- Email: [email protected]
Radiative corrections to eγ scattering
cu .__
2 May 1996
!!&I ‘Es
PHYSICS
LETTERS B
Physics Letters B 374 (1996) 217-224
EJ-SETIER
Radiative corrections to ey scattering Eric Laenen a~1,Gerhard A. Schuler bT2 a CERN TH-Division, 121 I-CH Geneva 23, Switzerland b Theoretische Physik, Universitiir Regensburg, D-93053 Regensburg, Germany Received 1 September 1995; revised manuscript received 22 January 1996 Editor: R. Gatto
Abstract We investigate tbe effects of photon radiation on deep-inelastic ey scattering. Depending on the set of variables chosen, we find appreciable effects in the kinematic region accessible at LEP2. Convenient analytic results for O(cu) corrected differential cross sections are presented.
1. Introduction In the period leading up to the start of the HERA program a substantial amount of work was done on radiative corrections to observables in deep-inelastic scattering (DIS) of electrons off protons (see e.g. [ 1 I for an overview). These corrections are substantial enough for both HERA experiments to correct for them. Part of the upcoming LEP2 physics programme involves the measurement of the photon structure function Fz (x, Q2). This structure function is extracted from deep-inelastic electron-photon scattering in the reaction e+e- + efe-X, where one of the leptons escapes undetected down the beam pipe, while the other is measured at rather large angle. It is therefore important to study radiative corrections to deep-inelastic electron-photon scattering. This is our purpose in this paper. Radiative corrections to e+e- + e+e-X have been calculated for X a (pseudo) scalar particle [2-41 and X = ,ucL+pu-[4,5]. It was found that they are very small (on the percent level) in the no-tag case, when neither the electron nor the positron is being measured. In such a kinematic configuration, when the momentum transfer between the incident and outgoing electron (or positron) is small, the vertex- and bremsstrahlung contributions effectively cancel each other, leaving a small correction dominated by vacuum polarization [ 3,4]. This implies that for the equivalent photon spectrum, which is essential to relate e*y with e+e- reactions, corrections are small. In contrast, radiative corrections can be sizeable in the case where one of the leptons scatters at a large angle - single tag - and one studies differential cross sections which depend strongly on the energy and angle of the tagged electron (or positron).
1E-mail address: [email protected]. 2 E-mail address: [email protected]. 0370-2693/96/$12.00
Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved
PII 0370-2693(96)00163-3
E. L.aenen, G.A. Schuler/ Physics Letters B 374 (1996) 217-224
218
fig. 1. Photon bremsstrahlung from the tagged deep-inelastic scattering off an equivalent photon.
lepton
line in
Surprisingly, no calculations of the size of radiative corrections for inclusive deep-inelastic electron-photon scattering, i.e. ey + eX with both large Q2 (the absolute value of the transferred momentum squared) and W (the mass of state X), have been performed yet. Correspondingly, in experimental analyses of the (hadronic) photon structure function Fz(x, Q2) radiative corrections have so far not been included (as usual, x = Q2/( Q2 + W2) ) . They are usually assumed to be negligible. Recently, the AMY collaboration [ 61 estimated the size of radiative corrections by comparing a Monte Carlo event generator based on the full cross section formula for e+e- + e+e-yp+p[5] (with the muon mass and electric charge changed to correspond with quark-antiquark pair production) with a generator for e’e- -+ e+e-qil where the cross section for ey -+ eqq with a real photon was convoluted with the equivalent photon spectrum. A (positive) correction of order 10% for the visible x distribution was found, which, however, cancelled effectively against the correction due to a non-vanishing target-photon mass. Hence no net correction was applied. Nevertheless, it is important to understand both corrections separately, and their behavior as a function of the kinematic variables chosen and phase space. Moreover, the hadronic structure of the photon must not be neglected. Here we therefore estimate the size of the radiative corrections to inclusive deep-inelastic ey scattering, using the full photon structure function, In the next section we describe the relevant kinematics and formalism and in Section 3 we present results.
2. Formalism We consider
the O(a)
corrections
to deep-inelastic
scattering
(DIS)
of electrons
on (quasi-real)
(1)
40 + Y(P) -+ 40 f y(k) + X(PX> . This process is depicted in Fig. 1, in which we indicate all momentum labels. The target photon y(p) is part of the flux of equivalent photons around the non-tagged that this flux has a momentum density given by the Weizsacker-Williams expression file(z)
= g
’ +(y-z)2 (
lnz
P2
photons:
-2?&(~
Piin
&)}’
lepton. We assume
(2)
2 = ( 1 - z ) (E&,& 2. Here z is the longitudinal momentum fraction of where P,$, = (z2m:)/( 1 -z) and Pm, the target photon with respect to its parent lepton, Eb = &/2 is the lepton beam energy, 8,, is the anti-tag3 angle and P2 = -p2. In the following we put P 2 = 0 and neglect electron masses everywhere except in (2). Moreover we substitute Pzax by Piax + PA,, so that we can easily extend the z range to 1, see 171. 3 i.e. all events in which the parent lepton scatters at an angle larger than timax are rejected.
E. Luenen, G.A. Schuler/ Physics Letters B 374 11996) 217-224
219
The DIS variables can be defined either from the leptonic or the hadronic momenta: q1 = 1 - 1’
)
qh=Px-P=qr-k,
W:=(P+qr)*,
Q:=-q;,
w;=(p+qh)*=p&
-v=Qf/Wa,
Xh=Q;/2P-qh,
Yl=P.ql/P*k
Q;=-4, yh=p*qh/p*l.
(3)
Note that both QF = xlyl+, and Q,” = XhyhSey,where ser = (p + Z)*, but that leptonic and hadronic variables agree only for nonradiative events, i.e. if k = 0. We will see that the size of the corrections strongly depends on which set of variables are used in the measurement. In practice one determines Q* from the tagged lepton, and x from the (visible) hadronic energy wh. The Born cross section (i.e. no y(k) in ( 1)) is given by
-j$$=fB(x,Q2A,
(4)
where fB(x,
Q2,
s)
= $$
F2(x,
Q*>
(5)
1
X
s
dz
_f,+(z> Y+
($) {l++Q*$-J}.
(6)
zrnin(~Q~~)
Here we have defined ztin(x, Q2, s) = Q*/xs, Y+(y) = 1 + (1 - y)* and R(x, Q*, y) = 1 +
-‘*- y)*
(1
FL(x, Q2> F2(x, Q*> ’
(7)
F2,L are the photon structure functions (we have dropped the superscript y). The above form E& (6) is useful because F2 can be factored out of the z integration. For comparison of ey with ep scattering we will also give the cross section in terms of x and y d2aB -dxdy =gB(x,y,s)
,
(8)
where
(9) The lower limit E = W&J< 1 - x) ys on the z-integration in I$ (9) arises if a lower cut is applied to the invariant hadronic mass W. At the Born level expressions (4) and (8) are equivalent and valid for both sets of variables in (3). For the ep case the full electroweak corrections have been calculated for both neutral current reactions [ 8,91, including elastic nucleon scattering [ 101, and charged-current reactions [ 111. It is well-known that the leading logarithmic approximation (LLA) (in In ( Q*/R& ) reproduces the exact results to within a few percent however [ 12,131. Here we will therefore work within this approximation and neglect furthermore Z exchange.
220
E. L..aenen, G.A. Schuler/Physics LettersB 374 (1996) 217-224
Fig. 2. a) Initial- and final state bremsstrahlung. b) Compton contribution.
Px
We consider first the radiative corrections to the differential cross section in (8) in terms of xl and yZ. The O(cr) corrections in LLA arise from coI1inea.r and soft bremsstrahlung from the initial and final electron that couple to the “probing photon” and from the Compton process4. The bremsstrahlung processes are represented in Fig. 2a. They can be written as the following convolution (for clarity we suppress the subscript I on x, y, and Q2): 1
d2aB’
-=
dxdy
dxiDe/,(xi,Q2) {O(xi-xQ)J(xl,xz)gS(A,y^,s^) -gB(X*YvS)}
s
*
(10)
0
Initial-state radiation corresponds to xi = xi and x2 = 1 in Eq. (10) and vice versa for final-state radiation. D,/,(v, Q*> are the structure functions for the initial- and final-state electron evaluated at the scale given by the squared momentum transfer Q2 = Q,“:
De/Au, Q2)
(11)
They represent the probability of finding, inside a parent electron, an electron with longitudinal momentum fraction v. The Born cross section g”(a, j, S) is written in terms of the reduced (“hatted”) variables R, 9, and 9, which are readily found from Fig. 2a (e.g. 9 = p . (XI I - ?/x2)/p - x1 Z) :=x,3,
R=xx1y/(x,xz+y-1),
j=yx/xzi.
(12)
Finally, the integration limits xe are fixed by the kinematics, x7 = ( 1 - y)/( 1 - xy) and ~(3 = xy + 1 - y, and .Z in Eq. ( 10) is the Jacobian of the transformation from the reduced to the external (i.e. leptonic) variables, the J(xI,x~) = detla(%9)/a(x,y)j = Y/( nlx$jt). Note that within the leading logarithmic approximation reduced variables in Eq. ( 12) coincide with the hadronic variables. The Compton process corresponds to the case in Fig. 1 where the exchanged photon ~(4) is nearly on-shell and collinear with the target photon, qh x xlp, but y(k) is radiated at a wide angle causing the lepton e( I’) to be tagged, see Fig. 2b. The Compton cross section is most easily derived in the parton model. It is the convolution of the cross section of e(Z) + y(qh) -+ e(P) + y(k) for a real y(qh) (i.e. qi = 0) with the probability distribution to find the (real) photon y(qh) in the target photon y(p). This distribution is simply given as the convolution of the quark-to-quark-plus-photon splitting function with the quark distribution in the target photon y(p). Since the latter are evaluated at the scale Q,“, the validity of the final result at low Qf is, however, questionable. Alternatively we can derive a model-independent result for the Compton contribution, valid also at low Q,“, by observing that the y* (qh) f y(p) -+ X(px> part of the e(Z) + y(p) -+ e(Z’) + y(k) + X(px) reaction is (for arbitary values of Ql) the same as that in ordinary inclusive ey --+ eX scattering. Hence the cross section of ey --+ eyX can be written [ 131 as 4 As stated above, radiative effects to the Weizslcker-Williams spectrum are small [4 ] and we therefore neglect them.
E. Laenen,
cT[ey
-3
eyX]
=
r?? ?I_
G.A. ScJzuler/ Phpics
Letters B 374 (1996) 217-224
J
dxldy~dxhdQ~{R*(x~,Yz,xh,Qh2)
221
FhhrQh2) +RL(xI,YI,x~.Q~~)FL(x~~Q~~)} . (13)
Ri behave as l/Q,” and in the limit of small Qi we find
For small values of Q,‘, the radiator functions
I$
da( ey -+ eyX)
dxdyl
= x;( 1 - yl)s
(14) The Qi-integration limits are (up to terms of O(m,)) given by Qi,ti,, = xhQf/xl and Qi,,,ax = $M*/( 1 - xl) where M denotes the minimum mass for inclusive, inelastic ey -+ eX scattering. In the following we take A4 to be the proton mass so that the visible hadronic energy wh is of order GeV, which is a typical experimental requirement in deep-inelastic ey scattering. Then the structure function Fi(xh, Q,“) are only weakly depending on Qi and we may expand them around Q,” = Qf. Folding back in the probability to find the target photon y(p) in the electron e(m) we finally obtain
I d*&
-= dx&
J
dz _&e(z) hc(xl,
YL, xwzs,
zs> ,
(1%
c
where
hC(x,y,Q2,s) =
a3
x2(1 - y)s
y+(Y)
ln$j?
[l+
(1-
t,‘]
&(x/&*)(1
+R(xh,Q2,x,nh)).
(16) Within the LLA we have also neglected subleading terms, e.g. proportiona to Inxl. Next we discuss the radiative corrections to the differential cross section expressed in hadronic variables. Now there are, in accordance with the KLN theorem [ 141, in LLA approximation neither corrections from final state radiation, nor from the Compton process because these process do not affect the kinematic variables. We find that the correction due to initial state radiation can simply be expressed as &COrr
___ d%dQ,2
1 2rrcY* - fi(xh,Q;) XhQ;
J
dz f+(z)
gh
zrni&,~>Q~,~)
(17)
where
(18)
222
E. Laenen, G.A. Schuler/
Physics Letters B 374 (1996) 217-224
and 2
hh(x,Q2,y)
= :
ln$
-y2ln(l-y)+~lny-;
(19)
e The corrections the result
are large at large y (soft region),
~~l(x~Q2~y) =Y+(y)
{exp [:( “1 ) lng-1
and can in fact be resummed
ln(l-y)
by simple exponentiation,
] -1 } +;lnZa “I{y (l--2 ‘) lny+y
with
( l-; ‘)} (20)
and
In the next section we use the formulae
listed in the above to estimate the size of the corrections.
3. Results Here we study the radiative correction to deep-inelastic electron-photon scattering numerically. For the results presented below we use fi = 175 GeV and We, = 2GeV. For the parton densities in the photon we use set 1 of [ 151. This set has an already low minimum Q2 of Q,’ = 0.36GeV2. However in radiative events even lower values of Q2 contribute. We therefore extrapolate below this value by
(22) (this expression correctly approaches the ~7 total cross section in the small Q2 limit) which vanishes as Q2 -+ 0, as required by gauge invariance. Furthermore we have neglected c, i.e. we put R in (6) and (9) to zero. We found that its inclusion has a negligible effect. For comparison with the ep case we now show in Fig. 3 the correction S(xr, yl), using (8), ( 10) and ( 15), defined by d2ti -++=dxldyl
d2ac
d2cn
dxzdyl
dxzdyl
6(R,Yl)
.
(23)
Note that we use here the leptonic variables to conform with the ep case. A closer examination reveals that initial and final state radiation are similar in order of magnitude throughout most of the X, y region, whereas the Compton contributions is appreciable only in the small and medium x, large y region. Also we note that if one freezes F2 at F2( Q,“) for Q2 < Q,‘, instead of extrapolating as in (22), we find that only the xl = 0.01 curve changes significantly. It decreases at small and medium yl by up to 50%. Note that inclusion of final state radiation implies a perfect measurement of the energy and momentum of the tagged lepton, even in the presence of collinear radiation. We see in Fig. 3 that radiative effects can in principle be large, around 40% for medium x and small y.
E. Laenen, G.A. Schuler / Physics Letters B 374 (1996) 217-224
-0.8' 0
' 0.1
' 0.2
' 0.3
' 0.4
' 0.5
' 0.6
' 0.7
' 0.8
223
' 0.9
Yl Fig. 3. The ratio 6(x,,y/) (23) vs. y, for various y values. Top curve is for xl = 0.01, the next for XI = 0.1. Each subsequent curve represents an increase of XI by 0.1.
Next we show in Fig, 4 the correction
Fig. 4. The ratio S(xn,Qi) vs. xh for three Q,’ values. Top curve: Qi = 1 GeV*. Middle curve: Qi = lOGeV*. Lower curve: Q; = lOOGeV*.
factor S(XJ,, Qf), defined by (24)
in terms of hadronic variables, as a function of xh for various choices of Qi, cf. ( 17). Here only initial state lepton bremsstrahlung is taken into account, because the scattered lepton is not used in constructing the kinematic variables. We see that the corrections are sizable only for large Q2. Using however the resummed version of (20) and (2 1) , we find that the corrections are reduced by about an order of magnitude. Thus we conclude that the size of the radiative corrections depends significantly on the set of variables chosen. These corrections can in principle be quite large. As noted before, in practice mixed variables are used. In view of the results obtained in this paper, we think a more careful study, involving Monte Carlo simulation of the full final state, is warranted.
References 111 Proc. of the HERA Workshop (Hamburg, Germany, 1988), ed. R.D. Peccei; Proc. of the Workshop on Physics at HERA (Hamburg, Germany, 1991), eds. W. Buchmiiller and G. Ingelman. [2] M. Defrise, 2. Phys. C 9 (1981) 41; M. Defrise, S. Ong, J. Silva and C. Carimalo, Phys. Rev. D 23 (1981) 663. [ 31 W.L. van Neerven and J.A.M. Vermaseren, Nucl. Phys. B 238 (1984) 73. [41 M. Landro, K.J. Mork and H.A. Olsen, Phys. Rev. D 36 (1987) 44. [5] EA. Berends, PH. Daverveldt and R. Kleiss, Nucl. Phys. B 253 (1985) 441; Comput. Phys. Comm. 40 (1986) 271. [6] S.K. Sahu et al. (AMY Collab.), Phys. Lett. B 346 (1995) 208. [71 S. Frixione, M. Mangano, P. Nason and G. Ridolfi, Phys. L&t. B 319 (1993) 339. [Sl D. Bardin, 0. Fedorenko and N. Shumeiko, J. Phys. G 7 (1981) 1331; D. Bardin, C. Burdik, P Christova and T. Riemmn, JINR Dubna preprint E 2-87-595 (1987); 2. Phys. C 42 ( 1989) 679;
224
E. Laenen, G.A. Schuler / Physics Letters B 374 (1996) 217-224
A. Akhundov, D. Bardin, L. Kalinovskaya and T. Riemann, Fortran program TERAD 91; a short write-up may be found in: Proc. of the Workshop on Physics at HERA (Hamburg, Germany, 1991), eds. W. Buchmtiller and G. Ingelman, p. 1285. [9] M. Bdhm and H. Spiesberger, Nucl. Phys. B 294 (1987) 1081; H. Spiesberger, in: Proc. of the HERA Workshop (Hamburg, Germany, 1988), ed. R.D. Peccei, p. 60.5. [lo] A. Akhundov, D. Bardin, C. Burdik, P Christova and L. Kalinovskaya, Z. Phys. C 45 (1990) 645. [ 111 M. Bohm and H. Spiesberger, Nucl. Phys. B 304 (1987) 749; D. Bardin, C. Burdik, P Christova and T. Riemann, Z. Phys. C 44 (1989) 149; H. Spiesberger, Nucl. Phys. B 349 (1991) 109. [ 121 M. Consoli and M. Greco, Nucl. Phys. B 186 (1981) 519; W. Beenakker, E Berends and W. van Neerven, in: Proc. of the Workshop on Radiative Corrections for e+e- Collisions (SchloB Ringberg, Tegernsee, Germany, 1989), ed. J. Kuhn, p. 3; J. Bliimlein, Z. Phys. C 47 ( 1990) 89; G. Montagna. 0. Nicrosini and L. Trentadue, Nucl. Phys. B 357 (1991) 390; .I. Kripfganz, H. Mijhring and H. Spiesberger, Z. Phys. C 49 (1991) 501. [ 131 H. Spiesberger et al., in: Proc. of the Workshop on Physics at HERA (Hamburg, Germany, 1991), eds. W. Buchmtlller and G. Ingelman, p. 798; A. Akhundov, D. Bardin, L. Kalinovskaya and T. Riemann, DESY preprint DESY 94-115 ( 1994) (hep-ph19407266). [ 141 T. Kinoshita, J. Math. Phys. 3 (1962) 650; T.D. Lee and M. Nauenberg, Phys. Rev. 133 (1964) B1549. [ 151 G.A. Schuler and T. Sjbstrand, preprint CERN-TH-95-62, hep-ph 9503384.
2 May 1996
!!&I ‘Es
PHYSICS
LETTERS B
Physics Letters B 374 (1996) 217-224
EJ-SETIER
Radiative corrections to ey scattering Eric Laenen a~1,Gerhard A. Schuler bT2 a CERN TH-Division, 121 I-CH Geneva 23, Switzerland b Theoretische Physik, Universitiir Regensburg, D-93053 Regensburg, Germany Received 1 September 1995; revised manuscript received 22 January 1996 Editor: R. Gatto
Abstract We investigate tbe effects of photon radiation on deep-inelastic ey scattering. Depending on the set of variables chosen, we find appreciable effects in the kinematic region accessible at LEP2. Convenient analytic results for O(cu) corrected differential cross sections are presented.
1. Introduction In the period leading up to the start of the HERA program a substantial amount of work was done on radiative corrections to observables in deep-inelastic scattering (DIS) of electrons off protons (see e.g. [ 1 I for an overview). These corrections are substantial enough for both HERA experiments to correct for them. Part of the upcoming LEP2 physics programme involves the measurement of the photon structure function Fz (x, Q2). This structure function is extracted from deep-inelastic electron-photon scattering in the reaction e+e- + efe-X, where one of the leptons escapes undetected down the beam pipe, while the other is measured at rather large angle. It is therefore important to study radiative corrections to deep-inelastic electron-photon scattering. This is our purpose in this paper. Radiative corrections to e+e- + e+e-X have been calculated for X a (pseudo) scalar particle [2-41 and X = ,ucL+pu-[4,5]. It was found that they are very small (on the percent level) in the no-tag case, when neither the electron nor the positron is being measured. In such a kinematic configuration, when the momentum transfer between the incident and outgoing electron (or positron) is small, the vertex- and bremsstrahlung contributions effectively cancel each other, leaving a small correction dominated by vacuum polarization [ 3,4]. This implies that for the equivalent photon spectrum, which is essential to relate e*y with e+e- reactions, corrections are small. In contrast, radiative corrections can be sizeable in the case where one of the leptons scatters at a large angle - single tag - and one studies differential cross sections which depend strongly on the energy and angle of the tagged electron (or positron).
1E-mail address: [email protected]. 2 E-mail address: [email protected]. 0370-2693/96/$12.00
Copyright 0 1996 Published by Elsevier Science B.V. All rights reserved
PII 0370-2693(96)00163-3
E. L.aenen, G.A. Schuler/ Physics Letters B 374 (1996) 217-224
218
fig. 1. Photon bremsstrahlung from the tagged deep-inelastic scattering off an equivalent photon.
lepton
line in
Surprisingly, no calculations of the size of radiative corrections for inclusive deep-inelastic electron-photon scattering, i.e. ey + eX with both large Q2 (the absolute value of the transferred momentum squared) and W (the mass of state X), have been performed yet. Correspondingly, in experimental analyses of the (hadronic) photon structure function Fz(x, Q2) radiative corrections have so far not been included (as usual, x = Q2/( Q2 + W2) ) . They are usually assumed to be negligible. Recently, the AMY collaboration [ 61 estimated the size of radiative corrections by comparing a Monte Carlo event generator based on the full cross section formula for e+e- + e+e-yp+p[5] (with the muon mass and electric charge changed to correspond with quark-antiquark pair production) with a generator for e’e- -+ e+e-qil where the cross section for ey -+ eqq with a real photon was convoluted with the equivalent photon spectrum. A (positive) correction of order 10% for the visible x distribution was found, which, however, cancelled effectively against the correction due to a non-vanishing target-photon mass. Hence no net correction was applied. Nevertheless, it is important to understand both corrections separately, and their behavior as a function of the kinematic variables chosen and phase space. Moreover, the hadronic structure of the photon must not be neglected. Here we therefore estimate the size of the radiative corrections to inclusive deep-inelastic ey scattering, using the full photon structure function, In the next section we describe the relevant kinematics and formalism and in Section 3 we present results.
2. Formalism We consider
the O(a)
corrections
to deep-inelastic
scattering
(DIS)
of electrons
on (quasi-real)
(1)
40 + Y(P) -+ 40 f y(k) + X(PX> . This process is depicted in Fig. 1, in which we indicate all momentum labels. The target photon y(p) is part of the flux of equivalent photons around the non-tagged that this flux has a momentum density given by the Weizsacker-Williams expression file(z)
= g
’ +(y-z)2 (
lnz
P2
photons:
-2?&(~
Piin
&)}’
lepton. We assume
(2)
2 = ( 1 - z ) (E&,& 2. Here z is the longitudinal momentum fraction of where P,$, = (z2m:)/( 1 -z) and Pm, the target photon with respect to its parent lepton, Eb = &/2 is the lepton beam energy, 8,, is the anti-tag3 angle and P2 = -p2. In the following we put P 2 = 0 and neglect electron masses everywhere except in (2). Moreover we substitute Pzax by Piax + PA,, so that we can easily extend the z range to 1, see 171. 3 i.e. all events in which the parent lepton scatters at an angle larger than timax are rejected.
E. Luenen, G.A. Schuler/ Physics Letters B 374 11996) 217-224
219
The DIS variables can be defined either from the leptonic or the hadronic momenta: q1 = 1 - 1’
)
qh=Px-P=qr-k,
W:=(P+qr)*,
Q:=-q;,
w;=(p+qh)*=p&
-v=Qf/Wa,
Xh=Q;/2P-qh,
Yl=P.ql/P*k
Q;=-4, yh=p*qh/p*l.
(3)
Note that both QF = xlyl+, and Q,” = XhyhSey,where ser = (p + Z)*, but that leptonic and hadronic variables agree only for nonradiative events, i.e. if k = 0. We will see that the size of the corrections strongly depends on which set of variables are used in the measurement. In practice one determines Q* from the tagged lepton, and x from the (visible) hadronic energy wh. The Born cross section (i.e. no y(k) in ( 1)) is given by
-j$$=fB(x,Q2A,
(4)
where fB(x,
Q2,
s)
= $$
F2(x,
Q*>
(5)
1
X
s
dz
_f,+(z> Y+
($) {l++Q*$-J}.
(6)
zrnin(~Q~~)
Here we have defined ztin(x, Q2, s) = Q*/xs, Y+(y) = 1 + (1 - y)* and R(x, Q*, y) = 1 +
-‘*- y)*
(1
FL(x, Q2> F2(x, Q*> ’
(7)
F2,L are the photon structure functions (we have dropped the superscript y). The above form E& (6) is useful because F2 can be factored out of the z integration. For comparison of ey with ep scattering we will also give the cross section in terms of x and y d2aB -dxdy =gB(x,y,s)
,
(8)
where
(9) The lower limit E = W&J< 1 - x) ys on the z-integration in I$ (9) arises if a lower cut is applied to the invariant hadronic mass W. At the Born level expressions (4) and (8) are equivalent and valid for both sets of variables in (3). For the ep case the full electroweak corrections have been calculated for both neutral current reactions [ 8,91, including elastic nucleon scattering [ 101, and charged-current reactions [ 111. It is well-known that the leading logarithmic approximation (LLA) (in In ( Q*/R& ) reproduces the exact results to within a few percent however [ 12,131. Here we will therefore work within this approximation and neglect furthermore Z exchange.
220
E. L..aenen, G.A. Schuler/Physics LettersB 374 (1996) 217-224
Fig. 2. a) Initial- and final state bremsstrahlung. b) Compton contribution.
Px
We consider first the radiative corrections to the differential cross section in (8) in terms of xl and yZ. The O(cr) corrections in LLA arise from coI1inea.r and soft bremsstrahlung from the initial and final electron that couple to the “probing photon” and from the Compton process4. The bremsstrahlung processes are represented in Fig. 2a. They can be written as the following convolution (for clarity we suppress the subscript I on x, y, and Q2): 1
d2aB’
-=
dxdy
dxiDe/,(xi,Q2) {O(xi-xQ)J(xl,xz)gS(A,y^,s^) -gB(X*YvS)}
s
*
(10)
0
Initial-state radiation corresponds to xi = xi and x2 = 1 in Eq. (10) and vice versa for final-state radiation. D,/,(v, Q*> are the structure functions for the initial- and final-state electron evaluated at the scale given by the squared momentum transfer Q2 = Q,“:
De/Au, Q2)
(11)
They represent the probability of finding, inside a parent electron, an electron with longitudinal momentum fraction v. The Born cross section g”(a, j, S) is written in terms of the reduced (“hatted”) variables R, 9, and 9, which are readily found from Fig. 2a (e.g. 9 = p . (XI I - ?/x2)/p - x1 Z) :=x,3,
R=xx1y/(x,xz+y-1),
j=yx/xzi.
(12)
Finally, the integration limits xe are fixed by the kinematics, x7 = ( 1 - y)/( 1 - xy) and ~(3 = xy + 1 - y, and .Z in Eq. ( 10) is the Jacobian of the transformation from the reduced to the external (i.e. leptonic) variables, the J(xI,x~) = detla(%9)/a(x,y)j = Y/( nlx$jt). Note that within the leading logarithmic approximation reduced variables in Eq. ( 12) coincide with the hadronic variables. The Compton process corresponds to the case in Fig. 1 where the exchanged photon ~(4) is nearly on-shell and collinear with the target photon, qh x xlp, but y(k) is radiated at a wide angle causing the lepton e( I’) to be tagged, see Fig. 2b. The Compton cross section is most easily derived in the parton model. It is the convolution of the cross section of e(Z) + y(qh) -+ e(P) + y(k) for a real y(qh) (i.e. qi = 0) with the probability distribution to find the (real) photon y(qh) in the target photon y(p). This distribution is simply given as the convolution of the quark-to-quark-plus-photon splitting function with the quark distribution in the target photon y(p). Since the latter are evaluated at the scale Q,“, the validity of the final result at low Qf is, however, questionable. Alternatively we can derive a model-independent result for the Compton contribution, valid also at low Q,“, by observing that the y* (qh) f y(p) -+ X(px> part of the e(Z) + y(p) -+ e(Z’) + y(k) + X(px) reaction is (for arbitary values of Ql) the same as that in ordinary inclusive ey --+ eX scattering. Hence the cross section of ey --+ eyX can be written [ 131 as 4 As stated above, radiative effects to the Weizslcker-Williams spectrum are small [4 ] and we therefore neglect them.
E. Laenen,
cT[ey
-3
eyX]
=
r?? ?I_
G.A. ScJzuler/ Phpics
Letters B 374 (1996) 217-224
J
dxldy~dxhdQ~{R*(x~,Yz,xh,Qh2)
221
FhhrQh2) +RL(xI,YI,x~.Q~~)FL(x~~Q~~)} . (13)
Ri behave as l/Q,” and in the limit of small Qi we find
For small values of Q,‘, the radiator functions
I$
da( ey -+ eyX)
dxdyl
= x;( 1 - yl)s
(14) The Qi-integration limits are (up to terms of O(m,)) given by Qi,ti,, = xhQf/xl and Qi,,,ax = $M*/( 1 - xl) where M denotes the minimum mass for inclusive, inelastic ey -+ eX scattering. In the following we take A4 to be the proton mass so that the visible hadronic energy wh is of order GeV, which is a typical experimental requirement in deep-inelastic ey scattering. Then the structure function Fi(xh, Q,“) are only weakly depending on Qi and we may expand them around Q,” = Qf. Folding back in the probability to find the target photon y(p) in the electron e(m) we finally obtain
I d*&
-= dx&
J
dz _&e(z) hc(xl,
YL, xwzs,
zs> ,
(1%
c
where
hC(x,y,Q2,s) =
a3
x2(1 - y)s
y+(Y)
ln$j?
[l+
(1-
t,‘]
&(x/&*)(1
+R(xh,Q2,x,nh)).
(16) Within the LLA we have also neglected subleading terms, e.g. proportiona to Inxl. Next we discuss the radiative corrections to the differential cross section expressed in hadronic variables. Now there are, in accordance with the KLN theorem [ 141, in LLA approximation neither corrections from final state radiation, nor from the Compton process because these process do not affect the kinematic variables. We find that the correction due to initial state radiation can simply be expressed as &COrr
___ d%dQ,2
1 2rrcY* - fi(xh,Q;) XhQ;
J
dz f+(z)
gh
zrni&,~>Q~,~)
(17)
where
(18)
222
E. Laenen, G.A. Schuler/
Physics Letters B 374 (1996) 217-224
and 2
hh(x,Q2,y)
= :
ln$
-y2ln(l-y)+~lny-;
(19)
e The corrections the result
are large at large y (soft region),
~~l(x~Q2~y) =Y+(y)
{exp [:( “1 ) lng-1
and can in fact be resummed
ln(l-y)
by simple exponentiation,
] -1 } +;lnZa “I{y (l--2 ‘) lny+y
with
( l-; ‘)} (20)
and
In the next section we use the formulae
listed in the above to estimate the size of the corrections.
3. Results Here we study the radiative correction to deep-inelastic electron-photon scattering numerically. For the results presented below we use fi = 175 GeV and We, = 2GeV. For the parton densities in the photon we use set 1 of [ 151. This set has an already low minimum Q2 of Q,’ = 0.36GeV2. However in radiative events even lower values of Q2 contribute. We therefore extrapolate below this value by
(22) (this expression correctly approaches the ~7 total cross section in the small Q2 limit) which vanishes as Q2 -+ 0, as required by gauge invariance. Furthermore we have neglected c, i.e. we put R in (6) and (9) to zero. We found that its inclusion has a negligible effect. For comparison with the ep case we now show in Fig. 3 the correction S(xr, yl), using (8), ( 10) and ( 15), defined by d2ti -++=dxldyl
d2ac
d2cn
dxzdyl
dxzdyl
6(R,Yl)
.
(23)
Note that we use here the leptonic variables to conform with the ep case. A closer examination reveals that initial and final state radiation are similar in order of magnitude throughout most of the X, y region, whereas the Compton contributions is appreciable only in the small and medium x, large y region. Also we note that if one freezes F2 at F2( Q,“) for Q2 < Q,‘, instead of extrapolating as in (22), we find that only the xl = 0.01 curve changes significantly. It decreases at small and medium yl by up to 50%. Note that inclusion of final state radiation implies a perfect measurement of the energy and momentum of the tagged lepton, even in the presence of collinear radiation. We see in Fig. 3 that radiative effects can in principle be large, around 40% for medium x and small y.
E. Laenen, G.A. Schuler / Physics Letters B 374 (1996) 217-224
-0.8' 0
' 0.1
' 0.2
' 0.3
' 0.4
' 0.5
' 0.6
' 0.7
' 0.8
223
' 0.9
Yl Fig. 3. The ratio 6(x,,y/) (23) vs. y, for various y values. Top curve is for xl = 0.01, the next for XI = 0.1. Each subsequent curve represents an increase of XI by 0.1.
Next we show in Fig, 4 the correction
Fig. 4. The ratio S(xn,Qi) vs. xh for three Q,’ values. Top curve: Qi = 1 GeV*. Middle curve: Qi = lOGeV*. Lower curve: Q; = lOOGeV*.
factor S(XJ,, Qf), defined by (24)
in terms of hadronic variables, as a function of xh for various choices of Qi, cf. ( 17). Here only initial state lepton bremsstrahlung is taken into account, because the scattered lepton is not used in constructing the kinematic variables. We see that the corrections are sizable only for large Q2. Using however the resummed version of (20) and (2 1) , we find that the corrections are reduced by about an order of magnitude. Thus we conclude that the size of the radiative corrections depends significantly on the set of variables chosen. These corrections can in principle be quite large. As noted before, in practice mixed variables are used. In view of the results obtained in this paper, we think a more careful study, involving Monte Carlo simulation of the full final state, is warranted.
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