(Warning! The equations have not been converted yet, proceed with caution.) Chapter 4 Quark Mass Effects 4.1 - Mass Corrections to the Order Parameter We would now like to include the effects of explicit chiral symmetry breaking due to non-zero current quark masses. At first, one might think that the only modification necessary is the inclusion of a quark mass term, yMy, in the model Lagrangian (eq. 3.2). However, the presence of quark masses in quantum chromodynamics (QCD) will, in general, introduce corrections in the solution of the Schwinger-Dyson equation for the quark self-energy (i.e., the order parameter). We will delay the discussion of such self-energy correction until section 4.4, and proceed to calculate the direct effects of a current quark mass term. Thus our minimal model Lagrangian with quark masses is: lM(x,y) = y(x)d(x-y)[]/+My(y) + y(x)Sp(x,y)y(y). (4.1) Note that the local mass term breaks the chiral symmetry since it contains no Goldstone boson (GB) couplings. Given the discussion of section 3.4, it is straightforward to calculate the symmetry breaking terms in the effective chiral Lagrangian. 4.2 - Mass Corrections to the Effective Chiral Lagrangian We now turn to calculating the effective chiral Lagrangian for the model, including explicit quark masses. The effective action, GM[p], is defined by exp()-GM[p] ??dydyexp????-d4xd4ylM(x,y) . (4.2) Thus GM[p] = - Tr log S-1Mp. (4.3) where now S-1Mp = -/yd(x-y) + Md(x-y) + Sp(x,y). Using the inverse quark propagator with a explicit mass S-1M(x,y) = - /yd(x-y) + Md(x-y) + S(x-y), (4.4) and simply repeating the analysis given in section 3.4, we find: GM[p] = - Tr log ????d(x-y)+d4zSM(x,z){}Sp(z,y)-S(z-y). (4.5) Thus, to find the mass corrections to the quadratic terms in the effective action, we can simply recalculate the graphs given in Figure 4.1 using the massive quark propagator (eq. 4.4). Figure 4.1; Quadratic terms in the effective action. Since, to fourth order in the energy expansion, we will only need corrections of order M2, it will be useful to expand the quark propagator in powers of M. The easiest way to do this is to treat M as a perturbation, and to construct the Feynman perturbation series. Thus, in momentum space we have that the massless propagator is given by S-1(p) = -ip/ + S(p), and S(p) = S(p) +S(p){-M}S(p) + S(p){-M}S(p){-M}S(p) + ... = ip/+S(p)p2+S2(p) - M -p2+2ip/S(p)+S2(p)(p2+S2(p))2 + M2 -ip/p2-3p2S(p)+3ip/S2(p)+S3(p)(p2+S2(p))3 + ... (4.6) We again write the coefficient of the papb term in the effective chiral Lagrangian as 12 Fab(q), and expand in powers of momentum and the quark mass matrix M: Fab(q) = FabM + q2F2dab + FabM2 + q2 Fab2,M + q4F4dab + ... (4.7) The slightly awkward notation is necessary since M will only appear inside traces. We have already calculated F2 in section 3.4. Expanding Fab(q) to first order in M and zeroth order in q gives us FabM = -8Ncf2 Tr(lalbM+lblaM)??d4p(2p)4S(p)p2+S2(p) = 2f2 Tr(lalbM+lblaM)0 | yy | 00 (4.8) where we have used 0 | yy | 00 = - 4 Nc ??d4p(2p)4S(p)p2+S2(p) , (4.9) which is the massless limit of the vacuum expectation value of yy in our model. Comparing the coefficient of the term quadratic in p in the lowest order chiral Lagrangian, l2 (eq. 2.10), with the order M term in equation 4.7 we find the relation -B0 Tr(lalbM+lblaM) = 12 FabM . (4.10) Therefore, using f = F0, the parameter B0 is determined to be B0 = -1F20 0|yy|00 . (4.11) Thus equation 4.8 reproduces the Gell-Mann-Oakes-Renner mass formulae [18]1 for the GB's. This is a check on our model assumption that the GB couplings did not involve the quark masses at first order. In general, if the GB couplings involved a term linear in quark masses then we would find an extra contribution to equation 4.8, and we would not recover this standard result. It should be noted that reproducing the factor of 0|yy|00 is non-trivial to the extent that it involves cancellations between the two graphs in Figure 4.1. Next we turn to fourth order terms in the energy expansion. By expanding the L8 term in l4 (eq. 2.13), and comparing the coefficient of the papb term with the order M2 term in equation 4.7 we find the relation 32 B20Ld8F20 Tr(M2lalb+MlaMlb) = 12 FabM2, (4.12) where we have introduced the superscript d (for direct) to distinguish this contribution to L8 from another contribution that we will discuss in the section 4.3. The term of order M2 in Fab(q) is FabM2 = - Tr(M2lalb+MlaMlb) Ncp2f2 ??dp2p2S(p)(p2+S2(p))2 , (4.13) thus (using f = F0) B20Ld8 = -Nc64p2 ??dp2p2S2(p)(p2+S2(p))2 . (4.14) Similarly for the L5 term we find 8 B0F20 L5 Tr(lalbM+lblaM) = 12 Fab2,M . (4.15) Expanding Fab(p) to first order in M and second order in q then gives B0 Ld5 = -Nc64p2 ??dp2p2I(p)(p2+S2(p))3 , (4.16) where I(p) = 4S3(p) + S4(p)S'(p) + p2()12S4(p)S''(p)-4S2(p)S'(p)-S3(p)S'2(p) + p4()3S(p)S'2(p)-S'(p) - 12 p6S''(p) (4.17) The three terms we have calculated are all the terms in the fourth order chiral Lagrangian (eq. 2.13) involving the quark mass matrix in a single trace, that is, these are all the terms that can arise from a single quark loop. 4.3 - The Effective Chiral Lagrangian to Fourth Order At this point we would like to discuss the parameters of the fourth order chiral Lagrangian, l4 (eq. 2.13). First of all we will mention the physical interpretation of the coupling constants L1 through L10. The terms in l4 multiplying L1, L2, and L3 contribute to pion-pion scattering at fourth order in momentum. The terms associated with L4, L6, and L7 are Zweig- rule [38] violating terms; L4 and L6 are thought to be close to zero [20], but L7 accounts for the effects of h-h' mixing [20] (the only process known to exhibit large Zweig-rule violation) and is important. The L5 and L8 terms give corrections to the lowest order formulae for meson masses and decay constants. The L9 term determines the charge radii of the mesons, and the L10 term contributes to radiative weak decays like pgen. From the discussion of the last paragraph, the attentive reader may have noticed that our model seems to run into problems with h-h' mixing. Our minimal model disposes of all Zweig-rule violating quark-GB interaction terms (i.e., terms that couple quarks to traces of GB fields), so we cannot obtain terms in the effective chiral Lagrangian that involve products of traces, since a single quark loop will only give one trace. Thus we should find that L4, L6, and L7 are identically zero. If L7 is zero, then our model can not account for h-h' mixing. Fortunately, there is a loop-hole in this argument: there are relations between matrices and traces of matrices. To explain this more fully, we recall that the expression given by Gasser and Leutwyler (G&L) for l4 (eq. 2.13) is not the most general Lagrangian at fourth order in the energy expansion. There are two additional possible terms [42]: l4,11 = L11 Tr (D2UD2U) (4.18) and l4,12 = L12 Tr(cD2U + cD2U), (4.19) where, as usual, c = 2B0M, and D2 = DmDm. To check whether such terms appear in the effective chiral Lagrangian for our model, we first return to the coefficient (in momentum space) of the papb term (eq. 3.50), 12 Fab(q). We can simply expand Fab(q) (eq. 3.53) out to fourth order in momentum to pick out the term corresponding to the leading term in l4,11. We find2, using equation 4.7, 2F20 L11 = - 12 F4 . (4.20) The Taylor expansion of equation 3.53 then yields: L11 = Nc32p2 dp2p2H(p), (4.21) where H(p) = S(p)(S'(p)+12p2S''(p))(p2+S2(p))2 - ??3S2(p)+6S3(p)S'(p)+ p2()5S3(p)S''(p)+7S2(p)S'2(p)+S(p)S'(p) ??+p4????23S3(p)S'''(p)+2S2(p)S'(p)S''(p) 1(p2+S2(p))3 + ??p2S2(p)()5+20[S(p)S'(p)+S2(p)S'2(p)] ??+4p2()S'2(p)+S(p)S''(p)+2S(p)S'3(p)+2S2(p)S''(p) 1(p2+S2(p))4 - 2p4S2(p)()1+6S(p)S'(p)+8S3(p)S'3(p)+12S2(p)S'2(p)(p2+S2(p))5 . (4.22) To calculate the coefficient of the l4,12 term we must resort to different tactics, since if there are no gauge fields present, this term reduces to a total derivative, and thus makes no contribution to the momentum space papb term. We may calculate L12 in two ways. Since the most obvious physical effect of the L12 term is a correction to the GB decay constant, we may calculate the order M corrections to the one quark loop amplitude for annihilation of a GB by an axial-vector current. This involves including all possible insertions of the mass matrix on quark lines in the GB decay graphs shown in Figure 3.4. We could then compare this calculation with the order M corrections to the pion decay amplitude3 that would follow from l4 (eq. 2.13). At order M this amplitude receives contributions from both L5 and L12 terms. A slightly simpler method (which gives the same result) is to temporarily allow for a position dependent mass matrix, that is to treat M as an external field to which momentum may be transferred. It is then convenient to rewrite the explicit mass term in our model as: y M y = y M 12(1-g5) y + y M 12(1+g5) y . (4.23) Then we may simply expand the effective action GM[p] (eq. 4.5) and pick out the term involving one GB field and one factor of M 12(1-g5). This term is simply minus the one quark loop amplitude for the external field M to annihilate a GB. The Feynman diagram for this amplitude is the same as that shown in Figure 3.3 with the axial-vector current replaced by M12(1-g5). Once we have obtained the coefficient of the Mp term in GM[p], we may compare it to that of the Mp term in l4,12. We find B0 L12 = -Nc32p2 ??dp2p2K(p)(p2+S2(p))3 , (4.24) where K(p) = p2S(p) + 2S3(p) + S4(p)S'(p) - 12 p4S'(p) + 3p4S(p)S'2(p) - p2S3(p)S'2(p) + p2S4(p)S''(p) - 72 p2S2S'(p) - p6S''(p) (4.25) G&L removed the l4,11 and l4,12 terms by using the classical equations of motion from l2 [20,41], and we will apply the same procedure to our effective chiral Lagrangian. This is consistent, to this order in the energy expansion, since the fourth order terms are only needed at the tree graph (classical) level. The second order equations of motion are: D2U U + DmUDmU + 12 (Uc - cU) - 12N Tr(Uc - cU) = 0, (4.26) where N is the number of flavours. Substituting equation 4.26 into equations 4.18 and 4.19, we find l4,11 = L11 Tr(DmUDmUDnUDnU) - 14 L11Tr(UcUc + cUcU) + 12 L11 Tr(cc) + 14N L11 []Tr(Uc-cU)2 , (4.27) and l4,12 = - L12 Tr(DmUDmU(Uc + cU)) + L12 Tr(cc) (4.28) - 12 L12 Tr(UcUc + cUcU) + 12N L12 []Tr(Uc-cU)2 Thus (for N = 3) L3, L5, L7, and L8 receive the following additional contributions: Lem13 = L11, Lem25 = - L12, Lem17 = 112 L11, Lem27 = 16 L12, Lem18 = - 14 L11, Lem28 = - 12 L12, (4.29) where the superscript em signifies "equation of motion", to distinguish from the direct contributions calculated previously. These contributions to L7 allows our model to account for some h-h' mixing. Numerical results for the L1 through L10 will be presented in chapter 6. 4.4 - Mass Corrections to the Quark Self-Energy As we noted at the beginning of this chapter, the presence of non-zero current quark masses will modify the solution of the Schwinger-Dyson equation for the self-energy, S(x-y). Thus it is plausible that there may be mass corrections to the GB couplings to quarks that should be incorporated in Sp(x,y). First we will remark on how such corrections are constrained by chiral symmetry. As in our discussion of explicit symmetry breaking effects in the chiral Lagrangian (section 2.3), we will (at first) pretend that M transforms under SU(N)LSU(N)R. Specifically, if the quarks transform as y' = exp(-ieg5)exp(-ia)y, where e = eala and a = aala, then, for the purposes of this discussion, we will temporarily assume the transformation law M' exp(ieg5) exp(-ia) M exp(ia) exp(ieg5); that is, M transforms in the same way as the exponential of GB fields, V. If M transformed in this manner, then the QCD Lagrangian (with a mass term) would be chirally invariant. It is now obvious how to construct symmetry breaking terms that could be included in a nonlocal model. Making a power series expansion in M, we can take any term that appears in the general form for Sp(x,y) (eq. 3.16 and 3.17), and replace any number of V's by M's, or V's by M's. Such terms should not necessarily involve S(x-y), in general these new terms should be multiplied by independent functions of x-y. Thus there is a large amount of arbitrariness if we only use chiral symmetry arguments. Since we are calculating the effective chiral Lagrangian only to fourth order in the energy expansion, we need terms only up to second order4 in M or M. Thus the possible corrections to Sp(x,y) that we would need to consider are proportional to M, M, MM, or MM. Terms linear in M or M, however seem to be ruled out. As we argued after equation 4.11, such couplings of one or two GB's which involved quark masses would in general destroy the agreement between our model and the Gell-Mann-Oakes-Renner result for the meson masses. This would be a high price to pay, since the Gell-Mann-Oakes-Renner result only relies on the assumed form of the symmetry breaking term in the underlying theory [18]. GB coupling terms that involve MM would contribute only to L8, so the comparison of our model prediction for L8 with the experimentally determined value can be considered as a test of the importance of such terms. This comparison will be made in chapter 6. As we will see, the agreement is quite good, which is circumstantial evidence that the coefficient of MM terms is small. Finally we consider what effect terms involving MM will have on the effective chiral Lagrangian. The answer is that to fourth order in the energy expansion, there is no contribution from the terms of order MM. This can be checked by explicit calculation, or by noting that there are no terms in l4 (eq. 2.13) which involve MM. This is because there is no hermitian SU(N)LSU(N)R invariant which can be constructed from one M, one M, and any number of U's and U's. Thus our model is consistent with mass dependence in GB couplings to quarks, as long as this mass dependence involves MM or higher powers of the quark mass matrix. To see whether this is actually the case in QCD is a highly non-trivial problem, that may only be resolved by lattice calculations. Footnotes for Chapter 4: 1 See the discussion following eq. 2.11. 2 The minus sign arises in four derivative terms from the Wick rotation from Euclidean to Minkowski space. 3 Without GB wavefunction renormalization factors. 4 See equation 2.13.