PRCPRVCANPhysical Review CPhys. Rev. C2469-99852469-9993American Physical Society10.1103/PhysRevC.98.065503ARTICLESElectroweak Interaction, SymmetriesCharged-current deep-inelastic scattering of muon neutrinos (νμ) off Fe56CHARGED-CURRENT DEEP-INELASTIC SCATTERING OF …GROVER, SARASWAT, SHUKLA, AND SINGHGroverDeepika^{1}^{*}SaraswatKapil^{1}^{†}ShuklaPrashant^{2,3}^{‡}SinghVenktesh^{1}^{§}Department of Physics, Institute of Science, Banaras Hindu University, Varanasi 221005, IndiaNuclear Physics Division, Bhabha Atomic Research Centre, Mumbai 400085, IndiaHomi Bhabha National Institute, Anushakti Nagar, Mumbai 400094, India

dgroverbhu@gmail.com

kapilsaraswatbhu@gmail.com

pshuklabarc@gmail.com

venkaz@yahoo.com

13December2018December20189860655031August2018Published by the American Physical Society2018authorsPublished by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Funded by SCOAP^{3}.

In this paper, we study charged-current deep-inelastic scattering of muon neutrinos off Fe56 nuclei by using the Hirai, Kumano, and Saito model. The LHA parton distribution functions (PDFs) - CT10 are used to describe the partonic content of hadrons. Modification of PDFs inside the nuclei is done by using EPPS16 parametrization at next-to-leading order. Target mass correction has also been incorporated in the calculations. We calculate the structure functions [F2(x,Q2) and xF3(x,Q2)], the ratios [R2(x,Q2)=F2Fe56/F2Nucleon and R3(x,Q2)=F3Fe56/F3Nucleon], and the differential cross sections of νμ deep-inelastic scattering off nucleons and Fe56 nuclei. We compare the results obtained with measured experimental data. The present theoretical approach gives a good description of data.

Department of Science and Technology, Ministry of Science and Technology10.13039/501100001409INTRODUCTION

In the standard model of physics, neutrinos are elementary particles with no electric charge, no magnetic moment, half-integral spin, and zero mass. However, several neutrino oscillation experiments [1–10] across the globe have confirmed that neutrinos oscillate from one flavor to another, leading to a small but nonzero neutrino mass and the possibility to go beyond the standard model. Being electrically neutral, neutrinos rarely interact with matter via the weak force. Neutrino interactions are classified into two categories: Charged current (CC) interactions via the exchange of W+/W− bosons and neutral current (NC) interactions via the exchange of Z bosons. There are many neutrino-scattering processes such as quasi-elastic-scattering (QES) [11], resonance pion production (RES) [12], and deep-inelastic scattering (DIS), at various neutrino energies; for a review see Refs. [13,14]. Low neutrino energies are dominated by QES whereas RES dominates at medium neutrino energies. As the neutrino energies become larger, DIS becomes more and more dominant [15]. In this scattering process, a highly energetic neutrino scatters off a quark in the nucleon producing a corresponding lepton, and many hadrons are produced:
νμ+N→μ−+XCC,νμ+N→νμ+XNC.

DIS is an important experimental tool for studying the hadronic matter where the final-state particles produced in the scattering are analyzed to probe hadronic properties. Several experiments planned worldwide, such as NuTeV [16], CHORUS [17], NOMAD [18], MINOS [19], MINERvA [20], etc., have analyzed neutrino deep-inelastic scattering off different targets to measure differential and integrated cross sections and structure functions. A review of the results from various experiments probing neutrino deep-inelastic scattering in presented in Ref. [21].

To describe the partonic content of the hadron, precise parton distribution functions (PDFs) are required. These PDFs are produced by several different groups, such as MRST [22–24], CTEQ [25], Alekhin [26,27], ZEUS [28], etc. PDFs are derived from fitting DIS and related hard-scattering data by using parametrization at low Q02 [1–7 (GeV/c)2] and evolving these to higher Q2. These PDFs are presented as grids in x-Q2 with codes given by PDF authors. The Les Houches Accord PDFs (LHAPDF) provides C++ code for these PDFs with an interpolation grid build into the pdflib[29]. We use LHAPDF (CT10) [30] parton distribution functions.

In this work, we study charged current νμ—nucleon and νμ—Fe56 deep-inelastic scattering by using the Hirai, Kumano, and Saito model [31]. Calculations of the structure functions [F2(x,Q2) and xF3(x,Q2)], the ratios [R2(x,Q2)=F2Fe56/F2Nucleon and R3(x,Q2)=F3Fe56/F3Nucleon], and the differential cross sections are presented and compared with the measured experimental data.

FORMALISM FOR DEEP-INELASTIC <inline-formula><mml:math><mml:mrow><mml:mi>ν</mml:mi><mml:mtext>-</mml:mtext><mml:mi>N</mml:mi></mml:mrow></mml:math></inline-formula> AND <inline-formula><mml:math><mml:mrow><mml:mi>ν</mml:mi><mml:mtext>-</mml:mtext><mml:mi>A</mml:mi></mml:mrow></mml:math></inline-formula> SCATTERING

The neutrino-nucleon (antineutrino-nucleon) deep-inelastic scattering process is
νlk+NP→l−(k′)+X(P′),ν¯l(k)+N(P)→l+(k′)+X(P′),where neutrino or antineutrino with four-momentum k=(ε,k⃗) scatters off a nucleon N with four-momentum Pμ=(E,P⃗) and E=(M2+P⃗2)1/2. The outgoing lepton l− or l+ (not neutrino) has four-momentum k′=(ε′,k′⃗). The hadronic final state X is left with a four-momentum P′=(E′,P′⃗). The schematic diagram of charged current ν-N deep-inelastic scattering is shown in Fig. 1.
10.1103/PhysRevC.98.065503.f11

Charged current ν-N deep-inelastic scattering.

Neutrino CC (charged current) interactions with nucleon are described by the matrix elements [31]M=GF2MW2MW2+Q2u¯(k′,λ′)×γμ(1−γ5)u(k,λ)〈X|JμCC(0)|P,λN〉,where GF is the Fermi coupling constant, MW is the W mass, Q2 is given by Q2=−q2=(k−k′)2 with the four-momentum transfer q, k (λ), and k′ (λ′) indicate initial and final lepton momenta (spins), P (λN) is the nucleon momentum (spin), and JμCC(0) is the weak charged current (CC) of the nucleon. The absolute value square |M|2 is calculated with an average over the nucleon spin for obtaining the differential cross section.

The neutrino-nucleon (antineutrino-nucleon) charged current differential scattering cross section is defined as [31]d2σCCνν¯dxdy=GF2MNEνπMW2MW2+Q22[F1(x,Q2)xy2+F2(x,Q2)1−y−MNxy2Eν±F3(x,Q2)xy1−y2],where ± indicates + for ν and − for ν¯, x is the Bjorken scaling variable defined as x=Q2/(2P·q), y is the inelasticity defined as y=P·q/(P·k), so Q2=2MNEνxy, Eν is the neutrino-beam energy, and MN is the nucleon mass. F1(x,Q2), F2(x,Q2), and F3(x,Q2) are the dimensionless structure functions. The Bjorken variable x and the inelasticity y are in the range 0≤x, y≤1.

In the Bjorken limit of scaling in the asymptotic region i.e., Q2→∞, x is finite. The structure functions Fi(x,Q2) are not Q2 dependent but depend only on x and satisfy the Callan–Gross relation [32]:
F2x=2xF1x.Using the Callan–Gross relation, the differential cross section can be expressed in terms of F2 and F3. In the quark parton model (QPM), F2(x,Q2) and F3(x,Q2) are determined in terms of PDFs for quarks and antiquarks.

The structure function F3(x,Q2) is defined as
F3N(x,Q2)=(ux,Q−u¯x,Q+dx,Q−d¯x,Q+sx,Q−s¯x,Q+cx,Q−c¯x,Q+bx,Q−b¯x,Q+tx,Q−t¯x,Q).

The structure function F2(x,Q2) is defined as
F2N(x,Q2)=x(ux,Q+u¯x,Q+dx,Q+d¯x,Q+sx,Q+s¯x,Q+cx,Q+c¯x,Q+bx,Q+b¯x,Q+tx,Q+t¯x,Q).

NUCLEAR MODIFICATIONS

There are some nuclear effects [33] in neutrino-nucleus deep-inelastic scattering processes. These effects were first pointed out in 1982 by the EMC collaboration at CERN, where they measured the ratio of the iron [F2A(x,Q2)] to deuterium [F2D(x,Q2)] structure functions, and found the results to be different from unity [34]. Several efforts since then have explored these effects in neutrino-nucleus DIS processes with the conclusion that, for a Bjorken variable x<0.1, the ratio is suppressed and the suppression increases with increase in the mass number of the target nucleus. This suppression is called shadowing effect. For 0.1<x<0.3, the ratio is more than unity. This increase in the ratio is called antishadowing effect. For 0.3<x<0.8, the ratio is again suppressed and this suppression is called EMC effect. For x>0.8, the ratio increases rapidly and this rapid increase in the ratio is due to the Fermi motion effect. For a review, see Ref. [35].

The neutrino-nucleus (antineutrino-nucleus) charged current differential scattering cross section is defined as [36]d2σCCνν¯AdxAdyA=GF2MAEνπMW2Q2+MW22[yA2xAF1νν¯A+1−yA−MAxAyA2EνF2νν¯A±xAyA1−yA2F3νν¯A],where MA is the mass of nucleus A. F1A, F2A, and F3A are the structure functions for the nucleus. The Bjorken variable in the nucleus is xA=x/A, where A is the mass number of the nucleus. The inelasticity in the nucleus is yA=y[36].

To correct for the nuclear effects in the structure functions F2A and F3A, we use the EPPS16 package [37]. EPPS16 is a package to obtain next-to-leading order (NLO) nuclear partonic distribution functions (nPDFs). The bound nucleon PDFs fiA(x,Q2) for each parton flavor i are given as
fiA(x,Q2)=RiA(x,Q2)fiCT10(x,Q2),where RiA(x,Q2) are the nuclear corrections to the free nucleon PDFs fiCT10(x,Q2). EPPS16 provides parametrization only in the kinematical domain 1e−7≤x≤1 and 1.3≤Q≤10000 GeV.

The neutrino structure function F3A(x,Q2) on nucleus A using EPPS16 [37] is calculated as
F3A(x,Q2)=uAx,Q−u¯Ax,Q+dAx,Q−d¯Ax,Q+sAx,Q−s¯Ax,Q+cAx,Q−c¯Ax,Q+bAx,Q−b¯Ax,Q+tAx,Q−t¯Ax,Q.

The neutrino structure function F2A(x,Q2) on nucleus A using EPPS16 [37] is calculated as
F2A(x,Q2)=x(uAx,Q+u¯Ax,Q+dAx,Q+d¯Ax,Q+sAx,Q+s¯Ax,Q+cAx,Q+c¯Ax,Q+bAx,Q+b¯Ax,Q+tAx,Q+t¯Ax,Q).

Target-mass correction

The target-mass correction (TMC) can be taken into account when partonic distribution functions evaluate at the Nachtmann variable ξ[38] rather than the Bjorken variable x as
ξ=2x1+1+4MN2x2Q2.At high Q2 (Q2≫MN2), ξ is equivalent to x.

RESULTS AND DISCUSSIONS

We calculated the structure functions F2(x,Q2) and xF3(x,Q2) for Fe56 with EPPS16 nuclear corrections [37] at next-to-leading order (NLO) and LHAPDF (CT10) parton distribution functions [30]. Figures 2 and 3 show present calculations of F2(x,Q2) as a function of the square of momentum transfer Q2 for different values of the Bjorken variable x (0.045, 0.080, 0.125, 0.175, 0.225, 0.275, 0.35, 0.45, 0.55, 0.65). Figures 4 and 5 show the present calculations of the structure function xF3(x,Q2) as a function of the square of momentum transfer Q2 for different values of the Bjorken variable x. In Figs. 2 and 4, panel (a) is for x=0.045, panel (b) is for x=0.080, panel (c) is for x=0.125, panel (d) is for x=0.175, panel (e) is for x=0.225, and panel (f) is for x=0.275. In Figs. 3 and 5, panel (a) is for x=0.35, panel (b) is for x=0.45, panel (c) is for x=0.55, and panel (d) is for x=0.65. The black solid lines show the calculations with the inclusion of shadowing effect whereas the red dashed lines show the calculations without the inclusion of shadowing effect. The obtained results are compared with measured data of the CDHSW [39] and CCFR [40] experiments. We can see that, for higher values of the Bjorken variable x, the present theoretical approach gives a good description of the data. Figures 3(a) and 5(a) show an excellent agreement between theoretical calculations and data for x=0.35.
10.1103/PhysRevC.98.065503.f22

F2(x,Q2) with EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30] for Fe56 as a function of Q2 for different values of Bjorken variable x. The results are compared with measured data of CDHSW [39] and CCFR [40] experiments.

10.1103/PhysRevC.98.065503.f33

F2(x,Q2) with EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30] for Fe56 as a function of Q2 for different values of Bjorken variable x. The results are compared with measured data of CDHSW [39] and CCFR [40] experiments.

10.1103/PhysRevC.98.065503.f44

xF3(x,Q2) with EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30] for Fe56 as a function of Q2 for different values of the Bjorken variable x. The results are compared with measured data of the CDHSW [39] and CCFR [40] experiments.

10.1103/PhysRevC.98.065503.f55

xF3(x,Q2) with EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30] for Fe56 as a function of Q2 for different values of the Bjorken variable x. The results are compared with measured data of the CDHSW [39] and CCFR [40] experiments.

Figure 6(a) shows the present calculations of the ratio R2(x,Q2)=F2Fe56/F2Nucleon as a function of Bjorken variable x for Q2=5.0GeV2 and Q2=50.0GeV2. Figure 6(b) shows present calculations of ratio R3(x,Q2)=F3Fe56/F3Nucleon as a function of Bjorken variable x for Q2=5.0GeV2 and Q2=50.0GeV2. The broken lines show the effect of target mass correction [38]. We can clearly see the effects discussed in Sec. III.
10.1103/PhysRevC.98.065503.f66

Ratios (a) R2(x,Q2)=F2Fe56/F2Nucleon and (b) R3(x,Q2)=F3Fe56/F3Nucleon as a function of Bjorken variable x with EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30] for Q2=5.0GeV2 and Q2=50.0GeV2.

We calculated 1Ed2σdxdy for Fe56 as a function of inelasticity y, for different values of Bjorken variable x and for different neutrino energies. Figures 7 and 8 show the present calculations of 1Ed2σdxdy for Eνμ=65 GeV. Figures 9 and 10 show the calculations of 1Ed2σdxdy for Eνμ=110 GeV. Figures 11 and 12 show the similar calculations of 1Ed2σdxdy for Eνμ=190 GeV. The results obtained are compared with measured data of CCFR [41] and CDHSW [41] experiments. In Figs. 7, 9, and 11, panel (a) is for x=0.045, panel (b) is for x=0.080, panel (c) is for x=0.125, panel (d) is for x=0.175, panel (e) is for x=0.225, and panel (f) is for x=0.275. In Figs. 8, 10, and 12, panel (a) is for x=0.35, panel (b) is for x=0.45, panel (c) is for x=0.55, and panel (d) is for x=0.65. The black solid lines show the calculations with the inclusion of shadowing effect, the red dashed lines show the calculations without the inclusion of shadowing effect, and the pink dotted lines show the calculations with the inclusion of shadowing effect and target-mass correction. We can see that the target-mass correction has negligible effect at low values of the Bjorken variable x, but the effect increases with the increasing values of x. Also, TMC effects are seen for low values of inelasticity y. Here also we can see that, with the increasing values of x, theoretical calculations show a better agreement with data.
10.1103/PhysRevC.98.065503.f77

1Ed2σdxdy for Fe56 as a function of inelasticity y, for different values of the Bjorken variable x and Eνμ=65 GeV using EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30]. Results are compared with measured data of the CCFR [41] and CDHSW [41] experiments.

10.1103/PhysRevC.98.065503.f88

1Ed2σdxdy for Fe56 as a function of inelasticity y, for different values of the Bjorken variable x and Eνμ=65 GeV using EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30]. Results are compared with measured data of the CCFR [41] and CDHSW [41] experiments.

10.1103/PhysRevC.98.065503.f99

1Ed2σdxdy for Fe56 as a function of inelasticity y, for different values of the Bjorken variable x and Eνμ=110 GeV using EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30]. Results are compared with measured data of the CCFR [41] and CDHSW [41] experiments.

10.1103/PhysRevC.98.065503.f1010

1Ed2σdxdy for Fe56 as a function of inelasticity y, for different values of Bjorken variable x and Eνμ=110 GeV using EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30]. Results are compared with measured data of the CCFR [41] and CDHSW [41] experiments.

10.1103/PhysRevC.98.065503.f1111

1Ed2σdxdy for Fe56 as a function of inelasticity y, for different values of Bjorken variable x and Eνμ=190 GeV using EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30]. Results are compared with measured data of the CCFR [41] and CDHSW [41] experiments.

10.1103/PhysRevC.98.065503.f1212

1Ed2σdxdy for Fe56 as a function of inelasticity y, for different values of Bjorken variable x and Eνμ=190 GeV using EPPS16 nuclear corrections [37] at NLO and LHAPDF (CT10) parton distribution functions [30]. Results are compared with measured data of the CCFR [41] and CDHSW [41] experiments.

CONCLUSION

We presented the structure functions [F2(x,Q2) and xF3(x,Q2)], the ratios [R2(x,Q2)=F2Fe56/F2Nucleon and R3(x,Q2)=F3Fe56/F3Nucleon] and the differential cross sections for charged current νμ—nucleon and νμ—Fe56 deep-inelastic scattering by using a formalism based on the Hirai, Kumao, and Saito model. We used the LHAPDF (CT10) parton distribution functions. Nuclear corrections inside the nuclei are applied to the PDFs at next-to-leading order by using EPPS16 parametrization. We also incorporated target-mass correction into our calculations. We studied the behavior of the structure functions F2(x,Q2) and xF3(x,Q2) as a function of the square of momentum transfer Q2 for different values of the Bjorken variable x. Differential cross sections are analyzed as a function of inelasticity y for different values of the Bjorken variable x and for different neutrino energies. The results obtained have been compared with measured experimental data. The present theoretical approach gives a good description of data. The agreement between theoretical calculations and data is even better for higher values of the Bjorken variable x.

ACKNOWLEDGMENTS

The authors are thankful to Department of Science and Technology (DST), New Delhi, Government of India for the financial support needed to pursue this work.

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