(Warning! The equations have not been converted yet, proceed with caution.) Chapter 5 Electroweak Effects 5.1 - Path Exponentials We would now like to gauge our minimal model, but since the model is nonlocal this will not be entirely straightforward. As a warm-up exercise we will gauge a local theory of free quarks in a slightly unusual way. In Euclidean space-time1, the Lagrangian for quarks interacting with photons can be written down directly (assuming minimal coupling) l = y(x)gm(m - i e Q Am(x))y(x) - 14 Fmn Fmn , (5.1) where Q is the charge matrix of the quarks. This Lagrangian is invariant under the infinitesimal transformations y(x) = (1 + i a(x)Q)y'(x), y(x) = y '(x)(1 - i a(x)Q), Am(x) = A'm(x) + 1e ma(x). (5.2) We will now try to obtain this result with a method that can be generalized to the nonlocal case. To gauge our model, we will use the path exponential introduced by Bloch [43], and the path-ordered exponential introduced by Wilson [44]. The path-ordering is not essential for electromagnetic gauge fields, since U(1) is an Abelian group, but it will be important for including external weak gauge fields. Consider the free quark action S = d4xd4yy(x)d(x-y)/y(y) = d4xd4yy(x)[]-y/d(x-y)y(y). (5.3) This action can be gauged by the introduction of an exponential of a line integral of the gauge field. The useful feature of the path exponential for nonlocal interactions is its transformation property. Under the gauge transformation defined in equation 5.2, the path exponential transforms as2 exp(-ieQxyAndwn) = (1 + i a(x)Q) exp(-ieQxyA'ndwn) (1 - i a(y)Q). (5.4) Thus, ignoring the Fmn Fmn term, we have the gauge invariant action Sg = ??d4xd4yy(x)[]-y/d(x-y)exp(-ieQxyAndwn)y(y) = ??d4xd4yy(x)d(x-y)y/exp(-ieQxyAndwn)y(y). (5.5) This looks very nice but the derivative of the line integral is ambiguous, and must be carefully defined. Following Mandelstam [45] we introduce the notation I(x,y,p) xyAndwn, (5.6) where p explicitly denotes the dependence on the path taken from x to y. Then the derivative may be defined by limdym0 dym ym I(x,y,p) = limdym0 I(x,y+dym,p') - I(x,y,p), (5.7) where p' is the path obtained from p by adding the extension dym to the y end, as shown in Figure 5.1. Figure 5.1; Extension of the path used in calculating the derivative of the line integral. Using the definition in equation 5.7 we have ym I(x,y,p) = Am(y), (5.8) the important point being that the derivative of the line integral does not depend on the path used in defining it. Expanding the exponential in equation 5.5 it is easy to see that if d(x-y)I(x,y,p) = 0, (i.e., if, as xy, the path p shrinks to a point, and not to a closed loop) then Sg = d4xd4yy(x)d(x-y)gm(m-ieQAm)y(y). (5.9) Thus, we recover minimal coupling from the introduction of the path exponential if we use the correct prescriptions for derivatives and for zero-length paths. We will use path-ordered exponentials, with these same prescriptions, to introduce external electroweak gauge fields to our model. 5.2 - The Gauged Nonlocal Mass First we will concentrate on coupling photons to quarks with a nonlocal dynamical mass. The action without photons is S = d4xy(x)gmmy(x) + d4xd4yy(x)S(x-y)y(y). (5.10) With the path-ordered exponential prescription (and ignoring the Fmn Fmn term again) the gauged action is Sg = d4xy(x)gm(m-ieQAm)y(x) (5.11) + ??d4xd4yy(x)S(x-y)exp(-ieQxyAndwn)y(y). We will refer to the local and nonlocal terms as SL and SNL respectively. The tricky part is how to actually use this action to perform calculations. Since Feynman rules for nonlocal gauge theories are not well known we will spend some time in deriving the amplitude for a photon to couple to two quarks as shown in Figure 5.2. Figure 5.2; The photon-quark-quark vertex, i e Gm(p,q,p'). The wavy line designates the photon. The local contribution to the amplitude is given by i e GmL(y,x,z) = - d3SLdAm(x)dy(y)dy(z) Am=0 . (5.12) Fourier transforming with ei(p'z-py-qx) and dropping the usual (2p)4 d(p'-p-q) we find i e GmL(p,q,p+q) = i e Q gm. (5.13) The nonlocal part is harder. First it will be advantageous to perform a derivative expansion of SNL. Let F(x,y) = exp(-ieQxyAndwn)y(y), then SNL = d4xd4yy(x)S(x-y)F(x,y). (5.14) Fourier transforming we have SNL = ??d4xd4yd4kd4pd4k'd4q(2p)16ei(kx-p(x-y)-k'x-qy)y(k)S(p)F(k',q) = ??d4kd4p(2p)8y(k)S(p)F(k-p,p), (5.15) then Taylor expanding S(p) gives SNL = ??d4kd4p(2p)8y(k)????n=01n!S(n)(0)(p2)nF(k-p,p). (5.16) Fourier transforming back to position space gives SNL =??d4xd4yd4zd4kd4p(2p)8ei(z(k-p)-xk)y(x)????Sn=01n!S(n)(0)(-2y)neipyF(z,y), (5.17) where the derivative acts only inside the square brackets. Finally integrating by parts we obtain SNL = ??d4xd4yd(x-y)y(x)????Sn=01n!S(n)(0)(-2y)nexp(- ieQxyAndwn)y(y)). (5.18) So, the nonlocal contribution to the amplitude is i e GmNL(y',x,z) = - d3SNLdAm(x)dy(y')dy(z) Am=0 = i e Q??d4yd(z-y)????n=01n!S(n)(0)(-2y)nzyd(x-w)dwmd(y'-y) i e Sn=0 GmNL,n(y',x,z) . (5.19) By Fourier transforming with ei(p'z-py'-qx) and performing some tedious differentiation we find i e GmNL,n(p,q,p+q) = - e Q 1n! S(n)(0) (2p + q)m fn , (5.20) where f0 = 0, f1 = 1, f2 = (q + p)2 + p2, f3 = (q + p)4 + (q + p)2p2 + p4 . (5.21) and in general, for n > 0, fn = fn-1 (q + p)2 +p2(n-1). (5.22) For this to be useful we must be able to sum the entire series. To do this we will proceed with some proofs by induction. First we wish to prove that (q + p)2n - p2n = (q2 + 2p.q)gn , (5.23) where gn is defined by gn = (q + p)2(n-1) + p2gn-1, (5.24) and g1 = 1. Note that (q + p)4 - p4 = (q2 + 2p.q)((q + p)2 + p2), so that g2 = (q + p)2 + p2 = f2. Suppose that equation 5.23 is true for some particular n, and consider (q + p)2(n+1) - p2(n+1) = (q2 + 2p.q + p2)(q + p)2n - p2(n+1) = (q2 + 2p.q)(q + p)2n + p2[](q+p)2n-p2n. (5.25) By assumption we then have (q + p)2(n+1) - p2(n+1) = (q2 + 2p.q)[](q+p)2n+p2gn, (5.26) and by the definition of gn+1 we have (q + p)2(n+1) - p2(n+1) = (q2 + 2p.q)gn+1, (5.27) which completes the proof. We would now like to prove that fn = gn, for n > 0. Suppose that this is true for a particular n (we have demonstrated this above for n = 1, and n = 2), then by the definition of fn+1 we have fn+1 = fn(q + p)2 + p2n. (5.28) Then by assumption we have fn+1 = gn(q2 + 2p.q + p2) + p2n. (5.29a) Use of the preceding lemma yields fn+1 = (q + p)2n - p2n + gnp2 + p2n, (5.29b) which, by the definition of gn+1, gives us fn+1 = gn+1, (5.30) which was to be proved. Inserting these results in equation 5.20, we find: i e GmNL(p,q,p+q) = - e Q n=11n!S(n)(0)(2p+q)mgn = - e Q (2p + q)m Sn=0 1n! S(n)(0) (q+p)2n-p2n2p.q+q2 = - e Q (2p+q)m2p.q+q2 ????n=01n!S(n)(0)(q+p)2n-n=01n!S(n)(0)p2n = - e Q (2p+q)m2p.q+q2 ()S(p+q)-S(p) . (5.31) We may check that the full vertex Gm = GmL + GmNLsatisfies the WT identity -iqmGm(p,q,p') = S-1(p+q) Q - Q S-1(p) . (5.32) Using the inverse quark propagator for our model, S-1(p) = -ip/ + S(p), both sides reduce to - i Q q/ + Q (S(p+q) - S(p)). 5.3 - Photons and Goldstone Bosons Emboldened by this result we may consider calculating the amplitudes which involve photons and Goldstone bosons (GB's). We have already noted (in section 3.1) how V, the exponential of GB fields, transforms under vector transformations. The gauge transformation associated with equation 5.2 is V = exp(ia) V' exp(-ia), which implies that the GB field transforms as pala = exp(ia) p'ala exp(-ia). Thus, we can easily write down the term in the gauged Lagrangian involving one GB: lgp(x,y) = -if y(x)g5S(x-y) ??p(x)exp(-ieQxyAndwn) ??+exp(-ieQxyAndwn)p(y)y(y). (5.33) Following a very similar analysis as that given above, we find that the vertex which couples one photon, one GB, and two quarks, depicted in Figure 5.3, is given by i e Gm,a(p,q,k,p+q+k) = (5.34) ieg5f ??laQ(2p+q)m2p.q+q2()S(p+q)-S(p) ??+Qla(2(p+k)+q)m2(p+k).q+q2()S(p+k+q)-S(p+k). Figure 5.3; The photon-GB-quark-quark vertex, i e Gm,a(p,q,k,p+q+k). The equations get cumbersome for two GB's, but introducing the notation E exp(-ieQxyAndwn), E exp(-ieQyxAndwn), (5.35) we find that the term in the gauged Lagrangian with two GB fields is lgp2(x,y) = -12f2 y(x)S(x-y){}p2(x)E+Ep2(y)+p(x)Ep(y)+Ep(y)Ep(x)Ey(y). (5.36) The vertex with one photon, two GB's, and two quarks, shown in Figure 5.4, is i e Gm,a,b(p,q,k1,k2,p+q+k1+k2) = (5.37) e2f2 ??(lalb+lbla)Q(2p+q)m2p.q+q2()S(p+q)-S(p) + Q(lalb+lbla) (2(p+k1+k2)+q)m2(p+k1+k2).q+q2 ()S(p+k1+k2+q)-S(p+k1+k2) + (Qlalb+lalbQ+lbQla-laQlb) (2(p+k1)+q)m2(p+k1).q+q2 ()S(p+k1+q)-S(p+k1) ??+(Qlbla+lblaQ+laQlb-lbQla)(2(p+k2)+q)m2(p+k2).q+q2()S(p+k2+q)-S(p+k2). Figure 5.4; The two-GB-photon-quark-quark vertex, i e Gm,a,b(p,q,k1,k2,p+q+k1+k2). 5.4 - Electromagnetic Form Factors We can now calculate the two-GB-photon coupling to fourth order in the energy expansion. This allows us to determine the electromagnetic form factors of the the GB's. We simply calculate the effective action for photons and GB's by integrating out the quarks, as before, but this time using the gauged Lagrangian, i.e., exp()-G[p,Am] ??dydyexp????-d4xd4ylg(x,y). (5.38) The effective action can be written as (dropping a constant term as before): G[p,Am] = - Tr log ????d(x-y)-d4zS(x-z)V(z,y), (5.39) where V(z,y) = -d2Sintdy(y)dy(z) . (5.40) and Sint is the interaction part of the gauged action. The quark interactions are represented by V(z,y) in equation 5.39, which is just minus the interaction terms in the Lagrangian, with the quarks fields left out. More explicitly V = Vp + 12 Vpp + VA + VAp + 12 VApp + . . . (5.41) where the subscripts indicate the particles involved in the interaction (the factors of 12 indicate that a term in the Lagrangian with two identical fields contributes twice to the corresponding vertex). Expanding out the log and only keeping terms with one gauge field and two GB fields we find simply G[p,Am] = Tr[]12SVApp+SVASVpSVp+12SVASVpp+SVApSVp. (5.42) If we calculate the amplitude which couples two GB's and one photon from this effective action (i.e. calculate -d3GdpadpbdAm ) we find that it is given by the diagrams in Figure 5.5, as we naively expected. It should be noted that in the calculation of the electromagnetic form factor by Pagels and Stokar [7], these authors only include the second and third Feynman diagram shown in Figure 5.5. Figure 5.5; The two-GB-photon amplitude, e Gm,a,b(p,q,p+q). The sum of the diagrams in Figure 5.5 gives: e Gm,a,b(p,q,p+q) = 4eNcf2 TrQ(lalb-lbla) ??d4k(2p)4Im(k,p,q)[]k2+S2(k)[](k-p)2+S2(k-p)[](k+q)2+S2(k+q) , (5.43) where Im(k,p,q) = (2p+q)m2k.q+q2 []S(k+q)-S(k) G(k,p,q) (5.44) + []S(k)+S(k-p) []S(k+q)+S(k-p) {km[](k+q).(k-p)+S(k+q)S(k-p) - (k-p)m[]k.(k+q)+S(k)S(k+q) + (k+q)m[]k.(k-p)+S(k)S(k-p)}, and G(k,p,q) = 2S(k-p)[]k2+S2(k)[](k+q)2+S2(k+q) (5.45) - []k.(k-p)+S(k)S(k-p)[]S(k)+S(k-p)[](k+q)2+S2(k+q) - [](k+q).(k-p)+S(k+q)S(k-p)[]S(k+q)+S(k-p)[]k2+S2(k) + []S(k)+S(k-p) []S(k+q)+S(k-p) {}k.(k-p)S(k+q)-k.(k+q)S(k-p)+(k+q).(k-p)S(k)+S(k)S(k-p)S(k+q). Taylor expanding to first order in the external momentum we find e Gm,a,b1 (p,q,p+q) = 4eNcf2 Tr Q[lb,la] (2p+q)m??d4k(2p)44S2(k)-2k2S(k)S(k)()k2+S2(k)2 . (5.46) Using equation 3.56 for f2 we find: e Gm,a,b1 (p,q,p+q) = 2e Tr Q[lb,la] (2p+q)m, (5.47) which is the same as the gauge vertex derived from the lowest order chiral Lagrangian, l2 (eq. 2.10) when Wick rotated to Euclidean space. The fact that to this order the amplitude comes out properly normalized and independent of S(p) is a consistency check on our gauging procedure. The contribution to the two-GB-photon amplitude at fourth order in the energy expansion is determined by the term proportional to L9 appearing in l4 (eq. 2.13). By Taylor expanding equation 5.43 to third order3 we find: e Gm,a,b3 (p,q,p+q) = -eNc4p2f2 Tr Q[lb,la] {}pm(I1p2+I2p.q+I3q2)+qm(I4p2+I5p.q+I6q2), (5.48) where the coefficients I1 through I6 are a series of quite messy integrals involving S(p). We will not reproduce these integrals here, but they will be given in Appendix B. The fourth order chiral Lagrangian (eqs. 2.13 and 4.18) gives e Gm,a,b3 (p,q,p+q) = -8eF02 Tr Q[lb,la] (5.49) {}pm(4L11p2+4L11p.q+(2L11+L9)q2)+qm(2L11p2+(2L11-L9)p.q+L11q2). By equating the right hand sides of equations 5.48 and 5.49 we obtain the following relations: L11 = I1Nc128p2 L11 = I2Nc128p2 L11 = I4Nc64p2 L11 = I6Nc32p2 L9 = I3Nc32p2 - 2L11 L9 = - I5Nc32p2 + 2L11 . (5.50) Since we have already determined L11 (in section 4.3) these equations provide several checks on our calculations. Having determined L9, we are in a position to also determine the charge radii (slope of the form factors) of the GB's. The form factor is defined by the coupling of a photon to two on-mass-shell GB's4: e Gm,a,b(p,q,p+q) = 2e Tr Q[lb,la] (2p+q)m Fa(q2). (5.51) In analogy with non-relativistic quantum mechanics, the Taylor expansion of the form factor is used to define the square of the charge radius, a, of particle a: Fa(q2) = 1 + 16 a q2 + . . . (5.52) Using the chiral Lagrangian, Gasser and Leutwyler [46] have calculated the GB charge radii to fourth order, including loop corrections. They found: p = 12L9F20 - 116p2F20 ????2lnMpm+lnMKm+32, K 0 = - 116p2F20 ln MKMp , K + = p + K 0. (5.53) where m is the renormalization scale. We will postpone a comparison with experimental values for the charge radii until chapter 6. 5.5 - Non-Abelian Gauge Fields In this section we discuss the inclusion of external non-Abelian vector gauge fields. The considerations of sections 5.1 through 5.3 still apply, with the only change being the substitution of P exp(-igTaxyVandwn) for exp(-ieQxyAndwn), where Van is the gauge field, g is the gauge coupling constant, Ta are the generators of the gauge group, and P denotes path- ordering of the exponential, starting with functions of x placed on the left, and working towards functions of y on the right. In sections 5.1 through 5.3 we only considered interactions that involved one gauge boson, so path- ordering the exponential makes no difference in any of the derivations given. However, path-ordering does make a difference when two gauge bosons are involved, and we will briefly discuss these differences in this section. We begin with the two gauge boson, two quark vertex (Figure 5.6): g2 Gma,nb(y',x,v,z) = - d4SNLdVam(x)dVbn(v)dy(y')dy(z) Vca=0 =g2??d4yd(z-y)????n=01n!S(n)(0)(-2y)n P ????zyTad(x-w1)dwm1zyTbd(v-w2)dwn2d(y'-y) g2 Sn=0 Gma,nbn(y',x,v,z) . (5.54) Figure 5.6; The vertex coupling two gauge bosons and two quarks, g2 Gma,nb(p,q1,q2,p+q1+q2). The path-ordering operator in equation 5.54 is taken to order Ta and Tb, according to the relative positions of their associated integration variables (w1 and w2) along the path from z to y. The important point is that a derivative with respect to y (the limit of integration) yields the integrand evaluated at y , so the path-ordering places the associated generator on the right. Thus, the ordering of Ta and Tb is determined by which integral is differentiated first, and repeated applications of the product rule will generate both orderings. By Fourier transforming equation 5.54 with ei(p'z-py'-q1x-q2v) and performing the derivatives we find Gma,nbn(p,q1,q2,p+q1+q2) = - 1n!S(n)(0) ??()TaTb+TbTagmndn(p,q1,q2)+TaTb(2p+q2)n(2(p+q2)+q1)mjn(p,q2,q1) ??+TaTb(2p+q1)m(2(p+q1)+q2)njn(p,q1,q2) (5.55) where dn dn(p,q1,q2) jn jn(p,q2,q1), (5.56) and d0 = 0 d1 = 1 d2 = (q1+q2+p)2+p2 d3 = (q1+q2+p)4+(q1+q2+p)2p2+p4 j0 = 0 j1 = 0 j2 = 1 j3 = (q1+q2+p)2+(q2+p)2+p2. (5.57) In general we find, for n > 0, dn = dn-1(q1+q2+p)2+p2(n-1) jn = jn-1(q1+q2+p)2+hn-1 , (5.58) where hn = hn-1(q2+p)2+p2(n-1), (5.59) and h0 = 0, h1 = 1. From equations 5.23 and 5.30, we know that dn = (q1+q2+p)2n-p2n2p.(q1+q2)+(q1+q2)2 (5.60) and jn = (q2+p)2n-p2n2p.q2+q22 (5.61) Note that d1 - h1 = 0, d2 - h2 = (q12+2(p+q2).q1)j2, and d3 - h3 = (q12+2(p+q2).q1)j3. We will again proceed by induction. Suppose that for some particular n, (q12+2(p+q2).q1)jn-1 = dn-1 - hn-1, then, by the definition of jn (eq. 5.58), (q12+2(p+q2).q1)jn = (q12+2(p+q2).q1)[]jn-1(q1+q2+p)2+hn-1. (5.62) By assumption we then have (q12+2(p+q2).q1)jn = []dn-1-hn-1(q1+q2+p)2 + (q12+2(p+q2).q1)hn-1 = dn-1(q1+q2+p)2 - hn-1(q2+p)2 = dn-1(q1+q2+p)2 + p2 - ()hn-1(q2+p)2+p2 = dn - hn , (5.63) where we have used the definitions of dn and hn (eqs. 5.58 and 5.59) Thus, jn = dn-hn2(p+q2)+q12 . (5.64) As in the case of a single gauge boson, we can sum the series and determine the full vertex. We find g2Gma,nb(p,q1,q2,p+q1+q2) = (5.65) - g2 ??(TaTb+TbTa)gmn2p.(q1+q2)+(q1+q2)2()S(p+q1+q2)-S(p)) +TaTb (2p+q2)n(2(p+q2)+q1)m2(p+q2).q1+q12 ????S(p+q1+q2)-S(p)2p.(q1+q2)+(q1+q2)2-S(p+q2)-S(p)2p.q2+q22 ??+TaTb(2p+q1)m(2(p+q1)+q2)n2(p+q1).q2+q22????S(p+q1+q2)-S(p)2p.(q1+q2)+(q1+q2)2-S(p+q1)-S(p)2p.q1+q12. As in the case of one gauge boson, the vertices involving two gauge bosons and GB's follow as simple generalizations of the vertex without GB's. The vertex for two gauge bosons, one GB, and two quarks (as shown in Figure 5.7) is g2Gma,nb,c(p,q1,q2,k,p+k+q1+q2) = (5.66) - ig5g2f {}lcGma,nb(p,q1,q2,p+q1+q2)+Gma,nb(p+k,q1,q2,p+k+q1+q2)lc, Figure 5.7; The vertex coupling two gauge bosons, two quarks, and one GB, g2 Gma,nb,c(p,q1,q2,k,p+k+q1+q2). Introducing the notation Gm,n(p,q1,q2,p+q1+q2) = (5.67) - ??2gmn2p.(q1+q2)+(q1+q2)2()S(p+q1+q2)-S(p)) + (2p+q2)n(2(p+q2)+q1)m2(p+q2).q1+q12 ????S(p+q1+q2)-S(p)2p.(q1+q2)+(q1+q2)2-S(p+q2)-S(p)2p.q2+q22 ??+(2p+q1)m(2(p+q1)+q2)n2(p+q1).q2+q22????S(p+q1+q2)-S(p)2p.(q1+q2)+(q1+q2)2-S(p+q1)-S(p)2p.q1+q12. and p' = p+q1+q2+k1+k2 p'' = p+q1+q2, (5.68) the vertex for two gauge bosons, two GB's, and two quarks (as shown in Figure 5.8) may be written as Gma,nb,c,d(p,q1,q2,p+q1+q2) = (5.69) - g22f2 ??lcldGma,nb(p,q1,q2,p'')+Gma,nb(p+k1+k2,q1,q2,p')lcld + ldGma,nb(p+k1,q1,q2,p''+k1)lc + Gma,nb(p+k1,q1,q2,p''+k1)lcld + lcldGma,nb(p+k1,q1,q2,p''+k1) + lcGmb,na(p+k1,q1,q2,p''+k1)ld - []lalclbld+lblclald-lalcldlb-lblcldla+lclaldlb+lclbldla Gm,n(p+k1,q1,q2,p''+k1) ??+cdk1k2 Note that in the sixth term, the indices a and b are interchanged. As before, all of these multi-gauge boson vertices can be checked using the WT identities. Figure 5.8; The vertex coupling two gauge bosons, two quarks, and two GB's, g2 Gma,nb,c,d(p,q1,q2,k1,k2,p+k1+k2+q1+q2). 5.6 - Radiative Weak Decays We now turn to considering radiative weak decays, specifically pgen. The standard technique [20,21] for describing these decays with chiral Lagrangians is to introduce external SU(N)LSU(N)R gauge fields5. Then the contributions to the pgen amplitude that actually correspond to SU(2)LU(1)Y electroweak gauge theory can be easily picked out. The leading contributions to the decay amplitude for pseudoscalar meson gen are determined by the lowest order chiral Lagrangian l2 (eq. 2.10) and by the Standard Model couplings of e and n to electroweak gauge bosons. These contributions are referred to as inner bremsstrahlung in the literature [47,48]. The remaining contributions are referred to as structure-dependent (SD) radiation. The W has vector and axial-vector couplings, so the SD terms are separated into vector and axial-vector terms. These SD terms in general involve form factors, but for pions the momentum dependence of the form factors can be neglected [48]. It is conventional to refer to the value of axial-vector form factor divided by the vector form factor as g. In the context of the chiral Lagrangian, the vector form factor is determined by the Wess-Zumino term, and the axial- vector form factor receives contributions from the L9 and L10 terms. Using the standard conventions [21,41,49] the chiral Lagrangian gives g = 32 p2 (L9 + L10) , (5.70) thus all that remains for us to do is to calculate L10. The term with fewest fields in the L10 term in the fourth order chiral Lagrangian (eq. 2.13) involves one GB, one vector gauge field, and one axial-vector gauge field. We could calculate the coefficient of this term in the effective chiral Lagrangian derived from our underlying model (and thus determine L10) by introducing vector and axial-vector gauge fields to our model. Since V(x) = exp ????-2ip(x)g5f transforms simply (linearly) under axial transformations, the introduction of axial gauge fields is a simple generalization of the methods used in sections 5.1, 5.2, and 5.5. However, the GB fields do not transform linearly, and an expansion in powers of GB fields proves to be awkward. An alternative method for calculating L10 can be seen by noting that there is a term involving two GB fields, and two vector gauge fields present in the L10 term. The coefficient of this term can be calculated using the techniques we have already developed. The effective action for GB's in the presence of external non-Abelian gauge fields is determined by integrating out the quarks. Dropping a constant term as usual we have G[p,Vam] = - Tr log ????d(x-y)-d4zS(x-z)V(z,y), (5.71) where (as before) V(z,y) = -d2Sintdy(y)dy(z) . (5.72) where Sint is the interaction part of the gauged action. Using our previous conventions, V(z,y) is given by V = Vp + 12 Vpp + VV + VVp + 12 VVpp + 12 VVV + 12 VVVp + 14 VVVpp +. . . (5.73) Expanding out the log and keeping only terms with two gauge fields and two GB's we find G[p,Vam] = Tr[14SVVVpp+12SVpSVVVp+14SVppSVVV+12SVVSVVpp + 12 SVVpSVVp + 12 SVVVSVpSVp + 12 SVppSVVSVV + SVVpSVpSVV ]+SVVpSVVSVp+SVpSVpSVVSVV+12SVpSVVSVpSVV. (5.74) Calculating the amplitude which couples two GB's and two gauge fields from this effective action (i.e. calculate -d4GdpcdpddVamdVbn ) we find that the amplitude is given by the diagrams in Figure 5.9 Figure 5.9; The amplitude coupling two gauge bosons and two GB, g2 Gma,nb,c,d(p,q1,q2,p+q1+q2). The crossed terms correspond to the 15 other ways to assign particles to the legs. Our calculations will be simplified a great deal if we consider the limit where the momenta of the GB's vanish. In this limit most of the contributions to the two GB, two gauge boson vertex vanish. For vanishing GB momenta, we Taylor expand the sum of diagrams in Figure 5.9 in the gauge boson momentum q. Keeping only terms of zeroth order in q we find g2 Gma,nb,c,d1(0,q,-q,0) = ig2Nc16p2f2 J1 gmn Tr[2lclaldlb+2lclbldla-lcldlalb ]-lclalbld-lcldlbla-lclblald. (5.75) where J1 represents a messy integral. The second order chiral Lagrangian (eq. 2.10) gives the amplitude g2 Gma,nb,c,d1(0,q,-q,0) = -2i g2 gmn Tr[2lclaldlb+2lclbldla-lcldlalb ]-lclalbld-lcldlbla-lclblald. (5.76) The requirement that f2 = - J1Nc32p2 (5.77) is another check on our calculations. The terms of second order in q give an amplitude of the form g2 Gma,nb,c,d2(0,q,-q,0) = g2Nc16p2f2 ()J2q2gmn+J3qmqn Tr[2lclaldlb+2lclbldla-lcldlalb ]-lclalbld-lcldlbla-lclblald. (5.78) where J2 and J3 are again some messy integrals which will be given in Appendix B. The fourth order chiral Lagrangian (eq. 2.13 and eq. 4.18) taking vanishing GB momentum gives g2 Gma,nb,c,d2(0,q,-q,0) = 8g2F20 ()L10q2gmn-(L11+L10)qmqn Tr[]2lclaldlb+2lclbldla-lcldlalb ]-lclalbld-lcldlbla-lclblald. (5.79) Thus, since we have already determined L11, we find two equations for L10: L10 = J2Nc128p2 L10 = -J3Nc128p2 - L11 . (5.80) The requirement that both equations give the same result for L10 is another check on our calculations. Numerical results for L10 will be presented in chapter 6. Footnotes for Chapter 5: 1 For a summary of the transformation to Euclidean space-time see the Appendix. 2 This simple transformation property is maintained in the non-Abelian case by path-ordering. 3 For the purposes of power counting in the energy expansion, gauge fields are treated as being the same order as derivatives [20,21]. 4 The GB's must be charge eigenstates. 5 Where N is, as always, the number of quark flavours.