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)|λ, Qα >= Cαβ(ξ)|λ, 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:

† U(ξ)ψαU (ξ) = Cαβ(ξ)ψβ (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

Q|Ω >6= 0 (3)

Therefore it is possible that there exists single particle states with the same quantum numbers of the charge, can be extracted from the vacuum:

< π|Q|Ω >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 = ψ¯ (iγµD − m )ψ − 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 integration 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 ∼ 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. We may say configurations with a non zero ν are solitons in the space-time sense, but a Euclidean soliton has more meaning than a spacial soliton because it describes a tunneling, between vacua with different topologies. But the first important fact is (7) which shows that the anomaly of axial U(1) is not merely formal, it really gives a non-vanishing integral, therefore we can say the symmetry is explicitly broken by the anomaly. However, certain questions remained unanswered: if we consider the symmetry breaking here as explicit, then do we have an effective symmetry-breaking term in the Lagrangian? The answer is yes, according to ’t Hooft, the fermion integral in the background of an instanton is equivalent to the integral not in gauge field background but with a determinant carrying chiral charge inserted[3]: Z Z ¯ ¯ ¯ ¯ ¯ DψDψ[exp(−SA,ψ − Jψψ)] ⇔ κ DψDψ[exp(−S0,ψ − Jψψ)] · det[ψR(x)ψL(x)] (8)

4 2 − 8π The constant κ is roughly e g2 , which suggests the eﬀective insertion is non-perturbative. When we sum over all possible numbers of instantons and anti-instantons, this determinant is lifted to the exponential:

iθ ¯ −iθ ¯ ∆Seff = κ[e det(ψR(x)ψL(x)) + e det(ψR(x)ψL(x))] (9) θ is an arbitrary phase that we can add in when we sum over instantons. ¯ From the perspective of symmetry, composite operator ψLψR can be eﬀectively repre- sented by scalar multiplets transforming under U(2)L × U(2)R[4]:

0 † φ = ULφUR (10) ¯ If we identify φ to ψLψR, then we have an eﬀective Lagrangian of the Goldstone bosons, it has U(2)L × U(2)R symmetry:

† L = −T r[∂µφ ∂µφ] − V (φ) (11)

Since φ is a general complex 2 × 2 matrix, we can expand it with Pauli matrices: 1 1 φ = (σ + iη) + (−→α + i−→π ) · −→τ (12) 2 2 With the Lagrangian above,we can see if we give σ ﬁeld a vacuum expectation value,then we will have axial U(2) broken and there would be 4 massless particles, η and π, however with the insertion (9) caused by the instantons, we would have an eﬀective term which explicitly break chiral U(1) symmetry and contributes only to the mass of η particle:

δV = κeiθdetφ + h.c. = κeiθ[(σ + iη)2 − (−→α + i−→π )2] + h.c. (13)

Since the Lagrangian has U(1) symmetry φ → eiωφ, we can rotate φ by ω and rotate θ by Lω, with L the number of ﬂavors.

φ → eiωφ, θ → θ + 2ω (14)

1 Choose ω = 2 (π − θ) we ﬁnd the insertion is:

δV = −2κ(σ2 + −→π 2 − η2 − −→α 2) (15)

The vacuum expectation value is shifted and η particle gets a mass proportional to 2 − 8π e g2 , while pions remain massless. This is the solution provided by ’t Hooft, the essence of this resolution are: (1)The axial U(1) symmetry is explicitly broken by the instanton, the eﬀect can be seen as an insertion. (2)Despite this explicit insertion, the symmetry is spontaneously broken and there is an U(1) Goldstone boson, but the mass is lifted by the explicit breaking term.

5 (3)As suggested by the appearence of κ the mass is non-perturbative, it is an instanton effect. There are two points worth of commenting.First, summing over instantons to approximate the path integral, is actually based on the assumption that semi-classical limit is good to be approximate description of QCD, since only in this limit the path integral is dominated by the configurations that are classical solutions of the field equation and making the action finite as well, or the path integral would be dominated by infinite-action solutions and then instantons are no longer a good description. But there remains a question that to what extent should we trust semi-classical approximation in QCD, if the real QCD cannot be described in the classical limit, then we should not trust instantons. Second, in ’t Hooft’s solution,the spontaneous breaking of axial U(1) is not included in the theory, it is an assumption just as the breaking of axial SU(2), the anomaly and instanton just serve to give an explicit breaking, there are no massless bosons. However, as was pointed out, when we go back to the anomaly equation we see the anomalous term is also a total divergence and if we incorporate it into the axial current we would have a conserved current:

J5µ = ψγ¯ µγ5ψ + δj5µ (16)

The new current J5µ is conserved and we expect to get the corresponding Ward identity and we will find a pole at zero momentum just as we did with a conserved current, if the axial U(1) is spontaneously broken by vacuum. But why we don’t see the pole? Kogut and Susskind[2] suggested a solution of the problem suggesting that the pole is not actually in the physical sector of the Hilbert space. Their resolution made no use of instantons and the long-range interaction that causes confinement is argued to be responsible for the absence of axial U(1) Goldstone boson, they called their resolution ”vacuum seizing”, the meaning of the name will be clarified below. Before closing the section we should mention an important parameter that suggested by ’t Hooft’s resolution of U(1) problem, it is vacuum angle[6][7]. We have seen its existence in (9), but its origin comes from the fact that the existence of Euclidean instanton solution suggests that a vacuum with different winding number n has a probability to tunnel into another vacuum with a different winding number, therefore the vacua labeled by a certain winding number are not good vacua in the sense that they are not stable, further it can be shown that it doesn’t satisfy cluster property[2], and the time evolution of such vacua fails to satisfy a multiplicative behavior in time, but all these properties are satisfied by θ vacua: X |θ >= eiνθ|ν > (17) ν The introduction of θ angle is certainly an outcome of instanton physics, but does it really rely on the existence of instantons? The answer is no, we will come back to this point later.

6 3 Seizing of vacuum and unphysical pole

The question that asked by Kogut and Susskind was: Do we have a way to circumvent Goldstone’s theorem, having a symmetry-breaking vacuum as well as avoiding massless bosons in physical states, without an explicit breaking? There is one, as found by Kogut and Susskind in Schwinger model, the theory exhibits spontaneous chiral symmetry breaking without a Goldstone boson. We say it is a spontaneous breaking, in the sense that there are degenerate vacua, and each of them is stable, there is no tunneling between any two of them. Below we will see how it happens, what Kogut and Susskind found is that the missing of Goldstone boson is related to the existence of long-range force. Schwinger model is quantum electrodynamics in two dimensional space-time[8], Z 1 S = dtdz(iψγ¯ µD − µ)ψ − F F µν (18) µ 4 µν

WhereDµ = ∂µ + igAµ is the gauge covariant derivative. What makes the model simple is that it can be mapped on a bosonic theory, the procedure is called bosonization[9], there is a series of duality relations between the quantities in the fermionic and bosonic theories:

jµ = ψγ¯ µψ = µν∂νφ (19)

j5µ = ψγ¯ 5γµψ = ∂µφ (20) ψψ¯ = K cos(2φ) (21)

ψγ¯ 5ψ = K sin(2φ) (22) 1 iψγ¯ µ∂ ψ = ∂ ∂ φ (23) µ 2 µ µ First we work in axial gauge, which is equivalent to Coulomb gauge in two dimensions, then we no longer have explicit Lorentz covariance and there appears a non-local interaction. The axial gauge is deﬁned by

Az = 0 (24) Then the equation of gauge ﬁeld becomes constraint, and the time-like component of the gauge ﬁeld can be solved, then there is only a non-local potential in the Lagrangian, with no photon degree of freedom. Z 1 Z S = dzdt(ψiγ¯ µ∂ ψ) − dzdz0dtψ†(z)ψ(z)V (z − z0)ψ†(z0)ψ(z0) (25) µ 4

0 1 0 Where V (z − z ) = 2 |z − z | is one-dimensional Coulomb potential. Through bosonization we ﬁnd the theory is equivalent to:

Z 2 Z 2 2 g 0 0 S = dzdt[φ˙ − (∂ φ) ] − dzdz dt∂ φ|z − z |∂ 0 φ (26) z 4 z z

7 The chiral transformation, acting on boson ﬁeld, is a shift in the ﬁeld strength:

eiαQ5 φe−iαQ5 = φ + α (27)

Since for the fermionic theory the chiral rotation with an angle which is an integer times π is identity, so the boson states should be invariant under φ → φ + nπ, which can be guaranteed if the chiral charge Q5 is even integer for these states, this also applies to vacuum states, so we see there are vacua that can be labeled by their chiral charge which is an integer. If we split the ﬁeld φ into a dynamical ﬁeld which is invariant under chiral transformation and a constant background which carries the integer chiral chargeφ = φˆ + θ,

eiαQ5 φeˆ −iαQ5 = φˆ (28)

eiαQ5 θe−iαQ5 = θ + α (29) then the Lagrangian can be reduced to a local form through integration by parts. We see there appears a parameter θ specifying the boundary condition at infinity, and it is decoupled from the dynamical field φˆ, once a boundary parameter is chosen, it is fixed and there is no way to change the parameter, therefore the state vector is a direct product|Ψ >= |Ψ)|α], |Ψ) labels the sector on which the dynamical field φˆ acts, α labels the sector on which the chiral charge and theta angle acts, we may call it vacuum sector since it is irrelevant of whether there are particles present—the mesons live in |Ψ) sector. From the discussions above we see that the vacuum states can be either labeled with the integer which specifies the chiral charge that the vacuum carries or by the theta angle that fixes the boundary condition at infinity. The two basis is related by a Fourier transformation: X |θ] = e−i2nθ|n] (30) n It is easily seen that applying a chiral rotation on |θ] results in a shift of the angle, therefore in the dual bosonic picture, chiral transformation shifts the boundary condition at infinity It can be verified that the integer labeled vacua don’t satisfy cluster property which is satisfied by theta vacua: lim < 0, n|Oˆ(z)Oˆ(0)|0, n >6= 0 (31) z→∞ Notice that in Schwinger model we can introduce theta vacua without introducing instantons, as one does in 3+1 dimensional non-Abelian theories, the θ angle is simply a parameter labeling the boundary condition of the dual boson field. Furthermore, we see that there are no reason to restrict the gauge field at infinity as a pure gauge, to see this ,notice the expression for the gauge field strength: 1 Z A = g dz0|z − z0|ψ†(z0)ψ(z0) (32) t 2

8 This cannot be gauged away and it gives a non-vanishing contribution to the field-strength at infinity for localized charge. Therefore we can conclude that in Schwinger model there are no instantons, at least we don’t consider their possibility of their existence, further the absence of a physical massless particle can be explained without the need of instantons, with the argument below. The basic idea is the existence of a long-range force prevents independent chiral rotations on distant points, there would be a mass gap of such local rotations. The origin of massless Goldstone boson lies in the possibility that one can rotate the vacuum state around each point independently and create long range gapless wave. So we try to chiral rotate the scalar field independently in different points. But we have shown that chiral transformation on the boson field is actually rotation of θ angle, therefore we expect a Goldstone boson is exactly the wave generated by degenerate vacua θ. The interaction term is: 2 Z g 0 0 dzdz ∂ φV (z − z )∂ 0 φ (33) 2π z z We chiral rotate the field on each point:

φ(z) → φ(z) + α(z) (34)

If we regularize the long range force with a mass parameter

1 0 V (z − z0) = (1 − e−m|z−z |) (35) 2m And assume the local rotation to be a plane wave form with momentum p, then the 1 shift in potential energy is proportional to − p2+m2 , when m is non-zero, the potential is short-range and the energy of local chiral transformation goes to zero as the momentum goes to zero, then we see a massless particle, but in the case of long-range force, m = 0, we see the energy goes to a non-zero value when momentum vanishes, this suggests that there is a gap between one excitation and the uniform vacuum, there is no massless particle when long-range force presents. However, we mentioned even with anomaly there should be a conserved current and a pole at zero momentum, where is the pole? And from the vacuum seizing point of view we don’t even see the role of anomaly, this is because we were working in a non-covariant gauge with non-local interaction. The anomaly and the pole can be seen in Lorentz gauge. When we quantize a gauge theory in a covariant gauge, there is negative norm sector in Fock space, this gives a way out from Goldstone’s theorem, but the same thing happens in Higgs mechanism and it solves the problems for the conserved current that couples to the gauge field. However, here we are treating a spontaneously broken global symmetry and there is no gauge field coupling to axial U(1) current. In fact, things happening here are more intricate: because the existence of anomaly, one cannot find a locally defined current, which is simultaneously conserved and gauge-invariant.

9 We deﬁne the axial current as an operator product:

¯ j5µ(x) = sym lim ψ(x)γµγ5ψ(x + ) (36) →0 This current is conserved but not gauge invariant since the gauge transformations are independent on diﬀerent points and a short distance singularity in the product prevents one to take a smooth limit. We have a gauge-invariant current:

R x+ µ g ig Aµdx ¯ ν ˆj5µ = lim ψγ¯ µγ5e x ψ(x + ) = lim ψ(x)γµγ5ψ(x + ) − µνA (37) →0 →0 π The divergence of gauge field is zero in Lorentz gauge, therefore we can express the gauge field with the gradient of a scalar field:

ν Aµ = µν∂ Φ (38)

Define a fermion field χ so that χ = eigγ5Φψ, and we haveχγ ¯ µχ conserved, further we have the conserved electric charge current, so we can define the gradients of scalar fields to represent them: ν ν χγ¯ µχ = µν∂ φ2, jµ = µν∂ φˆ (39) Then the scalar fields satisfy Klein-Gordon equations:

2 ˆ ¯ ¯ ( + m )φ = 2µψγ5ψ, φ2 = 2µψγ5ψ (40)

2 2 ˆ √g Where m = g is the mass of the vector meson in the model. Then we deﬁne φ1 = φ− π Φ and see φ1 satisﬁes: ¯ φ1 = 2µψγ5ψ (41)

We see this can be identified with the divergence of j5µ which is the conserved current in massless limit, therefore the conserved axial current can be expressed by the gradient of a free massless boson field φ1, this is the origin of the pole in the Green’s functions involving this current. ν j5µ = µν∂ φ1 (42) But this boson has a negative norm, as can be seen in the Lagrangian expressed totally in the dual boson language. 1 L = [(∂ φˆ)2 + (∂ φ )2 − (∂ φ )2] + µK cos 2(φˆ + φ − φ ) (43) 2 µ µ 2 µ 1 2 1 This effective Lagrangian is resulted from the equations of motion of the scalar fields, which are derived by conservation of currents. Notice that the massless particles φ1 and φ2 are exactly in equal footing despite the fact that their norms are opposite, therefore for any physical matrix elements, the internal lines mediated by these two particles exactly cancel, leaving no poles in gauge-invariant matrix elements, but for gauge-variant matrix elements,

10 the two massless particles don’t cancel and there are poles at zero momentum. This is how the expected Goldstone boson disappear in Schwinger model. This argument can be generalized to four dimensions, where it is possible for one to define a current which is conserved as well as gauge invariant, but such a current is not local, and it can be seen that the mesons that such a current creates always accompany a long-range dipole field, which is forbidden by the screening of long-range force in real quantum chromodynamics. We have seen two resolutions, and now it is necessary to compare them, they seem quite different. In ’t Hooft’s solution, the axial U(1) is explicitly broken by effective terms that resulted by instantons, however in the ”vacuum seizing” picture, there are no explicit breaking of axial U(1) symmetry, instead, long-range interaction gives a mass gap for local chiral transformation and forbids a massless particle appearing in physical process, further in a covariant gauge, one can see the Goldstone pole appears just in unphysical sector and get canceled in physical matrix elements, due to the fact that in a quantized U(1) gauge theory in 2 dimensions there are clashes between local gauge invariance and chiral symmetry—one cannot construct an axial current which is simultaneously conserved and gauge invariant. Then question comes: how a broken symmetry is simultaneously said to be explicitly broken and spontaneously broken? There is an answer to this question, that any explicit symmetry breaking can be treated as a spontaneous one by enlarging Hilbert space[4]. As an example, consider in a fermion theory we add in a mass term mψψ¯ , it breaks chiral symmetry explicitly, but if we promote parameter m to a field and include transformation m → me2iα, then the chiral symmetry becomes a symmetry of the Lagrangian and if we fix m to be real, we can say this symmetry is spontaneously broken. As for the case of axial U(1) in Schwinger model, we have seen the existence of θ vacua and its existence induces a background electric field which is independent of the dynamics, this gives an effective term in the Lagrangian that is proportional to θ:

µν θFµνF (44)

When we do chiral transformation,we shift theta as a variable and consider it to be a change from one vacuum to another, however another point of view is also applicable: we simply regard θ as a fixed parameter in the theory and the effective term is not invariant under chiral transformation due to anomaly. Therefore we see it doesn’t matter whether we call it ”explicit breaking” or ”spontaneously breaking”, our Lagrangian should be built on certain vacuum that satisfies cluster property, and if we change vacuum, it is necessary to change parameters in the theory. What is important is that, the resolution of axial U(1) actually just involve the existence of anomaly and θ parameter in the theory. If we have both, then even for a theory without instantons we may have a classical symmetry not realized either in multiplets or massless bosons.

11 Figure 1: FIG.1.The annihilation channel of a pair of ﬂavor singlet quarks, which contributes to the mass of the meson

4 Large N limit insights

There was a new proposal brought by Witten in 1979[10][11],starting from the fact that for two quarks forming SU(3) singlet, there is a channel in which quarks annihilate into gluons that is absent for the quark pairs that form SU(3) triplets, and η0 prime particle as a SU(3) singlet, its mass can split from pions by this annihilation channel. However this process is apparently what one can see in perturbation theory, though in rigorous sense it is not normal perturbation expansion—in normal perturbation expansion we don’t have bound states, but the question can be formulated when we introduce 1/N expansion—the above process can be described by 1/N expansion in a finite order as a perturbation theory, while instantons are non-perturbative effects and cannot be seen in large N limit. Notice 2 − 8π that in part one, the effective insertion caused by instanton is proportional to e g2 , large N limit is taken as N → ∞ while g2N = λ is fixed, therefore in large N limit, instanton insertion goes like e−N and decreases faster than any finite power of N, anything, if is well described by instantons, should not be well described by 1/N expansion, and if it were instantons that dominate the strong interaction in strong coupling regime, large N limit should always be a bad approximation. When instanton, as a classical solution, was introduced, it was believed to affect physics since it is in the leading contribution of semi-classical expansion. For a particle moving in certain potentials, there is a clear criteria of whether semi-classical expansion is a good approximation, that is when the characteristic wavelength of the particle is far shorter than the characteristic scale of the potential, then semi-classical approximation is good. However in field theory, since it involves fluctuations in all scales, it is hard to justify the semi-classical limit really dominates the path integral and qualitative features in the classical solutions will hold when loop diagrams are taken in. It is possible that in QCD the contribution from configurations with certain winding numbers is vanishingly small and the resolution of U(1) problem doesn’t rely on introducing instantons. There are actually examples in which quantum corrections drastically change the qualitative features of the theory, which will appear below. There are two kinds of classical theories, ones with instantons and ones

12 without: ∗ µ 2 ∗ 1 2 L = Dµφ D φ − M φ φ − 4 Fµν ∗ µ ∗ 2 1 2 L = Dµφ D φ − λ(φ φ − a ) − 4 Fµν Classically the first one, with an unbroken gauge symmetry, allows classical field configurations that vanish at infinity and no instantons, it has a topological charge Q = e R 2 µν 2π d xµνF , and so does the second theory. The difference is, the second theory has a spontaneous breaking of gauge symmetry, and thus the topological charge is quantized by non vanishing boundary condition of the scalar field, therefore classically there are instantons in the second theory but not in the first one. However such a qualitative feature may break down in some models that resemble the second kind, in the quantum level, this suggests that there are really field theories in which one cannot get good approximation from semi-classical limit. One good example is CP N−1 model. The model is: ∗ i L = (∂µ − iAµ)ni (∂µ + iAµ)n (45) ∗ i With constraint ni n = 1 The gauge field is auxiliary field and can be eliminated with its equation of motion. 1 R 2 ∗ i The theory possesses topological charge 2πi d x∂µ(ni µν∂νn ) and classically ,by de- manding finiteness of the action, the charge is quantized and there are instantons carrying the topological charges. When the theory is investigated in large N limit, in the leading order of 1/N, we sum over all diagrams with arbitrary loops in the same order to get their corrections to the effective Lagrangian, that means we are considering significant quantum fluctuations to see their effects. In this example, we will see the quantum corrections dramatically change the feature of the theory. The full Lagrangian we consider is: Z N θ S = d2x[ (∂ − iA )n∗(∂ + iA )ni − λ(n∗ni − 1) + ∂ A ] (46) g2 µ µ i µ µ i 2π µν µ ν Here |lambda is an auxiliary field we introduced to put the constraint into the Lagrangian, it can be eliminated with its equation of motion, equivalently, with a functional integral on λ field will also eliminate it. The θ term is proportional to the topological charge and it is the similar to the background electric field term in Schwinger model. The path integral of the theory Z Z = [dn∗][dn][dλ][dA]eiS (47) can be reduced to Z λg2 Z iθ Z Z = [dλ][dA]exp[−NT rln(−|∂ + iA |2 − ) + i d2xλ + d2x ∂ A ] (48) µ µ N 2π µν µ ν

If the integral over the ni ﬁeld is done ﬁrst. Evaluation of λ integral with saddle point method requires minimizing the action, we get an equation that determines the saddle

13 point value of λ ﬁeld, which can be called a ”gap equation”, since we see from the original

Lagrangian that if we have a positive stationary value for λ then we get a mass term for ni field—the field acquires a mass gap from quantum correction. Z d2k 1 i + g2 = 0 (49) 4π2 2 g2λ k − N + i Another effect brought by the quantum corrections is a kinetic term for the auxiliary vector field Aµ, the effective Lagrangian that have taken quantum corrections into account is:

1 ∗ 1 i 2 ∗ i 1 2 θ 1 Leff = (∂µ −iAµM √ )n (∂µ +iAµM √ )n −M n n − F + M √ µν∂µAν (50) N i N i 4 µν 2π N Here the constraint is removed by the integration on λ and the theory is, in quantum level, a theory with unbroken SU(N) symmetry, there are no instantons. Furthermore, the gauge field generates a confining Coulomb potential in 2 dimensions, there are no asymptotic n particles in the spectrum, instead there are neutral mesons. The θ dependence is not affected by the disappearance of instanton gas, the θ term is not induced by instantons and can generate an effective interaction between charged particles in quantum level. And the detailed behavior of the particle spectrum depends on θ in the leading order of 1/N expansion. The model above suggests that a resolution of U(1) problem based on instantons may be dangerous due to quantum effects, such effects may eliminate instantons. Actually we have seen an example of resolutions that not based on instantons—the vacuum seizing mechanism suggested by Susskind and Kogut, the only elements we need in this mechanism is the existence of long-range confining force, θ parameter and the axial anomaly, all present in QCD even without instantons. The simplest understanding here seems to be since it is possible that the gauge field would not be pure gauge at infinity due to quantum fluctuations that cause confinement, the θ term cannot be regarded as a vanishing boundary integral and eliminated, then the anomaly serves to change this term explicitly when we do chiral transformation. These are all independent of instantons, and are also the relevant part of ’t Hooft’s resolution. However in ’t Hooft’s theory, the η0 mass decreases exponentially in large N limit and the long-range force,θ dependence and anomaly all can appear in finite order of 1/N perturbation theory[10][13][14], therefore we expect the true resolution of the problems should be seen in 1/N expansion—we need a mass for an SU(3) singlet particle in 1/N limit. But this is exactly what the quark-anti-quark annihilation channel tells us, which is mentioned above. But the problem with the idea is that it is apparently not related to the anomaly. But we shall see from the massless Schwinger model that there are relations between them. Through bosonization, the Schwinger model can be cast in the form: 1 1 L = − F 2 + (∂ σ)2 − m cos 2σ + eσµν∂ A + θe∂ A (51) 4 µν 2 µ µ ν µ ν

14 If the fermion is massless then by simply redefining σ field we can eliminating the θ term and the physics is independent of θ, the coupling term between σ and Aµ is there because of the anomaly— with the effective term the anomaly equation becomes a canonical equation for σ

∂µ(∂µσ) = eµν∂µAν (52) Notice that this term is a mixing between the massless scalar field σ and the gauge field, as in Higgs mechanism, such a mixing produces a pole at zero momentum in the self-energy diagram of σ, resulting a mass of this particle, here σ particle is constructed as a fermion- anti-fermion pair, we identify it as the Goldstone boson associated with the spontaneous breakdown of the axial U(1) symmetry, the diagrams below shift the mass of the scalar field from zero to a finite value, notice that they are exactly the corresponding processes of quark-anti-quark pair annihilation diagrams in 4 dimensions. From above we see that, in Schwinger model the analog of axial U(1) problem can be resolved in perturbation theory, through bosonization procedure. In four dimensions with QCD, though we don’t have a dual boson theory, it is reasonable to believe that the axial anomaly in four dimensions of axial U(1) induces a coupling between SU(3) singlet quark-anti-quark pair with a pair of gluons—in the sense of quantum numbers, such a coupling is not forbidden for SU(3) singlet pair but is really forbidden for an SU(3) triplet pair. And we notice that the mass contribution to the scalar field has nothing to do with instantons, it is through the anomaly—which is an effect that can be seen in 1/N expansion. Therefore, it is reasonable to abandon the assumption that QCD phenomena are dominated by classical configurations—at least not in U(1) problem, we need a treatment of full quantum fluctuations, including fluctuations of boundary conditions, and fluctuating topological charge. This seems intractable, the QCD coupling constant is large below 1GeV, further it is not a free parameter in the sense of dimensional transmutation. However the way out again lies in 1/N expansion, which gives us a way to see how the existence of a massive SU(3) singlet scalar field is related to anomaly, even in chiral limit. And 1/N expansion also allows us to construct en effective Lagrangian which properly includes the anomaly and θ dependence, and describe the spectrum of U(3) boson nonet in closed form, which is the end of axial U(1) story.

5 Large N resolution and an eﬀective Lagrangian

Consider the θ dependence of QCD, it is assumed that in the leading order of 1/N expansion, i.e. the planar gauge ﬁeld diagrams, the θ dependence does not vanish[11]. However when there are massless fermions, the physics doesn’t depend on θ angle, as can be seen in the path integral and bosonized models in 1+1 dimensions. Considering this,

15 Figure 2: FIG.2.The insertion of the operator as a Feynman diagram which is in the leading order of 1/N expansion

Figure 3: FIG.3.The planar diagram made by gluon lines is in the leading order of 1/N the assumption that θ dependence can present in the leading order of 1/N expansion leads to a conundrum: The diagrams containing fermion loops are always in higher order of 1/N compared to diagrams made up by pure gauge ﬁelds, then if the θ dependence presents in pure gauge ﬁeld planar diagrams, how can diagrams with fermion loops cancel this dependence? To be concrete, consider the Lagrangian of QCD with θ term:

Z Z 1 g2θ Z = [dA ]exp[i d4xT r(− F F + F F˜ )] (53) µ 4 µν µν 16π2N µν µν We calculate the second derivative of vacuum energy on θ, it s equivalent to inserting operator F FF˜ F˜ in the path integral:

d2E 1 g2 Z ( ) = ( )2 d4x < T (F F˜(x)F F˜(0)) > (54) dθ2 θ=0 N 2 16π2 Introducing an infrared cutoﬀ, the average is turned into the zero momentum limit of a momentum space two-point function:

2 2 d E 1 g 2 ( )θ=0 = ( ) lim U(k) (55) dθ2 N 2 16π2 k→0 Where Z U(k) = d4xeik·x < T (F F˜(x)F F˜(0)) > (56)

The perturbation expansion of U(k) vanishes in each order in coupling constant, since F F˜

16 is a total divergence therefore its matrix elements vanish at zero momentum, but in large N limit, we are summing over inﬁnite diagrams that are in the leading order, here is N 2, as can be seen in the ont-loop diagram. The expansion of U(k) is like

U(k) = ak2 + bg2kk ln k2 + cg4k2 ln2 k2 + ... (57)

It is possible that we can get a non-vanishing summation though each term vanishes when k goes to zero. The assumption here is U(k) doesn’t vanish in the leading order of 1/N. Counting all the diagrams one gets an expansion in 1/N:

U(k) = U0(k) + U1(k) + U2(k) + ... (58)

The suﬃx is the order in the number of quark loops: U0 is the leading contribution, it is the sum of all diagrams without quark loops, U1 is the sum of all diagrams with one quark loop, and etc. Here we are just counting quark loops but not considering whether the diagrams are planar, because whatever the topology of the diagram, one quark loop always suppresses the diagram by another 1/N, and the problem we are considering has nothing to do with 2 the planarity of the diagrams. The U0 term is of order N , as the one-loop diagram above, and adding more gluon loops while keeping the topology doesn’t change the order of the diagrams. As every quark loop added in, the order of the diagrams goes up by one, so U1 is of order N, in large N limit, it is negligible compared to U0. In a gauge theory without massless fermions, the contribution to U(k) comes solely from U0, and by assuming theta dependence in this order, we are assuming the sum of these diagrams doesn’t vanish and it is θ dependent. However when we include massless fermions, the θ dependence should vanish, the only possibility is the θ dependence is canceled by U1, which is in the next order of 1/N. This seems impossible, but actually it is possible at zero momentum, provided the existence of a meson with mass square in the order of 1/N even in chiral limit. In large N picture, a two-point insertion like (56) equals to the inﬁnite sum of tree diagrams with all kinds of mesons and glueball states as internal propagators[19], we can express it as: 2 2 X N an X Nc U(k) = + n (59) k2 − M 2 k2 − m2 glueballs n mesons n

All the coeﬃcients an and cn’s are order 1.They are the amplitudes for the operatorF F˜to produce intermediate states.

Nan =< 0|F F˜|nthglueball > (60) √ Ncn =< 0|F F˜|nthmeson > (61) The problem becomes, how can diagrams with internal meson contributions, cancel the the glueball diagrams, at zero momentum? If there is at least a meson state with mass square

17 in the order of 1/N, we see that there is at least one term in meson summation that is enhanced by factor N at zero momentum, which is possible to cancel the contribution from the leading diagrams. Therefore, starting from the assumption that θ dependence exists in the leading 1/N expansion, we can see there must exists a meson whose mass doesn’t vanish even in the chiral limit in which pions are exactly Goldstone bosons. And if we assume there is only one such meson, the formula for its mass can be obtained, by steps similar to PCAC arguments. The cancelation leads directly to

Ncη02 2 = U0(0) (62) mη0

Since the η0 particle is the only meson state that may cancel the leading term in zero momentum. From (61) we see that √ ˜ 0 Ncη0 =< 0|F F |η > (63)

And the matrix element can be obtained from anomaly equation:

2L g2 ∂ J = F F˜ (64) µ 5µ N 16π2 L is the number of light quark ﬂavors.

In momentum space, the matrix element of ∂µJ5µ can be obtained by Lorentz invariance:

0 2 < 0|∂µJ5µ|η >= mη0 fη0 (65)

Combining (62),(63) and (65), and the fact that U0(0) gives the contribution of θ dependence when there are no quarks

d2E 1 g2 ( )gauge = ( )2U (0) (66) dθ2 θ=0 N 2 16π2 0 We get a formula for η0 mass: 2 2 4L d E gauge mη0 = 2 ( 2 )θ=0 (67) fπ dθ

Here fη0 is identified to fπ since they are the same in the lowest order of 1/N. Therefore we see, through the axial anomaly, the η0 meson receives mass from the matrix element of operator F F˜, which doesn’t vanish at zero momentum due to quantum fluctuations. The effect here is of order 1/N, in contrast to the instanton prediction that the η0 mass vanishes exponentially in large N limit, but the essential point is the same: the matrix elements of a total divergence F F˜ may not vanish at zero momentum limit, due to non-perturbative effects, which can be large fluctuations or instantons, but since the presence of confinement, it is hard to trust the validity of semi-classical expansion in QCD,

18 which underlies instanton theory, it is reasonable to believe that large N expansion gives a picture that is closer to real world. Later in the works of Veneziano and Di Veccia[15][16], it is proved that the anomalous Ward identities of axial U(1) current are satisfied without a massless particle and the mass formula (67) is rederived. The saturation of Ward identities without massless particles is in the way that is similar to Kogut-Susskind unphysical Goldstone bosons, but here what works are no longer ”Goldstone dipoles”, but a ghost vector field, for details see papers by Veneziano. As the picture becomes clear, we can ask the following question: could we build a phenomenological Lagrangian that allows us to describe the low energy physics of the mesons in the U(3) nonets? From such a Lagrangian, we expect to include all the Goldstone bosons of axial U(3), include the explicit symmetry breaking term of quark bare masses and axial anomaly, and reproduce the anomalous Ward identities, and the low energy scattering amplitudes. From the perspective of effective field theory, this is possible if we construct the most general Lagrangian that includes proper degrees of freedom and is compatible with all the relevant symmetries. The starting point are the assumptions that though chiral U(3)×U(3) symmetry is exact if N → ∞, it is spontaneously broken down to isospin U(3), producing four Goldstone bosons, one isoscalar η0 and an isotriplet, pions, and color confinement persists in large N limit. From the pattern of the chiral symmetry breaking, the degrees of freedom of the effective Lagrangian are η0 field and pions, from the color confinement assumption, there shall not be color quantum numbers in the effective Lagrangian.[17][18] The Goldstone bosons are described by 3 × 3 local unitary matrix that represents the spacial variation of vacuum state, the breaking is spontaneous, so the Lagrangian should be invariant under U(3) × U(3) without explicit breaking terms. There are two possible terms: −1 −1 2 I1 = T r∂µU∂µU and I2 = (T rU ∂µU) The second one, in the form of the square of a trace, is flavor disconnected thus is an OZI rule violating term, since in large N limit, OZI violating processes are always suppressed, in large N limit, we only consider the first term. And we don’t have terms without derivatives on U, since U −1U = 1 is fixed, such terms cannot be constructed. For the explicit breaking terms caused by quark bare masses, the only term with correct transformation property is T rMU˜ and its conjugate, therefore without anomaly, i.e. in N → ∞ limit we have the Lagrangian

f 2 Z f 2 S = π d4xT r∂ U∂ U −1 + π (T rMU˜ + T rM˜ †U †) (68) 2 µ µ 2 Formally this is the same as the sigma model of pions, but here the matrix U is in the group U(3) and there are 4 Goldstone bosons. Next consider the eﬀective term by axial U(1) anomaly. It is an explicit breaking from

19 U(3) × U(3) to SU(3) × SU(3) × U(1), therefore it depends only on the determinant of U, and from the large N counting rule in the leading order of 1/N, it should be quadratic in η0 ﬁeld, therefore the term is: f 2a π (i ln det U)2 (69) 2N The total eﬀective Lagrangian is:

f 2 Z f 2 f 2a S = π d4xT r∂ U∂ U −1 + π (T rMU˜ + T rM˜ †U †) − π (i ln det U)2 (70) 2 µ µ 2 2N It is the most general Lagrangian in large N limit that describes the low energy physics of U(3) meson nonet. An SU(3) × SU(3) transformation can bring the M˜ matrix into the form i θ M˜ = e 3 M (71)

− iθ M is real, positive and diagonal, θ is an arbitrary phase. Redeﬁne U → e 3 U and the Lagrangian becomes:

f 2 a L = π [T r∂ U∂ U −1 + T rMU + T rMU † − (i ln det U + θ)2] (72) 2 µ µ N We see the assumption that the physics depends on θ in the leading 1/N order is simply a 6= 0. Having obtained the effective Lagrangian (72), we may parametrize the U matrix with its phase identified with the linear superposition of Goldstone boson fields, and expand the Lagrangian around the minimum of the effective potential, then we will have a low energy effective theory describing the meson nonet, and the effects of anomaly is considered, the mass relations between the mesons and their θ dependence can be solved and their scattering amplitudes are determined, in large N approximation. This phenomenological theory, as the end of axial U(1) story, is called ”large N chiraldynamics”

6 End

The resolution of the axial U(1) problem is an especially controversial and frustrating one, many different resolutions seem to work but apparently they are contradicting to each other, and finally we see each of them contains certain key points in the final resolution. The axial U(1), like the axial SU(3), is spontaneously broken and there is a corresponding Goldstone boson, η0. The bare masses of quarks explicitly break the axial symmetries and contributes masses to the Goldstone bosons. However, for axial U(1), there is one more contribution from anomaly, the effect of anomaly doesn’t vanish since in the path integral the topologically non-trivial configurations are taken into account, this is the key insight provided by ’t Hooft and his instanton picture. The matrix element of the anomalous

20 contribution to the axial current at zero momentum, which can be described by a quark- anti-quark annihilation process contributes to the mass of the η0 particle, splits its mass from other pseudo-Goldstone bosons. As for the Ward identities, although with anomaly, there is still a conserved current with the contribution from the topological charges, the pole at zero momentum resulted by this current, however, is a ghost rather than a physical particle since the current is not gauge invariant, all the anomalous Ward identities can be satisfied without a physical massless particle, and there are no exact axial U(1) symmetry even in chiral limit. But in the large N limit, the anomaly is turned off and the chiral U(3) × U(3) would be exact, there would be massless Goldstone nonet in that limit. This is the whole picture of what happens to the chiral symmetry of the light quark flavors.

7 Final words (irrelevant of U(1))

The author is a first-year graduate student, this review is for the homework of the course ”Quantum Field Theory III” instructed by Professor Erick Weinberg. Limited by the author’s understanding of quantum field theory itself and the conceptions in low energy QCD, this work is little more than just listing the works of masters with some shallow interpretations. But it is still important to the author because it lets the author see a picture of what’s going on at the lower end of the energy ladder in QCD, with the existence of mesons fitted into symmetry principles. The author thanks Professor Erick Weinberg and Professor Alfred Mueller for valuable discussions, from the former the author learned the importance of instantons while the from latter the author knew the work of Witten and got inspired to read further.

References

[1] S.Weinberg, Phys.Rev.D, 11 (1975) 3583

[2] J.Kogut, L.Susskind, Phys.Rev.D, 11(1975) 3594

[3] G. ’t Hooft, Phys.Rev.Lett, 37(1976) 8

[4] G. ’t Hooft, Phys.Repts, 142, No.6 (1986) 357

[5] A.A.Belavin,A.M.Polyakov,A.S.Schwartz,Y.S.Tyupkin, Phys.Lett. B 59(1975) 85

[6] R.Jackiw, C.Rebbi, Phys.Rev.Lett, 37(1976) 172

[7] C.G.Callan, R.F.Dashen,D.J.Gross Phys.Lett, 63B (1976) 334

[8] J.Schwinger, Phys.Rev. 128 (1962), 2425

21 [9] S.Coleman, Phys.Rev.D 11 (1975),2088

[10] E.Witten, Nucl.Phys.B, 149 (1979) 285

[11] E.Witten, Nucl.Phys.B, 156 (1979) 269

[12] R.J.Crewther Phys.Lett, 70B (1977) 349

[13] S.Coleman, R.Jackiw, Ann.Phys, 93 (1975) 267

[14] S.Coleman, Ann.Phys, 101 (1976) 239

[15] G.Veneziano, Nucl.Phys. B 159 (1979) 213

[16] P. Di Veccia, Phys.Lett, 85B (1979) 357

[17] E.Witten, Ann.Phys, 128 (1980) 363

[18] P.Di Veccia, G.Veneziano Nucl.Phys.B 171 (1980) 253

[19] E.Witten, Nucl.Phys.B 160(1979) 57