Charged lepton production from iron induced by atmospheric neutrinos

DOI 10.1140/epja/i2004-10234-2

Eur. Phys. J. A 24, 459–474 (2005) THE EUROPEAN

PHYSICAL JOURNAL A

Charged lepton production from iron induced by atmosphericneutrinos

M. Sajjad Athar, S. Ahmad, and S.K. Singha

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

Received: 30 September 2004 / Revised version: 12 April 2005 /Published online: 8 June 2005 – c© Societa Italiana di Fisica / Springer-Verlag 2005Communicated by G. Orlandini

Abstract. The charged current lepton production induced by neutrinos in 56Fe nuclei has been studied. Thecalculations have been done for the quasielastic as well as the inelastic reactions assuming∆-dominance andtake into account the effect of Pauli blocking, Fermi motion and the renormalization of weak transitionstrengths in the nuclear medium. The quasielastic production cross-sections for lepton production arefound to be strongly reduced due to nuclear effects, while there is about 10% reduction in the inelasticcross-sections in the absence of the final-state interactions of the pions. The numerical results for themomentum and angular distributions of the leptons averaged over the various atmospheric-neutrino spectraat the Soudan and Gran Sasso sites have been presented. The effect of nuclear-model dependence and theatmospheric-flux dependence on the relative yield of µ to e has been studied and discussed.

PACS. 25.30.Pt Neutrino scattering – 13.15.+g Neutrino interactions – 23.40.Bw Weak-interaction andlepton (including neutrino) aspects – 21.60.Jz Hartree-Fock and random-phase approximations

1 Introduction

The study of neutrino physics with atmospheric neutri-nos 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 was nomore than suggestive due to low statistics of the experi-mental data and anticipated uncertainties in the flux cal-culations [2]. The clear evidence of a deficit in the at-mospheric muon neutrino flux was confirmed later whendata 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 phenomenaof neutrino oscillations [6] in which the neutrinos producedwith muon flavor after passing a certain distance throughthe atmosphere, manifest themselves as a different flavor.The implication of this phenomena of neutrino oscillationis that neutrinos possess a nonzero mass pointing towardsphysics beyond the standard model of particle physics.The evidence for neutrino oscillations and a nonzero massfor the neutrinos has also been obtained in the observa-tions made with solar [7] and reactor (anti)neutrinos [8].

It is well known that, in a two-flavor oscillation sce-nario involving muon neutrino, the probability for a muon

a e-mail: [email protected]

neutrino with energy Eν to remain a muon neutrino afterpropagating a distance L before reaching the detector isgiven by [6]

Pµµ = 1− sin2 2θ sin2(

1.27∆m2 (eV2)L (km)

Eν (GeV)

)

(1)

where ∆m2 = m21 − m2

2 is the difference of the squaredmasses of the two flavor mass eigenstates and θ is themixing angle between two states. The oscillation parame-ters ∆m2 and the mixing angle θ are determined by var-ious observations made in atmospheric-neutrino experi-ments. These include the flavor ratios of muon and elec-tron flavors, angular and L

E distributions of muons andelectrons produced by atmospheric neutrinos. The firstclaims of seeing neutrino oscillation in atmospheric neu-trinos came from the measurements of ratio of ratios Rν

defined as (µ/e)data(µ/e)MC

from the observations of fully con-

tained (FC) events [3,4], but there are now data availablefrom the angular and L

E distribution of the atmospheric-neutrino–induced muon and electron events from SK [9],MACRO [10] and Soudan [11] experiments which confirmthe phenomena of neutrino oscillations. These experimentsare consistent with a value of ∆m2 ≈ 3.2× 10−3 eV2 andsin2 2θ ≈ 1. The analysis of these data assuming a three-flavor neutrino oscillation phenomenology has also beendone by many authors [12].

460 The European Physical Journal A

The major sources of uncertainty in the theoreti-cal prediction of the charged leptons of muon and elec-tron flavor produced by the atmospheric neutrinos comefrom the uncertainties in the calculation of atmospheric-neutrino fluxes and neutrino nuclear cross-sections. Theatmospheric-neutrino fluxes at various experimental sitesof Kamioka, Soudan and Gran Sasso have been exten-sively discussed in the literature by many authors [13–16].The neutrino nuclear cross-sections have also been calcu-lated for various nuclei by many authors using differentnuclear models [17–24]. The aim of the present paper isto study the neutrino nuclear cross-section in iron nucleiwhich are relevant for the atmospheric-neutrino experi-ments performed at Soudan [11], FREJUS [25] and NU-SEX [26] and planned in the future with MINOS [27],MONOLITH [28] and INO [29] detectors. The uncertaintyin the nuclear production cross-section of leptons fromiron nuclei by the atmospheric neutrinos are discussed.For our nuclear model, we also discuss the uncertaintydue to the use of different neutrino fluxes for the sites ofSoudan and Gran Sasso which are relevant to MINOS,MONOLITH and 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 calculationsare done in a model which takes into account nuclear ef-fects like Pauli Blocking, Fermi motion effects and theeffect of renormalization of the weak transition strengthsin nuclear medium in local density approximation. Themodel has been successfully applied to describe variouselectromagnetic and weak processes like photon absorp-tion, electron scattering, muon capture and low-energyneutrino reactions in nuclei [24,30–32]. The model can beeasily applied to calculate the zenith angle dependenceand the L

E distribution for stopping and thorough goingmuon production from iron nuclei which is currently underprogress.

The plan of the paper is as follows. In sect. 2 we de-scribe the neutrino (antineutrino) quasielastic inclusiveproduction of leptons (e−, µ−, e+, µ+) from iron nuclei forvarious neutrino energies. In sect. 3 we describe the energydependence of the inelastic production of leptons throughthe ∆-dominance model and highlight the nuclear effectsrelevant to the energy of fully contained events. In sect. 4,we use the atmospheric-neutrino flux at Soudan and GranSasso sites as determined by various authors and discussthe flux-averaged momentum and angular dependence ofleptons corresponding to different flux calculations avail-able at these two sites.

2 Quasielastic production of leptons

Quasielastic inclusive production of leptons in nuclei in-duced by neutrinos has been studied by many authors [17–24], where nuclear effects have been calculated. Most of

these calculations have been done either for 16O relevantto IMB and Kamioka experiments [3,4] or for 12C rel-evant to LSND and KARMEN experiments [33]. Thesecalculations generally use a direct summation method(over many nuclear excited states) [17], closure approxi-mation [18], Fermi gas model [19,20], relativistic mean-field approximation [21], continuum random phase ap-proximation (CRPA) [22] and local density approxima-tion [23,24]. The calculations for 56Fe nucleus have beenreported by Bugaev et al. [17] in a shell model and in Fermigas model by Gallagher [34] and Berger et al. [35]. In thissection we briefly describe the formalism and results ofour calculations done for quasielastic inclusive productionof leptons for iron nuclei.

2.1 Formalism

In local density approximation the neutrino nucleus cross-section σ(Eν) for a neutrino of energy Eν scattering froma nucleus A(Z,N), is given by

σA(Eν) = 2

∫

d~rd~p

(2π)3nn(~p, ~r )σ

N (Eν) , (2)

where nn(~p, ~r ) is the local occupation number of the initialnucleon of momentum ~p (localized at position ~r in thenucleus) and σN (Eν) is the cross-section for the scatteringof neutrino of energy Eν from a free nucleon given by theexpression

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)iσµνqν2M

+FVA (q2)γµγ5

]

u(p) ; (6)

q2(q = k−k′) is the four-momentum transfer squared. Theform factors F V

1 (q2), FV2 (q2) and FV

A (q2) are isovectorelectroweak form factors written as

FV1 (q2) = F p

1 (q2)− Fn

1 (q2),

FV2 (q2) = F p

2 (q2)− Fn

2 (q2),

FVA (q2) = FA(q

2),

M. Sajjad Athar et al.: Charged lepton production from iron induced by atmospheric neutrinos 461

where

F p,n1 (q2) =

1(

1− q2

4M2

)

[

Gp,nE (q2)− q2

4M2Gp,nM (q2)

]

,

F p,n2 (q2) =

1(

1− q2

4M2

)

[

Gp,nM (q2)−Gp,n

E (q2)

]

,

GpE(q

2) =

(

1− q2

M2v

)−2

, (7)

GpM (q2) = (1 + µp)G

pE(q

2), GnM (q2) = µnG

pE(q

2);

GnE(q

2) =

(

q2

4M2

)

µnGpE(q

2)ξn;

ξn =1

1− λn q2

4M2

, 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) = −1.261.In a nuclear process the neutrons and protons are not

free and their momenta are constrained by the Pauli prin-ciple which is implemented in this model by requiring thatfor neutrino reactions initial-nucleon momentum p ≤ pFnand final-nucleon momentum p′ = (|~p+ ~q |) > pFp , where

pFn,p = [ 32π2ρn,p(r)]

13 , are the local Fermi momenta of

neutrons and protons at the interaction point in the nu-cleus defined in terms of their respective nuclear densitiesρn,p(r). These constraints are incorporated while perform-ing the integration over the initial-nucleon momentum ineq. (2) by replacing the energy conserving δ-function ineq. (3) by − 1

π ImUN (q0, ~q ), where UN (q0, ~q ) is the Lind-hard function corresponding to the particle-hole (ph) ex-citations induced by the weak interaction process throughW exchange shown in fig. 1(a). In the large mass limitof the W -boson, i.e. MW → ∞, fig. 1(a) is reduced tofig. 1(b) for which the imaginary part of the Lindhardfunction, i.e. ImUN (q0, ~q ), is given by

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 EF1 and EF2 are the local Fermi energy of initialand final nucleons and

A = Max

Mn, EF2 − q0,−q0 + |~q |

√

1− 4M2

q2

2

.

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 tothe νe + n→ e− + p process in nuclei. In the large mass limitof the W -boson (i.e. MW →∞) diagram (a) is reduced to (b)which is used to calculate |T |2 in eq. (5).

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(cos θ)

× 1

EνEl

∑∑

|T |2ImUN [Eν − El, ~q ]. (9)

Moreover, in the nucleus, the Q value of the nuclear re-action and the Coulomb distortion of the final lepton inthe electromagnetic field of the final nucleus should betaken into account. This is done by modifying the energyconserving δ-function δ(Eν − El + En − Ep) in eq. (3) toδ(Eν − Q − (El + Vc(r)) + En − Ep), where Vc(r) is theCoulomb energy of the produced lepton in the field of finalnucleus 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 the evaluation of the Lindhard func-

tion in eq. (8) at (q0 − (Q+ Vc(r)), ~q ) instead of (q0, ~q ).The implementation of this modification requires a ju-dicious choice of the Q value for inclusive nuclear reac-tions in which many nuclear states are excited in iron.We have taken a Q value of 6.8 MeV corresponding tothe transition to the lowest-lying 1+ state in 56Co for theνl+

56Fe→ l−+ 56Co? reaction and a Q value of 4.3 MeVcorresponding to the transition to the lowest-lying 1+

state in 56Mn for the 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 the final nucleus in our model is equiv-alent to the treatment of the Coulomb distortion ef-fect in the modified effective momentum approximation(MEMA). This approximation has been used in other cal-culations of charged current neutrino reactions [36] andelectron scattering at higher energies [37].

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


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 effects con-tributing to the process νe + n→ e− + p transitions.

given by

σA(Eν) = −4

π

∫ rmax

rmin

r2dr

∫ plmax

plmin

pl2dpl

∫ 1

−1

d(cos θ)

× 1

EνEl

∑∑

|T |2ImUN [q0 − (Q+ Vc(r)), ~q ]. (11)

It is well known that weak transition strengths are modi-fied in the nuclear medium due to the presence of stronglyinteracting nucleons. This modification of the weak transi-tions strength in the nuclear medium is taken into accountby considering the propagation of particle-hole (ph) exci-tations in the medium. While propagating through themedium, the ph excitations interact through the nucleonnucleon potential and create other particle-hole and ∆hexcitations as shown in fig. 2. The effect of these excita-tions are calculated in the random phase approximationwhich is described in refs. [23,32]. The effect of the nu-clear medium on the renormalization of weak strengthsis treated in a nonrelativistic framework. In the leadingorder, the nonrelativistic reduction of the weak hadroniccurrent defined in eq. (6), the F2(q

2) term gives a spin-

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

erator, while the FA(q2) term gives a spin-isospin transi-

tion operator ~σ · ~τ which has a longitudinal as well as atransverse part. This representation of the transition op-erators in the longitudinal and transverse parts is usefulwhen summing the diagrams in fig. 2 in random phaseapproximation (RPA) to calculate |T |2. While the chargecoupling remains unchanged due to nuclear medium ef-fects, the terms proportional to F 2

2 are affected by thetransverse part of the nucleon-nucleon potential, while theterms proportional to F 2

A are affected by transverse as wellas longitudinal parts. The effect is to replace the terms likeF 22 , 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)

0 200 400 600 800 1000

0.1

1

10

Eν(MeV)

σ(1

0 c

m )

-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 anddash-dotted line for νµ reactions.

where Vl and Vt are the longitudinal and transverse partsof the nucleon-nucleon potential calculated with π and ρexchanges and modulated by the Landau-Migdal param-eter g′ to take into account the short-range correlationeffects and 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 and mρ

are the pion and rho-meson masses and g′ is taken to be0.7 [31]. The effect of ∆h excitations are taken into ac-count by including the Lindhard function U∆ for the ∆hexcitations and replacing UN by UN +U∆ in eq. (12). Thecomplete expressions for UN and U∆ used in our calcula-tions are taken from [38]. The different couplings for Nand ∆ to the nucleon are incorporated in UN and U∆ andthen the same interaction strengths Vl and Vt are used forph and ∆h excitations [39,40].

2.2 Results

We present the numerical results for the total cross-section for the quasielastic processes νl(νl)+

56Fe →l−(l+)+ 56Co?(56Mn?) as a function of energy for neutrinoand antineutrino reactions on iron in the energy regionrelevant to the fully contained events of atmospheric neu-trinos, i.e. Eν < 3GeV. The cross-sections have been cal-culated using eq. (11) with the nuclear density ρ(r) given

by a two-parameter Fermi density [41]: ρ(r) = ρ(0)

1.+exp( r−cz

)

with c = 3.971 fm, z = 0.5935 fm, ρn(r) =(A−Z)A ρ(r) and

ρp(r) =ZAρ(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. The re-duction due to nuclear effects is large at lower energies but


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 neutrinonucleon cross-section for the reactions (a) νµ + n → µ− + p

(b) νµ + p → µ+ + n in iron nuclei in the present model withPauli suppression (solid line), with nuclear effects (dotted line)and in the Fermi gas model (dashed line) [20].

becomes small at higher energies. The energy dependencesof the cross-sections for muon- and electron-type neutrinosare similar except for the threshold effects which are seenonly at low energies (Eν < 500 MeV). This reduction in σis due to Pauli blocking as well as to the weak renormal-ization of transition strengths which have been separatelyshown in fig. 4(a) and (b) for neutrinos and antineutrinos,where we also show the results in the Fermi gas modelgiven by Llewellyn Smith [20]. We plot in fig. 4(a) and(b) for neutrinos and antineutrinos, the reduction factor

R = σnuclear(E)σnucleon(E) vs. E, where σnuclear(E) is the cross-section

per neutron (proton) for neutrino (antineutrino) reactionsin the nuclear medium. The solid lines show the reduc-tion factor R when only the Pauli suppression is takeninto account through the imaginary part of the Lindhardfunction given in eq. (8). This is similar to the results ofLlewellyn Smith [20] in the Fermi gas model shown bydashed lines. In this model the total cross-section σ is cal-culated by using the formula σ =

∫

dq2 R(q2)( dσdq2 )free,

where R(q2) describes the reduction in the cross-sectioncalculated in the Fermi gas model and includes the effect ofthe Pauli suppression only [20]. However, in our model weget further reduction due to renormalization of weak tran-

sition strengths in the nuclear medium when the effects offig. 2(a) and (b) are included. These are shown by dottedlines in figs. 4(a) and (b). We find that the reduction athigher energies (E > 1GeV) is around 20% for neutrinosand 40% for antineutrinos. It is worth noting that the en-ergy dependence of the reduction due to nuclear mediumeffects is different for neutrinos and antineutrinos. This isdue to the different renormalization of various terms likeF 2A, F2FA and F 2

2 in |T |2 which enter in different combina-tions for neutrino and antineutrino reactions. The resultsfor νe and νe cross-section are, respectively, similar to νµand νµ reactions except for the threshold effects and arenot shown here.

In figs. 5 and 6 we compare our results for σ(E)with the results of some earlier experiments which con-tain nuclear 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 nucleartargets considered here (except for Br in freon) are lighterthan Fe. Therefore, the reduction in the total cross-sectiondue to nuclear effects will be slightly overestimated. Forexample, for energies Eν ≥ 1GeV the reduction in neu-trino (antineutrino) cross-section in the case of 56Fe is 5%more than the reduction in the case of 12C [48]. In compar-ision to the neutrino (antineutrino) nuclear cross-sectionsas obtained in the Fermi gas model of Llewellyn Smith [20](shown by dashed lines in figs. 5 and 6) we get a smallerresult for these cross-sections. This reduction in the totalcross-section leads to an improved agreement with the ex-perimental results as compared to the Fermi gas model re-sults specially for antineutrino reactions (fig. 6). It shouldbe emphasized that the Fermi gas model has no specificmechanism to estimate the renormalization of weak tran-sition strengths in nuclei, while in our model this is incor-porated by taking into account the RPA correlations.

In fig. 7(a) and (b), we show the nuclear medium ef-fects on the momentum and angular distributions, i.e. dσ

dpl

and dσd cos θl

of leptons produced in νµ and νµ reactions. Wefind a large suppression in the results specially in the peakregion of momentum and angular distributions. Quantita-tively similar results are obtained for the case of νe andνe reactions and are not shown here.

We have also studied the effect of Coulomb distor-tion in the momentum distribution of leptons but foundno substantial effect around Eν = 1.0GeV. These arefound to affect the results only at low energies, i.e. Eν <500MeV, where the peak is slightly shifted to lower mo-mentum as shown in fig. 8(a) and (b).

3 Inelastic production of leptons

The inelastic production process of leptons is the pro-cess in which the production of leptons is accompaniedby one-pion (or more pions). There are many calcula-tions of one-pion production by neutrinos from free nu-cleons [49] but there are only a few calculations which


Fig. 5. Neutrino quasielastic total cross-section per nucleon in iron for the νµ + n → p + µ− reaction. The data are fromLSND [42] (ellipse), Bonnetti et al. [43] (squares), SKAT Collaboration [44] (triangle down), Pohl et al. [45] (circles) andBelikov et al. [46] (triangles up). The dashed line is the result of the cross-section in the Fearmi gas model [20] and the solidline is the result using the present model with nuclear effects.

Fig. 6. Antineutrino quasielastic total cross-section per nucleon in iron for the νµ + p → n + µ+ reaction. The data are fromBonnetti et al. [43], SKAT Collaboration [44] (triangle down), Pohl et al. [45] (circles), Belikov et al. [46] (triangle up). Thedashed line is the result of the cross-section in the Fermi gas model [20] and the solid line is the result using the present modelwith nuclear effects.

discuss the nuclear effects in these processes [50,52]. Inthis section we follow the method of ref. [51] to esti-mate the nuclear effects and nuclear-model dependenceof inelastic production cross-section of leptons induced byneutrinos from iron nuclei. The calculations are done as-suming ∆-dominance of one-pion production because thecontribution of higher resonances in the energy region ofatmospheric neutrinos leading to fully contained events isexpected to be small.

3.1 Formalism

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

νl(k) + p(p)→ l−(k′) +∆++(p′) , (14)

is given by eq. (5) where Jµ now defines the matrix ele-ment of the transition hadronic current between N and ∆

states. The most general form of JµCC is written as [51]

JµCC = ψα(p′)

[(

CV3 (q2)

M(gαµ 6 q − qαγµ)

+CV4 (q2)

M2(gαµq · p′ − qαp′µ)

+CV5 (q2)

M2(gαµq · p− qαpµ) + CV

6 (q2)

M2qαqµ

)

γ5

+

(

CA3 (q

2)

M(gαµ 6 q − qαγµ)

+CA4 (q

2)

M2(gαµq · p′ − qαp′µ)

+CA5 (q

2)gαµ +CA6 (q2)

M2qαqµ

)]

u(p) , (15)

where ψα(p′) and u(p) are the Rarita-Schwinger and Dirac

spinors for ∆ and nucleon of momenta p′ and p, respec-tively, q(= p′ − p = k − k′) is the momentum transferand CV

i (i = 3–6) are vector and CAi (i = 3–6) are ax-

ial vector transition form factors. The vector form factors


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

/(1

0 cm

/M

eV)

(10 cm

)

-40

22

-38

(MeV)

lco

s

θcos

θ

l(rad)

(a)

(b)

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

vs.

pl and (b) dσd cos θl

vs. cos θl at E = 1.0GeV in the present model

with Pauli suppression (solid line for νµ and dash-dotted linefor νµ) and with nuclear effects (dashed line for νµ and dottedline for νµ).

CVi (i = 3–6) are determined by using the conserved vec-

tor current (CVC) hypothesis which gives CV6 (q2) = 0

and relates CVi (i = 3, 4, 5) to the electromagnetic form

factors which are determined from photoproduction andelectroproduction of ∆’s. Using the analysis of these ex-periments [52,53] we take for the vector form factors

CV5 = 0, CV

4 = − M

M∆CV3 , and

CV3 (q2) =

2.05

(1− q2

M2V

)2, M2

V = 0.54GeV2 . (16)

The axial vector form factor CA6 (q

2) is related to C5A(q2)

using PCAC and is given by

CA6 (q

2) = C5A(q2)

M2

mπ2 − q2 . (17)

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

2) aretaken from the experimental analysis of the neutrino ex-periments producing ∆’s in proton and deuteron tar-gets [54,55]. These form factors are not uniquely deter-

100 200 300 400 5000

2

4

6

8

100 200 300 400 5000

2

4

6

8

dp

σ/

ld

dσ

dp

/l

(10 c

m /M

eV)

(10 c

m /M

eV)

-40

22

-40

(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 dash-dottedline for νl) and without Coulomb effect (dashed line for νl anddotted line for νl). (a) is for electron-type and (b) for muon-type neutrinos.

mined but the following parameterizations give a satisfac-tory fit to the data:

CAi=3,4,5(q

2) = CAi (0)

[

1 +aiq

2

bi − q2](

1− q2

MA2

)−2

(18)

with CA3 (0) = 0, CA

4 (0) = −0.3, CA5 (0) = 1.2, a4 = a5 =

−1.21, b4 = b5 = 2GeV2, MA = 1.28GeV. Using thehadronic current given in eq. (15), the energy spectrum ofthe outgoing leptons is given by

d2σ

dEk′dΩk′=

1

8π31

MM ′

k′

Eν

Γ (W )2

(W −M ′)2 + Γ 2(W )4.

LµνJµν ,

(19)

where W =√

(p+ q)2 and M ′ is the mass of ∆,

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

αk′β ,

Jµν = ΣΣJµ†Jν ,

and is calculated with the use of spin- 32 projection opera-tor Pµν defined as

Pµν =∑

spins

ψµψν


Table 1. Coefficients of eq. (25) for an analytical interpolationof 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

and given by

Pµν = − 6 p′ +M ′

2M ′

(

gµν − 2

3

p′µp′ν

M ′2

+1

3

p′µγν − p′νγµM ′

− 1

3γµγν

)

(20)

In eq. (19), the decay width Γ is taken to be anenergy-dependent 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

andM is the mass of nucleon. The step function Θ denotesthe fact that the width is zero for the invariant masses be-low the Nπ threshold. |qcm| is the pion momentum inthe rest frame of the resonance. When reaction (14) takesplace in the nucleus, the neutrino interacts with the nu-cleon moving inside the nucleus of density ρ(r) with itscorresponding momentum ~p constrained to be below itsFermi momentum. The produced ∆’s have no such con-straints on their momentum. These∆’s decay through var-ious decay channels in the medium. The most prominentdecay mode is ∆→ Nπ which produces pions. This decaymode in the nuclear medium is slightly inhibited due toPauli blocking of the final-nucleon momentum modifyingthe decay width Γ used in eq. (21). This modification ofΓ due to Pauli blocking of nucleus has been studied in de-tail in electromagnetic and strong interactions [56]. The

modified ∆ decay width, i.e. Γ , is written 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 additional

decay channels now open due to two-body and three-bodyabsorption processes like∆N → NN and∆NN → NNN

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

2-3

82

E (MeV)ν

E (MeV)ν

(a)

(b)

Fig. 9. The total scattering cross-section for the neutrino νl+N → ∆ + l− (shown by the dashed lines) and antineutrinoνl+N → ∆+l+ reactions (shown by dash-dotted lines) in 56Feand compared with the corresponding quasielastic cross-sectionin the present model with nuclear effects for the neutrino (solidlines) and antineutrino (dotted lines) reactions in 56Fe. (a) isfor electron-type and (b) is for muon-type neutrinos.

through which ∆’s disappear in the nuclear medium with-out producing a pion, while a two-body ∆ absorption pro-cess like∆N → πNN gives rise to some more pions. Thesenuclear medium effects on ∆ propagation are included byusing a ∆ propagator in which the ∆ propagator is writ-ten in terms of ∆ self-energy Σ∆. This is done by us-ing a modified mass and width of ∆ in nuclear medium,i.e. M∆ → M∆ + ReΣ∆ and Γ → Γ − ImΣ∆. Thereare many calculations of ∆ self-energy Σ∆ in the nuclearmedium [56–59] and we use the results of [56], where thedensity dependence of real and imaginary parts of Σ∆ areparametrized in the following form:

ReΣ∆ = 40ρ

ρ0MeV and

−ImΣ∆ = CQ

(

ρ

ρ0

)α

+ CA2

(

ρ

ρ0

)β

+ CA3

(

ρ

ρ0

)γ

. (24)

In eq. (24), the term with CQ accounts for the ∆N →πNN process, the term with CA2 for the two-body ab-sorption process ∆N → NN and the term with CA3


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, dash-dotted for νµ) and without nuclear effects (the solid line is theresult for the νµ reaction and the dotted one for νµ.)

for the three-body absorption process ∆NN → NNN .The coefficients CQ, CA2, CA3, α, β and γ (γ = 2β) areparametrized in the range 80 < Tπ < 320MeV (where Tπis the pion kinetic energy) as [56]

Ci(Tπ) = ax2 + bx+ c, for x =Tπmπ

. (25)

The values of the coefficients a, b and c are given in table 1.taken from ref. [56].

With these modifications, which incorporate the var-ious nuclear medium effects on ∆ propagation, thecross-section is now written as

σ =

∫ ∫

dr

8π3dk′

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 tothe charged pion production from the proton target, i.e.

νl+p→ l−+∆++ → l−+ p + π+, in the nucleus. In caseof antineutrino reactions, ρp+

13ρn is replaced by ρn+

13ρp.

3.2 Results

In this section we present results of inelastic lepton pro-duction cross-section due to ∆h excitations in iron in-duced by the charged current neutrino interactions us-ing eq. (26). In fig. 9(a) and (b), we show the resultsσ(Eν) ∼ Eν for νe(νe) and νµ(νµ) for the inelastic pro-duction of lepton accompanied by ∆ and compare thiswith the cross-section for the quasielastic production ofleptons discussed in sect. 2. We see that for Eν ≈ 1.4GeV

0 200 400 600 8000

1

2

3

4

5

-1 -0.5 0 0.5 10

10

20

30

40

dσ

/p

ld

dd

/l

cos

(a)

σ(1

0 cm

/M

eV)

(10 cm

)-4

0-3

82

2

(MeV)pl

cos θl

(b)

θ

Fig. 11. Differential scattering cross-section (a) dσdpl

vs. pl and

(b) dσd cos θl

vs. cos θl at Eν = 1.0GeV for the reaction νl(νl) +

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

the lepton production cross-section through quasielasticand inelastic production processes are comparable. Forenergies Eν ≤ 1.4GeV where the atmospheric-neutrinoenergies are important for fully contained events, the ma-jor contribution comes from the quasielastic events.

The effects 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 (b), we show the momentum dis-tribution dσ/dp and angular distribution dσ/d cos θ 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+ producedin neutrino and antineutrino reactions with νe, νµ, νe andνµ for Eν = 1GeV in fig. 12(a) and (b) in 56Fe nuclei.


0 200 400 600 8000

1

2

3

4

-1 -0.5 0 0.5 10

10

20

30

40

dσ

pl

/ dd

σd

/

(a)

(b)

(10 c

m /M

eV)

-40

2(1

0 c

m )

-38

2co

sθ

(MeV)p

l

l

lθcos

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

vs.

pl and (b) dσd cos θl

vs. cos θl with nuclear effects at E = 1.0GeV

for the reaction νl(νl)+N → ∆+ l−(l+) in 56Fe. The solid lineis the result for νe, the dashed one for νµ, the dotted one forνe and the dash-dotted one for νµ reactions.

Here, we would 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 nuclear in-teractions like multiple scattering and possible absorptionin iron nuclei while passing through the nucleus and an ap-propriate model has to be used for their description. Mod-els developed by Salcedo et al. [40] and Paschos et al. [52]have studied these effects but we do not consider these ef-fects here as they are beyond the scope of this paper. The∆ excitation process in the nuclear medium also gives riseto quasielastic-like events through two-body and three-body absorption processes like ∆N → NN and ∆N →∆NN , where only leptons are present in the final states.The quasielastic-like lepton events have been discussed byKim et al. [21] but no quantitative estimates have beenmade. We have discussed in an earlier paper [51] this pro-cess only qualitatively but make a quantitative estimateof these events in this paper. We find that around Eν =1GeV the contribution of these quasielastic-like events isabout 10–12% but its effect on the flux-averaged produc-

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

>l

p

(MeV)

l

l

< d

/

dp >

(10 cm

/M

eV)

-40

2

p

σ

l

(MeV)

(a)

(b)

σ

Fig. 13. Flux-averaged differential cross-section 〈 dσdpl〉 vs. pl at

the Soudan site simulated by Barr et al. [15]. (a) for νµ+N →∆ + µ− and (b) for νµ +N → ∆ + µ+ reactions in 56Fe. Thesolid line is for total ∆ events, the dashed line is for one π

production events and the dotted line is for quasielastic-likeevents (due to ∆ absorption in the nuclear medium).

tion of leptons for atmospheric neutrinos is not very signif-icant. This is discussed in some detail in the next section.

4 Lepton production by atmospheric

neutrinos

The energy dependences of the quasielastic and in-elastic lepton production cross-sections described insects. 2 and 3 have been used to calculate the leptonproduction by atmospheric neutrinos after averagingover the neutrino flux corresponding to the two sitesof Soudan and Gran Sasso, where iron-based detectorsare being used. There are quite a few calculations ofatmospheric-neutrino fluxes at these two sites. We usethe angle-averaged fluxes calculated by Honda et al. [14]and Barr et al. [15] for the Soudan site and the fluxes ofBarr et al. [15] and Plyaskin [16] for the Gran Sasso siteto calculate the flux-averaged cross-section 〈σ〉 and alsothe momentum and the angular distributions 〈 dσdpl 〉 and

〈 dσd cos θl

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


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 /M

eV )

-40

2d

σ

dp

l

><

p

2

l

//

-40

( 10 c

m /

MeV

)

)

dσ

dp

><

( MeV )

( MeV

lp

Fig. 14. Flux-averaged differential cross-section 〈 dσdpl〉 vs. pl

at the Gran Sasso site simulated by Barr et al. [15]. (a) forνµ +N → ∆+ µ− and (b) for νµ +N → ∆+ µ+ reactions in56Fe. The solid line is for total ∆ events, the dashed line is forone-π production events and the dotted line is for quasielastic-like events (due to ∆ absorption in the nuclear medium.)

4.1 Flux-averaged momentum and angulardistributions

In this section we present the numerical results for theflux-averaged momentum distributions 〈 dσdpl 〉 and angular

distributions 〈 dσd cos θl

〉 for various leptons produced fromνe, νe, νµ, νµ at the atmospheric-neutrino sites of Soudanand Gran Sasso. These leptons are produced through thequasielastic as well as inelastic processes. The quasielasticproduction has been discussed in sect. 2 and the inelas-tic production has been discussed in sect. 3. The inelasticproduction of leptons is accompanied by pions. However,in the nuclear medium where the production of ∆ is fol-lowed by ∆N → NN , ∆NN → NNN absorption pro-cesses, the leptons are produced without pions. These arequasielastic-like events. We show the momentum distribu-tion of the leptons produced in iron nuclei correspondingto one-pion and quasielastic-like events for atmosphericneutrinos relevant to Soudan and Gran Sasso sites. Weshow it for νµ and νµ for the Soudan site in fig. 13(a)and (b) and for the Gran Sasso site in fig. 14(a) and (b)corresponding to the flux of Barr et al. [15]. We see that atboth sites, the production cross-section of quasielastic-likeevents is quite small.

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

/M

eV)

<d /

dp >

(10 c

m /M

eV)

σl

p (MeV)l

p (MeV)l

l

-40

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 lineis for total quasielastic events and the dashed line is for theone-π production events for the νµ reaction in 56Fe. The corre-sponding results for νµ are shown by dotted and dash-dottedlines, (a) for muon-type and (b) for electron-type neutrinos.

We now present our final results for the momentumdistribution and angular distribution for various leptonsproduced by atmospheric neutrinos at Soudan and GranSasso sites in figs. 15-18. In these figures separate con-tributions from the quasielastic and inelastic processes tothe momentum and angular distributions of charged lep-tons are shown explicitly. The quasielastic events are thosewhere only a charge lepton is produced in the final stateeither by the quasielastic process described in sect. 2 or bythe inelastic process of ∆ production followed by its sub-sequent absorption in the nuclear medium as describedin sect. 3. The inelastic events are those events in whicha charged lepton in the final state is accompanied by acharged pion as a decay product of deltas excited in thenuclear medium.

We show the results for the momentum distri-bution 〈dσ/dpl〉 ∼ pl (l = e−, e+, µ+, µ−) for theatmospheric-neutrino fluxes of Barr et al. [15] at theSoudan site in fig. 15 and at Gran Sasso site in fig. 16.We see that in all cases the major contributions to thecharged lepton production comes from the quasielasticprocesses induced by neutrinos (solid line). The contribu-tion from the quasielastic processes induced by antineu-trinos (dotted line) and inelastic processes induced by


0 500 1000 15000

0.5

1

1.5

2

0 500 1000 15000

0.5

1

1.5

2

<d /

dp >

(10 cm

/M

eV)

<d /

dp >

(10 cm

/M

eV)

σ

p (MeV)

p (MeV)

σl

l

-40

2-4

0 2

(a)

(b )

l

l

Fig. 16. Flux-averaged differential cross-section 〈 dσdpl〉 vs. pl at

the Gran Sasso site simulated by Barr et al. [15]. The solid lineis for total quasielastic events and the dashed line is for theone-π production events for the νµ reaction in 56Fe. The corre-sponding results for νµ are shown by dotted and dash-dottedlines, (a) for muon-type and (b) for electron-type neutrinos.

neutrinos (dashed line) is small and is around 20% in thepeak region. The contribution due to inelastic processesinduced by antineutrinos is very small over the whole re-gion (dash-dotted lines).

The peak in quasielastic νµ reactions occurs aroundpl ≈ 200MeV. The peaks in the inelastic νµ and quasielas-tic νµ reactions are slightly shifted towards lower ener-gies. The momentum distribution of the leptons for thequasielastic reaction is peaked more sharply than the mo-mentum distribution of the inelastic reaction.

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

d cos θl〉 vs. cos θ for the

atmospheric-neutrino fluxes of Barr et al. [15] for theSoudan site in fig. 17 and for the Gran Sasso site infig. 18. The inelastic lepton production accompanied by pi-ons (dashed line for µ−(e−) production and dash-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 in allcases. The inelastic distributions due to pion productionare slightly more forward peaked than the quasielastic dis-

-1 -0.5 0 0.5 1

cos θ l

0

2

4

6

8

10

12

14

<d

σ/d

cos

θl>

(1

0-3

8cm

2)

-1 -0.5 0 0.5 1

cos θ l

0

2

4

6

8

10

12

<d

σ/d

cos

θl>

(10

-38

cm

2)

(a)

(b)

Fig. 17. Flux-averaged differential cross-section 〈 dσd cos θl

〉 vs.

cos θl at the Soudan site simulated by Barr et al. [15]. The solidline is for total quasielastic events and the dashed line is for theone-π production events for the νµ reaction in 56Fe. The corre-sponding results for νµ are shown by dotted and dash-dottedlines, (a) for muon-type and (b) for electron-type neutrinos.

tribution. The contributions of the inelastic lepton eventsare small compared to the quasielastic events and the an-gular distributions of the flux-averaged cross-sections aredominated by the quasielastic events.

At the Soudan site, we have also studied the mo-mentum and angular distributions for the atmospheric-neutrino fluxes of Honda et al. [14]. We find that the mo-mentum and angular distributions for the production ofmuons are similar to the distributions obtained for the fluxof Barr et al. [15]. In the case of electron production, theuse of the flux of Honda et al. [14], gives a slightly smallervalue for 〈dσ/dpl〉 and 〈 dσ

d cos θl〉 for electrons as compared

to the flux of Barr et al. [15].

Similarly, at the Gran Sasso site, the momentum andangular distributions for the atmospheric-neutrino fluxesof Plyaskin [16] have also been studied. Here, we also findthat the momentum and angular distributions for the pro-duction of muons are similar to the distributions obtainedfor the flux of Barr et al. [15]. In the case of electron pro-duction, the use of the flux of Plyaskin [16] gives a slightly


Table 2. Ratio R = Rµ/e =Yµ+YµYe+Ye

corresponding to quasielastic, inelastic and total production of leptons (FG refers to theFermi gas model, NM refers to the nuclear model, FN refers to the free nucleon, ∆N refers to ∆ in the nuclear model, ∆Frefers to ∆ free). RF shows the ratio of total lepton yields for muon to electron for the case of the free nucleon and RN showsthe ratio of total yields for muon to electron for the case of nucleon in the nuclear medium.

Site Soudan Soudan Soudan Gran Sasso Gran Sasso

FLUX 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 2.00RFG 1.81 1.66 1.89 1.95 2.09RFN 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 2.03

Total

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

-1 -0.5 0 0.5 1

cos θ l

0

3

6

9

12

15

<d

σ/d

cos

θl>

(10

-38

cm

2)

-1 -0.5 0 0.5 1

cos θ l

0

2

4

6

8

10

<d

σ/d

cos

θl>

(10

-38

cm

2)

(a)

(b)

Fig. 18. Flux-averaged differential cross-section 〈 dσd cos θl

〉 vs.

cos θl at the Gran Sasso site simulated by Barr et al. [15].The solid line is for total quasielastic events and the dashedline is for the one-π production events for the νµ reaction in56Fe. The corresponding results for νµ are shown by dotted anddash-dotted lines, (a) for muon-type and (b) for electron-typeneutrinos.

smaller value for 〈dσ/dpl〉 and 〈 dσd cos θl

〉 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 at the Gran Sasso site for muons aswell as for electrons.

4.2 Flux-averaged total cross-sections and leptonyields

The total cross-sections for the production of leptons andits energy dependence have been discussed in sect. 2 andsect. 3 for quasielastic and inelastic reactions. In this sec-tion we calculate the lepton yields Yl for lepton of flavor lwhich we define as

Yl =

∫

Φνl σ(Eνl) dEνl ,

where Φνl is the atmospheric-neutrino flux of νl andσ(Eνl) is the total cross-section for neutrino νl of energyEνl . We calculate this yield separately for the quasielasticand inelastic events. We define a relative yield of muon-

over electron-type events by R as R = Rµ/e =Yµ+YµYe+Ye

forquasielastic and inelastic events and present our results intable 2. We study the nuclear model dependence as wellas the flux dependence of the relative yield R.

The results for R are presented separately forquasielastic events, inelastic events and the total events intable 2. For quasielastic events νl(νl)+

56Fe→ l−(l+)+X,the results are presented for the case of the free nu-cleon by RFN , for the nuclear case with the 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 ,


Table 3. % ratio r = Y∆Yq.e.+∆

(N refers to the nuclear model, F refers to free case).

Sites Soudan Soudan Soudan Gran Sasso Gran Sasso

FLUX 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

RFG and RFN for the same fluxes at each site). Thisis also true for the inelastic production of leptons, i.e.

νl(νl)+56Fe → l−(l+) + π+(π−) +X for which the ratio

for the free-nucleon case (denoted by R∆F ) and the ra-tio for the nuclear case in our model (denoted by R∆N )are presented in table 2. It is, therefore, concluded thatthere is no appreciable nuclear-model dependence on theratio of total lepton yields for the production of muonsand electrons (compare the values of RF and RN , whereRF shows the ratio of total lepton yields for muon andelectron for the case of the free nucleon and RN shows theratio of total yield for muon and electron for the case ofthe nucleon in the nuclear medium).

However, there is some dependence of the ratio R onthe atmospheric-neutrino fluxes. The flux dependence ofR can be readily seen from table 2, for the two sitesof Soudan and Gran Sasso. At the Gran Sasso site, wesee that there is 4–5% difference in the value of RN forthe total lepton yields for the fluxes of Barr et al. [15]and Plyaskin [16]. At the Soudan site, the results for thefluxes of Honda et al. [14] and Barr et al. [15] are within4–5% but the flux calculation of Plyaskin [16] gives a re-sult which is about 10–11% smaller than the results ofHonda et al. [14] and 7–8% smaller than the results ofBarr et al. [15]. The flux dependence is mainly due to thequasielastic events. This should be kept in mind while us-ing the flux of Plyaskin [16] for making any analysis of theneutrino oscillation experiments.

In table 3, 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 tothe quasielastic and inelastic events, i.e. Yl = Yl

q.e. +Yl

q.e. + Yl∆ + Yl

∆. The relative yield for the case of thefree nucleon is shown by rl(F ) and for the case with thenuclear effects in our model is shown by rl(N). We see thatfor free nucleons, the relative yield of the inelastic eventsdue to ∆ excitation is in the range 12–15% for variousfluxes at the two sites. The ratio is approximately thesame for electrons and muons. When the nuclear effectsare taken into account this becomes 19–22%. This is dueto the different nature of the effect of nuclear structureon the quasielastic and inelastic production cross-sectionswhich 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].

5 Conclusions

We have studied the charged lepton production in iron in-duced by atmospheric neutrinos at the experimental sitesof Soudan and Gran Sasso. The energy dependence ofthe total cross-sections for the quasielastic and inelasticprocesses have been calculated in a nuclear model whichtakes into account the effect of the Pauli principle, Fermimotion effects and the renormalization of weak transitionstrengths in the nuclear medium. The inelastic processhas been studied through the ∆-dominance model whichincorporates the modification of mass and width of the∆-resonance in the nuclear medium. The numerical re-sults for the momentum and angular distributions of thecharged lepton production cross-section have been pre-sented for muons and electrons. The relative yield of muonto electron production has been studied. In addition tothe nuclear-model dependence, the flux dependence of thetotal yields, momentum distribution dσ

dpland dσ

d cos θlhave

been also studied. In the following we conclude this paperby summarizing our main results:

1. There is a large reduction due to nuclear ef-fects in the total cross-section for quasielastic productioncross-section specially at lower energies (40–50% around200 MeV) and the reduction becomes smaller at higherenergies (15–20% around 500 MeV).

2. For quasielastic reactions we find a larger reduc-tion in the total cross-section as compared to the Fermigas model. The energy dependence of this reduction incross-section at low energies (E < 500MeV) is found tobe different 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 the increase of energy. The effect of thenuclear 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 the Coulombdistortion 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 lep-tons produced by atmospheric neutrinos is peaked aroundthe momentum pl ∼ 200MeV for electrons and muons.The peak for the quasielastic production is sharper thanfor the inelastic production. The inelastic production of


leptons contributes about 20% to the total production ofleptons.

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

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

8. The nuclear-model dependence of the relative yieldof muons to electrons is negligible, even though the in-dividual yields for muons and electrons are reduced withthe inclusion of nuclear effects specially for the quasielasticproduction.

The work is financially supported by the Department of Scienceand Technology, Government of India under the grant DSTProject No. SP/S2K-07/2000. One of the authors (S. Ahmad)would like to thank CSIR for financial support.

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Charged lepton production from iron induced by atmospheric neutrinos

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