Heavy meson hyperfine splitting: a complete calculation

HEAVY MESON HYPERFINE SPLITTING: A COMPLETE 1/mQ CALCULATION. 1 N. Di Bartolomeo and R. Gatto arXiv:hep-ph/9411210v1 2 Nov 1994 Département de Physique Théorique, Univ. de Genève F. Feruglio Dipartimento di Fisica, Univ. di Padova I.N.F.N., Sezione di Padova G. Nardulli Dipartimento di Fisica, Univ. di Bari I.N.F.N., Sezione di Bari UGVA-DPT 1994/10-864 BARI/TH 190-94 hep-ph/9411210 October 1994 ∗ Partially supported by the Swiss National Foundation 1 Partially supported by the Swiss National Foundation. ABSTRACT The hyperfine splittings ∆D = (mDs∗ − mDs ) − (mD∗+ − mD+ ) and ∆B = (mBs∗ − mBs ) − (mB∗0 − mB0 ) are analyzed in the framework of an effective lagrangian possessing chiral, heavy flavour and spin symmetries, explicitly broken by a complete set of first order terms. Among these terms, those responsible for the difference between the couplings gP ∗ P ∗ π and gP ∗ P π are evaluated in the QCD sum rules approach. Their contribution to ∆D and to ∆B appears to quantitatively balance previously estimated chiral effects nice agreement with the experimental data, solving a suspected puzzle for heavy quark theory. 2 1 Introduction The spectroscopy of heavy mesons is among the simplest framework where the ideas and the methods of heavy quark expansion can be quantitatively tested. Recently, attention has been focused on the combinations [1, 2, 3, 4]: ∆D = (mDs∗ − mDs ) − (mD∗+ − mD+ ) (1.1) ∆B = (mBs∗ − mBs ) − (mB∗0 − mB0 ) (1.2) which are measured to be [5]: ∆D ≃ 1.0 ± 1.8 MeV (1.3) ∆B ≃ 1.0 ± 2.7 MeV (1.4) The above hyperfine splitting is free from electromagnetic corrections and it vanishes separately in the SU(3) chiral limit and in the heavy quark limit. In the combined chiral and heavy quark expansion, the leading contribution is of order ms /mQ and one would expect the relation [1]: mc ∆B = ∆D (1.5) mb In the so called heavy meson effective theory [6], which combines the heavy quark expansion and the chiral symmetry, there is only one lowest order operator contributing to ∆D,B : a λ2 j (mξ )b λ2 O2 = T r[H̄ai σµν Hjb σ µν ](m−1 (1.6) ) Q i 8 ΛCSB where i, j are heavy flavour indices and a,b light flavour indices. The 4 × 4 Dirac matrix Hai describes the spin doublet P , P ∗ , with P heavy meson composed by the heavy quark Qi and the light antiquark q̄a . The matrix mξ is mξ = (ξmq ξ + ξ † mq ξ † ) (1.7) Here mq is the light quarks mass matrix and ξ = exp(iM/f ), where M is the pseudoscalar 3 × 3 matrix and f the pseudoscalar decay constant (we take f = 132 MeV ). By taking ms /ΛCSB ≃ 0.15 and by taking λ2 ≃ Λ2QCD ≃ 0.1GeV 2 one would estimate: (2) ∆D ≃ 20 MeV (2) ∆B ≃ 6 MeV (1.8) (1.9) Given the present experimental accuracy, the above estimate is barely acceptable, as an order of magnitude, for ∆B , while it clearly fails to reproduce the data for ∆D . If the contribution from O2 were the only one responsible for the hyperfine splittings, agreement with the data clearly would require a rather small value for λ2 . In chiral perturbation theory, an independent contribution arises from one-loop corrections to the heavy meson self energies [3], evaluated from an initial lagrangian containing, 1 at the lowest order, both chiral breaking and spin breaking terms. The loop corrections in turn depend on an arbitrary renormalization point µ2 (e.g. the t’Hooft mass of dimensional regularization). This dependence is cancelled by the µ2 dependence of the counterterm λ2 (µ2 )O2 , as it should happen for any physical result. A commonly accepted point of view is that the overall effect of adding the counterterm consists in replacing µ2 in the loop corrections with the physical scale relevant to the problem at hand, Λ2CSB . Possible finite terms in the counterterm are supposed to be small compared to the large chiral logarithms. With this philosophy in mind, two classes of such corrections has been estimated in ref. [2] and they give (for the values of the parameters given by these authors): ∆0D ≃ +30 MeV, ∆1D ≃ +65 MeV, (1.10) ∆0B ≃ +10 MeV, ∆1B ≃ +22 MeV, (1.11) Here ∆0 represents the contribution of the chiral logarithm and ∆1 is a non analytic contribution, of order ms3/2 . This provides a rather uncomfortable situation since, to account for the observed data, one should require an accurate and innatural cancellation between (∆0 + ∆1 ) and the finite terms from ∆(2) , contrary to the usual expectation. The chiral computation giving ∆0 + ∆1 is however incomplete [4], because it does not include the spin breaking effect due to the difference between the P ∗ P ∗ π and the P ∗ P π couplings (P = D, B), defined by the relations: < π − (q) P o (q2 )|P ∗− (q1 , ǫ) > = 2 gP ∗ P π < π(q) P ∗ (q2 , ǫ2 )|P ∗(q1 , ǫ1 ) > = −i mP µ ǫ · qµ fπ 2 gP ∗ P ∗ π ǫµναβ ǫµ1 ǫν2 q α q1β fπ (1.12) (1.13) The scaling law 2gP ∗ P π mP /fπ for the strong D ∗ Dπ coupling constant was first proposed in [7, 8]. The splitting between the coplings (1.12) and (1.13) is of order 1/mQ , and therefore has to be taken into account in the chiral computation, to work consistently at the desired order. In the present paper we will provide an estimate of gP ∗ P ∗ π − gP ∗ P π based on a QCD sum rule, and, by including this additional spin breaking effect, we will complete the evaluation of ∆D,B coming from the chiral loops. 2 The Hyperfine Splitting To better clarify the importance of gP ∗ P ∗ π − gP ∗ P π for the problem at hand, we remind that the effective lagrangian for heavy mesons and light pseudoscalars, at first order in m−1 Q and in the light quark masses mq reads: L = L0 + Lq + LQ 2 (2.1) Here L0 represents the chiral, heavy flavour and spin symmetric term: L = −iT r[H̄ai vµ ∂ µ Hia ] + f2 T r[∂µ Σ† ∂ µ Σ] 8 i T r[H̄ai Hib ]v µ (ξ †∂µ ξ + ξ∂µ ξ † )ab 2 i gT r[H̄ai Hibγµ γ5 ](Aµ )ab + 2 + (2.2) where Σ = ξ 2 and: Aµ = ξ † ∂µ ξ − ξ∂µ ξ † (2.3) From the last term in eq. (2.2), one obtains the P ∗ P π and P ∗P ∗ π couplings defined in eqs. (1.12) and (1.13), in the limit mP → ∞: gP ∗ P π = gP ∗ P ∗ π = g (2.4) The leading chiral breaking corrections are given by: Lq = λ0 T r[mq Σ + Σ† mq ] + λ1 T r[H̄ai Hib ](mξ )ab + λ′1 T r[H̄ai Hia ](mξ )aa (2.5) The second term in eq. (2.5) is responsible for the mass splitting between strange and non-strange heavy mesons: (2.6) ∆s = 2λ1 ms One has approximately ∆s ≃ 100 MeV , λ1 ≃ 0.33. The third term, listed for completeness, gives an equal contribution to each heavy meson mass, it does not affect the hyperfine splitting, and it does not play any role in our analysis. Finally the terms of order 1/mQ , breaking either the heavy flavour or the spin symmetries, are given by: λ j LQ = − T r[H̄ai σµν Hja σ µν ](m−1 Q )i 8 ig (a + b) j µ a T r[H̄ai Hjb γµ γ5 ](m−1 + Q )i (A )b 2 2 ig (a − b) j µ a T r[H̄ai γµ γ5 Hjb](m−1 + Q )i (A )b 2 2 (2.7) The first term in eq. (2.7) is responsible for the splitting ∆ between the 1− and 0− heavy meson masses: 2λ ∆= (2.8) mQ 3 For the B, B ∗ system ∆ ≃ 46 MeV , whereas for D, D ∗ ∆ is approximately 141 MeV , so that one has: λ ≃ 0.10 − 0.11 GeV 2 (2.9) The second term in (2.7) breaks only the heavy flavour symmetry, making the B ∗ B (∗) π and D ∗ D (∗) π couplings different. The third term breaks also the spin symmetry and contributes differently to the P ∗ P π and to the P ∗ P ∗ π couplings. This is precisely the effect relevant to the hyperfine splitting. To this order in 1/mQ one has: gP ∗ P ∗ π a =g 1+ mQ ! gP ∗ P π b =g 1+ mQ and ∆g ≡ gP ∗ P ∗ π − gP ∗ P π = g ! a−b mQ (2.10) (2.11) The chiral and spin symmetry breaking parameters relevant to the hyperfine splitting are the light pseudoscalar masses mπ , mK and mη , ∆s , ∆ and ∆g . In terms of these quantities, one finds [3, 2, 4]: ∆P = g 2∆ h 2 Λ2CSB i Λ2CSB Λ2CSB 2 2 ) − 6m ln( ) ) + 2m ln( 4m ln( π η K 16π 2 f 2 m2k m2η m2π g 2∆ [24πmK ∆s ] 16π 2 f 2 g 2 ∆g 3 1 3 − (mK + m3η − m3π ) 2 6πf g 2 2 + (2.12) The dependence upon the heavy flavour P = D, B is contained in the parameters ∆ and ∆g . The first term in eq. (2.12) is the so called chiral logarithm [3]. In the ideal situation with pseudoscalar masses much smaller than ΛCSB , it would represent the dominant contribution to ∆P . For the case of D and B mesons the corresponding values have been listed in eqs. (1.10) and (1.11) as ∆0D and ∆0B , respectively. There a value g 2 = 0.5 has been used. The second term in eq. (2.12) represents a non analytic contribution of order ms3/2 [2], which, although formally suppressed with respect to the leading one, is numerically more important, because of the large coefficient 24π. It is given by ∆1D and ∆1B in eqs. (1.10) and (1.11). Finally, the last term in eq. (2.12) [4] is also of order m3/2 s . It can be numerically important as soon as ∆g /g is of order 10% and, if equipped with the right sign, it can cause a substantial cancellations of the previous two contributions. 4 3 QCD Sum Rules for gP ∗P π and gP ∗ P ∗π The coupling gP ∗ P π has already been calculated in [9] by means of QCD sum rules and here we proceed to a similar computation concerning the coupling gP ∗ P ∗ π . We start from the correlator: Aµν (q1 , q) = i Z dx < π(q)|T (Vµ(x)Vν† (0)|0 > e−iq1 x = A(q12 , q22, q 2 )ǫµναβ q α q1β + . . . (3.1) where Vµ = uγµ Q is the interpolating vector current for the P ∗ meson. We compute the scalar function A in the soft pion limit q → 0. This implies q1 = q2 forcing to use a single Borel transformation, and it is the origin of the so called parasitic terms [9]. The correlator in (3.1) can be calculated by an Operator Product Expansion: we keep all the operators with dimension up to five, arising from the expansion of the current Vµ (x) at the third order in power of x, and the heavy quark propagator to the second order. The result is: fπ 1 2 < uu > mb 8fπ m21 A(q12 , q12 , 0) = 2 + + q1 − m2b (q12 − m2b )2 3fπ 9 # " 2 2 2 10mb fπ m1 m0 < uu > mb 1 + − + + (q12 − m2b )3 9 3fπ m20 < uu > m3b 1 − (q12 − m2b )4 fπ " # (3.2) In eqs.(3.2) < uu > is the quark condensate (< uu >= −(240MeV )3 ), m0 and m1 are defined by the equations < ugs σ · Gu >= m20 < uu > (3.3) < π(q)|uD 2 γµ γ5 d|0 >= −ifπ m21 qµ (3.4) and their numerical values are: m20 = 0.8 GeV 2 , m21 = 0.2 GeV 2 [10, 11]. Proceeding in a standard way, we now compute the hadronic side of the sum rule. We can write down for A(q12 , q22 , 0) the following dispersion relation: A(q12 , q22 , 0) = ρ(s, s′ ) 1 Z ′ dsds . π2 (s − q12 )(s′ − q22 ) (3.5) It should be observed that we have not written down in (3.5) subtraction terms because, as proven in [12], only a subtraction polynomial P3 (q12 , q22 ) could be present in (3.5), but it would vanish after the Borel transform. We divide the integration region in three parts [9]. The first region (I) is the square given by m2b ≤ s, s′ ≤ s0 and it contains only the B ∗ pole, whose contribution is AI (q12 , q22 , 0) −2gB∗ B∗ π fB2 ∗ m2B∗ = fπ (q12 − m2B∗ )(q22 − m2B ) 5 (3.6) where fB∗ is defined by < 0|Vµ (0)|B ∗(ǫ, p) >= ǫµ fB∗ mB∗ (3.7) The second (II) integration region is defined as follows: m2b ≤ s ≤ s0 and s′ > s0 or m2b ≤ s′ ≤ s0 and s > s0 . Here we obtain a contribution coupling the vector current Vµ to the pion and the B ∗ . Introducing the form factor V as < π(q)|Vµ |B ∗ (q1 , ǫ) > = V (q22 )ǫµναβ ǫν q1α q β (3.8) where q2 = q1 − q, we get AII (q12 , q22 , 0) = fB∗ mB∗ V (q22 ) V (q12 ) + q12 − mB∗ 2 q22 − mB∗ 2 ! (3.9) In the previous formula one does not have to include the B ∗ pole contribution to V (q 2 ), being already taken into account in AI . We assume that, taken away B ∗ , a single higher resonance of mass m′ contribute to V Vres (q 2 ) = q2 k − m′2 (3.10) where k is an unknown constant. The third region is defined by s, s′ > s0 , and under the assumption of duality it should coincide with the asymptotic limit q12 = q22 → −∞ in (3.2). One gets: AIII (q12 , q12 , 0) = fπ q12 − s0 (3.11) The hadronic side of the sum rule is the sum of the contributions from the three regions: AI (q12 , q12 , 0) = −2gB∗ B∗ π fB2 ∗ m2B∗ k′ fπ + + 2 2 2 2 2 2 2 2 ′2 fπ (q1 − mB∗ )(q2 − mB ) (q1 − mB∗ )(q1 − m ) q1 − s0 (3.12) We have put q1 = q2 and k ′ = mB∗ fB∗ k. Equating now the hadronic and the QCD sides of the sum rule, respectively given by eq. (3.12) and (3.2), and taking the Borel transform with parameter M 2 we find: 2gB∗ B∗ π mB∗ 2 fB∗ 2 + k ′ + exp (−δ/M 2 )(fπ − k ′ ) = fπ M 2 ! " 1 2 < uu > mb 8fπ m21 2 = exp (Ω/M ) fπ − 2 + M 3fπ 9 ! # 1 m20 < uu > m3b 1 5m2b fπ m21 m20 < uu > mb − + 6 − M4 9 6fπ M 6fπ 2 2 = exp Ω/M S(M ) (3.13) 6 In the previous formula we have put m′2 ≃ s0 and we have introduced the parameters δ = s0 − mB∗ 2 and Ω = mB∗ 2 − m2b . Differentiating (3.13) respect to the variable 1/M 2 and combining the first and second derivatives in order to eliminate the unknown parameter k ′ , we obtain the following sum rule: gB ∗ B ∗ π = exp Ω/M 2 δ 2mB∗ 2 fB∗ 2 fπ h Ω(Ω + δ)S(M 2 )+ 2 2 +(2Ω + δ)∂1/M 2 S(M 2 ) + ∂1/M 2 S(M ) i (3.14) To eliminate the parameter k ′ one could also combine the first derivative with the original sum rule (3.13): we have checked that the two procedures give the same numerical results. We have used the second derivative to make an easy comparison with the sum rule for gB∗ Bπ [9]: gB∗ Bπ = × 4fπ m2b mB∗ exp (Ω′ /M 2 ) × 2m3B fB fB∗ (3mB∗ 2 + m2B ) (δ ′ − δ ′ ∆B∗ B /M 2 − ∆B∗ B ) h ′ 2 2 Ω′ (Ω′ + δ ′ )S ′ (M 2 ) + (2Ω′ + δ ′ )∂1/M 2 S ′ (M 2 ) + ∂1/M 2 S (M ) i (3.15) where ∆B∗ B = mB∗ 2 − m2B , δ ′ = s0 − m2B = δ + ∆B∗ B , Ω′ = m2B − m2b = Ω − ∆B∗ B and < uu > 3mb fπ # " 1 2 < uu > mb 10fπ m21 m20 < uu > + − + + M2 3fπ 9 3mb fπ # " 2 2 2 10mb fπ m1 m0 < uu > mb 1 + − + + 2M 4 9 6fπ m20 < uu > m3b + 6fπ M 6 S ′ (M 2 ) = fπ − (3.16) Eq. (3.15) differs slightly from the one given in [9], since it keeps track of the mass difference ∆B∗ B . The sum rules (3.14) and (3.15) have to be analyzed in the duality region, i.e. the region in M 2 where there exists a hierarchy among the different contributions of higher dimension operators (this fixes the lower bound for M 2 ); moreover we impose that the contribution of the parasitic term does not exceed that of the resonance term, which fixes the upper bound in M 2 . In this way we obtain for the B M 2 in the range 20−40 GeV 2 (for s0 = 33 − 36 GeV 2 ), and for the D M 2 = 4 − 7 GeV 2 (for s0 = 6 − 8 GeV 2 ). Using mb = 4.6 GeV and mc = 1.34 GeV one gets: fB2 ∗ gB∗ B∗ π = 0.0094 ± 0.0018 GeV 2 fD2 ∗ gD∗ D∗ π = 0.017 ± 0.004 GeV 2 ; (3.17) and for the gP ∗ P π coupling fB fB∗ gB∗ Bπ = 0.0074 ± 0.0014 GeV 2 fD fD∗ gD∗ Dπ = 0.0112 ± 0.0030 GeV 2 ; 7 (3.18) Once multiplied by 2mP /fπ the figures in (3.18) agree with those given in [9]. A recent calculation of the quantity reported in Eq. (3.18) has been given in Ref. [13]. When expressed in our units their results are as follows: fB fB∗ gB∗ Bπ = 0.0079 ± 0.0007 GeV 2 and fD fD∗ gD∗ Dπ = 0.018 ± 0.002 GeV 2 . The result for the B is only slightly larger than our outcome Eq. (3.18), whereas the result for the D is significantly (≃ 60%) larger. The origin of the discrepancy is in the different range of values for the Borel parameter M 2 , that, in the case of Ref. [13], are generally smaller. A possible origin of this difference is the fact that, while in this paper we use QCD sum rules in the soft pion limit, in [13] light cone sum rules are adopted, which results in an expansion in operators of increasing twist instead of increasing dimension. In particular we have included a dimension 5 contribution which is proportional to the m20 < uu > condensate. This term has no counterpart in [13]; since it has to be kept small, its inclusion in [13] might result in a more stringent constraint on the hierarchy among the different contributions of the Operator Product Expansion, and, therefore, in a more stringent lower limit on M 2 . We now expand the sum rules (3.14) and (3.15) in the parameter 1/mQ , keeping the leading term and the first order corrections. The leading term is the one surviving in the limit mb → ∞, and has already been calculated in [9] for gP ∗ P π . To extract the 1/mQ corrections we introduce the following parameters, finite in the large mass limit: M2 s0 − m2b E= ; y0 = ; ω = mB − mb (3.19) 2mb 2mb and the 1/mQ corrections to the leptonic decay constants, gP ∗ P π and gP ∗ P ∗ π fM A F̂ 1+ =√ mQ mQ ! fM ∗ F̂ A′ =√ 1+ mQ mQ ! (3.20) The coefficients A and A′ have been computed in [14] [15], but only for the B mesons. The corrections found are large and suffering of large uncertainties: there are significant numerical differences between [14] and [15]. In the limit mQ → ∞ the right hand sides of (3.14) and (3.15) coincide (notice that S = S ′ in this limit), confirming the result anticipated in (2.4) and giving: g F̂ 2 = i fπ exp (ω/E) h 2 y0 S0 (E) + (y0 + ω)∂1/E S0 (E)) + ∂1/E S0 (E) (y0 − ω) where S0 (E) = fπ − < uu > 5fπ m21 m20 < uu > − + 3fπ E 36E 2 48E 3 fπ (3.21) (3.22) The duality region extends for E = 4 − 6 GeV and y0 = 1.1 − 1.3 GeV . Numerically one obtains: F̂ 2 g = 0.040 ± 0.005GeV 3 8 (3.23) While this result agrees within the error with that given in [9], the central value reported here is 15 % larger due to a slightly different choices of the phenomenological parameters. We can then write the 1/mQ expansion for the function S and S ′ : S(E) = S0 (E) + S1 (E) ′ S ′ (E) S (E) = S0 (E) + 1 mQ mQ (3.24) where 4fπ m21 m20 < uu > S1 (E) = − + 9E 24E 2 fπ < uu > 5fπ m21 m20 < uu > ′ S1 (E) = − + + 3fπ 9E 48E 2 fπ (3.25) From (3.14) and (3.15), one gets the following sum rules for the parameters a and b: a = fπ exp ω/E h 2 (ω + 4λ)(y0S0 (E) + ∂1/E S0 (E))+ 2g F̂ 2(y0 − ω) i 2 + 2y0 ωS1(E) + 2(y0 + ω)∂1/E S1 (E) + 2∂1/E S1 (E) + + b = ω 2 + 4λ ω 2 + 4λ + − 2(A′ + ω) 2E 2(y0 − ω) fπ exp ω/E h 2 ω (y0 S0 (E) + ∂1/E S0 (E))+ 2g F̂ 2(y0 − ω) (3.26) i 2 + 2y0 ωS1′ (E) + 2(y0 + ω)∂1/E S1′ (E) + 2∂1/E S1′ (E) + + ω 2 + 4λ ω 2 + 4λ + − (A + A′ + 4ω) 2E 2(y0 − ω) (3.27) where λ has been given in (2.9). From the previous sum rules one gets: a + 2A′ = −0.15 ± 0.20 GeV b + A′ + A = −1.15 ± 0.20 GeV (3.28) and a − b + (A′ − A) = 0.99 ± 0.02 GeV (3.29) Notice that the difference has a quite smaller uncertainty, due to a partial cancellation of terms depending on the threshold. Neglecting radiative corrections, A and A′ are given by [14, 15]: A = −ω + GK + 3GΣ 2 A′ = − ω GK + − GΣ 3 2 (3.30) Notice that the splitting of the couplings depends on the quantity a − b that contains only the difference A′ − A given by: 2 A′ − A = ω − 4GΣ 3 9 (3.31) There is disagreement in the literature on the values of the parameter GΣ : at the b quark mass scale from ref.[14] one gets GΣ = (0.042 ± 0.034 ± 0.023 ± 0.030) GeV , while in ref. [15] the central value GΣ ≃ −(0.052) GeV is quoted. In view of this discrepancy, to provide an estimate of the difference (3.31), we will approximate A′ − A ≈ 2/3ω ≈ 0.4 GeV , obtaining a − b ≈ 0.6 GeV (3.32) 4 Discussion and conclusions From (2.11), (3.32) and from the formula (2.12) of the hyperfine mass splitting we obtain: ∆B ≈ g 2(27.3 + 61.4 − 75.8) MeV = 12.9g 2 MeV (4.1) Notice that we have used in eq. (2.12) f = fπ = 132 MeV for all the light pseudoscalar mesons of the octet. This is suggested by the sum rule for g which shows that g/f is flavour independent. In eq. (4.1) we have detailed the contributions ∆0 , ∆1 and the one from ∆g /g respectively. We have also taken ΛCSB = 1 GeV . It is evident that there is a large cancellation among the last term and the other ones. In order to be more quantitative we have to fix the value of g, which, on the basis of our result (3.23), depends on the value of F̂ . In Ref. [9] the range of values g ≃ 0.2 − 0.4 was found; therefore, putting g 2 = 0.1, we would obtain ∆B ≃ 1.3 MeV (4.2) The application of our results to the charm case is more doubtful, in view of the large values of the 1/mc correction (a − b)/mc . By scaling the result (4.2) to the charm case, one obtains mb ∆D = ∆B ≃ 4.4 MeV (4.3) mc In conclusion, our estimate of gP ∗ P ∗ π − gP ∗ P π allows to include a previously neglected term in the loop induced contribution to the hyperfine splitting. Although our estimate is affected by an uncertainty in the value of GΣ , nevertheless this new term tends to cause a substantial cancellation and to reconcile the chiral calculation with the experimental data. Acknowledgements We would like to thank P. Colangelo for useful discussions. 10 References [1] J.L. Rosner and M.B. Wise, Phys. Rev. D47 (1993) 343. [2] L. Randall and E. Sather, Phys. Lett. B303 (1993) 345. [3] E. Jenkins, Nucl. Phys. B412 (1994) 181. 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