has been resoved within covariant approach been proposed in [1]: the RC istobe nay free from dependence of any unphysica parameter ike "photon softnes

Transcription

1 Eectroweak Radiative Eects in Deep Ineastic Interaction of Poaried Leptons and Nuceons I. V. kushevich,.n. Iyichev and N. M.Shumeiko Nationa Scientic and Education Center of Partice and igh Energy Physics attached to Byeorussian State University bstract The resuts for one-oop correction to deep ineastic scattering of ongitudina poaried eptons on ongitudina poaried hadrons are obtained withing the framework of the standard theory of eectroweak interactions and ordinary uark-parton mode. The on-she renormaiation scheme in t'ooft-feynman gauge is appied. The numerica anaysis is carried out under conditions of modern partice physics experiments. Particuar emphasis is aid on contributions usuay ignored at RC procedure { eectroweak corrections to eectromagnetic asymmetry and RC to hadronic current. The structure of RC contribution to poaried asymmetries within the framework of QED and eectroweak theory is aso discussed. 1 Introduction Main sources of an information about spin properties of nuceons are experiments on deep ineastic scattering (DIS) of poaried epton by poaried targets [1]-[4]. Resuts of the ast ones []-[4] form an inconsistent picture of nature of nuceon spin (see [5] and reference therein). This stimuates reaiation of new experiments [6, 7] on measurement of proton and neutron spin-dependent SF g p;n 1 and g p;n and on testing of Bjorken [8], Eis-Jae [9] and Burkhardt-Cottingham [1] sum rues. n interpretation of experimenta data reuires an adeuate cacuation of background radiative eects (RE). The one-oop contribution to cross section of DIS of non-poaried partices within the framework of uantum eectrodynamics (QED) was widey discussed in iterature (see, for exampe, [11]-[13]). Resuts of modeindependent part of the one-oop QED contribution and mode-dependent RE in hadronic current (hadronic RE) in DIS of poaried partices can be found in [14, 15]. In the present report the resuts for one-oop radiative correction (RC) to DIS of poaried epton on poaried proton are obtained within the framework of the standard theory of eectroweak interaction and ordinary parton mode. e foow to the on-mass renormaiation scheme in t'ooft-feynman gauge [16] (see aso the reviews [17]-[]) and use the resuts of cacuation for the case of non-poaried partices [1, ] 1. e note, that the probem of infrared divergence canceation 1 Some estimates of the one-oop contribution in DIS of poaried partices can be found in artice [3] 1

2 has been resoved within covariant approach been proposed in [1]: the RC istobe nay free from dependence of any unphysica parameter ike "photon softness". In section we discuss one-oop RC to DIS of poaried partices. It is shown, that RC can be separated by eading-og contribution and contribution independent of partice masses. In this report we give the expicit form ony for eading og contribution. In section 3 infrared free one-oop RC is studied numericay. e study the structure of contributions in poariation asymmetry, cacuate the magnitudes of eects normay missed at RC procedure of experimenta data and anayse RC to SF g 1 (x; Q ) and to sum rues. One-oop RC e consider DIS of ongitudinay poaried eptons on ongitudinay poaried nuceons `(k 1 ;)+N(p; )! `(k )+ (1) with taking into account the one-oop RE within parton mode. The vectors in brackets designate a momentum and poariation of appropriate partices (k 1 = k = m, p = M ). compete set of Feynman graphs is presented in gure 1. The resut can be written in a form of sum of the Born contribution, R- and V-contributions (the rst ine of graphs in gure 1 and the ast ones) = + V + R ; () def where ;V;R = d ;V;R =dxdy, and x, y are scaing variabes. The on-she renormaiation scheme of eectroweak theory is submitted in the reviews [16, 17] and is advanced in [18]-[]. This scheme uses an eectrica charge and masses of partices as physica parameters. Both sef-energies together with a compete set of renormaiation constants and vertex functions of fermions were cacuated in [16]. Renormaiation formuae for sef-energies of vector bosons, vertex functions and graphs of two boson exchange were aso presented. resuts of [16] are given in a form convenient for further appications to RC cacuation for fermion processes, incuding DIS. e use these resuts for cacuation of V-contribution. V-contribution can be written as a sum of the contributions of sef-energies of vector boson and uark ( B S, S), f vertex functions of epton and hadron ( V, V ) and contributions of two boson exchange ( box ): V = B S + f S + V + V + box : (3) s usua wekeep the eading contributions (containing mass singuarity nm ) and next-to-eading ones (without any mass dependence). In this report we give the expicit form ony for eading contributions (for more detais see [4]).

3 y y y y J J J J J J J J Figure 1: compete set of eectroweak graphs, contributed to epton-uark scattering within uark-parton mode. The doube ine corresponds to the contribution of - or -exchange. possibe graphs, which give the contribution to vacuum poariation, are designated by symbo [18]. 3

4 The mass singuarity in the V-contribution due to smaness of mass of partices participating in a scattering, is contained ony in the correction to vertex: V = 1(Q ;m ) ; V = e 1 (Q ;m ) : ere e is uark charge, and = = ij. The function 1 (Q ;m ) is ij= given by the formua (B.3) of [16]. In our case it has the form! 1 (Q ;m )=n Q 1+n Q : (5) m m (4) Sef-energies of vector bosons B S = ( + ) + S (6) have mass singuarity due to smaness of partice mass appearing in oops of vacuum poariation. The uantities ;; are poariation operators and S was found in [4]. For simpicity wegive expressions for vacuum poariation by eptons ony. For the various poariation operators (6) we have = = = =e;; =e;; =e;; (v ) n Q ; v v c w s w m c w s w (v ) +3(a ) (v ) (a ) (v ) +3(a ) c w s w s w (v ) +(a ) n Q ; m n Q ; m where v and a are vector and axia couping constants of eptons, and c w and s w are cosine and sine of einberg ange. Resut for the contribution of process with radiation of rea photon `(k 1 ;)+(p; )! `(k )+(k)+ (8) (7) to observed cross section of DIS can be written in the form R = Q n J(Q ; ) + + R ; (9) where R = ij=; n ij +^ ij + e ij h + e ij h o +^ ij h : (1) 4

5 Low index (b = ; h; h) of cross sections in right side of the euation corresponds to contributions of radiation by eptons, by hadrons and their interference. Ony ij and ij h content eading contributions. The infrared divergence is extracted by the method of Bardin and Shumeiko [1]. It is competey contained in the rst term of expression (9) and is canceed with an appropriate term of V-contribution. e distinguish three kinds of mass singuarities: eptonic (n Q =m ), uark (n Q =m ) and nuceon (n Q =M ). Such singuarities are contained in uantity : where = 1 m + m ( v + sx +1)+e 1 ( ( v )); (11) 4 m =n Q m ; =n Q ; m v =n 1 x x ; sx =n y 1 y ; (1) and in ;h, ij which are considered beow. Firsty we consider dependence on eptonic mass m, which is contained ony in the contribution of radiation by eptons ij. Resut in standard eading og form is obtained by spitting the cross section by the contributions appearing from k 1 - and k -peaks [5]: and k 1 R k R = m = m ij ij = k 1 R + k R ; (13) ( yd k y 1 1 yd 1+y ( 1+ (1 ) k ere the cross sections k 1; are obtained from born one by repacements k 1 = 1 S; xy y ; 1 1+y 1 The ow imits of integration are eua to ) ; 1 1 ) : 1 (14) (S; x; y) (15) ; k = S; xy 1+y ; 1+y : (16) 1 =(1 y)=(1 xy); =1 y+xy: (17) The dependence on uark mass is retained ony in the contributions of radiation by hadrons and in the eptonic QED correction. For the contribution of hadronic radiation ij h we obtain = ij e ij h f (x)! f rad (x) ; (18) 5

6 where f rad (x) =e 1 + x f (x)( + v ) d 1+ 1 f (x=) 1 f (x) : In the case of eptonic eectromagnetic radiation the appearing of uark mass has a purey kinematic origin, and it can be repaced with proton mass in according to the rue of parton mode: m = M. It foows from comparison of resuts for eectromagnetic radiation in parton mode and ones obtained by a mode independent way. In this case we have (19) = 3 y 4 1 Q d n M x ft +MR V F V ()+T M R F ()g ; () where T M = 1 (1 y) Sy(1 y) 1 (1 x=) x : (1) Thus, sef-energies, vertex functions and the contributions of radiation by eptons and by hadrons have a mass singuarity and therefore are signicant. Neither the epton-hadron interference in bremsstrahung nor the contributions of two boson exchange contain any mass singuarities. In this sense we speak, that the eptonic and hadronic corrections are separated. Radiation by eptons contains both eptonic and nuceon mass singuarity. Extraction of contribution containing the eptonic mass in a separate term eads to peaking and eading og approximation. The contribution of a nuceon mass singuarity corresponds to t-peak when a rea photon is radiated parae to virtua one. This contribution is not extracted by methods of the eading ogarithms. The uark mass singuarity in purey hadronic radiation is reduced to the correction to parton distributions (19). In the eading og approximation this resut for unpoariation and poariation cases was received in refs. [6, 7]. 3 Numerica anaysis In this section compete one-oop RC to various observabes in DIS of poaried partices is anayed numericay in a wide range kinematic variabes. The specia attention is paid to the contributions, which are usuay negected at the data anaysis in modern poariation experiments: to the eectroweak corrections to eectromagnetic asymmetry and eects of radiation by hadrons. Important uestion on the structure of the contributions in poariation asymmetry in QED and eectroweak theory is discussed as we. 6

7 3.1 Structure of contribution to poariation asymmetry The cross section of DIS both at a born eve and at a eve of the radiative corrections can be written as foows: = a + P L + P N + P L P N : () ere a is unpoariation cross section, and three other terms give poariation contributions. P L and P N are poariation degrees of epton and hadron. Let us dene the foowing uantities: = a ; and consider poariation asymmetry = a ; = a (3) = "" "# "" + "# = + 1+ : (4) e note, that euas to in eectrodynamics. By using as an exampe we anayse magnitudes of the various one-oop contributions in asymmetry. Such anaysis is convenient to conduct by expansion with the account RC in a series over couping constants: = + P i i a + P i i a = + 1 a i ( i a a i )+O( ); (5) where uantities with an index "" are the born contributions. The sum on i corresponds to separating of the one-oop correction into the contributions: eects of vacuum poariation, corrections to vertex, two boson exchange and bremsstrahung of photon (9). >From (5) we see if for any cross section a; i a uantity i a a i (6) is eua to ero, then it does not give the contribution to poariation asymmetry. In QED the spin average and spin dependent parts of one-oop cross section can be written as a; = a; +( QED + F QED V + a; ) a; + a; R : (7) The uantities a; describe the contribution of two-photon exchange and are agreed with ones considered in [13, 15]. symbo "QED" indicates that ony QED eects are retained in corresponding uantities. Expansion (5) in this case gives = (1 a + )+( R a a R )= a + O( ): (8) 7

8 mong a terms of V-contribution ony sma (without mass singuarity terms) eects of two-photon exchange contribute to poariation asymmetry. R-contribution has a eading og term and dominates in (8). The ogarithmic correction to poariation asymmetry has been cacuated in [7]. In the eectroweak theory the contributions of vertex functions and sef-energies are not factoried in front of born section and do not vanish in combination (6). That is vaid for eading og contributions of poariation operators (7) and next-toeading terms of vertex functions. Thus, in contrast to QED, where ony bremsstrahung contributes to poariation asymmetry, in eectroweak theory V-contribution is aso signicant. 3. Eectroweak eects and radiation by hadrons Cross section of DIS with taking into account the compete one-oop correction can be presented by = bb : (9) b B The index B = ; ; I corresponds to the contributions -, -exchange and their interference, and b = ;;h;i to born contribution, eptonic, hadronic correction and epton - hadronic interference. e note, that in modern poariation experiments the procedure RC takes into account ony contribution, and the systematic error due to RC incudes ony an error of cacuation of the eptonic QED correction. Beow we consider the eectroweak eects at born and one-oop eves and contributions of radiation by hadrons. For convenience of the numerica anaysis we dene the next cross sections: = B ; had = + b ; B=;I; ep = + B=;I; B ; had b=;i;h = + b=;i;h B=;I; In each of them in addition to the contributions and/or one of the foowing eects are added: QED radiation by hadrons ( had ) and -exchange: at a born eve ( ), at bremsstrahung by epton ( ep ) and hadron ( had ). Using (3), we construct of asymmetry (4), had, ep and had. In the range of sma x the basis contribution at an one-oop eve is given by the QED correction, incuding correction to hadronic current. The contribution of the graphs with -exchange is insignicant both at a born eve ( ) and on a eve of RC ( ep, had had ). In the range of high x eectroweak eects dominate and the compete correction to asymmetry is dened basicay by an eectroweak interference ( ). The inuence of RE decreases with growth x: so, for exampe, the part of RE becomes smaer 1% for E 1 = 1 at x :8, and for E 1 = 1GeV aready at x :15. bb ; (3) 8

9 x y E 1 = 1 GeV E 1 = 1 GeV B ew h h B ew h h Tabe 1: Corrections B, ew, h, h in SMC kinematics [3]. e aso study numericay the next reative corrections: B = ; h = had ; ew = ep ; h = had : B is constructed from born asymmetries and gives the correction due to eectroweak interference to purey eectromagnetic born contribution. The other uantities are investigated in comparison with an one-oop mode independent part of correction. Thus, the corrections (31) (tab. 1) give insight on vaues of eects, not incuded by the usua QED procedure RC. The tabe aso iustrates a dependence of discussed uantities on energy of scattering epton. e note, that if at existing energies of the correction (31) does not exceed 3-5 %, these eects can not be ignored in future experiments with energies up to 1 TeV. 3.3 QED correction to g 1 (x; Q ) and sum rues RC procedure based on mode independent exact formuae for the owest order RE and described in [8] gives as a resut SF g 1 (x; Q ) with taking into account ony mode independent eects. Mode dependent eects (eectroweak eects, radiation by hadrons) are ignored. In this case the eects shoud be taking into account for subseuent anaysing of the SF in parton mode: where (31) g 1 (x; Q )=g 1(x)+g QCD 1 (x; Q )+g QED 1 (x; Q ); (3) g 1(x) = 1 e f( ) (x); (33) and g QCD 1 (x; Q ) and g QED 1 (x; Q ) are QCD and QED correction to it. In the report we discuss the QED eects which arise from rea photon radiation by hadrons and uark vertex function. naogousy we obtain for each uark avour j j = j+ QCD j 9 + QED j ; (34)

10 where each j = R 1 dx(f ( ) j (x) + f ( ) j (x)). e have to take into account the RC to the observabes g 1 (x; Q ), j and to obtain the uantities g1(x), j as a experimenta resuts. By consideration of eading and next-to-eading contribution we have for g QED 1 (x; Q ) g QED 1 (x; Q )= 4 and for QED j e 4 1 d + (1 ) x 3 + v v 7 v 5 =3 f ( ) (x) h (1 + )( n (1 ) 4) + 5 +n i f ( ) (x) 1 f ( ) ( x ) QED j = 9e 4 j : (36) The correction does not contain a eading contribution and is sma. By appying this resut to EMC resuts [] (35) u =:78; d = :47; s = :19 (37) obtained for Q = 1:7GeV and recacuated with taking into account QCD j we nd u =:78; d = :47; s = :19: (38) References [1] guard M.J. et a. Phys. Rev. Lett v.37. p.161, Phys. Rev. Lett v.41. p.7; Baum G. et a. Phys. Rev. Lett v.45. p., Phys. Rev. Lett v.51. p [] shman J. et.a. Nuc. Phys v.b38. p.1. [3] deva B. et a. Phys. Lett v.b3. p.533. [4] nthony P.L. et a. Determination of the Neutron Structure Function. // SLC- PUB-611(1993) [5] Jae R.L. and Manohar. Nuc. Phys v.b337. p.59. [6] Technica design report. The ERMES coaboration. // DESY-PRC 93/6, MPTI-V [7] Brock R., Brown S.N., Montgomery.E., Corcoran M.D. Fixed target eectroweak and hard scattering physics. // FERMILB-conf [8] Bjorken J.D. Phys. Rev v.148. p.1467, 197. v.d1. p [9] Eis J., Jae R.L., Phys. Rev v.d9. p.1444; v.d1. p (E). [1] Burkhardt. and Cottingham.N. nn. Phys v.56. p.453. [11] Mo L.. and Tsai Y.S. Rev. Mod. Phys v.41. p.5. 1

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