Gluons. ZZ γz z

Transcription

1 3 Gluons WW ZZ γz γγ

2 5 down quars 4 3 WW γγ ZZ γz

3 5 4 FF 3 ESF FF ESF

4 QCD STRUCTURE OF LEPTONS BNL-684 Wojciech S LOMI NSKI and Jery SZWED y Physics Department, Broohaven National Laboratory, Upton, NY 973, USA and Institute of Computer Science, Jagellonian University, Reymonta 4, 3-59 Kraow, Poland The QCD structure of the electron is dened and calculated. The leading order splitting functions are extracted, showing an important contribution from -Z interference. Leading logarithmic QCD evolution equations are constructed and solved in the asymptotic region where log behaviour of the parton densities is observed. Corrections to the naive evolution procedure are demonstrated. Possible applications with clear manifestation of 'resolved' photon and wea bosons are discussed. Wor supported by the Polish State Committee for Scientic Research (grant No. P3B 8 9) and the Volswagen Foundation. y J. William Fulbright Scholar. Permanent address.

5 The QCD component of the photon ('resolved' photon) calculated some time ago [] is now clearly visible in high energy experiments []. Theoretical analysis of the quar-gluon component has been recently extended to wea bosons [3] demonstrating new features of the parton densities of 'resolved' W and Z the calculated densities show strong spin and avour dependence. In the experiment however one has rarely real wea bosons at one's disposal and in most experiments even the high energy photons are 'nearly on-shell' only. In physical processes, where the structure of electrowea bosons may contribute signicantly, it is the lepton, initiating the process which is the source of the intermediate bosons. The standard procedure applied in such cases is to use the equivalent photon approximation [4] (extended also to the case of wea bosons [5]) and, as a next step, to convolute the obtained boson distributions with parton densities inside the bosons. For example, the parton density inside the electron F e would read: F e (; ^Q ; P ) = B B ( ^Q ) F B (P ) () where (F G)() R dx dy ( xy)f (y)g(x), P is the hard process scale, ^Q is the maximum allowed virtuality of the boson (usually taen to be proportional to P ) and - the momentum fraction of the parton with respect to the electron (detailed denitions follow). In such an approximation several questions arise: how far o-shell can the intermediate bosons be, how large are their interference eects, what are the energy scales governing the consecutive steps. It can be easier and more precise to answer the direct question: what is the quar and gluon content of the incoming lepton or, in other words, what is the lepton structure function? In this letter we address the above problem nding several corrections to the standard procedure. Let us consider inclusive scattering of a virtual gluon o an electron. In the lowest order in the electromagnetic and strong coupling constants ( and s ) the electron couples to q-q pair as shown in Figure. The incoming electron e carries 4-momentum l and the o-shell gluon G of 4-momentum p with large P p, serves here as a probe of the electron. In the nal state we have a massless quar q and antiquar q of 4-momenta and and lepton ` (electron or neutrino) of 4-momentum l. The exchanged boson B = f; Z; W g carries 4-momentum q (Q q ). The current matrix element squared for an unpolaried electron reads: where J (e G! ` q q ) = J (l; p) = Z d l d d () 4 4 ( + + l p l) 4 he jj()j`q y q ih`q q jj ()je i ; () d = d4 () 4 ( ) : (3) For massless quars we can decompose the current in the helicity basis: J (l; p) = () (p) J (l; p) ()(p) ; (4) where () (p) are polariation vectors of a spin- boson with momentum p = (p ; ; ; p ): = p (; ; i; ) ; (5) s (q) = jp j (p ; ; ; p ) : (6)

6 In a frame where ~q is antiparallel to ~p contributions from dierent helicities of exchanged bosons do not mix and the current reads: J (p; l)= s Z dy g Aq gbq y P A e B (y) Z Q max Q min A;B; Q dq (Q + M A)(Q + M B) H (x; Q ) ; (7) where PA e B (y) describes wea boson emission from the electron and H(x; Q ) q-q pair production by virtual gluon and electrowea boson. g Aq is the boson A to quar q coupling in the units of proton charge e. The sum runs over the electrowea bosons (A; B = ; W ; Z) and their polariations = ;. Note that although the sum is diagonal in the polariation index it is not in the boson type A; B. The o-diagonal terms in the sum arisefrom the -Z interference. As demonstrated below their contribution is substantial. To answer our main problem of 'an electron splitting into a quar' we tae the limit Q P and eep the leading terms only. Within this approximation the inematic variables x; y; read y = pq pl ; p = xy = pl ; (8) aquiring the parton model interpretation of the quar momentum fraction () and of boson momentum fraction (y), both with respect to the parent electron. The leading term of the hadronic part does not depend on quar helicity : H (x; Q ) = x log P Q ; (9) H (x; Q ) = ( x) log P Q ; () with other components nite for P =Q!. We also recognie PA e B (y) as a generaliation of the splitting functions of an electron into bosons [3]: with P e A B (y) = (gae gbe yy + g+ Ae gbe + yy ) () Y + (y) = y ; () Y (y) = Y (y) = ( y) y ; (3) ( y) y ; (4) where g Ae is the electron to boson A coupling in the units of proton charge e. From inematics y [; O(m e =P )] and the integration limits for Q read Q min = m e y y ; Q max = P + y y ; (5)

7 with m e being the electron mass. Although smaller than the already neglected quar masses, it is the electron mass which must be ept nite in order to regularie colinear divergencies. The upper limit of integration requires particular attention. In general it is a function of P, however integration up to the maximum inematically allowed value Q max would violate the condition Q =P. For our approximation to wor we integrate over Q up to ^Q max = P where and generally depends on y and. A similar condition is in fact used in phenomenological applications of the equivalent photon approximation [6]. Integrating Eq.(7) over Q within such limits and eeping only leading-logarithmic terms leads to J (p; l) = s 6 A;B; g Aq gbq A B (P ) [P q ; + P q ; ] log P ; (6) where (F P )() R dx dy ( xy)f (y)p (x). P q (x) and P q (x) are boson-quar (-antiquar) splitting functions [,3] P q (x) = P q (x) = 3x ; P q (x) = P q (x) = 3( x) (7) and A B (y; P ) is the density matrix of polaried bosons inside electron. Its transverse components read where W is the Weinberg angle and (y) = ( y) + y log ; FZ e +Z + (y) = tan W ( y) + W y log Z ; FZ e Z (y) = tan W + ( y) W y log Z ; +Z + (y) = tan W Z (y) = tan W W +W + (y) = W W (y) = W ( y) y W ( y) y ( y) 4 sin W y 4 sin W y log W ; W = log = log P m e log Z ; log Z ; log W ; (8a) (8b) (8c) (8d) (8e) (8f) (8g) sin W ; (9) ; log B = log P + M B MB : () All other density matrix elements (containing at least one longitudinal boson) do not develop logarithmic behaviour. It is natural to introduce at this point the splitting functions of an electron into a quar at the momentum scale P as 3

8 Pq e (P ) = AB g Aq gbq A B (P ) P q : () The expicit expressions for quars read P e q + (; P )= 3 4 fe q [ + () + ()] log +e q tan4 W h + () + W () i log Z e qtan W [ + () + W ()] log Z g ; Pq e (; P )= 3 4 fe q [ + () + ()] log + qtan 4 W h () + W + () i log Z (a) +e q q tan W [ () + W + ()] log Z +( + W ) + () qd log W g ; (b) where and + () = 3 ( + + ) + ( + ) log ; (3) ( )3 () = ; 3 (4) q = T q 3 sin W e q ; (5) with e q and T q 3 being the quar charge and 3-rd wea isospin component, respectively. The splitting functions for antiquar of opposite helicity can be obtained from Eq.() by interchanging + with. The splitting functions introduced above show two new features. The rst one, already mentioned before, is the contribution from the interference of electrowea bosons ( and Z only). The second is their P dependence, which arises from the upper integration limit ^Q max. The above splitting functions, when cast into the evolution equations, produce nontrivial eects in the P -dependence. We consider the evolution equations in rst order in electrowea couplings and leading logarithmic in QCD. Introducing t = log(p = QCD) one can write the standard evolution equations for the density of polaried QCD partons i (quars, antiquars and gluons): dfi e (t) = dt B; + s(t) P B i ; P i B (t) (t) ; (6) where P i are the standard QCD parton splitting functions [8]. The rst sum runs over electrowea bosons while the second one over QCD partons. We immediately recognie Eq.() to be the generaliation of the rst term in the above equation. Remembering that there is no direct coupling of the electrowea sector to gluons we arrive at the master equations for the parton densities inside the electron: 4

9 dfq e (t) dt d G (t) dt = Pe q (t) + s(t) = s(t) ; ; P q (t) (7a) P G F e (t): (7b) We stress that the convolution of the equivalent boson distributions and boson-quar splitting functions, Eq.(), occurs at the level of splitting functions. It is not equivalent to the usually performed convolution of the distribution functions Eq.() because of the P dependence of the boson distribution functions Eq.(8). Only in the case when the upper limit of integration ^Q max is ept xed (P -independent), e.g. by special experimental cuts, are the convolutions equivalent at both levels. The equations Eq.(7) can be solved in the asymptotic t region where we approximate the strong coupling constant s (t) ' bt ; (8) with b = = n f =3 for n f avours. The asymptotic (large t) solution to Eqs.(7) for the parton of polariation can be now parametried as F e (; t) ' f as () t (9) resulting in purely integral equations f as i = ^P e i + b ; P i f as ; (3) where ^P i e () are given by Eqs.() with all log A. Numerical solutions to the above equations, with the use of method described in Ref. [3], are presented in Figure for the unpolaried quar and gluon distributions. One notices signicant contribution from the W intermediate state in the d-type quar density (this would be even more pronounced if we looed at the left-handed d quars). The most surprising however is the -Z interference contribution which cannot be neglected, as it is comparable to the Z term. It violates the standard probabilistic approach where only diagonal terms are taen into account. This also stresses the necessity of introducing the concept of electron structure function in which all contributions from intermediate bosons are properly summed up. The asymptotic solutions to the polaried parton densities q = q + q and G = G + G are given in Ref. [7]. Due to the nature of wea couplings they turn out to be nonero, even in the case of gluon distributions. Again the -Z interference term is important and the W contribution dominates in the asymptotic region. One should eep in mind that at nite t the logarithms multiplying the photon contribution dier from the remaining ones (Eq.()). Being scaled by m e, they lead to the photon domination at presently available P. The importance of the interference term remains constant relative to the Z contribution, as they are both governed by the same logarithm. But even at presently available momenta, where the `resolved' photon dominates, one can see how the correct treatment of the scales changes the evolution. In Fig. 3 we present the asymptotic solutions of the evolution equations following from our procedure (ESF) compared to those following from naive application of the convolution Eq.() (FF). It is possible that the dierence can be traced in the analysis of presently available data. This question is currently under study. 5

10 To summarie we have presented a construction of the electron structure functions within leading logarihtmic approximation to QCD and leading order in electrowea interactions. The -Z interference, contrary to naive expectations, turns out to be important. Direct calculation of the splitting functions of an electron into quars allows for precise control of the momentum scales entering the evolution. It also shows that the convolution of leptons, electrowea bosons and quars should be made at the level of splitting functions rather than distribution functions. Unless forced otherwise by the experimental cuts, the electron splitting functions depend on the external scale P and inuence signicantly the parton evolution. Phenomenological applications of the above analysis require very high momentum scales in order to see the wea boson and interference contributions. Possible processes where these eects could show up include heavy avour, large p? jet and Higgs boson production in lepton induced processes. At lower momenta, where the photons dominate, the use of the electron structure function allows to treat correctly the parton evolution. Acnowledgements. The authors would lie to than the Theory Groups of Broohaven National Laboratory and DESY for their hospitality. References. [] E. Witten, Nucl. Phys. B, 89 (977); C.H. Llewellyn-Smith, Phys. Lett. 79B, 83 (978); R.J. DeWitt et al., Phys. Rev. D9, 46 (979); T.F. Walsh and P. Zerwas, Phys.Lett. 36B, 95 (973); R.L. Kingsley, Nucl. Phys. B 6, 45 (973). [] M. Derric et al., ZEUS Collaboration, Phys. Lett. B97, 44 (99); ibid. B3, 87 (994); ibid. B445, 47 (995); T. Ahmed et al., H Collaboration, Phys. Lett. B97, 5 (99); Nucl. Phys. B445, 95 (995); I. Abt et al., H Collaboration, Phys. Lett. B34, 436 (993); [3] W. S lominsi and J. Swed, Phys. Lett. B33, 47 (994); Phys. Rev. D5, 65 (995); The QCD Structure of W and Z Bosons at Very High Energies, Proc. of Int. Symp. on high Energy Spin Physics, Nagoya 99; [4] C. Weisacer and E.J. Williams, Z. Phys. 88, 6 (934). [5] G.L. Kane, W.W. Repo and W.B. Rolnic, Phys. Lett., 48B, 367 (984); S. Dawson, Nucl. Phys. B49, 4 (985). [6] see e.g. T. Sjostrand, PYTHIA at HERA, in Proc. of the Worshop on Physics at HERA, Hamburg, October 99, ed. W Buchmuller and G. Ingelman. [7] W. S lominsi and J. Swed, On the QCD Structure of Electron, BNL preprint, to be published in Acta Physica Polonica B. [8] G. Altarelli and G. Parisi, Nucl. Phys. B6, 98 (977). 6

11 l l l q p l q p FIG.. Lowest order graphs contributing to the process: e + G! ` + q + q. 7

12 5 down quars 4 3 WW 3 γγ ZZ γz up quars γγ 3 ZZ WW γz Gluons WW ZZ γz γγ FIG.. Unpolaried quar and gluon distributions f as () solid line. The other lines show contributions from dierent electrowea bosons. 8

13 5 4 FF 3 ESF FF ESF FIG. 3. Comparison of unpolaried d-quar distributions f as () calculated by ESF and FF methods. The upper two lines result from contributions from all electrowea bosons while the lower two from only. 9

14 l l l q p l q

15 3 up quars γγ ZZ WW γz