Forward-Backward Charge Asymmetry at Very High Energies

a r X i v :h e p -p h /0307238v 1 18 J u l 2003FORWARD-BACKWARD CHARGE ASYMMETRY AT

VERY HIGH ENERGIES

B.I.Ermolaev a b ,M.Greco c ,S.M.Oliveira a and S.I.Troyan d a CFTC,University of Lisbon Av.Prof.Gama Pinto 2,1649-003Lisbon,Portugal b Io?e Physico-Technical Institute,194021St.Petersburg,Russia c Dipartamento di Fisica and INFN,University of Rome III,Italy d St.Petersburg Institute of Nuclear Physics,188300Gatchina,Russia February 7,2008Abstract The impact of the electroweak radiative corrections on the value of the forward-backward asymmetry in e +e ?annihilation into a quark-antiquark pair is considered in the double-logarithmic approximation at energies much higher than the masses of the weak bosons.1Introduction The forward-bacward asymmetry for the charged hadrons produced in e +e ?annihilation at high energies in vicinity of the Z -boson mass has been the object of intensive theoretical and ex-perimental investigation.It is interesting to make a theoretical analysis of this phenomenon for much higher enegies where the leading,double-logarithmic (DL)contributions come not only from integrating over virtual photon momenta but also from virtual W and Z bosons.Indeed,at the annihilation energies much higher than 100Gev,when masses of virtual electroweak bosons are small compared to their momenta,the initial SU (2)x U (1)symmetry is restored in a certain sense and accounting for higher loop DL contributions involving the W,Z bosons is nonless imporatnt than the standard accounting for the photon DL contributions.The value of the asymmetry at

such energies would be expressed rather through the Cazimir operators of the electroweak gauge group than through electric charges.In a sense,studing the forward-backward asymmetry at such enegries is one of the simplest ways to see the total e?ects of contributions of the electroweak radiative corrections of higher orders.Our theoretical study can be useful in future when Next linear colliders will explore e +e ?annihilation at very high energies,probing further the Standard Model and eventually looking for New Physics.

In accordance with the present theoretical conceptions,we divide investigation of the annihila-tion into two stages:?rst we calculate the sub-process:e +e ?-annihilation into a quark-antiquark pair (which is studied with perturbative methods)and then,numerically,account for hadronization e?ects which are quite di?erent for converting the quarks into mesons and barions.We consider e +e ?annihilate into two hadronic jets in the kinematics when the leading particles of every jet goes in cmf close to the beam axis,so they are within the cones with opening angles θ?1and the axes around the e ?and e +directions.We obtain that the forward-backward asymmetry manifests itself as follows:the number of the hadrons with the positive electric charges,N +in the cone around the e +-direction exeeds the number of the negatively charged hadrons,N ?(see Ref.[1].It is depicted in Fig.1.The opposite e?ect is true for the other cone,around the e ?direction.The space outside of the cones is neutral.The numerical evaluations of the asymmetry are plotted in Fig.2separately for the produced mesons and barions.In particular,Fig.2shows that the value of the asymmetry is high enouph to be measured at energies ≈1TeV and grows steeply with energy.

Figure1:Relations between charged hadrons in diferent angular regions.

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In order to arrive at this result,let us consider?rst the basic sub-process of the annihilation: e+(p2)e?(p1)annihilation into a quark q(p3)and its antiparticleˉq.All these particles can belong either to the SU(2)doublets or to the singlets.Our?rst goal is to calculate the scattering amplitudes of the annihilation in the t-kinematics where

s=(p1+p2)2?t=(p3?p1)2(1) and in the u-kinematics:

s=(p1+p2)2?u=(p4?p1)2.(2) In order to account for DL contributions to all orders in the electroweak couplings,we use the evolution equations with respect to the infrared cut-o?.This cut-o?M is chosen in the transverse momentum space so that momenta of all virtual particles obey

k i⊥>M.(3) Although only the Feynman graphs vith virtual photons can have the infrared divergencies,it is convenient to keep the restriction(3)for momenta of all virtual particles,assuming that

M≥M Z≈M W.(4) With assumptions of Eqs.(34),one can neglect all masses and be safe of the infrared singu-larities at the same moment.On the other hand,the scattering amplitudes now depend on M.It makes possible to evolute them in M and to put M=M Z≈M W in the?nal expressions.As DL contributions appear in the regions where k i⊥obey the strong inequalities of the kind k i⊥?k j⊥, it is always possible to?nd the virtual particle with minimal(≡k⊥)transverse momentum in ev-ery such a region.Obviously,only integration over k⊥involves M as the lowest limit.Integrations over other transverse momenta are M-independent.DL contributions of the softest particles can be factorized.It allows to compose infrared evolution equations(IREE)for the scattering ampli-tudes.The most di?cult is the case when both the initial electron and the?nal quark belong to the SU(2)doublets.In order to simplify the IREE,one can use the SU(2)simmetry restored at such high energies and consider annihilation of lepton-antilepton pair into a quark-antiquark pair. After that,it is convenient to exapand the scattering amplitude into the sum of the irreducible SU(2)representations,using the standard projection operators multiplied by the invariant ampli-tudes A j,(j=1,2,3,4).At last,in order to calculate the invariant amplitudes in kinematics(1, 2),it is convenient to use the Mellin transform

A j= ?∞??∞dωκ ωF j(ω)(5)

whereκ=t,j=1,2for A j in kinematics(1)andκ=u,j=3,4when the kinematics is the u -inematics of Eq.(1).In the case of the collinear kinematics whereκ=M2,amplitudes F j obey (we consider amplitudes with the positive signatures only):

ωF j(ω)=a j+b j

dω

+

c j

4,a2=?g2+g′

2Y

1

Y2

4,a4=

g2+g′2Y1Y2

4,b2=

8g2+g′2(Y1?Y2)2

4,b4=

8g2+g′2(Y1+Y2)2

c 1=c 2=1,c 3=c 4=?1.(9)

Solutions to Eq.

(6)

can be

expressed in

terms of the

Parabolic cylinder functions D p :

F j (ω)=

a j

D p j (ω/λj )(10)

where p j =a j c j /b j and λj = 2π?e λj l (ρ?η)D p j ?1(l +λj η)

2

,h 2=?g 2+g ′2

Y l Y q 2,h 4=?g 2?g ′2Y l Y q 8π2 34g ′2 η′2

κ,is taken into account.

According to the results of Ref.[1],the amplitude for the forward e +e ?→u ˉu -annihilation,M F u is expressed in terms of amplitudes A 3,A 4of Eq.(12):

M F u =A 3+A 4

2

,(17)while the backward amplitude for this process is

M B d =A 2.(18)

We use the folloing terminology:by the forward kinematics for e +e ?→q ˉq -annihilation we mean that quarks with positive electric charges,u and ˉd

,are produced around the initial e +-beam,in the cmf,within a cone with a small opening angle θ,

1?θ≥θ0=2M

s

(19)

By backward kinematics we means just the opposite–the electric charge scatters backwards in a cone with the same opening angles.

The di?erential cross section dσF for the forward annihilation is

dσF=dσ(0) |M F u|2+|M F d|2 ≡dσ(0)[F u+F d],(20) and similarly,the di?erential cross section dσB for the backward annihilation is

dσB=dσ(0) |M B u|2+|M B d|2 ≡dσ(0)[B u+B d],(21) where dσ(0)stands for the Born cross section,though without couplings.We de?ne the forward-backward asymmetry as:

A≡dσF?dσB

F+B

(22)

where

F=F u+F d,B=B u+B d.(23) Contributions to the asymmetry from right leptons and quarks can be ealily obtained in a similar way.Then,using the standard programmes for hadronisation[3]and doing numerical cal-culations,we obtain the forward-backward asymmetry for e+e?annihilation into charged mesons and barions.The results are plotted in Fig.2.

Finally,we would like to note that when the annihilation energy is high enouph to produce,in addition to the two quark jets,W or Z bosons with energies?100GeV,it turnes out that the cross sectionσ(nγ)of n hard photon production,the cross sectionσ(nZ)of nZ boson production and the cross sectionσ(nW)of nW boson production obey very simple asymptotical relations(see Ref.[2]:

σ(nZ)

σ(nW)

～s?0.36.(25) The ratios of these cross sections as functions of the annihilation energy are plotted in Figs.3,4.

2Acknowledgement

The work is supported by grants POCTI/FNU/49523/2002,SFRH/BD/6455/2001and RSGSS-1124.2003.2.

References

[1]B.I.Ermolaev et al,Phys.Rev.D67,014017,(2003).

[2]B.I.Ermolaev et al,Phys.Rev.D66,114018,(2002).

[3]T.Sj¨o https://www.sodocs.net/doc/af4196240.html,puter Physics Commun.82,74,(1994).

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Figure3:Total energy dependence of W±to(Z,γ)rate in e+e?annihilation.

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Figure4:Total energy dependence of Z toγrate in e+e?annihilation.The dashed line shows the asymptotical value of the ratio:tan2θW≈0.28.