Landshoff-Nachtmann model

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Peter Landshoff (2008), Scholarpedia, 3(11):6812. doi:10.4249/scholarpedia.6812 revision #148548 [link to/cite this article]
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Curator: Peter Landshoff

Figure 1: Total cross sections for proton-proton and proton-antiproton collisions as a function of the centre-of-mass energy.

Experiment finds that, while at small centre-of-mass energy \(E\) total cross sections may decrease with increasing \(E\ ,\) at sufficiently high \(E\) they rise steadily. Figure 1 shows this for proton-proton and proton-antiproton collisions, but all other total cross sections behave similarly, for example pion-proton. Particles such as protons and pions are made of quarks, and the interactions between quarks that determine these total cross sections are known to be mediated by gluon exchange, in much the same way as the electromagnetic force is mediated by photon exchange. However, the electromagnetic force is weak and therefore electromagnetic interactions may be calculated as a power-series expansion in powers of the strength of the force, but this is not true for the exchanges that generate total cross sections. The Landshoff-Nachtmann model seeks to overcome this problem.


Theoretical background

In 1935 the Japanese physicist Hideki Yukawa predicted that there must be a particle, now known as the pion, which would transmit the strong interaction. The pion was duly discovered more than ten years later. However, we now know that although pion exchange is an important component of the static force, when the force acts between a pair of particles moving with high relative energy a very large number of particles collaborate in transmitting it. The theory needed to handle such particle exchanges is known as Regge theory, developed from the initial work of the Italian physicist Tullio Regge.

Regge theory is based on quite sophisticated mathematics. An account may be found in a book by Donnachie, Dosch, Landshoff and Nachtmann (2002). Although the theory is not simple, as far as total cross sections are concerned its output is simple: for centre-of-mass energies more than a few GeV the most important exchanges of particles listed in the data tables are \(\rho,\omega,f_2\) and \(a_2\ ,\) and all of these contribute to total cross sections terms that behave approximately as \(1/E\ .\) That is, their contributions decrease as \(E\) increases.

Therefore, to explain the rise of cross sections with increasing \(E\ ,\) such as is seen in Figure 1, we must introduce some additional exchange, not associated with any particle listed in the data tables. This new object was named after the Russian physicist Isaac Pomeranchuk. It was originally called the pomeranchukon, but this was later abbreviated to pomeron. During the 1960s it was found that, with the inclusion of the pomeron, Regge theory provides a very successful description of a huge quantity of experimental data. This was summarised by Collins (1977) in a classic book, which was published in 1977. However, the phenomenology appeared to be complicated. It was not until the 1980s that it became apparent that the reason for this was that the early data were at comparatively low energies. When higher-energy data became available, the phenomenology became much simpler (Donnachie et al. 2002). Meanwhile, quantum chromodynamics (QCD) had been discovered in the early 1970s. It was natural to try to explain the pomeron in terms of QCD, and first attempts to do so were made by Low (1975) and by Nussinov (1975).

These attempts were refined over the years, notably by Cheng and Wu (1987) and by Lipatov and his collaborators (1997). However, most of the recent work is based on what is known as perturbative QCD, which calculates physical quantities in powers of the strength of the force. But total cross sections are so large that the force must be too strong for such an expansion to be valid. So Landshoff and Nachtmann (1987) began, in the late 1980s the very difficult task of modelling it through nonperturbative QCD. Even now, we still cannot claim that we have more than a rough description of the pomeron in terms of QCD.

Properties of the pomeron

It is customary to introduce a variable \(s\ ,\) which is just the square of the centre-of-mass energy, so that \( E=\sqrt s. \)

So the exchanges \(\rho,\omega,f_2\) and \(a_2\) each contribute a term to total cross sections that behaves approximately as \(s^{-1/2}\ .\) To fit the rise seen in the total cross sections, we need the pomeron-exchange term to behave as a power of \(s\) a little greater than 0.08. See Figure 2.

Figure 2: Total cross sections fitted to contributions from pomeron exchange, together with \(\rho,\omega,f_2\) and \(a_2\) exchange.

Two features of the pomeron may be seen in these fits. The first is that when we replace a particle with its antiparticle, \(p\to\bar p\) or \(\pi ^+\to\pi ^-\ ,\) the contribution from the pomeron to the total cross section remains the same. The same is true also for the contributions from \(f_2\) and \(a_2\) exchange, while for \(\rho\) and \(\omega\) exchange the contribution changes sign when we go from particle to antiparticle. That is, the latter exchanges are responsible for the difference between the cross sections shown in each of the two plots in Figure 2. According to the theory, whether or not an exchange changes sign when we replace a particle with its antiparticle is determined by a property known as its charge parity \(C\ :\) if \(C=-\) it does, and if \(C=+\) it does not. So the pomeron has \(C=+\ .\)

According to the simple quark model, a proton consists essentially of three quarks \(uud\ ,\) and an antiproton of three antiquarks \(\bar u\bar u\bar d\ ,\) so the pomeron couples with equal strength to quarks and antiquarks. Data from deuteron-proton scattering give information on the neutron-proton total cross section, and these are consistent with having the same pomeron-exchange contribution as the \(pp\) total cross section (Donnachie et al. 2002). But in the quark model the neutron is \(ddu\ ,\) so the pomeron seems to couple equally to \(u\) and \(d\) quarks.

The second feature seen from the fits is that the contribution from pomeron exchange to the \(\pi^{\pm}p\) cross sections is close to \(2/3\) that for the \(pp\) or \(\bar pp\) cross sections. In the simple quark model, \(\pi ^+\) is \(u\bar d\) and \(\pi ^-\) is \(d\bar u\ .\) So the \(2/3\) finding amounts to saying that, to a first approximation, the pomeron couples to the separate single quarks and antiquarks, with only a small correction from couplings simultaneously to two or more quarks. This is known as the additive-quark rule (Levin and Frankfurt, 1965; Lipkin and Scheck, 1966; Kokkedee and Van Hove, 1966).

The Landshoff-Nachtmann Model

According to QCD, the basic mechanism for generating the force between quarks is the exchange of a gluon. As pomeron exchange does not correspond to the exchange of any of the particles listed in the data tables, a natural guess is that it is somehow related to gluon exchange. A single gluon does not have \(C=+\ ,\) nor indeed other properties of pomeron exchange that have not been explained here, such as neutral colour.

The simplest viable model for pomeron exchange is that it corresponds to the exchange of 2 gluons. As has been explained above, this exchange cannot be calculated from perturbative QCD. In the Landshoff-Nachtmann model a key assumption is that the gluons that are exchanged can only propagate a certain distance \(a\) through the nonperturbative vacuum. The origin of \(a\) is supposed to be the mysterious so-called confinement mechanism, which somehow prevents free quarks from escaping from a reaction.

To explain the additive-quark rule, one needs both of the two gluons that are exchanged to prefer to couple to the same quark in each proton, rather than to two different ones: see Figure 1. Calculation shows that this is achieved if

\( a^2\ll R^2 \)

where \(R\) is the radius of the proton. To understand this in fairly intuitive terms, work in a relativistic frame where the two initial colliding protons are moving in opposite directions. The two gluons can be exchanged only if the transverse separation between the two quarks to which they are attached is less than \(a\ .\) In each of the two protons, the quarks that are spectators in Figure 1 are separated from the active quark by a distance that on average is order \(R\ .\) So if \(R\) is large compared with \(a\) each spectator quark is likely to be at a distance of order \(R\) from any of the quarks belonging to the other proton and so a gluon is unlikely to be able to jump across from it.

Figure 3: Coupling of two gluons to the three-quark systems that make up two colliding protons.

In fact Figure 1 is too simple to be a realistic model for pomeron exchange, if only because it turns out to correspond to a constant contribution to the \(pp\) total cross section, rather than one that rises with \(s\ .\) To reproduce the observed \(s^{0.08}\ ,\) it must be that the two gluons interact with the vacuum, or with each other, on their way across from one proton to the other. Calculating such effects is extremely difficult (see Forshaw and Ross, 1997).

However, it turns out that the Landshoff-Nachtmann model makes the pomeron couple to single quarks similarly to a \(C=+\) photon, and this provides a very simple and successful description of the angular dependence of the differential cross section for \(pp\) elastic scattering (see Donnachie et al., 2002).

As a final remark, note the discrepancy (Amos et al., 1989; Abe et al., 1994) between the data points at the highest energy shown in Figure 1. As is seen in Figure 2, the simple fit described above favours the lower ones. If the upper data point should turn out to be correct, this indicates the onset of an additional rapidly-rising contribution to the total cross section. There is reason to believe (Landshoff, 2007) that this term rises with \(s\) as a power close to 0.4. It is given the name hard-pomeron exchange.


  • F Abe et al, Physical Review D50 (1994) 5550
  • N Amos et al, Physical Review Letters 63 (1989) 2784
  • H Cheng and TT Wu, Expanding protons, MIT Press (1987)
  • PDB Collins, An Introduction to Regge Theory, Cambridge University Press (1977)
  • S Donnachie, G Dosch, P Landshoff, and O Nachtmann, Pomeron physics and QCD, Cambridge University Press (2002)
  • JR Forshaw and DA Ross, Quantum Chromodynamics and the Pomeron, Cambridge University Press (1997)
  • JJJ Kokkedee and L Van Hove, Nuovo Cimento A42 (1966) 711
  • PV Landshoff and O Nachtmann, Zeitschrift f{\"u}r Physik, C35 (1987) 405.
  • EM Levin and L Frankfurt, Journal of Experimental and Theoretical Physics Letters 2 (1965) 65
  • LN Lipatov, in Perturbative QCD, AH Mueller, (editor), (World Scientific, Singapore, 1989)
  • HJ Lipkin and F Scheck, Physical Review Letters 16 (1966) 71
  • FE Low, Physical Review D12 (1975) 163
  • S Nussinov, Physical Review Letters 34 (1975) 1286

Recommended reading

  • S Donnachie, G Dosch, P Landshoff, and O Nachtmann, Pomeron physics and QCD, Cambridge University Press (2002)
  • JR Forshaw and DA Ross, Quantum Chromodynamics and the Pomeron, Cambridge University Press (1997)
  • PV Landshoff, (2007)

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