which formed the basis for the development of the Fritiof model . This model worked very well within that energy range, but at higher energies it could not in a satisfactory way reproduce the development of a high tail caused by hard parton-parton interactions. Nevertheless the wounded nucleon model works well for minimum bias events even at LHC energies, if the rising rapidity plateau in collisions is taken into account, although the production of high particles appear to scale better with the number of collisions. These features may be interpreted as signals for dominance of soft interactions, and were the basis for the development of the Fritiof model . This model worked very well within that energy range, but at higher energies, available at colliders at CERN and Fermilab, the effects of (multiple) hard parton–parton sub-collisions became increasingly important, and not so easily incorporated in the Fritiof model.
Today high energy collisions (above 100 GeV) are more often described as the result of multiple partonic sub-collisions, described by perturbative QCD. This picture was early proposed by Sjöstrand and van Zijl , and is implemented in the P YTHIA 8 event generator . This picture has also been applied in other generators such as HERWIG , S HERPA , DIPSY [18, 20], and others. The dominance of perturbative effects can here be understood from the suppression of low- partons due to saturation, as expressed e.g. in the Color Glass Condensate formalism .
2.2 Saturation and the transverse coordinate space
2.2.1 The eikonal approximation
The large cross sections in hadronic collisions imply that unitarity constraints are important, and the elastic amplitude has to satisfy the optical theorem, which with convenient normalisation reads
Here the sum runs over all inelastic channels . In high energy collisions the real part of the elastic amplitude is small, which indicates that the interaction is dominated by absorption into inelastic channels, with elastic scattering formed as the diffractive shadow of this absorption. This diffractive scattering is dominated by small , and the scattered proton continues essentially along its initial direction.
At high energies and small transverse momenta, multiple scattering corresponds to a convolution in transverse momentum space, which is represented by a product in transverse coordinate space. This implies that diffraction and rescattering is more easily described in impact parameter space. In a situation where all inelastic channels correspond to absorption (meaning no diffractive excitation), the optical theorem in eq. (1) implies that the elastic amplitude in impact parameter space is given by
Here represents the probability for absorption into inelastic channels.
If the absorption probability in the Born approximation is given by , then unitarity is restored by rescattering effects, which exponentiates in -space and give the eikonal approximation:
To simplify the notation we introduce the nearly real amplitude . The relation in eq. (2) then gives and . The optical theorem then gives
We note that the possibility of diffractive excitation is not included here. Therefore the absorptive cross section in eq. (3) is the same as the inelastic cross section.
How to include diffractive excitation and its relation to fluctuations will be discussed below in section 2.3. We then also note that diffractive excitation is very sensitive to saturation effects, as the fluctuations go to zero when saturation drives the interaction towards the black limit.
That rescattering exponentiates in transverse coordinate space also makes this formulation suitable for generalisations to collisions with nuclei.
2.2.2 Dipole models in transverse coordinate space
In this paper we will use our implementation of Mueller’s dipole model, called DIPSY , in order to have a model which gives a realistic picture of correlations and fluctuations in the colliding nucleons. In this way we can evaluate to what extent Glauber-like models are able to take such effects into account. The DIPSY model has been described in a series of papers [18, 19, 20] and we will here only give a very brief description. Mueller’s dipole model [21, 22] is a formulation of LL BFKL evolution in impact parameter space. A colour charge is always screened by an accompanying anti-charge. A charge–anti-charge pair can emit bremsstrahlung gluons in the same way as an electric dipole, with a probability per unit rapidity for a dipole ( to emit a gluon in the point , given by (c.f. figure 1)
The important difference from electro-magnetism is that the emitted gluon carries away colour, which implies that the dipole splits in two dipoles. These dipoles can then emit further gluons in a cascade, producing a chain of dipoles as illustrated in figure 1.
When two such chains, accelerated in opposite directions, meet, they can interact via gluon exchange. This implies exchange of colour, and thus a reconnection of the chains as shown in figure 2.
The elastic scattering amplitude for gluon exchange is in the Born approximation given by
BFKL evolution is a stochastic process, and many sub-collisions may occur independently. Summing over all possible pairs gives the total Born amplitude
The unitarised amplitude then becomes
and the cross sections are given by
2.2.3 The Lund dipole model Dipsy
Important non-leading effects in BFKL evolution.
Most essential are those related to energy conservation and running .
Saturation from Pomeron loops in the evolution.
Dipoles with identical colours form colour quadrupoles, which give Pomeron loops in the evolution. These are not included in Mueller’s model or in the BK equation.
Confinement via a gluon mass satisfies -channel unitarity.
It can be applied to collisions between electrons, protons, and nuclei.
Some results for total and elastic cross sections are shown in refs. [35, 36]. We note that there is no input structure functions in the model; the gluon distributions are generated within the model. We also note that the elastic cross section goes to zero in the dip of the -distribution, as the real part of the amplitude is neglected.
2.3 Diffractive excitation and the Good–Walker formalism
In his analysis of the Glauber formalism, Gribov considered low mass excitation in the resonance region, but experiments at high energy colliders have shown, that diffractive excitation is not limited to low masses, and that high mass diffraction is a significant fraction of the cross section also at high energies (see e.g. [5, 6, 7]). Diffractive excitation is often described within the Mueller–Regge formalism , where high-mass diffraction is given by a triple-Pomeron diagram. Saturation effects imply, however, that complicated diagrams with Pomeron loops have to be included, which leads to complicated resummation schemes, see e.g. refs. [38, 39, 40]. These effects make the application in Glauber calculations quite difficult.
High mass diffraction can also be described, within the Good–Walker formalism , as the result of fluctuations in the nucleon’s partonic substructure. Diffractive excitation is here obtained when the projectile is a linear combination of states with different absorption probabilities. This formalism was first applied to collisions by Miettinen and Pumplin , and later within the formalism for QCD cascades by Hatta et al.  and by Avsar and coworkers [43, 36]. When used in impact parameter space, this formulation has the advantage that saturation effects can easily be taken into account, and this feature makes it particularly suited in applications for collisions with nuclei. (For a BFKL Pomeron, the Good–Walker and the Mueller–Regge formalisms describe the same physics, seen from different sides .)
As an illustration of the Good–Walker mechanism, we can study a photon in an optically active medium. For a photon beam passing a black absorber, the waves around the absorber are scattered elastically, within a narrow forward cone. In the optically active medium, right-handed and left-handed photons move with different velocities, meaning that they propagate as particles with different mass. Study a beam of right-handed photons hitting a polarised target, which absorbs photons polarised in the -direction. The diffractively scattered beam is then a mixture of right- and left-handed photons. If the right-handed photons have lower mass, this means that the diffractive beam contains also photons excited to a state with higher mass.
2.3.1 A projectile with substructure colliding with a structureless target
For a projectile with a substructure, the mass eigenstates can differ from the eigenstates of diffraction. Call the diffractive eigenstates , with elastic scattering amplitudes . The mass eigenstates are linear combinations of the states :
The elastic scattering amplitude is given by
and the elastic cross section
The amplitude for diffractive transition to the mass eigenstate is given by
which gives a total diffractive cross section (including elastic scattering)
Consequently the cross section for diffractive excitation is given by the fluctuations:
We note in particular that in this case the absorptive cross section equals the inelastic non-diffractive cross section. Averaging over different eigenstates eq. (3) gives
2.3.2 A target with a substructure
If also the target has a substructure, it is possible to have either single excitation of the projectile, of the target, or double diffractive excitation. Let and be the diffractive eigenstates for the projectile and the target respectively, and the corresponding eigenvalue. (We here make the assumption that the set of eigenstates for the projectile are the same, for all possible target states. This assumption is also made in the DIPSY model discussed above.) The total diffractive cross section, including elastic scattering, is then obtained by taking the average of over all possible states for the projectile and the target. Subtracting the elastic scattering then gives the total cross section for diffractive excitation:
Here the subscripts and denote averages over the projectile and target respectively.
Taking the average over target states before squaring gives the probability for an elastic interaction for the target. Subtracting single diffraction of the projectile and the target from the total in eq. (17) will finally give the double diffraction. Thus we get the following relations:
where and is single diffractive excitation of the projectile and target respectively and is double diffractive excitation. Also here the absorptive cross section, which will be important in the following discussion of the Glauber model, corresponds to the non-diffractive inelastic cross section:
2.3.3 Diffractive eigenstates at high energies
In the early work by Miettinen and Pumplin , the authors suggested that the diffractive eigenstates correspond to different geometrical configurations of the valence quarks, as a result of their relative motion within a hadron. At higher energies the proton’s partonic structure is dominated by gluons. The BFKL evolution is a stochastic process, and it is then natural to interpret the perturbative parton cascades as the diffractive eigenstates (which may also depend on the positions of the emitting valence partons). This was the assumption in the work by Hatta et al.  and in the DIPSY model. Within the DIPSY model, based on BFKL dynamics, it was possible to obtain a fair description of both the experimental cross section [43, 36] and final state properties  for diffractive excitation. In the GG model two sources to fluctuations are considered; first fluctuations in the geometric distribution of valence quarks, and secondly fluctuations in the emitted gluon cascades, called colour fluctuations or flickering. In ref.  it was concluded that the latter is expected to dominate at high energies.
We here also note that at very high energies, when saturation drives the interaction towards the black limit, the fluctuations go to zero. This implies that diffractive excitation is largest in peripheral collisions, where saturation is less effective. This is true both for collisions and collisions with nuclei. (Although diffractive excitation of the projectile is almost zero in central collisions, this is not the case for nucleons in the target.)
3 Glauber formalism for collisions with nuclei
3.1 General formalism
High energy nuclear collisions are usually analysed within the Glauber formalism  (for a more recent overview see ). In this formalism, target nucleons are treated as independent, and any interaction between them is neglected333In the DIPSY model gluons with the same colour can interfere, also when they come from different nucleons. This so-called inter-nucleon swing mechanism was shown  to have noticeable effects in photon–nucleus collisions, but in , especially for heavy nuclei, the effects were less that 5%. We have therefore chosen to ignore such effects in this paper, but may return to the issue in a future publication. . The projectile nucleon(s) travel along straight lines, and undergo multiple diffractive sub-collisions with small transverse momenta. As mentioned in the introduction, multiple scattering, which in transverse momentum space corresponds to a convolution of the scattering -matrices, corresponds to a product in transverse coordinate space. Thus the matrices , for the encounters of the proton with the different nucleons in the target nucleus, factorise:
We denote the impact parameters for the projectile and for the different nucleons in the target nucleus by and respectively, and define . Using the notation in eq. (4), we then get the following elastic scattering amplitude for a proton hitting a nucleus with nucleons:
If there are no fluctuations, neither in the interaction nor in the distribution of nucleons in the nucleus, a knowledge of the positions and the elastic amplitude would give the total and elastic cross sections via the relations in eq. (4):
The inelastic cross section (now equal to the absorptive) would be equal to the difference between these two, in accordance with eq. (3).
Fluctuations in the interaction are discussed in the following subsection. Fluctuations and correlations in the nucleon distribution within the nucleus are difficult to treat analytically, and therefore most easily studied by means of a Monte Carlo, as discussed further in sections 3.4, 4 and 5 below. Valuable physical insight can, however, be gained in an approximation where all correlations between target nucleons are neglected. Such an approximation, called the optical limit, is discussed in section 3.5.
3.2 Gribov corrections. Fluctuations in the interaction
Gribov pointed out that the original Glauber model gets significant corrections due to possible diffractive excitation. In the literature it is, however, common to take only diffractive excitation of the projectile into account, disregarding possible excitation of the target nucleons. In this section we will develop the formalism to account for excitations of nucleons in both projectile and target. We will then see that in many cases fluctuations in the target nucleons will average out, while in other cases they may give important effects. (Fluctuations in both projectile and target will, however, be even more essential in nucleus–nucleus collisions, which we plan to discuss in a future publication.)
3.2.1 Total and elastic cross sections
When the nucleons can be in different diffractive eigenstates, the amplitudes in eq. (21) are matrices , depending on the states for the projectile and for the target nucleon . The elastic amplitude, , can then still be calculated from eq. (21), by averaging over all values for and , with running from 1 to . Thus
When evaluating the averages in these equations, it is essential that the projectile proton stays in the same diffractive eigenstate, , throughout the whole passage through the target nucleus, while the states, , for the nucleons in the target nucleus are uncorrelated from each other. This implies that for a fixed projectile state , the average of the -matrix over different states, , for the target nucleons factorise in eq. (20) or (24). Thus we have
Here () denotes average over projectile (target nucleon) substructures (), while denotes average over the target nucleon positions , an as before . We introduce the following notation for the average of the amplitude over target states:
The amplitude can then be written in the form
where the average is taken over the target nucleon positions and the projectile states, . The total and elastic cross sections in eqs. (24) and (25) are finally obtained from eq. (4). We want here to emphasise that these expressions only contain the first moment with respect to the fluctuations in the target states, , but also all higher moments of the fluctuations in the projectile states, .
To evaluate the -integrated cross sections, we must know both the distribution of the (correlated) nucleon positions, , and the -dependence of the amplitude . The distribution of nucleon positions is normally handled by a Monte Carlo, as will be discussed in section 3.4. When fluctuations and diffractive excitation was neglected in section 3.1, the -dependence of could be well approximated by a Gaussian distribution , corresponding to an exponential elastic cross section . With fluctuations it is necessary to take the unitarity constraint into account, which implies that a large cross section must be associated with a wider distribution. One should then check that after averaging the differential elastic cross section reproduces the observed slope.444In ref.  unitarity is satisfied assuming the slope to be proportional to the fluctuating total cross section .
3.3 Interacting nucleons
3.3.1 Specification of ”wounded” nucleons
The notion of “wounded” nucleons was introduced by Białas, Bleszyński, and Czyż in 1976 , based on the idea that inelastic or collisions can be described as a sum of independent contributions from the different participating nucleons555This idea was also the basis for the Fritiof model , which has been quite successful for low energies.. In ref.  diffractive excitation was neglected, and thus “wounded nucleons” was identical to inelastically interacting nucleons666It was also pointed out that for collisions the number of participant nucleons, , and the number of sub-collisions, , are related, , and a relation between particle multiplicity and the number of wounded nucleons, , is equivalent to a relation to the number of sub-collisions, . Only in collisions is it possible to distinguish a dependence on the number of participating nucleons from a dependence on the number of nucleon–nucleon sub-collisions..
Although the importance of diffractive excitation was pointed out by Gribov already in 1968 , it has, as far as we know, never been discussed whether or not diffractively excited nucleons should be regarded as wounded. These nucleons contribute to the inelastic, but not to the absorptive cross section, as defined in eq. (19).
Diffractive excitation is usually fitted to a distribution proportional to . A bare triple-Pomeron diagram would give , where is the intercept of the Pomeron trajectory, estimated to around 1.2 from the HERA structure functions at small . More complicated diagrams tend, however, to reduce . (In ref.  it is shown that the largest correction is a four-Pomeron diagram, which gives a contribution with .) Fits to LHC data [6, 7] give , but with rather large uncertainties.
If is small, diffractively excited target nucleons can contribute to particle production both in the forward and in the central region. If instead is large, diffraction would contribute mainly close to the nucleus fragmentation region. For , the experimentally favoured value, the contribution in the central region would be suppressed by a factor for collisions at LHC. We conclude that the definition of wounded nucleons should depend critically upon both the experimental observable studied in a certain analyses, and upon the still uncertain -dependence of diffractive excitation at LHC energies. (In section 5.1 we will show that a simple model, assuming similar contributions from absorbed and diffractively excited nucleons actually quite successfully describes the final state in collisions at LHC.)
Below we present first results for the absorbed, non-diffractive, nucleons, followed by results when diffractively excited nucleons are included.
3.3.2 Wounded nucleon cross sections
Absorptive cross section
We first assume that wounded nucleons correspond to nucleons absorbed via gluon exchange, which for large values of would be relevant for observables in the central region, away from the nucleus fragmentation region. Due to the relation , the absorptive cross section in eq. (19) can also be written . We here note that, as the -matrix factorises in the elastic amplitude in eqs. (20) and (24), this is also the case for . This implies that
In analogy with eq. (27) for , also here, when taking the average over the target states , the factors in the product depend only on the projectile state and the positions . We here introduce the notation
This quantity represents the probability that nucleon is absorbed by a projectile in state . Averaging over all values for and , it gives the total absorptive, meaning inelastic non-diffractive, cross section
This expression equals the probability that at least one target nucleon is absorbed.
Cross section including diffractively excited target nucleons
We now discuss the situation when also diffractively excited target nucleons should be counted as wounded. (The case with an excited projectile proton is discussed below.) The probability for a nucleon, , in the nucleus to be diffractively excited is obtained from eq. (18) by adding single and double diffraction:
Adding the absorptive cross section in eq. (19) we obtain the total probability that a target nucleon, , is excited or broken up by either diffraction or absorption,
and we will call such nucleons inclusively wounded (), as opposed to absorptively wounded ().
In analogy with eq. (30) we define by the relation
which gives the probability that the target nucleon is either absorbed or diffractively excited, by a projectile in state . Thus, if these target nucleons are counted as wounded, the cross section is also given by eq. (31), when is replaced by . We note that the expression for the wounded nucleon cross section resembles the total one in eqs. (28) and (25), with replaced by or . Note also that as is determined via eq. (34), when is known including its -dependence. This is not the case for , which contains the average over target states of the square of the amplitude .
Elastically scattered projectile protons
We should note that the probabilities given above include events, where the projectile is elastically scattered, and thus not regarded as a wounded nucleon. The probability for this to happen in an event with diffractively excited target nucleons, is given by the relation ( and denote averages over projectile and target states respectively)
In case these events do not contribute to the observable under study, this contribution should thus be removed. For a large target nucleus, this is generally a small contribution.
3.3.3 Wounded nucleon multiplicity
In the following we let denote either or , depending upon whether or not diffractively excited target nucleons should be counted as wounded.
Average number of wounded nucleons
As denotes the probability that target nucleon is wounded, the average number of wounded nucleons in the target is then (for fixed ) given by , obtained by summing over target nucleons , and averaging also over projectile states and all target positions . Averaging over impact parameters, , is only meaningful, when we calculate the average number of wounded target nucleons per event with at least one wounded nucleon, which we denote . This is obtained by dividing by the probability in eq. (31). Integrating over , weighting by the same absorptive probability, and normalising by the total absorptive cross section (also integrated over ) we get
Note that the total number of wounded nucleons is given by , as the projectile proton should be added, provided the projectile proton is not elastically scattered (in which case all wounded target nucleons have to be diffractively excited).
Multiplicity distribution for wounded nucleons
It is also possible to calculate the probability distribution in the number of wounded target nucleons . For fixed projectile states and target nucleon positions , the probability for target nucleon to be wounded, or not wounded, is and respectively. For fixed the probability distribution in the number of absorbed target nucleons is then given by
Here the sum goes over all subsets of wounded target nucleons, and is the set of the remaining target nucleons, which thus are not wounded. The states of the target nucleons can be assumed to be uncorrelated, and the averages could therefore be taken separately, as in eq. (30). The state and positions or give, however, correlations between the different factors, and these averages must be taken after the multiplication, which gives the result
The distribution in eq. (38) includes the possibility for . As for the average number of wounded nucleons above, to get the normalised multiplicity distribution for events, with , we should divide by the probability in eq. (31). The final distribution is then obtained by integrating over , with a weight given by the same absorption probability. This gives the result
We want here to emphasise that the quantity contains the average of the square of the amplitude , and is therefore not simply determined from the average , which appears in the expression for the total and elastic cross sections in eqs. (28) and (25). This contrasts to the situation for inclusively wounded nucleons, where in eq. (34) actually is directly determined by .
3.4 Nucleus geometry and quasi-elastic scattering
In a real nucleus the nucleons are subject to forces with a hard repulsive core, and their different points are therefore not uncorrelated. In Glauber’s original papers this correlation was neglected, and this approximation is discussed in the subsequent section.
In addition to the suppression of nucleons at small separations, the geometrical structure will fluctuate from event to event. These fluctuations are not only a computational problem, but have also physical consequences. Just as fluctuations in the nucleon substructure can induce diffractive excitation of the nucleon, fluctuations in the nucleus substructure induces diffractive excitation of the nucleus. If the projectile is elastically scattered these events are called quasi-elastic. The fluctuations in the target nucleon positions are also directly reproduced by the Monte Carlo programs mentioned above, and within the Good–Walker formalism the quasi-elastic cross section, , is given by (c.f. eq. (18)):
The average over the target states here includes averaging over all geometric distributions of nucleons in the nucleus, and all partonic states of these nucleons. Note that this expression includes the elastic proton–nucleus scattering (given by ). Some results for quasi-elastic collisions are presented in ref. [46, 9].
3.5 Optical limit – uncorrelated nucleons and large nucleus approximations
Even though the averages in eqs. (24) and (31) factorise, they are still complicated by the fact that all factors are different, due to the different values for the impact parameters. It is interesting to study simplifying approximations, assuming uncorrelated nucleon positions and large nuclei. This is generally called the optical limit. It was used by Glauber in his initial study , and is also described in the review by Miller et al. , for a situation when diffractive excitation is neglected. We here discuss the modifications necessary when diffractive excitation is included, also separating single excitation of projectile and target, and double diffraction.
3.5.1 Uncorrelated nucleons
Neglecting the correlations between the nucleon positions in the target nucleus, the individual nucleons can be described by a smooth density (normalised so that ). In this approximation all factors in eq. (26), which enter the total cross section in eq. (24), are uncorrelated and give the same result, depending only on projectile state and impact parameter and :
3.5.2 Large nucleus
If, in addition to the approximations in eqs. (41) and (42), the width of the nucleus (specified by ) is much larger than the extension of the interaction (specified by ), further simplifications are possible. For the amplitude in eq. (41) we can integrate over , and get the approximation
We have here introduced the notation for the total cross section for a projectile proton in state , averaged over all states for a target proton.
In the same way we get
where is either or , and is the corresponding cross section for a projectile in state .
3.5.3 Total cross section
The total cross section is then finally obtained by averaging over projectile states, , and integrating over impact parameters, :
We note here in particular, that in this approximation the -dependence of is unimportant, and the result depends only on its integral . We also note that to calculate the elastic cross section , which has a steeper -dependence, a knowledge about this dependence is also needed.
Proton-deuteron cross section
For the special case with a deuteron target we then get the result777Although the deuteron has only 2 nucleons, it is very weakly bound, and its wave function is extended out to more than 5 fm. Therefore the large nucleus approximation is meaningful also here.
and with the estimate describing the deuteron wavefunction, we recognise Glauber’s original result.
For a non-fluctuating amplitude, the optical theorem gives a direct connection between the total and elastic cross sections. As the integral over gives the Fourier transform at , we have
Here denotes the Fourier transform of the amplitude . For a Gaussian interaction profile we get
where the slope is a measure of the width of the interaction. As is determined by the squared amplitude, the ratio will be larger for a strong interaction with a short range, than for a weaker interaction with a wider range.
Here denotes the cross section for single diffractive excitation of the projectile proton (i.e. on one side only). For a fluctuating amplitude we then get instead of eq. (48)
The negative term in eq. (52) represents a shadowing effect, which for a deuteron target has one contribution from the elastic proton–nucleon cross section, and another from diffractive excitation. Note in particular, that it is only single diffraction which enters, with an excited projectile but an elastically scattered target nucleon. (This would be particularly important in case of a photon or a pion projectile.)
Larger target nuclei For a larger target higher moments, ( 1, 2, …, A), of the amplitude, averaged over target states, are needed. These moments cannot be determined from the total cross section and the cross section for diffractive excitation. They can be calculated if we know the full probability distribution, , for the amplitude averaged over target states, but for varying projectile states888The average for was estimated from diffractive proton-deuteron scattering in ref. .. In addition also higher moments of the nucleus density, , are needed.
We also note here that the factorisation feature in eq. (21) is not realised in collisions. This implies that also in the optical limit, the -results cannot be directly expressed in terms of the moments .
3.5.4 Wounded nucleon cross sections
Also for cross sections corresponding to wounded (absorptively or inclusively) nucleons, approximations analogous to eqs. (41) and (43) are possible. Integrating the expressions in eq. (44) over , and averaging also over projectile states gives, in analogy with eqs. (45) and (46), the following result
The average in eq. (53) includes averages of all possible powers . For this is just equal to the cross section for (with denoting either absorptively or inclusively wounded), but for higher moments a knowledge of the full probability distribution for is needed, in analogy with eq. (46) for the total cross section. Note, however, that a similar relation is not satisfied for the elastic or total inelastic cross sections, and , which as seen in eq. (18) contain the average over projectile states before squaring.
3.5.5 Average number of wounded nucleons
In eq. (53) represents the probability that a specific target nucleon is wounded, in a collision with a projectile in state at an impact parameter . In the optical limit this probability is the same for all target nucleons. Averaging over projectile states then gives the average number of wounded target nucleons for an encounter at this -value. Dividing by the probability for a “wounded” event, we get the average number of wounded target nucleons per wounded event for this :
Normalising by the probability for absorption in eq. (53), and integrating over with a weight given by the same probability, then gives
with given by eq. (53). As noted above, this needs knowledge of the full probability distribution for .
3.5.6 Multiplicity distribution for wounded nucleons
As in section 3.3, when calculating the full distribution in , it is important to take the average over projectile states after multiplication of the different nucleon absorption probabilities, which gives
Similar to the general result in eq. (38), this expression includes the probability for zero target participants. Normalising by the probability for absorption in eq. (53), and integrating over with a weight given by the same probability, gives finally