Nonlocal Models of Goldstone Bosons ...

Nonlocal Models of Goldstone Bosons in Asymptotically Free Gauge Theories


John Terning

A thesis submitted in conformity with the requirements
for the degree of Doctor of Philosophy at the
University of Toronto

© John Terning


We develop a simple model Lagrangian which couples the pseudo-Goldstone bosons of QCD (pions, kaons, and the eta) to up, down, and strange quark fields. The model is a nonlocal generalization of the nonlinear sigma model, which incorporates a momentum dependent dynamical mass S(p). S(p) is an order parameter for spontaneous chiral symmetry breaking, and the Goldstone bosons are included as fluctuations of this order parameter in a manner consistent with global chiral symmetry. No kinetic energy terms or self-interactions for the Goldstone bosons are introduced, instead these arise dynamically. Explicit chiral symmetry breaking effects are also introduced by including current quark masses and external electroweak gauge fields. By integrating out the quark fields we obtain a chiral Lagrangian, including the Wess-Zumino term. The dynamical quark mass acts as a natural regulator, thereby making the calculations finite. All low energy parameters are expressed in terms of integrals of S(p). The results are relatively insensitive to the precise form of S(p), as long as S(p) has the asymptotic behaviour required by QCD. We use physical values for masses and decay constants (m_pi^0, m_K^+, m_K^-, m_eta, f_pi, and f_K) to determine the five parameters of our model: the dynamical quark mass, the quark condensate, and the three current quark masses. We obtain the 14 parameters of the effective chiral Lagrangian and find that they are in good agreement with the experimental values.


I would like to thank John Moffat, Nathan Isgur, Kevin Dooley, Mark Sutherland, Daryl Scora, Colin Morningstar, and Karl Verbeek for helpful discussions. Karl also gave me invaluable computer tutorials and lent me his computational expertise. I would also like to thank Bob Holdom for being a great thesis advisor. Finally, I must thank Paul Geiger, Cathy Reader, and Eric Swanson for innumerable, lengthy, and inspiring discussions and for interminable proof-reading.

To Donna.

The only truths that are useful are instruments to be thrown away. - Umberto Eco

Table of Contents 

1.	Introduction	
		1.1 - Overview and Outline
2.	QCD and Spontaneous Chiral Symmetry Breaking
		2.1 - QCD	
		2.2 - Spontaneous Chiral Symmetry Breaking
		2.3 - QCD at Low Energies	
3.	A Model for Massless QCD
		3.1 - Motivation for the Model	
		3.2 - Ward-Takahashi Identities	
		3.3 - Comparison With Other Models	
		3.4 - The Effective Chiral Lagrangian	
		3.5 - Non-Minimal Models	
		3.6 - The Effective Chiral Lagrangian at Fourth Order
4.	Quark Mass Effects	
		4.1 - Mass Corrections to the Order Parameter
		4.2 - Mass Corrections to the Effective Chiral Lagrangian
		4.3 - The Effective Chiral Lagrangian to Fourth Order
		4.4 - Mass Corrections to the Quark Self-Energy
5.	Electroweak Effects
		5.1 - Path Exponentials	
		5.2 - The Gauged Nonlocal Mass	
		5.3 - Photons and Goldstone Bosons	
		5.4 - Electromagnetic Form Factors	
		5.5 - Non-Abelian Gauge Fields	
		5.6 - Radiative Weak Decays	
6.	Comparison with Experiment
		6.1 - Matching Conditions	
		6.2 - Parametrizing the Dynamical Mass
		6.3 - Fitting to Masses and Decay Constants
		6.4 - Pion Scattering Lengths	
7.	The Techni-Higgs	
		7.1 - Scale Invariance	
		7.2 - Technicolour	
		7.3 - A Local Dilaton Model	
		7.4 - A Simple Nonlocal Dilaton Model	
8.	Conclusions	
		8.1 - Summary and Speculations	
	Appendix A
		Euclidean Space-Time Conventions	
	Appendix B
		Computer Results	

Chapter 1


1.1 - Overview and Outline

The aim of this thesis is to study spontaneous chiral symmetry breaking in asymptotically free gauge theories, in particular, we focus on quantum chromodynamics (QCD) and technicolour theories. Specifically, a model is developed which makes use of the quark self-energy as a nonlocal order parameter associated with the spontaneous chiral symmetry breaking. The Goldstone bosons that must be present are then naturally included as local fluctuations of this order parameter. It is then possible, using the path integral formulation of quantum field theory, to functionally integrate out the quarks, thereby obtaining an effective action for the remaining degrees of freedom, the Goldstone bosons.

The idea that Goldstone bosons can be considered as fluctuations of the order parameter is common in condensed matter physics. A familiar example is that of the Heisenberg ferromagnet [1]. The model Hamiltonian written down by Heisenberg is invariant under a position-independent rotation of the spins, but none of the infinite set of ground states of the system are invariant: in any particular ground state all the spins point in some arbitrary direction (the z-axis). It can be shown explicitly that such a system has gapless (the condensed matter analogue of massless) excitations as expected from Goldstone's theorem [2]. If the spins are given a position dependent rotation that is slowly varying with respect to the lattice spacing, then locally each piece of the ferromagnet is approximately in some ground state. Since there are no long-range interactions in the Hamiltonian, the energy of such a state can be brought arbitrarily close to the ground state energy by letting the wavelength of the spin rotation become longer and longer. The collection of eigenstates with spatial variations in spin alignment and arbitrarily small energies are called spin-waves. An order parameter for this system is the expectation value of the z-component of the spin averaged over a group of spins. Local fluctuations of this order parameter about its ground state value correspond to superpositions of spin-waves. The introduction of Goldstone boson fields as fluctuations of an order parameter in our model is motivated by this analogy in condensed matter physics*.

Chapter 2 reviews chiral symmetry breaking in QCD and the phenomenological Lagrangian technique. Through the work of Weinberg [4], Coleman, Callan, Wess, and Zumino [5] one can easily write down a phenomenological Lagrangian which represents the low-energy behaviour of the Goldstone bosons associated with a particular spontaneously broken (or hidden) symmetry. For this Lagrangian to involve only Goldstone bosons requires that any other particles which interact strongly with the Goldstone bosons be heavier than the Goldstone bosons. The form of the phenomenological Lagrangian is then determined by symmetry considerations, while the coefficients of the various terms must be fit to experiment. If one had a chirally symmetric Lagrangian which modeled the underlying dynamics, the effective action obtained by integrating out the fields other than the Goldstone bosons must correspond to the appropriate phenomenological Lagrangian. However, the coefficients of the various terms would be determined by the properties of the underlying model.

For the case of QCD it is well known that the pseudoscalar octet mesons have the properties required of almost-Goldstone bosons of spontaneously broken chiral symmetry; this is borne out by lattice Monte Carlo simulations [6]. The appropriate phenomenological Lagrangian which describes the low energy behaviour of pi's, K's and the eta, is called a chiral Lagrangian. Chapter 3 explains our model for low energy QCD and our motivation for the specific form the Lagrangian takes. The model can be tested by comparing the experimentally determined coefficients (like the pion decay constant F_pi) in the chiral Lagrangian to those in the effective action derived from the model. However, the coefficients in the effective action depend on integrals involving the quark self-energy, which is unknown. Instead, we first compare the integral expression we obtain for F_0 (the pion decay constant in the limit that the current quark masses vanish) to another calculation of F_0 due to Pagels and Stokar [7]. We also compare our result for the coefficient of the Wess-Zumino term with the known coefficient [25].

To make further comparisons with QCD requires the introduction of explicit symmetry breaking, that is, quark masses and electroweak gauge fields. These are described in chapters 4 and 5 respectively. The introduction of quark masses is relatively simple. The major complication is that the self-energy (the order parameter) can depend on the quark masses. The introduction of an electromagnetic field (i.e., the introduction of local gauge invariance) in a nonlocal model can be done formally, but calculations become quite awkward. They are however doable, and give a prediction for the electromagnetic charge radius of the pion. Weak decays, like págen, can be determined in a similar manner.

In chapter 6 our results are compared with experiment. Using a crude approximation for the self-energy we can predict the 14 parameters of the chiral Lagrangian (to fourth order in an energy expansion) with the 5 parameters in our model. In addition, our model allows us to determine the current quark masses**.

Other questions that can be addressed using these techniques include those concerning the properties of the techni-Higgs. This is discussed in chapter 7. The techni-Higgs may be viewed as a dilaton, or Goldstone boson associated with spontaneously broken scale invariance. Of course quantum field theories are not scale invariant, as demonstrated by the case of massless QCD. Although this theory has no mass scale in the Lagrangian, the effective coupling constant runs with momentum, so there is indeed a scale. Such explicit scale invariance breaking would give a mass to the dilaton. In the case of technicolour theories there has been recent interest in walking theories, that is theories in which the coupling runs very slowly. It might be expected that such theories could possess a light dilaton. By techniques similar to those mentioned above we calculate the dilaton decay constant.

In chapter 8 we compare our results with those from other techniques, and summarize our conclusions.

Footnotes for Chapter 1:

* For other examples of Goldstone bosons in condensed matter physics see reference [3].

** A phenomenological chiral Lagrangian can only determine quark mass ratios.

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Chapter 2

QCD and Spontaneous Chiral Symmetry Breaking

2.1 - QCD

A major event in the history of particle physics was the development of quantum chromodynamics (QCD) circa 1972 [8]. QCD describes the interaction of various flavours of quarks (each of which come in three colours) with eight, non-Abelian, coloured gauge fields. The QCD Lagrangian is given by

L_QCD = psibar(x){gamma^µ   i  partial_µ +g  T^a  A^a_µ (x)  +  M}psi(x) - (1/4) F^a_µ nu F^a^µ nu, (2.1)

where the gluon field strength tensor is

F^a_µ nu = partial_µ A^a_nu - partial_nu A^a_µ + g f^abc A^b_µ A^c_nu , (2.2)

and M is the quark mass matrix (flavours and colours are implicitly summed over). The generators and structure constants of the gauge group SU(3)colour are denoted by Ta and fabc respectively. This Lagrangian (eq. 2.1) is invariant under local SU(3)colour gauge transformations. In the limit that the quark masses go to zero, it is also invariant under global transformations in which the left and right-handed quark flavours are independently rotated into other flavours by elements of SU(N), where N is the number of flavours. This SU(N)_L× SU(N)_R symmetry is called a chiral symmetry. This approximate chiral symmetry was built into QCD in the hope of successfully reproducing the current algebra results of the 60's.

The discovery of asymptotic freedom in 1973 [9] gave the new theory its first great qualitative success. By the start of the 80's, QCD was widely accepted as the theory of strong interactions. However, almost 20 years after its inception, QCD has not yet been quantitatively tested by experiment. This sad state of affairs is not the result of a lack of effort by experimentalists, but the inability of theorists to make predictions that can be tested with available machines^1. The problem with (or saving grace of) QCD is that non-perturbative effects dominate at energies where the most interesting data is available (~ 1 GeV). Despite this, and for lack of an alternative theory, QCD is now, more than ever, widely believed to be the correct theory of strong interactions. Discounting the possibility of exactly solving QCD, the avenues open to the theorist wishing to investigate strong interactions at low energies are to analyze QCD numerically with lattice simulations^2, or to construct models of QCD. In this thesis we will pursue the latter course.

A reasonable model of QCD is usually required to incorporate two non-perturbative effects: confinement and spontaneous chiral symmetry breaking. While (in the massless limit) confinement implies spontaneous chiral symmetry breaking, the reverse is not necessarily true. In fact lattice simulations have shown that chiral symmetry breaking in QCD-like theories can occur at much larger energy scales than the confinement scale [10]. This lends support to the idea that it may be possible to treat these two effects separately. For example, Georgi [11] has argued that the chiral symmetry breaking scale is roughly 1 GeV, while the conventional confinement scale is of order 100-300 MeV. In what follows we will concentrate almost entirely on chiral symmetry breaking.

The presently accepted (current) masses (renormalized at 1 GeV) for up, down, strange, charm, and bottom quarks are [12]:

m_u = 5.1 ± 1.5 MeV,
m_d = 8.9 ± 2.6 MeV,
m_s = 175 ± 55 MeV,
m_c = 1.35 ± 0.05 GeV, and
m_b = 5.3 ± 0.1 GeV. (2.3)

For most physics below 1 GeV, the heavy quarks (charm and bottom) should make only a negligible contribution, that is, the first order approximation is that they are infinitely heavy. The light quarks, on the other hand, have masses much lighter than the typical hadronic scales (m_rho, m_proton) and a reasonable first approximation (at least for up and down quarks) might be to set their masses equal to zero (i.e., take the the chiral limit). In this limit the QCD Lagrangian has, neglecting the heavy quarks, an SU(3)_L× SU(3)_R symmetry. Normally we would expect the spectrum of the theory to reflect the symmetries of the Lagrangian, which would imply that, for example, vector and axial-vector mesons should have the same masses, whereas in reality m_rho approx 770 MeV, and m_a1 approx 1270 MeV. The symmetry that is apparent in the spectrum is the diagonal subgroup SU(3)_V (the subscript indicates a vector symmetry, since the left and right-handed quarks are rotated by the same amount) or, as it is more commonly known SU(3)_flavour. There are three possibilities which explain this discrepancy: i) QCD is wrong, ii) taking the quark masses to be zero is a bad approximation, or iii) chiral symmetry is spontaneously broken. Stubbornly, we will examine the third possibility.

2.2 - Spontaneous Chiral Symmetry Breaking

Spontaneous symmetry breaking refers to the situation where the ground state (or vacuum) does not enjoy the symmetry of the Lagrangian. That is, the vacuum is not invariant under a symmetry operation. Through Goldstone's theorem [2] this implies the existence of massless particles, one for each broken generator, each with the quantum numbers of the associated generator. If a small explicit symmetry breaking term is added to the Lagrangian, the Goldstone bosons (GB's) obtain a small mass. In our case, there are eight broken axial generators lag5, where la is a generator of SU(3), so the GB's should be a flavour octet of pseudoscalar mesons. In reality, the eight lightest hadrons are the pseudoscalar mesons: pi^±, pi^0, K^±, K,Kbar, and eta. To see how chiral symmetry might be spontaneously broken in QCD, we will have to take a closer look at non- perturbative dynamics.

Let us consider the full quark propagator in massless QCD: Sf(p). Order by order in perturbation theory, chiral symmetry ensures that Sf(p) has the form

Sf(p) = i/(Z(p)  pslash) . (2.4)

However, at low energies we cannot trust perturbative arguments in QCD. The most general form for Sf(p) allowed by Lorentz and parity invariance is

Sf(p) = i/(Z(p)  pslash  -  S(p)) , (2.5)

where S(p) is the quark self-energy and S(p)/Z(p) is often referred to as the dynamical mass (it is often hoped that the dynamical mass is somehow related to the constituent mass used in non-relativistic quark models). To go beyond the confines of perturbation theory and determine which form the quark propagator takes, we turn to the Schwinger-Dyson equations. The Schwinger-Dyson equations are an infinite set of coupled integral equations that completely determine the theory^3. The Schwinger-Dyson equation for the quark propagator (or gap equation^4) is (in Minkowski space)

where G_µ nu(p-k) is the full gluon propagator, and Lambda^nu(p-k,k,p) is the full gluon-quark-quark vertex. This equation is shown pictorially in Figure 2.1. Since S(p) appears on both sides of this equation, it must be solved self-consistently; this allows for non-perturbative solutions.

Figure 2.1; The Schwinger-Dyson equation for the one particle irreducible (1PI) two point function. The straight line represents the full quark propagator.

Although simple approximations (eg., the ladder approximation [13], where G_µ nu(p-k) and Lambda^nu(p-k,k,p) are replaced by their tree-order counterparts) do indeed give non-trivial solutions for S(p), no one has succeeded in proving that S(p) is non-zero in massless QCD. However, lattice calculations should be able to determine whether or not S(p) = 0. The simplest way to do this is to note that the vacuum expectation value of psibar(x)psi(x) is determined by evaluating a single quark loop using the full propagator^5. Thus we find that, for one flavour,

where we have Wick rotated to Euclidean momentum, the 0 indicates the massless limit, and N_c is the number of colours ( N_c = 3). The importance of this formula is that S(p) is non-zero if and only if < 0|  psibar  psi|0> _0 is non-zero. Since psibar  psi is not invariant under chiral transformations, either of S(p) or < 0|  psibar  psi|0> _0 being non-zero implies that the QCD vacuum itself is not invariant, that is, chiral symmetry is spontaneously broken. A more concise way to say this is that S(p) and < 0|psibar  psi|0> _0 can be considered to be order parameters for chiral symmetry breaking.

Current lattice results do indeed suggest that < 0|psibar  psi|0> is non-zero in the chiral limit, and that chiral symmetry is spontaneously broken in QCD [14].

2.3 - QCD at Low Energies

The idea that chiral symmetry was spontaneously broken was in fact developed before QCD. Armed only with symmetry arguments, a small band of theorists in the sixties made an attack on strong interactions. They attempted to predict various scattering and decay amplitudes using a few measured parameters. Without a fundamental theory of strong interactions, one might have expected that they wouldn't get very far. However, they were remarkably successful: modern theorists have yet to calculate these amplitudes from QCD with any degree of accuracy. The point is that chiral symmetry forces Ward-Takahashi (WT) identities upon the Green functions of the theory, and Goldstone's theorem requires the existence of light pseudoscalar mesons with a special coupling to axial-vector currents. If there are no other light hadrons then the GB's dominate the low energy physics, and measuring a particular amplitude gave information, through WT identities, on amplitudes with additional low energy GB's. Thus with current algebra, the PCAC (partially conserved axial current) hypothesis, and soft-pion theorems^6, theorists were able to elucidate strong interaction effects without having to understand the underlying dynamics, indeed without having to assume that a fundamental Lagrangian even existed^7. Weinberg was the first to point out that a simpler method for obtaining these symmetry results was to use an effective Lagrangian [4]. In Weinberg's words [15]:
. . . if one writes down the most general possible Lagrangian, including all terms consistent with assumed symmetry principles, and then calculates matrix elements with this Lagrangian to any given order in perturbation theory, the result will simply be the most general possible S-matrix consistent with analyticity, perturbative unitarity, cluster decomposition and the assumed symmetry principles.

If we are only interested in the interactions at low energies, then we can concentrate on the GB's, and neglect interactions with more massive particles. Of course this approximation will break down at large energies, which suggests that we arrange the effective Lagrangian as an expansion in powers of the energy (momentum) of the GB's divided by some typical energy scale of the theory. Terms with higher powers of momentum will be suppressed if the momentum is less than the typical energy scale. Such effective Lagrangians based on chiral symmetry are referred to as chiral Lagrangians.

The only obstacle in the way of immediately writing down a low energy chiral Lagrangian is that there are an infinite number of Lagrangians that give the same low energy matrix elements. These Lagrangians are only distinguished by the manner in which the GB fields transform under the SU(3)_L× SU(3)_R symmetry. Given an effective Lagrangian of the GB fields pa, we can invent a new Lagrangian describing the same physics by defining new fields Pa = pa f(pa), with the only constraint being that f(0) = 1 (this result goes by the name of Haag's theorem [5,16]).

Obviously, the thing to do is to choose a GB field that transforms in some "nice" way. The "nicest" transformation properties were determined by Coleman, Callan, Wess, and Zumino [5]. They advocated the use of non-linear realizations of the SU(3)_L× SU(3)_R symmetry. This sounds extremely complicated, but the underlying idea is quite familiar in the context of classical mechanics. Given a point particle, we can choose to describe its position using Cartesian coordinates (x,y,z) or spherical coordinates (r,theta,phi). If the problem we are solving has a spherical symmetry, it is easier to use spherical coordinates. The Cartesian coordinates transform linearly under the rotation group, q and f transform non-linearly. Thus, spherical coordinates are termed a non-linear realization of the rotation group.

A standard choice for representing the GB fields is U(x) = exp (-2i pi(x)/F_0 ) where pi(x) equiv lambda^a pi^a(x) (the repeated index is implicitly summed over), and {lambda^a} are the generators of SU(3) with the normalization Tr(lambda^a  lambda^b) = (1/2) delta^ab. The matrix field U transforms under SU(3)_L× SU(3)_R as U' = LUR&dagger, where L and R are elements of SU(3)_L and SU(3)_R respectively. With this choice, the effective low energy Lagrangian for massless QCD is given by writing down all possible hermitian SU(3)_L× SU(3)_R invariants built out of U's. The invariants with no derivatives, like Tr(U&daggerU), are constants, so the first non-trivial term has two derivatives. Thus

L = (1/4) F_0^2 Tr(partial_µ U&daggerpartial^µ U) + . . . , (2.8)

where . . . indicates terms with more derivatives. Expanding in powers of pa, we find that the lowest order term is 12 partialmpapartialmpa. The factor 14 F_0^2 is present in equation (2.8) so as to obtain the standard normalization of this kinetic term. The SU(3)_L× SU(3)_R symmetry of this Lagrangian is spontaneously broken. A typical ground state is specified by < 0|U|0> = 1, which is only invariant under SU(3)_V.

External vector (V_µ) and axial-vector (A_µ) gauge fields can be included by replacing the derivative with a covariant derivative

D_µ U = partial_µ U - i[V_µ   +  A_µ ] U + iU[V_µ   -  A_µ ] . (2.9)

By examining the coupling of a pion to an external axial-vector field (i.e. the pion decay amplitude < 0|J^aµ _5(x)|pi^b(q)> equiv iq^µ   F_pi d^ab exp(-iq.x) ) one finds that, to this order, the pion decay constant F_pi equals F_0 (with our conventions F_pi approx 93 MeV).

One can also easily include the effects of explicit symmetry breaking, that is the effects of non-zero quark masses. To do this we consider the effect on the QCD Lagrangian (eq. 2.1) when the quark fields are transformed under SU(3)_L× SU(3)_R. The effect is to replace^8 the quark mass matrix M by L&daggerMR. If M transformed as M' = LMR&dagger, then the QCD Lagrangian (eq. 2.1), including mass terms, would have a global chiral symmetry. However, since M does not transform the mass term breaks the chiral symmetry. This argument shows us how to include explicit symmetry breaking in the chiral Lagrangian. To first order in M, we can write down one term that breaks the chiral symmetry, but would be invariant if M transformed in the prescribed manner [17]. Combining the two derivative term with this new term (allowing for an undetermined coefficient B_0) we obtain the chiral Lagrangian to lowest (second) order in the energy expansion^9:

L_2 = (1/4) F_0^2 {Tr(D_µ U&daggerD^µ U)  +  2B_0  Tr(MU&dagger +  M&daggerU)}. (2.10)

Since M doesn't transform, this additional term breaks the chiral symmetry and gives the GB's a mass (often they are then referred to as pseudo- or almost-Goldstone bosons.). To see this explicitly we can expand equation 2.10 in powers of pa and read off the mass terms. To this order we find (for simplicity we take mu = md = m ) :

M_pi^2 = 2 m B_0 ,

M_K^2 = (m + m_s) B_0 ,

M_eta^2 = (2/3) (m + 2 m_s) B_0 . (2.11)

where the mass eigenstates have been suggestively named based on their implied quark content. These are just the standard Gell-Mann-Oakes- Renner [18] mass formulae if we identify < 0|psibar  psi|0> _0 with -F_0^2 B_0. As expected the GB masses vanish as the quark masses approach zero. The functional form of this dependence has been verified in lattice calculations [6]. This result also shows why it was consistent to keep terms of second order in momentum and first order in quark masses. Near the GB mass shell p2 approx M2GB ~ mquark B_0, so in the energy expansion one quark mass is of the same order as two derivatives. The mass formulae also give the Gell-Mann-Okubo mass formula [19]

M_eta^2 = (1/3) (4 M_K^2 - M_pi^2) . (2.12)

This is all very reassuring. However the reason that chiral Lagrangians are useful is not that they can reproduce known current algebra results, but that they offer a systematic method for calculating the higher order corrections to these results. To go to the next order in the energy expansion one simply lists all possible terms with up to four derivatives or two powers of quark masses. In the notation of Gasser and Leutwyler (G&L) [20] the next to leading order (fourth order) terms are^10

L_4 = L1 {Tr(D_µ U&daggerD^µ U)}2 + L2 Tr(D_µ U&daggerD_nuU)Tr(D^µ U&daggerD^nU)
+ L3 Tr(D^µ U&daggerD_µ UD^nuU&daggerD_nuU)
+ L4 Tr(D_µ U&daggerD^µ U)Tr(chiU&dagger + chi&daggerU)
+ L5 Tr(D_µ U&daggerD^µ U(U&daggerchi + chi&daggerU))
+ L6 {Tr(chiU&dagger + chi&daggerU)}2 + L7 {Tr(chi&daggerU - chiU&dagger)}2
+ L8 Tr(chi&daggerUchi&daggerU + chiU&daggerchiU&dagger)
- iL9 Tr(F^R_µ nuD^µ UD^nuU&dagger + F^L_µ nuD^µ U&daggerD^nuU)
+ L10 Tr(U&daggerF^R_µ nuUF^L^µ nu) (2.13)

where Tr indicates a trace and

chi = 2 B_0 M,
F^R_µ nu = partial_µ R_nu - partial_nuR_µ - [[R_µ ,R_nu],
F^L_µ nu = partial_µ L_nu - partial_nuL_µ - [[L_µ ,L_nu],
R_µ = V_µ + A_µ ,
L_µ = V_µ - A_µ . (2.14)

We have also dropped GB independent terms which are necessary for renormalization.
Thus the chiral Lagrangian to fourth order in the energy expansion is given by

L_chi = L_2 + L_4 + L_WZ , (2.15)

where L_WZ is the Wess-Zumino term [25], which is required to reproduce the effects of anomalies.

The revival of interest in chiral Lagrangians in the 80's was largely due to the work of G&L [20,21] who determined the fourth order coefficients by comparison to experiment. They also calculated loop corrections to the same order. The loop corrections raise an interesting point: the Lagrangians L_2 and L_4 are not renormalizable which leads to the necessity of an infinite number of parameters to describe the theory. Fortunately, as Weinberg showed, the one-loop corrections involving vertices from L_2 are of order p^4 (i.e. the same order as tree graphs from L_4). In general each loop adds an additional power of p^2. Thus to renormalize one-loop graphs from l2, we will need counter-terms of order p^4, but since L_4 contains all the possible terms allowed by chiral symmetry, we must be able to absorb the counter-terms in redefinitions of the coefficients of L_4. This means that the chiral Lagrangian is renormalizable order by order in the energy expansion.

The result of the efforts of G&L is that more accurate predictions can be made using chiral perturbation theory for a variety of low energy phenomena. For example, they use information independent of the s-wave and p-wave scattering lengths to determine L1 through L10. This information allows them to calculate the corrections to the standard Weinberg (order p2) scattering lengths. The corrections bring the chiral Lagrangian scattering lengths into even closer agreement with experiment.

What does the success of chiral Lagrangians tell us about QCD? Most importantly it tells us that lQCD has the correct symmetry. Chiral perturbation theory also enables us to determine the light quark mass ratios^11, but unfortunately tells us nothing further about QCD. To see why, let us review the assumptions that lead to the chiral Lagrangian.

Consider a Lagrangian L = L_0 + L_SB which has the following properties:
  • 1. L_0 has an SU(3)_L× SU(3)_R symmetry;
  • 2. the ground state of L_0 is only invariant under SU(3)_V;
  • 3. the symmetry breaking term L_SB is parametrized by a 3x3 matrix M, which, if allowed to transform as M' = LMR&dagger, would render L_SB invariant; and
  • 4. in the limit M --> 0, the only massless particles in theory are the eight Goldstone bosons.

    Such a Lagrangian can be approximately described at low energies by the chiral Lagrangian^12 L_chi = L_2 + L_4. At low energies the only discernible differences between alternative underlying Lagrangians are the values of the low energy parameters F_0, B_0, L1, . . ., L10. Thus, short of calculating these coefficients from QCD, chiral perturbation theory cannot distinguish between QCD and other theories with spontaneously broken chiral symmetry. The next best thing would be to derive the low energy parameters from a QCD inspired model. This is what we endeavor to do in the succeeding chapters.

    Chapter 3
    Footnotes for Chapter 2:

    ^1 Although QCD will be tested at HERA.

    ^2 This entails waiting for much more powerful computers to achieve a reasonable accuracy, and the possibility of little physical insight into how the theory actually works.

    ^3 To demonstrate this, one can derive the Schwinger-Dyson eqs. by canonical quantization, and then derive the path integral formulation from the Schwinger-Dyson eqs. [22].

    ^4 This name refers to the analogous eq. found in the BCS theory of superconductivity [23].

    ^5 This is strictly true only in the massless limit. In the massive case < 0 |psibar  psi| 0> depends upon renormalization conventions.

    ^6 For a review of current algebra see reference [24].

    ^7 At the time, Lagrangians were out of fashion, and it was widely suspected that it was impossible to describe strong interactions using a Lagrangian formalism.

    ^8 To see this, rewrite the quark mass term in terms of left and right-handed quarks, i.e., as psibar_L M psi_R + psibar_R M&dagger psi_L.

    ^9 We follow the conventional parametrization due to Gasser and Leutwyler.

    ^10 G&L use the lowest order equations of motion to express an eleventh possible invariant in terms of the others [20].

    ^11 Note that chiral Lagrangians cannot determine the scale of the quark masses. Given values for the quark masses and B_0, we can scale the masses up by a factor of 2, say, and chiral perturbation theory will give the same results provided that B_0 is divided by 2.

    ^12 If the underlying theory has anomalies, then a Wess-Zumino term must be included.