5pt
GLAS-PPE/2000-081st September 2000
Structure Functions at High $Q^2$
Structure Functions at High Q2
Anthony T. Doyle*
Abstract
Progress in the study of structure functions in deep inelastic scattering
is reviewed. A brief introduction to the formalism and
the status of global analyses of parton distributions and their uncertainties
is given.
The review focuses on recent developments in the following areas:
HERA results on F2, FL and charm;
the resolution of the CCFR-NMC discrepancy;
Tevatron jet cross-sections at different energies; and
HERA high-Q2 cross-sections and the measurement of xF3.
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Plenary talk presented at the 7th Conference |
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on the Intersections of Particle and Nuclear Physics, |
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CIPANP2000 Quebec, May 2000. |
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Slides are available from |
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http://ppewww.ph.gla.ac.uk/%7Edoyle/ |
Introduction
The neutral current (NC) cross-section l(k)N(p)® l(k¢)X(p¢)
for a lepton (e, m)
with four-momentum k scattering off a nucleon with
four-momentum p can be expressed as
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d2s(l± N)
dx dQ2
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= |
2 pa2
x Q4
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·[ Y+ F2(x, Q2) -±Y- xF3(x, Q2) - y2 FL(x, Q2)] |
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where
Q2 is the four-momentum transfer squared,
x = Q2/2p·q is the Bjorken scaling variable,
y = p·q/p·k is the inelasticity variable and
Y± = 1 ±(1-y)2.
The contribution from F2 dominates the cross-section and is measured
directly.
To investigate sensitivity to FL at large y or xF3 at large Q2,
the reduced cross-section
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~
s
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º |
xQ4
2pa2
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1
Y+
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d2s
dxdQ2
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º F2 -± |
Y-
Y+
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xF3 - |
y2
Y+
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FL |
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is adopted. The contribution
from FL is a QCD correction which is important at large y:
the characteristic y dependence is utilised in the determination of FL
at low Q2.
The contribution from xF3, due to gZ interference and Z exchange,
violates parity conservation and enters for Q2 @ MZ2: the change
of sign for opposite charges of the incoming lepton beam enable xF3 to be
extracted.
In the quark parton model (or in the DIS scheme of NLO QCD)
F2/x is
the ``charge-weighted" sum of the quark densities
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F2(x, Q2) = x |
å
f
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Af(Q2) ·S(x,Q2) |
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where
Af(Q2) = ef2 - 2vlvfefPZ+(vl2+al2)(vf2+af2)PZ2
contains vector and axial terms due to g, gZ interference
and Z exchange respectively,
PZ = 1/(4sin2qWcos2qW) ·Q2/(Q2+MZ2)
is the Z propagator term and
S(x,Q2) = qf(x,Q2) + [`q]f(x,Q2)
is the singlet summed quark and anti-quark distributions.
Similarly,
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xF3(x, Q2) = x |
å
i
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Bf(Q2) ·qNS(x,Q2) |
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where
Bf(Q2) = 2alafeiPZ+4vlalvfafPZ2
contains vector and axial terms due to g-Z and Z exchange
respectively,
and
qNS(x,Q2) = qf(x,Q2) - [`q]f(x,Q2)]
is the non-singlet difference of the quark and anti-quark (valence quark)
distributions.
The charged current (CC) cross-section
l+(l-)N ® n([`(n)])X proceeds via W exchange
and can be expressed as
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d2sCC(l± N)
dx dQ2
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= |
GF2
4 px
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æ
ç
è
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MW2
Q2+MW2
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ö
÷
ø
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2
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|
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where GF is the Fermi constant
and a positive (negative) charged lepton
beam picks out valence d (u) quarks.
Given a phenomenological input as a function of x,
the parton distributions are evolved to different physical scales (Q2)
via the DGLAP evolution equations.
The non-singlet contribution evolves as
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¶qNS(x,Q2)
¶t
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= |
as(Q2)
2 p
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PqqNS ÄqNS(x¢,Q2) |
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where t = ln(Q2/L2) and
the Pij's represent the NLO DGLAP splitting probabilities for
radiating a parton with momentum fraction x from a parton with higher
momentum x¢.
Quantities such as xF3 provide a measure of as(Q2) which
is insensitive to the a priori unknown gluon distribution.
Similarly,
the singlet quark and gluon densities are coupled via the matrix equation
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¶
¶t
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æ
ç
è
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ö
÷
ø
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= |
as(Q2)
2 p
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é
ê
ë
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ù
ú
û
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Ä |
æ
ç
è
|
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ö
÷
ø
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and quantities such as F2 provide input for
S(x,Q2) as well as coupled knowledge of
as(Q2) and the gluon, g(x,Q2).
The gluon distribution, g(x,Q2), is determined through the
scaling violations of F2 at low x
or constrained via jet or direct photon production
at large x (where the gluon enters directly).
Figure 1: Parton momentum densities (full curves) at Q2 = 10 GeV2
including
uncertainties (shaded bands) compared to the CTEQ4M set (dashed curves).
At the starting scale, Qo2, the light valence quarks
(uv and dv) and the
sea of quark and anti-quarks (S)
as well as the gluon (g) are attributed a given functional form.
The measured structure functions are then described by the convolutions of the
parton densities with the appropriate NLO matrix elements.
In Fig. 1 the latest analysis of the
BCDMS, CCFR, E665, H1, NMC, SLAC and ZEUS
DIS measurements by Botje [1]
gives the parton densities and their associated uncertainties.
The uncertainties are determined from
a full treatment of the systematic uncertainties of the data,
varying the strong coupling constant (DaS(MZ) = ±0.005),
varying the renormalisation and
factorisation scale uncertainty from
Q2/2 to 2Q2, and assessing effects due to uncertainties on
the strange sea, nuclear effects and the charm threshold (treated here as a
light quark above threshold).
The analysis is noteworthy in that care has been taken
to provide the correlation coefficients for each of the parton densities.
The fit agress with the latest CTEQ and MRST fits, within the uncertainties
of up to 10% on each parton density at Q2 = 10 GeV2.
HERA Structure Function Measurements
Figure 2: ZEUS, E665 and NMC F2 -log10x values
versus Q2. Fixed x values are indicated by the full lines and on the
right hand side of the plot. Fixed W values are indicated by the
dashed lines and on the left hand side of the plot.
Figure 3: Derivatives of F2 with respect to log10Q2 for fixed W
values indicated by the symbols. In the upper plot the data are plotted as
a function of Q2 and in the lower plot as a function of x.
HERA F2(x,Q2)
measurements extend to:
low y (yHERA ~ 0.005) providing overlap
with the fixed-target experiments;
very low x (x [ < || ( ~ )]10-5) at low Q2 exploring the
transition region from soft to hard physics;
high y (y ® 0.8) giving
sensitivity to FL;
high x® 0.7 probing sensitivity to
electroweak effects in F2 and xF3 as well as
constraining the valence quarks at large Q2.
In Fig. 2 recent ZEUS F2 data, as well as fixed target data
from E665 and NMC, are plotted as function of Q2 for
fixed values of x.
The observed scaling violations are intimately coupled with the rise
of F2 with decreasing x via the gluon density
(in leading order dF2/dlnQ2 ~ xg(x)
neglecting sea quark contributions).
In order to study these scaling violations,
a compact parameterisation
F2(x) = A(x)+B(x)log10Q2+C(x)(log10Q2)2 has been fitted,
indicated by the
full line. The F2 points are observed to rise significantly at low x
with a tendency to flatten at the lowest x values.
Lines of fixed centre-of-mass energy
W2 = Q2(1/x-1) are indicated by the dashed lines.
The derivative dF2/dlog10Q2 = B(x)+C(x)log10Q2 is plotted in
Fig. 3 for constant W as a function of Q2 (upper) and x (lower plot),
motivated by similar theoretical plots in an
analysis of low-x data by Golec-Biernat and Wüsthoff [2],
incorporating saturation effects. The turnover of the derivative occurs
at higher Q2 (or x) for increasing W, consistent with predictions
from various models which go beyond the DGLAP formalism.
The DGLAP formalism itself is however
also sufficiently flexible to describe the data,
provided that the gluon (proportional to the derivative) decreases
significantly at low (x,Q2). The behaviour of the HERA low (x,Q2) data
remains a challenge to our understanding of QCD: this presentation of the
data represents one way in which to address the problem, although there are
various physical interpretations of the observed behaviour.
Figure 4: H1 determination of FL versus x for four values of Q2,
compared to the NLO QCD fit expectation and data from BCDMS, NMC and SLAC.
The contribution of FL enters as a QCD correction to the total DIS
cross-section. As such it provides a
method to calibrate the gluon at low x.
H1 [3]have used two methods to extract FL from the reduced
cross-section [(s)\tilde] = F2 - y2/Y+ ·FL
at high y. This is the region where the scattered electron energy
is low: in the H1 analysis scattered positrons are measured down to
Ee¢ @ 5 GeV
and backgrounds reduced by requiring the associated track to have correct
charge. FL is determined as a function of Q2 ³ 4 GeV2
by measuring local derivatives of
¶[(s)\tilde]/¶logy and observing deviations
from a straight line (or a QCD fit for Q2 > 10 GeV2)
at high y.
The extracted FL data are given in Fig. 4,
compared to the H1 NLO QCD fit to H1,
NMC and BCDMS F2 data as well as to FL data from SLAC and NMC.
The data are in agreement with the QCD expectations although the
precision is limited by statistics and uncertainties due to the input
F2.
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Figure 5: ZEUS ratio of F2c[`c]/F2 as a function of x for
four values of Q2, compared to NLO QCD expectations.
Another method to calibrate the gluon distribution is to extract the
contribution due to charm.
D*® (K p) ps measurements in DIS provide a
significant test of the gluon density of the proton
determined from the scaling violations of F2.
They also help to constrain theoretical uncertainties in the
fits to F2 where different prescriptions for charm production
are adopted. The D* data are extrapolated outside the measured
range of h, pT to obtain the charm cross-section and hence
F2c[`c].
The ratio F2c[`c]/F2 is plotted in Fig. 5
as a function of x for four Q2 values, compared to the NLO QCD
expectations, where the dashed curves correspond to the uncertainty due
to the parton distributions from the NLO fit.
There is an overall normalisation uncertainty of approximately 10% (not
shown) due (mainly) to the charm hadronisation fraction to D*.
At high W (or low x), the charm contribution to F2 is significant
(up to 30%) and therefore a careful treatment of charm production is
required in any analysis of low (x,Q2) data.
The threshold region of of x @ 10-2, where the charm
cross-section is changing rapidly, is also important in consideration
of the CCFR-NMC discrepancy.
Comparison of nN and lN Data
To leading order the muon (NMC) and neutrino (CCFR)
structure functions are related by the ``5/18ths rule"
from quark counting.
Here the
strange sea enters as a correction,
since s® c production is reduced in nN scattering due to
the charm mass.
The original CCFR analysis, where data were
corrected using the dimuon result to constrain the strange sea,
lay significantly ( @ 20%) above the NMC data for all values of
Q2 at x < 0.1.
Advances in the H1 and ZEUS analyses of low y data, lead
to a region of overlap where the HERA and fixed-target experiments
can be compared with a precision of @ 5%.
In Fig. 6, a comparison of the CCFR
(nN), H1, NMC and E665 (lN) data indicates that the discrepancy lies
with the treatment of the CCFR data which is significantly above the lN data
at low x.
Figure 6: Original CCFR analysis compared to H1, E665 and NMC F2
data. The F2 data are plotted as a function of Q2 for various x
values. The full line is the H1 NLO QCD fit.
Figure 7: Model-independent CCFR analysis of the
ratio of 5/18 F2nN/F2lN
as a function of Q2 in various x intervals, compared to NLO
calculations incorporating various treatments of charm.
The CCFR data is divided by NMC, BCDMS and SLAC data.
The HERA comparisons were made in parallel with a re-analysis of CCFR data by
Yang and Bodek, reported at this conference [4].
The previous CCFR analysis used a slow rescaling variable
x = x(1-mc2/Q2), where mc is the effective charm mass,
in order to suppress charm production and correct F2.
In the latest CCFR analysis no explicit correction is made to the F2
data for charm (or heavy target) effects.
Various NLO treatments of charm are now available which allow data to
be compared more precisely.
The variable flavour scheme, adopted by CTEQ and MRST with different
approaches to the threshold behaviour and fixed-order treatment,
handle charm as another parton in the proton,
a good approximation for Q [ > || ( ~ )]mc.
The fixed flavour scheme (FFS), adopted by GRV,
treats charm as generated dynamically in NLO QCD,
a good approximation for Q @ mc.
The ratio of
5/18 F2nN/F2lN is plotted in Fig. 7.
A clear departure from unity is observed at the lowest x values.
The data are
compared to NLO charm calculations from MRST, CTEQ and GRV which differ
somewhat in shape, but provide a good description of the data ratios
at all values of x.
The difference in xF3 from n and [`(n)]
data (DxF3n-[`(n)], not shown)
is also sensitive to charm production and provides an additional constraint
on the calculations.
The significant discrepancy between F2nN and F2l N data
is thus resolved. The area is now open for more detailed comparisons
of NLO charm methods, constraints on the strange sea from the ratio of
CCFR to NMC and HERA data compared to CCFR dimuon data as well as an
understanding nuclear shadowing corrections for nFe (CCFR) data.
In addition
new data from NuTeV tagging n and [`(n)]
beams will provide more information on
DxF3n-[`(n)], as well as improving the
measurement of sin2qW.
Inclusive Jet Production at the Tevatron
The Tevatron collider provides a unique opportunity to study the properties
of hard interactions in p[`p] collisions at short distances.
CDF and DØ have measured jet cross-sections at Ös = 1800 GeV over ten
orders of magnitude in d2s/dETdh up to ET = 500 GeV, half-way to
the kinematic limit.
These data indicate that the gluon should be enhanced at large x compared to
direct photon measurements. One open area for further experimental and
theoretical investigation is in this measurement at different energies.
The ratio of the inclusive jet cross-section at Ös = 630 GeV
over Ös = 1800 GeV is plotted in Fig. 8 as a function
of scaled jet transverse energy xT = 2ET/Ös.
The ratio should lead to a partial cancellation of experimental and
theoretical systematics [5].
However, it is clear that there is an experimental discrepancy
at the lowest values of xT [ < || ( ~ )]0.15. In addition, the data lie below the
NLO QCD predictions at all values of xT. Two explanations have been proposed:
first that different renormalisation scales could be used for the
theoretical calculations at the two energies; second that
a relatively small shift in jet energies by approximately 3 GeV
(due to non-perturbative jet pedestal or underlying event effects)
brings the theory in agreement.
The forthcoming Run II will provide higher energy data in order to
clarify these issues.
Figure 8: Ratio of CDF and DØ inclusive jet cross sections
at Ös = 1800 GeV
to Ös = 630 GeV, as a function of xT = 2ET/Ös,
compared to NLO QCD. The shaded band indicates the DØ energy scale
uncertainty
High Q2 Cross-Sections at HERA
The HERA collider provides a unique window to explore ep interactions
at the highest energies, extending the range of momentum transfer Q2
by about two orders of magnitude compared to fixed-target experiments.
As the HERA luminosity increases we explore the region of
Q2 ~ 104 GeV2, where electroweak effects play a rôle.
It is in this unexplored kinematic region that we are
sensitive to deviations from the standard model (SM).
The theoretical uncertainties on the SM e± p NC and CC cross-sections
were determined as discussed with respect to the parton densities of Fig. 1
and correspond to
@ 6-8% on the NC cross-sections and
@ 6-12% on the CC cross-sections at
the highest accessible Q2 values.
The measured cross-sections generally
agree with theory within these uncertainties
and therefore represent a benchmark for the standard model.
The example cross-sections, discussed below, are corrected to the electroweak
Born level.
The e+p charged-current cross-section [7,8]
is sensitive to the valence d-quark
distribution in the proton
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d2 se+p
dx dQ2
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@ |
GF2
2 p
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æ
ç
è
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MW2
Q2 + MW2
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ö
÷
ø
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2
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[ |
u
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+ |
c
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+ (1-y)2 (d + s)]. |
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Figure 9: ZEUS charged current cross-section
dsCC(e+p)/dx versus x for Q2 > 400 GeV2
and ratio
with respect to the standard model prediction using the CTEQ4D PDF (lower plot).
The curves and shaded band in the lower plot correspond to the NLO PDF fits
described in the text.
In the upper plot of Fig. 9 the ZEUS cross-section is described by the SM,
falling over four orders of magnitude.
The ratio of the data to the SM, adopting the CTEQ4D PDF,
is shown in the lower plot of Fig. 9 where
good agreement is observed as a function of x for Q2 > 400 GeV2.
Comparison of the data with the PDF uncertainties
(shaded band) indicates that the data will help to
constrain the d-quark densities at large x.
The uncertainties on the data are large in this region, but an increase of
the ratio of d/u quarks, such as that proposed by Yang and Bodek [6]
results in better agreement with the data than the standard PDFs.
The CC cross-section is suppressed
relative to NC exchange due to the W
propagator term: this characteristic
dependence on Q2 has been fitted to yield values for the mass of the
exchanged space-like W-boson.
The `propagator-mass' fit value (with GF fixed) of
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MW = 81.4+2.7-2.6(stat.) ±2.0 (sys.) +3.3-3.0 (PDF) GeV |
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agrees with the PDG value of
MW = 80.41±0.10 GeV obtained from
time-like production of W bosons at LEP and the Tevatron.
The consistency of the data with the SM has been checked using the
PDG values for GF, a, MZ and Mt, yielding a consistent value
of
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MW = 80.50+0.24-0.25 (stat.) +0.13-0.16 (sys.) ±0.31 (PDF)+0.03-0.06 ( DMt, DMH, DMZ ) GeV. |
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This result is in agreement with the value of MW = 80.35±0.21 GeV
obtained by NuTeV [9].
The NC cross-sections are particularly sensitive to the valence u-quark
distribution in the proton
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d2se± p
dx dQ2
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@ |
2 pa2
x Q4
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[ Y+ F2(x, Q2) -±Y- xF3(x, Q2)]. |
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Here, F2 is the generalised structure function,
incorporating g and Z terms, which is sensitive to the singlet
sum of the quark distributions
(xq + x[`q]) and
xF3 is the parity-violating (Z-contribution) term which is
sensitive to the non-singlet valence quark distributions
(xq - x[`q]).
The e± p data are now becoming sensitive to
the difference in sign of the xF3 contribution (due mainly to
electroweak gZ interference effects)
which suppresses (enhances) the e+p (e-p) NC cross-section by
~ 30% for Q2 > 10,000 GeV2.
This is illustrated in Fig. 10 where the H1 reduced cross-section
for e+ p (820 GeV) and e-p (920 GeV) data is compared.
A further step has been taken in the ZEUS analysis where the
difference of the cross-sections (modified slightly due to the
different beam energies) yields a first measurement of xF3 [10].
In Fig. 11 the scaling of xF3 is shown as a
function of Q2 compared to the CTEQ4D and MRST(99) PDFs:
the statistical precision is currently limited,
although the systematic precision on data and theory is
good, boding well for a future determination of aS(MZ).
Figure 10: H1 neutral current reduced cross-section as a function of Q2
for various values of x. The e+p and e-p data are compared to the
NLO QCD fit expectations (full line) obtained from fitting the e+p data.
Figure 11: ZEUS xF3 data extracted from the difference
of the e-p and e+p neutral current cross-sections, compared to the
CTEQ4D and MRST(99) PDF expectations.
Outlook:
The first phase of HERA running from 1992-2000 is now complete.
An upgrade of HERA during 2000-2001 will enable luminosities to be increased
by a factor of 4-5. In the longer term, DESY has recognised a
future programme for ep physics by ensuring that the proposed
TESLA 500 GeV e linear collider is capable of combining with the HERA
proton ring to provide the next generation of deep inelastic scattering
measurements.
The historical perspective of such an endeavour is illustrated in Fig. 12,
where the resolved dimension (d [fm] = 0.2/Qmax [GeV])
is plotted as a function of year.
Progress on the path of DIScovery started with
Rutherford, observed to be 30 years ahead of his time,
and is planned to reach down to
sub-attometre distances with the TERA collider.
Figure 12: Resolved dimension as a function of year including the proposed
TERA (=TESLAÄHERA) collider.
Acknowledgements
It is a pleasure to thank the organisers for
an excellent conference.
Many thanks to Ian Bertram, Chris Cormack, Doris Eckstein, Brian Foster,
Sergey Levonian, Jason Webb, Un-Ki Yang, Rik Yoshida
and all the speakers in the structure functions parallel session
for their help and advice.
References
- [1]
- M. Botje, Eur. Phys. J. C14 (2000) 285.
- [2]
- K. Golec-Biernat and M. Wüsthoff, Phys. Rev. D59 (1999) 014017.
- [3]
- S. Levonian, CIPANP2000 proceedings.
- [4]
- U.K. Yang, CIPANP2000 proceedings.
- [5]
- R. Hirosky, CIPANP2000 proceedings.
- [6]
- U.K. Yang and A. Bodek, Phys. Rev. Lett. 82 (1999) 2467.
- [7]
- H1 Collab., C. Adloff et al., Eur. Phys. J. C13 (2000) 609.
- [8]
- ZEUS Collab., J. Breitweg et al., Eur. Phys. J. C12 (2000) 411.
- [9]
-
CCFR/NuTeV Collab., K.S. McFarland et al., Eur. Phys. J. C1 (1998) 509.
- [10]
- C. Cormack, CIPANP2000 proceedings.
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