The Standard Model is a consistent, renormalizable quantum field theory that accounts for a wide variety of experimental data over an energy range that encompasses a fraction of an electron volt to about 100 GeV, a range of over twelve orders of magnitude. Initially, the SM was tested at the tree level, but the remarkable agreement between SM predictions and the precision measurements at the CERN LEP collider have tested the SM to at least a part per mille and, more importantly, have established that radiative corrections as given by the SM are essential for agreement with these data. Quite aside from this, the SM also qualitatively explains why baryon and lepton numbers appear to be approximately conserved: with the particle content of the SM, it is not possible to write renormalizable interactions that do not conserve baryon and lepton numbers, so that these interactions (if they exist) must be suppressed by (powers of) some new physics scale.

The SM is nevertheless incomplete. Experimental arguments in support of this are:

• E1 The solar and atmospheric neutrino data, interpreted as neutrino oscillations, strongly suggest neutrinos have mass.

• E2 Observations, starting with Zwicky in 1933 and continuing to this day with studies of the fluctuations in the spectrum of the relic microwave background from the Big Bang, have established the existence of cold dark matter in the Universe for which there is no candidate in the SM.

• E3 Observations of type Ia supernovae at large red shifts as well as the cosmic microwave background radiation both suggest that the bulk of the energy of the Universe resides in a novel form dubbed “dark energy”. This could be the cosmological constant first introduced by Einstein, or something more bizarre.

• E4 Gravity exists.

We saw in Chapter 3 how the Wess–Zumino model could be formulated in terms of the fields S, φL, and the auxiliary field F, which transform into each other under a supersymmetry transformation. Here, we simply “pulled a Lagrangian out of a hat”, and verified by brute force that (at least the free part of) this Lagrangian led to a supersymmetric action. While this example was instructive, it provided no guidance as to how to write down other more complicated supersymmetric theories. We alluded, however, to the fact that we could think of the fields, S, φL, and F as the components of a single entity, a chiral superfield.

The superfield formalism provides a convenient way to formulate general rules for the construction of supersymmetric Lagrangians, even for theories with non-Abelian gauge symmetry that are the foundation of modern particle physics. The superfield calculus that we develop in this and succeeding chapters will provide us with a constructive procedure for writing down theories that are guaranteed to be supersymmetric. This procedure will ultimately be used to write down the simplest supersymmetric extension of the Standard Model. This theory, augmented with suitable soft supersymmetry breaking terms, is known as the Minimal Supersymmetric Standard Model, or MSSM.

Superfields

To begin, we would like to somehow combine the fields S, φL, and F into a single “superfield”, in much the same way that the neutron and proton fields are combined into a single “nucleon” field in the isospin formalism.

The 1970s witnessed the emergence of what has become the Standard Model (SM) of particle physics. The SM describes the interactions of quarks and leptons that are the constituents of all matter that we know about. The strong interactions are described by quantum chromodynamics (QCD) while the electromagnetic and the weak interactions have been synthesized into a single electroweak framework. This theory has proven to be extremely successful in describing a tremendous variety of experimental data ranging over many decades of energy. The discovery of neutral currents in the 1970s followed by the direct observation of the W and Z bosons at the CERN SpS collider in the early 1980s spectacularly confirmed the ideas underlying the electroweak framework. Since then, precision measurements of the properties of the W and Z bosons at both e+e− and hadron colliders have allowed a test of electroweak theory at the 10−3 level. QCD has been tested in the perturbative regime in hard collision processes that result in the breakup of the colliding hadrons. In addition, lattice gauge calculations allow physicists to test non-perturbative QCD via predictions for the observed properties of hadrons for which there is a wealth of experimental information.

Gauge invariance

One of the most important lessons that we have learned from the SM is that dynamics arises from a symmetry principle. If we require the Lagrangian density to be invariant under local gauge transformations, we are forced to introduce a set of gauge potentials with couplings to elementary scalar and fermion matter fields that, apart from an overall scale, are completely determined by symmetry principles.

In the present chapter we shall discuss the consequences the finite number of particles have in the phenomenon of pairing phase transition in atomic nuclei. Finite size effects give rise to fluctuations of the pairing gap and thus of the correlation length (order parameter) ξ. Because ξ is much larger than the size of the nucleus, it comes as no surprise that in describing the phase transition in the small-particle superconductors one doesn't need the non-analytic functions necessary to account for the condensation in infinite systems. On the other hand, the phenomena in both systems are closely related and, in a system like the nucleus, we have the possibility of studying the transition in terms of the spectrum of individual states. Thus the transition from a pair-correlated to a normal system with increasing angular momentum involves the coupling between rotational bands associated with the ground state and with excited (few quasiparticle) states.

Because all the transitions we shall treat are connected with level crossings at zero temperature, it is more appropriate to talk about quantal phase transitions (see Sachdev (1999)).

The variation of the moment of inertia I of rotational bands with angular momentum provides one clue to the variation of the pairing gap with angular momentum. This is because the moment of inertia has a simple monotonic dependence on Δ. In characterizing a superfluid nucleus the moment of inertia I of the rotational bands and the energy of the lowest non-collective excitations 2Ev play a central role.

The Hartree–Fock mean field is represented by a static potential. Non-locality may be approximated by a momentum dependence but the potential is energy independent. In general, independent particle motion is renormalized by coupling to more complicated degrees of freedom. Such couplings often involve a time delay and introduce an energy dependence into the single-particle motion. For example the effective interaction of two nucleons mediated by coupling to a surface vibration has an energy dependence related to the frequency of the vibrational mode.

The state of motion of a nucleon in a nucleus may change by a core polarization process where it promotes a nucleon from a state in the Fermi sea to a state above the Fermi surface as illustrated in Fig. 9.1 (see also Fig. 8.3(b)), or by an inelastic collision as illustrated in Fig. 8.10. This is an example of the doorway phenomenon, the states containing a nucleon and a vibration being the doorway states. The original formulation of the concepts of doorway state can be found in Block and Feshbach (1963). The review by Feshbach (1974) contains details of subsequent developments.

Doorway states

Through the coupling introduced in equation (8.24) a particle can set the nuclear surface into vibration. Such process can be repeated, the particle interacting a second time with the surface and reabsorbing the vibration (see Fig. 9.2).

Pairing in nuclei, superconductors, liquid 3He and neutrons stars

If one sweeps a magnetic field through a metallic ring (e.g. a ring made out of lead) immersed in liquid helium (T ∼ 4 K) it induces a current which does not show any measurable decrease for a year, and a lower bound of 105 years for its characteristic decay time has been established using nuclear resonance to detect any slight decrease in the field produced by the circulating current (File and Mills (1963)). If a torus-shaped vessel filled with liquid helium below the critical temperature Tc = 2.17 K (known as He II) and packed with porous material, which provides very narrow capillary channels, is rotated around its axis of symmetry and then brought to rest, the liquid continues to flow (Reppy and Depatie (1964)), showing no reduction in the angular velocity over a twelve-hour period, and indicating that He II can flow without dissipation. Using an adiabatic cooling apparatus, Osheroff et al. (1972a,b) found two anomalies in the pressure–time curve of liquid 3He, when the volume was changed at a constant rate. At the critical temperature Tc = 2.7 mK the slope of the curve suffered a discontinuity, and at about Tc = 1.8 mK there was a singularity involving hysteresis (see also Osheroff (1997) and Lee (1997)).

In some circumstances the nucleus acts as a liquid and in others like an elastic solid. In general it responds elastically to sudden forces, and it flows plastically over longer periods of time (Bertsch (1980, 1988)). Examples of this behaviour are giant resonances and low-lying collective surface vibrations respectively. In the first case, as we shall see in Section 8.3, pairing plays no role, at least in the case of nuclei lying along the valley of stability. The nuclear single-particle states change their shape but the occupation numbers do not change. The energy of a giant resonance in a nucleus is of the order of the energy difference between major shells (ħω ≈ 41/A1/3 MeV, ≈ 7 MeV, for medium heavy nuclei), a quantity which is much larger than the pairing gap Δ ≈ 1–1.5 MeV. Giant resonances are fast modes, the collective motion is dominated by mean-field effects and the rigidity is provided by the mean field (Bortignon, Bracco and Broglia (1998)). On the other hand, low-energy surface modes are associated with particle–hole excitations which are of the order of the pairing gap. Pairing plays a dominant role and the collective states are coherent linear combinations of two-quasiparticle excitations. The situation is, however, different in the case of exotic nuclei, where the last nucleons are very weakly bound. Nucleon spill out makes these systems particularly polarizable leading to ‘pigmy resonances’, whose properties can be influenced by pairing (Frascaria et al. (2004), see also last paragraph of Chapter 6).

General background

As already mentioned in previous chapters, the nuclear structure exhibits many similarities with the electron structure of metals. In both cases, one is dealing with systems of fermions which may be characterized in a first approximation in terms of independent particle motion. However in both systems, important correlations in the particle motion arise from the action of the forces between particles. In particular, it is well established that nucleons moving close to the Fermi energy in time-reversal states have the tendency to form Cooper pairs which eventually condense (Bohr, Mottelson and Pines (1958), Bohr and Mottelson (1975)). This phenomenon, which has its parallel in low-temperature superconductivity, modifies the structure of nuclei in an important way. In particular it influences the occupation numbers of single-particle levels around the Fermi surface (Chapter 3), the moment of inertia of deformed nuclei (Chapter 3), the lifetime of alpha and cluster decay and fission processes (Chapter 7), the depopulation of superdeformed configurations (Chapter 6) and the cross-sections of two-nucleon transfer reactions (Chapter 5).

While one does not expect the transition between the normal and the superfluid phases of the atomic nucleus to be sharp because of finite size effects and the central role played by fluctuations (see Chapter 6), there is a strong analogy between phenomena in nuclei and the corresponding phenomena in bulk superconductors. Spontaneous symmetry breaking is important in both nuclei and superconductors.