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Jun

2005

Nuclear Astrophysics

Charged Lepton Production from Iron Induced by Atmospheric Neutrinos

M. Sajjad Athar, S.Ahmad and S.K.Singh∗

Department of Physics, Aligarh Muslim University, Aligarh-202 002, India.

(Dated: February 9, 2008)

The charged current lepton production induced by neutrinos in 56Fe nuclei has been studied.The calculations have been done for the quasielastic as well as the inelastic reactions assuming ∆dominance and take into account the effect of Pauli blocking, Fermi motion and the renormalizationof weak transition strengths in the nuclear medium. The quasielastic production cross section forlepton production are found to be strongly reduced due to nuclear effects while there is about 10%reduction in the inelastic cross sections in the absence of the final state interactions of the pions.The numerical results for the momentum and angular distributions of the leptons averaged over thevarious atmospheric neutrino spectra at the Soudan and Gransasso sites have been presented. Theeffect of nuclear model dependence and the atmospheric flux dependence on the relative yield of µ

to e has been studied and discussed.

PACS numbers: 25.30Pt neutrino scattering - 13.15.+g Neutrino interactions - 23.40.Bw Weak-interactionand lepton (including neutrino) aspects - 21.60.Jz Hartee-Fock and random-phase approximations

I. INTRODUCTION

The study of neutrino physics with atmospheric neu-trinos has a long history with first observations of muonsproduced by atmospheric muon neutrinos in deep un-derground laboratories of KGF in India and ERPM inSouth Africa[1]. The indications of some deficit in theatmospheric neutrino flux was known to exist from theearly days of these experiments but the evidence wasno more than suggestive due to low statistics of theexperimental data and anticipated uncertainties in theflux calculations[2]. The clear evidence of a deficit inthe atmospheric muon neutrino flux was confirmed laterwhen data with better statistics were obtained at IMB[3],Kamiokande[4] and Soudan[5] experiments. The mostlikely cause of this deficit is believed to be the phenom-ena of neutrino oscillations[6] in which the neutrinos pro-duced with muon flavor after passing a certain distancethrough the atmosphere, manifest themselves as a differ-ent flavor. The implication of this phenomena of neu-trino oscillation is that neutrinos possess a nonzero masspointing towards physics beyond the standard model ofparticle physics. The evidence for neutrino oscillationsand a nonzero mass for the neutrinos has also been ob-tained in the observations made with solar[7] and reactor(anti)neutrinos[8].

It is well known that, in a two flavor oscillation scenarioinvolving muon neutrino, the probability for a muon neu-trino with energy Eν to remain a muon neutrino afterpropagating a distance L before reaching the detector isgiven by[6].

Pµµ = 1 − sin22θsin2

(

1.27∆m2(eV 2)L(km)

Eν(GeV )

)

(1)

where ∆m2 = m21 −m2

2 is the difference of the squared

∗Electronic address: pht13sks@rediffmail.com

masses of the two flavor mass eigenstates and θ is themixing angle between two states. The oscillation param-eters ∆m2 and the mixing angle θ are determined byvarious observations made in atmospheric neutrino ex-periments. These include the flavor ratios of muon andelectron flavors, angular and L

E distributions of muonsand electrons produced by atmospheric neutrinos. Thefirst claims of seeing neutrino oscillation in atmosphericneutrinos came from the measurements of ratio of ra-tios Rν defined as (µ/e)data

(µ/e)MCfrom the observations of

fully contained (FC) events[3]-[4], but there are nowdata available from the angular and L

E distribution ofthe atmospheric neutrino induced muon and electronevents from SK[9], MACRO[10] and Soudan[11] exper-iments which confirm the phenomena of neutrino oscil-lations. These experiments are consistent with a valueof ∆m2 ≈ 3.2 × 10−3eV 2 and sin22θ ≈ 1. The analysisof these data assuming a three flavor neutrino oscillationphenomenology have also been done by many authors[12].

The major sources of uncertainty in the theoretical pre-diction of the charged leptons of muon and electron flavorproduced by the atmospheric neutrinos come from theuncertainties in the calculation of atmospheric neutrinofluxes and neutrino nuclear cross sections. The atmo-spheric neutrino fluxes at various experimental sites ofKamioka, Soudan and Gransasso have been extensivelydiscussed in literature by many authors[13]-[16]. Theneutrino nuclear cross sections have also been calculatedfor various nuclei by many authors using different nuclearmodels[17]-[24]. The aim of the present paper is to studythe neutrino nuclear cross section in iron nuclei whichare relevant for the atmospheric neutrino experimentsperformed at Soudan[11], FREJUS[25] and NUSEX[26]and planned in future with MINOS[27], MONOLITH[28]and INO[29] detectors. The uncertainty in the nuclearproduction cross section of leptons from iron nuclei bythe atmospheric neutrinos are discussed. For our nu-clear model, we also discuss the uncertainty due to useof different neutrino fluxes for the sites of Soudan and

2

Gransasso which are relevant to MINOS, MONOLITHand INO detectors[27]-[29].

The momentum and angular distribution of muonsand electrons relevant to fully contained events producedby atmospheric neutrinos in iron nuclei are calculated.These leptons of muon and electron flavor characterizedby track and shower events include the leptons producedby quasielastic process as well as the inelastic processesinduced by charged current interactions. The calcula-tions are done in a model which takes into account nu-clear effects like Pauli Blocking, Fermi motion effectsand the effect of renormalization of the weak transi-tion strengths in nuclear medium in local density ap-proximation. The model has been successfully appliedto describe various electromagnetic and weak processeslike photon absorption, electron scattering, muon captureand low energy neutrino reactions in nuclei[24],[30]-[32].The model can be easily applied to calculate the zenithangle dependence and the L

E distribution for stoppingand thorough going muon production from iron nucleiwhich is currently under progress.

The plan of the paper is as follows. In section-II wedescribe the neutrino(antineutrino) quasielastic inclusiveproduction of leptons (e−, µ−, e+, µ+) from iron nucleifor various neutrino energies. In section-III we describethe energy dependence of the inelastic production of lep-tons through ∆-dominance model and highlight the nu-clear effects relevant to the energy of fully containedevents. In section-IV, we use the atmospheric neutrinoflux at Soudan and Gransasso sites as determined by vari-ous authors and discuss the flux averaged momentum andangular dependence of leptons corresponding to differentflux calculations available at these two sites.

II. QUASIELASTIC PRODUCTION OF

LEPTONS

Quasielastic inclusive production of leptons in nucleiinduced by neutrinos has been studied by many au-thors where nuclear effects have been calculated. Mostof these calculations have been done either for 16Orelevant to IMB and Kamioka experiments[3]-[4]or for12C relevant to LSND and KARMEN experiment[33].These calculations generally use direct summationmethod (over many nuclear excited states)[17], clo-sure approximation[18], Fermi gas model[19]-[20], rel-ativistic mean field approximation[21], continuum ran-dom phase approximation (CRPA)[22] and local densityapproximation[23, 24]. The calculations for 56Fe nucleushave been reported by Bugaev et al.[17] in a shell modeland in Fermi gas model by Gallagher[34] and Berger etal.[35]. In this section we briefly describe the formal-ism and results of our calculations done for quasielasticinclusive production of leptons for iron nuclei.

A. Formalism

In local density approximation the neutrino nucleuscross section σ(Eν) for a neutrino of energy Eν scatteringfrom a nucleus A(Z,N), is given by

σA(Eν) = 2

∫

d~rd~p

(2π)3nn(~p,~r)σN (Eν) (2)

where nn(~p,~r) is the local occupation number of the ini-tial nucleon of momentum ~p (localized at position ~r inthe nucleus) and σN (Eν) is cross section for the scatter-ing of neutrino of energy Eν from a free nucleon given bythe expression

σN (Eν) =

∫

d3k′

(2π)3mν

Eν

me

Ee

Mn

En

Mp

Ep

× ¯∑∑

|T |2δ(Eν − El + En − Ep) (3)

where T is the matrix element for the basic process

νl(k) + n(p) → l−(k′) + p(p′), l = e, µ (4)

written as

T =GF√

2cos θc u(k

′)γµ(1 − γ5)u(k) (Jµ)CC

(5)

(Jµ)CC

is the charged current(CC) matrix element of thehadronic current defined as

(Jµ)CC = u(p′)[

FV1 (q2)γµ + FV

2 (q2)iσµν qν2M

(6)

+FVA (q2)γµγ5

]

u(p)

q2(q = k−k′) is the four momentum transfer square. Theform factors FV

1 (q2), FV2 (q2) and FV

A (q2) are isovectorelectroweak form factors written as

FV1 (q2) = F p

1 (q2) − Fn1 (q2), FV

2 (q2) = F p2 (q2) − Fn

2 (q2),

FVA (q2) = FA(q2)

where

F p,n1 (q2) =

1

(1 − q2

4M2 )

[

Gp,nE (q2) − q2

4M2Gp,n

M (q2)

]

F p,n2 (q2) =

1

(1 − q2

4M2 )[Gp,n

M (q2) −Gp,nE (q2)]

GpE(q2) =

(

1 − q2

M2v

)−2

(7)

GpM (q2) = (1 + µp)G

pE(q2), Gn

M (q2) = µnGpE(q2);

GnE(q2) = (

q2

4M2)µnG

pE(q2)ξn; ξn =

1

1 − λnq2

4M2

3

µp = 1.79, µn = −1.91,Mv = 0.84GeV, and λn = 5.6.

The isovector axial vector form factor FA(Q2) is given by

FA(Q2) =FA(0)

(1 − q2

M2A

)2

where MA = 1.032GeV ; FA(0)=-1.261In a nuclear process the neutrons and protons are not

free and their momenta are constrained by the Pauliprinciple which is implemented in this model by requir-ing that for neutrino reactions initial nucleon momentump ≤ pFn

and final nucleon momentum p′ = (|~p+~q|) > pFp

where pFn,p= [32π

2ρn,p(r)]13 , are the local Fermi mo-

menta of neutrons and protons at the interaction pointin the nucleus defined in terms of their respective nu-clear densities ρn,p(r). These constraints are incorpo-rated while performing the integration over the initial nu-cleon momentum in Eqn.(2) by replacing the energy con-serving δ-function in Eqn.(3) by − 1

π ImUN (q0, ~q) whereUN (q0, ~q) is the Lindhard function corresponding to theparticle hole(ph) excitations induced by the weak inter-action process through W exchange shown in Fig.1(a).In the large mass limit of W boson i.e.MW → ∞, thisFig.1(a) is reduced to Fig.1(b) for which the imaginarypart of the Lindhard function i.e. ImUN (q0, ~q) is givenby

ν

ν

n p

M

k

k’

k

pn

q

(a) (b)

W

W

e−

W

ν

νe

e e

e

e−

FIG. 1: Diagrammatic representation of the neutrino self-energy diagram corresponding to the ph-excitation leading toνe + n → e− + p process in nuclei. In the large mass limit ofthe W boson (i.e.MW → ∞) the diagram 1(a) is reduced to1(b) which is used to calculate |T |2 in Eqn.(5).

ImUN (q0, ~q) = − 1

2π

MpMn

|~q| [EF1−A] with (8)

q2 < 0, EF2− q0 < EF1

and−q0 + |~q|

√

1 − 4M2

q2

2< EF1

where EF1and EF2

are the local Fermi energy of initialand final nucleons and

A = Max

Mn, EF2− q0,

−q0 + |~q|√

1 − 4M2

q2

2

The expression for the neutrino nuclear cross sectionσA(Eν), is then given by:

σA(Eν) = − 4

π

∫ rmax

rmin

r2dr

∫ plmax

plmin

pl2dpl

∫ 1

−1

d(cosθ)

× 1

EνEl

¯∑∑

|T |2ImUN [Eν − El, ~q]. (9)

Moreover in the nucleus, the Q value of the nuclear reac-tion and the Coulomb distortion of the final lepton in theelectromagnetic field of the final nucleus should be takeninto account. This is done by modifying the energy con-serving δ-function δ(Eν − El + En − Ep) in Eqn.(3) toδ(Eν − Q − (El + Vc(r)) + En − Ep) where Vc(r) is theCoulomb energy of the produced lepton in the field offinal nucleus and is given by

Vc(r) = ZZ ′α4π(1

r

∫ r

0

ρp(r′)

Zr′

2dr′ +

∫ ∞

r

ρp(r′)

Zr′dr′)

(10)This amounts to evaluation of Lindhard function in

Eqn.(8) at (q0 − (Q+ Vc(r)), ~q) instead of (q0, ~q). Theimplementation of this modification requires a judiciouschoice of Q value for inclusive nuclear reactions in whichmany nuclear states are excited in iron. We have takenQ-value of 6.8MeV corresponding to the transition to lowestlying 1+ state in 56Co for νl+

56Fe→ l−+56Co⋆ reactionand Q-value of 4.3MeV corresponding to the transition tolowest lying 1+ state in 56Mn for νl+

56Fe→ l++56Mn⋆

reaction.The inclusion of Vc(r) to modify energy and cor-

responding momentum of the charged lepton in theCoulomb field of final nucleus in our model is equivalentto the treatment of Coulomb distortion effect in mod-ified effective momentum approximation(MEMA). Thisapproximation has been used in other calculations ofcharged current neutrino reactions[36] and electron scat-tering at higher energies[37].

With these modifications, the final expression for thequasielastic inclusive production from iron nucleus isgiven by

σA(Eν) = − 4

π

∫ rmax

rmin

r2dr

∫ plmax

plmin

pl2dpl

∫ 1

−1

d(cosθ)

× 1

EνEl

¯∑∑

|T |2ImUN [q0 − (Q+ Vc(r)), ~q].(11)

It is well known that weak transition strengths are mod-ified in the nuclear medium due to presence of stronglyinteracting nucleons. This modification of the weak tran-sitions strength in the nuclear medium is taken into ac-count by considering the propagation of particle-hole(ph)excitations in the medium. While propagating throughthe medium, the ph-excitations interact through the nu-cleon nucleon potential and create other particle hole and∆-h excitations as shown in Fig.2. The effect of theseexcitations are calculated in random phase approxima-tion which is described in Ref.[23, 32]. The effect of nu-clear medium on the renormalization of weak strengths is

4

pn

ν

ν

n p pn

n

ν

ν

π π, ,ρ ρ

∆

(a) (b)e

e e

e

e

e+...... +......

FIG. 2: Many body Feynman diagrams (drawn in the limitMW → ∞) accounting for the medium polarization effectscontributing to the process νe + n → e− + p transitions.

treated in a non-relativistic frame work. In leading order,the non-relativistic reduction of the weak hadronic cur-rent defined in Eqn.6, F2(q

2) term gives a spin-isospin

transition operator ~σ×~q2M ~τ which is a transverse opera-

tor while the FA(q2) term gives a spin-isospin transi-tion operator ~σ · ~τ which has a longitudinal as well asa transverse part. This representation of the transi-tion operators in the longitudinal and transverse partsis useful when summing the diagrams in Fig.(2) in ran-dom phase approximation(RPA) to calculate |T |2. Whilethe charge coupling remains unchanged due to nuclearmedium effects, the terms proportional to F 2

2 are affectedby the transverse part of the nucleon nucleon potentialwhile the terms proportional to F 2

A are affected by trans-verse as well as longitudinal parts. The effect is to re-place the terms like F 2

2 , F 2A, F2FA etc. in the following

manner[23, 32]

(F 22 , F2FA) → (F 2

2 , F2FA)1

|1 − UNVt|2

F 2A →

[

1

3

1

|1 − UNVl|2+

2

3

1

|1 − UNVt|2

]

(12)

where Vl and Vt are the longitudinal and transverse partsof the nucleon nucleon potential calculated with π and ρexchanges and modulated by Landau-Migdal parameterg′ to take into account the short range correlation effectsand are given by

Vl(q) =f2

m2π

[

q2

−q2 +m2π

(

Λ2π −m2

π

Λ2π − q2

)2

+ g′

]

,

Vt(q) =f2

m2π

q2

−q2 +m2ρ

Cρ

(

Λρ2 −m2

ρ

Λρ2 − q2

)2

+ g′

(13)

with Λπ = 1.3GeV , Cρ = 2.0, Λρ = 2.5GeV , mπ andmρ are the pion and rho meson masses and g′ is takento be 0.7[31]. The effect of ∆h excitations are takeninto account by including the Lindhard function U∆ for

the ∆h excitations and replacing UN by UN + U∆ inEqn.(12). The complete expression for UN and U∆ usedin our calculations are taken from [38]. The differentcouplings for N and ∆ to the nucleon are incorporatedin UN and U∆ and then the same interaction strengthsVl and Vt are used for ph and ∆h excitations[39]-[40].

B. Results

We present the numerical results for the total crosssection for the quasielastic processes νl(νl) +56 Fe →l−(l+) +56 Co⋆(56Mn⋆) as a function of energy for neu-trino and anti neutrino reactions on iron in the energyregion relevant to the fully contained events of atmo-spheric neutrinos i.e. Eν < 3GeV . The cross sectionshave been calculated using Eqn.(11) with the nucleardensity ρ(r) given by a two parameter Fermi density[41]:

ρ(r) = ρ(0)

1.+exp( r−cz

)with c = 3.971fm, z = 0.5935fm,

ρn(r) = (A−Z)A ρ(r) and ρp(r) = Z

Aρ(r).In Fig.3 we show the numerical results of σ(E) vs E,

for all flavors of neutrinos i.e. νµ, νµ, νe and νe. Thereduction due to nuclear effects is large at lower ener-gies but becomes small at higher energies. The energydependence of the cross sections for muon and electrontype neutrinos are similar except for the threshold effectswhich are seen only at low energies (Eν < 500MeV ).This reduction in σ is due to Pauli blocking as well as dueto the weak renormalization of transition strengths whichhave been separately shown in Fig.4(a) and Fig.4(b)for neutrinos and antineutrinos, where we also showthe results in the Fermi gas model given by LlewellynSmith[20]. We plot in Fig.4(a) and Fig.4(b) for neutrinos

and antineutrinos, the reduction factor R = σnuclear(E)σnucleon(E)

vs E where σnuclear(E) is the cross section per neu-tron(proton) for neutrino(antineutrino) reactions in thenuclear medium. The solid lines show the reduction fac-tor R when only the Pauli suppression is taken into ac-count through the imaginary part of the Lindhard func-tion given in Eqn.8. This is similar to the results ofLlewellyn Smith[20] in Fermi gas model shown by dashedlines. In this model the total cross section σ is calculatedby using the formula σ =

∫

dq2 R(q2)( dσdq2 )free where

R(q2) describes the reduction in the cross section calcu-lated in the Fermi gas model and includes the effect ofPauli suppression only[20]. However, in our model we getfurther reduction due to renormalization of weak transi-tion strengths in the nuclear medium when the effectsof Fig.2(a) and Fig.2(b) are included. These are shownby dotted lines in Figs.4(a) and 4(b). We find that thereduction at higher energies (E > 1GeV ) is around 20%for neutrinos and 40% for antineutrinos. It is worth not-ing that the energy dependence of the reduction due tonuclear medium effects is different for neutrinos and an-tineutrinos. This is due to the different renormalizationof various terms like F 2

A, F2FA and F 22 in |T |2 which

enter in different combinations for neutrino and antineu-

5

trino reactions. The results for νe and νe cross sectionare respectively similar to νµ and νµ reactions except forthe threshold effects and are not shown here.

In Fig.5 and Fig.6 we compare our results for σ(E) withthe results of some earlier experiments which containnuclear targets like Carbon[42], Freon[43]-[44], Freon-Propane[45] and Aluminum[46], where the experimentalresults for the deuteron targets[47] are not included asthey are not subject to the various nuclear effects dis-cussed here. It should be kept in mind that the nu-clear targets considered here (except for Br in Freon)are lighter than Fe. Therefore, the reduction in the totalcross section due to nuclear effects will be slightly over-estimated. For example, for energies Eν ≥ 1GeV the re-duction in neutrino(antineutrino) cross section in case of56Fe is 5% more than the reduction in case of 12C[48]. Incomparision to the neutrino(antineutrino) nuclear crosssections as obtained in the Fermi gas model of LlewellynSmith[20] (shown by dashed lines in Fig.5 and Fig.6) weget a smaller result for these cross sections. This reduc-tion in the total cross section leads to an improved agree-ment with the experimental results as compared to theFermi gas model results specially for antineutrino reac-tions(Fig.6). It should be emphasized that the Fermi gasmodel has no specific mechanism to estimate the renor-malization of weak transition strengths in nuclei while inour model this is incorporated by taking into account theRPA correlations.

In Fig.7(a) and Fig.7(b), we show the nuclear mediumeffects on the momentum and angular distributions i.e.dσdpl

and dσdcosθl

of leptons produced in νµ and νµ reactions.

We find a large suppression in the results specially inthe peak region of momentum and angular distributions.Quantitatively similar results are obtained for the case ofνe and νe reactions and are not shown here.

We have also studied the effect of Coulomb distortionin the momentum distribution of leptons but find no sub-stantial effect around Eν = 1.0GeV . These are found toaffect the results only at low energies i.e. Eν < 500MeVwhere the peak is slightly shifted to lower momentum asshown in Figs.8(a) and 8(b).

III. INELASTIC PRODUCTION OF LEPTONS

The inelastic production process of leptons is the pro-cess in which the production of leptons is accompanied byone pion (or more pions). There are many calculations ofone pion production by neutrinos from free nucleons[49]but there are only a few calculations which discuss thenuclear effects in these processes[50]-[52]. In this sec-tion we follow the method of Ref.[51] to estimate thenuclear effects and nuclear model dependence of inelasticproduction cross section of leptons induced by neutrinosfrom iron nuclei. The calculations are done assuming∆-dominance of one pion production because the contri-bution of higher resonances in the energy region of at-mospheric neutrinos leading to fully contained events is

0 200 400 600 800 1000

0.1

1

10

Eν (MeV)

σ(

10

cm

)

-38

2

FIG. 3: Total quasielastic cross sections in the present modelfor the neutrino and antineutrino reactions in 56Fe are shownby solid line for νe, dashed line for νe, dotted line for νµ anddashed-dotted line for νµ reactions.

expected to be small.

A. Formalism

The matrix element for neutrino production reactionof ∆ on proton targets leading to one pion events i.e.

νl(k) + p(p) → l−(k′) + ∆++(p′) (14)

is given by Eqn.5 where Jµ now defines the matrix ele-ment of the transition hadronic current between N and∆ states. The most general form of Jµ

CC is written as[51]:

Jµcc = ψα(p′)

[(

CV3 (q2)

M(gαµ 6 q − qαγµ)

+CV

4 (q2)

M2(gαµq · p′ − qαp′µ)

+CV

5 (q2)

M2(gαµq · p− qαpµ) +

CV6 (q2)

M2qαqµ

)

γ5

+

(

CA3 (q2)

M(gαµ 6 q − qαγµ) +

CA4 (q2)

M2(gαµq · p′ − qαp′µ)

+ CA5 (q2)gαµ +

CA6 (q2)

M2qαqµ

)]

u(p)

(15)

where ψα(p′) and u(p) are the Rarita Schwinger andDirac spinors for ∆ and nucleon of momenta p′ and p re-spectively, q(= p′−p = k−k′) is the momentum transferand CV

i (i=3-6) are vector and CAi (i=3-6) are axial vector

transition form factors. The vector form factors CVi (i=3-

6) are determined by using the conserved vector cur-rent(CVC) hypothesis which gives CV

6 (q2)=0 and relatesCV

i (i=3,4,5) to the electromagnetic form factors whichare determined from photoproduction and electroproduc-tion of ∆’s. Using the analysis of these experiments[52]-

6

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

0 500 1000 1500 20000

0.2

0.4

0.6

0.8

1

R

E (MeV)ν

R

νE (MeV)

(a) (b)

FIG. 4: Ratio of the total cross section to the free neutrino nucleon cross section for the reactions (a) νµ + n → µ− + p (b)νµ + p → µ+ + n in iron nuclei in the present model with Pauli suppression (solid line) and with nuclear effects (dotted line)and in the Fermi gas model (dashed line)[20].

0 1 2 3 40

0.2

0.4

0.6

0.8

1

1.2

1.4

σ(1

0

cm

)-3

82

E (GeV)ν

FIG. 5: Neutrino quasielastic total cross section per nucleon in iron for νµ + n → p + µ− reaction. The data are fromLSND[42](Ellipse), Bonnetti et al.[43](squares), SKAT collab.[44](Triangle Down), Pohl et al.[45](Circle) and Belikov etal.[46](Triangle Up). The dashed line is the result of the cross section in the Fermi gas model[20] and solid line is the re-sult using the present model with nuclear effects.

0 1 2 3 40

0.2

0.4

0.6

0.8

σ(1

0

cm

)-3

82

E (GeV)ν

FIG. 6: Anti-neutrino quasielastic total cross section per nucleon in iron for νµ + p → n + µ+ reaction. The data are fromBonnetti et al.[43], SKAT collabn.[44](Triangle Down), Pohl et al.[45](Circle), Belikov et al.[46](Triangle Up). The dashed lineis the result of the cross section in the Fermi gas model[20] and solid line is the result using the present model with nucleareffects.

7

200 400 600 800 10000

2

4

6

8

10

-1 -0.5 0 0.5 10

20

40

60

80

dσ

p/

ld

dσ

d

pl

/

(10

cm

/M

eV)

(10

cm

)

-40

2

2-3

8

(MeV)

lco

s

θcos

θ

l (rad)

(a) (b)

FIG. 7: The results for the differential cross section Fig.(a) dσdpl

vs pl and Fig.(b) dσdcosθl

vs cosθl at E = 1.0GeV in the present

model with Pauli suppression (solid line for νµ and dashed-dotted line for νµ) and with nuclear effects (dashed line for νµ anddotted line for νµ).

100 200 300 400 5000

2

4

6

8

100 200 300 400 5000

2

4

6

8

dp

σ/

ld d

σd

p/

l

(10

cm

/M

eV)

(10

cm

/M

eV)

-40

2 2-4

0

(MeV)pl

pl

(MeV)

(a) (b)

FIG. 8: Differential scattering cross section dσdpl

vs pl for Eν = 500MeV , with Coulomb effect (solid line for νl and dashed-dotted

line for νl) and without Coulomb effect (dashed line for νl and dotted line for νl) Coulomb effect. Fig.(a) is for electron typeand Fig.(b) for muon type neutrinos.

[53] we take for the vector form factors

CV5 = 0, CV

4 = − M

M∆CV

3 , and

CV3 (q2) =

2.05

(1 − q2

M2V

)2, M2

V = 0.54GeV 2 (16)

The axial vector form factor CA6 (q2) is related to C5

A(q2)using PCAC and is given by

CA6 (q2) = C5

A(q2)M2

mπ2 − q2

(17)

The remaining axial vector form factor CAi=3,4,5(q

2)are taken from the experimental analysis of the neu-trino experiments producing ∆’s in proton and deuterontargets[54]-[55]. These form factors are not uniquely de-termined but the following parameterizations give a sat-

isfactory fit to the data.

CAi=3,4,5(q

2) = CAi (0)

[

1 +aiq

2

bi − q2

](

1 − q2

MA2

)−2

(18)with CA

3 (0) = 0, CA4 (0) = −0.3, CA

5 (0) = 1.2, a4 = a5 =−1.21, b4 = b5 = 2GeV 2, MA = 1.28GeV . Using thehadronic current given in Eqn.(15), the energy spectrumof the outgoing leptons is given by

d2σ

dEk′dΩk′

=1

8π3

1

MM ′

k′

Eν

Γ(W )2

(W −M ′)2 + Γ2(W )4.

LµνJµν

(19)

where W =√

(p+ q)2 and M ′ is mass of ∆,

Lµν = kµk′ν + k′µkν − gµνk · k′ + iǫµναβk

αk′β ,

Jµν = ΣΣJµ†Jν

8

and is calculated with the use of spin 32 projection oper-

ator Pµν defined as

Pµν =∑

spins

ψµψν

and given by:

Pµν = − 6 p′ +M ′

2M ′

(

gµν − 2

3

p′µp′ν

M ′2

+1

3

p′µγν − p′νγµ

M ′− 1

3γµγν

)

(20)

In Eqn.(19), the decay width Γ is taken to be an energydependent P-wave decay width given by

Γ(W ) =1

6π

(

fπN∆

mπ

)2M

W|qcm|3Θ(W −M −mπ) (21)

where

|qcm| =

√

(W 2 −m2π −M2)2 − 4m2

πM2

2W

and M is the mass of nucleon. The step function Θ de-notes the fact that the width is zero for the invariantmasses below the Nπ threshold. |qcm| is the pion mo-mentum in the rest frame of the resonance. When the re-action(14) takes place in the nucleus, the neutrino inter-acts with the nucleon moving inside the nucleus of densityρ(r) with its corresponding momentum ~p constrained tobe below its Fermi momentum. The produced ∆’s haveno such constraints on their momentum. These ∆′s de-cay through various decay channels in the medium. Themost prominent decay mode is ∆ → Nπ which producespions. This decay mode in the nuclear medium is slightlyinhibited due to Pauli blocking of the final nucleon mo-mentum modifying the decay width Γ used in Eqn.(21).This modification of Γ due to Pauli blocking of nucleushas been studied in detail in electromagnetic and stronginteractions[56]. The modified ∆ decay width i.e. Γ iswritten as[56]:

Γ =1

6π

(

fπN∆

mπ

)2M |qcm|3

WF (kF , E∆, k∆)Θ(W−M−mπ)

(22)where F (kF , E∆, k∆) is the Pauli correction factor givenby

F (kF , E∆, k∆) =k∆|qcm| + E∆E

′pcm

− EFW

2k∆|q′cm| (23)

where kF is the Fermi momentum, EF =√

M2 + k2F , k∆

is the ∆ momentum and E∆ =√

W + k2∆.

Moreover, in the nuclear medium there are addi-tional decay channels now open due to two body andthree body absorption processes like ∆N → NN and∆NN → NNN through which ∆′s disappear in the nu-clear medium without producing a pion while a two body

∆ absorption process like ∆N → πNN gives rise to somemore pions. These nuclear medium effects on ∆ propa-gation are included by using a ∆ propagator in whichthe ∆ propagator is written in terms of ∆ self energyΣ∆. This is done by using a modified mass and widthof ∆ in nuclear medium i.e. M∆ → M∆ + ReΣ∆ andΓ → Γ − ImΣ∆. There are many calculations of ∆ selfenergy Σ∆ in the nuclear medium [56]-[59] and we use theresults of [56], where the density dependence of real andimaginary parts of Σ∆ are parametrized in the followingform:

ReΣ∆ = 40ρ

ρ0MeV and

−ImΣ∆ = CQ

(

ρ

ρ0

)α

+CA2

(

ρ

ρ0

)β

+CA3

(

ρ

ρ0

)γ

(24)

In Eqn.24, the term with CQ accounts for the ∆N →πNN process, the term with CA2 for two-body absorp-tion process ∆N → NN and the term with CA3 forthree-body absorption process ∆NN → NNN . Thecoefficients CQ, CA2, CA3, α, β and γ (γ = 2β) areparametrized in the range 80 < Tπ < 320MeV (whereTπ is the pion kinetic energy) as [56]

Ci(Tπ) = ax2 + bx+ c, for x =Tπ

mπ(25)

The values of coefficients a, b and c are given in Table-I.taken from ref.[56].

TABLE I: Coefficients of Eqn.(25) for an analytical interpo-lation of ImΣ∆

CQ(MeV) CA2(MeV) CA3(MeV) α β

a -5.19 1.06 -13.46 0.382 -0.038b 15.35 -6.64 46.17 -1.322 0.204c 2.06 22.66 -20.34 1.466 0.613

With these modifications, which incorporate the vari-ous nuclear medium effects on ∆ propagation, the crosssection is now written as

σ =

∫ ∫

dr

8π3

dk′

EνEl

1

MM ′

×Γ2 − ImΣ∆

(W −M ′ −ReΣ∆)2 + ( Γ2. − ImΣ∆)2

×[

ρp(r) +1

3ρn(r)

]

LµνJµν (26)

The factor 13 in front of ρn comes due to suppression of

charged pion production from neutron targets i.e. νl +n→ l−+∆+ → l−+n+π+, as compared to the chargedpion production from the proton target i.e.νl + p→ l− +∆++ → l−+p+π+, in the nucleus. In case of antineutrinoreactions ρp + 1

3ρn is replaced by ρn + 13ρp.

9

B. Results

In this section we present results of inelastic leptonproduction cross section due to ∆h excitations in ironinduced by the charged current neutrino interactions us-ing Eqn.(26). In Fig.9(a) and Fig.9(b), we show theresults σ(Eν) ∼ Eν for νe(νe) and νµ(νµ) for the in-elastic production of lepton accompanied with ∆ andcompare this with the cross section for the quasielas-tic production of leptons discussed in section-II. We seethat for Eν ≈ 1.4GeV the lepton production cross sec-tion through quasielastic and inelastic production pro-cesses are comparable. For energies Eν ≤ 1.4GeV wherethe atmospheric neutrino energies are important for fullycontained events the major contribution comes from thequasielastic events.

The effect of nuclear effects on the ∆ production areshown in Fig.10 for νµ and νµ reactions. The results forthe νe and νe are similar to Fig.10 and are not shownhere. We see that the nuclear medium effects reduce the∆ production cross section by 5-10%.

In Fig.11(a) and Fig.11(b), we show the momentumdistribution dσ/dp and angular distribution dσ/dcosθ formuon type neutrinos where we also show the effects ofnuclear effects. The nuclear medium effects reduce thecross sections mainly in the peak region of the momen-tum and angular distributions for muons. The results forthe momentum distribution and angular distribution forelectrons is similar to the muon distributions. We showour final results on momentum and angular distributionsfor all charged leptons i.e. e−, e+, µ− and µ+ produced inneutrino and antineutrino reactions with νe, νµ, νe andνµ for Eν = 1GeV in Fig.12(a) and Fig.12(b) in 56Fenuclei.

Here we will like to make some comments about thelepton events produced through the ∆ excitation. The∆ excitation process gives rise to leptons accompanied byone pion events produced by ∆ → Nπ and ∆N → πNNprocesses in the nuclear medium. The pions producedthrough these processes will undergo secondary nuclearinteractions like multiple scattering and possible absorp-tion in iron nuclei while passing through the nucleus andan appropriate model has to be used for their description.Models developed by Salcedo et al.[40] and Paschos etal.[52] have studied these effects but we do not considerthese effects here as they are beyond the scope of thispaper. The ∆ excitation process in the nuclear mediumalso gives rise to quasielastic like events through two bodyand three body absorption processes like ∆N → NN and∆N → ∆NN where only leptons are present in the finalstates. The quasielastic-like lepton events are discussedby Kim et al.[21] but no quantitative estimates have beenmade. We have in an earlier paper[51] discussed thisprocess only qualitatively but make a quantitative esti-mate of these events in this paper. We find that aroundEν = 1GeV the contribution of these quasielastic likeevents is about 10 − 12% but its effect on the flux aver-aged production of leptons for atmospheric neutrinos is

not very significant. This is discussed in some detail inthe next section.

IV. LEPTON PRODUCTION BY

ATMOSPHERIC NEUTRINOS

The energy dependences of the quasielastic and inelas-tic lepton production cross sections described in sectionsII and III have been used to calculate the lepton produc-tion by atmospheric neutrinos after averaging over theneutrino flux corresponding to the two sites of Soudanand Gransasso, where iron based detectors are beingused. There are quite a few calculations of atmosphericneutrino fluxes at these two sites. We use the angle av-eraged fluxes calculated by Honda et al.[14] and Barret al.[15] for the Soudan site and the fluxes of Barr etal.[15] and Plyaskin[16] for the Gransasso site to calcu-late the flux averaged cross section < σ > and also themomentum and the angular distributions < dσ

dpl> and

< dσdcosθl

> for leptons produced by νe, νe, νµ and νµ.

A. Flux averaged Momentum and Angular

distributions

In this section we present the numerical results for theflux averaged momentum distributions < dσ

dpl> and an-

gular distributions < dσdcosθl

> for various leptons pro-duced from νe, νe, νµ, νµ at the atmospheric neutrinosites of Soudan and Gransasso. These leptons are pro-duced through the quasielastic as well as inelastic pro-cesses. The quasielastic production has been discussedin section-II and the inelastic production has been dis-cussed in section-III. The inelastic production of lep-tons is accompanied by pions. However, in the nu-clear medium where the production of ∆ is followed by∆N → NN , ∆NN → NNN absorption processes, theleptons are produced without pions. These are quasielas-tic like events. We show the momentum distribution ofthe leptons produced in iron nuclei corresponding to onepion and quasielastic like events for atmospheric neutri-nos relevant to Soudan and Gransasso sites. We showit for νµ and νµ for the Soudan site in Figs. 13(a) and13(b) and for the Gransasso site in Figs.14(a) and 14(b)corresponding to the flux of Barr et al.[15]. We see thatat both sites, the production cross section of quasielastic-like events is quite small.

We now present our final results for the momentumdistribution and angular distribution for various lep-tons produced by atmospheric neutrinos at Soudan andGransasso sites in Figs. 15-18. In these figures sep-arate contributions from the quasielastic and inelasticprocesses to the momentum and angular distributionsof charged leptons are shown explicitly. The quasielas-tic events are those where only a charge lepton is pro-duced in the final state either by a quasielastic process

10

described in section-II or by an inelastic process of ∆-production followed by its subsequent absorption in thenuclear medium as described in section-III. The inelasticevents are those events in which a charged lepton in thefinal state is accompanied by a charged pion as a decayproduct of deltas excited in the nuclear medium.

We show the results for the momentum distribution< dσ/dpl >∼ pl(l = e−, e+, µ+, µ−) for the atmosphericneutrino fluxes of Barr et al.[15] at Soudan site in Fig.15and at Gransasso site in Fig.16. We see that in all casesthe major contributions to the charged lepton productioncomes from the quasielastic processes induced by neu-trinos(solid line). The contribution from the quasielas-tic processes induced by antineutrinos(dotted line) andinelastic processes induced by neutrinos(dashed line) issmall and is around 20% in the peak region. The contri-bution due to inelastic processes induced by antineutrinosis very small over the whole region (dashed-dotted lines).

The peak in quasielastic νµ reactions occurs aroundpl ≈ 200MeV . The peaks in the inelastic νµ andquasielastic νµ reactions are slightly shifted towards lowerenergies. The momentum distribution of the leptons forthe quasielastic reaction is peaked more sharply than themomentum distribution of the inelastic reaction.

We have also presented the numerical results for an-gular distributions of leptons < dσ

dcosθl> vs cosθ for

the atmospheric neutrino fluxes of Barr et al.[15] for theSoudan site in Fig.17 and for the Gransasso site in Fig.18.The inelastic lepton production accompanied by pions(dashed line for µ−(e−) production and dashed-dottedline for µ+(e+) production) and the quasielastic leptonproduction events (solid line for µ−(e−) production anddotted line for µ+(e+) production) have been explicitlyshown in these figures. We see from these figures that thelepton production cross sections are forward peaked inall cases. The inelastic distribution due to pion produc-tion are slightly more forward peaked than the quasielas-tic distribution. The contribution of the inelastic leptonevents are small compared to the quasielastic events andthe angular distribution of the flux averaged cross sec-tions are dominated by the quasielastic events.

At Soudan site, we have also studied the momentumand angular distributions for the atmospheric neutrinofluxes of Honda et al.[14]. We find that the momentumand angular distributions for the production of muons aresimilar to the distributions obtained for the flux of Barret al.[15]. In the case of electron production, the use ofthe flux of Honda et al.[14], gives a slightly smaller valuefor < dσ/dpl > and < dσ

dcosθl> for electrons as compared

to the flux of Barr et al.[15].Similarly at Gransasso site, the momentum and an-

gular distributions for the atmospheric neutrino fluxesof Plyaskin[16] have also been studied. Here, also wefind that the momentum and angular distributions forthe production of muons are similar to the distributionsobtained for the flux of Barr et al.[15]. In the case of elec-tron production, the use of the flux of Plyaskin[16], givesa slightly smaller value for < dσ/dpl > and < dσ

dcosθl>

for electrons as compared to the flux of Barr et al.[15].We further find that for the flux of Barr et al.[15],

the lepton production cross section at the Soudan site isslightly larger than the Gransasso site for muons as wellas for electrons.

B. Flux averaged Total cross sections and lepton

yields

The total cross sections for the production of leptonsand its energy dependence have been discussed in sectionII and section III for quasielastic and inelastic reactions.In this section we calculate the lepton yields Yl for leptonof flavor l which we define as

Yl =

∫

Φνlσ(Eνl

) dEνl

where, Φνlis the atmospheric neutrino flux of νl and

σ(Eνl) is the total cross section for neutrino νl of energy

Eνl. We calculate this yield separately for the quasielas-

tic and inelastic events. We define a relative yield ofmuon over electron type events by R as R = Rµ/e =Yµ+Yµ

Ye+Yefor quasielastic and inelastic events and present

our results in Table-II. We study the nuclear model de-pendence as well as the flux dependence of the relativeyield R.

The results for R are presented separately forquasielastic events, inelastic events and the total eventsin Table-II. For quasielastic events νl(νl) +56 Fe →l−(l+) +X , the results are presented for the case of freenucleon by RFN , for the nuclear case with Fermi gasmodel description of Llewellyn Smith[20] by RFG andfor the case of nuclear effects within our model by RNM .We see that there is practically no nuclear model depen-dence on the value of R (compare the values of RNM ,RFG and RFN for the same fluxes at each site). Thisis also true for the inelastic production of leptons i.e.νl(νl)+56 Fe→ l−(l+)+π+(π−)+X for which the ratiofor free nucleon case (denoted by R∆F ) and the ratio forthe nuclear case in our model (denoted by R∆N) are pre-sented in Table II. It is, therefore, concluded that there isno appreciable nuclear model dependence on the ratio oftotal lepton yields for the production of muons and elec-trons(compare the values ofRF andRN , where RF showsthe ratio of total lepton yields for muon and electron forthe case of free nucleon and RN shows the ratio of totalyield for muon and electron for the case of nucleon in thenuclear medium).

However, there is some dependence of the ratio R onthe atmospheric neutrino fluxes. The flux dependence ofR can be readily seen from Table-II, for the two sites ofSoudan and Gransasso. At the Gransasso site, we seethat there is 4-5% difference in the value of RN for thetotal lepton yields for the fluxes of Barr et al.[15] andPlyaskin[16]. At Soudan site, the results for the fluxes ofHonda et al.[14] and Barr et al.[15] are within 4-5% butthe flux calculation of Plyaskin[16] gives a result which

11

is about 10-11% smaller than the results of Honda etal.[14] and 7-8% smaller than the results of Barr et al.[15].The flux dependence is mainly due to the quasielasticevents. This should be kept in mind while using the fluxof Plyaskin[16] for making any analysis of the neutrinooscillation experiments.

In Table-III, we present a quantitative estimate of the

relative yield of inelastic events rl defined by rl = Yl∆

Yl,

where Yl∆ = Yl

∆ + Yl∆ is the lepton yield due to the

inelastic events and Yl is the total lepton yield due to thequasielastic and inelastic events i.e.Yl = Yl

q.e. + Ylq.e. +

Yl∆ +Yl

∆. The relative yield for the case of free nucleonis shown by rl(F ) and for the case with the nuclear ef-fects in our model is shown by rl(N). We see that forfree nucleons, the relative yield of the inelastic eventsdue to ∆ excitation is in the range of 12-15% for vari-ous fluxes at the two sites. The ratio is approximatelysame for electrons and muons. When the nuclear effectsare taken into account this becomes 19-22%. This is dueto different nature of the effect of nuclear structure onthe quasielastic and inelastic production cross scetionswhich gives a larger reduction in the cross section for thequasielastic case as compared to the inelastic case. Thisquantitative estimate of rl(N) in iron may be comparedwith the results in oxygen where the experimental resultsat Kamiokande give a value of 18%[60].

V. CONCLUSIONS

We have studied the charged lepton production in ironinduced by atmospheric neutrinos at the experimentalsites of Soudan and Gransasso. The energy dependenceof the total cross sections for the quasielastic and inelasticprocesses have been calculated in a nuclear model whichtakes into account the effect of Pauli principle, Fermimotion effects and the renormalization of weak transi-tion strengths in nuclear medium. The inelastic processhas been studied through the ∆ dominance model whichincorporates the modification of mass and width of ∆ res-onance in the nuclear medium. The numerical results forthe momentum and angular distributions of the chargedlepton production cross section have been presented formuons and electrons. The relative yield of muon to elec-tron production has been studied. In addition to the nu-clear model dependence, the flux dependence of the totalyields, momentum distribution dσ

dpland dσ

dcosθlhave been

also studied. In the following we conclude this paper bysummarizing our main results:

1.There is a large reduction due to nuclear effects in the

total cross section for quasielastic production cross sec-tion specially at lower energies(40-50% around 200MeV)and the reduction becomes smaller at higher energies(15-20% around 500MeV ).

2.For quasielastic reactions we find a larger reductionin the total cross section as compared to the Fermi gasmodel. The energy dependence of this reduction in crosssection at low energies (E < 500MeV ) is found to bedifferent for neutrino and antineutrino reactions.

3.The inelastic production of leptons where a chargedlepton is accompanied by a pion becomes comparable tothe quasielastic production of leptons for Eν ∼ 1.4GeVand increases with increase of energy. The effect of nu-clear medium on the inelastic production cross section(in absence of pion re-scattering effects) in not too large(∼ 10%).

4. For quasielastic reactions the effect of Coulomb dis-tortion of the final lepton in the total cross section issmall except at very low energies (E < 500MeV ) andbecomes negligible when averaged over the flux of atmo-spheric neutrinos.

5.The flux averaged momentum distribution of leptonsproduced by atmospheric neutrinos is peaked around themomentum pl ∼ 200MeV for electrons and muons. Thepeak for the quasielastic production is sharper than theinelastic production. The inelastic production of leptonscontributes about 20% to the total production of leptons.

6.The flux averaged angular distribution of leptons foratmospheric neutrinos for quasielastic as well as inelasticproduction is sharply peaked in the forward direction.

7.There is a very little flux dependence on the relativeyield of muons and electrons at the site of Gransasso.However, at the Soudan site, the atmospheric flux as de-termined by Plyaskin[16] gives a value of relative yieldwhich is smaller than the relative yield using the flux ofHonda et al.[14] and Barr et al.[15].

8.The nuclear model dependence of the relative yield ofmuons to electrons is negligible, even though the individ-ual yields for muons and electrons are reduced with theinclusion of nuclear effects specially for the quasielasticproduction.

VI. ACKNOWLEDGMENT

The work is financially supported by the Departmentof Science and Technology, Government of India underthe grant DST Project No. SP/S2K-07/2000. One ofthe authors (S.Ahmad) would loke to thank CSIR forfinancial support.

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12

TABLE II: Ratio R = Rµ/e =Yµ+Yµ

Ye+Yecorresponding to quasielastic, inelastic and total production of leptons [FG refers to

Fermi Gas Model, NM refers to Nuclear Model, FN refers to Free Nucleon, ∆N refers to ∆ in Nuclear Model, ∆F refers to ∆Free]. RF shows the ratio of total lepton yields for muon to electron for the case of free nucleon and RN shows the ratio oftotal yields for muon to electron for the case of nucleon in the nuclear medium.

Site Soudan Soudan Soudan Gransasso GransassoFLUX Barr et al.[15] Plyaskin[16] Honda et al.[14] Barr et al.[15] Plyaskin[16]

Quasielastic

RNM 1.80 1.65 1.89 1.95 2.00RF G 1.81 1.66 1.89 1.95 2.09RF N 1.82 1.68 1.90 1.95 2.08

Inelastic

R∆N 1.84 1.81 1.95 2.02 2.05R∆F 1.82 1.80 1.94 2.01 2.03Total

RN 1.81 1.68 1.90 1.96 2.01RF 1.82 1.69 1.90 1.96 2.07

TABLE III: % ratio r = Y∆

Yq.e.+∆[N refers to Nuclear Model, F refers to free case]

Sites Soudan Soudan Soudan Gransasso GransassoFLUX Barr et al.[15] Plyaskin[16] Honda et al.[14] Barr et al.[15] Plyaskin[16]rµ(F ) 14 12 14 15 12re(F ) 14 11 14 14 12rµ(N) 22 20 22 23 19re(N) 22 19 22 22 19

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13

0 500 1000 1500 2000 2500 3000

0.1

1

10

0 500 1000 1500 2000 2500 3000

0.1

1

10

σ(1

0

cm )

σ(1

0

cm )

-38

2

-38

2

E (MeV)νE (MeV)ν

(a) (b)

FIG. 9: The total scattering cross section for the neutrino νl + N → ∆ + l− (shown by dashed line) and anti-neutrinoνl + N → ∆ + l+ reactions (shown by dashed-dotted line) in 56Fe and compared with the corresponding quasielastic crosssection in the present model with nuclear effects for the neutrino (solid line) and anti-neutrino(dotted line) reactions in 56Fe.Fig.(a) is for electron type and Fig.(b) is for muon type neutrinos.

0 500 1000 1500 2000 2500 30000

5

10

15

20

25

30

35

Eν (MeV)

σ(

10

cm

)

-38

2

FIG. 10: Total cross section at Eν = 1.0GeV for the reactionνl(νl) + N → ∆ + l−(l+) in 56Fe with (dashed line for νµ,dashed-dotted for νµ) and without nuclear effects(solid line isthe result for νµ reaction and dotted for νµ.)

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14

0 200 400 600 8000

1

2

3

4

5

-1 -0.5 0 0.5 10

10

20

30

40

dσ/

pl

d

dd

/l

cos

(a)

σ

(10

cm

/M

eV)

(10

cm

)

-40 -3

8

2

2

(MeV)pl cos θl

(b)

θFIG. 11: Differential scattering cross section Fig.(a) dσ

dplvs pl and Fig.(b) dσ

dcosθlvs cosθl at Eν = 1.0GeV for the reaction

νl(νl) + N → ∆ + l−(l+) in 56Fe with (dashed line for νµ, dotted for νµ) and without nuclear effects(solid line is the result forνµ reaction and dashed-dotted for νµ).

0 200 400 600 8000

1

2

3

4

-1 -0.5 0 0.5 10

10

20

30

40

dσ

pl

/d dσ

d/

(a) (b)

(10

cm

/M

eV)

-40

2

(10

cm

)

-38

2co

sθ

(MeV)p

l

l lθcos

FIG. 12: The results for the differential cross section Fig.(a) dσdpl

vs pl and Fig.(b) dσdcosθl

vs cosθl with nuclear effects at

E = 1.0GeV for the reaction νl(νl) + N → ∆ + l−(l+) in 56Fe. Solid line is the result for νe, dashed for νµ, dotted for νe anddashed-dotted for νµ reactions.

0 500 1000 15000

0.1

0.2

0.3

0.4

0 500 1000 15000

0.01

0.02

0.03

0.04

0.05

0.06

0.07

(10

cm

/M

eV)

2-4

0 <

d

/dp

>

σ

l

p (MeV)

l <

d

/dp

>

l

< d

/d

p >

< d

/d

p >

< d

/d

p >

(10

cm

/M

eV)

-40

2

p

σ

l(MeV)

(a) (b)

FIG. 13: Flux averaged differential cross section < dσdpl

> vs pl at the Soudan site simulated by Barr et al.[15]. Fig.(a) for

νµ + N → ∆ + µ− and Fig.(b) for νµ + N → ∆ + µ+ reactions in 56Fe. The solid line is for total ∆ events, dashed line is forone π production events and dotted line is for quasielastic like events(due to ∆ absorption in the nuclear medium)

15

0 500 1000 15000

0.1

0.2

0.3

0.4

0.5

0 500 1000 15000

0.01

0.02

0.03

0.04

0.05

0.06

0.07(a) (b)

l

( 10

c

m

/MeV

)-4

02

dσ

dp l >

<

p

2

l

//

-40

( 10

c

m

/MeV

)

( MeV )

dσ

dp

>

<

( MeV ) ( MeV )lpp

FIG. 14: Flux averaged differential cross section < dσdpl

> vs pl at the Gransasso site simulated by Barr et al.[15]. Fig.(a) for

νµ + N → ∆ + µ− and Fig.(b) for νµ + N → ∆ + µ+ reaction in 56Fe. The solid line is for total ∆ events, dashed line is forone π production events and dotted line is for quasielastic like events(due to ∆ absorption in the nuclear medium)

.

0 500 1000 15000

0.5

1

1.5

2

0 500 1000 15000

0.5

1

1.5

2

-40

2

<d

/dp

> (

10

cm /

MeV

)

<d

/dp

> (

10

cm /

MeV

)σ

l

p (MeV)lp (MeV)l

l-4

0

2σ

(a) (b)

FIG. 15: Flux averaged differential cross section < dσdpl

> vs pl at the Soudan site simulated by Barr et al.[15]. The solid line is

for total quasielastic events and dashed line is the one π production events for νµ reaction in 56Fe. The corresponding resultsfor νµ are shown by dotted and dashed-dotted lines, Fig.(a) for muon type and Fig.(b) for electron type neutrinos.

0 500 1000 15000

0.5

1

1.5

2

0 500 1000 15000

0.5

1

1.5

2

<d

/dp

> (

10

cm /

MeV

)

<d

/dp

> (

10

cm /

MeV

)

σ

p (MeV)p (MeV)

σ

l l

-40

2

-40

2

(a) (b)

l l

FIG. 16: Flux averaged differential cross section < dσdpl

> vs pl at the Gransasso site simulated by Barr et al.[15]. The solid

line is for total quasielastic events and dashed line is the one π production events for νµ reaction in 56Fe. The correspondingresults for νµ are shown by dotted and dashed-dotted lines, Fig.(a) for muon type and Fig.(b) for electron type neutrinos.

16

-1 -0.5 0 0.5 1

cos θl

0

2

4

6

8

10

12

14

<dσ

/dco

s θ l>

(10

-38 cm

2 )

-1 -0.5 0 0.5 1

cos θl

0

2

4

6

8

10

12

<dσ

/dco

s θ l>

(10

-38 cm

2 )(a) (b)

FIG. 17: Flux averaged differential cross section < dσdcosθl

> vs cosθl at the Soudan site simulated by Barr et al.[15]. The solid

line is for total quasielastic events and dashed line is the one π production events for νµ reaction in 56Fe. The correspondingresults for νµ are shown by dotted and dashed-dotted lines, Fig.(a) for muon type and Fig.(b) for electron type neutrinos.

-1 -0.5 0 0.5 1

cos θl

0

3

6

9

12

15

<dσ

/dco

s θ l>

(10

-38 cm

2 )

-1 -0.5 0 0.5 1

cos θl

0

2

4

6

8

10

<dσ

/dco

s θ l>

(10

-38 cm

2 )(a) (b)

FIG. 18: Flux averaged differential cross section < dσdcosθl

> vs cosθl at the Gransasso site simulated by Barr et al.[15]. The solid

line is for total quasielastic events and dashed line is the one π production events for νµ reaction in 56Fe. The correspondingresults for νµ are shown by dotted and dashed-dotted lines, Fig.(a) for muon type and Fig.(b) for electron type neutrinos.

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