QCD PRECISION MEASUREMENTS AND STRUCTURE
FUNCTION EXTRACTION AT A HIGH STATISTICS, HIGH
ENERGY NEUTRINO SCATTERING EXPERIMENT:
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Citation
Adams, T. et al. “QCD PRECISION MEASUREMENTS AND
STRUCTURE FUNCTION EXTRACTION AT A HIGH
STATISTICS, HIGH ENERGY NEUTRINO SCATTERING
EXPERIMENT: NuSOnG.” International Journal of Modern
Physics A 25.05 (2010) : 909. Copyright c2010 World Scientific
Publishing
As Published
http://dx.doi.org/10.1142/s0217751x10047828
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World Scientific
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Wed May 25 15:22:49 EDT 2016
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QCD Preision Measurements and Struture Funtion Extration at a
High Statistis, High Energy Neutrino Sattering Experiment: NuSOnG
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T. Adams , P. Batra , L. Bugel , L. Camilleri , J.M. Conrad , A. de Gouvêa , P.H. Fisher , J.A. Formaggio ,
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J. Jenkins , G. Karagiorgi , T.R. Kobilarik , S. Kopp , G. Kyle , W.A. Loinaz , D.A. Mason ,
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R. Milner , R. Moore , J. G. Morfín , M. Nakamura , D. Naples , P. Nienaber , F.I Olness ,
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J.F. Owens , S.F. Pate , A. Pronin , W.G. Seligman , M.H. Shaevitz , H. Shellman , I. Shienbein ,
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, T. Takeuhi , C.Y. Tan , R.G. Van de Water , R.K. Yamamoto , J.Y. Yu
M.J. Syphers , T.M.P. Tait
1
arXiv:0906.3563v1 [hep-ex] 19 Jun 2009
2
5
Amherst College, Amherst, MA 01002
Argonne National Laboratory, Argonne , IL 60439
3
Central College, Pella IA 50219
4
Columbia University, New York, NY 10027
Fermi National Aelerator Laboratory, Batavia IL 60510
6
Florida State University, Tallahassee, FL 32306
7
Los Alamos National Aelerator Laboratory,
Los Alamos, NM 87545
8
LPSC, Université Joseph Fourier Grenoble 1,
38026 Grenoble,
9
Cambridge,
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11
14
Frane
Massahusetts Institute of Tehnology,
MA 02139
Nagoya University, 464-01, Nagoya, Japan
New Mexio State University, Las Crues, NM 88003
12
Northwestern University, Evanston, IL 60208
13
University of Pittsburgh, Pittsburgh, PA 15260
Saint Mary's University of Minnesota, Winona, MN 55987
15
Southern Methodist University, Dallas, TX 75205
16
University of Texas, Austin TX 78712
17
Virginia Teh, Blaksburg VA 24061
(Dated: Deember 27, 2010)
We extend the physis ase for a new high-energy, ultra-high statistis neutrino sattering experiment, NuSOnG (Neutrino Sattering On Glass) to address a variety of issues inluding preision
QCD measurements, extration of struture funtions, and the derived Parton Distribution Funtions (PDFs). This experiment uses a Tevatron-based neutrino beam to obtain a sample of Deep
Inelasti Sattering (DIS) events whih is over two orders of magnitude larger than past samples.
We outline an innovative method for tting the struture funtions using a parameterized energy
shift whih yields redued systemati unertainties. High statistis measurements, in ombination
with improved systematis, will enable NuSOnG to perform diserning tests of fundamental Standard Model parameters as we searh for deviations whih may hint of Beyond the Standard Model
physis.
2
Contents
I. INTRODUCTION
I. Introdution
2
A. NuSOnG: Preision Struture Funtions and
Inisive QCD Measurements
A. NuSOnG: Preision Struture Funtions and
Inisive QCD Measurements
2
II. Deep Inelasti Sattering and Parton
Distribution Funtions
2
III. Experimental Extration of Struture
Funtions in NuSOnG
4
The searh for new physis at the Terasale energy
∼ 1 TeV and beyond is the highest priority for
sales of
partile physis.
NuSOnG is a proposed high energy, high statistis
neutrino sattering experiment that an searh for new
A. Desription of NuSOnG
4
physis from the keV through TeV energy sales via pre-
B. Desription of NuSOnG Calibration Beam
4
ision eletroweak and QCD measurements.
5
SOnG experiment ould reord almost one hundred thou-
C. Experimental Extration of Struture
Funtions in NuSOnG
D. Fitting for
∆xF3
IV. Nulear Eets
A. Nulear eets at small
6
10
x
B. Nulear Eets in Neutrino Interations
10
12
C. Measuring Nulear Eets with the Minerva
and NuSOnG Detetors
V. QCD Fits
14
14
A. PDFs
15
B. Struture funtions
15
C. Constraints on the PDFs
16
VI. Isospin (Charge Symmetry) Violation
and ∆xF3
∆xF3 and Isospin Violations
B. Measurement of
5
F2CC (x, Q2 ) − F2N C (x, Q2 )
∆F2 ≡ 18
C. Other Measurements of CSV
A.
17
During its ve-year data aquisition period, the Nusand neutrino-eletron elasti satters, and hundreds of
millions of Deep Inelasti Sattering (DIS) events, exeeding the urrent world data sample by more than an
order of magnitude.
This experiment an address onerns related to extration of struture funtions and their derived Parton
Distribution Funtions (PDFs), investigate nulear orretions, onstrain isospin violation limits, and perform
inisive measurement of heavy quarks.
II. DEEP INELASTIC SCATTERING AND
PARTON DISTRIBUTION FUNCTIONS
17
18
Experiment
18
D. Conlusions on Charge Symmetry Violation 19
VII. Measurements of the Heavy Quarks
19
A. Measurement of the Strange Sea
19
B. Strange Quark Contribution to the Proton
Spin
C. Measurements of the Charm Sea:
ν
DIS
ν̄
DIS
main
isosalar
events events target orretion
CCFR
0.95M 0.17M
iron
5.67% [4℄
NuTeV
0.86M 0.24M
iron
5.74% [5℄
606M
glass
isosalar
NuSOnG
34M
Table I: Comparison of statistis and targets for parton distri-
20
bution studies in NuSOnG ompared to the two past highest
21
statistis DIS neutrino sattering experiments.
1. Charm Prodution
21
2. Bakgrounds
22
3. Intrinsi Charm
22
Obtaining a high quality model of the parton distribution funtions in neutrino and antineutrino sattering
VIII. Summary and Conlusions
23
is ruial to the NuSOnG eletroweak measurements [3℄.
Aknowledgments
23
NuSOnG will go a step beyond past experiments in ad-
24
(PDFs) by making high statistis measurements for neu-
Referenes
dressing the systematis of parton distribution funtions
trino and antineutrino data separately. Table I shows the
large improvement in statistis for NuSOnG ompared
to NuTeV and CCFR, the previous highest statistis experiments. Issues of unertainties on the nulear orretions are avoided by extrating PDFs on SiO2 diretly, in
similar fashion to the NuTeV Pashos-Wolfenstein (PW)
analysis.
The dierential ross setions for neutrino and antineutrino CC DIS eah depend on three struture funtions:
F2 , xF3
and
RL .
They are given by:
3
"
G2F M Eν
d2 σ ν(ν)N
ν(ν)N
(x, Q2 )
=
F2
2 )2
dxdy
π (1 + Q2 /MW
!
ν(ν)N
±xF3
y i
, (1)
y 1−
2
x
harged lepton sattering rather than neutrino sattering
[7℄. Therefore, neutrino experiments have used harged
funtions are diretly related to the PDFs.
2
The funtion xF3 (x, Q ) is unique to the DIS ross
and
is for
ν(ν)
M xy
+1−y−
ν(ν)N
2Eν
2 + 2RL
(x, Q2 )
is the Bjorken saling variable, y the inelastiity, and
Q2 the squared four-momentum transfer. The struture
where
+(−)
y 2 + (2M xy/Q)2
sattering.
In this equation,
RL as an input to the measurements of xF3
[8℄. This, however, is just a matter of the statis-
lepton ts to
F2
tis needed for a global t to all of the unknown struture
x and Q2 bins [9℄. With the high statistis
setion for the weak interation. It originates from the
funtions in
parity-violating term in the produt of the leptoni and
of NuSOnG, preise measurement of
hadroni tensors. For an isosalar target, in the quarkparton model, where
s = s̄
and
c = c̄,
As an improvement on past experiments, the high
xF3νN (x) = x (u(x) + d(x) + 2s(x)
¯ − 2c̄(x) ,
−ū(x) − d(x)
(2)
xF3ν̄N (x) = xF3νN (x) − 4x (s(x) − c(x)) .
In past experiments, the average of
xF3
xF3,LO =
2
(3)
statistis of NuSOnG allows measurement of up to six
ν
ν̄
ν
ν̄
ν
ν̄
struture funtions: F2 , F2 , xF3 , xF3 , RL and RL . This
is done by tting the neutrino and antineutrino data sep2
arately in x, y and Q as desribed in Eq. (1). The rst
steps toward tting all six struture funtions indepen-
for neutrinos
and antineutrinos has been measured. Dening
1
νN
+ xF3ν̄N ), at leading order in QCD,
2 (xF3
X
RL will be possible
from neutrino sattering for the rst time.
xF3 =
2
xqi (x, Q ) − xqi (x, Q ).
dently were made by the CCFR experiment [10℄, however
xF3ν , xF3ν̄ , and F2 -average
and R-average ould be measured, where the average is
statistis were suh that only
over
(4)
i=u,d..
x dependene, and thus anel, xF3
an be thought
of as probing the valene quark distributions. The dierene between the neutrino and antineutrino parity violatνN
ing struture funtions, ∆(xF3 ) = xF3
− xF3ν̄N , probes
the strange and harm seas. (Cf. Se. VI.)
2
The funtion F2 (x, Q ) appears in both the ross setion for harged lepton (e or
for
ν
µ) DIS and the ross setion
DIS. At leading order,
F2,LO =
X
ei
ν̄ .
A global t of up to six struture funtions
the underlying PDFs whih an aount for the nulear
e2i (xqi (x, Q2 ) + xqi (x, Q2 )),
In addition to tting to the inlusive DIS sample,
neutrino sattering an also probe parton distributions
through exlusive samples. A unique and important ase
is the measurement of the strange sea through harged
urrent (CC) opposite sign dimuon prodution.
the neutrino interats with an
The harmed hadron's semi-leptoni deay (with branhing ratio
Bc ∼ 10%) produes a seond muon of
opposite
sign from the rst:
(5)
νµ + N −→ µ− + c + X
֒→ s + µ+ + νµ .
is the harge assoiated with the interation. In
the weak interation, this harge is unity. For hargedlepton sattering mediated by a virtual photon,
When
s or d quark, it an produe
a harm quark that fragments into a harmed hadron.
i=u,d..
where
and
and isospin violation issues disussed below.
To the level that the sea quark distributions have the
same
ν
in NuSOnG would allow separate parameterizations of
ei
(6)
is
the frational eletromagneti harge of the quark avor.
e(µ)N
νN
Thus F2
and F2
are analogous but not idential
Similarly, with antineutrinos, the interation is with an
s
or
d,
and omparison yields useful information about spei
parton distribution avors [6℄ and harge symmetry violation as disussed below. In past neutrino experiments,
F2ν and F2ν̄ have been taken to be idential and an average
F2 has been extrated, although this is not neessarily
ν µ + N −→ µ+ + c + X
֒→ s + µ− + ν µ .
(7)
true in nulear targets, as disussed below.
2
Similarly, RL (x, Q ), the longitudinal to transverse vir-
The opposite sign of the two muons an be determined
tual boson absorption ross-setion ratio, appears in both
trometer. Study of these events as a funtion of the kine-
for those events where both muons reah the toroid spe-
the harged-lepton and neutrino sattering ross setions.
mati variables allows extration of the strange sea, the
RL from the ross setion, one must bin in the
2
variables x, Q and y . This requires a very large data
set. To date, the best measurements for RL ome from
harm quark mass, the harmed partile branhing ratio
To extrat
(Bc ), and the Cabibbo-Kobayashi-Maskaka matrix element, |Vcd |.
4
muon spetrometer to measure outgoing muon momenta.
The target alorimeter will be omposed of 2,500 2.5 m
×5m×
5 m glass planes interspersed with proportional
tubes or sintillator planes. This gives a target whih is
made of isosalar material with ne 1/4 radiation length
sampling. The detetor will be omposed of four target
setions eah followed by muon spetrometer setions and
low mass deay regions to searh for long-lived heavy neutral partiles produed in the beam. The total length of
the detetor is
∼200 m
and the duial mass for the four
target alorimeter modules will be 3 kiloton whih is 6
times larger than NuTeV or CHARM II. Figure 2 shows
Figure
(E
1:
The
6
dφ/dE/10 P OT )
assumed
based
on
energy-weighted
the
NuTeV
ux
a simulated
νµ
harged urrent event in the detetor.
experiment
in a) neutrino mode (left) and b) antineutrino mode (right).
a) In neutrino mode the uxes are ordered: upper (blak),
muon neutrino; middle (blue), eletron neutrino and antineutrino; lower (red), muon antineutrino.
mode the uxes are ordered: upper (red), muon antineutrino;
middle (blue),
B. Desription of NuSOnG Calibration Beam
b) In antineutrino
eletron neutrino and antineutrino;
lower
(blak), muon neutrino.
The requirements for NuSOnG alibration beam would
be similar to those of NuTeV. Tagged beams of hadrons,
eletrons,
and muons over a wide energy range (5-
200 GeV) would be required. The alibration beam will
have the ability to be steered over the transverse fae of
III. EXPERIMENTAL EXTRACTION OF
STRUCTURE FUNCTIONS IN NUSONG
the detetor in order to map the magneti eld of eah
toroid with muons.
Steering for hadrons and eletrons
would be less ruial than it was in NuTeV's ase, but
A. Desription of NuSOnG
would still be useful.
The alibration beam an be onstruted with a sim-
The NuSOnG detetor was designed to be sensitive
ilar design to NuTeV. Upstream elements were used to
ν -eletron
selet hadrons, eletrons, or muons. An enhaned beam
sattering, though this
of eletrons was produed by introduing a thin lead radi-
paper fouses mainly on the latter proess. The design
ator into the beam and detuning the portion of the beam
has been desribed in detail in Refs. [3, 11℄ and so we
downstream of the radiator.
provide only a brief summary here.
in the nominal beam tune to remove eletrons. Partile
to a wide range of neutrino interations from
sattering as well as
ν -nuleon
A radiator was also used
The neutrino beam would be produed via a high en-
ID (a threshold Cerenkov and TRDs) was inorporated
ergy external proton beam from a high intensity aeler-
in the spetrometer and used to tag eletrons when run-
∼1
TeV. The Fermilab Tevatron is
ning at low energy. A pure muon beam was produed by
an existing example. The CERN SPS+ [12, 13℄, whih
introduing a 7 m long beryllium lter in the beam as an
is presently under onsideration beause of its value to
absorber.
ator with energy of
the LHC energy and luminosity upgrades and to a future
beta beam, is a seond example.
The NuTeV alibration spetrometer determined inoming partile momenta with a preision of better than
The NuSOnG beam design will be based on the one
0.3% absolute [14℄.
The NuSOnG goal for alibration-
used by the NuTeV experiment, whih is the most reent
beam preision would be to measure energy sales to a
high energy, high statistis neutrino experiment.
The
preision of about 0.5%, and we demonstrate (in later
experiment would use 800 GeV protons on target fol-
text of this paper) that this an be improved with ts to
lowed by a quad-foused, sign-seleted magneti beam-
neutrino data.
line. The beam ux, shown in Fig. 1, has very high neutrino or antineutrino purity (∼98%) and small
νe
on-
For omparison, using the alibration beam, NuTeV
ahieved 0.43% preision on absolute hadroni energy
tamination (∼2%) from kaon and muon deay. Using an
sale and 0.7% on absolute muon energy sale (dominated
upgraded Tevatron beam extration it is expeted that
19
NuSOnG ould ollet 5 × 10
protons/yr, an inrease
by the ability to aurately determine the toroid map).
by a fator of 20 from NuTeV. With this high intensity,
important in order to ahieve high measurement auray
Preise knowledge of the muon energy sale is espeially
suh a new faility would also produe a neutrino beam
on the neutrino uxes using the low-ν method. For ex-
from the proton dump having a sizable fration of tau
ample, a 0.5% preision on muon energy sale translates
neutrinos for study.
into about a 1% preision on the ux. Both energy sales
The
baseline
detetor
design
is
omposed
of
a
are important for preision struture funtion measure-
ne-grained target alorimeter for eletromagneti and
ments, and were the largest ontributions to struture
hadroni shower reonstrution followed by a toroid
funtion measurement unertainties in NuTeV [15℄.
5
Figure 2: A simulated muon neutrino, harged urrent event in the NuSOnG detetor.
C. Experimental Extration of Struture Funtions
in NuSOnG
The high statistis of the NuSOnG experiment makes
it possible to extrat the struture funtions diretly
2
from the y -distributions within bins of x, Q . Previous
lower-statistis high-energy neutrino experiments either
extrated struture funtions by omparing the number
2
of ν versus ν events in an x, Q
bin [4℄, or by extrating
2
bin and tthe ross-setions dσ/dy within the x, Q
ting for the struture funtions using Equation (1) [15℄.
Either method assumes a value for
RL = σL /σT
as mea-
Let us denote Eq. (1) as a funtion of the stru
dσ ν(ν) (xF3 , F2 , R), where the x, Q2 dependene is assumed and where the struture funtions
ture funtions by
an be dierent for neutrinos and antineutrinos. A samν(ν)
NMC,gen , is generated using an
assumed set of struture funtions for the ross-setion:
dσ ν,ν (xF3gen , F2gen , Rgen ). One
an then t for the stru2
bin by minimizing
ture funtions in eah x, Q
ple of Monte Carlo events,
χ2 =
sured by other experiments [16℄, and depends on a mea-
ν,ν y−bins
surement of the strange sea from dimuon events [17, 18℄.
With suient statistis, we an explore
the possibility
2
2
2
ν
ν
of measuring xF3 x, Q , xF3 x, Q , F2 x, Q , and
R x, Q2 from the same data [10℄.
ν(ν)
NMC,pred (SF f it ) =
X
ν(ν) events in
(x,y,Q2 ) bin
X X
where
events
2
ν(ν)
ν(ν)
Ndata − NMC,pred (SF f it )
ν(ν)
Ndata
ν(ν)
NMC,pred (SF f it ), the reweighted
2
in an x, Q , y bin, is given by
dσ ν(ν) xF3f it , F2f it , Rf it
dσ ν(ν) (xF3gen , F2gen , Rgen )
ν(ν)
NMC,gen (SF gen ) ,
,
(8)
Monte-Carlo
(9)
6
Figure 3: Frational hange in number of events for two harateristi
(x, Q2 )
bins as a funtion of
y.
The frational hange
omes from saling the energy of eah event by a fator of 1.005.
ν(ν)
2
is the number of ν or ν data events in the x, y, Q
ν(ν)
bin, and NMC,gen is the number of Monte-Carlo
f it
2
x, Q2 ,
bin. xF3
events generated in the x, y, Q
F2f it x, Q2 , and Rf it x, Q2 are the t parameters in
2
the χ -minimization of Eq. (8). In priniple they an be
t separately for
D. Fitting for ∆xF3
Ndata
ν
and
ν
struture funtions. Here we
will onentrate on the measurement ox f up to four sep
2
arate struture funtions, ∆xF3 x, Q
= xF3ν x, Q2 −
Q2 ,
xF3ν x, Q2 , xF3avg = (xF3ν + xF3ν )/2, F2 x,
2
2
where we assume that F2 x, Q
and
and R x, Q
2
R x, Q are the same for neutrinos and antineutri
F2 x, Q2 = F2ν x, Q2 = F2ν x, Q2 and
nos i.e.
R x, Q2 = Rν x, Q2 = Rν x, Q2 .
We have studied the extration of the struture funtion from the 600 million neutrino and 33 million antineutrino deep inelasti sattering events expeted in the
full NuSOnG data set.
The dominant systemati error
omes from the measurement of the muon momentum
in the toroidal spetrometer.
At NuTeV, the system-
ati unertainty was 0.7% and we assume NuSOnG will
ahieve 0.5%. Our studies are arried out by tting the y 2
distribution in eah x, Q bin for F2 , the average value of
avg
ν
xF3 = xF3 = (xF3 + xF3ν )/2, ∆xF3 = xF3ν − xF3ν ,and
R. In the rst set of studies, R(x, Q2 ) is set equal to the
measured value[16℄ and ts are done to the three struF2 , xF3avg , and ∆xF3 .
ture funtions,
Our tting proedure begins with a sample of Monte
N gen (x, Q2 , y), sampled from
Carlo generated events,
7
∆xF3 in dierent x bins as a funtion of Q2 . The t is to multiple x and Q2
avg
bins extrating the three struture funtions, F2 ,xF3
and ∆xF3 . For eah of the ts, a global set of energy sale parameters
is also determined from the t. The dotted lines show the frational error for the NuTeV 2µ measurement.
Figure 4: Frational unertainty for the t value of
8
the CCFR struture funtions and the nominal value for
∆xF3 from NuTeV. We t in bins of (x, Q2 ) as a funtion
N f it (x, Q2 , y) =
of
y
and obtain the t spetra by reweighting the original
sample:
F2f it (x, Q2 )(2 − 2y + y 2 /(1 + R)) ± xF3f it (x, Q2 )(1 − (1 − y)2 )
N gen (x, Q2 , y).
F2nom (x, Q2 )(2 − 2y + y 2 /(1 + R)) ± xF3nom (x, Q2 )(1 − (1 − y)2 )
where the upper sign is for neutrinos and the lower for
between 50 and 70 GeV. This tting tehnique renders
anti-neutrinos. In order to study the eets of the sys-
the systemati error from the sale shift to be small in
temati energy sale shift, we produe a Monte Carlo
omparison with the statistial error. For example, in
2
the bin x, Q
= (0.275, 32 GeV2 ) bin, the systemati
sample where the muon energy sale is shifted by 0.5%,
Eµmeas = 1.005Eµtrue, for eah event. The frational
hange in the number of events in eah bin due to the
energy sale shift is shown in Fig. 3.
This shifted event distribution,
about 10%.
N shif t (x, Q2 , y),
is
then used to arry out a three parameter t to Eq. 8
F2 , xF3avg , and ∆xF3 are varied. Large shifts in
where
∆xF3
in the
result. For example, the shift from the input value
(x, Q2 ) = (0.08, 12.6GeV 2 ) bin is 19.01% and the
shift in other bins is even larger.
The eets of the energy sale unertainty an be pratially eliminated by inluding energy sale shift parameters in the t. A muon energy sale hange shifts the
events in the various y-bins by an amount that is not onsistent with that expeted from hanges in the struture
funtions. Therefore, ts to the y-distributions an isolate the eets of an energy sale shift and signiantly
redue the struture funtion unertainty from this systemati error.
rameters are introdued in the t to the y-distributions.
These three parameters are used to produe an energy
2
sale shift parameterization in eah x, Q , y bin given
by
Eµscale = Eµscale1 + Eµscale2 Eµ + Eµscale3 Eµ2 .
The updated predition
for the number of events in a
2
given x, Q , y bin is
ν(ν)
ν(ν)
Npred (SFf it ) = NMC,pred (SF f it )
+Eµscale
N shif t (x, Q2 , y) − N gen (x, Q2 , y) ,
χ2
used in the minimization similar to Eq.
∆xF3 for dierent x bins as a funtion of
In general, we believe NuSOnG an measure ∆xF3
2
over most of the (x, Q ) range to better than 10%; in
tional error on
Q2 .
many ases around 3%.
Typial values for NuTeV are
shown in two x bins in Fig. 4. Sine more than one
x, Q2 bin is being used to determine the energy sale
Eµscale1 parameter an
also determined to about 3% from these ts.
shift parameters, the value of the
ν,ν y−bins
2
ν(ν)
ν(ν)
Ndata − Npred (SF f it )
∆xF3
and
Rlong
stru-
Rlong measurements are shown in Fig. 7
along with previous measurements.[16℄ As indiated from
The simulated
Rlong
is muh more preise that any previous experiment.
In summary, due to the very high statistis of a Nu-
extrated from the data without introduing theoretial or experimental approximations. Further, systemati
ν(ν)
Ndata
Eµscale3
Eµscale2
+
.
+ Eµscale1 +
2
(0.02)
(0.0002)2
sale parameters. In this ase, the
ture funtions an be determined to between 5% and 20%
2
for most of the x and Q range as shown in Figs. 5 and 6.
SOnG type experiment, an almost omplete set of stru2
ture funtions over a broad range of x and Q an be
energy sale parameters
X X
Simulation studies have also been made to estimate the
unertainties assoiated with doing ts to extrat the four
avg
, ∆xF3 ,and R. The proestruture funtions, F2 , xF3
dure is the same as used for the three struture funtion
2
ts where the χ in Eq. 11 is minimized simultaneously
2
over a number of x and Q bins with one set of energy
8
with the addition of pull terms assoiated with the three
χ2 =
ergy sale parameters. We have studied this using eight
x bins and six to eight Q2 bins. Figure 4 shows the fra-
this gure, the apabilities of the NuSOnG to measure
(10)
and the
In the ultimate analysis, the t will be arried out six and Q2 bins with one set of en-
multaneously over all
To estimate the systemati error redu-
tion for this tehnique, three additional energy sale pa-
.
error for ∆xF3 is 0.3% while the statistial error is 10%;
the value of the Eµscale1 parameter is also determined to
unertainties that have limited the preision of previous
struture funtion measurements an be eliminated by in(11)
luding ts to these unertainties in the extration proe-
These pull terms orrespond to an energy sale uner-
funtion measurements will be statistis limited even for
tainties of about 0.5% for muon energy values averaging
NuSOnG.
dure. We believe that with these tehniques the struture
9
2
2
1
3
Fx
Fx
6
4
∆
3
∆
1
/ ) Fx (
σ ∆
σ ∆
0.1
6
4
x = 0.045
2
8
2
4
6
1
8
2
Q
10
2
4
6
8
8
2
4
6
1
8
2
[GeV ]
Q
10
2
4
6
8
2
100
2
[GeV ]
2
1
6
Fx
4
∆
Fx
∆
x = 0.100
6
3
3
/ ) Fx (
2
6
4
2
NuTeV
3
3
/ ) Fx (
0.1
σ ∆
σ ∆
4
2
1
6
4
x = 0.175
2
0.1
6
4
x = 0.250
2
0.01
0.01
6
8
2
4
6
1
8
2
Q
10
2
4
6
8
2
6
8
100
2
2
4
6
1
8
2
[GeV ]
Q
2
10
2
4
6
8
2
100
2
[GeV ]
2
1
1
3
Fx
Fx
6
4
∆
3
∆
6
100
2
2
/ ) Fx (
2
6
4
2
3
3
/ ) Fx (
0.1
σ ∆
σ ∆
NuTeV
0.01
6
6
4
x = 0.325
2
0.1
6
4
x = 0.400
2
0.01
0.01
6
8
2
4
6
1
8
2
Q
10
2
4
6
8
2
6
8
100
2
2
4
6
1
8
2
[GeV ]
Q
2
10
2
4
6
8
2
100
2
[GeV ]
2
1
1
3
Fx
Fx
6
4
∆
3
∆
2
0.1
2
0.01
/ ) Fx (
2
6
4
2
3
3
/ ) Fx (
0.1
σ ∆
σ ∆
4
3
3
/ ) Fx (
2
6
6
4
x = 0.475
2
0.1
6
4
x = 0.550
2
0.01
0.01
6
8
2
4
1
6
8
2
Q
10
2
2
[GeV ]
4
6
8
100
2
6
8
2
1
4
6
8
2
Q
10
2
2
4
6
8
2
100
[GeV ]
2
Figure 5: Frational unertainty for the t value of ∆xF3 in dierent x bins as a funtion of Q . The t is to multiple x
avg
2
and Q bins extrating the four struture funtions, F2 ,xF3
, ∆xF3 , and R. For eah of the ts, a global set of energy sale
parameters is also determined.
10
A. Nulear eets at small x
IV. NUCLEAR EFFECTS
Historially, neutrino experiments have played a major
We expet to nd a dierene between harged-lepton
role in expanding our understanding of parton distribu-
nuleus and neutrino nuleus sattering at small-x be-
tion funtions through high statistis experiments suh as
ause the spae-time pitures for the two proesses are
CCFR [8℄, NuTeV [8, 15, 19℄, and CHORUS [20℄. How-
dierent in this region. The underlying physial meha-
ever, the high statistis extrat a prie sine the large
nism in the laboratory referene frame an be skethed
event samples require the use of nulear targets iron in
as a two-stage proess. At the rst stage, the virtual
∗
∗
∗
photon γ , or W or Z in ase of neutrino interations,
the ase of both CCFR and NuTeV and lead in the ase
of the Chorus experiment.
The problem is that if one
utuates into a quark-gluon (or hadroni) state. In the
wants to extrat information on nuleon PDFs, then the
seond stage, this hadroni state then interats with the
eets of the nulear targets must rst be removed. Nu-
target. The unertainty priniple allows an estimate of
SOnG an provide key measurements whih will improve
the average lifetime of suh hadroni utuation as
these orretions.
Charged lepton deep inelasti sattering has been measured on a wide range of targets. The most simplisti ex-
τ=
petation for the struture funtions might be that they
m2
2ν
Q2
1
=
,
2
2
+Q
x M m + Q2
(12)
would simply be given by an average of the appropriate
number of proton and neutron results as in
where
m
is invariant mass of hadrons into whih the vir-
tual boson onvert, and
F2A (x, Q2 ) =
sale
A−Z n
Z P
F (x, Q2 ) +
F2 (x, Q2 ).
A 2
A
τ
M
is the proton mass. The same
also determines harateristi longitudinal dis-
tanes involved in the proess. At small
x, τ
exeeds the
average distane between bound nuleons. For this reason oherent multiple interations of this hadroni u-
However, the results from a wide range of experiments
show a muh more omplex behavior for the struture
funtions on nulei. The typial behavior of the ratio
F2A (x, Q2 ) to F2d (x, Q2 ) where d denotes a deuterium
target shows four distint regions as skethed in Fig. 8.
of
At small
x
At somewhat larger values of
x
the ratio rises above one in the antishadowing region. At
still larger values of
the EMC region.
x
the ratio again falls below one in
Finally, as
x
approahes one, Fermi
nulear interations of hadroni utuations of virtual
intermediate boson.
This behavior shows only a modest dependene on
A
for values above beryllium, with the shape remaining
qualitatively the same and the amount of the suppression
inreasing slowly with
log(A).
Furthermore,
2
there is little, if any, observed dependene on Q . These
features are summarized niely in the results shown in
Ref. [21℄.
The mehanisms of nulear sattering have also been
studied theoretially.
1
For neutrino interations whih are mediated by the
axial-vetor urrent, the utuation time
τ
is also given
by Eq. 12. However, as was argued in Ref. [24℄, the utuation and oherene lengths are not the same in this
ase. In partiular, the oherene length is determined
mπ in Eq. 12 beause of the dominane
of o-diagonal transitions like a1 N → πN in nulear in-
by the pion mass
motion smearing auses a signiant rise in the ratio.
x ≈ 0.6
matial region. It is well known that the nulear shadowing eet for struture funtions is a result of oherent
the ratio dips below one in what is alled
the shadowing region.
at
tuation in a nuleus are important in the small-x kine-
terations. Sine the pion mass is muh smaller than typial masses of intermediate hadroni states for the vetor urrent (mρ , mω , et.), the oherene length Lc of
2
intermediate states of the axial urrent at low Q will
be muh larger than Lc of the vetor urrent. A diret
onsequene of this observation is the early onset of nulear shadowing in neutrino sattering at low energy and
2
and low Q as ompared with the shadowing in harged-
These mehanisms appear to be
lepton sattering. The basi reason for this earlier onset
x as viewed from the
2
Bjorken x is dened as x = Q /2M ν ,
and dierent behavior in the transition region is the differene in the orrelation lengths of hadroni utuations
are energy and momentum transfer to
2
2
2
the target and Q = q − ν . The physial quantity
of the vetor and axial-vetor urrents. This is also illus2
trated by observing that for a given Q , the ross-setion
dierent for small and large Bjorken
laboratory system.
where
ν
and
q
whih is responsible for the separation between large and
small
x
regions is a harateristi sattering time, whih
2
is also known as Ioe time (or length) τI = ν/Q [22℄.
If
τI
suppression due to shadowing ours for muh lower energy transfer (ν ) in neutrino interations than for harged
leptons.
is smaller than the average distane between bound
nuleons in a nuleus then the proess an be viewed as
inoherent sattering o bound nuleons. This happens
at larger
x(> 0.2).
1
For a reent review of nulear shadowing see, e.g., [23℄.
11
2
2
1
6
4
4
R / )R(
2
0.1
σ
R / )R(
σ
1
6
6
6
4
4
x = 0.045
2
0.01
6
8
2
4
6
1
8
2
Q
10
2
4
6
8
2
6
2
4
6
1
8
2
[GeV ]
Q
10
2
4
6
8
2
100
2
[GeV ]
2
1
1
6
6
4
4
R / )R(
2
0.1
σ
R / )R(
σ
8
100
2
2
2
0.1
6
6
4
4
x = 0.175
2
x = 0.250
2
0.01
0.01
6
8
2
4
6
1
8
2
Q
10
2
4
6
8
2
6
8
100
2
2
4
6
1
8
2
[GeV ]
Q
2
10
2
4
6
8
2
100
2
[GeV ]
2
1
1
6
6
4
4
R / )R(
2
0.1
σ
R / )R(
σ
x = 0.100
2
0.01
6
2
0.1
6
4
4
x = 0.325
2
x = 0.400
2
0.01
0.01
6
8
2
4
6
1
8
2
Q
10
2
4
6
8
2
6
8
100
2
2
4
6
1
8
2
[GeV ]
Q
2
10
2
4
6
8
2
100
2
[GeV ]
2
1
1
6
6
4
4
R / )R(
2
0.1
σ
R / )R(
σ
2
0.1
6
2
0.1
6
4
4
x = 0.475
2
x = 0.550
2
0.01
0.01
6
8
2
1
4
6
8
2
Q
10
2
2
[GeV ]
4
6
8
100
2
6
8
2
1
4
6
8
2
Q
10
2
2
4
6
8
2
100
[GeV ]
2
2
Figure 6: Frational unertainty for the t value of R in dierent x bins as a funtion of Q . The t is to multiple x and Q bins
F2 ,xF3avg , ∆xF3 , and R. For eah of the ts, a global set of energy sale parameters is
also determined.
extrating the four struture funtions,
12
B. Nulear Eets in Neutrino Interations
parison is shown in Fig. 9.
This gure shows some results from Ref. [29℄ in the
Q2 and presented
As there has been no systemati experimental study
of
ν
and
ν
nuleus interations, one must then rely on
form of data/theory averaged over
x.
theoretial models of the nulear orretions. This is an
versus
unsatisfatory situation sine one is essentially measuring
without the model-dependent nulear orretions whih
quantities sensitive to the onvolution of the the desired
were used in the t (the neutrino data were not used in
The results are from a global t but are plotted
PDFs and unknown or model dependent nulear or-
the referene t.) It is notable that the overall pattern of
retions.
deviations shown in Fig. 9 are, in general, similar to that
As noted above, theoretially there are substantial differenes between harged lepton and neutrino interations on the same nuleus. There are other expeted differenes for neutrinos. For example, the relative nulear
shadowing eets for the struture funtion
F3
dited to be substantially dierent from that for
This is beause the struture funtion
F3
is pre-
F2
[25℄.
desribes the
orrelation between the vetor and the axial-vetor urrent in neutrino sattering.
In terms of heliity ross
setions, the struture funtion F3 is given by the ross
setion asymmetry between the left- and right-polarized
states of a virtual
W
boson. It is known that suh a dif-
seen in harged lepton DIS as skethed in Fig.8. However,
the deviations from unity are perhaps smaller. At high
x,
the eet of Fermi smearing is lear. At moderate
the EMC eet is observable.
that there is no lear indiation of a turnover at low
ν
the shadowing region for
similarity between the
ν
x
It is interesting to note
x
in
data. Also, note the striking
ν
and
results. This appears to
imply that the dierenes in the nulear eets between
neutrino and antineutrino DIS are small.
later, when we onsider
∆xF3
As disussed
and isospin violation, it is
ruial to model dierenes in the nulear eets between
ν
and
ν̄
sattering as a funtion of
x.
ferene of ross setions is strongly aeted by Glauber
[26, 27, 28℄
To make progress in understanding nulear orretions
This auses an enhaned nulear shadowing eet for the
in neutrino interations, aess to high statistis data on
multiple sattering orretions in nulei.
struture funtion
a variety of nulear targets will be essential.
F3 .
It is important to experimentally address the question
A-dependene to
x and Q2 , as has been
allow the
This will
be studied as a funtion of
of nulear eets in neutrino sattering so that the neu-
both
trino data an be used in proton ts without bringing
inelasti sattering.
in substantial nulear unertainties.
neutrino data an then be used to make preditions to be
For example, in a
done in harged lepton deep
PDFs from global ts without the
A-dependent ν
ν
reent analysis [29℄ the impat of new neutrino data on
ompared with the
global ts for PDFs was assessed. The onlusion reahed
thereby allowing the nulear orretions to be mapped
in this analysis was that the unertainties assoiated with
out for omparison with theoretial models.
and
ross setions,
nulear orretions preluded using the neutrino data to
onstrain the nuleon PDFs.
The primary target of NuSOnG will be SiO2 . However,
If NuSOnG an address
A-values by replaing a few
these unertainties, then the neutrino data an play an
we an investigate a range of
even more prominent role in the global ts to the proton
slabs of glass with alternative target materials: C, Al,
Fe, and Pb.
PDF.
Furthermore, nulear eets are interesting in their
own right.
Parameterizations of nulear PDFs on vari-
ous targets exist in the literature. However, there is no
This range of nulear targets would both
extend the results of Minerva to the NuSOnG kinemati
region, and provide a hek (via the Fe target) against
the NuTeV measurement.
universally aepted model whih desribes these nulear
orretions over the entire range of
x from rst priniples.
This makes it diult to generalize the above behavior
observed in harged lepton DIS to DIS with
ν
or
ν
beams.
Models suh as that in Ref. [30℄ exist, but to date there
have been no high statistis studies of
ν
or
ν
DIS over a
wide range of nulear targets with whih to test them.
A study presented in Ref. [29℄ examined the role of new
lepton pair prodution data from E-866 and new neutrino
58k ν 30k ν̄ -indued CC DIS events per ton of material. A single ton would be suient to extrat F2 (x)
2
and xF3 (x) averaged over all Q ; a single 5 m×5 m×2.54
Given the NuSOnG neutrino ux, we antiipate
indued and
m slab of any of the above materials will weigh more
than that. The use of additional slabs would permit further extration of the struture funtions into separate
(x, Q2 ) bins as was done in the NuTeV analysis, at the
DIS data from the NuTeV and CHORUS ollaborations
potential expense of ompliating the shower energy res-
in global ts for nuleon PDFs. For the atual tting of
olution in the sub-detetors ontaining the alternative
the PDFs it was neessary to inlude nulear orretions
targets; this issue will be studied via simulation.
for the neutrino and antineutrino ross setions and the
model of Ref. [30℄ was used. As a byprodut of that anal-
Table II shows that two 50-module staks would be
ysis, it was possible to ompare a referene t, obtained
suient to aumulate enough statistis on alternative
without using data on nulear targets, to the neutrino
nulear targets for a full struture-funtion extration for
and antineutrino data in order to obtain an estimate of
eah material. However, for basi ross-setion ratios in
what the nulear orretions should look like. This om-
x,
a single slab of eah would sue.
13
0.6
0.6
x = 0.100
0.5
NuSOnG
0.4
SLAC
NuSOnG
0.4
gnoL
R
R
gnoL
0.3
0.2
0.3
0.2
0.1
0.1
0.0
6
8
2
4
6
8
1
2
4
6
10
2
Q
8
0.0
2
6
100
8
2
4
6
1
2
8
2
[GeV ]
Q
0.6
10
2
4
6
8
2
100
2
[GeV ]
0.6
x = 0.250
0.5
x = 0.325
0.5
SLAC
NuSOnG
0.4
SLAC
NuSOnG
0.4
gnoL
R
R
gnoL
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
6
8
2
4
6
1
8
2
Q
2
10
4
6
8
2
6
8
100
2
2
4
6
1
8
2
[GeV ]
Q
0.6
10
2
4
6
8
2
100
2
[GeV ]
0.6
x = 0.400
0.5
x = 0.475
0.5
SLAC
NuSOnG
0.4
SLAC
NuSOnG
0.4
gnoL
R
R
gnoL
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
6
8
2
4
6
1
8
2
Q
2
10
4
6
8
2
6
8
100
2
2
4
6
1
8
2
[GeV ]
Q
0.6
10
2
4
6
8
2
100
2
[GeV ]
0.6
x = 0.550
0.5
x = 0.625
0.5
SLAC
NuSOnG
0.4
SLAC
NuSOnG
0.4
gnoL
R
R
gnoL
0.3
0.3
0.2
0.2
0.1
0.1
0.0
0.0
6
8
2
4
6
1
8
2
Q
Figure 7:
x = 0.175
0.5
SLAC
Extrated values of
2
[GeV ]
R
measurements[16℄ (labeled SLAC).
2
10
4
6
8
100
2
6
8
2
1
4
6
8
2
Q
10
2
2
4
6
8
2
100
[GeV ]
(labeled NuSOnG) from the four struture funtion ts as ompared to previous
14
Material
Mass of
Number of slabs needed
2.54 m slab (tons)
for NuTeV-equivalent statistis
C
1.6
33
Al
1.9
27
Fe
5.5
10
Pb
7.9
7
Table II: Alternative target materials for ross-setion analysis
1020
12 × 1020
hek for onsisteny. For a run onsisting of 4.0 x
POT in the NuMI Low Energy (LE) beam and
POT in the NuMI Medium Energy (ME) beam, Minerva
would ollet over 2 M events on Fe, 2.5 M events on Pb,
600 K on helium and 430 K events on C as well as 9.0 M
events on the sintillator within the duial volume.
Studying nulear eets with the NuSOnG detetor
Figure 8: Typial behavior of the ratio of the struture funtion on a nulear target
A
to that on a deuterium target.
will involve fewer nulear targets but onsiderably more
statistis on eah. In addition, the muh higher energy
of the inoming neutrinos with NuSOnG means a muh
wider kinemati range of study. In partiular, NuSOnG
2
will have a muh higher Q for a given low-x to study
shadowing by neutrinos and will be able to measure the
shadowing region down to muh smaller x for the same
2
Q range as Minerva. A signiant addition to the study
of nulear eets with neutrinos would be the addition of
a large, perhaps ative ("Bubble Chamber"), ryogeni
target ontaining hydrogen or deuterium. With the intense NuSOnG neutrino beam, a signiant sample of
neutrino-hydrogen and neutrino-deuterium events ould
provide the normalization we need to further unfold nulear eets in neutrino-nuleus interations.
V. QCD FITS
The extration of up to six struture funtions from
the ross setions of neutrino and anti-neutrino DIS disFigure 9: Comparison between the referene t and the un-
ussed so far (f, Eq. (1)) has been ompletely model-
shifted CHORUS and NuTeV neutrino data without any nu-
independent relying only on some fundamental priniples
suh as Lorentz-invariane of the ross setion and gauge-
lear orretions.
invariane of the hadroni tensor whih is expanded in
C. Measuring Nulear Eets with the Minerva
and NuSOnG Detetors
terms of the struture funtions whih parameterize the
unknown hadroni physis.
More an be said about the struture funtions in
QCD. While it is still not possible to aurately om-
The Minerva experiment will also be studying neutrino
pute the
x-dependene
of the struture funtions from
indued nulear eets and will be starting its initial
rst priniples, QCD allows us to derive renormalization
physis run in early 2010.
group equations (RGEs) whih relate the struture fun-
To study nulear eets in
Q.
Minerva, a ryogeni vessel ontaining liquid helium (0.2
tions at dierent (perturbative) sales
ton duial mass)will be installed upstream of the Min-
struture funtions at the sale
erva detetor.
to struture funtions at a dierent sale
Within the Minerva detetor, solid ar-
Q
Note that the
an be diretly related
Q0
(see, e.g.,
bon, iron and lead targets will be installed upstream of
Eqs. (5.58) and (5.76) in [31℄). However, it is more onve-
the pure sintillator ative detetor.
nient to work in the QCD-improved parton model where
The total mass is
0.7 ton of Fe, 0.85 ton of Pb, 0.4 ton of He and some-
the RGEs governing the sale-dependene of the parton
what over 0.15 ton of C. Sine the pure sintillator ative
distribution funtions (PDFs) are the familiar DGLAP
detetor essentially ats as an additional 3-5 ton arbon
evolution equations; these an also be used to ompute
target (CH), the pure graphite (C) target is mainly to
the struture funtions at Q given the PDFs at that sale.
15
[32, 33, 34℄ Furthermore, this approah has the ruial
advantage that the universal PDFs allow us to make preditions for other observables as well.
the
Q-dependene,
In addition to
the QCD alulations provide ertain
(approximate) relations between dierent struture funtions as will be visible from the parton model expressions
below.
In this setion we will disuss the analysis of the ross
setion data within the framework of the QCD-improved
parton model. Already in the past, high statistis measurements of neutrino deeply inelasti sattering (DIS) on
heavy nulear targets (NuTeV, ...) have attrated muh
assume
sp/A = sn/A
Deviations from isospin symmetry an be parameterized in the following way:
δup/A
= up/A
− dn/A
, δdp/A
= dp/A
− un/A
,
v
v
v
v
v
v
p/A
p/A
n/A
p/A
p/A
n/A
¯
¯
¯
δū
= ū
−d
, δd
=d
− ū
.
symmetry manifest:
p/A
2uA
+ dp/A
− δdp/A
v = [uv
v
v ]−
∆[up/A
− dp/A
+ δdp/A
v
v
v ],
the extration of free nuleon PDFs, beause modeldependent orretions must be applied to the data (f
2ū
for extrating the nulear parton distribution funtions
¯A
2d
(NPDFs) and for suh an analysis no nulear orreConversely, the NPDFs an
proton PDFs, universal nulear PDFs are needed for the
This involves physis at other neutrino experi-
∆[up/A
− dp/A
− δup/A
v
v
v ],
p/A
p/A
p/A
= [ū
+ d¯ − δ d¯ ] −
p/A
∆[ū
− d¯p/A + δ d¯p/A ] ,
= [ū
p/A
∆[ū
be utilized to ompute the required nulear orretion
state.
A
Of ourse, these same data are also useful
desription of many proesses with nulei in the initial
(17)
p/A
2dA
+ dp/A
− δup/A
v = [uv
v
v ]+
use of nulear targets is unavoidable; this ompliates
Similar to
(16)
These denitions allow us to write the PDFs in a way
Due to the weak nature of neutrino interations, the
fators within the QCD parton model [37℄.
(15)
whih makes deviations from isosalarity and isospin
formation for global ts of PDFs [35, 36℄.
tion fators are required.
s̄p/A = s̄n/A .
Also, we neglet
A
A
any possible harm asymmetry, i.e., we use c = c̄ suh
−,A
A
A
+,A
A
A
= c − c̄ = 0 and c
= c + c̄ = 2cA .
that c
interest in the literature sine they provide valuable in-
Se. IV).
and
where
∆ = (N − Z)/A
¯p/A
+d
p/A
− δū
¯p/A
−d
p/A
− δū
(18)
(19)
]+
p/A
],
(20)
parameterizes the deviation from
isosalarity. We have written Eqs.(17)(20) so that the
RHS is expressed expliitly in terms of proton PDFs
p/A
p/A
δ -terms {δuv , δdv , δūp/A , δ d¯p/A }; the
and the four
ments, heavy ion olliders (RHIC, LHC), and a possible
δ -terms
future eletron-ion ollider (EIC).
served.
vanish individually if isospin symmetry is pre-
The NuSOnG experiment will have two orders of magnitude higher statistis than the NuTeV and CCFR ex-
B. Struture funtions
periments (over an extended kinemati range), and so
it will be possible to study small eets suh as the
strangeness asymmetry with better preision, or to establish for the rst time isospin violation in the light quark
setor. Better understanding these eets is relevant for
improving the extration of the weak mixing angle in a
PashosWolfenstein type analysis.
The struture funtions for a nulear target
(A, Z) are
given by
FiA (x, Q) =
Z p/A
(A − Z) n/A
Fi (x, Q) +
Fi (x, Q)
A
A
(21)
suh that they an be omputed in next-to-leading order
as onvolutions of the nulear PDFs with the onven-
A. PDFs
tional Wilson oeients, i.e., generially
FiA (x, Q) =
NuSOnG will perform measurements on dierent nulear targets.
The PDFs for a nuleus
(A, Z)
are on-
struted as
fiA (x, Q) =
X
Cik ⊗ fkA .
(22)
k
In order to disuss whih information an be extrated
from a high statistis measurement of neutrino and anti-
Z p/A
(A − Z) n/A
(x, Q) +
f
fi (x, Q) .
A i
A
neutrino DIS ross setions we briey review the parton
(13)
model expressions for the 6 struture funtions. For simpliity, we rst restrit ourselves to leading order, neglet
In the following disussion we take into aount devia-
heavy quark mass eets (as well as the assoiated pro-
tions from isospin symmetry, a non-vanishing strangeness
dution thresholds), and assume a diagonal CKM matrix.
asymmetry and the possibility to have non-isosalar tar-
In our numerial results, these eets are taken into a-
gets. For this purpose we introdue the following linear
ount.
The neutrinonuleus struture funtions are given by
x and Q2 ):
ombinations of strange quark PDFs:
(suppressing the dependene on
s+,A = sA + s̄A ,
s−,A = sA − s̄A ,
(14)
where the strangeness asymmetry is desribed by a non−
vanishing PDF s . Note however that we ontinue to
F1νA = dA + sA + ūA + c̄A + . . . ,
F2νA = 2xF1νA ,
F3νA = 2 dA + sA − ūA − c̄A + . . . .
(23)
(24)
(25)
16
C. Constraints on the PDFs
The dierential ross setion in Eq. (1) an be written
as:
dσ
= K[A + B(1 − y)2 + Cy 2 ]
dxdy
K=
with
and
C =
(32)
G2F ME
2 )2 , A = F2 ± xF3 , B = F2 ∓ xF3 ,
2π(1+Q2 /MW
2x2 M 2
Q2 F2 − FL where the upper sign refers to
neutrino and the lower one to anti-neutrino sattering.
This form of
dσ
shows that the (anti-)neutrino ross se-
tion data naturally enodes information on the four struνA
ν̄A
νA
ν̄A
ture funtion ombinations F2 ±xF3
and F2 ±xF3
in separate regions of the phase spae. In addition, at
large
y the struture funtion ombination C ontributes.
C ν = C ν̄ so that C drops out
However, to good auray
in the dierene of neutrino and anti-neutrino ross setions.
Figure 10: Struture funtion ratio
RL
for neutrino and antin)/2)) sattering at Q2 = 20
neutrino nuleon (N = (p +
2
GeV . The solid and dashed lines show NLO results obtained
with the GRV98 PDFs [38℄, while the dotted line shows the
LO result of Eq. (31).
The struture funtions for anti-neutrino sattering are
obtained by exhanging the quark and anti-quark PDFs
in the orresponding neutrino struture funtions:
ν̄A
νA
F1,2
= F1,2
[q ↔ q̄] ,
F3ν̄A = −F3νA [q ↔ q̄] .
(26)
(27)
(28)
(29)
The longitudinal struture funtion an be obtained with
the help of the following relation:
FLνA = r2 F2νA − 2xF1νA =
where
r2 = 1 + 4x2 M 2 /Q2 .
4x2 M 2 νA
F2 ,
Q2
(30)
Finally, it is ustomary to
introdue the ratio of longitudinal to transverse struture
funtions:
νA
RL
=
r2 F2νA
4x2 M 2
FLνA
.
=
−
1
=
Q2
2xF1νA
2xF1νA
1 νA
A
F2 = 2 dA
,
v +Σ
x
1 ν̄A
A
F2 = 2 uA
.
v +Σ
x
(33)
(34)
Furthermore, we have
Expliitly this gives
F1ν̄A = uA + cA + d¯A + s̄A + . . . ,
F2ν̄A = 2xF1ν̄A ,
F3ν̄A = 2 uA + cA − d¯A − s̄A + . . . .
A
Assuming s
= s̄A and cA = c̄A , the struture funνA
ν̄A
tions F2
and F2
onstrain the valene distributions
A
A
A
A
¯
dv = d − d , uv = uA − ūA and the avor-symmetri
A
A
¯A + s̄A + c̄A + . . . via the relations:
sea Σ := ū + d
(31)
Similar equations hold for anti-neutrino sattering. As
ν
ν̄
an be seen, in leading order RL = RL . As is shown
ν
in Fig. 10, also in NLO, the dierenes between RL and
ν̄
RL are tiny suh that the dierene between these two
funtions an be negleted in the following disussion.
1 νA
F + F3νA = 4(dA + sA ) ,
x 2
1 ν̄A
F − F3ν̄A = 4(d¯A + s̄A ) .
x 2
(35)
(36)
Sine we onstrain the strange distribution utilizing the
dimuon data, the latter two struture funtions are useful
dA and d¯A distributions.
to separately extrat the
For an isosalar nuleus we enounter further simpliA
A
A
¯A =: q̄ A whih
ations. In this ase, u = d and ū = d
A
A
A
implies uv = dv =: v . Hene, the independent quark
A A A
A A
A
distributions are {v , q̄ , s = s̄ , c = c̄ , . . .}. In parνA
ν̄A
tiular, we have F2
= F2 for an isosalar target suh
that our original set of 6 independent struture funtions
νA
νA
ν̄A
redues to 3 independent funtions (say F2 , F3 , F3 )
under the approximations made.
In a more rened analysis, allowing for a non-vanishing
strangeness asymmetry and isospin violation we an evalν
νA
−
uate the non-singlet struture funtion ∆F2 ≡ F2
ν̄A
F2 with the help of the relations in Eqs. (17) (20):
∆F2ν = 2xs−,A + x δdp/A
− x δup/A
v
v
+∆x[2up/A
− 2dp/A
+ δdp/A
− δup/A
v
v
v
v ] . (37)
For a nulear isosalar target (Z
= N = A/2, ∆ = 0)
this expression simplies to
∆F2ν = 2xs−,A + x δdp/A
− x δup/A
.
v
v
(38)
17
As one an see,
∆F2ν
will be small and sensitive to the
strangeness asymmetry and isospin violating terms for
the valene quarks.
{uv (x), dv (x)}, and betransitions are not subjet to slow-
to enhaned valene omponents
ause the
d → u
resaling orretions whih suppress the
The dierene of the neutrino and anti-neutrino rosssetions provides, in priniple, aess to this quantity:
d2 σ νA
d2 σ ν̄A
−
≃ K[∆F2ν + xF3ν
dxdy
dxdy
+(1 − y)2 (∆F2ν − xF3ν )]
bution to
∆xF3
F3ν = F3νA + F3ν̄A .
It should be noted, however, that in a global t to
with
extrat struture funtions we do not make diret use
2
of these equations [the (1 − y) -dependene℄ but simply
2
perform a χ -analysis of all neutrino and anti-neutrino
ross setion data.
ontri-
Here the ability of NuSOnG to
ν
ν̄
separately measure xF3 and xF3 over a broad kinemati
range will provide powerful onstraints on the sensitive
struture funtion ombination
(39)
s → c
[41℄.
∆xF3 .
∆F2
≡
Separately,
the
measurement
of
5
CC
2
NC
2
in
Charged
Current
F
(x,
Q
)
−
F
(x,
Q
)
2
18 2
±
(CC) W
exhange and Neutral Current (NC) γ/Z
exhange proesses an also onstrain CSV [43℄; beause
CC
NuSOnG will measure F2
on a variety of targets, this
will redue the systematis assoiated with the heavy
nulear target orretions thus providing an additional
avenue to study CSV.
In the following, we provide a detailed analysis of CSV
whih also investigates the various experimental systematis assoiated with eah measurement. We shall nd it
VI. ISOSPIN (CHARGE SYMMETRY)
VIOLATION AND ∆xF3
is important to onsider all the systematis whih impat
the various experimental measurements to assess the disriminating power.
The question of isospin violation is entral to the
PW eletroweak measurement. In the NuTeV analysis,
A.
isospin symmetry was assumed. As disussed in Ref. [3℄,
∆xF3
and Isospin Violations
various models whih admit isospin violation an pull the
NuTeV
sin θW
measurement toward the Standard Model.
However it would take signiantly larger isospin viola-
We reall the leading-order relations of the neutrino
struture funtion
tion to bring NuTeV into agreement with the rest of the
be ruial to the interpretation of the NuSOnG results.
of isospin symmetry where we assume that the proton
and neutron PDFs an be related via a
u ↔ d
inter-
hange. While isospin symmetry is elegant and well motivated, the validity of this exat harge symmetry must
ultimately be established by experimental measurement.
There have been a number of studies investigating isospin
symmetry violation [39, 40, 41, 42, 43, 44, 45℄; therefore,
it is important to be aware of the magnitude of potential
violations of isospin symmetry and the onsequenes on
the extrated PDF omponents. For example, the naive
parton model relations are modied if we have a violation
p ↔ n isospin-symmetry, or harge symmetry vin
p
p
n
(CSV); e.g., u (x) 6≡ d (x) and u (x) 6≡ d (x).
of exat
olation
It is noteworthy that a violation of isospin symmetry is
automatially generated one QED eets are taken into
aount [46, 47, 48℄. This is beause the photon ouples
p
to the up quark distribution u (x) dierently than to the
n
down quark distribution d (x). These terms an be as
muh as a few perent in the medium
x
on a general nulear target:
1 νA
F (x) = dA + sA − ūA − c̄A + ...,
2 3
1 ν̄A
F (x) = uA + cA − d¯A − s̄A + ...
2 3
world's data. Better onstraints of isospin violation will
When we relate DIS measurements from heavy targets
56
207
suh as 26 Fe (used in NuTeV) or 82 Pb (Chorus) bak
to a proton or isosalar target, we generally make use
F3
where
A
represents the nulear target
and the ...
(40)
(41)
A = {p, n, d, ...},
represent higher-order ontributions and
terms from the third generation
{b, t}
quarks. Note that
to illustrate the general features of these proesses, we
use a shemati notation as in Eq. (40) and Eq. (41);
for the numerial alulations, the full NLO expressions
are employed inluding mass thresholds, slow-resaling
variables, target mass orretions, and CKM elements
where appropriate.
For a nulear target
A
we an onstrut
∆xF3A
as:
∆xF3A = xF3νA − xF3ν̄A
1
= 2x∆ up/A − dp/A + ūp/A − d¯p/A + δI A
2
+ 2x s+,A − 2xc+,A + x δI A + O (αS )
where
O (αS )
(42)
represents the higher order QCD orreδI A :
tions, and the isospin violations are given by
range, see e.g.
δI A = δd − δu + δ d¯ − δū .
Fig. 1 in Ref. [48℄.
(43)
Combinations of struture funtions an be partiu-
F3ν
For a ux-weighted linear ombination of
suited to measure some of these observables. For examu, d-ontributions to ∆xF3 = xF3ν − xF3ν̄
from harge symmetry violation would be amplied due
enter Eq. (42), f. Refs. [39, 43, 44℄. For a sign-seleted
ple, residual
and
F3ν̄ ,
larly sensitive to isospin violations, and NuSOnG is well
terms proportional to the strange quark asymmetry an
ν/ν̄
beam as for NuTeV or NuSOnG, this ompliation is
18
not neessary. We have dened s
±,A
and c
(x) = [cA (x) ± c̄A (x)].
±,A
(x) = [sA (x)±s̄A (x)]
with the denitions:
F2CC,A =
In the limit of isospin symmetry, all four terms on
the RHS of Eq. (43) vanish individually. For a nulear
isosalar target,
Z = N = A/2,
we an onstrut
∆xF3
from the above:
1 νA
F2 + F2ν̄A
2
F2N C,A = F2ℓA
∆xF3 = xF3νA −xF3ν̄A = 2xs+,A −2xc+,A +x δI A +O (αS ) .
Note in Eq. (42) that for a nulear target
lose to isosalar we have
Z ∼N
A
In Eq. (45), the rst term is proportional to
(N − Z)/A
(44)
whih vanishes for an isosalar target. The seond and
whih is
third terms are proportional to the heavy quark distri+,A
+,A
butions s
and c
. The next term is the CSV onA
tribution whih is proportional to δI
given in Eq. (43).
suh that the up and
down quark terms are suppressed; this is a benet of
the NuSOnG glass (SiO2 ) target whih is very nearly
SiO2 we have Z(O) = 8,
Z(Si) = 14, m(O) = 15.994, m(Si) = 28.0855. Using A =
Z +N we have (N −Z)/A = (A−2Z)/A for the prefator
in Eq. (42) whih yields (N − Z)/A ∼ −0.000375 for O
and (N − Z)/A ∼ 0.00304 for Si.
p/A p/A
In Eq. (42) the PDFs {u
, d , ...} represent quark
distributions bound in a nuleus A. With a single nulear
A
target, we an determine the CSV term δI
for this spei A; measurements on dierent nulear targets would
A
be required in order to obtain the A dependene of δI
isosalar. More speially, for
if we need to sale to a proton or isosalar target.
It is urious that this has the same form as the CSV ontribution for
∆xF3
of Eq. (42).
Finally, the last term
represents the higher-order QCD orretions.
While the harater of the terms on the LHS of Eq. (44)
and Eq. (45) are quite similar, the systematis of mea-
∆F2 may dier substantially from that of ∆xF3 .
For example, the measurement of ∆F2 requires the subsuring
tration of struture funtions from two entirely dierent experiments. The CC neutrinonuleon data are extrated from heavy nulear targets (to aumulate suient statistis); as suh, these data are generally subjet
quires a areful separation of these ontributions from
to large nulear orretions so that the heavy targets an
1
be related to the isosalar N =
2 (p + n) limit. Conversely, the NC harged-leptonnuleon proess proeeds
the strange, harm, and higher order terms. Theoretial
via the eletromagneti interation. Therefore suient
Thus, an extration of any isospin violation
NLO alulations for
∆xF3
δI A
are available; thus the
re-
O (αS )
orretions an be addressed. Additionally, NuSOnG an
+ −
use the dimuon proess (νN → µ µ X) to onstrain the
statistis an be obtained for light targets inluding
and
sary.
Therefore, we must use the appropriate nulear
F2CC and F2N C , and
this will introdue a systemati unertainty.
In onlusion we nd that while this is a hallenging
Separately, the heavy quark prodution mehanism is
measurement, NuSOnG's high statistis measurement of
should provide a window on CSV whih is rela-
tively free of large experimental systematis. We emphasize that
H
and no large heavy target orretions are nees-
orretion fators when we ombine
strange sea.
∆xF3
D
∆xF3
may be extrated from a single target,
dierent in the CC and NC proesses. Speially, in the
+
→ c where the
CC ase we enounter the proess s + W
harm mass threshold kinematis must be implemented.
c+γ → c
thereby avoiding the ompliations of introduing nu-
On the other hand, the NC proess is
lear orretions assoiated with dierent targets. This
proportional to the harm sea distribution and has dif-
is in ontrast to the other measurements disussed below.
A
However, if we desire to resale the δI eets to a dier-
though the harm prodution proess is modeled at NLO,
ent nuleus
A,
then multiple targets would be required.
whih is
ferent threshold behavior than the CC proess.
Even
the theoretial unertainties whih this introdues an
dominate preision measurements.
B. Measurement of ∆F2 ≡
5
18
F2CC (x, Q2 ) − F2NC (x, Q2 )
C. Other Measurements of CSV
A separate determination of CSV an be ahieved using
the measurement of
F2
in CC and NC proesses via the
relation:
We very briey survey other measurements of CSV in
omparison to the above.
The measurement of the lepton harge asymmetry in
W
5 CC,A
F
(x, Q2 ) − F2N C,A (x, Q2 )
18 2
i
1 (N − Z) h p/A
u
− dp/A + ūp/A − d¯p/A
≃ x
6
A
1 +,A
1
1 N
+ xs
(x) − x c+,A (x) + x δI A
6
6
6 A
+ O(αs )
(45)
∆F2 ≡
deays from the Tevatron an onstrain the up and
down quark distributions [49, 50℄. In this ase, the extration of CSV onstraints is subtle; while isospin symmetry is not needed to relate
p
and
p̄,
this symmetry is
typially used in a global t of the PDFs to redue data
on heavy targets to
p.
In the limit that all the data in the analysis were from
proton targets, CSV would not enter; hene this limit
only arises indiretly from the mix of targets whih enter
19
a global t.
At present, while muh of the data does
ome from proton targets (H1, ZEUS, CDF, D0), there
are some data sets from both
p
and
d
(BCDMS, NMC,
E866), and some that use heavier targets (E-605, NuTeV)
[29, 51℄. Thus, an outstanding question is if CSV were
present, to what extent would this be absorbed into a
global t. The ideal proedure would be to parameterize
the CSV and inlude this in a global analysis. While this
step has yet to be implemented, there is a reent eort
to inlude the nulear orretions as a dynami part of a
global t [37℄.
Additionally, NMC measures
F2n /F2p data whih has an
unertainty of order a few perent [52℄. There are also
xed-target Drell-Yan experiments suh as NA51 [45℄ and
E866 [53℄ whih are sensitive to the ratio
0.04 < x < 0.27.
¯ in the range
d/ū
We will soon have LHC data (pp) to add
to our olletion, thus providing additional onstraints in
a new kinemati region.
D. Conlusions on Charge Symmetry Violation
NuSOnG will be able to provide high statistis DIS
measurements aross a wide
x
range. Beause the target
material (SiO2 ) is nearly isosalar, this will essentially
allow a diret extration of the isosalar observables.
∆xF3
is one of the leaner measurements of CSV in
terms of assoiated experimental systemati unertainties as this measurement an be extrated from a single
target. The hallenge here will be to maximize the event
−
2
Figure 11: NuTeV measurement of xs (x) vs x at Q =
2
16 GeV . Outer band is ombined errors, inner band is with-
out
Bc
unertainty.
sample.
Hene, the reent high statistis dimuon mea-
surements [17, 18, 55, 56, 57℄ play an essential role in
onstraining the strange and anti-strange omponents of
the proton. On NuSOnG, the dimuon data will be used
in the same manner.
Distinguishing the dierene between the
samples.
The measurement of
∆F2
s(x) and s̄(x)
distributions,
xs− (x) ≡ xs(x) − xs(x),
is more ompliated as this
(46)
must ombine measurements from both CC and NC ex-
is neessary for the PW style analysis. This analysis is
periments whih introdues nulear orretion fators
sensitive to the integrated strange sea asymmetry,
[37, 54℄.
Sine NuSOnG will provide high statistis
F2CC measurements for a variety of A targets, this will
yield an alternate handle on the CSV and also improve
S− ≡
external onstraints, will yield important information on
this fundamental symmetry.
1
s− (x)dx,
through its eet on the denominator of the PW ratio, as
has been reognized in numerous referenes [58, 59, 60,
61, 62℄).
The highest preision study of
the NuTeV experiment [55,
VII. MEASUREMENTS OF THE HEAVY
QUARKS
A. Measurement of the Strange Sea
(47)
0
our understanding of the assoiated nulear orretions.
The ombination of these measurements, together with
Z
63℄.
s−
to date is from
The sign seleted
beam allowed measurement of the strange and antistrange seas independently,
reording 5163 neutrino-
indued dimuons, and 1380 antineutrino indued dimuon
events in its iron target. Figure 11 shows the t for asymmetry between the strange and anti-strange seas in the
NuTeV data.
Charged urrent neutrino-indued harm prodution,
(ν/ν̄)N → µ+ µ− X , proeeds primarily through the sub+
−
proesses W s → c and W s̄ → c̄ (respetively), so
The integrated strange sea asymmetry from NuTeV
has a positive entral value: 0.00196 ± 0.00046 (stat)
±0.00045 (syst) +0.00148
−0.00107 (external). In NuSOnG, as in
this provides a unique mehanism to diretly probe the
NuTeV, the statistial error will be dominated by the an-
s(x)
tineutrino data set and is expeted to be about 0.0002.
and
s̄(x)
distributions. Approximately 10% of the
time the harmed partiles deay into
µ + X,
adding a
The systemati error is dominated by the
π and K
deay-
seond oppositely signed muon to the CC event's nal
in-ight subtration. This an be addressed in NuSOnG
state. These dimuon events are easily distinguishable,
through test-beam measurements whih will allow a more
and make up approximately 1% of the total CC event
aurate modeling of this bakground, as well as applying
20
the tehniques of CCFR to onstrain this rate [64, 65, 66℄.
We expet to be able to redue this error to about 0.0002.
The ombination of these redues the total error by about
10%, beause the main ontributon omes from the external inputs.
The external error on the measurement is dominated
by the error on the average harm semi-muoni branhing
ratio,
Bc :
Bc = Σi
Z
φ(E)fi (E)Bµ−i dE,
(48)
where φ is the neutrino ux in energy bins, fi is the
energy dependent prodution fration for eah hadron,
Bµ−i is the semi-muoni branhing ratio for eah
hadron. In the NuTeV analysis, this is an external input,
and
with an error of about 10%. To make further progress,
this error must be redued.
Fig. 12 shows the world measurements of
from referenes [17, 56, 57, 67, 68, 69, 70℄.
Bc
Bc , taken
Measuring
diretly requires the apability to resolve the indi-
vidual harmed partiles reated in the interation. The
best diret measurements are from emulsion. This kind
Figure 12: World measurements of
of measurement has been performed in past experiments
67, 68, 69, 70℄.
Bc .
See refs. [17, 56, 57,
(E531, Chorus) using emulsion detetors [69, 70℄, where
the deay of the harmed meson is well tagged.
Sine
the ross setion for harmed meson prodution is energy dependent, it is important to make a measurement
near the energy range of interest. The NuTeV strange sea
asymmetry study used a re-analysis of 125 harm events
measured by the FNAL E531 experiment [69℄ in the energy range of the NuTeV analysis (Eν
> 20 GeV). Bc
has
also been onstrained through indiret measurement via
ts.
For NuSOnG, our goal is to redue the error on
Bc
[71, 72℄ or a similar detetor. This is a 45 kg silion vertex
detetor whih ran in front of the NOMAD experiment.
The target was boron arbide interleaved with the silion.
This detetor suessfully measured 45 harm events in
+
0
that beam, identifying D , D and Ds . A similar detetor of this size in the NuSOnG beam would yield about
900
ν
events and 300
ν̄
events. This has the advantage
of being a low-Z material whih is isosalar and lose in
mass to the SiO2 of the detetor.
using an in situ measurement on glass by at least a fator
of 1.5. One method is to inlude Bc as a t parameter
in the analysis of the dimuon data. The unpreedentedly
B. Strange Quark Contribution to the Proton Spin
high statistis will allow a t as a funtion of neutrino
x
neutrino
An investigation of the strange quark ontribution to
DIS almost exlusively result from sattering o valene
the elasti vetor and axial form fators of the proton
quarks, suh that the dimuon ross setion in that region
is possible in NuSOnG, by observing NC elasti and
energy for the rst time. Dimuons from high
may be
CC quasi-elasti sattering events; namely νp → νp and
νn → µ− p events in neutrino mode, and ν̄p → ν̄p and
ν̄p → µ+ n events in antineutrino mode. The motivation
Unfortunately, antineutrino harm prodution is not
for making this measurement omes from a number of
isolates
Bc
from the strange sea.
assumption is then taken that
In dimuon ts, the
Bc−ν = Bc−ν , Bc
measured diretly from the dimuon data.
well measured by past experiments. This leads to on-
Bc−ν = Bc−ν .
reent (and not so reent) studies in proton struture.
An
Over the last 15 years a tremendous eort has been
example of a potential soure of dierene in neutrino
−
and antineutrino mode, onsider that νn → µ Λc has no
made at MIT-Bates, Jeerson Lab, and Mainz to mea-
erns about the assumption that
analogous reation in the antineutrino hannel.
sure the strange quark ontribution to the vetor form
fators (that is, the eletromagneti form fators) of the
These arguments provide the motivation for inluding
a high resolution target/traker in the NuSOnG design
proton via parity-violating eletron sattering from pro4
tons, deuterons, and He [73, 74, 75, 76, 77, 78, 79, 80, 81,
that an diretly measure the semileptoni branhing ra-
82, 83℄. The tehnique is to observe the parity-violating
tio to harm in both
ν
ν̄
There
beam spin asymmetry in elasti sattering of longitudi-
are two feasible detetor tehnologies. The rst is to use
nally polarized eletrons from these unpolarized targets;
emulsion, as in past experiments. This is proven tehnol-
this asymmetry is aused by an interferene between the
and
running modes.
ogy and sanning ould be done at the faility in Nagoya,
one-photon and one-Z exhange amplitudes [84℄.
Japan. The seond is to use the NOMAD-STAR detetor
result, the weak neutral urrent analog of the eletro-
As a
21
magneti form fators of the proton may be measured
Sine the neutrino experiments will undoubtedly be
and this gives aess to the strange quark ontribution.
arried out on nulear targets (perhaps arbon or argon),
This worldwide experimental program will soon be om-
then the extration of the properties of the proton from
plete. The results available to date (from global analy-
these data needs to be done with are. Reent theoretial
ses [85, 86, 87℄) indiate a small (and nearly zero) ontri-
investigations point to the idea of measuring the ratio of
bution of the strange quarks to the elasti eletri form
s
fator, GE ; this is not surprising, as the total eletri
harge in the proton due to strange quarks is zero. At
in this ratio [93℄, leaving behind the ratio that would have
NC to CC yields; nulear eets appear to largely anel
been obtained on nuleon targets.
the same time, these same data point to a small but likely
The only available data on neutrino NC elasti sat-
positive ontribution of the strange quarks to the elasti
s
magneti form fator, GM , indiating a small positive
ontribution of the strange quarks to the proton mag-
tering is from the BNL E734 experiment [94℄; the uner-
neti moment. Due to the prominent role played by the
Z -exhange
amplitude, these experiments are also sensi-
tive to the strange quark ontribution to the elasti axial
tainties reported from that experiment are onsiderable
s
and limit the preiseness of any extration of GA based
on them. If NuSOnG an provide more preise measureQ2 then
ments of NC elasti sattering extended to lower
the promise of this analysis tehnique an be fullled.
form fator, whih is related to the proton spin struture.
It is now well established by leptoni deep inelasti
C. Measurements of the Charm Sea:
sattering experiments that the spins of the valene and
sea quarks in the proton together ontribute about 30% of
the total proton intrinsi angular momentum of
h̄/2.
The
1.
Charm Prodution
strange quark ontribution is estimated to be about -10%
in inlusive DIS (an analysis whih makes use of SU(3)avor symmetry) [88℄, but is found to be approximately
zero in semi-inlusive DIS (an alternative analysis whih
makes no use of SU(3) but needs fragmentation funtions
instead) [89℄. A reent global analysis [90℄ whih made
use of both inlusive and semi-inlusive DIS and whih
allowed for the possibility of SU(3)-avor violation found
We an also study the harm sea omponent of the
proton whih an arise from the gluon splitting proess
g → cc̄
the above strange sea extration, the harm sea,
νµ + c → νµ + c
.
֒→ s + µ+ + νµ
no need in the data for any violation of SU(3) and into the proton spin.
In the deep inelasti ontext, the
enapsulated in the heliity-dierene strange quark parton distribution funtion,
∆s(x) = s→ (x) − s← (x)
where
s→ (x) [s← (x)℄
is the probability density for nd-
ing a strange quark of momentum fration
x with its spin
parallel [anti-parallel℄ to the proton spin. The axial urrent relates the rst moment of this parton distribution
funtion to the value of the strange quark ontribution
s
to the elasti axial form fator of the proton [91℄, GA , at
Q2 = 0 :
c(x, µ),
an be measured using the following proess:
diated a small negative ontribution of strange quarks
ontribution strange quarks make to the proton spin is
produing harm onstituents inside the
proton.[95, 96, 97℄ In a measurement omplementary to
In this proess, we exite a onstituent harm quark in
the proton via the NC exhange of a
Z
boson; the -
nal state harm quark then deays semi-leptonially into
sµ+ νµ . We refer to this proess as Wrong Sign Muon
(WSM) prodution as the observed muon is typially the
−
opposite sign from the expeted νµ d → µ u DIS proess.
For antineutrino beams, there is a omplementary pro−
ess ν̄µ + c̄ → ν̄µ + c̄ with a subsequent c̄ → s̄ + µ + ν̄µ
deay with yields a WSM with respet to the onven+
tional ν̄µ u → µ d proess. Here, the ability of NuSOnG
to have sign-seleted beams is ruial to this measurement as it allows us to distinguish the seondary muons,
and thus extrat the harm-sea omponent.
In the onventional implementation of the heavy quark
Z
0
1
PDFs, the harm quark beomes an ative parton in the
dx∆s(x) = GsA (Q2 = 0).
proton when the sale
mc ;
The strange quark ontribution to the elasti axial form
fator an be measured by ombining data from neutrino
i.e.
fc (x, µ)
µ
is greater than the harm mass
µ > mc . Additionally, we
variable as we have a mas-
is nonzero for
must resale the Bjorken
x
In this way
sive harm in the nal state. The original resaling proex → x(1+m2c /Q2 ) whih
provides a kinemati penalty for produing the heavy
the strange quark spin ontribution to the proton spin
harm quark in the nal state.[98℄ As the harm is pair-
an be measured in a ompletely independent way using
2
2
low-Q elasti sattering instead of high-Q deep inelas-
harm quarks in the nal stateone whih is observed in
NC elasti sattering from the proton with data from
parity-violating elasti
~ep
sattering [92℄.
ti sattering. An analysis done using this method [85℄
GsA may in fat be negative at Q2 = 0 but
this onlusion is not denitive due to the limitations of
indiates that
the urrently available neutrino data.
dure is to make the substitution
produed by the
g → c c̄
proess, there are atually two
the semi-leptoni deay, and one whih deays hadronially and is part of the hadroni shower. Thus, the ap2
2
propriate resaling is not x → x(1 + mc /Q ) but instead
2
2
x → χ = x(1 + 4mc /Q ); this resaling is implemented
22
in the ACOTχ sheme, for example.[99, 100, 101℄ The
2
2
fator (1 + 4mc /Q ) represents a kinemati suppression
fator whih will suppress the harm proess relative to
the lighter quarks.
The dierential ross setion for NC neutrino sattering is
2
G MN Eν 2 2
dσ
(νp → νc) = F
RZ (Q ) ×
dξdy
π
1
MN
2
2
× gL
+ gR
(1 − y)2 − (2gL gR )
ξ c(ξ, µ),
2
Eν
gL = t3 − Q2c sin2 θW , gR = −Q2c sin2 θW , and for
2
harm t3 = 1/2 and Qc = 2/3. The fator RZ (Q ) =
2
2
1/(1 + Q /MZ ) arises from the Z -boson propagator. The
where
orresponding result for the anti-harm is given with the
substitutions
gL ↔ gR
and
c ↔ c̄.
In the limit we an neglet the MN /Eν term we
have the approximate expressions for the total ross
R1
x fi (x, Q) of
0
harm (upper urve) and bottom (lower urve) PDFs (in perFigure 13:
ent) vs.
Integrated momentum frations
Q
in GeV. Both the quark and antiquark ontri-
butions are inluded. Horizontal lines at 0.5% and 1.0% are
indiated as this is the typial size of postulated intrinsi ontributions.
setion:[96℄
σ(νp → νc) ∼
G2F MN Eν
(0.129) C
π
(49)
massive harm ross setion is down a fator of
∼ 0.005
ompared to the total inlusive ross setion.
As the
muon from the NC harm proess is a seondary muon,
and
we must additionally fold in the semi-leptoni branhing
ratio
G2 MN Eν
σ(νp → νc̄) ∼ F
(50)
(0.063) C̄
π
R1
R1
with C =
ξmin ξ c(ξ, µ) dξ and C̄ = ξmin ξ c̄(ξ, µ) dξ . We
2
2
2
take ξ = x(1 + 4mc /Q ) and ξmin = mc /(2MN ν).
We will be searhing for the WSM signal ompared to
the onventional harged-urrent DIS proess; therefore
it is useful to benhmark the rate for WSM prodution
Bc ∼ 10%,
and the aeptane fator of observ-
ing the seondary muon in the detetor (Aµ ∼20%).[17℄
Combining the relevant fators, we estimate the rate for
−4
NC harm prodution is approximately a fator of
10
as ompared to the CC DIS proess. Thus, for an antiipated design of 600M νµ CC events, one would expet on
the order of 60K NC harm events. This estimate is also
onsistent with a diret saling from the NuTeV result of
Ref. [97℄.
by omparing this to the the usual harged-urrent DIS
proess,
2.
2
G MN Eν 2
dσ
(νp → µ− X) = F
RW (Q2 ) ×
dx dy
π
× q(x) + (1 − y)2 q̄(x)
Bakgrounds
Extrapolating from investigations by CCFR [96℄, and
NuTeV [97℄, the dominant bakground for the measure(51)
ment of the harm sea omes from
ν̄µ
ontamination.
In these studies, it was determined that by demanding
2
RW (Q2 ) = 1/(1+Q2 /MW
). We an again integrate
x and y to obtain an estimate of the total ross
Evis > 100 GeV, the bakground rate ould be redued
−4
. Other bakground proesses inlude νe into 2.3 × 10
setion in terms of the integrated PDFs as in Eq. (49)
dued dilepton prodution, mis-identied dimuon events,
with
over
and NC interations with a
and Eq. (50):
2
G MN Eν 2
σ(νN → µ− X) ∼ F
RW (Q2 ) ×
π
1
1
×
U + D + 2S + (Ū + D̄ + 2C̄)
2
3
where
{U, D, S} are dened analogously to C ,
N = 12 (p + n) for an isosalar target.
π/K
deay in the hadron
shower; these proesses ontribute approximately an ad−4
ditional 1.5 × 10
to the bakground rate. As ompared
to CCFR and NuTeV, the NuSOnG design has a number
of improvements suh as lower mass density for improved
shower measurement; hene, omparable bakground re(52)
dutions should be ahievable.
and we
have used
3.
Intrinsi Charm
The relative rate for NC harm prodution is determined by the above fators together with a ratio of inte-
In the above disussion we have assumed that the
grated PDFs. For a mean neutrino energy of 100 GeV, the
harm omponent of the proton arises perturbatively
23
g → cc̄;
has foused mainly on the QCD physis whih an be
in this senario the harm PDF typially vanishes at
aessed with this new high energy, high statistis neu-
sales below the harm mass (fc (x, µ
< mc ) = 0), and for
trino sattering experiment. During its ve-year data a-
all the harm partons arise from gluon splitting.
quisition period, the NuSOnG experiment ould reord
There is an alternative piture where the harm quarks
almost one hundred thousand neutrino-eletron elasti
from gluons splitting into harm quark pairs,
µ > mc
are taken to be intrinsi to the proton; in this ase there
satters and hundreds of millions of deep inelasti sat-
µ < mc .
tering events, exeeding the urrent world data sample
are intrinsi harm partons present at sales
For
µ > mc ,
the harm PDF is then a ombination of
this intrinsi PDF and the extrinsi PDF omponent
arising from the
g → cc̄
proess.
by more than an order of magnitude.
With this wealth of data, NuSOnG an address a wide
variety of topis inluding the following.
A number of analyses have searhed for an intrinsi
harm omponent of the proton, and this intrinsi om-
•
NuSOnG an inrease the statistis of the Elas-
ponent is typially onstrained to have an integrated mo-
ti Sattering (ES) and Deeply Inelasti Sattering
mentum fration less than a perent or two [102, 103℄.
(DIS) data sets by nearly two orders of magnitude.
In Figure 13 we display the integrated momentum fraR1
tion,
0 x fi (x, µ), for harm and bottom as a funtion of
µ due to the extrinsi PDF omponent arising from
the
g → cc̄
or
g → bb̄
•
possibility to perform separate extrations of the
ν
ν
ν
ν̄
ν̄
ν̄
struture funtions: {F2 , xFe , RL , F2 , xFe , RL }.
This allows us to test many of the symmetries
proess. These momentum fra-
tions start from zero at the orresponding quark mass,
and assumptions whih were employed in previous
and inrease slowly as the partoni omponents pik up
struture funtion determinations.
momentum from the gluon splitting proess.
If we are searhing for an additional intrinsi ompo-
∼ 1%,
The unpreedented statistis of NuSOnG allow the
•
NuSOnG will help us to disentangle the nulear
we will be
eets whih are present in the PDFs. Furthermore,
most sensitive to suh a omponent in the threshold re-
this may help us address the long-standing tensions
nent with a momentum fration of
gion where the intrinsi omponent is not overwhelmed
between the NC harged-lepton and CC neutrino
by the extrinsi ontribution. In this regard, NuSOnG
DIS measurements.
is well suited to searh for these intrinsi terms as it will
provide good statistis in the threshold region. Measur-
•
High preision NuSOnG measurements are sensi-
ing the harm prodution proess desribed above, Nu-
tive to Charge Symmetry Violation (CSV) and
SOnG an attempt to extrat the harm PDF as a fun-
other new physis proesses. Suh eets an sig-
tion of the
µ
sale, and then evolve bak to
µ = mc .
niantly inuene preision Standard Model pa-
1.
fc (x, µ = mc ) < 0,
sin θW . In partiular,
is a sensitive probe of both the heavy quark
rameter extrations suh as
Three outomes are possible:
∆xF3
whih would imply the data are
omponents, and CSV eets.
2
inonsistent with the normal QCD evolution.
2.
fc (x, µ = mc ) = 0,
whih would imply the data is
•
tional probe of the strange quark PDFs, and the
onsistent with no intrinsi harm PDF.
3.
sign-seleted beam an separately study
s̄(x).
fc (x, µ = mc ) > 0, whih would imply the data is
inonsistent with an intrinsi harm PDF.
By making aurate measurements of harm indued proesses in the threshold region, NuSOnG an provide a
disriminating test to determine whih of the above possibilities is favored. Hene, the high statistis of NuSOnG
in the threshold region are well suited to further onstrain
the question of an intrinsi harm omponent.
VIII. SUMMARY AND CONCLUSIONS
The NuSOnG experiment an searh for new physis
from the keV through TeV energy sales.
NuSOnG dimuon prodution provides an exep-
s(x) and
s-quark
Additionally, NuSOnG an probe the
ontribution to the proton spin.
•
The high statistis of NuSOnG may allow the measurement of the harm sea and an method to prove
the intrinsi-harm ontent of the proton.
While
this is a diult measurement, the NuSOnG kinematis allow the measurement of harm-indued
proesses in the threshold region where the intrinsi harater an most easily be diserned.
While the above list presents a very ompelling physis
ase for NuSOnG, this is only a subset of the full range
of investigations that an be addressed with this faility.
This artile
Aknowledgments
2
If we work at NLO,
fc (x, µ = mc ) should
be stritly greater than
or equal to zero; at NNLO and beyond the boundary onditions
yield a negative PDF of order
∼ α2s
We thank the following people for their informative
disussions regarding neutrino-nuleus interations. and
their thoughtful omments on the development of this
24
physis ase:
Andrei Kataev, Sergey Kulagin, P. Lan-
Deutshe Forshungsgemeinshaft, The Kavli Institute
gaker, Roberto Petti, M. Shaposhnikov, F. Vannui,
for Theoretial Physis, The United States Department
and J. Wells.
We aknowledge the support of the fol-
lowing funding agenies for the authors of this paper:
[1℄ E. A. Pashos and L. Wolfenstein.
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