On the measurement of sin2θw at HERA

Nuclear Physics B233 (1984) 365-374 © North-Holland Publishing Company

ON THE MEASUREMENT OF sinZ0w AT HERA J.F. WHEATER Department of Theoretical Pt~vsics, Unit,ersiO,of Oxford, 1, Keble Road, Oxford OX1 3NP, UK

Received 12 August 1983

The potential of the HERA accelerator project for the measurement of sin20,~is considered. The likely errors from uncertainties in Mw, Mz, beam flux determinations and sea quark distributions are estimated together with the statistical errors. It seems that a measurement of sin20,~ from data at a mean spacelike q2 _ 4000 GeV2 with a combined theoretical and statistical error of +0.007 or less is practical. This is small enough to allow meaningful numerical comparison of weak-interaction physics at HERA and other present and future experiments at very different values of q2.

1. Introduction In this paper we investigate the accuracy with which sin20,v may be measured in experiments at H E R A and therefore the potential of this project for testing the numerical consistency of the standard SU(2)L X U(1) model. Deep inelastic ep scattering at H E R A will probe the electroweak interaction in a kinematic regime untouched by any other accelerator; high spacelike q2 ( _ 4000 GeV2). LEP probes the very different region of high timelike q2. If the standard model is correct, it is important that experiments in all these regimes produce consistent results not only at the tree-graph level but also when electroweak radiative corrections are included [1]. The radiative corrections are very different for spacelike and timelike processes; agreement between sinZ0w measured at H E R A and that deduced from, for example, measurements at LEP would be strong evidence for the correctness of the field theory at the one-loop level. The value of sin20w deduced from present experiments is typically shifted by about 0.01 by the inclusion of radiative corrections (a comprehensive list of references is given in ref. [2]). Therefore, in order to make a meaningful comparison of the values of sin20w measured in different experiments (and with radiative corrections included), residual uncertainties less than the shift due to radiative corrections are needed. In a recent paper on neutrino hadron scattering it was argued that an error in sin20,~, of less than + 0.005 was desirable [3]. To find the likely statistical and theoretical errors in sin20,,, measured at HERA, we use the tree-graph approximation. As we will show, 365

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J.F. Wheater / Measurement of sin-'O,,

one should obtain a combined theoretical and statistical error of _< _+0.007. It is therefore worth making a great effort to minimize experimental systematic errors. This paper is organized as follows; we first givea very brief description of HERA itself followed by a discussion of sin20w. We then discuss possible ways of measuring sinZ0w and the likely errors from various sources. Our conclusions are given in sect. 10.

2. HERA HERA will be a machine for colliding 800 GeV unpolarized protons with polarized 30 GeV electrons or positrons ( S = 96000 GeV2). By decreasing the lepton energy to 20 GeV, a five-fold increase in luminosity is expected; for investigations of the standard electroweak SU(2)L × U(1) model this higher luminosity is of great value and throughout this paper we assume operation at S = 64000 GeV 2. At this value of S events at spacelike q2 up to about 1 0 4 GeV 2 are obtained with a reasonable rate. For further details see ref. [4]. At this point we wish to establish a notational convention about polarizations; because the beams are not 100% polarizable, it is not possible to measure a cross section for purely left- or fight-handed leptons. Throughout this paper we will assume 80% polarization, and O(XL.R) is to mean the cross section measured with an 80% L / R polarized beam of X's. The error in the polarization measurement is expected to be - 0.5% which has a negligible effect on all the quantities examined in this paper.

3. sin20~ We will consider the measurement of z = sin20 where 0 is the mixing angle between the neutral U(1) and SU(2)L gauge fields of the standard model. The SU(2) coupling g and %.m. are then related by g2 = 4Trae.m/z which we use to eliminate g from the expressions for cross sections (these have been recorded in, for example, ref. [4]; they are repeated in the appendix of this paper solely for the readers' convenience). The masses of W ± , Z ° bosons appear in the propagators in 1 / 2 • we will consider their measured values as input to the analysis of ( q 2 + M w,z), HERA experiments. The effect of uncertainties in these values is considered in sect. 7. We note that the above procedure of using the known W +, Z masses is in contrast to low-energy experiments where, at the tree-graph level, only the combinations g2/M2 and Mw/Mzcos 0 appear. The former is proportional to G F which is very accurately measured while the latter is either used as a parameter in fitting to the data or put equal to its theoretical value (which is 1 at the tree-graph level in a model with one Higgs isodoublet). As far as the model is concerned, using the measured

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J.F. Wheater / Measurement of sin:O,.

values for Mw, z is equivalent to not specifying that the Higgs structure should be a weak isodoublet.

4. Neutral and charged currents The n.c. scattering amplitude for e + p ~ e-+X has contributions from y and Z exchange. The rate is dominated by events at low x and y because of the q2 = 0 pole in the e.m. part of the amplitude which we will use to determine the relative fluxes of different beams (see sect. 5). We note that y = 1 - E~ cos2½(0 + a)

q2 = S x y ,

Ee

cos2½ot

where the energies E~ and E e and angles c~ and 0 are measured in the laboratory frame, see fig. 1. N L - fN R

A'exp = NL + f N R "

It turns out that low x and y cuts are always necessary so o should be interpreted as

o=

fl

fl

d2o

dx

~,nin

dY dxdy. rain

With four different beams available there are altogether 6 possible asymmetries to measure. In this paper A will be used for illustrative purposes; some of the others are as good but none is significantly better. The parity violating asymmetries, such as A, have the advantage that pure e.m. effects do not contribute to the numerator at any order in perturbation theory (e.m. being parity conserving). Asymmetries with apparent C violation are caused by e.m. box diagrams and bremsstrahlung as well as the weak interactions and these e.m. contributions can be rather large at lower q2 values [5]. All the asymmetries require a method for the accurate determination of the relative flux in different beams which is discussed in sect. 5.

(E'e, p~)

S : (p~ ÷pp)2

Pp

Fig. 1. Kinematics in the laboratory frame for ep scattering.

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J.F Wheater / Measurementof sin20.

At HERA, the c.c. event rate (e p --* vX etc.) will be of the order of a hundred events per day. Therefore, in addition to the n.c. asymmetries, we may also attempt to use the c.c. data in the measurement of z. The quantity D

anc(eL) -- °nc(e~) Occ(ec) -- acc(e~ )

(analogous to the Paschos-Wolfenstein relation in neutrino physics) has the advantage that sea quarks cancel provided that q ( x ) = q(x). However, like A, an accurate determination of the relative flux of two beams is needed. On the other hand, the quantities R - - %c(ec) Oct(e;)

R+

%c(ep" )

'

Occ(e

) '

involve data from only one beam but are sensitive to sea quark distributions. Of course, it is not necessary to have polarized beams to measure R -+ although the c.c. event rate is improved by polarization. In the following sections we will consider the utility of A, D and R +- in the measurement of z, assuming that the value of z is - 0.22. To estimate the various errors we will use the parton distributions of Buras and Gaemers [6] with A QCD = 0.3 GeV. The presence of the b-quark in the sea will require modifications of these distributions for the actual experimental analysis but for the estimation of the size of possible errors it does not matter if our distributions are not quite right. We do, in any case, consider potential errors from uncertainties in parton distributions. In practice we expect that the experimental analysis will be done in the framework of QCD as for present semileptonic experiments (eg. see ref. [7]).

5. Relative flux

To determine the relative flux f of different beams in the machine we propose to exploit the high-event rate at low ( - 30-100 GeV 2) q2. If we are interested in A then f = e c f l u x / e R flux is needed; suppose that in the range q2 = 50 --, 51 (corresponding to a scattering angle of 20°), N c e [ events and N R e~ events are observed and form the asymmetry NI_ f N R NL + f NR " -

A'exp We can calculate the value of

A ' = d°nc (e{) - dane (eR) done (e~) + done (e R ) '

where do = do ) dq2 '

J.F. Wheater / Measurement of sin:O..

369

assuming the standard model. At q2 = 50, A' is (2.7 +_ 0.4) × 10 -3, including electroweak effects and bremsstrahlung, where the error comes from uncertainties in the sea quark distributions (see sect. 8). By equating A'exp to the calculated value of A' we get a theoretical error in f of 0.08%. This is swamped by the statistical errors in NLR; assuming a luminosity of 5 - 1 0 31 c m - 2 s-1 and running for 100 days with each beam we find an error in f of 0.5%. Using events between q 2 = 45 and q 2 = 55 should allow a measurement of f accurate to 0.2% provided systematic experimental errors can be controlled at this level. (We note that the use of different q2 bins in this region may help to minimize systematics.) Similar procedures will work with other beam combinations; the value of A' varies from case to case but the theoretical error is always less than the statistical error.

6. Choice of kinematic range The determination of f discussed above is not accurate enough to allow the inclusion of very low x, y ( < 0.05 say) events in the data sample used to determine z from A or D; the imperfect cancellation of the purely electromagnetic 1 / q 4 terms in the cross section causes huge errors. Practically, this will not bother us because our aim is to measure z at very high values of q2. If we restrict the data to x, y > 0.1 2 (qmin = 640) we find that a 0.2% error in f causes an error in z of _+0.0075 if D is used or _+0.001 if A is used; raising the x cut to 0.15 reduces the errors to +_0.006 and +_0.0007 respectively (see table 1). The cuts of Xmin 0.15 and Yr~n = 0.1 are also convenient for R +; the x cut moderates the influence of the sea while the y cut removes a large number of very low q2 events leaving an event sample with high average q2. The average q2 of events contributing to neutral currents, charged currents and the difference of two neutral currents are shown in table 2 (for the difference of two charged currents it is the same as for a single charged current). Table 2 shows that the weak interactions are studied at a q2 at least a factor one hundred higher than in :

TABLE 1 V a l u e s o r t h e s h i f t in z = sin20,~, as d e d u c e d f r o m A, D a n d R • g i v e n v a r i o u s c h a n g e s o f i n p u t p a r a m e t e r s a n d a s s u m i n g x a n d y c u t s o f 0.15 a n d 0.1 r e s p e c t i v e l y

Source of error

8f/f= 6M z =

A

D

+0.002

+0.0007

-0.006

+ 1 GeV

+ 0.0013

6M w = + 1 GeV

0.009 +0.019

R - 0.0007

R~ 0.0002

+0.004

+0.0039

sea quarks: (see t e x t sect. 8) (i)

- 0.0001

-

- 0.0007

+ 0.0015

(ii)

- 0.0001

-

+ 0.0057

0.011

+ 0.001

0.003

(iii)

0.0006

+ 0.016

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370

TABLE 2 Average q2 of events contributing to A, D and R + with the cuts shown Cuts

On~(eL)--%c(eR)

On~(e )

O~c(e )

x, y > 0.1, 0.1 x, y > 0.15, 0.1

4000 5100

2700 3700

6400 7300

present experiments. For the rest of this discussion we adopt the cuts Xmin =

0.15,

Ymin = 0.1.

7. M w and M z As discussed in sect. 3, the values of M w and M z are needed for the measurement of z. At present, their masses are given [8] as M w = 81 + 2 and M z = 95.2 + 2.5 GeV (not including systematic errors). However, by the time H E R A is running, the value of M z should be known much more accurately from LEP and SLC. In addition, the data from p~ colliders can be analysed to give M w - M z with small systematic errors which we anticipate will be combined with M z to give the most accurate value for M w. Ultimately, we should know M w and M z with an error of no more than _+1 GeV. The effect of such an error on the value of z is shown in table 1.

8. Sea quark distributions Sea quarks make substantial contributions to cross sections for x ~< 0.2. The high values of q2 will ensure that all of u, d, s, c, and b are excited in the sea and measuring all of the distribution functions at H E R A will be a considerable task. The x cut of 0.15 ensures that no more than 10% of any cross section is contributed by the sea; this value of x roughly coincides with the peak of the valence quark distribution. Starting with the predicted distributions of ref. [6], we have investigated the effects of various changes to the sea distributions on the value of z deduced from A, D and R + as follows. (i) A 20% decrease in the magnitude of xqsea(x). (ii) Changing from a soft sea quark distribution to a much harder one; this is accomplished by changing from a gluon distribution x g ( x ) - (1 - x) 1° at q2 = 1.8 to one x g ( x ) - ( 1 x) 5 at the same q2 [6]. At q 2 2000 this changes the u,d,s sea from x q ° ( x ) - (1 - x ) 20"65 t o x q ( x ) - (1 - x ) 16"85. (iii) It is quite possible that x s ( x ) 4 : x g ( x ) (for example, scattering off the low mass K + in fig. 2 tends to crowd ~ towards smaller x while scattering off the strange baryon hardens the s distribution) but this would be very hard to detect. We

J.F. Wheater / Measurementofsin20w

371

AI Z°__ ° Fig. 2. A possible contribution to s(x) =/=g(x).

consider the effect of varying the distributions so that xs and xg change from xq ° at q 2 2000 to X S ( X ) - (1- X)t6, X g ( X ) - ( 1 - X)24 (SO that f x 2 ( x ) d x / f x s ( x ) d x = 0.66, a rather large difference which is not ruled out experimentally). Any such effect for u, d quarks is absorbed in the valence quark distributions and is not expected for c , b . . , quarks because of the very high masses of the hadrons which would be involved. The results of all these changes are shown in table 1. We note that our starting point was an SU(3) symmetric sea (after ref. [6]); if the present result that the strange sea carries less m o m e n t u m than the u, d sea [9] persists to very high q2, then our errors from changes in s(x) will be correspondingly reduced. Other possible discrepancies, for example ~(x)/d(x) v~ 1, will have effects smaller than those of the large variations considered in (i) and (ii).

9. Thresholds and mass effects

Uncertainties in the threshold factors and the associated Kobayashi-Maskawa angles for heavy quark production in c.c. processes are a major source of error in low-energy experiments. The high q2 available at H E R A ensures that the d + w + - , c vertex contributes fully. For q2 >> mc2 the suppression may be reliably calculated from the tree-level diagram and is found to be

m2 o = 0o(1- 2~ sin201COS202) ~ where 0, are the usual K M angles and oo the cross section for production of u and c quarks if the c were massless. Even at the lowest q2 included in R + this suppression is - (1-0.0002). The production of third generation quarks is negligible; u + W - , b has a cross section - 00 sin201sin203 which, assuming sin 03 - sin 01, gives a possible error due to the threshold of = 2.5 × 10 3M~/q200 < 0.0001%. The t quark is likewise irrelevant. Finally, we note that higher-twist effects are most unlikely to be a problem at HERA.

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10. Conclusions All the likely sources of error which we have discussed are tabulated in table 1 together with their consequences. In addition, in table 3 we give the expected statistical errors arising from 100 days of running (with each beam in the case of A and D) at two different polarizations and a luminosity of 5 × 1031 c m - 2 s-1. As can be seen, a drop in polarization to 50% greatly increases the errors in A; R -+ are little affected (as already noted, we do not need polarization to measure R±: a change from 0% to 100% decreases the statistical error by a factor - 1.2). The change in the errors from D arises since the value of D decreases with increasing polarization because of the increasing charged current rate; this decreases dD/dz so that the error in D, which is about the same in both cases, translates into a larger error in z. It is clear that, for every source of error, D is much worse than the other three quantities; this is because it is much less sensitive to z and we will not discuss it any further. The asymmetry A has quite large statistical errors but is very insensitive to variations in the sea quark distributions. This insensitivity is a numerical fluke; the proportion of o ( e [ ) and o(eR) contributed by sea quarks is very similar so that the effects tend to cancel in A. It is tempting to exploit this insensitivity by decreasing Xmin to increase the number of events, but the value of A falls and the error in z due to statistical errors in the number of events actually gets larger. The cuts we have chosen seem to be the optimum for deducing z from A. Combining the errors in quadrature we see that it should be possible to deduce z from A with a theoretical uncertainty of no more than _+0.002. In addition, it seems not unreasonable to hope for a statistical error of _+0.005 yielding a combined error of < _+0.007. R + have better statistical errors than A but are rather vulnerable to the sea quark distributions; increasing Xmin to 0.2 would repair this at the expense of slight increase in the statistical errors by a factor of about 1.2. However, we note that the errors arising from sea quarks are generally opposite in sign between R + and R while those from other sources are of the same sign; this could be exploited to minimize the uncertainties from sea quarks. The largest error in determination of z from R ± will clearly arise from uncertainties in the W mass. A statistical error of +_0.002 should be achieved easily yielding a combined error in z of < _+0.006.

TABLE 3 Statistical errors in z = sin20w for 100 days r u n n i n g at a l u m i n o s i t y 5 × 1031 cm this m e a n s 100 days with each b e a m Polarization

A

D

80% 50%

± 0.005 ± 0.008

+ 0.015 ± 0.01

R +_0.0025 + 0.0025

2 s l: for A and D

R+ + 0.0035 ± 0.0035

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It is clear that there are at least two ways of measuring z at HERA. The accuracy of one of them, A, is dominated by statistical errors while that of the other, R +, by the error in the W boson mass. However, the combined theoretical and statistical errors in z are very similar. This error is to be compared with the target of _+0.005 which is theoretically possible for low-energy neutrino scattering [3,10] and is sufficiently small to allow a test of the general consistency of the theory, including radiative corrections. The measurement of z to an accuracy of at least _+0.007 is theoretically possible at H E R A and so no effort should be spared in minimizing experimentalsystematic errors to achieve this. I would like to thank Chris Llewellyn Smith for reading the first draft of this paper and, in particular, Roger Cashmore for constructive criticism and advice on matters experimental.

Appendix The charged current cross section for an unpolarized lepton beam due to single W exchange is given by d2o.

dxdy

g4

S

64rr (q2 + M 2

)2(xf(x, q2)+(l_y)2xf(x, q2)),

where, for e - scattering,

f=u+c+ ..., )=

~7+~+

...

,

while for e +,

f=d+s+ ..., f=a+g+

''-

.

The SU(2) coupling g is related to G F by GF =

2-S/2g2/M2

and to a e.... by

g2 = 4~.ae.m./Sin20w"

The neutral current cross section for a polarized lepton beam due to coherent ~, and Z exchange is, d20

2rro~2.m. [ (xqi+ xgli)(IA]le+lAi212)[½(l+(1-y)2)] ]

dxdy- Sx2y2 ~ +(xqi_xgli)(lA~12_lA~[2)[½(l_(l_y)2)]

'

J.F. Wheater / Measurement ofsin:O,

374 where for L, R h a n d e d e -

g2

q2

A~ = - Q,-~ 4,;,rOte.m (q2 + M ~ ) gL'RGL/coS2Ow'

g2 q2 i 2 Ai2 = - Qi -~ 47r~Xe.m" (q2 + M 2) gL, R G R / c ° s Ow" F o r L, R h a n d e d e +, gt. a n d gR should be swopped. G i a n d g are given by

GR _

Qisin20w '

G [ = I j - QisinZ0w, gR = sinZ0w, gL = -- ½ + sinZ0w, Qi is the q u a r k charge a n d I j is the third c o m p o n e n t of weak isospin.

References [1] M. Veltman, Nucl. Phys. B123 (1977) 89 [2] J.F. Wheater, J. Phys. 43 (1982) 305 C3 [3] C,H. Llewellyn Smith, Nucl. Phys. B228 (1983) 205 [4] Study on the proton electron storage ring project HERA, DESY HERA 80/01 [5] D. Yu Bardin et al., J. Phys. G7 (1981) 1331 [6] A.J. Buras and K.J.F. Gaemers, Nucl. Phys. B132 (1978) 249 [7] J.E Kim et al., Rev. Mod. Phys. 53 (1981) 211 [8] G. Arrfison et al., CERN preprint EP/83-73 [9] H. Abramowicz et al., Z. Phys. C15 (1982) 19 [10] E.A. Paschos and M. Wirbel, Nucl. Phys. B194 (1982) 189