MPI-PhT/94-96 TUM-T31-82/94 hep-ph/9501281 December 1994 5 9 9 1 n a J 2 Effective Hamiltonian for B X e+e Beyond Leading 1 s − → 1 Logarithms in the NDR and HV Schemes v ∗ 1 8 2 Andrzej J. BURAS1,2 and Manfred MU¨NZ1 1 0 1 Physik Department, Technische Universita¨t Mu¨nchen, D-85748 Garching, Germany. 5 9 2 Max-Planck-Institut fu¨r Physik – Werner-Heisenberg-Institut, / h F¨ohringer Ring 6, D-80805 Mu¨nchen, Germany. p - p e h : Abstract v i X Wecalculate thenext-to-leadingQCDcorrections totheeffective Hamiltonian forB r X e+e intheNDRandHVschemes. Wegiveforthefirsttimeanalyticexpressionsf→or a s − theWilsonCoefficientoftheoperatorQ = (s¯b) (e¯e) intheNDRandHVschemes. 9 V A V − Calculating the relevant matrix elements of local operators in the spectator model we demonstrate the scheme independence of the resulting short distance contribution to the physical amplitude. Keeping consistently only leading and next-to-leading terms, we find an analytic formula for the differential dilepton invariant mass distribution in the spectator model. Numerical analysis of the m , Λ and µ (m ) dependences t MS ≈ O b of this formula is presented. We compare our results with those given in the literature. ∗ Supported by the German Bundesministerium fu¨r Forschung und Technologie under contract 06 TM 743 and the CEC Science project SC1-CT91-0729. 1 Introduction The rare decay B X e+e has been the subject of many theoretical studies in the frame- s − → work of the standard model and its extensions such as the two Higgs doublet models and models involving supersymmetry [1–8]. In particular the strong dependence of B X e+e s − → on m has been stressed by Hou et. al. [1]. It is clear that once B X e+e has been ob- t s − → served, it will offer an useful test of the standard model and of its extensions. To this end the relevant branching ratio, the dilepton invariant mass distribution and other distributions of interest should be calculated with sufficient precision. In particular the QCD effects should be properly taken into account. The central element in any analysis of B X e+e is the effective Hamiltonian for s − → ∆B = 1 decays relevant for scales µ (m ) in which the short distance QCD effects b ≈ O are taken into account in the framework of a renormalization group improved perturbation theory. These short distance QCD effects have been calculated over the last years with increasing precision by several groups [2,9,10] culminating in a complete next-to-leading QCD calculation presented by Misiak in ref. [11] and very recently in a corrected version in [12]. The actual calculation of B X e+e involves not only the evaluation of Wilson coeffi- s − → cients of ten local operators (see (2.1)) which mix under renormalization but also the cal- culation of the corresponding matrix elements of these operators relevant for B X e+e . s − → The latter part of the analysis can be done in the spectator model, which, according to heavy quark effective theory, for B-decays should offer a good approximation to QCD. One can also include the non-perturbative (1/m2) corrections to the spectator model which en- O b hance the rate for B X e+e by roughly 10% [13]. A realistic phenomenological analysis s − → should also include the long distance contributions which are mainly due to the J/ψ and ψ ′ resonances [14–16]. Since in this paper we are mainly interested in the next-to-leading short distance QCD corrections to the spectator model we will not include these complications in what follows. It is well known that the Wilson coefficients of local operators depend beyond the leading logarithmic approximation on the renormalization scheme for operators, in particular on the treatment of γ in D = 4 dimensions. This dependence must be cancelled by the 5 6 scheme dependence present in the matrix elements of operators so that the final decay amplitude does not depend on the renormalization scheme. In the context of B X e+e s − → this point has been emphasized in particular by Grinstein et. al. [2]. Other examples such as K ππ, K π0e+e , B X γ can be found in refs. [17–19]. The interesting L,S − s → → → feature of B X e+e as compared to decays such as K ππ, is the fact that due to the s − → → ability of calculating reliably the matrix elements of all operators contributing to this decay, the cancellation of scheme dependence can be demonstrated in the actual calculation of the short distance part of the physical amplitude. Now all the existing calculations of B X e+e use the NDR renormalization scheme s − → (anticommuting γ in D = 4 dimensions). Even if arguments have been given, in particular 5 6 in [2] and [11], how the cancellation of the scheme dependence in B X e+e would s − → take place, it is of interest to see this explicitly by calculating this decay in two different renormalization schemes. In addition, in view of the complexity of next-to-leading order (NLO) calculations and the fact that the only complete NLO analysis of B X e+e has s − → 1 been done by a single person, it is important to check the results of refs. [11,12]. Herewewillpresent thecalculationsoftheWilsoncoefficientsandmatrixelementsrelevant for B X e+e in two renormalization schemes (NDR and HV [20]) demonstrating the s − → scheme independence of the resulting amplitude. Beside this the main results of our paper are as follows: We give for the first time analytic NLO expressions for the Wilson coefficient of the • operator Q = (s¯b) (e¯e) in the NDR and HV schemes. 9 V A V − Calculating the matrix elements of local operators in the spectator model we fully • agree with Misiak’s result for the dilepton invariant mass distribution very recently given in [12]. Wefind, that intheHVscheme thescheme dependent terminthematrixelements (the • socalledξ-term)receivesinadditiontocurrent-current contributionsalsocontributions from QCD penguin operators which are necessary for the cancellation of the scheme dependence in the final amplitude. This should be compared with the discussion of the scheme dependence given in refs. [2] and [11] where the ξ-term received only contributions from current-current operators. We stress that in a consistent NLO analysis of the decay B X e+e , one should s − • → on one hand calculate the Wilson coefficient of the operator Q = (s¯b) (e¯e) in- 9 V A V − cluding leading and next-to-leading logarithms, but on the other hand only leading logarithms should be kept in the remaining Wilson coefficients. Only then a scheme independent amplitude can be obtained. This special treatment of Q is related to the 9 fact that strictly speaking in the leading logarithmic approximation only this operator contributes to B X e+e . The contributions of the usual current-current opera- s − → tors, QCD penguin operators, magnetic penguin operators and of Q = (s¯b) (e¯e) 10 V A A − enter only at the NLO level and to be consistent only the leading contributions to the corresponding Wilson coefficients should be included. In this respect we differ from the original analysis of Misiak [11] who in his numerical evaluation of B X e+e also s − → included partiallyknown NLOcorrections toWilson coefficients of operatorsQ (i = 9). i 6 These additional corrections are, however, scheme dependent and are really a part of stillhigherorder intherenormalizationgroupimproved perturbationtheory. Themost recent analysis of Misiak [12] does not include these contributions and can be directly compared with the present paper. Keeping consistently only the leading and next-to-leading contributions to B • → X e+e we are able to give analytic expressions forall Wilson coefficients which should s − be useful for phenomenological applications. Our paper is organized as follows: In sect. 2 we collect the master formulae for B X e+e in the spectator model which s − → include consistently leading and next-to-leading logrithms. In sect. 3 we describe some details of the NLO calculation of the Wilson coefficient C (µ) and of the relevant one-loop 9 matrix elements in NDR and HV schemes. In sect. 4 we present a numerical analysis. We end our paper with a brief summary of the main results. 2 2 Master Formulae 2.1 Operators Our basis of operators is given as follows: Q = (s¯ c ) (c¯ b ) 1 α β V A β α V A − − Q = (s¯c) (c¯b) 2 V A V A − − Q = (s¯b) (q¯q) 3 V A q V A − − Q = (s¯ b ) P (q¯ q ) 4 α β V A q β α V A − − Q = (s¯b) P(q¯q) 5 V A q V+A − (2.1) Q = (s¯ b ) P (q¯ q ) 6 α β V A q β α V+A − Q = e m s¯ σµPν(1+γ )b F 7 8π2 b α 5 α µν Q = g m s¯ σµν(1+γ )Ta b Ga 8 8π2 b α 5 αβ β µν Q = (s¯b) (e¯e) 9 V A V − Q = (s¯b) (e¯e) 10 V A A − where α and β denote colour indices. We omit the colour indices for the colour-singlet currents. Labels (V A) refer to γ (1 γ ). Q are the current-current operators, Q µ 5 1,2 3 6 ± ± − the QCD penguin operators, Q “magnetic penguin” operators and Q semi-leptonic 7,8 9,10 electroweak penguin operators. Our normalizations are as in refs. [18] and [19]. 2.2 Wilson Coefficients The Wilson coefficients for the operators Q –Q are given in the leading logarithmic approx- 1 7 imation by [18,21–23] 8 C(0)(µ) = k ηai (j = 1,...6) (2.2) j ji i=1 X 8 8 C7(0)eff(µ) = η2136C7(0)(MW)+ 3 η2134 −η1263 C8(0)(MW)+ hiηai, (2.3) (cid:16) (cid:17) Xi=1 with α (M ) s W η = , (2.4) α (µ) s 1 (0) C (M ) = A(x ), (2.5) 7 W −2 t 1 (0) C (M ) = F(x ), (2.6) 8 W −2 t 3 where x = m2/M2 and A(x) and F(x) are defined in (2.14) and (2.19). The numbers a , t t W i k and h are given by ji i a = ( 14, 16, 6 , 12, 0.4086, 0.4230, 0.8994, 0.1456 ) i 23 23 23 −23 − − k = ( 0, 0, 1, 1, 0, 0, 0, 0 ) 1i 2 −2 k = ( 0, 0, 1, 1, 0, 0, 0, 0 ) 2i 2 2 k = ( 0, 0, 1 , 1, 0.0510, 0.1403, 0.0113, 0.0054 ) 3i −14 6 − − k = ( 0, 0, 1 , 1, 0.0984, 0.1214, 0.0156, 0.0026 ) 4i −14 −6 k = ( 0, 0, 0, 0, 0.0397, 0.0117, 0.0025, 0.0304 ) 5i − − k = ( 0, 0, 0, 0, 0.0335, 0.0239, 0.0462, 0.0112 ) 6i − − h = ( 2.2996, 1.0880, 3, 1 , 0.6494, 0.0380, 0.0186, 0.0057 ). i − −7 −14 − − − − (2.7) The first correct calculation of the two-loop anomalous dimensions relevant for (2.3) has been presented in [21,22] and confirmed subsequently in [12,24,25]. The coefficient C(0)eff(µ) does not enter the formula for B X e+e at this level of 8 → s − accuracy. An analytic formula is given in ref. [18]. The coefficient of Q is given by 10 α Y(x ) t C (M ) = C (M ), C (M ) = (2.8) 10 W 2π 10 W 10 W −sin2Θ W e e withY(x)givenin(2.13). SinceQ doesnotrenormalizeunder QCD,itscoefficient doesnot 10 depend on µ (m ). The only renormalization scale dependence in (2.8) enters through b ≈ O the definition of the top quark mass. We will return to this issue in sect. 4. Finally, including leading as well as next-to-leading logarithms, we find α CNDR(µ) = CNDR(µ) (2.9) 9 2π 9 Y(x ) CNDR(µ) = PNDeR + t 4Z(x )+P E(x ) (2.10) 9 0 sin2Θ − t E t W e with π 8 PNDR = ( 0.1875+ p ηai+1) 0 α (M ) − i s W i=1 X 8 +1.2468+ ηai[rNDR +s η] (2.11) i i i=1 X 8 P = 0.1405+ q ηai+1 (2.12) E i i=1 X 1 Y(x) = C(x) B(x), Z(x) = C(x)+ D(x). (2.13) − 4 Here x(8x2 +5x 7) x2(2 3x) A(x) = − + − lnx, (2.14) 12(x 1)3 2(x 1)4 − − 4 x x B(x) = + lnx, (2.15) 4(1 x) 4(x 1)2 − − x(x 6) x(3x+2) C(x) = − + lnx, (2.16) 8(x 1) 8(x 1)2 − − 19x3 +25x2 x2(5x2 2x 6) 4 D(x) = − + − − lnx lnx, (2.17) 36(x 1)3 18(x 1)4 − 9 − − x(18 11x x2) x2(15 16x+4x2) 2 E(x) = − − + − lnx lnx, (2.18) 12(1 x)3 6(1 x)4 − 3 − − x(x2 5x 2) 3x2 F(x) = − − + lnx. (2.19) 4(x 1)3 2(x 1)4 − − The coefficients p , rNDR, s , and q are found to be as follows: i i i i p = ( 0, 0, 80 , 8 , 0.0433, 0.1384, 0.1648 0.0073 ) i −203 33 − rNDR = ( 0, 0, 0.8966, 0.1960, 0.2011, 0.1328, 0.0292, 0.1858 ) i − − − − s = ( 0, 0, 0.2009, 0.3579, 0.0490, 0.3616, 0.3554, 0.0072 ) i − − − − q = ( 0, 0, 0, 0, 0.0318, 0.0918, 0.2700, 0.0059 ). i − (2.20) P is (10 2) and consequently the last term in (2.10) can be neglected. We keep it E − O however in our numerical analysis. In the HV scheme only the coefficients r are changed. They are given by i rHV = ( 0, 0, 0.1193, 0.1003, 0.0473, 0.2323, 0.0133, 0.1799 ). (2.21) i − − − − Equivalently we can write 4 PHV = PNDR +ξHV 3C(0) +C(0) C(0) 3C(0) (2.22) 0 0 9 1 2 − 3 − 4 (cid:16) (cid:17) with 0, NDR ξ = (2.23) 1, HV. ( − We note that 8 8 p = 0.1875, q = 0.1405, (2.24) i i − i=1 i=1 X X 8 4 8 16 (r +s ) = 1.2468+ (1+ξ), p (a +1) = . (2.25) i i i i − 9 −69 i=1 i=1 X X In this way for η = 1 we find P = 0, PNDR = 4/9 and PHV = 0 in accordance with the E 0 0 initial conditions in (3.3). Moreover, the second relation in (2.25) assures the correct large logarithm in PNDR, i. e. 8/9 ln(M /µ). The derivation of (2.9)–(2.22) is given in sect. 3. 0 W 2.3 The Differential Decay Rate Introducing (pe+ +pe−)2 mc sˆ= , z = (2.26) m2 m b b 5 and calculating the one-loop matrix elements of Q using the spectator model in the NDR i scheme we find d Γ(b se+e ) α2 V 2 (1 sˆ)2 R(sˆ) dsˆ → − = ts − (1+2sˆ) Ceff 2 + C 2 + ≡ Γ(b ceν¯ ) 4π2 V f(z)κ(z) · | 9 | | 10| → (cid:12)(cid:12) cb(cid:12)(cid:12) (cid:20) (cid:16) (cid:17) 4 1(cid:12)(cid:12)+ 2(cid:12)(cid:12) C(0)eff 2 +12C(0)eff ReeCeff e (2.27) sˆ | 7 | 7 9 (cid:18) (cid:19) (cid:21) e where Ceff = CNDRη˜(sˆ)+h(z,sˆ) 3C(0) +C(0) +3C(0) +C(0) +3C(0) +C(0) 9 9 1 2 3 4 5 6 1 (cid:16) (cid:17) e e h(1,sˆ) 4C(0) +4C(0) +3C(0) +C(0) −2 3 4 5 6 1 (cid:16) 2 (cid:17) (0) (0) (0) (0) (0) (0) h(0,sˆ) C +3C + 3C +C +3C +C . (2.28) −2 3 4 9 3 4 5 6 (cid:16) (cid:17) (cid:16) (cid:17) Here 8 m 8 8 4 b h(z,sˆ) = ln lnz + + x (2.29) −9 µ − 9 27 9 2 ln √1 x+1 iπ , for x 4z2 < 1 − (2+x) 1 x 1/2 √1 x 1 − ≡ sˆ −9 | − |  (cid:16)2ar(cid:12)cta−n −1(cid:12) , (cid:17) for x 4z2 > 1,  (cid:12)(cid:12) √x−(cid:12)(cid:12)1 ≡ sˆ 8 8 m 4 4 b h(0,sˆ) = ln lnsˆ+ iπ. (2.30) 27 − 9 µ − 9 9 f(z) = 1 8z2 +8z6 z8 24z4lnz, (2.31) − − − 2α (µ) 31 3 κ(z) = 1 s (π2 )(1 z)2 + (2.32) − 3π − 4 − 2 (cid:20) (cid:21) α (µ) s η˜(sˆ) = 1+ ω(sˆ) (2.33) π with 2 4 2 5+4s ω(sˆ) = π2 Li (s) lnsln(1 s) ln(1 s) 2 −9 − 3 − 3 − − 3(1+2s) − − 2s(1+s)(1 2s) 5+9s 6s2 − lns+ − . (2.34) 3(1 s)2(1+2s) 6(1 s)(1+2s) − − Here f(z) and κ(z) are the phase-space factor and the single gluon QCD correction to the b ceν¯ decay [26,27] respectively. η˜ on the other hand represents single gluon corrections → to the matrix element of Q with m = 0 [12,28]. For consistency reasons this correction 9 s should only multiply the leading logarithmic term in PNDR. 0 In the HV scheme the one-loop matrix elements are different and one finds an additional explicit contribution to (2.28) given by 4 ξHV 3C(0) +C(0) C(0) 3C(0) . (2.35) − 9 1 2 − 3 − 4 (cid:16) (cid:17) However CNDR has to be replaced by CHV given in (2.10) and (2.22) and consequently Ceff 9 9 9 is the same in both schemes. e e e 6 The first term in the function h(z,sˆ) in (2.29) represents the leading µ-dependence in the matrix elements. It is cancelled by the µ-dependence present in the leading logarithm in C . 9 The µ-dependence present in the coefficients of the other operators can only be cancelled by e going to still higher order in the renormalization group improved perturbation theory. To this end the matrix elements of four-quark operators should be evaluated at two-loop level. Also certain unknown three-loop anomalous dimensions should be included in the evaluation of Ceff and C [18,19]. Certainly this is beyond the scope of the present paper and we will 7 9 only investigate the left-over µ-dependence in sect. 4. The fact that the coefficient C should include next-to-leading logarithms and the other 9 coefficients should be calculated in the leading logarithmic approximation is easy to un- derstand. There is a large logarithm in C represented by 1/α in P in (2.11). Conse- 9 s 0 quently the renormalization group improved perturbation theory for C has the structure 9 (1/α ) + (1) + (α ) + ... whereas the corresponding series for the remaining coeffi- s s O O O cients is (1)+ (α )+.... Therefore in order to find the next-to-leading (1) term, the s O O O full two-loop renormalization group analysis for the operators in (2.1) has to be performed in order to find C , but the coefficients of the remaining operators should be taken in the 9 leading logarithmic approximation. This is gratifying because the coefficient of the mag- netic operator Q is known only in the leading logarithmic approximation. Q does not mix 7 7 with Q and has no impact on the coefficients C –C . Consequently the necessary two-loop 9 1 6 renormalization group analysis of C can be performed independently of the presence of the 9 magnetic operators, which was also the case of the decay K π0e+e presented in ref. [19]. L − → Letusfinallycompareourmainformulae(2.27)–(2.35)withtheonesgivenintheliterature: i) The general expression (2.27) with κ(z) = 1 is due to Grinstein et. al. [2] who in their approximate leading order renormalizationgroup analysis kept only the operators Q ,Q ,Q ,Q ,Q . 1 2 7 9 10 ii) Inserting C(0) and CNDR in (2.2)and (2.8) into (2.28) we find an analytic expression i 9 for Ceff which agrees with a recent independent calculation of Misiak [12]. 9 e iii) Theesign of iπ in (2.29) differs from the one given in [2] and [11] but agrees with [12] and also with the work of Fleischer [29]. iv) The “ξ-term” given in (2.35) contains in the HV scheme also contributions from the operators Q and Q , which are however negligible. The discussion of the “ξ-term” in 3 4 refs. [2] and [11] does not apply then to the HV scheme. 3 Technical Details 3.1 Wilson Coefficients In order to calculate the coefficient C including next-to-leading order corrections we have 9 to perform in principle a two-loop renormalization group analysis for the full set of operators given in (2.1). However, Q is not renormalized and the dimension five operators Q and 10 7 Q have no impact on C . Consequently only a set of seven operators, Q and Q , has to 8 9 1 6 9 be considered. This is precisely the case of the decay K π0e+e conside−red in [19] except L − → 7 for an appropriate change of quark flavours and the fact that now µ (m ) instead of b ≈ O µ (1GeV) should be considered. Because our detailed NLO analysis of K π0e+e L − ≈ O → has already been published we will only discuss very briefly an analogous calculation of B X e+e , referring the interested reader to [19]. We should stress that Misiak [11,12] s − → used different conventions for the evanescent operators than used in [19] and here. The agreement on Ceff is therefore particularly satisfying. 9 Integrating out simultaneously W,Z and t we construct first the effective Hamiltonian for e ∆B = 1 transitions relevant for b se+e with the operators normalized at µ = M . − W → Dropping the operators Q , Q and Q for the reasons stated above and using the unitarity 7 8 10 of the CKM matrix we find G 6 F (∆B = 1) = V V C (M )Q +C (M )Q Heff −√2 t∗s tb" i W i 9′ W ′9# i=1 X G F (u) (u) + V V C (M )(Q Q )+C (M )(Q Q ) . (3.1) √2 u∗s ub 1 W 1 − 1 2 W 2 − 2 h i (u) Here Q are obtained from Q through the replacement c u. In order to make all the el- 1,2 1,2 → ements of the anomalous dimension matrix be of the same order in α , we have appropriately s rescaled C and Q : 9 9 α α (µ) s Q = Q , C (µ) = C (µ) (3.2) ′9 α (µ) 9 9′ α 9 s NotethatbecauseofGIMcancellationtherearenopenguincontributionsinthetermpropor- tional to V V . They would appear only at scales µ < m as was the case in K π0e+e . u∗s ub c L → − Since V V /V V < 0.02 we will drop the second term in what follows. | u∗s ub t∗s tb| The initial conditions at µ = M for the coefficients C –C in NDR and HV schemes have W 1 6 been given in sect. 2.4 and in the appendix A of ref. [19] respectively. Here it suffices to give only the initial condition for the coefficient C (denoted by C in [19]) which reads: 9′ 7′V α (M ) Y(x ) 4 s W t C (M ) = 4Z(x )+ (1+ξ) , (3.3) 9′ W 2π "sin2ΘW − t 9 # where ξ has been defined in (2.23). The x dependence originates in box diagrams and in t the γ- and Z-penguin diagrams [30]. With C~T (C ,...,C ,C ) (3.4) ≡ 1 6 9′ ˆ one can calculate the coefficients C (µ) by using the evolution operator U (µ,M ) relevant i 5 W for an effective theory with f = 5 flavours: C~(µ) = Uˆ (µ,M )C~(M ). (3.5) 5 W W An explicit expression for Uˆ is given in sect. 2 of [19] where also the relevant expressions for 5 one- and two-loop anomalous dimensions can be found. One only has to set f = 5, u = 2 and d = 3 in the formulae given in [19]. Using (3.5) and rescaling back the operator Q we find at µ (m ) 9 b ≈ O G 6 F (∆B = 1) = V V C (µ)Q +C (µ)Q (3.6) Heff −√2 t∗s tb" i i 9 9# i=1 X 8 with the coefficient C (µ) given in (2.10) and (2.22) for NDR and HV schemes respectively. 9 The result for HV can either be found directly using (3.5) or by using the relation α (µ) C~HV(µ) = ˆ1 s ∆rˆT C~NDR(µ) (3.7) − 4π ! with the matrix ∆rˆ given in appendix A of ref. [19]. 3.2 One-Loop Matrix Elements The operators Q and Q contribute at this level of accuracy only through tree level matrix 7 10 elements. Q contributes only through the renormalization of Q and its impact is only felt 8 7 (0)eff in C . The four-quark operators Q , contribute at one-looplevel through the diagrams 7 1 6 − in fig. 1 where “ ” denotes the operator insertion. Finally at next-to-leading level (α ) s ⊗⊗ O corrections to the matrix element Q have to be calculated. 9 h i γ γ e e e e a) b) Figure 1: The two possibilites for insertion of a four-quark operator into a penguin diagram. Let us begin with Q . As usual two types of insertions of the operators into the 1 6 h − i penguin diagrams have to be considered. As already discussed in ref. [31] the appearance of a closed fermion loop in fig. 1a does not pose any problems in the NDR scheme because nowhere in the calculation one has to evaluate Tr[γ γ γ γ γ ]. The diagrams in fig. 1 have µ ν ρ λ 5 been evaluated for the operators Q and Q by Grinstein et. al. [2] and by Misiak [11] for the 1 2 full set Q –Q . These calculations have been done in the NDR scheme. Calculating these 1 6 diagrams in the NDR and HV schemes we find α 4 Q = 3h(z,sˆ) ξ Q 1 9 0 h i 2π − 3 h i (cid:18) (cid:19) α 4 Q = h(z,sˆ) ξ Q 2 9 0 h i 2π − 9 h i (cid:18) (cid:19) α 1 2 4 Q = 3h(z,sˆ) 2h(1,sˆ) h(0,sˆ)+ + ξ Q (3.8) 3 9 0 h i 2π − − 2 3 9 h i (cid:18) (cid:19) α 3 2 4 Q = h(z,sˆ) 2h(1,sˆ) h(0,sˆ)+ + ξ Q 4 9 0 h i 2π − − 2 9 3 h i (cid:18) (cid:19) α 3 2 Q = 3h(z,sˆ) h(1,sˆ)+ Q 5 9 0 h i 2π − 2 3 h i (cid:18) (cid:19) α 1 2 Q = h(z,sˆ) h(1,sˆ)+ Q 6 9 0 h i 2π − 2 9 h i (cid:18) (cid:19) 9