Neutral current coherent pion production

0.15 MB·

by  L. Alvarez-Ruso

Neutral current coherent pion production
L. Alvarez-Ruso, L. S. Geng, and M. J. Vicente Vacas
Departamento de F´ısica Te´orica and IFIC, Universidad de Valencia - CSIC;
Institutos de Investigaci´on de Paterna, Aptdo. 22085, 46071 Valencia, Spain.
We investigate the neutrino induced coherent pion production reaction at low and intermediate
energies. Themodelincludespion,nucleonand∆(1232)astherelevanthadronicdegreesoffreedom.
Nuclear medium effects on the production mechanisms and pion distortion are taken into account.
Weobtainthatthedominanceofthe∆excitationholdsduetolargecancellationsamongtheback-
groundcontributions. Weconsidertwosets ofvectorandaxial-vector N-∆ transition form-factors,
evidencingthestrongsensitivityoftheresultstotheaxialcouplingCA(0). Thedifferencesbetween
5
neutrinoand antineutrino cross sections, emerging from interference terms, are also discussed.
8 PACSnumbers: 25.30.Pt,13.15.+g,23.40.Bw
0
0
2 I. INTRODUCTION termsandthedressingofthe∆verticesduetotherescat-
n tering. Later, this model has been applied to incoherent
a pion production in nuclei [18]. In Ref. [17] the back-
Thestudyofweakcoherentpionproductionincharged
J ground terms were derived using the non-linear σ model
current (CC) and neutral current (NC) processes, is
8 for pions and nucleons. The work of Ref. [17] and some
in the agenda of several current and future experi-
2 recentlattice [19] and quarkmodel [20] calculationssug-
ments[1,2,3,4]. Forinstance,theSciBooNEdetector[2]
] shouldbe able toidentifyπ0’semitted inthe forwarddi- gest smaller values for the axial-vector N-∆ transition
h form factors (FF) than those traditionally found in the
rection, where most of the coherent fraction of the total
t literature1. Therefore, we explore the sensitivity of the
- NCπ0 production is concentrated. MINERνA [3] will
l weak coherent pion production cross sections to differ-
c collect data with high statistics, allowingfor a clear sep-
entchoicesofFF inthe energyrangeofMiniBooNEand
u aration between coherent and incoherent processes, and
n K2K experiments.
the comparison between neutrino and antineutrino cross
[
sections. The measurements will be performed on C, Fe

2 and Pb targets, so that the A dependence can be es-
II. THEORETICAL MODEL
v tablished. The NC reaction ν +A ν +π0 +A is of
2 particular importance as a source o→f background in ν
e
7 We have applied the model of Ref. [17] for weak pion
appearance oscillation experiments [4, 5]. The presently
1 available data on coherent NCπ0 production are scarce. production on the nucleon to the description of the NC
2 coherent reaction ν(k) + A ν(k′) + π0(p ) + A on
They were obtained at E = 2 GeV [6] or higher [7], π
7. h νi (nearly) isospin symmetric nu→clei such as 12C, 27Al and
often with large error bars that complicate the discrimi-
0 56Fe. Besides the dominant direct ∆ excitation [21],
nation between theoretical predictions.
7 this modelincludes the crossed∆termandnucleonpole
0 Many theoretical calculations used PCAC extrapo-
terms both direct and crossed shown in Fig. 1. The ini-
: lated to q2 = 0 [8, 9, 10]. Other studies focused on
v 6
the neutrino-energy region around 1 GeV, where the
i
X excitation of the ∆(1232) resonance gives the domi- Z0 π0
r nant contribution to weak pion production on the nu-
a
cleon. They have stressed the relevance of the mod-
ification of the ∆ spectral function inside the nuclear
medium [11, 12, 13, 14]. Another important ingredient N, ∆
Z0 π0
is the pion distortion. In Refs. [8, 10], it is taken into
account by factorizing the pion-nucleus elastic cross sec-
tion. Inamoregeneralfashion,itcanbeincorporatedin
the amplitude by means of the distorted wave Born ap- N, ∆
proximation, using a pion wave function obtained in the
eikonallimit[13]orbysolvingtheKlein-Gordonequation
FIG. 1: Mechanisms contributing to coherent π0 production
with a realistic optical potential [12, 14].
in isospin symmetric matter.
In this article we extend our model of Ref. [14] by
addinglowenergybackgroundtermsto∆excitation,and
apply it to the NC case. Backgroundterms for the weak
pion production on the nucleon were considered in the
1 Note, however,that adirectcomparisonoftheoretical withem-
past in Refs. [15], and more recently in Refs. [16, 17]. In
pirical FF is not straightforward due to the dressing of the ∆
Ref. [16] itwas shownthe importance of the background vertices asobtainedinRef.[16].
2
tial and final nucleons are in the same quantum state as FF,andthestandarddipoleF =g (1 q2/M2)−2 with
A A − A
requiredby coherence. Other possible contributions (see M = 1 GeV. PCAC has been applied to relate F to
A P
Fig. 2ofRef.[17]forthecompletesetofdiagrams)cancel F . βµ carries the information about the weak N-∆
A
for isospin symmetric nuclei and will not be considered. transiAtion ( ˜βµ(p′,q)=γ [ βµ(p′, q)]†γ )
0 0
A A −
The cross section for this reaction is given by
C˜V C˜V
dσ 1 ~k′ p~ 1 G2 βµ = 3 (gβµ/q qβγµ)+ 4 (gβµq p′ qβp′µ)
dΩνdEνdΩπ = 8| |~k| π|(2π)5 2 |lµJAµ|2, (1) A (mN − m2N · −
| | C˜V C˜A
TwhheerenutchleeanreucutrrirneontcuJrµre,notbltµai=neud¯νa(ks′)tγhµe(1co−heγr5e)nutνs(ku)m. + m52N(gβµq·p−qβpµ))γ5+(m3N(gβµ/q−qβγµ)
A
over all nucleons, for all four mechanisms is C˜A C˜A
+ 4 (gβµq p′ qβp′µ)+C˜Agβµ+ 6 qβqµ ,(8)
Jµ = i f∗ d3reiq~·~rρ(r)F (p )D (p ) m2N · − 5 m2N )
∆d √6mπ Z ∆ d ∆ d where C˜V =(1 2sin2θ )CV, C˜A =CA.
× pˆαπTr u¯(0)ΛαβAβµu(0) φ∗>(p~π,~r), (2) ForthieFFCV−,AweconWsideri twoidiffereintparametriza-
i
i f(cid:2)∗ (cid:3) tions. The first one, adopted in our recent study of CC
J∆µc = √6mπ d3reiq~·~rρ(r)F∆(pc)D∆(pc) coherent pion production [14], assumes the M1+ dom-
Z inance of the N-∆ electromagnetic transition [26], and
× pˆαπTr u¯(0)A˜βµΛβαu(0) φ∗>(p~π,~r), (3) CthAe =Ad0l,erwimthodel [27] for the axial FF, C4A = −C5A/4,
h i 3
g
JNµd =× −pˆαπiT8rfAπ[u¯Z(0)dγ3αrγe5i(q~p·/~rdρ+(rm)FNN)B(pVµdu)D(0N)](φp∗>d)(p~π,~r),(4) C5A =C5A(0)(cid:18)1+ 2 G1e.2V12q−2 q2(cid:19)(cid:18)1− MqA22∆(cid:19)−2. (9)
JNµc = −i8gfA d3reiq~·~rρ(r)FN(pc)DN(pc) WGoeldtabkeergCer5A-T(0r)ei=ma1n.2,reinlataigornee[2m6e],ntanwditMhtAh∆e=off1-d.2ia8gGoneaVl
π Z as extracted from BNL data [28]. CA is related to CA
pˆαTr[u¯(0) µ(p/+m )γ γ u(0)]φ∗(p~ ,~r).(5) 6 5
× π BV d N α 5 > π by PCAC.
Here,g =1.26istheaxialcouplingofthenucleon,f = The second set of FF is taken from Ref. [17]. In the
A π
92.4 MeV, the pion decay constant and f∗ = 2.13, the vector sector, it adopts the parametrizations of Ref. [29]
∆ Nπ decaycoupling;m andm standforthe pion obtainedfromanew analysisofelectron-scatteringdata,
π N
and→nucleonmasses. Thelocalnucleardensityisdenoted which allows to go beyond the M1+ approximation. In
by ρ(r). D (p′)Λ (p′) and D (p′)(p/′+ m ) are the the axialsector,the Adler modelisalsoused, butwith a
∆ αβ N N
∆ and nucleon propagators with the momentum of the different ansatz for C5A [30]
intermediate state p′ being p =p +q =p +p for the
d i f π q2 −1 q2 −2
directdiagrams,andpc =pi pπ =pf qforthecrossed CA =CA(0) 1 1 . (10)
ones;pi andpf aretheinitia−landfinal−nucleonmomenta 5 5 (cid:18) − 3MA2∆(cid:19) (cid:18) − MA2∆(cid:19)
while q is the momentum transfered by the lepton q = Hernandezet al. [17]fitthe freeparametersaboveto the
k k′. The effect of hadron internal structure on the
− ANL data [31] with a πN invariant mass constraint of
∆Nπ and NNπ vertices has been parametrized as [22]
W < 1.4 GeV finding M = 0.985 0.082 GeV and
A∆
F∆(N)(p′)= Λ4+(p′2Λ4m2 )2, (6) aCb5Aly(0s)m=al0le.8r6t7ha±n0t.h0e75o.nTehoibstavianluede forfoCm±5At(h0e)oisff-cdoinasgiodnear-l
− ∆(N) Goldberger-Treimanrelation. Thecoherentpionproduc-
with Λ = 1 GeV. In the evaluation of the spin traces, tion process is more forward-peakedthan the incoherent
we neglect corrections of the order (p~ /m ) which is one, and filters part of the background terms. For these
i(f) N
equivalenttosettingnucleonthree-momentatozero. On reasons, it is specially sensitive to CA(0) and, therefore,
5
theotherside,intheevaluationofD (p′),weassume better suited to constrain its value than the incoherent
∆(N)
that the three-momentum transfered to the nucleus is reaction, provided that the experimental separation of
equally shared by the initial and final nucleons, so that these two contributions is possible.
p~ = (p~ ~q)/2 [23, 24]. The modification of the ∆ properties inside the nuclei
i(f) π
The f±unctio−n µ is given in terms of the vector and [32]causesalargereductionofthecoherentpionproduc-
BV
axial-vector nucleon form factors as follows tion cross section, as shown in Refs. [13, 14]. Therefore,
q we take it into account for the direct ∆ mechanism by
µ =F˜Vγµ+iF˜Vσµν ν F˜Vγµγ F˜Vqµγ , (7)
BV 1 2 2m − A 5− P 5 adding a density dependent ∆ selfenergy and modifying
N
the free width in D (p ), as explained in SectionII B of
∆ d
where F˜1V,2 =(1−2sin2θW)(F1p,2−F1n,2), F˜AV,P =FA,P. Ref. [14]. One would also expect some in-medium cor-
WeusetheBBA-2003parametrization[25]forthevector rections to the amplitudes of the other diagrams but, as
3
shown below, these mechanisms have only a very small not negligible, as can be observed in Fig. 3. The for-
impactontheobservables. Thus,weneglectfurthermod- malism described in the previous section is applicable to
ifications which would have a negligible effect. antineutrinos replacing (1 γ ) by (1+γ ) in the lep-
5 5
−
Finally, the operator pˆα in Eqs. (2-5) acts on φ∗ as tonic current. The contributions of interference terms
π >
with q2 = 0 are apparent in these curves, which would
6
pˆ φ∗(p~ ,~r)=(ω φ∗(p~ ,~r),i~φ∗(p~ ,~r)). (11) be otherwise identical for a given nucleus.
π > π π > π ∇ > π
φ∗(p~ ,~r) is the distorted wave function of the outgoing 1.2
> π
pionwithasymptotic momentump~π, obtainedasthe so- ν (ν-) + A-->ν (ν-) + A + π0 with FF set I
lutionoftheKlein-Gordonequationwithamicroscopical 1.0
12C (ν)
opticalpotentialbasedonthe∆-holemodel. Detailscan
befoundinSectionIIDofRef.[14]andinRefs.[33,34]. 2cm]0.8 1526CFe ( (ν-ν))
-400 0.6 56Fe (ν-)
1
III. RESULTS A [
0.4
σ/
The integrated cross section as a function of the neu- 0.2
trinoenergyisshowninFig.2. Thecomparisonbetween
the solid and the dashed lines, both obtained with the 0.0
choice of FF denoted as set I, indicates that the direct 0.0 0.4 0.8 1.2 1.6 2.0 2.4
∆ excitation is the dominant mechanism. The crossed Eν [GeV]
∆ term accounts for about 0.6% of the cross section at
E =1 GeV. The direct and crossed nucleon-pole terms FIG. 3: (Color online) ν vs. ν¯ cross section for NC coherent
ν
are approximately six times bigger than the crossed ∆, π0 production on 12C and 56Fe as a function of energy.
but there is a large cancellation between them, so that
their sum gives less than 0.2% of the total cross sec- A σ(ν¯) < σ(ν) is compatible with the findings of the
tion. The more realistic description of the vector form Aachen-Pa∼dova collaboration on 27Al [6]. Our results,
averaged over the CERN PS (anti)neutrino fluxes [36],
for both sets I and II are shown in Table I together with
0.14
the experimental ones [6]. The results with set I are be-
0.12 ν + 12C-->ν + 12C + π0 low the central experimental values but within the large
FF set I error bars, while with set II the data are clearly under-
0.10 FF set I only ∆ estimated. In any case, these theoretical values should
]
2
m FF set II be taken with care because these fluxes extend to high
c0.08
8 neutrino energies, where our model is less reliable.
3
-00.06 We alsoshow in Table I our neutrino CC crosssection
1
σ [ on 12C averaged over the K2K flux [37], and the (anti)
0.04 neutrinoNC onesaveragedoverthe correspondingMini-
BooNE fluxes [38]. In the case of K2K, the experimen-
0.02
tal threshold for the muon momentum p > 450 MeV/c
µ
0.00 is taken into account. The result with set I is slightly
0.0 0.4 0.8 1.2 1.6 2.0 2.4 larger than the one given in our recent publication [14]
Eν [GeV] (10 10−40 cm2) due to the inclusion of background
×
mechanisms. This number is above the upper bound
FIG. 2: (Color online) Cross section for coherent π0 produc- obtained by K2K. While possible reasons for that dis-
tion on 12C as a function of theneutrino energy. crepancy were discussed in Ref. [14], here we find that
the smaller CA(0) value obtained in Ref. [17] (set II) is
5
factors of set II has a negligible effect on the result. On compatible with that limit. The situation is not so clear
the contrary, the reduction of CA(0) brings down the forMiniBooNEbecausesetII predictsaverysmallcross
5
cross section by about a factor two with respect to set section although still within the large error bars of the
I. These features can be easily understood from the fact preliminaryexperimentalresult. Intheν¯mode,thevalue
thatinthe forwarddirection(q2 =0),wheremostofthe issmallerthanintheν mode,duetoboththesmalleran-
strength of this reaction is concentrated, the only form tineutrinocrosssectionsthatweobtainandthe different
factor that contributes is CA [35]. Therefore, one can energy distribution of the fluxes.
5
2 In summary,we have calculatedthe NC coherentpion
infer that σ(I)/σ(II) CA (0)/CA (0) 1.9.
∼ 5(I) 5(II) ≈ production cross sections at low energies and compared
On the basis of thishargument one alsoi expects that them with the still scarce data. Our model includes all
interference plays a minor role. It is small indeed, but mechanisms relevant for pion production at these ener-
4
uncertainty of our results comes from the N-∆ FF. In
TABLE I: Cross sections for weak coherent pion production
inunitsof10−40cm2,averagedovertheAachen-Padova(AP) order to estimate it we have presented results for two
representative sets. Actually, we find that the coherent
[36], K2K [37] and MiniBooNE [38] spectra. Columns σ
I(II) π cross section is more sensitive to the N-∆ axial FF
correspond to theform factor sets I and II respectively.
than the incoherent one. This is due to the cancellation
Reaction Experiment σ σ σ Exp. of the isovector terms in isospin symmetric nuclei and
I II
NCν+ 27Al AP 19.9 10.1 29± 10 [6] to the forward peaked kinematics driven by the nuclear
NCν¯+ 27Al AP 19.7 9.8 25± 7 [6] form factor. We obtain smaller NC coherent π produc-
CC ν+ 12C K2K 10.8 5.7 <7.7 [1]a tion cross sections for neutrinos than for antineutrinos.
NCν+ 12C MiniBooNE 5.0 2.6 7.7±1.6±3.6[39]b Interference terms in the production amplitudes are re-
sponsible for this difference.
NCν¯+ 12C MiniBooNE 4.6 2.2 -
aObtained using the ratio between coherent and σCC, the total
CCcrosssection,andthevalueforσCC oftheK2KMCsimulation.
Acknowledgments
bPreliminary.
We thank M. SorelandG.Zellerfor usefuldiscussions
gies. Wefindthattheprocessisdominatedbytheexcita- and valuable communications. This work was partially
tionofthe∆resonancewhilethebackgroundtermsplay supportedbytheMECcontractFIS2006-03438,theGen-
only a minor role. As in the CC case, nuclear effects, eralitat Valenciana ACOMP07/302, and the EU Inte-
mainly the ∆ selfenergy and the pion distortion, pro- grated Infrastructure Initiative Hadron Physics Project
duce a large reduction of the cross section. The largest contract RII3-CT-2004-506078.
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