(Warning! The equations have not been converted yet, proceed with caution.) Chapter 3 A Model for Massless QCD 3.1 - Motivation for the Model We will now construct a simple model for the low energy behaviour of massless QCD. The model will contain quarks with a nonlocal, nonlinear, and chirally symmetric coupling to a Goldstone boson (GB) field. We want a model with an SU(N)L x SU(N)R symmetry which contains N flavours of quarks, each with Nc "colours", although the SU(Nc) "colour" symmetry will not be gauged. Of course, in QCD the colour symmetry is gauged: there are gluons. To make a reasonable model of QCD we have to reproduce non-perturbative effects due to gluons. To account for spontaneous chiral symmetry breaking we will include a bilocal dynamical mass S(x-y) (the Fourier transform of S(p)) which is "colour" and flavour symmetric. Thus, in Euclidean space-time1, our model Lagrangian can be written as L(x,y) = y(x)delta(x-y)/y(y) + y(x)S(x-y)y(y) + Lint(x,y), (3.1) where Lint(x,y) contains the interactions of the quarks and GB's, and flavours and colours are implicitly summed over. To obtain the action we must integrate L(x,y) over x and y. Ideally we would determine S(x-y) by solving the Schwinger-Dyson equation for the self-energy, or perhaps by measuring S(x-y) in a lattice simulation. Unfortunately, accurate determinations by either of these methods is not possible at the present time, so we will have to parametrize2 our ignorance about S(x-y). It should be noted that we have not included any confinement effects; part of the motivation for this model is to test whether or not GB dynamics are sensitive to confinement physics. It is plausible that the properties of pions are insensitive to long distance confinement effects if, for example, pions are tightly bound systems. To develop our model further, we focus on the fact that S(x-y) is an order parameter for spontaneous chiral symmetry breaking. If S(x-y) is zero, then the chiral symmetry is not spontaneously broken, and there should be no GB's present in the theory. A non-zero S(x-y) implies that the chiral symmetry is spontaneously broken down to the diagonal subgroup SU(N)V, and in analogy to condensed matter systems [1] the GB's should appear as fluctuations of the order parameter in the hidden symmetry directions. It is natural then to introduce the GB's as fluctuations in the orientation of S(x-y) in the coset space SU(N)L x SU(N)R/SU(N)V. We therefore re-write our model as L(x,y) = y(x)d(x-y)/y(y) + y(x)Sp(x,y)y(y), (3.2) where Sp(x,y) is a nonlinear functional, to be determined, of the GB field p(z) lapa(z)) (where the repeated index is implicitly summed over, and {la} are the generators of SU(N), with Trlalb = 12 dab). Sp(x,y) is required to be diagonal in "colour" space, and to reduce to S(x-y) for p(z) = 0. Our model thus has quarks interacting with GB's, but the GB's have no kinetic terms and no interactions among themselves; these will be generated dynamically. To determine Sp(x,y), we require that the Lagrangian be invariant under global SU(N)L x SU(N)R transformations. To do this we must first discuss how p(x) behaves under such transformations; we will require the standard transformation for p(x). We choose the quarks to transform globally as y' = exp(-ieg5)exp(-ia)y, (3.3) where e = eala and a = aala. To specify how p transforms we introduce V(x) = exp ????-2ip(x)g5f (3.4) (where f is a constant, with dimensions of mass, determined by the normalization of the p field) and define the GB transformation by V' exp(ieg5) exp(-ia) V exp(ia) exp(ieg5). (3.5) This is equivalent to the more usual definition, U' exp(ie) exp(-ia) U exp(ia) exp(ie), (3.6) where U(x) = exp ????-2ip(x)f (3.7) is the field that appears in the standard chiral Lagrangian. It is perhaps instructive at this point to consider a special case of the general model which is easier to understand. If we consider the case S(x-y) = m d(x-y), and S(p) = m, then an invariant Lagrangian is obviously given by a familiar nonlinear s model: l(x,y) = y(x)d(x-y)/y(y) + y(x)md(x-y)V(y)y(y). (3.8) Of course, this is not the most general invariant Lagrangian. We could replace the kinetic term by y(x)d(x-y)V(x)/ [V(y)y(y)], but terms like this do not involve the order parameter m, and hence do not represent fluctuations of the order parameter. It is interesting to note that the Lagrangian in equation 3.8 is not really consistent as it stands, since quantum effects, specifically quark loop graphs, will produce divergences that can only be absorbed by counter-terms which have the form of kinetic and self-interaction terms for the p fields. This will not be a problem for the type of model that we will consider, since if S(p) represents a quark self-energy, then the known asymptotic behaviour of S(p) will render the quark loop integrals finite, as we shall see. It can be seen from equation 3.2 that our nonlocal model Lagrangian can be chirally invariant under global transformations for a large class of possible forms for Sp(x,y). However, there is a standard method [89] for coupling the GB's to fermions that has a simple generalization for the nonlocal case. Consider the field x(x), which is the square root of V(x): x(x) = exp ????-ip(x)g5f . (3.9) Given the global transformation law for V (eq. 3.5), we find that x transforms in an unusual manner: x' = exp(ieg5) exp(-ia) x w(e,a,p(x)) = w(e,a,p(x)) x exp(ia) exp(ieg5), (3.10) where w(e,a,p(x)) is a complicated unitary matrix which must be solved for using equations which define V, x, and their transformation properties (eqs. 3.4, 3.5, 3.9, and 3.10). Although we are only discussing global transformations of the GB field p, the field x transforms in a space-time position dependent fashion that is reminiscent of a local gauge transformation. In fact, we can define a "gauge" field [11] Am = i2 ()mxx+ȍmxx , (3.11) which transforms as a real gauge field would, Am' = w Am w + i (mw) w . (3.12) This allows us to simply generalize equation 3.8 to the case of a nonlocal mass term l(x,y) = y(x)d(x-y)/ y(y) + y(x)S(x-y)x(x)E(x,y)x(y)y(y), (3.13) where E(x,y) is the path-ordered exponential [43,44] of the "gauge" fieldAm, E(x,y) = P exp????i??xyAm(w)dwm . (3.14) The operator P is the path-ordering operator. Note that there are no terms in equation 3.13 which involve traces of V's and V's. This result is phenomenologically pleasing, since such trace terms could produce amplitudes which would violate3 Zweig's rule [38] . The Lagrangian given in equation 3.13 is invariant under the global symmetry transformations (eqs. 3.3, 3.10, and 3.12) and obviously reduces to the nonlinear s model (eq. 3.8) when S(x-y) = m d(x-y). The method used for imposing the global symmetry on the nonlocal mass term may be said to be minimal in the sense that Sp(x,y) reduces to a very simple form, Sp(x,y) = x(x) S(x-y) x(y) , (3.15) when Am vanishes. The "gauge" field Am will vanish when []mx,x = 0, or when []p(x),p(y) = 0, for all x and y. The appearance of a path-ordered exponential in the Lagrangian given in equation 3.13 makes the study of such a model somewhat awkward. In order to simplify matters we will attempt to develop a model Lagrangian that can be expressed in a power series in the GB field p. If Sp(x,y) is first written as a power series in V and V, then the class of possible forms which will make the Lagrangian globally invariant is characterized by the requirement that each term have one more power of V than of V. We note that hermiticity requires that g0Sp(x,y) = Sp(y,x)g0. Thus, we write Sp(x,y) as Sp(x,y) = S(x-y) G[]p(x),p(y) (3.16) where G[]p(x),p(y)= (3.17) 12{}[]a1V(x)+a2V(x)V(y)V(x)+a3V(x)V(y)V(x)V(y)V(x)+...+[]xy. In order that Sp(x,y) = S(x-y) for p(x) = 0, the constants ai must satisfy i=1ai = 1 . (3.18) That Sp(x,y) depends only on the values of the GB fields at space-time points x and y can be seen to be consistent with the requirement that the GB fields represent fluctuations of the order parameter S(x-y). Any dependence on p(z) (zx,zy) could be rewritten by Taylor expanding around x (or y). These terms would then be seen to be nonlocal generalizations of the derivative terms that we ignored in the case of the nonlinear s model. If we require that when the GB fields commute our general form for Sp(x,y) (eqs. 3.16, 3.17) reduces to the simple form given in equation 3.15 then we obtain further restrictions on the coefficients ai. By simply equating equations 3.15 and 3.16, and commuting GB fields, we find k=1akcos((2k-1)(p(x)-p(y))) = 1 , (3.19) which is just the cosine series for the square wave. The solution for ak is: ak = 4p (-1)k-12k-1 cos((2k-1)f) , f < p2 . (3.20) This is slightly disappointing, since f is undetermined. However, if one expands equation 3.17 in powers of p(x) and p(y), one finds [39] that f drops out, and the coefficients of the p's are uniquely determined. Therefore, the value of f is physically irrelevant, at least in perturbation theory. To fourth order in p we have: G[]p(x),p(y) = 1 - if g5 []p(x)+p(y) - 12f2 []p2(x)+p2(y)+p(x)p(y)+p(y)p(x) + i6f3 g5 []p3(x)+p3(y)+3p(x)p(y)p(x)+3p(y)p(x)p(y) + 124f4 [p4(x)+p4(y)-p3(x)p(y)-p(x)p3(y)-p3(y)p(x)-p(y)p3(x) +3p2(x)p(y)p(x)+3p(x)p(y)p2(x)-3p2(x)p2(y)-3p(x)p2(y)p(x) +3p2(y)p(x)p(y)+3p(y)p(x)p2(y)-3p2(y)p2(x)-3p(y)p2(x)p(y) ]+9p(x)p(y)p(x)p(y)+9p(y)p(x)p(y)p(x) + O(p5). (3.21) This gives us a minimal model for low-energy massless QCD which is completely determined up to the functional form of S(x-y). To see whether this model is at all reasonable we will now consider how quarks couple to the GB's in massless QCD. It will be useful to compare the generalized axial Ward-Takahashi (WT) identities for our model with the analogous QCD WT identities. Since our model Lagrangian is nonlocal, the easiest way to derive the WT identities is by functional methods4, but this will lead us into a substantial digression. 3.2 - Ward-Takahashi Identities; We recall the massless QCD Ward-Takahashi identity for the one- particle-irreducible (1PI) vertex Gma5, which couples an axial-vector current, an incoming quark with momentum p, and an outgoing quark with momentum p+q, -iqmGma5 (p,q,p+q) = S-1f(p+q)lag5 + lag5S-1f(p). (3.22) From Lorentz and parity invariance, the mostgeneral form for S-1f(p), the full inverse quark propagator in QCD, can be written (in Euclidean space) as S-1f(p) = -iZ(p)p/ + S(p). The inclusion of a self-energy term in the massless theory allows for the possibility of spontaneous chiral symmetry breaking. If we take the limit as q0 of equation 3.22, we find limq0 -iqmGma5 (p,q,p+q) = 2S(p)lag5 . (3.23) If chiral symmetry is spontaneously broken, implying that S(p)0, then equation 3.23 implies that Gma5 has a pole in q, and hence that the quark antiquark scattering amplitude has a pole in q. The pole is due to the presence of a massless GB, as shown in Figure 3.1. In fact this analysis is one way to derive Goldstone's theorem [26]. Figure 3.1; The GB pole contribution to the axial-vector vertex. The solid lines denote quarks, and the dashed line denotes the GB. Examining Figure 3.1, we see that the pole part of Gma5 gets a factor from the axial current coupling to the GB; the GB propagator contributes a factor of 1q2 , and there is a factor from the GB coupling to the quarks. The most general form allowed for the coupling of the axial current to the GB is 0|Jam5(x)|pb(q) -qm F0 dab e-iq.x. (This is equivalent to the Minkowski space definition 0|Jam5(x)|pb(q) iqm F0 dab e-iq.x.) Since the GB is on its mass-shell (q2=0), F0 is the pion decay constant in massless QCD. Thus from equation 3.23 the coupling of a zero momentum GB to a quark and antiquark is determined to be Gap(p,0,p) = 2iF0 S(p)lag5 . (3.24) Now, let us consider the coupling of a GB to a quark and an antiquark in our model. In our model, this coupling is determined for arbitrary GB momentum. Using only the normalization constraint (eq. 3.18) the terms linear in p in the general form of the model (eq. 3.17) are determined: Sp(x,y) = S(x-y)????1-ifg5[]p(x)+p(y) + O(p2), (3.25) where f is the parameter introduced in the definition of V(x) (eq. 3.4). Thus the single GB vertex is completely specified and is given by Gap(p,q,p+q) = if []S(p)+S(p+q)lag5 . (3.26) This reduces to the correct functional form given by the QCD result (eq. 3.24) for zero GB momentum (i.e., q = 0). Comparing equations 3.24 and 3.26, it will not be surprising to find that f turns out to be the GB decay constant in our model. We will now derive the axial WT identities for our model. We begin with the action, S[y,y,pa], for our model, including sources for the quarks and GB's. S[y,y,pa] = d4xd4yl(x,y) - d4xyh+hy+sapa. (3.27) In Euclidean space the vacuum to vacuum transition amplitude, in the presence of sources is Z = exp()-W[h,h,sa] = dydydpaexp()-S[y,y,pa], (3.28) where W[h,h,sa] is the generating functional for connected Green functions and d implies functional integration. This allows us to simply define the expectation value of any function or functional of the fields: F[y,y,pa] Z-1dydydpaF[y,y,pa]exp()-S[y,y,pa]. (3.29) The quark and GB fields in equation 3.28 are merely integration variables, so we can perform the following local infinitesimal (axial) change of variables without affecting W[h,h,sa]: y(x) = (1 + i ea(x)lag5)y'(x), y(x) = y '(x)(1 + i ea(x)lag5), pa(x) = p'a(x) + f ea(x) . (3.30) The change in the functional measure can be ignored if we do not consider WT identities for processes related to anomalies [91]. Retaining terms of order ea the action becomes S[y,y,pa] = S[y ',y',p'a] + d4xd4ydl(x,y) - d4x(y'iealag5h+hiealag5y'+fsaea) + O(ea)2. (3.31) The change in the action arises from the change in the Lagrangian and from the change in the source terms. Since the Lagrangian is invariant under global axial transformations, the integral of the change in the Lagrangian under a local transformation is related to the divergence of the conserved current associated with the symmetry. This is just the definition of the Noether current. Thus d4xd4ydl(x,y) = -id4xea(x)mJma5(x) + O(ea2), (3.32) where Jma5(x) is the axial-vector current. For the local s model, that is S(x-y) ~ d(x-y), Jma5(x) reduces to ygmlag5y. In the nonlocal model there are additional terms involving GB fields since the dynamical mass term is no longer invariant under local transformations. We can now insert our expression for the action into equation 3.28, and expand in powers of ea. We note that W[h,h,sa] is independent of ea, so that terms on the right hand side of (the transformed) equation 3.28 proportional to ea must vanish. Thus, to order ea, dropping primes, and dividing by Z, we have d4x[]imJma5+iylag5h+ihlag5y+fsaea(x) = 0 . (3.33) Now, ea(x) is an arbitrary function, so we can let it approach d(x-z). Thus imJma5(z) + iy(z)lag5h(z) + ih(z)lag5y(z) + f sa(z) = 0 . (3.34) To get the WT identity in a form that is easy to apply to amplitudes, we must introduce the effective action, which is a functional of expectation values of fields. For notational simplicity we define Y(x) y(x) = dWdh(x) , Y(x) y(x) = -dWdh(x) , Pa(x) pa(x) = - dWdsa(x) , Jma5(x) Jma5(x) . (3.35) Then the effective action, G[Y,Y,Pa], is defined by a Legendre transform of W[h,h,sa]. Thus G[Y,Y,Pa] W[h,h,sa] + d4xYh+hY+saPa. (3.36) It follows from this definition that dGdY(x) = - h(x), dGdY(x) = h(x), dGdPa(x) = sa(x). (3.37) Then equation 3.34 can be rewritten as m Jma5(x) + Y(x)lag5dGdY(x) - dGdY(x)lag5Y(x) - i f dGdPa(x) = 0 . (3.38) This is the basic axial WT identity for our model; all axial WT identities follow from this equation. Setting the external sources to zero, this equation simply tells us that the expectation value of the axial-vector current is conserved. More useful results can be obtain by taking functional derivative of this equation with respect to the fields. Functional derivatives of the effective action, G, are just the one-particle irreducible (1PI) amplitudes, with (amputated) legs corresponding to each field differentiation. The functional derivatives of Jma5(x) with respect to fields are 1PI amplitudes with one insertion of the current. By differentiating equation 3.38 with respect to Y(z) and Y(y), setting sources to zero, and Fourier transforming we arrive at the long awaited WT identity for the axial-vector vertex Gma5 in our model, -iqmGma5 (p,q,p+q) = S-1(p+q)lag5 + lag5S-1(p) + i f Gap(p,q,p+q), (3.39) where S(p) is the quark propagator and Gap(p,q,p+q) is the GB-quark- quark vertex in our model. This equation must, of course, be true to all orders. At tree order, using equations 3.26 and 3.32, both sides simplify to -iq/ lag5 . (3.40) The pole in the QCD axial-vector vertex (eq. 3.22) is not present in the model axial-vector vertex, but the relevant information is still retained. The GB bound state that was responsible for the pole in QCD is treated as a field in its own right in our model, and cannot contribute to Gma5 since this vertex is 1PI. However, the GB now contributes the extra term on the right hand side of equation 3.39, so that the WT identities are satisfied. At this point we would like to make a comparison of the results of our model with those of "dynamical perturbation theory" (DPT). DPT was introduced by Pagels and Stokar [7] as an approximation scheme for massless QCD which would allow them to study spontaneous chiral symmetry breaking. Formally DPT is a double perturbation expansion in the gauge coupling, g, and l = exp ????-1bg2 . Terms of order l are non- perturbative effects, like S(p), that vanish order-by-order in perturbation theory. To lowest order, DPT keeps terms of order l, and drops terms of order g. Thus the quark propagator in lowest order DPT is given by S-1(p) = -ip/ + S(p), (3.41) which is also the propagator in our model (for vanishing background GB field). The non-pole piece of the axial-vector vertex, Gma5, in lowest order DPT is simply lagmg5. Using the DPT results for the quark propagator and the non-pole part of Gma5, Pagels and Stokar were able to determine the GB- quark-antiquark coupling from the QCD WT identity for the axial-vector vertex. By simply repeating the argument leading up to equation 3.23 and using the DPT results one finds: Gap(p,q,p+q) = iF0 []S(p)+S(p+q)lag5 . (3.42) This has the same form as the GB coupling in our model, with the (as yet) undetermined parameter f replaced by the pion decay constant F0. We will show later on that f and F0 are indeed equal in our model. 3.3 - Comparison With Other Models At this point it may be useful to compare our model with other models of spontaneous chiral symmetry breaking. The first such model was the s- model of Gell-Mann and Levy [32]. The interactions of the (scalar) s and (pseudoscalar) p fields, leads to a vacuum expectation value for s, which gives a mass to the nucleons. The p field then plays the role of a GB (although Gell-Mann and Levy actually introduced a symmetry breaking term which made the p's massive). The s-model can be turned into the non-linear s model by letting the mass of the s particle approach infinity [33]. The next historically significant model of spontaneous chiral symmetry breaking was presented in the classic paper of Nambu and Jona- Lasinio [34]. Their model is based on the analogy with spontaneous symmetry breaking in superconductivity. The model contains a nucleon with an attractive, chirally symmetric, four fermion coupling. Chiral symmetry is spontaneously broken by a solution to a Schwinger-Dyson equation (with a cutoff) for the nucleon self-energy. The pion in this model arises as a massless bound state of the nucleons, it is not introduced as a fundamental field. Nambu and Jona-Lasinio were the first to discuss the fact that spontaneous symmetry breaking requires the presence of massless particles (the GB's) for consistency of the theory. Since the advent of QCD, quarks have replaced nucleons as the presumed constituents of the pion. Accordingly, a variety of local models [28,29] involving quarks interacting with GB's have been proposed. Often these models are developed with the intent of integrating out the quarks and thereby deriving effective chiral Lagrangians. Some of these local models can be considered as equivalent to a special case of our model when S(x-y) = m d(x-y), and S(p) = m, that is the nonlinear s model. These models are usually re-written in a slightly different fashion from the nonlinear s model, by performing a local axial transformation of the quark fields which shifts all the p dependence from the mass term onto the kinetic term. Unfortunately, such local models produce divergent results for GB decay constants, and the second order chiral Lagrangian l2 (eq. 2.10) must be explicitly introduced into the model to absorb these infinities. Thus the decay constants cannot be predicted by these models; they are input parameters. This problem is usually avoided by ignoring contributions to l2, and only considering loop graphs that are convergent in the local limit, which then give predictions for some of the parameters of the fourth order chiral Lagrangian l4. For example, Balog [28] has obtained values for L1, L2, L3, L9, and L10 which, incidentally, turn out to be independent of m. The most sophisticated treatment of local models is that of Espriu, De Rafael, and Taron [42]. These authors include the effects of current quark masses, and attempt to calculate gluonic corrections. In addition to the results of Balog, they obtain L5 and L8. However, since they are representing the quark self-energy by a constant (S(p) = m), their results for L5 and L8 are cutoff dependent. Some nonlocal quark-GB models have also been considered [30,31]. The nonlocal model that is most similar to ours is that of Roberts, Cahill, and Praschifka [30]. Their model may be considered as a nonlocal generalization of the Nambu-Jona-Lasinio model [34]. A major difference between our approaches is that they make model assumptions about interactions between quark currents; these interactions are supposed to model the effects of gluon exchange between two quarks. Roberts et. al. do not calculate any of the parameters5 of l4. 3.4 - The Effective Chiral Lagrangian With our nonlocal model for quarks and GB's firmly in hand, we can now derive an effective action [27,35] for the GB's alone. The effective action6, G[p] = d4xleff()p(x), for the GB fields is defined by exp()-G[p] ??dydyexp????-d4xd4yl(x,y) . (3.43) The effective action is useful for energies below the "threshold" for quark pair production (roughly twice the quark "mass") or for calculating processes that only involve GB's in the initial and final states. Because our model is chirally symmetric, the effective Lagrangian leff is guaranteed to take the form of a chiral Lagrangian, hence we can calculate the couplings of the chiral Lagrangian that reproduces the low-energy behaviour of our effective Lagrangian. We will refer to the chiral Lagrangian determined in this manner as the effective chiral Lagrangian. By integrating out the quark fields we obtain G[p] = - Tr log S-1p. (3.44) where S-1p = -/yd(x-y) + Sp(x,y), and Tr implies a trace over all available indices, including, if necessary, a functional trace over space-time indices. It is convenient to expand G[p] using S-1(x,y) = -/yd(x-y) + S(x-y), the quark propagator in the absence of external p fields. Inserting a factor of S-1 S gives G[p] = - Tr log S-1 - Tr log S S-1p. (3.45) The first term on the right hand side of this equation is independent of p(x), i.e. it is a constant term in the effective chiral Lagrangian, so we will drop it. The second term can be written as G[p] = - Tr log ????d4zS(x,z){}S-1(z,y)+Sp(z,y)-S(z-y) = - Tr log ????d(x-y)+d4zS(x,z){}Sp(z,y)-S(z-y). (3.46) We will Taylor expand the functional log in powers of the interaction Sp(z,y) - S(z-y), and expand the interaction in powers of p(x). We then collect terms of the same order in p, and denote terms of order p, p2, ... by G1[p], G2 [p], ... respectively. Thus Gn [p] is determined by a set of Feynman graphs with one quark loop and n external GB legs, with GB fields attached to the legs and the space-time coordinates of the fields integrated over. We can now proceed to calculate terms in the effective action. G1[p] vanishes, since it contains one factor of g5 p and one quark propagator, so the trace over Dirac matrices will vanish, i.e. Tr g5 = 0 and Tr g5 gm = 0. G2 is given by G2 [p] = 12 Tr d4zS(x,z)S(z-y){}[]b(z)2+b(y)2+[]b(z)b(y)+b(y)b(z) - 12 Trd4zd4wd4vS(x,z)S(x-w)[]b(z)+b(w)S(w,v)S(v-y)[]b(v)+b(y). (3.47) where b(x) = p(x)f g5. Fourier transforming this equation we obtain G2 [p] = -??d4pd4q(2p)8TrS(q)????S(q)+12(S(q+p)+S(q-p))b(p)b(-p) + 12 ??d4pd4q(2p)8TrS(q)[]S(q)+S(q-p)b(p)S(q-p)[]S(q-p)+S(q)b(-p). (3.48) We have followed the standard convention7 for Fourier transforms of Euclidean fields: f(x) = i ??d4p(2p)4e-ip.xf(p). (3.49) To make G2 [p] look more like an effective Lagrangian we rewrite it as G2 [p] = ??d4p(2p)412pa(p)pb(-p)Fab(p). (3.50) Using S(p) = igmpm+S(p)p2+S2(p) , (3.51) and performing the traces we find Fab(p) = 2Ncdabf2 ??d4q(2p)4()q.(q-p)+S(q)S(q-p)()S(q)+S(q-p)2()q2+S2(q)()(q-p)2+S2(q-p) - 4Ncdabf2 ??d4q(2p)4S(q)[]S(q)+S(q-p)q2+S2(q) . (3.52) The two terms in equation 3.52 are given by the diagrams (a) and (b) respectively in Figure 3.2. Figure 3.2; Quadratic terms in the effective action. It can be seen from equation 3.52 that Fab(p=0) vanishes. From equation 3.50 we see that a non-zero value for Fab(p=0) corresponds to a mass term for the GB. Chiral symmetry assures us that a constant GB field, p(x) = constant, is equivalent to p(x) = 0. This is evident from the Lagrangian (eqs. 3.2, 3.16, and 3.17), since a constant GB field can be absorbed into a redefinition of the quark fields: y = exp ????ipg5fy', y = y ' exp ????ipg5f. Thus, the effective chiral Lagrangian should only depend on derivatives of p(x), and hence the p field should be massless. One may have worried that using a nonlocal Lagrangian might invalidate Goldstone's theorem, so it is reassuring to find that the GB's are indeed massless in this model, as required. It should be noted that the vanishing of the GB mass is non-trivial to the extent that it involves a cancellation between the two graphs in Figure 3.2. To get further information we will perform a Taylor expansion of Fab2(p) in momentum around p=0. Upon Fourier transforming equation 3.50 we would then have the quadratic piece in the effective Lagrangian given as a series of terms with increasing numbers of derivatives, i.e. a derivative expansion of the nonlocal theory. Of course a Lagrangian with an infinite number of terms is usually not very useful, but if we are only interested low energy physics, then we only need the first few terms of such a derivative expansion. It will be convenient to rewrite equation 3.52 as Fab(p) = Ncdabf2 ??d4q(2p)44q.pS2(q)-2p2()2S2(q)+S(q)S(q-p)()q2+S2(q)()(q-p)2+S2(q-p) . (3.53) Writing Fab(p) as Fab(p) = p2 F2 dab + O(p4), (3.54) we find8 F2 = 2Ncf2 ??d4q(2p)4q2S(q)S(q)-2S2(q)()q2+S2(q)2 . (3.55) where denotes ddq2 . In massless QCD one can calculate the asymptotic behaviour of S(q). The result is that for q2 large and Euclidean, S(q) ~ 1q2 , up to powers of logarithms of q2 [37]. With this asymptotic behaviour for S(q), the integral in equation 3.55 converges. Note that with S(q) = m, the integral diverges logarithmically. If we require the standard normalization of the kinetic term in the Lagrangian, 12 mpa mpa, then with our Fourier transform convention in equation 3.49 we must require F2 = -1. This determines the normalization constant f. Performing the angular integration we obtain f2 = Nc4p2??0dq2q2S2(q)-12q2S(q)S(q)()q2+S2(q)2 . (3.56) This result was previously obtained by Pagels and Stokar in their calculation of the pion decay constant [7,36]. Pagels and Stokar derived their formula for the pion decay constant by calculating the amplitude for the annihilation of a GB by an axial-vector current in "dynamical perturbation theory". The Feynman diagram for this amplitude is shown in Figure 3.3. Pagels and Stokar used equation 3.41 for the quark propagator, equation 3.42 for the GB vertex, and the usual tree-order axial-vector current vertex lagmg5. Figure 3.3; The Pagels and Stokar pion decay amplitude. Our model, however, indicates that there are additional GB dependent contributions to the axial-vector current. Recall from Noether's theorem that the non-invariance of the standard kinetic term under local axial transformations leads to the usual contribution to the divergence of the axial-vector current: m(ygmlag5y). As was noted in the derivation of the axial WT identities, since the nonlocal term Sp(x-y) is also not invariant under local axial transformations there are necessarily additional contributions to the divergence of the axial-vector current. These additional contributions are nonlocal, GB dependent, and gm independent. The contribution that involves one GB also contributes to the amplitude for the annihilation of a GB by an axial-vector current. Thus it may seem somewhat surprising that our model gives the same result as the Pagels- Stokar calculation. However, consideration of the WT identities for our model show that this is no accident. Returning to the general WT identity in equation 3.38, we differentiate with respect to Pb(y) and then set the sources equal to zero to obtain the identity xm Jma5(x)dPb(y) = i f d2GdPa(x)dPb(y) , (3.57) which relates the amplitude for the annihilation of a GB by the divergence of the axial-vector current to the inverse GB propagator. This identity is shown to one loop in Figure 3.4. Figure 3.4; WT identity for the nonlocal model. The left hand side is the GB decay amplitude, the terms in brackets on the right are the inverse GB propagator. The box in the second term on the left hand side denotes the nonlocal contribution to the axial-vector current. Using the definition of the pion decay constant9 0|Jam5(x)|pb(q) -qm F0 dab e-iq.x , and the lowest order propagator from the effective Lagrangian obtained above we obtain i q2 F0 dab = i f q2 dab , (3.58) which demonstrates that f is indeed the GB decay constant. The WT identity in equation 3.57 also demonstrates that the method of calculating the decay constant by normalizing the kinetic term in the effective Lagrangian is equivalent to calculating the annihilation amplitude, as long as the full nonlocal axial-vector current is used. The question that remains is why our model produces the same result as the Pagels-Stokar calculation of the GB annihilation amplitude when Figure 3.4 shows that there is an additional graph that they did not consider. The answer is that although the additional nonlocal contribution to the axial-vector current does not vanish, the additional graph does vanish. It vanishes because the nonlocal contribution to the axial-vector vertex involving one GB is proportional to the commutator of the generators associated with the current and GB. When we calculate the Feynman graph, the fermion loop is traced over, and the trace of a commutator vanishes. It can easily be seen that the appearance of a commutator in the axial current vertex generalizes to more complicated vertices. If we perform a local SU(N)L x SU(N)R transformation on the quark and GB fields, we find that the minimal Lagrangian is locally invariant, provided that the generator of the transformation, p(x), and p(y) all commute with each other for arbitrary x and y. Another way to say this is that amplitudes involving left-handed or right-handed currents, quarks, and GB's must be proportional to commutators involving the generators of the GB fields and the currents. 3.5 - Non-Minimal Models In this section we will consider results for non-minimal models, that is we will consider a more general form for Sp(x,y): Sp(x,y) = S(x-y)??1-if'g5[]p(x)+p(y) - af'2 []p2(x)+p2(y) - bf'2 []p(x)p(y)+p(y)p(x) ??-cf'2[]pa(x)pa(x)+pa(y)pa(y)-2df'2pa(x)pa(y)+O(p3) (3.59) (Recall p(z) lapa(z)). By considering the possibility that S(x-y) ~ d(x-y), in which case the model should reduce to the nonlinear s model, it is established that a+b = 1, and d= -c. This general form reduces to our minimal model if we choose a=b= 12 and c = d = 0. We note that the form of Sp(x,y) given in equation 3.59 is so general as to be phenomenologically unpleasant since the terms proportional to c and d violate Zweig's rule [38]. If we now repeat the analysis given in section 3.4, using equation 3.59 for Sp(x,y). We find (using b = 1 - a, and d = - c) the coefficient of the quadratic term in the effective chiral Lagrangian to be: Fab(p) = Ncdabf'2 ??d4q(2p)44q.pS2(q)-2p2()2S2(q)+S(q)S(q-p)()q2+S2(q)()(q-p)2+S2(q-p) - 8Nc(a-12+2cN)dabf'2 ??d4q(2p)4S2(q)()S(q)-S(q-p)q2+S2(q) . (3.60) Taylor expanding in momentum, and requiring that the GB field be properly normalized we obtain f'2 = Nc4p2??0dq2q2S2(q)-12q2S(q)S(q)()q2+S2(q)2 - Nc(a-12+2cN)2p2??0dq2q2S(q)S(q)+12q2S(q)S(q)q2+S2(q) . (3.61) Of course, the second term vanishes for the minimal model where a = 12 and c=0. As before, using the WT identity (eq. 3.57) we can show that in a model with the more general form of Sp(x,y) (eq. 3.59) the GB decay constant is given by f'. The WT identity also shows where the difference between the general model and the Pagels-Stokar calculation arises. The WT identity can be checked explicitly to one loop and the result for the GB decay constant agrees with equation 3.61. Now, the nonlocal contribution to the axial-vector current does not vanish, it provides the term proportional to a - 12 + 2cN. The conclusion of this analysis of the GB decay constant, F0, is that the requirement of chiral symmetry is not enough to uniquely determine F0 in a more general model. The more general form of the model contains the same quark propagator and one GB vertex as Pagels and Stokar's DPT, however they arrive at a unique result by omitting the nonlocal piece of the axial current, which is, in general, required for consistency. 3.6 - The Effective Chiral Lagrangian at Fourth Order Recall that to second order in the energy expansion (for the case of zero quark mass), the chiral Lagrangian contained one parameter, F0, which we determined in terms of our model in section 3.4. At fourth order, there are three extra parameters (L1, L2, and L3) corresponding to terms that do not involve quark masses or gauge fields. When the U's are expanded, these terms start at fourth order in p fields, and have four derivatives; thus they contribute to pion-pion scattering at fourth order in momentum. Following the procedure of section 3.4, we must first calculate G4 [p]. This term in the effective action is determined by the Feynman graphs in Figure 3.5. Figure 3.5; Quartic terms in the effective action. Writing G4 [p] as G4 [p] = ??d4p1d4p2d4p3d4p4(2p)12pa(p1)pb(p2)pc(p3)pd(p4)Fabcd(p1,p2,p3,p4) d(p1+p2+p3+p4) , (3.62) and Taylor expanding Fabcd(p1,p2,p3,p4) to fourth order in momentum, we can pick out contributions to L1, L2, and L3 by looking for terms with the appropriate trace and momentum structures10. We use the identity [20] Tr(DmUDnUDmUDnU) = - 2 Tr(DmUDmUDnUDnU) + 12 {Tr(DmUDmU)}2 + Tr(DmUDnU)Tr(DmUDnU), (3.63) to write the effective chiral in the standard form (eq. 2.13). Equation 3.63 allows us to obtain terms involving products of traces even though single quark loops can give only single traces. The Taylor expansion of Fabcd(p1,p2,p3,p4) produces some very long and messy integrals which we will not reproduce here, but which are relegated to Appendix B. For completeness we mention that another fourth order term (in momentum) in the chiral Lagrangian is needed in order to reproduce the effects of anomalies. We refer to, of course, the Wess-Zumino term [25]. Verbeek has shown [39,40] that our model reproduces the Wess-Zumino term. The appropriate term in the effective action is generated by the graph in Figure 3.6. Figure 3.6; Wess-Zumino term in the effective action. To fourth order in momentum Verbeek finds: lWZ = 8Nc5p2f5 Tr{lalblcldle} e mntsmpanpbtpcspdpe ??dq2q2S4(q)S2(q)-2q2S(q)S'(q)(q2+S2(q))5 . (3.64) The functional derivative with respect to S(q) of the integral in equation 3.63 vanishes, and the integral equals 112 for any non-zero S(q) which allows the integral to converge. This gives the correct coefficient for the Wess-Zumino term. Since the graph in Figure 3.6 only involves single GB couplings, this result is independent of the choice of the ak's. That is, this result relies only on the normalization condition (eq. 3.18). We have now gone as far as we can without introducing explicit chiral symmetry breaking effects. We will introduce these effects in our minimal model in chapters 4 and 5. Footnotes for Chapter 3: 1 For a summary of the transformation to Euclidean space-time see Appendix A. 2 This parametrization is discussed in chapter 6. 3 For example, such terms would give equal amplitudes for two neutral kaons coupling to u-u and for two pions coupling to u-u. 4 For a review of functional methods see references [22] and [27]. 5 Roberts is presently calculating the pion form factor in this model [90]. 6 Unfortunately, this effective action (which is a functional of fields which have not been integrated over) bears the same name as the effective action discussed in section 3.2 (which is a functional of expectation values of fields). 7 For a summary of the transformation to Euclidean space-time see the Appendix. 8 The integrand in the formula for F2 is not unique, since one can always add terms to the integrands which are total derivatives. Such terms are easily generated by simply shifting the loop momentum in eq. 3.53 by a function of p. A particularly simple form (the one in eq. 3.55) can be obtained by letting qm = (q'+ 12 p)m in eq. 3.53. 9 For massless QCD. 10 Four derivative terms in the effective chiral Lagrangian (like zero derivative terms) pick up an extra minus sign in the Wick rotation from Euclidean to Minkowski space.