# Confinement and the Mass Gap

##### 5 minutes • 954 words

## Table of contents

## Confinement and the Mass Gap

Asymptotic freedom gave a dynamical reason to believe that Yang-Mills was likely responsible for the strong force.

Earlier arguments that quarks should have three colour degrees of freedom meant that attention quickly focussed on the gauge group SU (3) [84, 65].

But the infrared puzzles still remained.

Why are the massless particles predicted by Yang-Mills not seen? Why are individual quarks not seen?

Here things were murky.

Was the SU (3) gauge group broken by a scalar field? Or was it broken by some internal dynamics?

Or perhaps the gauge group was actually unbroken but the flow to strong coupling does something strange. This latter possibility was mooted in a number of papers [84, 207, 208, 65].

This from Gross and Wilczek in 1973:

This idea was slowly adopted over the subsequent year. The idea of dimensional transmutation, in which dimensionless constants combine with the cut-off to give the a physical scale, was known from the 1973 work of Coleman and E. Weinberg [27].

Although they didn’t work with Yang-Mills, their general mechanism removed the most obvious hurdle for a scale-invariant theory to develop a gap.

A number of dynamical explanations were mooted for confinement. But the clearest came only in 1974 with Wilson’s development of lattice gauge theory [214]. This paper also introduced the “Wilson line”.

The flurry of excitement surrounding these developments highlights the underlying confusion, as some of the great scientists of the 20th century clamoured to disown their best work.

For example, in an immediate response to the discovery of asymptotic freedom, and six years after his construction of the electroweak theory [205], Steven Weinberg writes [208]

Not to be outdone, in the same year Gell-Mann offers [65]

It’s not easy doing physics.

## Semi-Classical Yang-Mills

In these lectures, we first described the classical and semi-classical structure of Yang-Mills theory, and only then turned to the quantum behaviour.

This is the logical way, not the historical way.

Our understanding of the classical vacuum structure of Yang-Mills theory started in 1975, when Belavin, Polyakov, Schwartz and Tyupkin discovered the Yang-Mills instanton [14].

Back then, Physical Review refused to entertain the name “instanton”, so they were referred to in print as “pseudoparticles”.

’t Hooft was the first to perform detailed instanton calculations [101, 102], including the measure K(⇢) that we swept under the carpet in Section 2.3.3.

- Among other things, his work clearly showed that physical observables depend on the theta angle.

Motivated by this result, Jackiw and Rebbi [113], and independently Callan, Dashen and Gross [23], understood the semi-classical vacuum structure of Yang-Mills in Section 2.2.

Jackiw’s lectures [115] give a very clear discussion of the theta angle and were the basis for the discussion here. Reviews covering a number of different properties of instantons can be found in [182, 191, 197].

## Magnetic Yang-Mills

Monopoles in SU (2) gauge theories were independently discovered by ’t Hooft [99] and Polyakov [158] in 1974.

The extension to general gauge groups was given in 1977 by Goddard, Nuyts and Olive [80].

This paper includes the GNO quantisation condition that we met in our discussion of ’t Hooft line, and offers some prescient suggestions on the role of duality in exchanging gauge groups. (These same ideas rear their heads in mathematics in the Langlands program.)

The Bogomolnyi trick was introduced in [20]. Prasad and Sommerfeld then solved the resulting equations of motion for the monopole [162], and the initials BPS are now engraved on all manner of supersymmetric objects which have nothing to do with monopoles. (A more appropriate name for BPS states would be Witten-Olive states [217].) Finally, Witten’s Witten effect was introduced in [216].

Excellent reviews of ’t Hooft-Polyakov monopoles, both with focus on the richer BPS sector, can be found in Harvey’s lecture notes [89] and in Manton and Sutcliffe’s book [133]. There are also some TASI lectures [191].

The Nielsen-Olesen vortex was introduced in 1973 [145]. Their motivation came from string theory, rather than field theory. The fact that such strings would confine magnetic monopoles was pointed out by Nambu [142] and the idea that this is a useful analogy for quark confinement, viewed in dual variables, was made some years later by Mandelstam [130] and ’t Hooft [100].

The ’t Hooft line as a magnetic probe of gauge theories was introduced in [103].

This paper also emphasises the importance of the global structure of the gauge group. A more modern perspective on line operators was given by Kapustin [120].

A very clear discussion of the electric and magnetic line operators allowed in different gauge groups, and the way this ties in with the theta angle, can be found in [4].

Towards the end of the 1970s, attention began to focus on more general questions of the phases of non-Abelian gauge theories [103, 104].

The distinction, or lack thereof, between Higgs and confining phases when matter transforms in the fundamental of the gauge group was discussed by Fradkin and Shenker [63] and by Banks and Rabinovici [9]; both rely heavily on the lattice.

The Banks-Zaks fixed point, and its implications for the conformal window, was pointed out somewhat later in 1982 [10].