Free The Higgs Boson: Searching for the God Particle by Scientific American Editors

renormalization would not solve it. New techniques had to be devised.
    An important idea was introduced in 1963 by Feynman: it is the notion of a
"ghost" particle, a particle added to a theory in the course of a calculation that vanishes when the calculation is finished.
It is known from the outset that the ghost particle is fictitious, but its use can be justified if it never appears in the final state. This can be ensured by making certain the total probability of producing a ghost particle is always zero.
    Among theoretical groups that continued work on the Yang-Mills theory the ghost-particle method was taken seriously only at the University of Utrecht, where I was then a student.
Martin J. G. Veltman, my thesis adviser,
together with John S. Bell of the E uropean Organization for Nuclear Research
(CERN) in Geneva, was led to the conclusion that the weak interactions might be described by some form of the Yang-Mills theory. He undertook a systematic analysis of the renormalization problem in the modified Yang-Mills model (with massive charged fields), examining each class of Feynman diagrams in turn. The diagrams having no closed loops were readily shown to make only finite contributions to the total interaction probability. The diagrams with one loop do include infinite terms, but by exploiting the properties of the ghost particles it was possible to make the positive infinities and the negative ones cancel exactly.
    As the number of loops increases, the number of diagrams rises steeply; moreover,
the calculations required for each diagram become more intricate. To assist in the enormous task of checking all the two-loop diagrams a computer program was written to handle the algebraic manipulation of the probabilities. The output of the program is a list of the coefficients of the infinite quantities remaining after the contributions of all the diagrams have been summed. If the infinities are to be expunged from the theory,
the coefficients must without exception be zero. By 1970 the results were known and the possibility of error had been excluded; some infinities remained.
    The failure of the modified Yang-Mills theory was to be blamed not on any defect in the Yang-Mills formulation itself but rather on the modifications.
The masses of the charged fields had to be put in "by hand" and as a result the invariance with respect to local isotopic-spin rotations was not quite perfect. It was suggested at the time by the Russian investigators L. D. Faddeev,
V. N. Popov, E. S. Fradkin and I. V.
Tyutin that the p ure Yang-Mills theory,
with only massless fields, could indeed be renormalized. The trouble with this theory is that it not only is unrealistic but also has long-range fields that are difficult to work with.
    In the meantime another new ingredient for the formulation of gauge theories had been introduced by F. Englert and Robert H. Brout of the University of Brussels and by Peter Higgs of the University of Edinburgh. They found a way to endow some of the Yang-Mills fields with mass while retaining exact gauge symmetry. The technique is now called the Higgs mechanism.
    The fundamental idea of the Higgs mechanism is to include in the theory an extra field, one having the peculiar property that it does not vanish in the vacuum.
One usually thinks of a vacuum as a space with nothing in it, but in physics the vacuum is defined more precisely as the state in which all fields have their lowest possible energy. For most fields the energy is minimized when the value of the field is zero everywhere, or in other words when the field is "turned off. " An electron field, for example, has its minimum energy when there are no electrons. The Higgs field is unusual in this respect. Reducing it to zero costs energy; the energy of the field is smallest when the field has some uniform value greater than zero.
    The effect of the Higgs field is to provide a frame of reference in which the orientation of the isotopic-spin arrow can be