Table Of ContentCan One Distinguish τ−Neutrinos from Antineutrinos in Neutral-Current Pion
Production Processes?
E. Hern´andez,1 J. Nieves,2 and M. Valverde2
1Grupo de F´ısica Nuclear, Departamento de F´ısica Fundamental e IUFFyM,
Facultad de Ciencias, E-37008 Salamanca, Spain.
2Departamento de F´ısica At´omica, Molecular y Nuclear,
Universidad de Granada, E-18071 Granada, Spain
Apotentialwaytodistinguishτ−neutrinosfromantineutrinos,belowtheτ−productionthreshold,
but above the pion production one, is presented. It is based on the different behavior of the
neutral current pion production off the nucleon, depending on whether it is induced by neutrinos
or antineutrinos. This procedure for distinguishing τ−neutrinos from antineutrinos neither relies
on any nuclear model, nor it is affected by any nuclear effect (distortion of the outgoing nucleon
7 waves,etc...). Weshowthatneutrino-antineutrinoasymmetriesoccurbothinthetotallyintegrated
0 crosssectionsandinthepionazimuthaldifferentialdistributions. Todefinetheasymmetriesforthe
0 latterdistributionswejustrelyonLorentz-invariance. Alltheseasymmetriesareindependentofthe
2 leptonfamily and canbeexperimentally measured byusingelectron ormuonneutrinos, duetothe
lepton family universality of theneutralcurrentneutrinointeraction. Neverthelessand toestimate
n
their size, we have also used the chiral model of hep-ph/0701149 at intermediate energies. Results
a
J are really significant since the differences between neutrino and antineutrino induced reactions are
always large in all physicalchannels.
8
1
PACSnumbers: 12.15.Mm,25.30.Pt,13.15.+g
2
v
9 I. INTRODUCTION
1
1
Distinguishing τ−neutrinos from antineutrinos,canbe easilydone abovethe τ−production thresholdenergy,since
8
then the charged–current(CC) channelis open. Conservationofleptonnumber implies thatneutrinos generatea tau
0
lepton, while anantineutrino producesan antilepton. The chargeofthe taulepton unambiguouslyrevealsthe nature
6
0 of the incident neutrino beam. Below the τ−threshold, τ−neutrinos or antineutrinos only interact with matter via
/ neutral–current (NC) driven processes, and the outgoing lepton is also a neutrino or antineutrino, and the difference
h
does not manifest itself in this clear way.
p
Nonetheless, it is out of any doubt the great interest of distinguishing between neutrinos and antineutrinos in
-
p different scenarious: CP–violationin ν –ν oscillations [1], supernova–neutrino physics [2]1, ...
µ τ
e
Very recently, Jachowicz and collaborators suggested that in neutrino–nucleus processes, the helicity–related dif-
h
ferences between neutrino and antineutrinos induce some asymmetries in the polarizationof the ejected nucleons [3].
:
v Fromthis fact, the authorsof this referenceconclude thatthese asymmetries representa potentialway to distinguish
i
X neutrinos from antineutrinos in NC neutrino scattering on nuclei.
We present in this letter an alternative manner to distinguish τ−neutrinos from antineutrinos, below the
r
a τ−production threshold (≈ 3.5 GeV in the Laboratory (LAB) frame for production off the nucleon), but above
the pion production one (≈ 0.15 GeV in the LAB frame). The method is based on the different behavior of the
NC pion production reaction in the nucleon, depending on whether it is induced by neutrinos or antineutrinos. This
procedure for distinguishing τ−neutrinos from antineutrinos neither relies on any nuclear model, nor it is affected
by any nuclear effect (distortion of the outgoing nucleon waves, etc...). We show that neutrino-antineutrino asym-
metries occur both in the totally integrated cross sections and in the pion azimuthal differential distributions. To
define the asymmetries for the latter distributions we just rely on Lorentz-invariance, and experimentally it requires
determining the neutrino scattering plane, detecting either a neutral or a positively charged pion and measuring its
momentum. Since the outgoing neutrino will not likely be detected, to fix the neutrino scattering plane would also
require measuring the outgoing nucleon momentum and some knowledge of the incoming neutrino momentum.
All these asymmetries are independent of the lepton family and can be experimentally measured by using electron
ormuonneutrinos,due totheleptonfamilyuniversalityoftheNC neutrinointeraction. Neverthelessandtoestimate
their size, we have used the chiral model of Ref. [4] to compute them up to neutrino/antineutrino energies of 2 GeV.
1 Attypical supernovaenergies,ντ neutrinosdonotparticipateinCCreactions
2
II. KINEMATICS AND CROSS SECTIONS
Y
φ
π X
kπ
θ’ θ
k’ π
(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1)(cid:0)(cid:0)(cid:1)(cid:1) Z
(cid:0)(cid:1)(cid:0)(cid:1)(cid:0)(cid:1)
k q
FIG. 1: Definitionofthedifferentkinematical variablesusedthroughthiswork.
We will focus on the neutrino–pion production reaction off the nucleon driven by neutral currents,
′ ′
ν(k)+ N(p)→ν (k )+N(p)+ π(k ) (1)
l l π
though the generalization of the obtained expressions to antineutrino induced reactions is straightforward.
The unpolarized differential cross section, with respect to the outgoing neutrino and pion kinematical variables is
given in the LAB frame (kinematics is sketched in Fig 1) by2
d5σ |~k′| G2 +∞ dk k2
ν = π πL(ν)(Wµσ )(ν) (2)
dΩ(kˆ′)dE′dΩ(kˆπ) |~k |16π2 Z0 Eπ µσ NCπ
with~k and~k′ theLABneutrinomomenta,E′ =|~k′|,theenergyoftheoutgoingneutrino,~k andE ,themomentum
π π
andenergyofthe pioninthe LABsystem,G=1.1664×10−11MeV−2,theFermiconstantandLandW theleptonic
and hadronic tensors, respectively. The leptonic tensor is given by (in our convention, we take ǫ = +1 and the
0123
metric gµν =(+,−,−,−)):
L(ν) = (L(ν)) +i(L(ν)) =k′k +k′k −g k·k′+iǫ k′αkβ (3)
µσ s µσ a µσ µ σ σ µ µσ µσαβ
and it is orthogonalto qµ =(k−k′)µ for massless neutrinos, i.e, L(ν)qµ =L(ν)qσ =0.
µσ µσ
The hadronic tensor includes all sort of non-leptonic vertices and it reads
1 d3p′ 1
(Wµσ )(ν) = δ4(p′+k −q−p)hN′π|jµ (0)|NihN′π|jσ (0)|Ni∗ (4)
NCπ 4M Z (2π)32E′ π nc nc
sXpins N
withM thenucleonmassandE′ theenergyoftheoutgoingnucleon. Inthesumoverinitialandfinalnucleonspins3,
N
the bar over the sum denotes the average over the initial ones. As for the one particle states they are normalized so
that hp~|p~′i=(2π)32p δ3(p~−p~′), and finally for the neutral current we take
0
8 4 4
jµ =Ψ γµ(1− sin2θ −γ )Ψ −Ψ γµ(1− sin2θ −γ )Ψ −Ψ γµ(1− sin2θ −γ )Ψ (5)
nc u 3 W 5 u d 3 W 5 d s 3 W 5 s
with Ψ , Ψ and Ψ quark fields, and θ the Weinberg angle (sin2θ =0.231). Note that with all these definitions,
u d s W W
thematrixelementhN′π|jµ (0)|Niisdimensionless. Both,leptonandhadrontensorsareindependentoftheneutrino
nc
2 ToobtainEq.(2)wehaveneglectedthefour-momentumcarriedoutbytheintermediateZ bosonwithrespecttoitsmass(MZ),and
have used the existing relationbetween the gauge weak coupling constant, g=e/sinθW, a−nd the Fermi constant: G/√2=g2/8MW2 ,
withetheelectroncharge,θW theWeinbergangle,cosθW =MW/MZ andMW theW bosonmass.
3 Sinceright-handedneutrinos aresterile,onlyleft-handedneutrinos areconsidered. −
3
leptonfamily, andthereforethe crosssectionfor the reactionofEq.(1)is the samefor electron,muonortauincident
neutrinos. As the quantity L(ν)(Wµσ )(ν) is a Lorentz scalar, to evaluate it we take for convenience ~q in the Z
µσ NCπ
direction (see Fig. 1). Referring now the pion variables to the outgoing πN pair center of mass frame (as it is usual
in pion electroproduction) would be readily done by means of a boost in the Z direction. Note that the azimuthal
angle φ is left unchanged by such a boost.
π
For antineutrino induced reactions we have
L(ν¯) =L(ν), (Wµσ )(ν¯) =(Wµσ )(ν) (6)
µσ σµ NCπ NCπ
For the sake of simplicity, from now on, we will omit in the hadronic tensor the explicit reference to ν or ν¯ and the
NC π label. By construction, the hadronic tensor accomplishes
Wµσ =Wµσ+iWµσ (7)
s a
with Wµσ (Wµσ) real symmetric (antisymmetric) tensors. The hadronic tensor is completely determined by up to a
s a
total of 19 Lorentz scalar and real, structure functions W (q2, p·q, p·k , k ·q),
i π π
Wµν = Wµν PC+ Wµν PV (8)
s,a s,a s,a
(cid:0) (cid:1) (cid:0) (cid:1)
(Wµν)PC = W gµν +W pµpν +W qµqν +W kµkν +W (qµpν +qνpµ)+W (qµkν +qνkµ)
s 1 2 3 4 π π 5 6 π π
+ W (pµkν +pνkµ) (9)
7 π π
(Wµν)PV = W qµǫν kαpβqγ +qνǫµ kαpβqγ +W pµǫν kαpβqγ +pνǫµ kαpβqγ
s 8 .αβγ π .αβγ π 9 .αβγ π .αβγ π
(cid:16) (cid:17) (cid:16) (cid:17)
+ W kµǫν kαpβqγ +kνǫµ kαpβqγ (10)
10 π .αβγ π π .αβγ π
(cid:16) (cid:17)
(Wµν)PV = W (qµpν −qνpµ)+W (qµkν −qνkµ)+W (pµkν −pνkµ) (11)
a 11 12 π π 13 π π
(Wµν)PC = W ǫµναβp q +W ǫµναβp k +W ǫµναβq k +W qµǫν kαpβqγ −qνǫµ kαpβqγ
a 14 α β 15 α πβ 16 α πβ 17 .αβγ π .αβγ π
(cid:16) (cid:17)
+ W pµǫν kαpβqγ −pνǫµ kαpβqγ +W kµǫν kαpβqγ −kνǫµ kαpβqγ (12)
18 .αβγ π .αβγ π 19 π .αβγ π π .αβγ π
(cid:16) (cid:17) (cid:16) (cid:17)
Though Wµν is not orthogonal to qµ, the W , W , W , W , W , W and W terms do not contribute to the
3 5 6 8 11 12 17
differential cross section since the leptonic tensor is orthogonalto the four vector qµ for massless neutrinos.
Thetensor(Wµν)PV =(Wµν)PV+ i(Wµν)PV whencontractedwiththeleptonicone,L(ν),providesapseudo-scalar
s a µν
quantity, i.e., such contraction is not invariant under a parity transformation. Indeed, under a parity transformation
we have,
L(ν) →(Lνµ)(ν), (W )PV →−(Wνµ)PV (13)
µν µν
whereasthetensor(Wµν)PC =(Wµν)PC+ i(Wµν)PC transformsas(Lµν)(ν). Thisexplainsthe originofthe adopted
s a
labels PC and PV, which stand for parity violating and conserving contributions to the fifth differential cross section
d5σ/dΩ(kˆ′)dE′dΩ(kˆ ), respectively4. The triple differential cross section d3σ/dΩ(kˆ′)dE′ is a scalar, up to the factor
π
|~k′|/|~k |. Thus all parity-violating contributions must disappear when performing the pion solid angle integration.
Note that the coordinate systemused to define dΩ(kˆ ) involvesthe pseudo-vector~k×~k′ to setup the Y−axis,which
π
induces the non-parity invariant nature of d5σ/dΩ(kˆ′)dE′dΩ(kˆ ). In electropion production processes, the leptonic
π
tensor is purely symmetric, and the symmetric part of the hadronic one can not contain terms involving the Levi-
Civita tensor, since the electromagnetic interaction preserves parity. Hence in that case d5σ/dΩ(kˆ′)dE′dΩ(kˆ ) turns
π
out to be a scalar under parity.
A final remark concerns the time-reversal (T) violation effects apparently encoded in the decomposition of the
hadronic tensor in Eqs. (8–12). Under a time reversal transformation, and taking into account the antiunitary
4 Similarconclusions canbealsoreachedbyjustnotingthat thesymmetricpartoftheleptonictensordoes notcontain anyLevi-Civita
ǫµναβ pseudo–tensor,whereasitsantisymmetricpartisjustgivenbyaLevi-Civitatensor. Thus,togetaparityinvariantquantity,the
symmetricpartofthehadronictensorshouldnothaveanyLevi-Civitatensor,whilealltermsofitsantisymmetricpartshouldcontain
aLevi-Civitatensor.
4
character of the T−operator,we have
L(ν) →(Lµν)(ν), (W ) →(Wµν)PC, (W )PV →−(Wµν)PV (14)
µν µν PC µν
and therefore L(ν)Wµν is not T−invariant either, because of the presence of the PV terms in the hadronic tensor5.
µν
This does not necessarily means that there exists a violation of T−invariance in the process [5]. Time reversal is
the transformation that changes the direction of time: t → −t. Thus invariance under time reversal means that the
description of physical processes does not show any asymmetry if we look backward in time. The invariance under
time reversalis equivalent to
|Mi→f|2 =|MTf→Ti|2 (15)
wherei→f denotesthetransitionfromtheinitialstateitothefinalstatef,beingMi→f itscorrespondingtransition
amplitude,andTf andTidenotethestatesobtainedfromf andi,respectively,byreversingthemomenta,spins,etc..
When one is dealing with electromagneticor weakinteractions,which may be treated to first orderin the interaction
Hamiltonian (H ), the transition matrix operator can be approximated by H , being it then hermitian. In these
I I
circumstances |MTf→Ti|≈|MTi→Tf| and the invariance under time reversalmay be written in the following form
|Mi→f|2 =|MTf→Ti|2 ≈|MTi→Tf|2 (16)
Theaboveequationimpliesthatthetransitioni→f cannotshowupcorrelationsthatchangesignundertimereversal
(T−odd correlations). Thus at first order in the interaction Hamiltonian, the tensor (Wµν)PV should vanish, since
it would lead to violations of time reversal invariance. However, here we have strong interacting final states (pion
and nucleon) and in fact the transition matrix operator is not hermitian, since it cannot be approximated by H .
I
Therefore Eq. (16) does not hold and time reversal invariance does not forbid a non-vanishing (Wµν)PV tensor. In
summary, besides genuine time reversal violations [6]6, the existence of strong final state interaction effects7 allows
for the existence of T−odd correlations in (L(ν)) Wµν, induced by the (Wµν)PV term of the hadronic tensor.
µν
After this discussion,we arein conditions ofstudying the pionazimuthal angledependence ofthe differential cross
section. With our election of kinematics (~k,~k′ in the XZ plane), we find (L(ν)) =(L(ν)) =(L(ν)) =(L(ν)) =
s 0y s xy s zy a 0x
(L(ν)) =(L(ν)) =0, and then
a 0z a xz
+∞ dk k2 +∞ dk k2
π π(L(ν)) Wµν = π π (L(ν)) W00+2(L(ν)) W0x+2(L(ν)) W0z+(L(ν)) Wxx
Z0 Eπ s µν s Z0 Eπ n s 00 s s 0x s s 0z s s xx s
+ (L(ν)) Wyy+(L(ν)) Wzz+2(L(ν)) Wxz
s yy s s zz s s xz s
o
= A +B cosφ +C cos2φ +D sinφ +E sin2φ (17)
s s π s π s π s π
+∞ dk k2 +∞ dk k2
π π (L(ν)) Wµν = 2 π π (L(ν)) W0y+(L(ν)) Wxy+(L(ν)) Wyz
Z0 Eπ a µν a Z0 Eπ n a 0y a a xy a a yz a o
= −A −B cosφ −D sinφ (18)
a a π a π
whereweexplicitlyshowtheφ dependence8. ThePVtermofthehadronictensorhasledtotheparityviolatingsinφ
π π
and sin2φ contributions (all of them proportional to k ). They disappear when the pion solid angle integration is
π πy
5 NotethattransformationsgiveninEqs.(6), (13)and(14)implythattheleptonictensor,byitself,isinvariantunderCPT.
6 Oneisfacedwiththeproblemofeliminatingfinal-state-interaction effects,oratleastofhavingthemundercontrol,inordertohavea
significanttestofT violation[5,6].
7 Forinstance,atintermediateenergiesandforCCdrivenprocesses,the∆(1232)resonanceplaysacentralrole[7]. Theinclusionofthe
resonancewidthaccounts partiallyforthestrongfinalstateinteractioneffects.
8 Allstructurefunctions,Wi=1,···19,dependontheLorentzscalarp kπandkπ qfactors,whicharefunctionsoftheangleformedbetween
· ·
theq~and~kπ vectors,andthustheyareindependent ofφπ,whenq~istaken alongtheZ axis.
−
5
performed, as anticipated. Thanks to Eq. (6), we obtain for neutrino and antineutrino reactions
d5σ |~k′| G2
ν
(ν,ν)→ = A +A +(B +B )cosφ +C cos2φ +(D +D )sinφ
dΩ(kˆ′)dE′dΩ(kˆπ) |~k |16π2n s a s a π s π s a π
+ E sin2φ (19)
s π
o
d5σ |~k′| G2
ν¯
(ν¯,ν¯)→ = A −A +(B −B )cosφ +C cos2φ +(D −D )sinφ
dΩ(kˆ′)dE′dΩ(kˆπ) |~k |16π2n s a s a π s π s a π
+ E sin2φ (20)
s π
o
III. NEUTRINO-ANTINEUTRINO ASYMMETRIES
From Eqs. (19) and (20), we see that neutrino-antineutrino asymmetries may occur both in the totally integrated
cross sections and in the pion azimuthal differential distributions. For the neutrino and antineutrino induced total
cross sections we have
G2
(ν,ν)→σ = dΩ(kˆ′)dE′dΩ(kˆ )|~k′| A +A (21)
ν 16π2|~k |Z π (cid:16) s a(cid:17)
G2
(ν¯,ν¯)→σ = dΩ(kˆ′)dE′dΩ(kˆ )|~k′| A −A , (22)
ν¯ 16π2|~k |Z π (cid:16) s a(cid:17)
while for the azimuthal distributions, relations of the type
d5σ d5σ |~k′| G2
(φ )− (φ +π) = {(B +B )cosφ +(D +D )sinφ } (23)
dΩ(kˆ′)dE′dΩ(kˆπ) π dΩ(kˆ′)dE′dΩ(kˆπ) π (cid:12)ν s a π s a π |~k |8π2
(cid:12)
d5σ d5σ (cid:12) |~k′| G2
(φ )− (φ +π) = {(B −B )cosφ +(D −D )sinφ } (24)
dΩ(kˆ′)dE′dΩ(kˆπ) π dΩ(kˆ′)dE′dΩ(kˆπ) π (cid:12)ν¯ s a π s a π |~k |8π2
(cid:12)
(cid:12)
enhanceneutrino-antineutrinoasymmetries. Toincreasethestatisticsintheseazimuthalasymmetries,itisinteresting
to study the relations of Eqs. (23) and (24) for integrated cross sections. From the discussion in the Appendix A it
follows
1 dσ(φ ) dσ(φ +π)
π π
− = (B +B )cosφ +(D +D )sinφ (25)
s a π s a π
2(cid:18) dφπ dφπ (cid:19)(cid:12)ν
(cid:12)
1 dσ(φ ) dσ(φ +π) (cid:12)
π π
− = (B −B )cosφ +(D −D )sinφ (26)
s a π s a π
2(cid:18) dφπ dφπ (cid:19)(cid:12)ν¯
(cid:12)
(cid:12)
where for each event, φ is the angle formed by the neutrino and pion scattering planes. Since the outgoing neutrino
π
will not likely be detected, if the incoming neutrino momentum is known, the neutrino scattering plane will be set
up by detecting also the outgoing nucleon. Indeed, the neutrino scattering plane is also determined by the incoming
neutrino and the transferred momenta, and in the LAB system this latter one is given by the sum of the outgoing
nucleon and pion momenta (~q =~p′+~k ). Thus the strategy will be detecting in coincidences the outgoing pion and
π
nucleonmomenta,usethosetodeterminetheneutrinoscatteringplaneandthenmeasurethecorrespondingφ angle.
π
All these relations can be used to distinguish τ−neutrinos from antineutrinos, below the τ−production threshold,
but above the pion production one. Thus and using a proton target,the total cross section or the dependence on the
angle formed by the neutrino and pion planes in either of the channels,
νp→νnπ+, νp→νpπ0
ν¯p→ν¯nπ+, ν¯p→ν¯pπ0, (27)
may be used to determine the nature of the incident τ−neutrino beam below the τ−production threshold.
Moreover,wewouldliketopointoutthatthe(A ±A ),(B ±B )and(D ±D )responsefunctions,theintegrated
s a s a s a
cross sections σ , σ and the partially integrated quantities (B ±B ) and (D ±D ) are independent of the lepton
ν ν¯ s a s a
familyandcanbeexperimentallymeasuredbyusingelectronormuonneutrinos,duetotheleptonfamilyuniversality
oftheNCneutrinointeraction. Thusonewouldnothavetorelyonanytheoreticalmodeltodeterminetheasymmetry
relations proposed in this section.
6
A. Neutrino-antineutrino asymmetries from the chiral model of Ref. [4]
The above suggestion for distinguishing neutrino from antineutrino in NC pion production reactions will be of
no value if the terms A , B and D are much smaller than A , B and D . To estimate these response functions
a a a s s s
and address this issue, we have used the model developed in Ref. [4] to study the weak pion production off the
nucleon at intermediate energies. In this model, besides the Delta pole mechanism ∆P (weak excitation of the
∆(1232)resonanceandits subsequentdecayinto Nπ), somebackgroundterms,requiredby chiralsymmetry,arealso
considered. These chiral background terms are calculated within a SU(2) non-linear σ model involving pions and
nucleons, which implements the pattern of spontaneous chiral symmetry breaking of QCD. The model gives a fair
description of the neutrino and antineutrino CC and NC available data. Details can be found in Ref. [4].
In Fig. 2, we show neutrino and antineutrino NC pion production total cross sections predicted by the model of
Ref.[4],asafunctionoftheincomingneutrinoorantineutrinoenergy. There,68%CLbands(externallines),inferred
fromuncertainties of the model of Ref. [4] are alsodisplayed. Those uncertainties come fromre-adjusting the CA(q2)
5
form–factorthatcontrolsthe largesttermofthe ∆−axialcontribution. Onthe otherhand, sincethe maindynamical
ingredients of the model are the excitation of the ∆ resonance and the non-resonant contributions deduced from the
leading SU(2) non-linearσ lagrangianinvolvingpions andnucleons,we will concentrateinthe pion-nucleoninvariant
massW ≤1.4GeVregion. Forlargerinvariantmasses,thechiralexpansionwillnotwork,oratleastthelowestorder
used in Ref. [4] will not be sufficient and the effect of heavier resonances will become much more important [8]. For
this reasoninthe figure,wealwaysplot crosssections calculatedwith aninvariantpion-nucleonmasscutof1.4 GeV,
1.4 GeV
it is to say we compute dW dσ/dW. Such cut is commonly used in the CC pion production experimental
mπ+M
analyses at intermediateRenergies. See for instance [9] where it can also be seen that up to incoming neutrino LAB
energies of the order of 1 GeV the implementation of this cut hardly changes the measured cross section. In the
−
figure, we also display low energy data measuredin the Argonne NationalLaboratory(ANL) [10] for the νn→νpπ
reaction, which do not include any cut in W. As shownin Ref. [4], including or not in the theoretical calculation the
cut in W only influences significantly the result for the highest energy data–point, leading to variations of the order
of 20%, which are around half the experimental error.
Results of Fig. 2 are really significant since the differences between neutrino and antineutrino induced reactions
are always large, about a factor of two, in all physical channels. This confirms that this observable can be used to
distinguish neutrino from antineutrino above the pion production threshold.
Next we show results, within this model, for B ±B and D ±D defined in Eqs. (25) and (26). In Fig. 3 we plot
s a s a
−
theseresponsefunctionsforapπ finalstate. Therestofchannelsleadtosimilarresults. Clearneutrino-antineutrino
asymmetries canbe appreciatedin both panels of the figure. In particularD +D and D −D haveopposite signs.
s a s a
The difference between the number of events for which the pion comes out above the neutrino plane (N ) and
abv
those where the pion comes out below this plane (N ) is 4 (D +D ) [4 (D −D )] for a neutrino [antineutrino]
blw s a s a
induced process. Thus, the sign of this difference would determine whether one has a neutrino or an antineutrino
beam. Nevertheless, we should mention that the neutrino-antineutrino asymmetry based in the total cross section
discussed above (Fig. 2) would provide a signal with much better statistical significance. This is because the ratio
Nabv−Nblw would be of the order of 10−2 or smaller. Asymmetries based on the B response function will not be as
Nabv+Nblw
much unfavorable, from the statistical point of view.
These results point out that the neutrino–antineutrino asymmetries based on the total cross sections are certainly
much more useful than those based on the the pion azimuthal response functions, for intermediate neutrino energies.
The situation could be different for energies larger than those explored with the model of Ref. [4].
Acknowledgments
We thank M.J. Vicente-Vacas for useful discussions. JN acknowledges the hospitality of the School of Physics &
Astronomy at the University of Southampton. This work was supported by DGI and FEDER funds, under contracts
FIS2005-00810, BFM2003-00856 and FPA2004-05616, by Junta de Andaluc´ıa and Junta de Castilla y Le´on under
contracts FQM0225 and SA104/04, and it is a part of the EU Integrated Infrastructure Initiative Hadron Physics
Project contract RII3-CT-2004-506078.
APPENDIX A: PARTIALLY INTEGRATED AZIMUTHAL DISTRIBUTION
To define an azimuthal pion distribution, by integrating the outgoing neutrino variables and pion polar angle in
Eqs.(19) and(20), requires for eachevent a rotationto guaranteethat~q is in the Z−directionand that~k and~k′ are
7
0.08 0.06
ν ν
0.07 ν¯ ν¯
Z0n→nπ0 0.05 Z0p→nπ+
0.06
2m) 0.05 2m) 0.04
c c
38− 0.04 38− 0.03
0 0
1 1
( 0.03 (
σ σ 0.02
0.02
0.01
0.01
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
E(GeV) E(GeV)
0.08 0.07
ν ν
0.07 ν¯ 0.06 ν¯
Z0p→pπ0 ANL
0.06
) ) 0.05 Z0n→pπ−
2m 0.05 2m
38c− 0.04 38c− 0.04
0 0
1 1 0.03
( 0.03 (
σ σ
0.02
0.02
0.01 0.01
0 0
0 0.5 1 1.5 2 0 0.5 1 1.5 2
E(GeV) E(GeV)
FIG.2: NeutrinoandantineutrinoNCpionproductiontotalcrosssectionspredictedbythemodelofRef.[4],asafunctionoftheincoming
neutrino or antineutrino energy. An invariant pion-nucleon mass cut of W 1.4 GeV has been implemented. Central lines stand for
≤
the resultofthe model. 68% CLbands (external lines), inferredfromuncertainties of the model, arealsodisplayed. Experimental cross
sectionsforneutrinoπ− productionaretakenfromRef.[10].
contained in XZ−plane. The totally integrated unpolarized neutrino cross section reads
G2 d3k′d3k
σ = πL(ν)(k,k′)Wµσ(p,q,k ) (A1)
ν 16π2|~k |Z |~k′| Eπ µσ π
Toperformtheaboveintegrals,wetakeafixsystemofcoordinatesXYZ,andsinceL(ν)(k,k′)Wµσ(p,q,k )isascalar
µσ π
under spatial rotations, it can be evaluated as
L(ν)(Rk,Rk′)Wµσ(Rp,Rq,Rk ) (A2)
µσ π
where for each~k′, R is the spatial rotation which brings ~q and ~k×~k′ into the Z− and Y−axes, respectively. Such
rotation depends on~k′ but it is independent of~k . Then
π
G2 d3k′d3k
σ = πL(ν)(Rk,Rk′)Wµσ(Rp,Rq,Rk )
ν 16π2|~k |Z |~k′| Eπ µσ π
G2 d3k′d3k
= πL(ν)(Rk,Rk′)Wµσ(Rp,Rq,k ) (A3)
16π2|~k |Z |~k′| Eπ µσ π
where in the last step, for a fix ~k′, we have made a change of variables in the pion integral, ~k → ~kR ≡ R~k ,
π π π
and for simplicity we have called the dummy variable ~kR again ~k . Finally using spherical coordinates ~k =
π π π
8
0 8e-05
ν ν
-5e-05 Z0n→pπ ν¯ 6e-05 ν¯
−
2m) -1e-04 2m) 4e-05 Z0n→pπ−
38c− -0.00015 38c− 2e-005
(10 -0.0002 (10 -2e-05
Ba -0.00025 Da -4e-05
± -0.0003 ±
Bs Ds -6e-05
-0.00035 -8e-05
-0.0004 -0.0001
-0.00045 -0.00012
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
E (GeV) E (GeV)
FIG. 3: Predictions of the chiral model of Ref. [4] for the s a and s a (Eqs. (25) and (26)) response functions, as function of
B ±B D ±D
theneutrinoorantineutrinoincomingenergy.
|~k |(sinθ cosφ ,sinθsinφ ,cosθ ) for the pion integral, we get
π π π π π
dσ G2 d3k′d|~k ||~k |2d(cosθ )
ν = π π π L(ν)(Rk,Rk′)Wµσ(Rp,Rq,k )
dφπ 16π2|~k |Z |~k′| Eπ µσ π
G2 d3k′
= d(cosθ ) A +A +(B +B )cosφ +C cos2φ +(D +D )sinφ +E sin2φ
16π2|~k |Z |~k′| π n s a s a π s π s a π s πo
= A +A +(B +B )cosφ +C cos2φ +(D +D )sinφ +E sin2φ (A4)
s a s a π s π s a π s π
since we have R~q in the Z−direction and R~k and R~k′ are contained in XZ−plane, and thus the results of Eq. (19)
can be used. In the above equations, φ is the azimuthal angle of the rotated pion momentum in the XYZ system
π
of coordinates, or equivalently, it is the angle formed by the neutrino and pion planes. The first of these planes
is determined by the incoming (~k) and outgoing (~k′) neutrino momenta, while the transferred (~q) and pion (~k )
π
momenta fix the latter one. Thus, φ is the angle formed by the vectors (~k×~k′)×~q and ~k −(~k ·~q)~q/|~q|2 ,
π π π
(cid:16) (cid:17) (cid:16) (cid:17)
and it is therefore invariant under spatial rotations.
The antineutrino cross section can be obtained in a similar way, and thus it follows the neutrino–antineutrino
asymmetry relations given in Eqs. (25) and (26) similar to those in Eqs. (23) and (24), but for integrated cross
sections.
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