Λ b pπ, pk decays in k T factorization approach Yu-Ming Wang

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1 Λ b pπ, pk ecays in k T factorization approach Yu-Ming Wang Theoretische Physik I, Universität Siegen Base on work one in collaboration with Ali, Kramer, Lü an Zou Phys.Rev.D80:034011,2009 Talk given at Euroflavour 2009, Bari, Italy 1

2 Outline 1. Motivation an introuction 2. An introuction to k T factorization approach 3. Calculations of Λ b pπ, pk amplitues in PQCD 4. Discussions an conclusions Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 2

3 I. Motivation an introuction b quark ecays offer an ieal platform to test the SM with great achievable precision, explore the origin of CP violation, probe the signals of new physics, -unerstan the ynamics of QCD. Bottom baryon ecays provie important clues on the flavor-changing currents beyon the SM in a complementary fashion to the B-meson ecays. Avantages of the bottom baryons over B mesons: Heavy baryon ecays allow the stuy of spin correlations, proviing valuable information on the chirality of the short-istance transition. Baryon ecays are flavor self-tagging processes, which shoul make their experimental reconstructions easier. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 3

4 Experimental sie: Nonleptonic two-boy ecay of Λ b pπ, pk have been observe at Tevatron an their branching ratios an irect CP asymmetries are also measure by CDF collaboration. BR ( 10 6 ) CP Λ b pπ Λ b pk 3.5 ± 0.6 ± ± 0.8 ± ± 0.17 ± ± 0.17 ± 0.03 The b b cross section is expecte to be 500µb proucing b b pairs in a stanar (10 7 s) year of running at the LHCb operational luminosity of cm 2 sec 1. Hence, Λ b baryon will be even more copiously prouce at the LHC. Theoretical sie: various theoretical frameworks have been propose: - phenomenological moels, ynamical approaches, methos base on classifications in terms of flavour topologies CDF collaboration, arxiv: M. P. Altarelli an F. Teubert, arxiv: [hep-ph] Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 4

5 2. An introuction to k T factorization approach Concept of facrotization Motivation of k T factorization: Removing en-point singularities Factorization formulae in PQCD approach Comparison of PQCD, QCDF an SCET Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 5

6 Concept of facrotization Factorization is an important concept in QCD perturbation theory. Factorization amounts to the separation of physics at ifferent energy scales. Collinear factorization an k T factorization are two funamental tools to analyze the exclusive heavy flavor haron ecays. Collinear factorization: QCD factorization (Beneke, Buchalla, Neubert, Sachaja,...), Soft-collinear effective theory(bauer, Pirjol, Rothstein, Stewart,..). k T factorization: PQCD approach(keum, Li, Sana, Lü, Ukai, Yang,...). Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 6

7 Motivation of k T factorization: Removing en-point singularities Absorbe into form factors an introuce haronic parameters in QCDF: 1 0 x x X A = (1 + ρ A e iφ A ln m b Λ ), (ρ A 1, Λ 500MeV). Zero-bin subtraction in SCET (Stewart: SCET 08, FPCP 09): The annihilation singularity has to o with a potential ouble counting an a rapiity cut off is neee. Singularity regulate by k T in PQCD: A parton transverse momentum k T is prouce by gluon raiation before har scattering occurs. Singularities imply that the parton transverse momentum can not be neglecte. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 7

8 b 1 xm 2 B +k2 T w ū B 1 0 xφ π(x) x 2 m 2 B 1 0 x φ π (x) x(xm 2 B + k2 T ). There is not en-point singularity anymore! k T istribution is governe by the Suakov factor. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 8

9 Factorization formulae in PQCD approach Taking the two-boy nonleptonic ecay B M 1 M 2 as an example, the amplitue can be factorize into the convolution of the six-quark har kernel, the Wilson coefficient, the jet function an the Suakov factor with the bounstate wave functions. A = φ B H (6) J S φ M1 φ M2, all of which are well-efine an gauge-invariant. J enotes the jet function from threshol resummation, which organizes the ouble logarithms ln 2 x ue to the raiative corrections to the har kernel. S enotes the Suakov factor from k T resummation, which organizes the ouble logarithms ln 2 k T ue to the raiative corrections to the meson wave function. H. n. Li, Prog. Part. Nucl. Phys. 51 (2003) 85. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 9

10 Graphics representation of PQCD factorization theorem: M 3 Φ M3 e S B M 2 e S H e S Φ B Φ M2 k T Λ k T Λm b Comments on PQCD factorization: The factorization limit of PQCD approach at large m b is consistent with that obtaine in QCDF, but the reasonings for achieving the same power counting are quite ifferent (Li an Ukai, 02). The gluon invariant mass q 2 in the BSS mechanism is clearly efine an relate to the parton momentum fractions. The penguin annihilation amplitue is almost imaginary in PQCD, whose mechanism is similar to the BSS one: the loop is forme by the internal particles in the LO har kernel an by infinitely many Suakov gluons exchange between two partons in a light meson (Li, FPCP 09). Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 10

11 Comparison of PQCD, QCDF an SCET: strong phase Accoring to the QCDF power counting rules, the factorizable emission iagram gives the leaing contribution of O(α 0 s ), which is real. The strong phase arises from the factorizable annihilation iagram, being of O(α s m 0 /m B ), an from the vertex correction to the leaing iagram, being of O(α s ). For m 0 /m B slightly smaller than unity, the vertex correction is the leaing source of strong phases. In k T factorization the power counting rules change. The factorizable emission iagram is calculable an of O(α s ). The factorizable annihilation iagram has the same power counting as in QCDF. The vertex correction becomes of O(α 2 s ). Therefore, the annihilation iagram contributes the leaing strong phase. This is the reason the strong phase erive from PQCD an from QCDF coul be opposite in sign, an the former has a large magnitue. At leaing power, no large source of strong phases in SCET, since there is no annihilation iagrams. Long-istance O(1/m b ) charming penguin is introuce, parameterize as A c c, which provie the main strong phase in SCET. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 11

12 Comparison of PQCD, QCDF an SCET: Charm loop In QCDF, charming penguinin is perturbative an the resonance contributions to the charming-penguinin iagrams for exclusive haronic B M 1 M 2 ecays is parametrically suppresse in the heavy-quark limit, irrespective of whether the charm quark is treate as being heavy or light (BBNS, 09). In SCET, charm is relatively heavy an may be more sensitive to nonperturbative effects. Charming penguinins coul give long-istance effects at leaing power, which must be parameterize as an arbitrary amplitue A c c. In PQCD, charming penguinin is perturbative an only contribute at the nextto-leaing orer. The long-istance charming-penguin is power-suppresse accoring to QCDF, PQCD an LCSR. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 12

13 3. Λ b pπ, pk ecays in PQCD Effective Hamiltonian for nonleptonic two-boy ecays of Λ b Haronic istribution amplitues Factorization formulae of Λ b pπ, pk ecays Numerical analysis of Λ b pπ, pk ecays Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 13

14 Effective Hamiltonian for nonleptonic two-boy ecays of Λ b The weak effective Hamiltonian are specifie as below: H eff = G F 2 {V ub V uq[c 1 (µ)q u 1 (µ) + C 2(µ)Q u 2 (µ)] V tbv tq[ The functions Q i (i = 1,..., 10) are the local four-quark operators: current current (tree) operators Q u 1 = (ū αb β ) V A ( q β u α ) V A, Q u 2 = (ū αb α ) V A ( q β u β ) V A, QCD penguin operators Q 3 = ( q α b α ) V A q Q 5 = ( q α b α ) V A ( q β q β ) V A, Q 4 = ( q β b α ) V A q ( q β q β ) V +A, Q 6 = ( q β b α ) V A q q 10 i=3 } C i (µ)q i (µ)] + H.c.. ( q α q β ) V A, ( q αq β ) V +A, electro-weak penguin operators Q 7 = 3 2 ( q αb α ) V A e q ( q β q β ) 3 V +A, Q 8 = 2 ( q βb α ) V A e q ( q α q β ) V +A, q q Q 9 = 3 2 ( q αb α ) V A e q ( q β q β ) 3 V A, Q 10 = 2 ( q βb α ) V A e q ( q α q β ) V A. q q Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 14

15 Distribution amplitue of Λ b : (Y Λb ) αβγ (k i, ν) = Haronic istribution amplitues N c l=2 w + l w l (2π) 3 eik lw l ε abc 0 T [b a α (0)ub β (w 2) c γ (w 3)] Λ b (p) = f Λb 8 2N c [(p/ + M Λb )γ 5 C] βγ [Λ b (p)] α Ψ(k i, ν). Distribution amplitues of proton: (Y p ) αβγ (k i, µ) = 1 2 2N c 3 = l=2 w + l w l (2π) 3 eik l wl ɛ abc 0 T [u a α (0)ub β (w 2) c γ (w 3)] P (p ), f N 8 2N c { (p /C) αβ [γ 5 N(p )] γ Φ V (k i, µ) + ( p γ 5 C) αβ [N(p )] γ Φ A (k i, µ) (σ µν p ν C)αβ [γ µ γ 5 N(p )] γ Φ T (k i, µ)}. The light-cone istribution amplitues for the pseuoscalar meson: P (P ) q 2β (z)q 1α (0) 0 = i = i [ xe ixp z γ 5 P Π A (y, µ) + m 0 γ 5 Π P (y, µ) m 0 σ µν Π σ (y, µ) γ 5 P µ z ν 6 xe ixp z [ γ 5 P Π A (y, µ) + γ 5 m 0 Π P (y, µ) + m 0 γ 5 ( n v 1)Π T (y, µ) ] αβ. ] αβ Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 15

16 Moel of haronic wavefunctions The phenomenological moel for the istribution amplitue of the Λ b baryon (Schlumpf, 92) ψ CQM (x 1, x 2, x 3 ) = Nx 1 x 2 x 3 exp [ M 2 Λ b 2β 2 x 1 m2 l 2β 2 x 2 m2 l 2β 2 x 3 ]. Twist-3 istribution amplitues of proton (Braun et al, 00): ] φ V (x i, µ) = 120x 1 x 2 x 3 [φ 0 3(µ) + φ + 3 (µ)(1 3x 3), φ A (x i, µ) = 120x 1 x 2 x 3 (x 2 x 1 )φ 3 (µ), φ T (x i, µ) = 120x 1 x 2 x 3 [φ 0 3(µ) 1 ( ) φ φ 3 (µ)(1 3x 3 )]. Distribution amplitues of pseuoscalar mesons: φ A π (x) = 3f π x(1 x)[ C 3/2 2 (t)], φ P π (x) = f π 6 2 [ C1/2 2 (t)], 6 φ T π (x) = f π 2 6 [C1/2 1 (t) C 1/2 3 (t)], φ A K (x) = 3f K x(1 x)[ C 3/2 1 (t) C 3/2 2 (t)], 6 φ P K (x) = f K 2 [ C1/2 2 (t)], φ T K 6 (x) = f K 2 6 [C1/2 1 (t) C 1/2 3 (t)]. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 16

17 Comparisons of Λ b DAs in quark moel an QCD QCD moel for the twist-2 istribution amplitue of the Λ b baryon (Ball et al, 08): ψ QCD (ω, u) = ω 2 u(1 u) [ 1 ɛ 4 e ω ɛ 0 + a 2 C 3/2 2 (2u 1) 1 0 ɛ 4 1e ] ω ɛ 1. (1) ω is the total energy carrie by the light quarks in the rest frame of Λ b baryon an the imensionless parameter u escribes the momentum fraction carrie by the u quark in the iquark system. Define two inverse moments involving negative powers of the variables ω an u, (ωu) 1 (Λ UV ) ω 1 (Λ UV ) ΛUV 0 ΛUV 0 ω ω u u ψqcd/cqm (ω, u), (2) uψ QCD/CQM (ω, u),. Typical inverse moments for the two LCDAs ψ QCD (ω, u) an ψ CQM (ω, u): ω 1 [GeV 1 ] (ωu) 1 [GeV 1 ] ψ QCD (ω, u) ψ CQM (ω, u) Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 17

18 Factorization formula of the ecay amplitue The Λ b pπ, pk ecay amplitue M can be ecompose as M = p(p )[f 1 +f 2 γ 5 ]Λ b (p). The factorization formula for the coefficients f i (i = 1, 2) is f j i = G F π f Λ b f p n=a,p,t m=v,a,t [Dx] [Db] j [α s (t j )] 2 a j (t j )ψ Λb (x)ψ m p (x )φ n M(y) H mnj i (x, x, y)ω j (b, b, b q ) exp[ S j ], (3) Suakov factor: S j (x, x, b, b ) = 3 i=2 s(w, k + i ) tj κw µ µ γ q(α s ( µ)) + 3 i=1 s(w, k i ) + 3 tj κw µ µ γ q(α s ( µ)) (4) S j (x, x, y, b, b, b q ) = 3 i=2 + s(w, k + i ) i=1 tj s(w q, q + i ) + 2 κw µ µ γ q(α s ( µ)) + tj w q 3 i=1 s(w, k i ) + 3 tj κw µ µ γ q(α s ( µ)) µ µ γ q(α s ( µ)). (5) Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 18

19 Typical Feynman iagrams in the Λ b ecays b u u u ū b u u u ū b u u u ū b u ū u u b u ū u u (T ) (C) (E) (B) (P ) Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 19

20 Part of lowest-orer har amplitues for Λ b ecays in PQCD b u ū u Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 20

21 Numerical analysis on Λ b pπ, pk ecays The coefficients f 1 an f 2 contribute by the Feynman iagrams with efinite topologies in Λ b pπ ecays: f 1 f 2 T i i C i i E i i B i i P i i G i i Factorizable an nonfactorizable contributions to Λ b pπ, pkecays: factorizable non-factorizable f 1 (Λ b pπ) i i f 2 (Λ b pπ) i i f 1 (Λ b pk) i i f 2 (Λ b pk) i i Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 21

22 Some remarks on the factorizable contributions: (1) Three-gluonvertex iagrams are comparable to the C iagrams, as can be seen the above Table, an hence they may inuce significant corrections to the color-suppresse moes, such as the Λ b ΛJ/ψ ecay. (2)The factorizable contribution is approximately 2 orers of magnitue smaller than the nonfactorizable contribution. This is also the reason that the the conventional pqcd preictions for the semileptonic ecay Λ b pl ν are much smaller than those evaluate in other theoretical frameworks. (3) Interpretation on the large nonfactorizable contributions: The Suakov factor can only suppress the region with large b corresponing to small k T, an has almost no effect in the region where the transverse momentum k T is large. Taking the non-factorizable iagram T 25 as an example, the two virtual quarks can be on the mass shell even in the region with large k T. Therefore, this iagram is not subjecte to the suppression from the Suakov factor. It is then expecte that the amplitues for the non-factorizable iagrams shoul be much larger than those from the factorizable iagrams, where the two virtual quarks can be on the mass shell only in the small k T region. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 22

23 Preictions of transition form factors The form factor g 1 of Λ b p ecay calculate in the NRQM, LCSR, an PQCD approaches. NRQM (Mohanta, 00) LCSR (Huang, 04) pqcd (Li, 99) pqcd (this work) g It inicates that the soft ynamics in the heavy-to-light transition form factors is the ominant effect, in all likelihoo overwhelming the mechanism of the har gluon exchange for the baryonic transitions. Unlike QCD sum rules, soft contribution can not be inclue into the PQCD formalism in a consistent way: if there is no har gluon exchange to provie a large characteristic scale, twist expansion oes not hol. Therefore, soft contribution can not be estimate using the same meson istribution amplitues resulting from twist expansion. (Li, 03) Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 23

24 Hybri pqcd scheme The smallness of the factorizable contributions for Λ b pπ, pk in the conventional pqcd approach is not natural. Consequently, the Λ b p transition form factors may be ominate by nonperturbative soft contributions. An alternative approach, calle hybri PQCD scheme, is to treat the Λ b p form factors as nonperturbative inputs. The complete ecay amplitues for Λ b pπ, pk in hybri PQCD scheme are given by M(Λ b pπ) = M f (Λ b pπ) + M nf (Λ b pπ) M(Λ b pk) = M f (Λ b pk) + M nf (Λ b pk), where M nf (Λ b pπ) an M nf (Λ b pk) enote the contributions from the non-factorizable iagrams an have been compute in the conventional pqcd approach. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 24

25 It is straightforwar to write the factorizable amplitues M f (Λ b pπ) an M f (Λ b pk) as M f (Λ b pπ) = G {[ ][ ] F f π p(p ) V ub Vu a 1 V tb Vt (a 4 + a 10 + R1 π (a 6 + a 8 )) g 1 (m 2 π)(m Λb M p ) + g 3 (m 2 π)m 2 π 2 + [ V ub V u a 1 V tb V t (a 4 + a 10 R π 2 (a 6 + a 8 )) ][ G 1 (m 2 π)(m Λb + M p ) G 3 (m 2 π)m 2 π] γ5 } Λb (p), (6) M f (Λ b pk) = G {[ ][ ] F f K p(p ) V ub Vus a 1 V tb Vts (a 4 + a 10 + R1 K (a 6 + a 8 )) g 1 (m 2 K )(M Λ b M p ) + g 3 (m 2 K )m2 K 2 + [ V ub V usa 1 V tb V ts(a 4 + a 10 R K 2 (a 6 + a 8 )) ][ G 1 (m 2 K )(M Λ b + M p ) G 3 (m 2 K )m2 K] γ5 } Λb (p). (7) The efinitions of Λ b p form factors are p(p ) ūγ µ b Λ b (p) = p(p )(g 1 γ µ + g 2 iσ µν q ν + g 3 q µ )Λ b (p), Λ(P ) sγ µ γ 5 b Λ b (P + q) = Λ(P )(G 1 γ µ + G 2 iσ µν q ν + G 3 q µ )γ 5 Λ b (P + q). In HQET, the form factors satisfy the relations g 1 = G 1 = ξ 1 + m Λ m Λb ξ 2, (8) g 2 = G 2 = g 3 = G 3 = ξ 2 m Λb. (9) Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 25

26 Numerical values of the form factors g 1 an m Λb g 2 responsible for the Λ b p transition, estimate in the LCSR (Huang, 04) form factors g 1 m Λb g 2 Λ b p The coefficients f 1 an f 2 in the ecays Λ b pπ, pk contribute by the factorizable an non-factorizable external W emission (T ) iagrams in the hybri pqcd scheme. factorizable non-factorizable f 1 (Λ b pπ) i i f 2 (Λ b pπ) i i f 1 (Λ b pk) i i f 2 (Λ b pk) i i It can be seen that the factorizable contributions are now much larger than in the conventional pqcd approach, though they are still smaller than the corresponing non-factorizable contributions. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 26

27 Results of ecay rate, CP asymmetry an polarization asymmetry pqcd (conventional) pqcd (hybri scheme) Exp. B(Λ b pπ) ± 0.6 ± B(Λ b pk) ± 0.8 ± A CP (Λ b pπ) A CP (Λ b pk) α(λ b pπ) α(λ b pk) ± 0.17 ± ± 0.17 ± The errors for these entries correspon to the uncertainties in the input haronic quantities, from the scale-epenence, an the CKM matrix elements, respectively. The asymmetry parameter α associate with the anisotropic angular istribution of the proton emitte in the polarize Λ b baryon ecays is efine as follows: Γ = Γ 0 (1 + αp s Λb ) with p being a unit vector along the irection of proton momentum in the rest frame of the Λ b baryon. The preictions for the parameter α in the Λ b pπ ecay are relatively stable with respect to the variations of haronic parameters, the CKM matrix elements an the har scale, an therefore it serves as a goo quantity to test the stanar moel. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 27

28 Some remarks on the numerical results The ratio of pπ an pk moes: R πk (Λ b ) BR(Λ b pπ) BR(Λ b pk) = (10) Experimental ata: R πk (Λ b ) = 0.66 ± 0.14 ± Whether this iscrepancy reflects the inaequacy of the current theoretical formalism embee in the stanar moel, or the stanar moel itself, or requires improve ata remains to be seen. For the tree-ominate processes., The parameter α in B i ( ) Bf ( )P (V ) ecays approaches 1 in the soft pseuoscalar meson or vector meson limit. As for the Λ b pk ecay, the contributions from the QCD penguin operators are comparable to that of the tree amplitue. The operator O 6 contributes to the Λ b p transition via the (V + A) current an the Wilson coefficient a 6 is very sensitive to the energy scale. Hence, the asymmetry parameter α can flip its sign for the Λ b pk ecay ue to the large penguin contributions. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 28

29 4. Discussions an conclusions A brief review of PQCD approach an comparison of PQCD, QCDF an SCET. PQCD analysis of non-leptonic ecays Λ b pπ, pk: BR, irect CP, Λ b polarization asymmetry. The non-factorizable contributions are important in baryonic ecays. The asymmetry parameter α in the polarize Λ b pπ ecay is insensitive to the haronic inputs an higher-orer corrections. The contribution of three-gluon-vertex iagrams is comparable to that from the color suppresse tree. Theoretical preictions presente here can be systemically improve by incluing the higher twist DAs contributions as well as raiative corrections to the har amplitues. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 29

30 Thanks very much for your attentions! Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 30

31 BACKUP! Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 31

32 Physical interpretation of Suakov factor In orer to unerstan the Suakov factor physically (Ref: Y. Y. Keum, M. Matsumori an A. I. Sana,, Phys. Rev. D 72 (2005) ), first we consier QED. When a charge particle is accelerate, infinitely many photons must be emitte by the bremsstrahlung. A similar phenomenon occurs when a quark is accelerate: infinitely many gluons must be emitte. Accoring to the feature of strong interaction, gluons cannot exist freely, so haronic jet is prouce. Then we observe many harons in the en if gluonic bremsstrahlung occurs. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 32

33 The amplitue for an exclusive ecay H b h 1 h 2 is proportional to the probability that no bremsstrahlung gluon is emitte. This is the Suakov factor an it is epicte in the following figure. exp[-s(q,b)] b Q 1 0 Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 33

34 As can be seen, the Suakov factor is large for small b an Q. Large b implies that the quark an antiquark pair is separate, which in turn implies less color shieling (see the figure below). Similar absence of shieling occurs when b quark carries most of the momentum while the momentum fraction of spectator quark x in the B meson is small. Then the Suakov factor suppresses the long istance contributions for the ecay process an gives the effective cutoff about the transverse irection. In short, the Suakov factor correspons to the probability for emitting no photons. Accoring to this factor, the property of short istance is guarantee. Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 34

35 The explicit form for the function s(q, b) is: s(q, b) = A(1) ˆq ln 2β 1 (ˆqˆb + A(1) β 2 4β1 3 ˆq ) A(1) 2β 1 (ˆq ˆb ) + A(2) [ ln(2ˆq) + 1 ˆq 4β 2 1 ln(2ˆb) + 1 ˆb where the variables are efine by ) (ˆqˆb 1 ] + A(1) β 2 8β 3 1 [ A (2) 4β 2 1 ( A(1) ln 4β 1 [ ln 2 (2ˆq) ln 2 (2ˆb) ], e 2γ E 1 2 )] (ˆqˆb) ln ˆq ln[q/( 2Λ)], an the coefficients A (i) an β i are ˆb ln[1/(bλ)], β 1 = 33 2n f 12 A (1) = 4 3, A(2) = 67 9 π2, β 2 = n f n f β 1ln( 1 2 eγ E), n f is the number of the quark flavors an γ E is the Euler constant. We will use the one-loop running coupling constant, i.e. we pick up the four terms in the first line of the expression for the function s(q, b)., Yu-Ming Wang Talk given at Euroflavour 2009, Bari, Italy 35