Quadratic Mass Corrections of Order O(α 3 sm 2 q/s) to the Decay Rate of Z- andw- Bosons

Size: px

Start display at page:

Download "Quadratic Mass Corrections of Order O(α 3 sm 2 q/s) to the Decay Rate of Z- andw- Bosons"

Imogen Page
5 years ago
Views:

1 MPI/PhT/96-84 hep-ph/96090 Augut 1996 Quadratic Ma Correction of Order Oα 3 m q/) to the Decay Rate of Z- andw- Boon K.G. Chetyrkin a,b,j.h.kühn c, a b c Intitute for Nuclear Reearch, Ruian Academy of Science, 60th October Anniverary Propect 7a Mocow 11731, Ruia Max-Planck-Intitut für Phyik, Werner-Heienberg-Intitut, Föhringer Ring 6, Munich, Germany Intitut für Theoretiche Teilchenphyik, Univerität Karlruhe D-7618 Karlruhe, Germany Abtract We analytically compute quadratic ma correction of order Oα 3 m q/) to the aborptive part of the non-diagonal) correlator of two axial vector current. Thi allow u to find the correction of order Oα 3 m q/m W )toγw hadron) a well a imilar correction to ΓZ hadron). chet@mppmu.mpg.de johann.kuehn@phyik.uni-karlruhe.de

2 1 Introduction Preciion meaurement of the total and a well a partial Z decay rate have provided one of the the mot important and, from the theoretical viewpoint, clean determination of the trong coupling contant α with a preent value of α =0.10 ± [1. Theoretical ingredient were the knowledge of QCD correction to order α 3 in the limit of male quark plu charm and bottom quark effect ee, e.g. [ and reference therein). Thee ma correction which indeed are relevant at the preent level of accuracy have been calculated up to the order α 3 m q/ for the vector and α m q/ for the axial current induced decay rate. In thi hort note the prediction i extended to include α 3 m q/ term for the non-inglet part) of the axial current induced rate. At the ame time reult are obtained for the non-diagonal current correlator with two different mae a cae of relevance e.g. for the W decay rate into charmed and bottom quark. The ame formulae can alo be applied to a ubcla of correction which enter ingle top production in the Drell-Yan like reaction qq tb far above threhold. The calculation i baed on an approach introduced in Ref. [3, 4. Knowledge of the polarization function to order α, the appropriate anomalou dimenion at order α3, combined with the renormalization group equation allow one to predict the correponding logarithmic term of order α 3 and hence the contant term of the imaginary part. The firt of thee ingredient ha been available ince ome time [5, 6, 7, 8 while the anomalou dimenion can been obtained from Ref. [9 in a traightforward way. In thi hort note only the theoretical framework and the analytical reult are preented numerical tudie will preented elewhere. Renormalization Group Analyi In analogy to the vector cae, we take a a tarting point the generic vector/axial quark current correlator Π V/A µν which i defined by Π V/A µν q,m u,m d,m,µ,α ) = i dxe iqx T [ jµ V/A x)jν V/A ) 0) 1) = g µν Π 1) V/A Q )q µ q ν Π ) V/A Q )). with Q = q, m q = f m f and jv/a µ = qγ µ γ 5 )q.hereqand q are jut two generically different) quark with mae m u and m d repectively. Note that the vector and axial correlator are related through Π A µνq,m u,m d,m,µ,α )=Π V µνq,m u, m d,m,µ,α ) ) The polarization function Π 1) V/A and the pectral denity RV/A ) which in turn govern the Z and W decay rate obey the following diperion relation Π 1) A Q )= 1 d RV/A, m u,m d,m,µ,α ) mod ub. 3) 1 m um d ) Q Wherea R V/A a a phyical quantity i invariant under renormalization group tranformation, the function T [ jµ V/A x)jν V/A ) 0) contain ome non-integrable ingularitie 1

3 in the vicinity of the point x = 0. Thee cannot be removed by tandard quark ma and coupling contant renormalization, but mut be ubtracted independently. A a reult the relevant renormalization group equation aume the form [ µ d dµ ΠV/A µν =q µ q ν g µν q )γ q ± 1 α ) 16 m u m d ) g µν γ mα ± 1 ) 16, 4) where µ d dµ = µ µ βα ) γ m α ) m f. 5) α m f f Here and below the upper and lower ign give the reult for vector and axial vector correlator repectively. From the identity ) we infer that both anomalou dimenion and γ m, ± being not dependent on any mae, alo do not depend on the ign. In what γ ± q follow we will denote γ q ± = γq VV and γ m ± = γm VV. The β-function and the quark ma anomalou dimenion γ m are defined in the uual way µ d dµ ) α µ) = α βα ) i 0 µ d dµ mµ) = mµ)γ mα ) m γm i i 0 ) α i1 β i, 6) ) α i1. 7) Their expanion coefficient up to order Oα 3 ) are well known [10, 11, 1, 13 and read n f i the number of quark flavour) β 0 = 11 ) 3 n f /4, β 1 = ) 3 f /16, 857 β = n f 35 ) 54 n f /64, 8) 0 γm 0 =1, γm 1 = 3 0 ) 9 f /16, [ 16 γm = ζ3) n f 140 ) 81 f /64. 9) Another ueful and cloely related object i the correlator of the peudo)calar quark current Π S/P Q,m u,m d,m,µ,α )= e iqx 0 T [ j S/P x)j S/P) 0) 0. 10) Scalar and peudocalar current correlator are alo related in a imple manner: Π S Q,m u,m d,m,µ)=π P Q,m u, m d,m,µ). 11) For vanihing quark mae calar and peudocalar correlator are, therefore, identical: Π S =Π P and meet the following RG equation µ d ) dµ γ mα ) Π S/P =Q γq SS 1 α ) 16. 1)

4 The axial) vector and peudo)calar correlator are connected through a Ward identity [14 q µ q ν Π V/A µν =m u m d ) Π S/P m u m d ) ψ q ψ q ψ q ψ q ), 13) where the vacuum expectation value on the r.h.. are undertood within the framework of perturbation theory and the minimal ubtraction. Equation 13) lead to the following relation between the correponding anomalou dimenion [4: γ VV m γ SS q. 14) Thi relation wa ued in Ref. [4 in order to find the anomalou dimenion γm AA order tarting from the reult of Ref. [5. at the α In what follow we will be intereted in quadratic ma correction to the polarization which i convenient to repreent in the form m = m u,m d,m}): operator Π 1) A Π 1) V/A Q,m,µ,α )= 3 16 Π1) V/A,0 µ Q,α ) 3 16 Π1) V/A, µ Q,m,α )Om 4 ). 15) Here the firt term on the rh correpond to the male limit while the econd term tand for quadratic ma correction. Note that Π 1) V/A, i a econd order polynomial in quark mae: a logarithmic dependence on quark mae may appear tarting from m 4 term only 1. From the RG equation 4) we arrive at the following equation for Π 1) V/A, : or, equivalently, L q =ln µ Q ) µ d dµ Π1) V/A, = 1 3 m u m d ) γ VV m α ) 16) Π 1) V/A, L = 1 q 3 m u m d )γm VV ) βα γ m α Π 1) V/A,. 17) The lat relation explicitly demontrate that R V/A the aborptive part of Π 1) V/A, depend in order α n on the very function Π 1) V/A, which i multiplied by at leat one factor of α. Thi mean that one need to know Π 1) V/A, recontruct all Q-dependent term in Π 1) V/A, to αn and anomalou dimenion γ m and γ VV up to order αn 1 only to unambiguouly, provided, of coure, the beta function m are known to α n. Thi obervation wa made firt in [3 where it wa ued to find the aborptive part R V in order α 3 for the cae of the diagonal vector current that i for the cae of m u = m d ). In the preent paper we will ue the reult of a recent calculation of γq SS [9 to order α 3 to determine the aborptive part R V/A to the ame order in the general cae of non-diagonal current. 1 Provided of coure that one ue a ma independent renormalization cheme like the MS-cheme employed in thi work. 3

5 3 Calculation and reult The reult for the function Π ) V/A, in the general non-diagonal cae to order α wa firt publihed in Ref. [6. On the other hand, the Ward identity 13) expree the combination Π 1) V/A, /Q Π ) V/A, in term of the male polarization operator ΠS known from Ref. [5, 7. A um of thee two function lead u to the following reult for Π 1) V, [ Π 1) V, = m ln µ Q Q m α Q m Q m Q m Q α m Q [ [ ζ3) µ µ [ ln 3 Q ln m α 16 Q Q 3 4lnµ Q ) [ ζ3) ζ4) 55 ζ5) 108 n f 3 9 ζ3) n f 81 7 µ µ ln 39 ζ3) ln Q Q n f ln µ Q 4 3 ζ3) n f ln µ Q ln µ Q 8 9 n f ln µ Q 19 6 ln3 µ Q 1 9 n f ln 3 µ Q ) [ α ζ3) ζ5) n f 53 µ ln 6 Q 13 9 n f ln µ Q 19 µ ln Q 1 3 n f ln µ Q ) α [ ζ3). 18) Here m = m u m d and m = m u m d, Q = q,allmaeawellaqcd coupling contant α are undertood to be taken at a generic value of the t Hooft ma µ. All correlator are renormalized within MS-cheme. We have alo checked 18) by a direct calculation with the help of the program MINCER [16 written for the ymbolic manipulation ytem FORM [17. In a particular cae of m u = m d Eq. 18) i in agreement with Ref. [15, 8. Now, a wa hown in [3 the anomalou dimenion γm AA the reult of [9 we have: γ VV m = γq SS = α ) α α [ ζ3) 1 3 n f ) 3 [ ζ3) ζ4) ζ5) 1 = γq SS, and, thu, from 7776 n f } 13 9 ζ3) n f 11 1 ζ4) n f n f 1 9 ζ3) n f. 19) At lat, integrating eq. 17) we find the pectral denity R V in general cae to order α 3 : } m R V =3 rv, m rv, m rv,0, 0) 4

6 where the function r V are r, V =3 α α ) α 3 [ 161 r, V = 3 [ α α ) [ n f 57 4 lnµ 1 n f ln µ ζ3) ζ5) n f 17 1 n f 1) ζ3) n f ζ5) n f n f 1 36 n f lnµ n f ln µ n f lnµ 855 µ 16 ln 17 4 n f ln µ 1 µ 1 n f ln 11 3lnµ ) [ ζ3) 4 48 n f 1 1 n f ζ3) n f lnµ 4 3 n f ln µ 57 µ 8 ln 1 4 n f ln µ ) α 3 [ ζ3) ζ5) n f n f 54 ζ3) n f 5 4 ζ4) n f 55 4 ζ5) n f n f 7 n f ζ3) n f 4693 ln µ ln µ 1755 ζ3) ln µ n f ln µ 17 1 n f ln µ 59 4 ζ3) n f ln µ n f ln µ 1 36 n f lnµ 1 3 ζ3) n f lnµ 335 µ ln n f ln µ 9 n f ln µ, 85 µ 16 ln n f ln 3 µ 1 36 n f ln 3 µ ) r,0 V α 3 [ = ζ3) 3 9 n f 8 3 ζ3) n f. 3) The expreion for the hadronic decay rate of the intermediate boon read: ΓZ hadron) = Γ Z ) 0 [ g f V ) ga) f ) R 0 )R V, 0, 0, m b m c f g f V ) R V, m f,m f,0) f=b,c g f A ) R V, m f,m f,0), 4) f=b,c, ) 5

7 ΓW hadron) = Γ W 0 [ R 0 )R V, 0, 0, m b m c )) ) 1 i = u, c j = d,, b 5) V i,j R V, m i,m j,0) R A, m i,m j,0)), with Γ 0 = G F M Z/W 3 6, g f V = I f 3 Q f in θ w, g f A = I f 3 and V ij being the CKM matrix. Here R 0 ) i the non-inglet part) of the ratio R)inmaleQCD;itwacomputed to α 3 in [18, 19 and confirmed in [0; it read: R 0 ) = 3 1 α ) [ α ζ3) n f ) [ α ) 3 ζ3) ) 6 n f ln µ ζ3) 4 6 ζ5) n f ζ3) 5 ) ζ5) n f ) 7 ζ3) [ 785 ζ3) n f 7 [ 3 ζ3) n f ) 9 ζ3) ln µ W n f 1 ) 36 n f ln }. µ 6) In deriving Eq. 4) and 5) we have aumed that i) the top quark i completely decoupled the power uppreed correction to thi approximation tart from the order m t α and have been tudied in Ref. [, 3, 4); ii) all other quark except for the charmed and bottom one are male. Note that for the cae of diagonal current there exit alo o-called inglet contribution to R). We will ignore thee contribution in what follow a they are abent for the cae of non-diagonal current relevant for the W -decay a detailed dicuion of the Z-decay rate including inglet contribution can be found in [). Taking into account the peculiar tructure of the general reult 0), the lat formula can be written in a impler form, viz. Here ) ΓW hadron) = Γ W 0 [ R 0 )R V, 0, 0, m b m c)) m ef f = R V, m ef f, 0, 0). 7) i = u, c j = d,, b and we have taken into account the fact that V i,j m i m j ) R V, m i,m j,0) R A, m i,m j,0) = R V, m i m j, 0, 0) = R A, m i m j, 0, 0) 6

8 A a direct conequence of Eq.,0) we obtain the following expreion for particular function entering into 4,5) R V, m, m, 0) = 4m 3rV,, 8) R A, m, m, 0) = 4m 3rV,, 9) R V, m, 0, 0) = R A, m, 0, 0) = m 3rV, r, ), V 30) R V, 0, 0,m) = R A, 0, 0,m)= m 3r,0. 31) At lat, with n f =5andµ =the above formula are implified to R V, m, m, 0) = m 3 α 1 α 69 6 ) 3 [ ) α 3) } ζ3) ζ5) 7 54 R A, m, m, 0) = m 3 6 α ) α [ ζ3) ) α 3 [ } ζ3) 5 ζ4) 995 ζ5) 1 33) m, m, 0, 0) = ) α 3 [ ζ3) R V or, in the numerical form, R V, m, m, 0) = m 3 ) α α [ ζ3) ζ4) ζ5) } 34) 1 α ) α ) } α , 35) R A, m, m, 0) = m 3 6 α ) α α R V, m, 0, 0) R A, m, 0, 0) = m α α ) 3 } ) α, 36) ) 3 }. 37) 4 Acknowledgment Thi work wa upported by BMFT under Contract 057KA9P0) and INTAS under Contract INTAS

9 Reference [1 A. Blondel, plenary talk preented at the 8th International Conference on High Energy Phyic, 5 31 July 1996, Waraw, Poland. [ K.G. Chetyrkin, J.H. Kuehn and A. Kwiatkowki, QCD Correction to the e e Cro-Section and the Z Boon Decay Rate, In the Report of the Working Group on Preciion Calculation for the Z 0 Reonance, CERN Yellow Report 95-03, ed. D. Yu. Bardin, W. Hollik and G. Paarino, page 313. [3 K.G. Chetyrkin, J.H. Kühn, Phy. Lett. B ) 359. [4 K.G. Chetyrkin, J.H. Kühn, A. Kwiatkowki, Phy. Lett. B 8 199) 1. [5 S.G. Gorihny, A.L. Kataev, S.A. Larin, L.R. Surguladze, Mod. Phy. Lett. A5 1990) 703. [6 K.G. Chetyrkin and A. Kwiatkowki, Z. Phy. C591993) 55. [7 K.G. Chetyrkin, C.A. Dominguez, D. Pirjol and K. Schilcher, Phy. Rev. D ) [8 L. R. Surguladze, Phy. Rev. D541966) 118. [9 K.G. Chetyrkin, Preprint MPI/PhT/96-61, hep-ph/ , Augut [10 O.V. Taraov, A.A. Vladimirov, A.Yu. Zharkov, Phy. Lett. B931980) 49. [11 S.A. Larin and J.A.M. Vermaeren, Phy. Lett. B ) 334. [1 O.V.Taraov, preprint JINR P ). [13 S.A. Larin, Preprint NIKHEF-H/9-18, hep-ph/ ); In Proc. of the Int. Bakan School Particle and Comology April -7, 1993, Kabardino-Balkaria, Ruia) ed. E.N. Alexeev, V.A. Matveev, Kh.S. Nirov, V.A. Rubakov World Scientific, Singapore, 1994). [14 D. Broadhurt, Nucl. Phy. BB851975) 189. [15 S.G. Gorihny, A.L. Kataev, S.A. Larin, Nouvo Cim. 9A 1986) 117. [16 S.A. Larin, F.V. Tkachov, J.A.M. Vermaeren, Preprint NIKHEF-H/ ). [17 J.A. M. Vermaeren, Symbolic Manipulation with FORM, Verion, CAN, Amterdam, [18 S.G. Gorihny, A.L. Kataev and S.A. Larin, Phy. Lett. B ) 144. [19 L.R. Surguladze and M.A. Samuel, Phy. Rev. Lett ) 560; erratum ibid,

10 [0 K.G. Chetyrkin, Preprint MPI/PhT/96-83, hep-ph/ Augut [1 B.A. Kniehl, J.H. Kühn, Nucl. Phy. B ) 547. [ B.A. Kniehl, Phy. Lett. B ) 17. [3 K.G. Chetyrkin, Phy. Lett. B ) 169. [4 D.E. Soper and L.R. Surguladze, Phy. Rev. Lett )

Coordinate independence of quantum-mechanical q, qq. path integrals. H. Kleinert ), A. Chervyakov Introduction

vvv Phyic Letter A 10045 000 xxx www.elevier.nlrlocaterpla Coordinate independence of quantum-mechanical q, qq path integral. Kleinert ), A. Chervyakov 1 Freie UniÕeritat Berlin, Intitut fur Theoretiche