P. CH. CHRISTOVA a Faculty of Physics, Bishop Konstantin Preslavsky Univ., Shoumen, Bulgaria

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1 HARD-PHOTON EMISSION IN e + e ff WITH REALISTIC CUTS P. CH. CHRISTOVA a Faculty of Physics, ishop Konstantin Preslavsky Univ., Shoumen, ulgaria penka@main.uni-shoumen.acad.bg M. JACK, T. RIEMANN DESY Zeuthen, Platanenallee 6, D Zeuthen, Germany s: jack@ifh.de, riemann@ifh.de We derive compact analytical formulae of the onneau-martin type for the reaction e + e ffγ cuts on minimal energy and acollinearity of the fermions, where the photons may be emitted both from the initial or final states. Soft-photon exponentiation is also taken into account. One of the cleanest scattering processes at elementary particle accelerators is fermion-pair production in e + e annihilation, potentially accompanied by one or few photons: e + e ff +n)γ. 1) Initial-state corrections may be written as an integral over the normalized) invariant mass squared R = s /s of the final-state fermion pair: σt ini s) = dr σ 0 s ) ρ ini T R), ) where σ 0 s ) is an effective orn cross-section. The radiator function ρ ini T R) for the initialstate first-order corrections to the total cross-section σ T, soft-photon exponentiation, is 1: ρ ini T R) = 1+ S ) β1 R) β 1 + ini H T R), 3) and S = 3 4 β + α π Q e π 3 1 ) + h.o., β = α π Q e ln s ) m 1, 4) e H T ini R) = H M R) β + h.o., 5) 1 R where h.o. stands for higher orders, and : H M R) = 1 β. 6) 1 R Experiments at LEP1, SLC, LEP, and those planned at a linear collider aim at precisions well below a per cent and need theoretical predictions an accuracy of the order of 0.1 % a Supported by ulgarian foundation for Scientific Research grant Φ 60/1996.

2 or better. A basic ingredient of the predictions is the complete photonic Oα) correction including initial and final state radiations and their interferences: σs) =σ 0 s)+σ ini s)+σ int s)+σ fin s). 7) These corrections have to be determined for two basic quantities: The total cross-section σ T s) and the forward-backward asymmetry A F =σ F /σ T ; other asymmetries may then easily be derived. asically, there are two experimental set-up s to be treated: i) a lower cut on s, s min 4m f, often applied to quark-pair production; ii) combined cuts on acollinearity ξ, ξ max 180, and minimal energy E min, E min m f, of the fermions; often applied to lepton-pair production. oth cut settings may be combined an acceptance cut c, c 1, on the cosine of the fermionic production angle cos θ. For case i), a generalization of the onneau-martin formula, including the complete first-order photonic corrections together soft-photon exponentiation, may be found in 3 out and in 4 acceptance cut. The extremely compact expressions for case i) c = 1 get quite more involved when the acceptance cut is applied. In this article, we give the complete first-order photonic corrections together softphoton exponentiation for case ii). A three-fold analytical integration of the squared matrix elements is performed in order to get the integrand of the last one, the s -integration over R, which is assumed to be performed numerically. One may use the phase-space parameterization derived in 5. The kinematical regions of two variables are shown in figure 1: the normalized) invariant mass x of photon + anti-fermion) in their rest system, and R as introduced above. The first and third analytical integrations are over the full production angle of the photon, φ γ, in the photon + fermion) rest system, and over the cosine of the anti-fermion in the center-of-mass system, cos ϑ. oth angles are completely independent of x and R. Asfigure1shows,wehavetodetermine radiators ρ b = T,F and b = ini, int, fin) in three phase-space regions: σs) b = + dφ γ dx ds dσ b d cos ϑ A) I II III dφ γ dxds d cos ϑ, 8) Region I applies to the simple s -cut. The integration over R extends from R min to 1, sin R min = R E 1 ξ max /) 1 R E cos. 9) ξ max /) In each of the three regions, the boundaries for the integration over x are, for a given value of R: x max,min = 1 1 R) 1±A), 10) where the parameter A = AR) depends in every region on only one of the cuts applied: A I = 1 R m R, 11) A II = R E, 1) 1 R A III = 1 R1 R ξ) R ξ 1 R), 13) )

3 Figure 1: Phase space cuts on acollinearity and lower) fermion energy. : R m = 4m f s, 14) R E = E min, s 15) R ξ = 1 sinξ max/) 1+sinξ max /). 16) Initial State Radiation Here, for σt ini s) the onneau-martin function gets replaced by: HT ini R, A) = 3α ) 4π Q e A + A3 ln s ) 3 1 R m 1 +A A 3 ) R. 17) e 1 R In σf ini s), the corresponding hard radiator part is: H ini F R; A A 0)= α π Q e { 1 R 4R 1 + R) ln ) s1 + R) 4m e R R) y +y ln y + y +4Rln4R) 1 A s ) } ) ln 4m e 1 + A) R 1 4A1 A)R +, 18) 1 R HF ini R; A<A 0 )= α { π Q e y +y 1 R 1 + R) ln y + +1 A )ln 1+A 1 A + 8AR 1 + R)1 R) 3 } y. 19)

4 These functions have to be used in 3) and its analogue, equal S, forσf ini. The following definitions are used: A 0 = 1 R, 0) y ± =1 R)±A1 + R). 1) In region I, the above expressions 17) and 18) reduce to those known from and 3. In this region the radiators diverge for R 1, and soft-photon exponentiation and the subtraction β/1 R) is applied there and only there) in order to get R); see 5). Initial-Final State Interferences In the initial-final state interferences, the effective orn cross-sections depend on both s and s as well as on the type of exchanged vector particles V i e.g. photon and or Z): σ int s) = dr V i,v j =γ,z H ini σ 0 s, s,i,j) ρ int R, A, i, j). ) For =T it is =F and vice versa. We give as examples simple model-independent orn expressions in order to fix the overall normalization: σ 0 T s, s )= σ 0 F s, s )= V i,v j =γ,z V i,v j =γ,z σ 0 T s, s,i,j)= 4πα 3s V, 3) σ 0 F s, s,i,j)= πα s A, 4) V = Q eq f + Q e Q f v e v f Reχs)+χs ) + ve + a e)vf + a f)reχs)χ s ), 5) A = Q e Q f a e a f Reχs)+χs ) + 4v e a e v f a f Reχs)χ s ), 6) and the following Z boson propagators: The radiator functions are: χs) = G µmz 8 s πα s MZ + im. 7) ZΓ Z ρ int R, A; i, j) =δ1 R)S +b i, j) + θ1 R ɛ)h int R, A). 8) The box contributions b T i, j) may be taken from equations 116) and 118) to be multiplied by 4/3) of 6 and the b F i, j) from equations 13) and 16). For convenience, we give the soft corrections explicitly: ST int =8 α π Q eq f 1 ln ɛ ), 9) λ SF int = α π Q eq f ln ) ln ɛ λ +4ln +ln π Finally, the hard radiator parts are:. 30) H int T R, A) = α π Q eq f 4AR1 + R) 1 R, 31) 4

5 and H int F R, A A 0 )= α π Q eq f { 3R ln z + + R R ln R z 1 R 1 R) 5 R 1 + R)1 + A) +5R )ln 1 1 4R + R )A1 + R) 1 R) + 41 R) } +A1 A)1 + R 3 ), 3) HF int R, A < A 0)= 3α { π Q eq f R ln z + R R ln 1+A } +A1 R), 33) z 1 R 1 A z ± =)±A1 R). 34) Again, for A 1theHT int R, A) andhint F R, A A 0) approach the known expressions of the s cut given in 3 b. Final State Radiation The final state corrections to order Oα) are: σ fin s)=σ0 s) ρ fin dr ρ fin R, A), 35) R, A) =δ1 R)S f + θ1 R ɛ)h fin R, s, A), 36) S f = S f + β f ln ɛ, 37) where S f and β f are the final state s analogues of S and β. The hard radiators are: H fin T R, s, A) =α 1+A π Q f ln 1 R 1 A 8Am f /s 1 A A1 R), 38) )1 R) H fin F R, s, A) =Hfin T R, A)+α π Q f A1 R) 1 + R)ln z +. 39) z Common initial- and final state soft-photon exponentiation may be performed as follows 7,6: σ ini+fin s)= ρ fin R, s,a)=1 R E ) β f1 + S f )+ 1 dr σ 0 s ) ρ ini R, A) ρfin R, s,a), 40) R min /R du H fin u, s,a ) β f 1 u θr R E).41) The soft part of ρ fin u, s,a ), A = Au), and β f are derived from 3) by replacing there Q e by Q f and s/m e by s /m f. We mention here that in the hard radiators the integration over u may also be performed analytically. In region III, one has to interchange for this the order of integration over u and x 8. b We realized a misprint in eq. ) of 3; the non-logarithmic terms have to be multiplied there by 1/1 + R). 5

6 Conclusions We recalculated the photonic corrections acollinearity cut having applications in the Fortan program ZFITTER in mind 9. When the code was created in , an accuracy of 0.5 % at LEP1 was assumed to be needed c. We performed several numerical applications of the above formulae. For this purpose, the package acol.f was added. As a result, we conclude that ZFITTER until version 5.0 treats the Oα) photonic corrections acollinearity cut a numerical accuracy for σ T of not less than about 0.4 % near the Z resonance LEP1 energy region, s in M Z ± 3GeV)or better at resonance). The coding acol.f gave a numerical agreement of σ T for leptons, ξ max 10 and E min = 1 GeV, predictions from TOPAZ0 v at LEP1 of 0.01 % at the wings) or better at resonance) 14. For A F, we estimate at LEP1 energies the accuracy of the Oα) photonic corrections acollinearity cut in ZFITTER until v.5.0 to be about 0.0 % or better at the resonance and about 0.13 % or better at the wings. The numerical limitations at LEP1 are due to the initial-final state interference. For applications at higher energies, the accuracy of ZFITTER acollinearity cut was not dedicatedly controlled until recently, although there was reason to suspect that it comes out much worse than at LEP1. With the cuts mentioned, the accuracy of v.5.0 at LEP is again limited by the initial-final interference but not less than roughly 1 %. A higher accuracy in the acollinearity mode is prevented in ZFITTER until v.5.0 by several reasons. The main reason is a neglect of a certain class of ordinary, angular dependent Oα) terms in the initial state and in the initial-final state interference hard radiator parts. Some numerical approximations in the treatment of final state radiation may also be influential; this has not been studied in detail so far. Finally we should like to mention that the correct Oα) hard radiators for the angular distributions and for the integrated cross-sections cuts on both acollinearity and angular acceptance also have been determined in this project and will be published elsewhere together a scetch of the calculations and more numerical results. Acknowledgments We would like to thank D. ardin, S. Vasileva, M. Grünewald, G. Passarino, and S. Riemann for numerical comparisons, and D. ardin for a careful discussion of the conclusions. References 1. M. Greco, G. Pancheri-Srivastava, and Y. Srivastava, Nucl. Phys ) G. onneau and F. Martin, Nucl. Phys ) D. ardin, M. ilenky, A. Chizhov, A. Sazonov, Y. Sedykh, T. Riemann, and M. Sachwitz, Phys. Lett ) D. ardin, M. ilenky, A. Sazonov, Y. Sedykh, T. Riemann, and M. Sachwitz, Phys. Lett ) G. Passarino, Nucl. Phys ) D. ardin, M. ilenky, A. Chizhov, A. Sazonov, O. Fedorenko, T. Riemann, and M. Sachwitz, Nucl. Phys ) c For other recent comparisons see e.g. 11,1 and for the influence of higher order corrections e.g. 1, and references therein. 6

7 7. O. Nicrosini and L. Trentadue, Z. Physik C ) G. Montagna, O. Nicrosini, and G. Passarino, Phys. Lett ) D. ardin et al., Fortran program ZFITTER v.4.6, CERN-TH. 6443/9 199), hepph/94101; v Feb 1999) and many earlier versions are available from /afs/ cern.ch/user/b/bardindy/public and M. ilenky and A. Sazonov, ZFITTER Fortran routines for photonic corrections acollinearity and acceptance cuts, unpublished 1989). 11. P. Christova, M. Jack, S. Riemann, and T. Riemann, Predictions for fermion-pair production at LEP, DESY ), to appear in the proceedings of RADCOR 98, Sep 8-1, 1998, arcelona, Spain, hep-ph/ D. ardin, M. Grünewald, and G. Passarino, Precision Calculation Project Report, hep-ph/ G. Montagna et al., Fortran program TOPAZ0 v.4.0, FNT/T 98/0 1998), hepph/980411, v.4.4 available from D. ardin, M. Grünewald, and G. Passarino, TOPAZ0 values by private communication. 7