2p-2h excitations in neutrino scattering: angular distribution and frozen approximation

0.15 MB·English

by  I. Ruiz Simo

2p-2h excitations in neutrino scattering: angular
distribution and frozen approximation
5
1
0
2

n
a I. RuizSimo∗†,a C.Albertus,a J.E.Amaro,a M.B.Barbaro,b J.A.Caballero,c andT.W.
J
Donnellyd
8
2 a DepartamentodeFísicaAtómica,MolecularyNuclear,andInstitutodeFísicaTeóricay

ComputacionalCarlosI,UniversidaddeGranada,Granada18071,Spain
]
h b DipartimentodiFisica,UniversitàdiTorinoandINFN,SezionediTorino,ViaPietroGiuria1,
t
- 10125Torino,Italy
l
c cDepartamentodeFísicaAtómica,MolecularyNuclear,UniversidaddeSevilla,Apartado1065,
u
41080Sevilla,Spain
n
d CenterforTheoreticalPhysics,LaboratoryforNuclearScienceandDepartmentofPhysics,
[

MassachusettsInstituteofTechnology,Cambridge,Massachusetts02139,USA
1
E-mail: [email protected]
v
0
2
We studythephase-spacedependenceof2p-2hexcitationsinneutrinoscatteringusingtherela-
1
7 tivistic Fermi gas model[1]. We follow a similar approachto Refs. [2, 3], but focusing in the
0
phase-spaceproperties,comparingwiththenon-relativisticmodelof[4]. Acarefulmathematical
.
1
analysisoftheangulardistributionfunctionfortheoutgoingnucleonsisperformed.Ourgoalsare
0
5 tooptimizetheCPUtimeofthe7Dintegraltocomputethehadrontensorinneutrinoscattering,
1
andtoconciliatethedifferentrelativisticandnonrelativisticmodelsbydescribinggeneralprop-
:
v
ertiesindependentlyofthetwo-bodycurrent. Forsomeemissionanglestheangulardistribution
i
X becomesinfiniteintheLabsystem, andwederiveamethodtointegrateanalyticallyaroundthe
r
divergence.Ourresultsshowthatthefrozenapproximation,obtainedbyneglectingthemomenta
a
of the two initial nucleons inside the integral of the hadron tensor, reproduces fairly the exact
responsefunctionsforconstantcurrentmatrixelements.
16thInternationalWorkshoponNeutrinoFactoriesandFutureNeutrinoBeamFacilities
25-30August,2014
UniversityofGlasgow,UnitedKingdom
∗Speaker.
†SpecialthankstoL.Alvarez-Rusoforinvitation
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
2p-2hexcitationsinneutrinoscattering I.RuizSimo
1. Introduction
Theanalysis of neutrino oscillation experiments requires having under control all the nuclear
effects, which are inherent to any n -nucleus scattering event. These effects play a major role in
modifyingthefreen -nucleoncrosssection,andagoodunderstandingofthemisclearlymandatory
inordertoreducethesystematic uncertainty ofthemodelsemployedintheexperimental analysis.
There is strong theoretical evidence [5, 6, 7] about a significant contribution from multi-nucleon
knockout to the inclusive Charged Current (CC) cross section around and above the quasielastic
(QE)peakregion.
Thereare,atleast,threedifferentmicroscopicmodels[5,6,7,8]whicharebasedintherelativistic
FermiGas. Buttheydifferfromeachotherinseveralassumptionsanddifferentnuclearingredients
for the interaction. Therefore, it is really difficult to disentangle the origin of any discrepancy in
thefinalresults.
Other models [9, 10] are based on the available phase space just assuming a constant transition
matrixelementandfittingittotheexperimental crosssection.
Finally, ourgoalistoreducethecomputational timeneeded inthiskindofcalculations, oratleast
to establish what assumptions previously made by other authors are really good enough in order
to estimate accurately and fast the contribution of these multi-nucleon processes to the inclusive
channel.
2. 2p-2h phasespaceinthe RelativisticFermiGasmodel
Thehadrontensorforthe2p-2hchannel inafullyrelativistic frameworkisgivenby:
Wmn = V d3p′ d3p′ d3h d3h m4N rmn (p′,p′,h ,h )d (E′+E′−E −E −w )
2p2h (2p )9 Z 1 2 1 2E E E′E′ 1 2 1 2 1 2 1 2
1 2 1 2
Q (p′,p′,h ,h )d 3(p′ +p′ −h −h −q) (2.1)
1 2 1 2 1 2 1 2
where m is the nucleon mass,V is the volume of the system and we have defined the product of
N
stepfunctions,
Q (p′,p′,h ,h )≡q (p′ −k )q (p′ −k )q (k −h )q (k −h ) (2.2)
1 2 1 2 1 F 2 F F 1 F 2
whichencodes thenuclearmodel.
Finally, the function rmn (p′,p′,h ,h ) is the elementary “hadron” tensor for the transition of a
1 2 1 2
nucleon pair with given initial (h ,h ) and final (p′,p′) momenta, summed up over spin and
1 2 1 2
isospin, givenschematically intermsoftheantisymmetrized two-body currentsby:
rmn (p′,p′,h ,h )= 1(cid:229) jm (1′,2′;1,2)∗ jn (1′,2′;1,2) (2.3)
1 2 1 2 4 A A
s ,t
Theabovemultidimensional integral(2.1)canbedoneeithernumericallyoritsdimensionscanbe
further reduced under some approximations [4, 11, 12]. Wedo not know exactly the origin of the
discrepancies betweentheavailable models. Thesecouldbeduetodifferent two-body currents, to
localFermiGasmodelratherthanglobalone,orothereffects. Itisdifficulttomakeanyconcluding
2
2p-2hexcitationsinneutrinoscattering I.RuizSimo
assessmentrightnow,butallthosemodelsshouldagreeatthelevelofphasespacefunctionF(w ,q),
mn
obtained assumingaconstantr :
m4
F(w ,q)≡ d3h d3h d3p′ N Q (p′,p′,h ,h )d (E′+E′−E −E −w ) (2.4)
Z 1 2 1E E E′E′ 1 2 1 2 1 2 1 2
1 2 1 2
where rmn =1and p′ =h +h +q−p′ integrating out thedelta function of momentum conser-
2 1 2 1
vation.
Theremainingdeltafunction enablesanalytical integration overthemodulusofp′:
1
F(w ,q)=2p d3h d3h dq ′ sinq ′ m2N (cid:229) m2Np′12 Q (p′,p′,h ,h ) (2.5)
Z 1 2 1 1E1E2a =±|E2′p′1−E1′p′2·pˆ′1| 1 2 1 2 (cid:12)(cid:12)(cid:12)p′=p′(a )
1 1
(cid:12)
(cid:12)
′(±)
and the sum inside the integral sign runs over the two solutions p of the energy conservation
1
deltafunction (seeappendix ConRef. [1]).
2.1 Frozennucleonapproximation
Thefrozennucleonapproximationisjustaparticularcaseofthemean-valuetheoreminseveral
variables
b
f(x)dx = f(c)(b−a) with c∈[a,b] (2.6)
Za
f(r)dnr = f(c) dnr= f(c)V with c∈V (2.7)
ZV ZV
In our case we are going to skip the two 3D integrations over the holes momenta (h ,h ) by
1 2
fixing them to some, in principle unknown, values inside the Fermi sphere, while keeping the
integration over the emission angle and constraining it by energy conservation. Of course, the
obvious particularization ofthemean-valuetheorem (2.7)is:
4 2
F(w ,q) = d3h d3h d3p′ f(h ,h ,p′)= p k3 d3p′ f(hh i,hh i,p′) (2.8)
Z 1 2 1 1 2 1 (cid:18)3 F(cid:19) Z 1 1 2 1
where(hh i,hh i)arethetwounknownholemomentainsidetheFermisphere.
1 2
Up to now the whole discussion is exact. What makes the difference between an excellent ap-
proximation to the true result or a poor one is just the selection of the “average” hole momenta
(hh i,hh i). Our choice to call it frozen nucleon approximation will be (~0,~0). There are mainly
1 2
twoargumentswhichfavorthischoice:
• Forhighq>>k >h,onecanassumebothinitialnucleons atrest.
F i
• Ifonedoesnotassumetheabovestatement, onehastodetermine theaverage angles forthe
twoholes. Andthiscannotbedonewithoutcomputingthefull7Dintegral(seesection2.2).
Now,wecandefinethefrozenapproximation phase-space functionF(w ,q):
4 2 m2
F(w ,q)= p k3 d3p′ d (E′+E′−w −2m )Q (p′,p′,0,0) N (2.9)
(cid:18)3 F(cid:19) Z 1 1 2 N 1 2 E′E′
1 2
3
2p-2hexcitationsinneutrinoscattering I.RuizSimo
wherenowp′ =q−p′.
2 1
The above phase-space function (2.9) allows us to define the angular distribution in the frozen
approximation:
4 2 p
F(w ,q) = p k3 2p dq ′F (q ′) (2.10)
(cid:18)3 F(cid:19) Z0 1 1
m2
F (q ′) = sinq ′ dp′ p′2d (E′+E′−w −2m )Q (p′,p′,0,0) N (2.11)
1 1Z 1 1 1 2 N 1 2 E′E′
1 2
0.45 5
0.4
0.35 90MeV 4 1800MeV
2GeV] 0.02.53 2GeV] 3
′θΦ()[1 0.01.52 ′θΦ()[1 2
0.1 1
0.05
0 0
0.25 5
0.2 120MeV 4 2000MeV
2GeV] 0.15 2GeV] 3
[ [
′θΦ()1 0.1 ′θΦ()1 2
0.05 1
0 0
0.25 1
0.2 200MeV 0.8 2200MeV
2GeV] 0.15 2GeV] 0.6
[ [
′θΦ()1 0.1 ′θΦ()1 0.4
0.05 0.2
0 0
0 20 40 60 80 100120140160180 0 20 40 60 80 100120140160180
θ1′ [deg] θ1′ [deg]
(a) (b)
Figure 1: Angular distributions F (q ′) for q=500 MeV/c (left panel) and for q=3 GeV/c (right
1
panel) in the frozen nucleon approximation, for three different values of w as a function of the
emissionangleq ′.
1
Theproblemstandsonthedivergence oftheangulardistribution forsomekinematics. Thiscanbe
seen in some of the panels of figure 1 for the frozen nucleon approximation. For other values of
the holes momenta, the position of the pole is different, but it is still present. This makes crucial
todetermine analytically theangular interval where theintegration hastobeperformed. Thiswas
doneexplicitly onsection VIofRef. [1].
Once the problem of the divergence has been correctly addressed, the results for the phase
space function in the frozen approximation can be compared with the full integral in figure 2.
Here we can appreciate the quality of the approximated result for a wide range of momentum
transfers. Themaindiscrepanciesariseonthelowenergy-transferregionineachplot. Butthephase
space function is well-reproduced in the rest of the interval and, furthermore, we have skipped 6
additionalintegrals(overholesmomenta),thusreducingsignificantlythecomputationtime. These
4
2p-2hexcitationsinneutrinoscattering I.RuizSimo
0.014 0.014
0.012 0.012
q=500MeV/c q=1.5GeV/c
8GeV] 00.0.0018 fraoszyremenl 8GeV] 0.00.0081 fraoszyremenl
[ [
ω) 0.006 ω) 0.006
q,( 0.004 q,( 0.004
F F
0.002 0.002
0 0
0 50100150200250300350400450500 0 200 400 600 800 100012001400
0.014 0.014
0.012 0.012
q=700MeV/c q=2000MeV/c
8V] 0.01 8V] 0.01
Ge 0.008 Ge 0.008
[ [
ω) 0.006 ω) 0.006
q,( 0.004 q,( 0.004
F F
0.002 0.002
0 0
0 100 200 300 400 500 600 700 600 800 100012001400160018002000
0.014 0.014
0.012 0.012
q=1000MeV/c q=2.5GeV/c
8V] 0.01 8V] 0.01
Ge 0.008 Ge 0.008
[ [
ω) 0.006 ω) 0.006
q,( 0.004 q,( 0.004
F F
0.002 0.002
0 0
0 1002003004005006007008009001000 10001200140016001800200022002400
ω[MeV]
(a) (b)
Figure 2: Comparison between frozen nucleon approximation and full integral for low and inter-
mediatemomentumtransfers (leftpanel)andforhighmomentumtransfers(rightpanel).
results indicate that this approximation is especially well-suited for Montecarlo event generators,
particularly ifthegoalistoestimatequite accurately thecrosssection formulti-nucleon knockout
intheshortesttimeaspossible.
2.2 Otherinitialconfigurations
We have chosen up to now the frozen nucleon approximation, but other initial (hh i,hh i)
1 2
configurations could have been selected as well. We have considered six configurations depicted
infig. 3a. Theconfigurations withtotalinitial momentum ofthepairequal tozero(U,DorT,−T)
would,inprinciple, yieldaresultveryclosetothatofthefrozenapproximation. Thiscanbeeasily
observed infigure4a.
In the first comparison, Fig. 3b, we show the contribution of two pairs of nucleons with the
samemomentumh =h =200MeV/c,andbothparallel,pointingupward(U)anddownward(D)
1 2
with respect to the z axis, that is, the direction of q. The contribution of the UU configuration is
smaller than average, while the DD is larger. This is so because in the UU case the total momen-
tum p′ inthefinalstateislarge. Bymomentumconservation, themomenta p′ and p′ mustalsobe
1 2
large. Therefore, thesestatesneedalargeexcitation energy, andtheystarttocontribute forhighw
transfer. In the DD configuration, the total momentum p′ is smaller, so the final momenta p′ and
1
5
2p-2hexcitationsinneutrinoscattering I.RuizSimo
0.014
0.012
x x rel q=500MeV/c
8V] 0.01 UU
p~1′ p~1′ ω)[Ge 00..000086 asDymD
DD q,( 0.004
UU F
~q ~q 0.002
z z 0
100 200 300 400 500
0.014
0.012
q=1GeV/c
x x 8V] 0.01
T Ge 0.008
p~1′ p~1′ ω)[ 0.006
D U −T Fq,( 0.004
0.002
~q ~q
z z 0 200 400 600 800 1000
0.014
0.012
q=1.5GeV/c
x x 8V] 0.01
Ge 0.008
T′ T p~1′ −T p~1′ Fq,ω()[ 00..000064
0.002
~q ~q 0
z −T′ 0 300 600 900 1200 1500
ω[MeV]
(a)
(b)
Figure 3: Geometry employed for emission of a pair of nucleons with initial parallel momenta
(UU,DD), anti-parallel (UD,T-T) and perpendicular (TT’,-T-T’) on the left panel. In the right
panel acomparison between average momentum approximation and full integral is shown. Initial
momentaareboth200MeV/c.
p′ can also be small, with small required excitation energy. Therefore, they start to contribute at
2
lowerw .
IntheexampleofFig. 4a,twoanti-parallel configurations areshown. IntheUDcase,onenucleon
is moving upward and the other downward the z axis with total momentum zero of the pair. This
situation is similar to that of a pair of highly correlated nucleons with large relative momentum
[13]. Since the total momentum is zero, the final 2p-2h state has total momentum q, exactly the
same that it would have in the frozen nucleon approximation. Therefore, the contribution of this
configuration issimilartotheaverage. Thesameconclusions canbedrawninthecaseofthecon-
figuration T, −T, with one nucleon moving along the x axis (transverse direction) and the other
along −xwithopposite momentum. Thecontribution ofthispairisexactlythesameasthatofthe
UDconfiguration inthetotalphase-space function.
Finally, we show in Fig. 4b two intermediate cases that are neither parallel nor anti-parallel con-
figurations. Theyconsistoftwopairsoftransversenucleonsmovingalongmutuallyperpendicular
directions. Inthefirstcase,weconsideraT nucleon andasecondT′ nucleonmovingintheyaxis
6
2p-2hexcitationsinneutrinoscattering I.RuizSimo
0.014 0.014
0.012 0.012
rel q=500MeV/c rel q=500MeV/c
8V] 0.01 UD 8V] 0.01 TT’
Ge 0.008 aTsy,-mT Ge 0.008 -Tas,y-Tm’
[ [
ω) 0.006 ω) 0.006
q,( 0.004 q,( 0.004
F F
0.002 0.002
0 0
100 200 300 400 500 100 200 300 400 500
0.014 0.014
0.012 0.012
q=1000MeV/c q=1000MeV/c
8V] 0.01 8V] 0.01
Ge 0.008 Ge 0.008
[ [
ω) 0.006 ω) 0.006
q,( 0.004 q,( 0.004
F F
0.002 0.002
0 0
200 400 600 800 1000 200 400 600 800 1000
0.014 0.014
0.012 0.012
q=1500MeV/c q=1500MeV/c
8V] 0.01 8V] 0.01
Ge 0.008 Ge 0.008
[ [
ω) 0.006 ω) 0.006
q,( 0.004 q,( 0.004
F F
0.002 0.002
0 0
300 600 900 1200 1500 300 600 900 1200 1500
ω[MeV] ω[MeV]
(a) (b)
Figure 4: Comparison between average momentum approximation and full integral. In the left
panel the initial momenta are both 200 MeV/c with total momentum of the pair equal to zero. In
therightonetheinitialmomentaareboth200MeV/cpointing inorthogonal directions.
outofthescattering plane. Thecontribution oftheTT′ pairislarge, whiletheoneoftheopposite
case,−T,−T′,issmall. Ontheaverage, theyareclosetothetotalresult.
3. Conclusions
Wehaveperformedadetailedstudyofthetwo-particle-two-holephase-spacefunction,which
is proportional to the nuclear two-particle emission response function for constant current matrix
elements. Our final goal was to obtain accurate enough results without calculating the 7D inte-
gral. Thefrozennucleonapproximation(1Dintegral),thatis,neglectingthemomentaoftheinitial
nucleons forhighmomentum transfer, seemstobeaquite promising approach toreduce thecom-
putation time without missing significant accuracy. The CPU time of the 7D integral has been
reduced significantly. Wearepresently working onanimplementation ofthepresent method with
acompletemodeloftheMECoperators.
4. Acknowledgements
This work was supported by DGI (Spain), Grants No. FIS2011-24149 and No. FIS2011-
7
2p-2hexcitationsinneutrinoscattering I.RuizSimo
28738-C02-01; by the Junta de Andalucía (Grants No. FQM-225 and No. FQM-160); by the
Spanish Consolider-Ingenio 2010 programmed CPAN, in part (M. B. B.) by the INFN project
MANYBODYandinpart(T.W.D.)byU.S.DepartmentofEnergyunderCooperativeAgreement
No. DE-FC02-94ER40818. C.A.issupported byaCPANpostdoctoral contract.
References
[1] I.RuizSimo,C.Albertus,J.E.Amaro,M.B.Barbaro,J.A.Caballero,T.W.Donnelly,Relativistic
effectsintwo-particleemissionforelectronandneutrinoreactions,Phys.Rev.D90,033012(2014)
[nucl-th/1405.4280].
[2] A.DePace,M.Nardi,W.M.Alberico,T.W.Donnelly,A.Molinari,The2p-2helectromagnetic
responseinthequasielasticpeakandbeyond,Nucl.Phys.A726(2003)303-326
[nucl-th/0304084].
[3] J.E.Amaro,C.Maieron,M.B.Barbaro,J.A.Caballero,T.W.Donnelly,Pioniccorrelationsand
meson-exchangecurrentsintwo-particleemissioninducedbyelectronscattering,Phys.Rev.C82,
044601(2010)[nucl-th/1008.0753].
[4] J.W.VanOrden,T.W.Donnelly,MesonicProcessesinDeepInelasticelectronScatteringfromNuclei,
AnnalsPhys.131(1981)451-493.
[5] M.Martini,M.Ericson,G.Chanfray,J.Marteau,AUnifiedapproachfornucleonknock-out,coherent
andincoherentpionproductioninneutrinointeractionswithnuclei,Phys.Rev.C80(2009)065501
[nucl-th/0910.2622].
[6] J.Nieves,I.RuizSimo,M.J.VicenteVacas,InclusiveCharged–CurrentNeutrino–Nucleus
Reactions,Phys.Rev.C83(2011)045501[hep-ph/1102.2777].
[7] J.E.Amaro,M.B.Barbaro,J.A.Caballero,T.W.Donnelly,C.F.Williamson,Meson-exchange
currentsandquasielasticneutrinocrosssectionsintheSuperScalingApproximationmodel,Phys.
Lett.B696(2011)151[nucl-th/1010.1708].
[8] J.E.Amaro,M.B.Barbaro,J.A.CaballeroandT.W.Donnelly,Meson-exchangecurrentsand
quasielasticantineutrinocrosssectionsintheSuperScalingApproximation,Phys.Rev.Lett.108
(2012)152501[nucl-th/1112.2123].
[9] O.Lalakulich,K.GallmeisterandU.Mosel,Many-BodyInteractionsofNeutrinoswithNuclei-
Observables,Phys.Rev.C86(2012)014614[Erratum-ibid.C90(2014)2,029902]
[nucl-th/1203.2935].
[10] J.T.Sobczyk,MultinucleonejectionmodelforMesonExchangeCurrentneutrinointeractions,Phys.
Rev.C86(2012)015504[hep-ph/1201.3673].
[11] W.M.Alberico,M.EricsonandA.Molinari,TheRoleofTwoParticles-TwoHolesExcitationsinthe
Spin-IsospinNuclearResponse,AnnalsPhys.154(1984)356.
[12] A.Gil,J.NievesandE.Oset,Manybodyapproachtotheinclusive(e,e-prime)reactionfromthe
quasielastictotheDeltaexcitationregion,Nucl.Phys.A627(1997)543[nucl-th/9711009].
[13] I.Korover,N.Muangma,O.Hen,R.Shneor,V.Sulkosky,A.Kelleher,S.Giladand
D.W.Higinbothametal.,ProbingtheRepulsiveCoreoftheNucleon-NucleonInteractionviathe
4He(e,e’pN)Triple-CoincidenceReaction,Phys.Rev.Lett.113(2014)022501
[nucl-ex/1401.6138].
8