Mass and the Higgs field
The standard model of particle physics is an application of quantum field theory, and the latter holds that the fundamental structure of the physical world consists of quantum fields on space-time. Within quantum field theory, particles are represented as localised excitation states of the underlying quantum fields.
One of the fields postulated by the standard model is the Higgs field. The excitation states of the Higgs field are Higgs bosons. Whilst the Higgs field permeates all of space, Higgs bosons are localised excitations of that field, and at the energy levels available in a universe 14 billion years old, these are difficult to produce.
According to modern cosmology, the Higgs field dropped into its 'vacuum' state (i.e., its lowest energy state) when the universe was only 10-11 s old. However, the potential energy function of the Higgs field is such that its lowest energy state corresponds to a non-zero value of the Higgs field. This value is referred to as the vacuum expectation value of the Higgs field.
The Higgs field is represented to interact with all the quarks and all the leptons (e.g. electrons) in the universe. When the universe was younger than 10-11 s, the quarks and leptons were believed to be massless. Since the time at which the Higgs field dropped into its vacuum state, the non-zero vacuum expectation value of the Higgs field is considered to be responsible for the masses of the quarks and leptons.
The Higgs field is also a self-interacting field, so the Higgs field is considered to be responsible for the mass of the Higgs boson itself.
The statistics of Higgs detection
For example, Figure 1 depicts the number of gamma-gamma detection events as a function of their energy. The red dotted line plots the ‘background’, which in this context is the number of expected gamma-gamma events, as a function of energy, if the Higgs boson hadn’t been produced.
Assuming there is no Higgs production, at each energy level there is a normal (‘Gaussian’) distribution over the number of detection events (see Figure 2). This distribution has a standard deviation (‘sigma’), and by taking integer multiples of sigma, confidence bars can be plotted either side of the red dotted line in Figure 1.
This approach enables one to estimate the probability of a false positive. Thus, if the number of detection events at a particular energy is outside the 3sigma bars, it means that the probability of that result being produced by the play of chance alone is less than 0.3%. By requiring a result to be established at the 5sigma level, this means that the probability of it being a false positive is less than 0.0001%.
A couple of other points should be noted from Figure 1. Firstly, particle physicists use the term ‘luminosity’ to refer to the flux, and 'integrated luminosity' to refer to the ‘fluence’. The latter is the total number of incident particles per unit area over the course of the experiment.
The standard unit of integrated luminosity in use at CERN is the inverse femtobarn (fb-1). A barn (b) is 10-24 cm2, and a femtobarn is 10-15 b. Thus, an integrated luminosity of 5.3 fb-1 means that there was a fluence of 5.3 particles per femtobarn of area.
The number of detected events, (in the case of Figure 1 the bump at 125 GeV), is dubbed the ‘signal strength’. Now, in general, the number of reactions per unit fluence is called the ‘cross-section’ of a reaction, and is specified in units of area. Thus, by multiplying the fluence (integrated luminosity) with the cross-section for Higgs production, the number of Higgs particle production events can be estimated.
However, only a fraction of the Higgs particles will decay into pairs of gamma-rays, and this fraction is specified by the so-called ‘branching ratio’. Thus, the number of detection events N in a particular channel will be the product of the fluence F with the Higgs production cross-section C and the branching ratio R for that channel:
N = F x C x R
Given the experimentally ascertained signal strength, and the known fluence, the quantum field theory for the Higgs field must supply a consistent Higgs production cross-section and branching ratio.