Note on Axial U(1) Problem XIAO XIAO May 5, 2011 Abstract The process of the resolution of a long-standing axial U(1) problems is reviewed, and works done by various authors are introduced. I chose this topic because it really links many things that I've learned in this semester together and there are really deep insights and sharp questions in the explorations made by the authors, making a summary of these insights would be valuable. 1 Introduction The idea of symmetry and symmetry breaking has become the organizing principle of quantum field theory, through the developments in mid-20th century, people recognized that the dynamics of the physical world{four fundamental interactions are ruled by gauge symmetries. On the other hand, it was discovered that symmetry principles are powerful in strongly interacting many-body problems, such as QCD phenomena near or below 1GeV, the existence of pions and some of other mesons can be explained by the break down of chiral symmetry, such theories determine the properties of their interactions. And the development of current algebra method and the introduction of the concept ”effective field theory" further illustrated that just with a certain symmetry group and a proper choice of degrees of freedom one can construct the most general and correct theory in low energy. It took a long time to realize the fore-mentioned successes, among the confusions that appeared along the way, those with the understanding of symmetry breaking are surely the most intriguing and interesting ones, the resolution of axial U(1) problem is one of such confusions. People have understood that the way a symmetry is realized in the nature is not unique, the most common one would be like below: the space of particle states is spanned by multiplets labeled by the eigenvalues Qα of a Hermitian operator Q, Q generates unitary transformations U that mix the components of the multiplets, while keeping the expectation values of all other observables invariant. U(ξ)jλ, Qα >= Cαβ(ξ)jλ, Qβ > (1) 1 Here λ denotes the set of the eigenvalues of all other observables other than Q and ξ is the parameter that determines the transformation. Such a transformation on the particle states must also transform the operators that create these states: y U(ξ) αU (ξ) = Cαβ(ξ) β (2) This is the Wigner-Weyl realization of a symmetry, particle states and field operators are transformed within the multiplets and we say a theory is invariant under this symmetry if the action is a singlet under such transformations. If a symmetry is realized in this way, then we expect to see degenerate particles in the spectrum|particles with the same mass and spin could be grouped into multiplets, and their interactions are determined by couplings that satisfy the symmetry, but in reality people usually see particles that are "nearly degenerate", they have different masses and the difference is much smaller than the masses themselves. Historically for example, grouping proton and neutron into an SU(2) doublet was the idea in the seminal paper of Yang and Mills, and grouping pions into SU(2) triplet is important in understanding the partially conserved axial current. The breaking of degeneracies is understood as the existence of explicit symmetry-breaking perturbations, if such perturbations are turned off, then we have perfect degeneracies. Another way of realizing symmetries is adopted by nature in both strong and weak interactions, when we have a symmetry realized in this way, we cannot see degenerate multiplets in the spectrum, but particles seemingly ungrouped and some massless particles. This is well-known Nambu-Goldstone realization, here the vacuum is not singlet under a continuous symmetry{there are degenerate vacua, so if the charge of the symmetry is Q, then the charge cannot annihilate the vacuum QjΩ >6= 0 (3) Therefore it is possible that there exists single particle states with the same quantum num- bers of the charge, can be extracted from the vacuum: < πjQjΩ >6= 0 (4) It is proved that for a Lorentz covariant and conserved current in a quantum theory with only positive norm, this must be accompanied by the appearance of massless bosons, that can be grouped into a subgroup of the symmetry of the Lagrangian. In this situation we say the symmetry is "spontaneously broken", actually it is hidden. Again if in the theory there are interactions that explicitly break the symmetry, then what is spontaneously broken is an approximate symmetry, the corresponding Goldstone bosons will have small but non-zero masses. Pions are considered to be such "approximate Goldstone bosons". QCD is described by the Lagrangian: 1 L = ¯ (iγµD − m ) − F · F µν (5) f µ f f 4 µν 2 For QCD below 1GeV, the flavor index f run from 1 to 3, including up down and strange quarks in the model. The masses of the quarks are small compared to 1GeV, ranging from ∼ 1MeV to ∼ 100MeV . Originally since the mass of strange is still much larger than up and down, the model only contains up and down quarks. The important fact is, when we take the limit that all quark masses vanish, the left handed and right handed fermions decouple and the model has an U(2)L × U(2)R symmetry, if we take strange into account, the symmetry would be U(3)L × U(3)R. But what we can see in the spectrum are just up and down quarks in approximate SU(2)v multiplet, approximate in the sense that quarks masses lift the degeneracy and "v" denotes "vector" means the diagonal part of the group SU(2)L × SU(2)R, and three light pseudoscalar mesons|pions, therefore it is proposed that the axial SU(2) is spontaneously broken and pions are just the corresponding Goldstone bosons, again if the third flavor is taken in, we have axial SU(3) spontaneously broken and the identities of K mesons and η meson are explained. However, there is still an axial U(1) symmetry seemingly not realized in nature|neither explicitly in the spectrum nor with a Goldstone boson. We cannot associate η0 particle with this symmetry since the mass of η0 in considerably higher than K's η and pions|if they are lifted by the same quark masses, their masses should be more or less the same. This is axial U(1) problem[1]. During the period of 1970's, different approaches for its resolution were proposed and there were heavy discrepancies among physicists, here I try to describe the ideas that led people towards an understanding of the problem and explain these discrepancies. 2 Inspiration by instantons Though U(1) problem was widely recognized, it seems that the problem was actually not perfectly well defined—discrepancies arise even in the way of asking the question. One way of considering the problem is to consider the possibility that the Goldstone boson somehow gets an anomalous large mass, the other way, is to consider the possibility that there are no Goldstone bosons, at least in physical states. The second formulation of the question seems likely to bring a third way of realizing a symmetry|without multiplets or light mesons, however similar thing happens in gauge theory, in which we have hidden symmetry but no Goldstone bosons|the Goldstone theorem is evaded because when quantizing a gauge theory either one should introduce non-covariant gauge condition or there are negative norm ghost fields. And in tree diagrams the Goldstone boson generates a pole at zero momentum in gauge boson self-energy diagrams, shifting gauge boson mass from zero, this is so-called "Higgs mechanism". Here it seems the key still lies in gauge field, because the first thing that people noticed 3 was the axial U(1) bears an anomaly 2 Nf g a a @µJ5µ = −i F Fe (6) 16π2 µν µν With Nf flavors and here all the indices are Euclidean. One may recognize that the right hand side is actually a total divergence,thus its inte- gration over Euclidean space would give a surface integral which vanishes if we assume that the gauge field vanishes at infinity. 't Hooft points out that[3][4], if we assume QCD can be described by the semi-classical approximation of the path integral, then U(1) problem can be solved by a kind of configuration called instanton. Instanton configurations were discovered by Belavin, Polyakov et.al.[5] in the classical solutions of the field equations. Suppose we approximate the QCD path integral with the limit of ~ ! 0, then the path integral would be dominated by the action evaluated at the classical solution of the field equation, there are solutions with infinite value for the action, and the measure that such solutions take in configuration space is infinite compared to the measure taken by the finite-action solutions, however, in the classical limit, the integral is dominated by solutions with a finite action since they are less suppressed by the exponential in the path integral. Take a look at (5) we see that such solutions must satisfy: 1. the fermion field vanishes at infinity, 2. the gauge field becomes a pure gauge at infinity. This amounts to a map from the Euclidean infinity S3 to the gauge group G = SU(3), and such maps are characterized by an integer: Z 4 a a d xFµνFeµν s ν (7) ν is an integer, it is the winding number characterizing the map with a certain topology. Notice that a special case of finite action solutions is the classical QCD vacuum, they are also classified by the winding number.