1 Deep-inelastic Electron-Photon Scattering at High Q 2 Neutral and Charged Current Reactio

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We present the results of a calculation of deep inelastic electron-photon scattering at a linear collider for very high virtuality of the intermediate gauge boson up to NLO in perturbative QCD. The real photon is produced unpolarized via the Compton back s

1 TTP99-30Deep-inelastic Electron-Photon Scattering at High Charged Current Reactions

Q2: Neutral and

A. Gehrmann{De Ridder a a Institut fur Theoretische Teilchenphysik, Universitat Karlsruhe, D-76128 Karlsruhe, Germany and Deutsches Elektronen-Synchrotron DESY, D-22603 Hamburg, Germany.We present the results of a calculation of deep inelastic electron-photon scattering at a linear collider for very high virtuality of the intermediate gauge boson up to NLO in perturbative QCD. The real photon is produced unpolarized via the Compton back scattering of laser light of the incoming beam. For Q2 values close to the masses squared of the Z and W gauge bosons, the deep inelastic electron-photon scattering process receives important contributions not only from virtual photon exchange but also from the exchange of a Z-boson and a W-boson. We nd that the total cross section for center of mass energies above 500GeV is at least of O(pb) and has an important charged current contribution.

1. IntroductionProcesses induced by initial state photons provide us with an interesting testing ground for QCD. As a photon can interacts directly through a pointlike coupling with quarks or through its parton content like a hadron it has a twofold nature. Its point-like interaction gives rise to perturbatively calculable short-distance contributions 1] while its hadron-like or resolved part cannot be described with perturbative methods. It is described in terms of the parton distribution functions inside the photon. These parton distributions obey a perturbative evolution equation with a non-perturbative boundary condition usually parameterized in the form of an initial distribution at some low starting scale 0 . The pointlike and resolved processes contribute to the various real structure functions entering the cross section for e+ e? ! e?+ ! l+ X, the object of this study.

The di erential Compton cross section for the process e+ ! e+ 0, 0where the polarization of the nal state photon is not observed takes the following form 2,3], d dy2 n 1= x m2 1? y+ (1? y)? 4r(1? r) 0 e o+Pe P x0 r (1? 2r)(2? y): (1)

2. The photon spectrumThe process of Compton backscattering of laser beams o highly energetic electrons/positrons offers an e cient mechanism to transfer a large fraction of the lepton energy to the photon which structure can be investigated in deep inelastic electron-photon scattering.

The energy transferred from the electron to the backscattered photon is denoted by y . Pe; P are the helicities of the incoming lepton and laser photon with?1 Pe; P 1. In the above formula, the ratio r= y= (1? y)x0] while the fractional energy of the nal state photon y is x bounded by y 1+x . The parameter x0 is de4Ew where E and w0 are the energies ned as me of the incoming electron and photon. By tuning the energy of the incoming photon, the parameter x0 can be chosen close to 4.83, just below the threshold for production of e+ e? pairs from the collision

of laser and nal state photons. To give an order of magnitude, for an energy of the incoming lepton of 250 GeV, the laser energy is of the order of 1 eV. It is worth noting that the photon luminosity for backscattered photons is of the same order as the initial electron luminosity. It is therefore enhanced compared to the photon luminosity ob0 2 0 0

2 tained for photon produced by bremsstrahlung o the lepton which is of O( ). Furthermore, by choosing the helicities of laser and electron beams to be opposite, the photon spectrum is peaked at high energies. In this region, about 80% of the energy of the electron/positron beam can be transferred to the backscattered photon. In the following we shall give our predictions for the most advantageous photon spectrum and will therefore consider the backscattered photon obtained for opposite polarizations of electron and laser photon, with x0= 4:83. Our results will be obtained for the cross section e+ e? ! l+X which is related to the cross section for e? ! l+ X as follows, Z d (2) d e e? !l+X= dy 1 dy c d e !l+X: c++

while in the charged current di erential cross section enter the x and Q2 dependent structure functions F2;CC, FL;CC and F3;CC multiplied by weak propagator and coupling factors as follows? d2 CC (e+ !+X )= dxdQ2 Q4 (1 P ) 1 2 2 4 Q2+ MW]2 sin4 w 4 xQ h (1+ (1? y)2 )F2;CC( )

(1? (1? y)2 )xF3;CC? y2FL;CC: (4)

MW and W are the W-mass and the weak Wein-

At momentum transfer squared Q2 close to the Z and W masses squared, the deep inelastic cross section for e+ ! l+ X becomes sensitive to contributions from these exchanges 4]. Neutral current deep inelastic events are characterized by the exchange of a photon or a Z boson, and have an electron and a jet (or jets) in the nal state. Charged current events arise via the exchange of a W-boson and contain an undetected neutrino and a jet (or jets) in the nal state. The presence of an neutrino is as usual detected as missing transverse momentum. Having these signatures, those processes constitute potential background sources for searches for new physics at the future e+ e? linear collider. To give a prediction for the size of these contributions is precisely the aim of this study.

3. Deep-Inelastic Electron-Photon scattering

berg angle. P denotes the degree of left-handed longitudinal polarization (P=?1 for left-handed electrons, P= 1 for right-handed electrons). The kinematics of these neutral and charged current reactions is described by the Bjorken variables x and y. Those can be expressed in terms of the four-momentum transfer, Q2, the hadronic energy W, the lepton energies and scattering angles. Indeed we have,2 Q2 x= 2q:p= Q2 Q W 2+ q:p= 1? E cos2 (=2): y= k:p E0

(5)

3.1. Kinematics

The neutral current di erential cross section is parameterized by the x and Q2 dependent structure functions F2;NC and FL;NC, d2 NC (e+ ! e+ X )= dxdQ2 2 2 h(1+ (1? y)2 )F;NC? y2 F;NC i (3) 2 L xQ4

e initial state photon is real and on-shell and the lepton masses can be set to zero. Experimentally, one needs to constrain the lepton scattering angle e to be larger than some minimum angle min . Equivalently, one can also require, 4 2 (1 o )xse Q2> xs (1E c )? c4E 2 (1? c ) e+ o+ o 2> Q2 (x; se; co ) Q (6) with co= cos min and se the electron-photon center-of-mass energy squared. This constraint on the 4-momentum transfer Q2 is shown as a p function of x in Fig. 1 for se= 500 GeVand for min= 5; 10; 20 degrees. The corresponding allowed kinematical regions are then situated between Q2= Q2 (x; se; co) and Q2= xse (corresponding to y= 1).

Q2106105√seγ = 500 GeV

104y = 1

Θ0

min = 20

Θ0

min = 10

103

Θmin = 50

102

y = 0.01

10-410-310-210-11

4 For the scattering of an e? with the photon, the charged current structure functions F2;CC and F3;CC at the lowest order are respectively given by, F2;CC= x u(x; Q2 )+ c(x; Q2 )+d(x; Q2 )+ s(x; Q2 ) F3;CC= u(x; Q2 )+ c(x; Q2 )?d(x; Q2 )? s(x; Q2 ): (11) For e+ scattering, one has to make the following replacements: u(x; Q2 )+ c(x; Q2 ) ! u(x; Q2 )+ c(x; Q2 ) and d(x; Q2 )+ s(x; Q2 ) ! d(x; Q2 )+ s(x; Q2 ). The CKM matrix is approximated by the unity matrix, avour mixing e ects can here be neglected. At the next-to-leading order, corrections proportional to s log(Q2 ) have to be included in all the above leading-order expressions. To the terms proportional to the parton distributions themselves, terms involving convolution of these with quark (Cq ) and gluon (Cg ) coe cient functions have to be taken into account. For example, for the light avours, F2;NC given at leading order in eq.( 9) becomes in the MS scheme at the nextto-leading order

4. Results and conclusionAs mentioned before, we shall present our results for xed values of Q2, namely Q2= min min 10000 GeV2 and Q2= 1000 GeV2 . Since min the leading and next-to-leading order results are found to be very close to each other, all results will be given at the next-to-leading order level only. The NLO corrections being at most of the percent level indicates furthermore that the obtained results are perturbatively stable. In Figs. 2 and 3 we present the di erent contributions to the total NC and CC cross sections as a function of the electron-positron p center of mass energy, the latter varying between s= 200 GeV p and s= 2000 GeV for virtualities of the intermediate gauge boson equal to 10000 GeV2 and 1000 GeV2 respectively. The cross sections being inversely proportional to Q4, their value is dominated by the smallest Q2 values. As can be seen in these gures, for Q2= 10000 GeV2, the largest cross section, min the total charged current cross section is of O(pb) while for Q2= 1000 GeV2, the dominant neumin tral current cross section is about 10 times larger. For this latter choice of Q2 the charged curmin rent cross section is approximately a third of the neutral current cross section. By these high virtualities of the intermediate gauge boson, the exchange of a W-boson gives always rise to signi cantly high cross sections. The contribution arising from

the exchange of a Z-boson is the smallest for both choices of Q2 . min The di erential cross p sections with respect to x, which are shown for s= 500 GeV in Figs. 4 and 5 di er not only in magnitude but also in shape. This can be understood as follows. For a given value of x, the allowed phase space regions in the (x; Q2 ) plane corresponding to the two values of Q2 chosen here, di er signi cantly from min each other. As can be seen in Fig. 1, this results in an enhanced importance of the small x region for smaller Q2 . min Finally, we also present the di erential cross section with respect to Q2 for Q2 values varying between 1000 GeV2 and the electron-positron p center-of-mass energy squared s, for s= 500 GeV. As can be seen in Fig. 6 the neutral

F2;NC= x+ 2

s (Q2 ) h

e2 q(x; Q2 )+ q (x; Q2 )^q q(x; Q2 )+ q (x; Q2 )i

+Cg g(x; Q2 )+ C;2:

(12)

At this order, we use the beyond-leadinglogarithmic (BLL) GRV 5] massless parton distributions q(x; Q2 ) and g(x; Q2 ). These are given in the DIS factorization scheme, de ned by, (13) q(x; Q2 )DIS= q(x; Q2 )MS+ 2 C;2: As a consequence, in this DIS factorization scheme the direct term C;2 is absent from the expression of F2;NC . The gluon distribution g(x; Q2 ) remains una ected by the change of fac-

torization scheme. The precise expressions of the other neutral and charged structure functions at the next-to-leading order will be given in 6].

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1.5b

pNLO

[

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2NCV

G1NC (Z0 only)

2Q0.5

(

totσ

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