Heavy flavour mass corrections to the longitudinal and transverse cross sections in e+e−-collisions

31 December 1998

Physics Letters B 445 Ž1998. 206–213

Heavy flavour mass corrections to the longitudinal and transverse cross sections in eqey-collisions V. Ravindran, W.L. van Neerven

1

DESY-Zeuthen, Platanenallee 6, D-15738 Zeuthen, Germany Received 14 October 1998; revised 4 November 1998 Editor: P.V. Landshoff

Abstract We present the heavy flavour mass corrections to the order a s corrected longitudinal Ž sL . and transverse Ž s T . cross sections in eqey-collisions. Its effect on the value of the running coupling constant extracted from the longitudinal cross section is investigated. Furthermore we make a comparison between the size of these mass corrections and the magnitude of the order a s2 contribution to s T and sL which has been recently calculated for massless quarks. Also studied will be the changes in the above quantities when the fixed pole mass scheme is replaced by the running mass approach. q 1998 Elsevier Science B.V. All rights reserved.

Experiments carried out at electron positron colliders like LEP and LSD have provided us with a wealth of information about the constants w1x appearing in the standard model of the electroweak and strong interactions. One of them is the strong coupling constant a s which can be extracted from various quantities. Examples are the hadronic width of the Z-boson, the total hadronic cross section, event shapes of jet distributions and jet rates Žfor a review on this subject see w2x.. Another quantity from which a s can be extracted is the longitudinal cross section sL which is measured in the semi inclusive reaction eyq eq™ V ™ P q ‘‘X’’ .

Ž 1.

Here V represents the intermediate vector bosons Z and g and ‘‘ X ’’ denotes any inclusive final state. Furthermore P stands for a hadron which is produced via fragmentation by a quark or a gluon. As we will see below the perturbation series in QCD for sL starts in order a s provided the quark which couples to the vector boson is massless. Therefore this quantity will become very sensitive to the value of the strong coupling constant. Some time ago the perturbation series was only known up to order a s Žsee w5,6x.. However it turned out that the result for the longitudinal cross section was much smaller than its experimental value measured at LEP w3,4x which indicated that the higher order corrections may become rather large. The latter was confirmed by the second order corrections, computed very recently in w7x, which amount to about 35% with respect to the

1

On leave of absence from Instituut-Lorentz, University of Leiden,P.O. Box 9506, 2300 RA Leiden, The Netherlands.

0370-2693r98r$ - see front matter q 1998 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 0 - 2 6 9 3 Ž 9 8 . 0 1 4 4 1 - 5

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lowest order contribution to sL . If the second order corrections are included the longitudinal cross section agrees now very well with experiment so that it can be used to extract a s Žsee w8x.. Until now sL was only evaluated for massless quarks. This will be a correct approximation for the light quarks like u,d, s and probably also for c but it will certainly fail for the b-quark, leave alone for the top quark. Therefore one has to compute the mass corrections to this quantity at least up to first order in a s . This contribution will be presented in this paper and we discuss its effect on the value of a s when extracted from experiment via sL . The cross section corresponding to process Ž1. where the hadron P emerges directly, or indirectly via the gluon, from the heavy quark H is given by d 2s H ,P Ž x ,Q 2 . dx dcos u

s 38 Ž 1 q cos 2u .

d s TH ,P Ž x ,Q 2 . dx

q 34 sin2u

d sLH ,P Ž x ,Q 2 . dx

q 34 cos u

d sAH ,P Ž x ,Q 2 .

.

dx

Ž 2.

From the equation above one can extract the transverse Ž s T ., the longitudinal Ž sL . and the asymmetric Ž sA . cross sections. Further u denotes the angle between the outgoing hadron P and the incoming electron. The Bjørken scaling variable is defined by xs

2 pq Q2

q 2 s Q 2 ) 0,

,

0-xF1 ,

Ž 3.

where the momenta p and q correspond with the outgoing hadron P and the virtual vector boson Ž Z,g . respectively. To obtain the total transverse and longitudinal cross sections one has to multiply Eq. Ž2. with x. After integration over x and summation over all hadron species P emerging from the heavy flavour H one obtains

s kH Ž Q 2 . s 12 Ý P

1

H0

dx x

d s kH ,P dx

Ž x ,Q 2 .

ksT ,L .

Ž 4.

The result above can be written as follows

s kH Ž Q 2 . s s V V Ž Q 2 . hÕk Ž r . q sA A Ž Q 2 . h ak Ž r .

with

rs

4 m2 Q2

,

Ž 5.

where m denotes the heavy quark mass. The above definitions for the total cross sections should not be confused with the ones which are obtained by integrating d s krd x without multiplication by x. The heavy flavour contributions to the latter quantities were computed for the first time in w9x Žsee also w10–12x.. The difference between these two definitions will be pointed out below. Finally the total cross section for heavy flavour production is given by

stotH Ž Q 2 . s s TH Ž Q 2 . q sLH Ž Q 2 . .

Ž 6.

In Eq. Ž5. the pointlike cross sections are defined by

sV V Ž Q

sA A Ž Q

2

2

4pa 2

.s

3Q 2

4pa 2

.s

3Q 2

2

2 Q 2 Ž Q 2 y MZ2 .

N

N

Ž Q2 . Ž CV2 , l q CA2 , l . CA2 , q 2 2 ZŽQ .

ZŽ Q2 .

2

2

Ž Q2 . e l e q CV , l CV , q q Ž C 2 q CA2 , l . CV2 , q < ZŽ Q2 . < 2 V , l

e l e q2 q

,

Ž 7.

2

,

Ž 8.

where N denotes the number of colours corresponding to the gauge group SUŽ N . Žin the case of QCD one has N s 3.. Furthermore we adopt for the Z-propagator the energy independent width approximation Z Ž Q 2 . s Q 2 y MZ2 q iMZ GZ .

Ž 9.

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In Eqs. Ž7. and Ž8. the charges of the lepton and the quark are denoted by e l and e q respectively and the electroweak constants are given by CA , l s

1 2sin2 u W

C V , l s yCA , l Ž 1 y 4sin2u W . ,

,

s CA , l Ž 1 y 83 sin2u W . ,

CA ,u s yCA , d s yCA , l ,

C V ,u

C V , d s yCA , l Ž 1 y 43 sin2u W . .

Ž 10 .

The functions h lk Ž k s T, L, l s Õ,a. in Eq. Ž5. can be obtained order by order in perturbative QCD from the singlet quark and the gluon coefficient functions in the following way h lk Ž r . s

1

l ,S k,q

H'r dx x C

1y r

Ž x , r ,Q 2rm2 . q 12 H

0

dx x Ckl,,Sg Ž x , r ,Q 2rm2 . .

Ž 11 .

Here m stands for the factorization as well as the renormalization scale. The result in Eq. Ž11. has been derived from the differential cross section d s kH ,Prd x Žsee e.g. w6x.. The latter can be written as a convolution of the fragmentation densities DiP and the coefficient functions Ck,i Ž i s q, g . Žsee Eq. Ž2.4. in w14x.. Because of the momentum sum rule satisfied by the fragmentation densities Žsee Eq. Ž2.9. in w6x. expression Ž11. follows immediately. Notice that the functions h lk differ from the functions f kl computed in w12x Žsee also the functions Hi Ž i s 2,6. in w10x.. The latter were derived from the first moment of the non-singlet quark coefficient function and they differ from the former which are given by the second moment of the singlet quark and gluon coefficient functions presented in Eq. Ž11.. However because of Eq. Ž5. the sum of the transverse and longitudinal components of both functions have to lead to the same perturbation series of the total cross section so that one has the relation hTl q h lL s f Tl q f Ll Ž l s Õ,a.. The functions h lk can be expanded in the strong coupling constant a s as follows `

h lk

Ž r. s

Ý ns0

ž

asŽ m. 4p

n

/

h lk,Ž n. Ž r . .

Ž 12 .

The lowest order contributions corresponding to the Born reaction V™HqH ,

Ž 13 .

with V s g ,Z are given by hÕT,Ž0. Ž r . s 1 y r ,

'

hÕL,Ž0. Ž r . ,s

r 2

'1 y r ,

hTa ,Ž0. Ž r . s Ž 1 y r .

3r2

,

h aL,Ž0. Ž r . s 0 ,

Ž 14 .

where r is defined in Eq. Ž5.. The next-to-leading order ŽNLO. contributions originate from the one-loop virtual corrections to reaction Ž13. and the gluon bremsstrahlungs process V™HqHqg .

Ž 15 .

We have computed the order a s contributions to the quark Ck,l,Sq and gluon Ck,l,Sg coefficients functions for the case m / 0 in Eq. Ž11. and confirmed the result obtained in appendix A of w6x. Notice that the quark coefficient functions are derived from the process where the hadron P emerges from the quark H. The gluon coefficient function corresponds with the process where the hadron is emitted by the gluon. After performing the integral in Eq. Ž11. we obtain the following results for k s T, L hÕT,Ž1. Ž r . s CF

1 2

r Ž 1 y 3 r . F1 Ž t . q r 3r2 Ž 1 q r . F2 Ž t . q Ž 32 y 392 r q 212 r 2 . Li 2 Ž t .

q Ž 16 y 10 r q 6 r 2 . F3 Ž t . q 2 1 y r F4 Ž t . q Ž 8 y 6 r q 6 r 2 . ln Ž t . ln Ž 1 q t . q Ž y12 q 3 r y 32

' r 2 . ln Ž t . y 5 r'1 y r

,

Ž 16 .

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hÕL,Ž1. Ž r . s CF y 12 r Ž 1 y 3 r . F1 Ž t . y r 3r2 Ž 1 q r . F2 Ž t . q Ž 392 r y 372 r 2 . Li 2 Ž t . q 10 r Ž 1 y r . F3 Ž t . qr 1 y r F4 Ž t . q Ž 6 r y 8 r 2 . ln Ž t . ln Ž 1 q t . q Ž yr q 134 r 2 . ln Ž t . q Ž 3 q 192 r . 1 y r ,

'

'

Ž 17 . hTa ,Ž1. Ž r . s CF

1 2

2 r Ž 1 y 4 r . F1 Ž t . q r 3r2 Ž 1 q 2 r . F2 Ž t . q Ž 32 y 103 2 r q 30 r . Li 2 Ž t .

q Ž 16 y 26 r q 16 r 2 . F3 Ž t . q 2 Ž 1 y r .

3r2

F4 Ž t . q Ž 8 y 14 r q 12 r 2 . ln Ž t . ln Ž 1 q t .

q Ž y12 q 9r y 92 r 2 q 34 r 3 . ln Ž t . y Ž 12 r q 32 r 2 . 1 y r ,

'

Ž 18 .

h aL,Ž1. Ž r . s CF y 12 r Ž 1 y 4 r . Ž F1 Ž t . y 4 F3 Ž t . y 7Li 2 Ž t . y 4ln Ž t . ln Ž 1 q t . . y r 3r2 Ž 1 q 2 r . F2 Ž t . q Ž 2 r q 134 r 2 y 89 r 3 . ln Ž t . q Ž 3 q 3 r q 49 r 2 . 1 y r ,

'

Ž 19 .

with ts

' 1 q '1 y r 1y 1yr

Ž 20 .

and the colour factor CF is given by CF s Ž N 2 y 1.r2 N. The functions Fi Ž t . appearing above are defined by F1 Ž t . s Li 2 Ž t 3 . q 4z Ž 2 . q 12 ln2 Ž t . q 3ln Ž t . ln Ž 1 q t q t 2 .

Ž 21 .

F2 Ž t . s Li 2 Ž yt 3r2 . y Li 2 Ž t 3r2 . q Li 2 Ž yt 1r2 . y Li 2 Ž t 1r2 . q 3z Ž 2 . q 2ln Ž t . ln Ž 1 q 't . y 2ln Ž t . ln Ž 1 y 't . q 32 ln Ž t . ln Ž 1 q t y 't . y 32 ln Ž t . ln Ž 1 q t q 't .

Ž 22 .

F3 Ž t . s Li 2 Ž yt . q ln Ž t . ln Ž 1 y t .

Ž 23 .

F4 Ž t . s 6ln Ž t . y 8ln Ž 1 y t . y 4ln Ž 1 q t . ,

Ž 24 .

where z Ž n., which occurs in the formulae of this paper for n s 2,3, represents the Riemann z-function and Li 2 Ž x . denotes the dilogarithm. Using Eqs. Ž16. – Ž19. one can check that the order a s contribution to the total cross section in Ž6. is in agreement with the literature w13x. The next-to-next-to-leading order ŽNNLO. contributions come from the following processes. First one has to calculate the two-loop vertex corrections to the Born process Ž13. and the one-loop corrections to Ž15.. Second one has to add the radiative corrections due to the following reactions V™HqHqgqg

Ž 25 .

V™HqHqHqH

Ž 26 .

V™HqHqqqq

Ž 27 . h l,Ž2. k

where q denotes the light quark. The results for presented below are only computed for those contributions containing Feynman graphs where the vector boson V is always coupled to the heavy quark H so that the light quarks can be only produced via fermion pair production emerging from a gluon splitting. Because we are only interested in the ratios Rk Ž Q2 . s

s kH Ž Q 2 . stotH Ž Q 2 .

for

ksT ,L

Ž 28 .

we do not consider other contributions which drop out in the expression above provided we put the mass of H equal to zero in the second order correction. The contributions which can be omitted are given by one- and two-loop vertex corrections which contain the triangular quark-loop graphs w15,16x. They only show up if the

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quarks are massive and are coupled to the Z-boson via the axial-vector vertex. Notice that one has to sum over all members of one quark family in order to cancel the anomaly. Further we exclude all contributions from reaction Ž27. which involve Feynman graphs where the light quarks q are coupled to the vector boson V. Notice that interference terms of the latter with diagrams where heavy quark are attached to the vector boson vanish if the heavy quark is taken to be massless provided one sums over all members in one family. The order a s2 contributions to the quark and gluon coefficient functions have been calculated in w7,14x. Because of the complexity of the calculation of these functions the heavy quark mass was taken to be zero. This approximation is good for the charm and bottom quark but not for the top quark as we will see below. Substituting these coefficient functions in the integrand of Eq. Ž11. Žhere r s 0. we obtain 196 8 hÕT,Ž2. s hTa ,Ž2. s CF2  6 4 q CA CF  y 89 15 y 5 z Ž 3 . 4 q n f C F Tf  3 q 16 z Ž 3 . 4 ,

Q2

½ ž /

hÕL,Ž2. s h aL,Ž2. s CF2  y 152 4 q CA CF y11ln

m2

5

Ž 29 . Q2

½ ž / 5

24 q 2023 30 y 5 z Ž 3 . q n f C F Tf 4ln

m2

y 743 ,

Ž 30 .

where the colour factors are given by CA s N and Tf s 1r2 Žfor CF see below Eq. Ž20... Further n f denotes the number of light flavours which originate from process Ž27.. Finally m appearing in the strong coupling constant a s and the logarithms in Eq. Ž30. represents the renormalization scale. Notice that the coefficient of the logarithm is proportional to the lowest order coefficient of the b-function. The logarithm does not appear in and h a,Ž2. for hTl,Ž2. because hTl,Ž1. s 0 for m s 0 and l s Õ,a. Since m s 0 there is no distinction between hÕ,Ž2. k k k s T, Lanymore unlike in the case of the first order corrections in Eqs. Ž16. – Ž19. where the heavy quark was taken to be massive. Furthermore one can check that substitution of Eqs. Ž29. and Ž30. into Eq. Ž6. provides us with the order a s2 contribution to the total cross section which is in agreement with the results obtained in w17x. When the quark is massless we get from Eqs. Ž5. and Ž6. the following perturbation series for the ratio in Eq. Ž28. up to order a s2 R HL Ž Q 2 . s

asŽ m. 4p

CF  3 4 q

ž

Q2

asŽ m. 4p

2

/

½ ž / 5

qn f CF TF 4ln R TH Ž Q 2 . s 1 y R HL Ž Q 2 . .

m2

y 743

Q2

½ ž /

CF2  y 332 4 q CA CF y11ln

,

m2

q 123 2 y 44z Ž 3 .

5 Ž 31 . Ž 32 .

We will now show how the results for the above ratios will be modified when the Born and the first order contribution are computed for massive quarks. Further we discuss the consequences of our findings for the extraction of a s . In order to make the comparison between the massless and massive approach to Eq. Ž28. we have chosen the following parameters Žsee w18x.. The electroweak constants are: MZ s 91.187 GeVrc 2 , GZ s 2.490 GeVrc 2 and sin2u W s 0.23116. For the strong parameters we choose: L5 MS s 237 MeVrc Ž n f s 5. which implies a s Ž MZ . s 0.119 Žtwo-loop corrected running coupling constant.. Further we take for the renormalization scale m s Q unless mentioned otherwise. Notice that we study R Hk for H s c,b at the CM energy Q s MZ . For the heavy flavour masses the following values are adopted: m c s 1.5 GeVrc 2 , m b s 4.50 GeVrc 2 and m t s 173.8 GeVrc 2 . The results for the bottom quark can be found in Table 1. A comparison between the first and second column reveals that on the Born level the difference between the massive and massless approach to R Tb is very small Žabout two promille.. For R Lb it is more conspicuous but the correction 2 due to mass effects, which equals 0.0014 for R Ž1. L , is still much smaller than the order a s contribution which amounts to 0.0121. The corrections due to mass effects are also smaller than the changes caused by a different Ž2. choice of the renormalization scale. If we choose m s Qr2 or m s 2 Q one gets R Ž2. L s 0.0509 and R L s 0.0461 Ž2. respectively which differ by 0.0024 from the central value R L s 0.0485. We also studied the effect of the

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Table 1 The ratio R k s s k r stot Ž k sT, L. for bb-production R Tb R Ž0. T R Ž1. T R Ž2. T

m b s 0.0 GeVrc 2

m b s 4.50 GeVrc 2

m b Ž MZ . s 2.80 GeVrc 2

1.0 0.962 0.950

0.999 0.964 0.952

0.999 0.963 0.951

0.0016 0.0364 0.0485

0.0006 0.0374 0.0495

R Lb R Ž0. L R Ž1. L Ž2. RL

0.0 0.0378 0.0499

running quark mass on the ratio in Eq. Ž28.. For this purpose one has to change the on-mass shell scheme used in Eqs. Ž16. – Ž19. into the MS-scheme. This can be done by substituting in all expressions the fixed pole mass m by the running mass mŽ m .. Moreover one has to add to the first order contributions Ž16. – Ž19. the finite counter term

D h lk,Ž1. s m Ž m . CF 4 y 3ln

ž

m2 Ž m .

m2

/ž

d h lk, Ž 0 . Ž r . dm

/

,

Ž 33 .

msm Ž m .

where m stands for the mass renormalization scale for which we choose m s Q. Further we adopt the two-loop corrected running mass with the initial condition mŽ m 0 . s m 0 . Using the relation between the MS-mass and the fixed pole mass, as is indicated by the first factor on the righthand side in Eq. Ž33., we have taken for bottom production m 0 s 4.10 GeVrc 2 which corresponds with a pole mass m b s 4.50 GeVrc 2 . This choice leads to m b Ž MZ . s 2.80 GeVrc 2 which is 5% above the experimental value 2.67 GeVrc 2 measured at LEP w19x. The results are presented in the third column of Table 1. Comparing the second with the third column we observe that the mass corrections decrease because the running mass is smaller than the fixed pole mass. It is now interesting to see how the mass terms contributing to R Lb affect the extraction of the running coupling constant at m s Q s MZ . To that order we equate the mass corrected formula for R Lb with the massless result presented in Ž . Eq. Ž31.. The latter yields R Ž2. L s 0.0499 for a s s 0.119 see first column, third row of Table 1 . Using the same number for the massive expression of R Ž2. L we obtain a s s 0.122 which amounts to an enhancement of 2.5% for m b s 4.50 GeVrc 2 . In the case of a running mass i.e. m b Ž MZ . s 2.80 GeVrc 2 we get a s s 0.120 so that the enhancement becomes 1.0%. However as we have said before these mass effects are smaller than those caused by a variation in the renormalization scale. Following the same procedure we equate R L at different values of Ž2. m. Choosing m s Q one infers from Table 1 that R Ž2. L s 0.0499 for a s s 0.119. This value of R L can also be obtained at m s Qr2 but then one has to take a s s 0.127. If we choose m s 2 Q the result is a s s 0.112. Therefore the uncertainty is about 6% which is larger than the mass correction. The latter becomes even smaller when we also add the part coming from the bottom quark to the longitudinal cross section consisting of the contributions of the light quarks and the charm quark. Concerning the latter quark we want to remark that its mass is so small that the massive approach will become indistinguishable from the massless results. Since the mass effects up to the order a s level are rather small even for the bottom quark we can assume that the second order corrections, derived for m s 0, also apply for m / 0 at least as long as Q 4 m. This will be correct at large collider energies for all quarks except for the top quark as we will see below. For tt production we will study the mass effects up to order a s at a CM energy Q s 500 GeVrc. Here we have chosen m 0 s 166.1 GeVrc 2 which corresponds with a fixed pole mass of m t s 173.8 GeVrc 2 . This choice leads to m t Ž Q . s 153.5 GeVrc 2 . From Table 2 we infer that for this process the mass corrections are

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Table 2 The ratio R k s s k r stot Ž k sT, L. for tt-production at Qs 500 GeVrc R Tt R Ž0. T R Ž1. T R Ž2. T

m t s 0 GeVrc 2

m t s173.8 GeVrc 2

m t Ž Q . s153.5 GeVrc 2

1.0 0.970 0.962

0.826 0.828 0.821

0.862 0.834 0.826

0.175 0.172 0.180

0.138 0.167 0.174

R Lt R Ž0. L R Ž1. L Ž2. RL

0.0 0.030 0.038

huge and they are much larger than the first and second order corrections. Therefore the numbers in the last row, presented for R Tt and R Lt , are unreliable because they only hold if the mass corrections in the order a s2 contributions can be neglected as was done in Eqs. Ž29. and Ž30.. This implies that for the top quark R Ž2. T and R Ž2. have to be computed for m / 0 which will be an enormous enterprise. We also studied the effect of the L running mass presented in the third column of Table 2. Here the difference between the fixed pole mass and the running mass approach is larger than the one observed for the bottom quark in Table 1. Another feature is that the order a s correction increases when the running mass scheme is used and its sign is reversed with respect to the correction obtained for the fixed pole mass approach. Notice that a study of the change in R tk Ž k s T, L. under variation of the renormalization scale makes no sense because of the missing exact result in second order for m / 0. Summarizing our findings we have computed the mass corrections up order a s for the transverse and longitudinal cross sections which are due to heavy flavours. In the case of charm there is no observable difference between the massless and massive approach. For the bottom quark we see a difference which leads to an enhancement of the extracted value of a s which is of the order of 2.5% Žfixed pole mass m b s 4.50 GeVrc 2 .. or 1.0% Žrunning mass m b Ž MZ . s 2.80 GeVrc 2 .. These numbers become much smaller when the light flavour contributions are added to the cross sections. A variation in the renormalization scale introduces larger effects on the value for a s which are about 6.0% Žsee e.g. also w8x.. In the case of top-quark production the mass terms cannot be neglected anymore since they are much larger than the second order corrections computed for massless quarks.

Acknowledgements The authors would like to thank J. Blumlein for reading the manuscript and giving us some useful remarks. ¨ This work is supported by the EC-network under contract FMRX-CT98-0194.

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