Hadron Reactions

By defining the structure of a hadron as its tendency to undergo various reactions, S-matrix theory gives the concept of structure an essentially dynamic connotation. At the same time, this notion of structure is in perfect agreement with the experimental facts. Whenever hadrons are broken up in high-energy collision experiments, they disintegrate into combinations of other hadrons; thus they can be said to ‘consist’ potentially of these hadron combinations. Each of the particles emerging from such a collision will, in turn, undergo various reactions, thus building up a whole network of events which can be photographed in the bubble chamber. The picture on page 267 and the ones in Chapter 15 are examples of such net- works of reactions.

Although it is a matter of chance which network will arise in a particular experiment, each network is nevertheless structured according to definite rules. These rules are the conservation laws mentioned before; only those reactions can occur in which a well-defined set of quantum numbers is conserved. To begin with, the total energy has to remain constant in every reaction. This means that a certain combina- tion of particles can emerge from a reaction only if the energy carried into the reaction is high enough to provide the required masses. Furthermore, the emerging group of .particles must collectively carry exactly the same quantum numbers that have been carried into the reaction by the initial particles. For example, a proton and a X-, carrying a total electric charge of zero, may be dissolved in a collision and rearranged to emerge as a neutron plus a n”, but they cannot emerge as a neutron and a z+, as this pair would carry a total charge of +1.

The hadron reactions, then, represent a flow of energy in which particles are created and dissolved, but the energy can only flow through certain ‘channels’ characterized by the quantum numbers conserved in the strong interactions. In S-matrix theory, the concept of a reaction channel is more fundamental than that of a particle. It is defined as a set of

quantum numbers which can be carried by various hadron combinations and often also by a single hadron. Which com- bination of hadrons flows through a particular channel is a matter of probability but depends, first of all, on the available energy. The diagram opposite, for example, shows an inter- action between a proton and a n- in which a neutron is formed as an intermediate state. Thus, the reaction channel is made up first by two hadrons, then by a single hadron, and finally by the initial hadron pair. The same channel can be made up, if more energy is available, by a A-K0 pair, a Z–K+ pair, and by various other combinations.

The notion of reaction channels is particularly appropriate to deal with resonances, those extremely short-lived hadron states which are characteristic of all strong interactions. They are such ephemeral phenomena that physicists were first reluctant to classify them as particles, and today the clarification of their properties still constitutes one of the major tasks in experimental high-energy physics. Resonances are formed in hadron collisions and disintegrate almost as soon as they come into being. They cannot be seen in the bubble chamber, but can be detected due to a very special behaviour of reaction probabilities.

The probability for two colliding hadrons to undergo a reaction-to interact with one another-depends on the energy involved in the collision. If the amount of this energy is modified, the probability will also change; it may increase or decrease with increasing energy, depending on the details of the reaction. At certain values of energy, however, the reaction probability is observed to increase sharply; a reaction is much more likely to occur at these values than at any other energy. This sharp increase is associated with the formation of a short-lived intermediate hadron with a mass corresponding to the energy at which the increase is observed.

The reason why these short-lived hadron states are called resonances is related to an analogy that can be drawn to the well-known resonance phenomenon encountered in connection with vibrations. In the case of sound, for example, the air in a cavity will in general respond only weakly to a sound wave coming from outside, but will begin to ‘resonate’, or vibrate very strongly, when the sound wave reaches-a certain frequency called the resonance frequency. The channel of a

hadron reaction can be compared to such a resonant cavity, since the energy of the colliding hadrons is related to the frequency of the corresponding probability wave. When this energy, or frequency, reaches a certain value the channel begins to resonate; the vibrations of the probability wave suddenly become very strong and thus cause a sharp increase in the reaction probability. Most reaction channels have several resonance energies, each of them corresponding to the mass of an ephemeral intermediate hadron state which is formed when the energy of the colliding particles reaches the resonance value.

In the framework of S-matrix theory, the problem of whether one should call the resonances ‘particles’ or not does not exist. All particles are seen as intermediate states in a network of reactions, and the fact that the resonances live for a much shorter period than other hadrons does not make them funda- mentally different. In fact, the word ‘resonance’ is a very appro- priate term. It applies both to the phenomenon in the reaction channel and to the hadron which is formed during that phenomenon, thus showing the intimate link between particles and reactions. A resonance is a particle, but not an object. It is much better described as an event, an occurrence or a happening.

This description of hadrons in particle physics recalls to mind the words of D. T. Suzuki quoted above:* ‘Buddhists have conceived an object as an event and not as a thing or substance.’ What Buddhists have realized through their mystical experience of nature has now been rediscovered through the experiments and mathematical theories of modern science.

In order to describe all hadrons as intermediate states in a network of reactions, one has to be able to account for the forces through which they mutually interact. These are the strong-interation forces which deflect, or ‘scatter’, colliding hadrons, dissolve and rearrange them in different patterns, and bind groups of them together to form intermediate bound states. In S-matrix theory, as in field theory, the interaction forces are associated with particles, but the concept of virtual particles is not used. Instead, the relation between forces and particles is based on a special property of the S matrix known as ‘crossing’. To illustrate this property, consider the following diagram picturing the interaction between a proton and a II -

If this diagram is rotated through 90°, and if we keep the convention adopted previously,* that arrows pointing down-

wards indicate antiparticles, the new diagram will represent a reaction between an antiproton (j3 and a proton (p) which emerge from it as a pair of pions, the n+ being the antiparticle of the n- in the original reaction. The ‘crossing’ property of the S matrix, now, refers to the fact that both these processes are described by the same S-matrix element. This means that the two diagrams represent merely two different aspects, or ‘channels’, of the same reaction.** Particle physicists are used to switching from one channel to the other in their calculations, and instead of rotating the diagrams they just read them upwards or across from the

In fact, the diagram can be rbtated further, and individual lines can be ‘crossed’ to obtain different processes which are still described by the same 5matrix element. Each element represents altogether six different processes, but only the two mentioned above are relevant for our discussion of interaction forces.

left, and talk about the ‘direct channel’ and the ‘cross channel’. Thus the reaction in our example is read as p+7t+p+n- in the direct channel, and as p+p-+z-+z+ in the cross channel.

The connection between forces and particles is established through the intermediate states in the two channels. In the direct channel of our exampl.e, the proton and the 7c- can form an intermediate neutron, whereas the cross channel can be made up by an intermediate neutral pion (x0). This pion

the intermediate state in the cross channel-is interpreted as the manifestation of the force which acts in the direct channel binding the proton and the rc- together to form the neutron. Thus both channels are needed to associate the forces with particles; what appears as a force in one channel is manifest as an intermediate particle in the other. Although it is relatively easy to switch from one channel to the other mathematically, it is extremely difficult-if at all possible-to have an intuitive picture of the situation. This is because ‘crossing’ is an essentially relativistic concept arising in the context of the four-dimensional formalism of relativity theory, and thus very difficult to visualize. A similar situation occurs in field theory where the interaction forces are pictured as the exchange of virtual particles. In fact, the diagram showing the intermediate pion in the cross channel is reminiscent of the Feynman diagrams picturing these particle exchanges,* and one might say, loosely speaking, that the proton and the Z- interact ‘through the exchange of a fl. Such words are often used by physicists, but they do not fully describe the situation. An adequate description can only be given in terms of direct and cross channels, that is, in abstract concepts which are almost impossible to visualize. In spite of the different formalism, the general notion of an interaction force in S-matrix theory is quite similar to that in field theory. In both theories, the forces manifest themselves as particles whose mass determines the range of the force,** and in both theories they are recognized as intrinsic properties of the interacting particles; they reflect the structure of the particles’ virtual clouds in field theory, and are generated by bound states of the interacting particles in S-matrix theory. The parallel to the Eastern view of forces discussed previously*** applies thus to both theories. This view of interaction forces, furthermore, implies the important conclusion that all known particles must have some internal structure, because only then can they interact with the observer and thus be detected. In

• It should be remembered, however, that S-matrix diagrams are not space-time diagrams but symbolic representations of particle reactions. The switching from one channel to the other takes place in an abstract mathematical space.

the words of Geoffrey Chew, one of the principal architects of S-matrix theory, ‘A truly elementary particle-completely devoid of internal structure-could not be subject to any forces that would allow us to detect its existence. The mere knowledge of a particle’s existence, that is to say, implies that the particle possesses internal structureY2