(Warning! The equations have not been converted yet, proceed with caution.) Chapter 6 Comparison with Experiment 6.1 - Matching Conditions We would now like to apply our minimal model to the case of three flavours of quarks (up, down, and strange) with three colours, in the hope of comparing the predictions of our model with experiment. First, however, it is necessary to be more specific about the relation between the parameters of our model (these include the quark masses, and the parameters we will use to model S(p)) and the parameters (coupling constants) of the effective chiral Lagrangian derived from the model. Specifically, to compare the predictions of the chiral Lagrangian (to fourth order in the energy expansion) with experiment one must include renormalization effects. As mentioned in section 2.3, although the chiral Lagrangian is non-renormalizable in the usual sense, it is renormalizable order by order in the energy expansion. The renormalization procedure entails introducing counter-terms of the same form as the bare terms in the Lagrangian in order to absorb the usual infinities encountered in loop graphs. Thus one might suspect that the renormalized parameters of the chiral Lagrangian are input parameters that can only be determined by experiment, just as the electron mass and charge are input parameters of quantum electrodynamics. This, however, is not the case. To explain why this is so we will make use of what Georgi refers to as "effective-field- theory language" and "matching conditions" [50]. We begin our discussion by considering a theory which contains some heavy particles. For definiteness we will discuss the Standard Model and use the W and Z0 as examples of the heavy particles, but the arguments we will make can be easily generalized [50]. At low energies real W's and Z's cannot be produced, so it is useful to integrate them out of the theory by performing the functional integration over the W and Z0 fields in the path integral representation of the generating functional for the Standard Model. The resulting effective action is, of course, nonlocal. At low energies it is well approximated by the lowest order term in its derivative expansion1, that is, by a local theory whose only non-renormalizable interactions are the familiar four-Fermi terms. To describe physics at energies of the order of MW, a series of higher derivative terms in the effective action are required. At low energies these terms are unimportant, since they are suppressed by appropriate inverse powers of MW or MZ (this is an example of the decoupling theorem of Appelquist and Carrazone [51]). The effective action obtained by integrating out the W and Z0 describes an effective field theory without any W and Z0 particles. The effective field theory is non-renormalizable, but the coupling constant of the non- renormalizable interactions (the Fermi constant GF) can be calculated in terms of the parameters of the underlying renormalizable theory (the Standard Model). The relation between the low-energy couplings and the underlying parameters may be determined by the requirement that amplitudes for specific processes (like electron-neutrino scattering) calculated in the effective theory with momenta of order MW must be equal2 to the corresponding amplitude calculated in the underlying theory and renormalized at the scale3 m = MW. Such relations between the parameters of the effective theory and those of the underlying theory are referred to as matching conditions. Thus, although the effective field theory is non-renormalizable and generates an infinite series of infinite counter-terms, the matching conditions ensure that we do not lose any predictive power. Once the matching conditions have been met, we can use the renormalization group [20] of the effective theory to obtain the appropriate coupling constants at lower energy scales. We will use the same type of effective field theory arguments in the analysis of our model in order to justify the way in which we integrate out the quarks and determine the parameters of the effective chiral Lagrangian. First let us consider how such arguments could be applied to real quantum chromodynamics (QCD). In the case of QCD, we could consider introducing some auxiliary fields [52] to the path integral, and then integrating out the quarks, which (it is assumed) can never appear as asymptotic states. If quarks are confined it would seem that we could integrate out the quarks and impose a matching condition at any scale we wanted to choose. However, if the quarks develop a dynamical mass4 then we could hope that the effects of quark loops are suppressed for external momentum much smaller than the dynamical quark mass. Since quarks are fermions, and are thus produced only in pairs, we would actually expect the relevant scale for a matching condition to be twice the dynamical quark mass. It should now be fairly clear how we will apply the effective field theory concept to our model. We integrate out the quarks and obtain an effective chiral Lagrangian, the coupling constants of which are expressed in terms of integrals over the self-energy S(p). These expressions for the coupling constants will give the "underlying" side of the matching condition; they are to be identified with the renormalized coupling constants of the chiral Lagrangian. The renormalization scale (the matching scale) is taken to be twice the quark "mass". Since it is conventional to use the coupling constants renormalized at the scale Mh [20], we will use the renormalization group equations for the chiral Lagrangian to "run" the coupling constants down from the matching scale to Mh. The actual value of the matching scale is important in that the values of low-energy coupling constants depend on this scale. The parameters of our underlying model do not run (since we do not have any gluons), as they would in QCD, but the coupling constants (L1 to L10) of the chiral Lagrangian do run. For example, if we renormalize at the matching scale, our model determines L4 and L6 to be zero. In the real world there is some specific energy scale where L4 vanishes when renormalized at that particular scale. Thus we should be more specific by what we mean by the statement that the matching scale is twice the quark "mass". To do this we have to know a little more about S(p). 6.2 - Parametrizing the Dynamical Mass By using the operator product expansion, Politzer [37] derived an expression for the asymptotic form of the dynamical quark mass as a function of Euclidean momentum. Taking the full propagator for quark flavour i in (massive) QCD as5 Smi(q) = 1-iAi(q)q/+Bi(q) , (6.1) Politzer found the dynamical mass to be Bi(q)Ai(q) = miR(m) ????g2(q)g2(m) d + 4g2(q)q2 ????g2(q)g2(m) -d 0 | yiyi(m) | 0, (6.2) where g2(q) is the running coupling of QCD, mR(m) is the perturbatively renormalized current quark mass, m is the renormalization scale, and d depends on the number of flavours and colours. In our model (including quark masses) the quark propagator for flavour i is: Si(q) = 1-iq/+mi+S(q) . (6.3) The model obviously approximates Ai(q) by 1. In QCD we can always choose a renormalization convention such that Ai(q=m) = 1 for some particular m. If it is possible to choose a gauge such that Ai(q) is slowly varying over the relevant range of momentum6, then Ai = 1 may not be such a bad approximation. In keeping with the spirit of keeping non- perturbative effects and throwing away perturbative effects we can pick out the points of correspondence between our model and QCD. The perturbative effects in equation 6.2 are contained in the implicit logarithmic variation of g2(q). If we ignore this variation we are lead to identify the quark mass parameter in our model, mi, with the renormalized quark mass parameter in QCD miR(m) (in light of the effective field theory argument given above, m should be the matching scale). This is not too surprising. We are also led to identify our S(q) with the second term in equation 6.2, that is, S(q) should fall-off asymptotically as 1q2 . As a simple extrapolation of this asymptotic form, which does not behave wildly as q2 approaches zero, we will use S (q) = ()A+1m3m2+Aq2 . (6.4) The normalization is chosen such that S(q = m) = m. By varying the value of A, we can test how sensitive the results of the model are to the assumed functional form of S(q). All but one of the loop integrals we will consider are insensitive to ultraviolet momentum, since the integrands are rapidly damped by the 1q2 behaviour of S(q). Also, these integrals are infrared insensitive, since the dynamical mass plays the role of an infrared cutoff, and the factors of q in d4q will suppress the integrands at low momentum. Thus, the range of loop momentum that makes the dominant contribution should be q m, and the typical value of S(q) for this range of momentum is S(qm) m (we will refer to m as the dynamical mass). This suggests that in determining the matching scale, we take the quark "mass" to be the dynmical mass plus the current mass (m + mi for quark flavour i). Thus the matching scale is not an extra free parameter of the model. We will not attempt to first integrate out the (presumably heavier) strange quark, renormalize down to the up and down mass scales, and then integrate out the up and down quarks. We will simply integrate out all the quarks at a common scale. Since the matching scale enters only in a logarithm this approximation should not be too bad. We will take the matching scale to be twice the average of the up and down "masses" (as determined by our model), i.e., m = 2m + mu + md. We mentioned above that one of the integrals which determine the coupling constants of the effective chiral Lagrangian is sensitive to large loop momentum. Using the form for S(q) given in equation 6.4, it is in fact divergent. The integral in question is 0 | yy | 00 = - 4 Nc ??d4q(2p)4S(q)q2+S2(q) , (6.5) that is the massless limit of the vacuum expectation value of yy as determined by our model. This problem could be alleviated somewhat by including in S(q) the logarithmic dependence suggested in equation 6.2, that is, modifying the asymptotic behaviour of S(q) to be S(q) ~ Cq2[]ln(q2)1-d , (6.6) where C is a constant. Then 0|yy|00 could be multiplicatively renormalized by a known renormalization factor [37,53]. However, all the other integrals involving S(q) receive important contributions only from low momenta, so they are insensitive to the logarithmic behaviour given in equation 6.6. Including the logarithmic behaviour in equation 6.6 in order to calculate 0|yy|00 would mean that the value of C would have to be adjusted in order to obtain an appropriate value for 0| yy|00, thus introducing another parameter into the model. We will simply bypass the matter of finding the correct value of C by retaining the form of S(q) given in equation 6.4, and taking 0|yy|00 as a parameter in our model. Thus our final model including quark masses and external gauge fields contains five parameters: mu, md, ms, m, and 0|yy|00. We will determine these parameters by fitting to meson masses and decay constants. By way of comparison, the lowest order chiral Lagrangian, l2, contains four parameters (the parameter B0 cannot be determined independently of quark masses, since only the product B0M appears in the chiral Lagrangian), so our model does not gain us any new information at this order in the energy expansion. However the fourth order chiral Lagrangian, l4 (eq. 2.13), contains 10 additional coupling constants, giving 14 coupling constants which can be determined in terms of the five parameters of our model. In addition to these coupling constants, we also determine the scale of the current quarks masses. Therefore, we obtain 10 predictions. It should be noted that except for L5, L7, and L8, the values of the parameters of the fourth order chiral Lagrangian, l4, renormalized at the matching scale are independent of the five parameters of our model7. This is true for L4 and L6 since these Zweig rule [38] violating terms vanish at the matching scale. The coupling constant L7 is parameter dependent since in our model it is determined by the value of L11 and L12 (eq. 4.29), and the value of L12 (eq. 4.24) is parameter dependent. L11 and the remaining Li's are all dimensionless numbers determined by quark loop integrals that depend only on S(q). Thus the values of these parameters (at the matching scale) are completely determined once the functional form of S(q) is known. Of course, the value of the matching scale does depend on our parameters. Nevertheless, for the sake of completeness, and for comparison with local models, we will give the values of the Li's at the matching scale. If we write B0L5 m g5, B20Ld8 m2g8, and B0L12 m g12, then g5, g8, and g12 are dimensionless numbers determined by quark loop integrals (eqs. 4.16, 4.14, and 4.24 respectively) that are also independent of the parameters of our model. These results are given in Table 6.1. For comparison we also include the values8 extracted from experiment by Gasser and Leutwyler (G&L) [20], and the values predicted by a local quark-GB model [28]. G&L cannot quote a value for g5, g8, and g12, since phenomenological chiral Lagrangians cannot set the scale of the parameter B0. Table 6.1 Gasser & Minimal Minimal Local Leutwyler Model Model Model x 10-3 A = 1 A = 13 L1 0.9 0.3 0.85 0.80 0.8 L2 1.7 0.7 1.7 1.6 1.6 L3 -4.4 2.5 -4.2 -3.9 -3.2 L4 0 0.5 0 0 g5 -6.7 -7.6 L6 0 0.3 0 0 g8 -2.9 -4.0 L9 7.4 0.7 5.8 5.9 6.3 L10 -6.0 0.7 -6.8 -5.0 -3.2 L11 3.0 3.0 g12 -14.4 -17.9 In our minimal model (and in the local models) the relation L2 = 2L1 follows from Zweig's rule [20]. It may be noted that the largest difference between our model and the local version (for the cases when the local model makes a prediction) occurs for L10. For a more accurate comparison with the experimental values, we should renormalize the values predicted by our model at the same scale thatG&L used, however to do this we need to know the parameters of our model in order to determine the matching scale. This will be done in the next section. 6.3 - Fitting to Masses and Decay Constants In chapter 2 we discussed the GB masses and decay constants to second order in the energy expansion. To fourth order there are two types of corrections: tree order corrections from l4, and loops containing vertices from l2. G&L have completed the analysis of the chiral Lagrangian to this order, and we will merely quote the relevant results [20]. For the GB masses they found: M2p = 2 m^ B0{}1+mp-13mh+2m^K3+K4, M2K = (m^ + ms) B0{}1+23mh+(m^+ms)K3+K4, M2h = 23 (m^ + 2 ms) B0{}1+2mK-43mh+23(m^+ms)K3+K4 + 2 m^ B0{}-mp+23mK+13mh + K5. (6.7) where K3 = 8B0F20 ()2L8-L5, K4 = (mu + md + ms) 16B0F20 ()2L6-L4, K5 = (ms - m^)2 128B09F20 ()3L7+L8, K6 = 4B0F20 L5 , K7 = (mu + md + ms) 8B0F20 L4 , m^ = 12 ()mu+md, mP = M2P16p2F20 ln MPm , (6.8) where P specifies a particular pseudoscalar meson, and m is the renormalization scale (i.e., L1 through L10 are to be renormalized at the scale m). G&L also calculated the GB decay constants to this order. Recall that to lowest order nothing distinguished the pion decay constant from the kaon decay constant (i.e., Fp = FK = F0). To fourth order they found: Fp = F0{}1-2mp-mK+2m^K6+K7, FK = F0{}1-34mp-32mK-34mh+(m^+ms)K6+K7, Fh = F0{}1-3mK+23(m^+2ms)K6+K7, (6.9) In all of the above mentioned G&L formulas, isospin splittings were neglected (i.e., the approximation mu = md = m^ has been made). However, the up-down mass difference does contribute to the differences in meson masses (eg., the MK0 - MK+ splitting). To the order that we are working, G&L found the following relation for isospin violation: Rm ()M2K0-M2K+QCDM2K-M2p M2pM2K = md-mums-m^ 2m^ms+m^ {}1+O(M2), (6.10) where the subscript QCD indicates that electromagnetic isospin violating contributions to the masses have been subtracted out. This can be done to first order in e2 by using Dashen's theorem [20,54]: ()M2K0-M2K+QCD = M2K0 - M2K+ - M2p0 + M2p+ + O(e2M). (6.11) We are now ready to explain our fitting procedure. We will first discuss the approximation in which we neglect the effect of theoretical errors. To determine the five parameters of our model (mu, md, ms, m, and 0|yy|00) we will need five pieces of input data. We may use Mp0 , MK0 , MK+ , Fp , and FK. Since these masses and decay constants are physical observables, they must be renormalization group invariant, so we can do the fitting with the coupling constants (L1 through L10) renormalized at any scale. Since our derivation determines the coupling constants at the matching scale, m, and the matching scale depends on the (as yet) undetermined parameters m, mu, and md, it is convenient to renormalize at the matching scale. This choice has the additional benefit that L4 and L6 are zero when renormalized at this scale, so the equations are somewhat simplified. The parameters of our model can be determined by searching in the three parameter space (m,mc = m^ + ms,m). The combination m^ + ms is convenient since this is the only current quark mass dependence in M2K (eq. 6.7), when L4 and L6 are zero. Given a value for m, then the pion decay constant in the massless limit, F0 = f, is determined by equation 3.56. Given m, we also have that equation 4.16 determines B0L5 m g5, equation 4.14 determines B20Ld8 m2g8, and equations 4.21 and 4.29 determine Lem18. Note that g5, g8, and Lem18 are dimensionless parameters determined by quark loop integrals. We also note that using equation 4.29 in equations 6.7 and 6.8 we find that the L12 contributions to the meson masses completely cancel. This had to be true, since in the absence of external gauge fields the L12 term (eq. 4.19) is a total derivative. Using equation 6.7, we find a quadratic equation for B0: 0 = 16 m2cF20 Lem8 B20 + ????mc{1+23mh}-8m2cmF20g5 B0 + 16 m2cm2F20 g8 - M2K , (6.12) Thus we can solve for B0 (we take the positive root), which then determines 0|yy|00 through the relation 0|yy|00 = - F20 B0. This also determines L5, and L8. Next, we determine m^ . Using equation 6.7 we obtain 0 = 32 B20F20 ()2L8-L5 m^ 2 + 2B0()1+mp-13mh m^ - M2p . (6.13) We solve for m^ (taking the positive root) using Mp = Mp0 = 134.96 MeV [48], which then also determines ms = mc - m^ . The fit proceeds by searching through the parameter space (m,mc,m) to obtain an exact fit to the remaining constraints. We scan through different values of m to satisfy the requirement m = 2m + 2m^ . For each value of m, we scan through values of mc to fit FK using equation 6.9. For each value of mc, we scan through values of m to fit to Fp, using equation 6.9 again. We use the latest Particle Data Group value9 Fp = 93.1 MeV [48]. When this fitting procedure has converged to a sufficient accuracy, we end up with values for m^ =12 (mu+md), ms, m, and 0|yy|00. It should be noted that although we did not use G&L's equation for M2h in the fit, it was necessary to know Mh in order to include h loop effects (represented by the mh terms) in the formulae we did use. To determine mu and md we rearrange equation 6.10 to find mumd = 1-Rm4????m2sm^2-11+Rm4????m2sm^2-1 . (6.14) To find md we use md = 2m^1+mumd , (6.15) and mu follows trivially. So far we have neglected the theoretical errors that result from the approximations we have made. We recall that we have not solved our model exactly; we have integrated the quarks out of our model and resorted to an energy expansion of the resulting effective theory of GB's. While the quark integration is in principle exact, the energy expansion which leads to the effective chiral Lagrangian was only carried out to fourth order in energy, so we are neglecting sixth order terms. Without explicitly calculating these sixth order terms, we can only estimate their relative size. An oft-used rule of thumb [21] is that leading corrections to SU(2)SU(2) chiral symmetry (involving only up and down quarks) are typically 2%, while leading corrections to SU(3)SU(3) (involving the more massive strange quark) are typically 20%. For example in the chiral SU(3)SU(3) limit FK = Fp, while in reality FK is 22% larger. These leading corrections correspond to fourth order terms in the energy expansion. Thus we might expect the next (sixth) order terms to give 0.022 = 0.4% to SU(2)SU(2) symmetry, and 0.22 = 4% corrections for SU(3)SU(3) symmetry. To go further than such a rule of thumb we need some quantitative method of estimating our theoretical errors. Fortunately this can be done using a c2 fit. Since we expect that the theoretical errors in observables relating to strange quarks (i.e., MK,Mh, FK) are much larger than the theoretical errors in observables relating to up and down quarks (i.e., Mp, Fp) we can save ourselves considerable effort by the ignoring the latter entirely. To do this we may take MKT and FKT (where the subscript T stands for theoretical) to be the independent parameters to be determined by our c2 fit. Using the procedure outlined above we can (given values for MKT and FKT, and using Mp = 134.96 MeV, Fp = 93.1 MeV) we can numerically determine values for our model parameters (m^ = 12 ()mu+md, ms, m, and 0|yy|00). Once we have determined the model parameters for arbitrary values of MKT and FKT we can use equation 6.7 to calculate a theoretical value for MhT. We then proceed to search through the space (MKT, FKT) trying to minimize the difference between the theoretical and observed values of MK,Mh, and FK. More specifically we search for a minimum in c2, where c2 = ????MK-MKTsMK2 + ????FK-FKTsMK2 + ????Mh-MhTsMh2 (6.16) and the s's are the total (combined in quadrature) experimental and theoretical errors. For the results we will quote we have used a conservative value of 6% for the three theoretical errors. The physical value of FK comes from the experimentally determined [55] ratio FKFp = 1.22 0.01, and we use 495 MeV for MK, which is determined by the average of M2K0 and M2K+ with electromagnetic effects subtracted out. Once we have found the minimum in c2 we can use equations 6.14 and 6.15 to determine mu and md. There is a standard procedure for estimating errors using the c2 fitting procedure [48]. Once we have found the point in the space (MKT, FKT) corresponding to the minimum in c2 (which we will refer to as c2min), we then search for a closed curve in the space (MKT, FKT) which surrounds the minimum and on which c2 = c2min + 1. The extreme values of MKT and FKT on this equi-c2 boundary give an estimate of the total errors in these quantities. We will use the extreme values of our model parameters and predicted coupling constants on the equi-c2 boundary to quote estimates of the theoretical errors in these quantities as well. The fitting procedure we have described above determines all five parameters of our model. We should make a special note of how our model manages to obtain the quark masses, while phenomenological chiral Lagrangians can only determine quark mass ratios. To explain this fully we first note that the chiral Lagrangian possesses a reparametrization invariance. That is, the Lagrangian is invariant under the scaling transformation M a-1M, B0 aB0, with all the other coupling constants held fixed. Thus the scale of the quark masses cannot be determined. We now try to find the corresponding transformation of our model parameters. Using the result 0|yy|00 = - F20 B0, and requiring F0 be held fixed, gives M a-1M, 0|yy|00 a0|yy|00, and m must be held fixed since it determines the scale of F0. While this is a symmetry of the effective chiral Lagrangian to second order in the energy expansion, it is not a symmetry at fourth order. This is because there are extra relations imposed on the coupling constants by our model. Specifically, the scales of B0L5, B20Ld8, B0L12, and F0 are all determined by m, and thus do not change under the scaling transformation. Therefore the L5 term, for example, is not invariant under the reparametrization, and there should be some scale for the quark masses which gives the best fit to experimental data. We point out that it is crucial to this argument that the scale of F0 is fixed by m. If we could not predict F0, then we would be allowed to rescale m like B0, and the scaling transformation would be a symmetry of the effective chiral Lagrangian. This is precisely what happens in the case of local models where S(q) = m, since for constant S(q) the integral which determines F0 (eq. 3.56) diverges, and F0 must be taken as an additional parameter in such models. Of course, even though we can determine the quark masses, there is still an approximate symmetry if L5, L8, and L12 are small, so we should expect that a large range of values for the quark masses will be consistent (within errors) with the experimental data. In order to test how sensitive the results of our model are to the specific form chosen for S(p), we compare the numerical results of our fit for two different values of A (A = 1, A = 13) in Table 6.2. The value of c2min is found to be (0.95, 0.92) for A = (1, 13 ). With one degree of freedom (as is the case here) a model which gives a fit with c2min larger than 2.7 can be rejected with 90% confidence. Our model could be rejected with roughly 70% confidence. The fitting procedure gives the best fit values FKTFp = 1.27+0.05-0.03 and MKT = 501 +22-23 MeV for A = 1, and FKTFp = 1.26+0.05-0.04 and MKT = 502 +22-22 MeV for A = 13. Table 6.2 A = 1 A = 13 mu 2.8+2.3-1.7 MeV 2.7+2.0-1.6 MeV md 7.7+2.3-1.2 MeV 7.3+2.1-1.3 MeV ms 174+38-25 MeV 164+35-28 MeV m 332+1-3 MeV 298+2-2 MeV 0| yy|00 -(226+23-26 MeV)3 -(236+25-26 MeV)3 The values in Table 6.2 determine the matching scale, m, to be (674, 607) MeV for A = (1, 13 ). From Table 6.2 we see that the dynamical mass m is the parameter most sensitive to the form of S(p). It should be noted that the values given in Table 6.2 compare quite well with values obtained in strong interaction phenomenology. In particular, the current quark masses are within the errors of the values given in equation 2.3; m + mu and m + md are comparable10 to constituent quark masses in common use [56]; and 0|yy|00 is of the order of - L3QCD (sum rules give 0| yy|00= (1948 MeV)3 [12c]). In Figure 6.1 we show our results for the quark mass ratios, as well as the constraints on the ratio R = ms-m^md-ms derived by baryon mass splittings [57] and the K+-K0 mass difference (eq. 6.10). Figure 6.1; The graph (after Leutwyler [58]) shows the allowed regions for the quark mass ratios as determined by baryon mass splittings [57] and meson mass splittings [20] (eq. 6.10). The cross shows the value determined by our model for A= 13. The fitting procedure also determines the coupling constants of the effective chiral Lagrangian, renormalized at the matching scale m. We can now use the renormalization group equations [20] for the chiral Lagrangian to run these values down the the scale Mh = 548.8 MeV, which G&L have used as a renormalization point in their work. In Table 6.3 we give the values determined by G&L from comparison with experiment, the values11 predicted by our minimal model for two values of A, the values predicted by a modern version [59] of vector meson dominance (VMD), and the values predicted by a local quark-GB model [28]. In the VMD model, the usual GB's of the chiral Lagrangian are given couplings to vector and axial-vector mesons. The vector and axial-vector mesons are integrated out, giving an effective chiral Lagrangian. The values shown in Table 6.3 (in the VDM column) are obtained by assuming that the only contribution to the coupling constants comes from vector and axial-vector meson exchange. It should be noted that this type of model can not predict the parameters of the second order chiral Lagrangian. Table 6.3 Gasser & Minimal Minimal VMD Local Leutwyler Model Model Model Model A = 1 A = 13 F0 (85.5 MeV) 81.4+0.3-0.6 MeV 83.7+0.4-0.6 MeV ????2m^B0 (136.5 MeV) 136.2+0.3-0.5 MeV 136.8+0.3-0.4 MeV mumd 0.56 0.06 0.36+0.16-0.22 0.37+0.15-0.21 msmd 20.0 2.9 22.5+1.4-2.0 22.3+1.4-1.9 in units of 10-3: L1 0.9 0.3 0.972 0.001 0.860 0.001 1.1 0.8 L2 1.7 0.7 1.943 0.001 1.720 0.001 2.2 1.6 L3 -4.4 2.5 -4.205 -3.916 -5.5 -3.2 L4 0 0.5 0.162 0.001 0.080 0.001 0 L5 2.2 0.5 1.94+0.62-0.34 1.88+0.64-0.41 0 L6 0 0.3 0.099 0.001 0.049 0.001 0 L7 -0.4 0.15 -0.20+0.11-0.19 -0.22+0.12-0.20 0 L8 1.1 0.3 0.64+0.47-0.27 0.64+0.47-0.31 0 L9 7.4 0.7 6.267 0.001 5.999 0.001 7.8 6.3 L10 -6.0 0.7 -7.084 0.001 -5.123 0.001 -6.0 -3.2 Examining the table we find that ourminimal model is in surprisingly good agreement with experiment. By far, the largest change between A=1 and A = 13 occurs in the value of L10. Aside from L10 our model seems fairly insensitive to the assumed form of S(p). However, some comments are in order. The combination ????2m^B0 is a convenient way to display the parameters, since it represents the lowest order value for Mp. G&L do not quote values for F0 or ????2m^B0, but they can be easily obtained using their central values for the coupling constants L4 through L8 and their formulae for M2p, and Fp (eqs. 6.7 and 6.9). It should be noted that G&L obtain their value of L9 by fitting to the charge radius of the pion; however they make the approximation12 F0Fp= 93.3 MeV, which inflates their estimate of L9 slightly. If they had used the value F0 = 85.5 MeV, they would have obtained L9=(6.10.7) x 10-3, which is much closer to the values predicted by our model with A = 1 or 13. To say this in another way, the experimental value [20,60] for the charge radius squared of the pion is p = 0.439 0.03 fm2, while our model predicts p = 0.498 fm2 and 0.449 fm2 for A = 1 and 13 respectively. G&L also use their value of L9 to determine L10 through their relation to the ratio of axial and vector structure dependent form factors, g = 32 p2 (L9 + L10), so the approximation made for F0 also slightly inflates the magnitude of L10. Also, when G&L did their fits, the experimental value of the form factor ratio was g = 0.44 0.12. Using the more recent experimental result of g = 0.52 0.06 [41,61], they would have obtained L10 = -(5.8 0.7) x 10-3, which is still between the values predicted by the model. Using F0 = 85.5 MeV and the more recent value of g they would have obtained L10 = -(4.5 0.7) x 10-3, which is somewhat smaller than the values predicted by our model (the A = 13 value for L10 is still within the errors of this value). To look at this in another way, we can use our model values to predict g. However, we note that small errors in L9 and L10 lead to large errors in g, since L9 and L10 have roughly equal magnitudes and opposite signs, so there is a large cancellation involved in determining g. Our model predicts g = - 0.27 for A = 1 for and g = 0.28 for A = 13 . It should also be noted that our predicted value for the magnitude of L7 is slightly smaller than the experimental value (but within the errors). We should not expect to get L7 exactly right, since it represents the effects of h-h' mixing, and if any of the parameters of the chiral Lagrangian should be sensitive to gluon effects (which our model does not directly account for) it should be L7. This sensitivity arises since not only does quark- antiquark annihilation into gluons contribute to the mixing, but the mass of the h' receives a contribution (because of the QCD axial anomaly) from the winding number density e mnab Fmn Fab [62]. We note that the model predicts Mh = (526+20-22 MeV, 524 +18-21MeV) for A = (1,13 ) instead of the experimental value [48] Mh = 548.8 MeV. As G&L have pointed out [20], the apparent success of the second order (in the energy expansion) prediction for Mh (i.e. the Gell-Mann-Okubo formula), which is only off by 20 MeV, is something of a coincidence since individual contributions at fourth order can shift the h mass by around 80 MeV. The value of L7 also contains information about the h-h' mixing angle. Leutwyler [58] has obtained a formula relating L7 and the mixing angle by assuming that h-h' mixing is the sole contribution to the value of L7. Leutwyler finds the relation F20 (M2h' - M2h ) sin2q = - 24 L7 (M2h - M2p )2 . (6.17) Using our predicted values of Mh and L7, and the experimental value13 [48] of Mh' = 958 MeV, we find a mixing angle |q|=16o for both A = 1 and13 , whereas the phenomenological value14 [58,63] is q=-22o4o. Using G&L's value for L7 and the experimental h mass one would obtain |q| = 24o. Finally, we note that the relation L2 = 2L1 is a direct result of the absence of Zweig rule violating interactions in our model. 6.4 - Pion Scattering Lengths In order to get a feel for how well our model can reproduce experimental data we compare our predicted values for pion scattering lengths with the values calculated by Weinberg [66] (i.e., the lowest order chiral Lagrangian results) and the experimental values [21,67] in table 6.4. The scattering lengths are defined through the scattering amplitude TIJ, which is the partial wave amplitude for pions scattering with total angular momentum J and total isospin I in the s channel. Near threshold the partial wave amplitude is parametrized as Re TIJ = ????q2M2p+J ????aIJ+q2M2p+bIJ+O(q4) (6.18) where it is conventional to insert the factors of M2p+ so as to make aIJ and bIJ dimensionless. G&L have given the expressions for aIJ and bIJ in terms of the parameters of the chiral Lagrangian in references [20] and [21]. Table 6.4 Weinberg Model (A = 1) Model (A = 13 ) Experiment a00 0.16 0.20 0.20 0.26 0.05 b00 0.18 0.28 0.27 0.25 0.03 2a00-5a20 0.54 0.59 0.59 0.614 0.028 b20 -0.090 -0.067 -0.069 -0.082 0.008 a11 0.030 0.040 0.040 0.038 0.002 b11 0 0.0068 0.0064 a02 0 27 x 10-4 24 x 10-4 (17 3) x 10-4 a22 0 3.3 x 10-4 2.9 x 10-4 (1.3 3) x 10-4 On the whole, the corrections (given by our model) to the lowest order (Weinberg) results improve the agreement with experiment. Footnotes for Chapter 6: 1 That is, when the momentum flowing through the W and Z0 propagators can be taken to be much less than MW or MZ. 2 When calculated to the same order in external momentum. 3 By renormalizing at the scale of the external momentum, we minimize the logarithms of perturbation theory. 4 In section 2.2, we argued that quarks do indeed develop dynamical masses. 5 In slightly different notation. 6 To one-loop, there are an infinite number of gauges which give Ai(q) = 1, see reference [64]. 7 There is also one combination of L52, L7, and L8 that is independent of the parameters of the model. 8 Renormalized at Mh = 548.8 MeV. 9 The Particle Data Group use a different convention to normalize Fp. To convert from their convention to ours, one may simply divide by????2. For an explanation see reference [65]. 10 Comparisons such as this can only be made qualitatively, since meaning of the phrase "constituent quark mass" is ambiguous. 11 We keep an extra decimal in some parameters in order to show the small variation between the two cases with different values of A. 12 This is an older value for Fp. 13 The model cannot predict the mixing angle since it does not predict Mh' 14 As noted by G&L [20], the standard argument, as given by the Particle Data Group [48], which arrives at a value of q = - 10o, is inconsistent in that it keeps mixing effects of order M2, but neglects other contributions of the same order.