Effects of nuclear medium and nonisoscalarity in extracting sin^{2}\theta_{W} using Paschos-Wolfenstein relation

# Effects of nuclear medium and nonisoscalarity in extracting sin2θW using Paschos-Wolfenstein relation

H. Haider Department of Physics, Aligarh Muslim University, Aligarh-202 002, India    I. Ruiz Simo Departamento de Física Atómica Molecular y Nuclear, Universidad de Granada, E-18071 Granada, Spain Dipartimento di Fisica, Università degli studi di Trento, I-38123 Trento, Italy    M. Sajjad Athar Department of Physics, Aligarh Muslim University, Aligarh-202 002, India
###### Abstract

We study nuclear medium effects and nonisoscalarity correction in the extraction of weak mixing angle sin using Paschos-Wolfenstein(PW) relation. The calculations are performed for the iron nucleus. The nuclear medium effects like Fermi motion, binding, shadowing and antishadowing corrections and pion and rho meson cloud contributions have been taken into account. Calculations have been performed in the local density approximation using a relativistic nuclear spectral function which includes nucleon correlations. The results are discussed along with the experimental result inferred by the NuTeV Collaboration. These results may be useful for the ongoing MINERA experiment as well as for the proposed NuSOnG experiment.

###### pacs:
13.15.+g, 24.10.-i, 24.85.+p, 25.30.-c, 25.30.Mr, 25.30.Pt

## I Introduction

MINERminerva1 () is presently taking data using neutrinos from NuMI Lab., and their aim is to perform cross section measurements in the neutrino energy region of 1-20 GeV with different nuclear targets like helium, carbon, oxygen, iron and lead. Among the various goals of the MINERA experiment, one of them is to measure in the deep-inelastic scattering (DIS) region using these nuclear targets, obtaining the ratio of the structure functions between the different target materials and also to study structure functions in the DIS and transition region minerva2 (); minerva3 (). Neutrino Scattering On Glass (NuSOnG) nusong () is another experiment proposed at Fermi lab to study the neutrino and antineutrino charged current deep-inelastic scattering events to precisely measure the structure functions , , etc. Furthermore, they plan to measure from -nucleon scattering using the Paschos-Wolfenstein(PW) relation Paschos (), a similar type of analysis was performed by the NuTeV group Zeller:Thesis (); Zeller:2001hh (); McFarland (). Recently, we studied nuclear medium effects on the electromagnetic structure function (,Sajjad1 (), and the weak structure functions (,) and (,Sajjad2 (); Sajjad3 (). For (,Sajjad1 (), we found that our results are reasonably in good agreement with the recent results from JLab Seely () as well as with some of the older experiments like SLAC Gomez (). In the case of deep inelastic scattering induced processes, the results were compared with the available data from NuTeV, CDHSW and CHORUS experiments Berge (); Tzanov (); Onengut () for the weak structure functions (,) and (,) in iron Sajjad2 () and lead Sajjad3 (), as well as with the results of the differential scattering cross section . In this work, we study the effect of nuclear medium and nonisoscalarity correction in extracting using PW relation taking iron as the nuclear target.

Paschos and Wolfenstein Paschos () suggested that the ratio of neutral current to charged current neutrino interaction cross sections on nucleon targets may be used to measure :

 RPW=σ(νμ N→νμ X) − σ(¯νμ N→¯νμ X)σ(νμ N→μ− X) − σ(¯νμ N→μ+ X)=12 − sin2θW (1)

The above relation is valid when there is no contribution from heavy quarks, neglecting the charm quark mass, assuming the strange quark and anti-strange quark symmetry, considering no medium effect and no contribution from outside the standard model. This equation is valid for both the total cross section as well as differential cross sections , because the neutral current (NC) differential cross section can be expressed in terms of the charged current (CC) ones, and these cancel out in the quotient Bilenky (). For the total cross sections, the Paschos-Wolfenstein (PW) relationship is true under more general assumptions (see for instance Paschos (), Bilenky () and McFarland:2003jw ()).

The above relation is also valid for an isoscalar nuclear target(N=Z) for both the total cross sections and differential cross sections and the above Eq. (1) may be written as

 RPW=σ(νμ A→νμ X) − σ(¯νμ A→¯νμ X)σ(νμ A→μ− X) − σ(¯νμ A→μ+ X) (2)

where is the neutral current induced neutrino(antineutrino) cross section, is the charged current induced neutrino(antineutrino) cross section. The condition of pure isoscalarity includes the requirement of the cancellation of different strong interaction effects which also include the nuclear medium effects in the ratio of the neutral current to the charged current scattering cross sections.

NuTeV Collaboration Zeller:Thesis (); Zeller:2001hh (); McFarland () has measured the ratio R of neutral current to charged current total cross sections in iron, for which they took the ratio of charged current antineutrino to neutrino cross sections i.e. as , and obtained the value for the weak mixing angle using Eq. (2) as

 RPW = σ(νμ A→νμ X)σ(νμ A→μ− X)−σ(¯νμ A→¯νμ X)σ(¯νμ A→μ+ X)σ(¯νμ A→μ+ X)σ(νμ A→μ− X)1−σ(¯νμ A→μ+ X)σ(νμ A→μ− X) (3) = Rν − r R¯ν1−r

where and  Zeller:Thesis (); Zeller:2001hh (). The reported value of is  Zeller:Thesis (); Zeller:2001hh (); McFarland () which is 3 standard deviations above the global fit of  Abbaneo:2001ix () and this is known as NuTeV anomaly. To resolve this anomaly, explanations within and outside the standard model of electroweak interactions have been looked for Davidson:2001ji ()-Thomas2 ().

Paschos and Wolfenstein Paschos () relation is valid for an isoscalar target while iron is a nonisoscalar target(N=30,Z=26), therefore, nonisoscalar corrections are required. Furthermore, nuclear dynamics may also play an important role in the case of neutrino nucleus scattering. Various corrections made by the NuTeV Collaboration have been discussed in literature, but still the reported deviation could not be accounted for McFarland (). Theoretically, Kulagin Kulagin () has investigated the effect of nuclear medium on the PW ratio and pointed out that the shadowing effect being a low x and low phenomenon is small in the region of NuTeV experiment Zeller2002 () and observed the effects of Fermi motion and binding energy correction to be small but significant isoscalar correction. KumanoKumano:2002ra () in a phenomenological analysis pointed out the difference between the nuclear effects in the valence and quark distributions may be a reason for this anomaly. However, this effect is too small to explain the anomaly. Recently Thomas Thomas2 () has discussed various possible corrections and concluded that charge symmetry violation and the isovector EMC effect together may explain this anomaly.

In the present work, we used the results of our earlier study of nuclear medium and nonisoscalarity correction on the weak structure functions and the differential scattering cross sections Sajjad2 (); Sajjad3 (), on the extraction of using PW relation. We have obtained the modified PW relation for a nonisoscalar nuclear target. This study has been performed using a relativistic nucleon spectral function FernandezdeCordoba:1991wf (); FernandezdeCordoba:1995pt (), which is used to describe the momentum distribution of nucleons in the nucleus. We define everything within a field-theoretical approach where nucleon propagators are written in terms of this spectral function. The spectral function has been calculated using the Lehmann’s representation for the relativistic nucleon propagator and nuclear many body theory is used for calculating it for an interacting Fermi sea of nuclear matter. The local density approximation is then applied to translate these results to finite nuclei Sajjad1 (); Sajjad2 (); Sajjad3 (); Marco:1995vb (); Sajjad (). The contributions of the pion and rho meson clouds are taken into account in a many-body field-theoretical approach which is based on Refs. Marco:1995vb (); GarciaRecio:1994cn (). We have taken into account the target mass correction (TMC) following Ref. schienbein (), which has a significant effect at low , and at moderate and high Bjorken . To take into account the shadowing effect, which is important at low and low , and which modulates the contribution of pion and rho cloud contributions, we have followed the works of Kulagin and Petti Kulagin1 (); Petti (). All the formalism is the same for neutral current scattering as done in the case of charged current neutrino/antineutrino induced reactions. For the numerical calculations, parton distribution functions for the nucleons have been taken from the parametrization of CTEQ Collaboration (CTEQ6.6) cteq ().

The paper is organised as follows. In Sec. II, we present the formalism where we write the expression for the -nucleon differential scattering cross section in subsection-II.1, the expressions for -nucleus differential scattering cross section for the isoscalar as well as nonisoscalar nuclear targets are given in subsection-II.2. In subsection-II.3, we explicitly show the construction of the nuclear hadronic tensor for non-symmetric nuclear matter and in subsection-II.4, the nuclear corrections to PW ratio are presented. In Sec. III, we present and discuss the results of our calculations and finally our conclusions are summarised in Sec. IV.

## Ii Formalism

### ii.1 Deep Inelastic Neutrino Nucleon Scattering

The cross section for the charged current(CC) neutrino interaction with a nucleon target i.e.

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

is given by:

 σ = 1vrel2mν2Eν(k)2M2E(p)∫d3k′(2π)32ml2El(k′)N∏i=1∫d3p′i(2π)3∏lϵf(2M′l2E′l)∏jϵb(12ω′j)¯∑∑|T|2 (5) ×  (2π)4δ4(p+k−k′−N∑i=1p′i)

where stands for fermions and for bosons in the final state . The index is split into and for fermions and bosons respectively, four momentum of the particles involved in the process are represented as (incomig neutrino), (outgoing lepton), (target nucleon) and (jet of hadrons).

is the invariant matrix element for the above reaction and is, written as

 −iT=(iGF√2)¯ul(k′)γα(1−γ5)uν(k)(m2Wq2−m2W)⟨X|Jα|N⟩. (6)

After performing the phase space integration in Eq. (5), the double differential scattering cross section evaluated for a nucleon target in its rest frame is expressed as,

 d2σNν,¯νdΩ′dE′=GF2(2π)2|k′||k|(m2Wq2−m2W)2Lαβν,¯νWNαβ, (7)

where is the Fermi coupling constant, is the mass of the W boson, is a lepton, is the four momentum transfer and refer to the outgoing lepton. is a nucleon, is a jet of n hadrons consisting of fermions(f) and bosons(b) in the final state.

The leptonic tensor for antineutrino(neutrino) scattering is given by

 Lαβ=kαk′β+kβk′α−k.k′gαβ±iϵαβρσkρk′σ, (8)

and the hadronic tensor is defined as

 WNαβ = 12π¯∑sN∑X∑sin∏i=1∫d3p′i(2π)3∏lϵf(2M′l2E′l)∏jϵb(12ω′j)⟨X|Jα|N⟩⟨X|Jβ|N⟩∗ (9) ×  (2π)4δ4(p+q−n∑i=1p′i),

where the spin of the nucleon and the spin of the fermions in . In the case of antineutrino is replaced by .

The most general form of the hadronic tensor is expressed as:

 WNαβ= (qαqβq2−gαβ)Wν(¯ν)1+1M2(pα−p.qq2qα)(pβ−p.qq2qβ)Wν(¯ν)2−i2M2ϵαβρσpρqσWν(¯ν)3+ (10) 1M2qαqβWν(¯ν)4+1M2(pαqβ+qαpβ)Wν(¯ν)5+iM2(pαqβ−qαpβ)Wν(¯ν)6,

where is the nucleon mass and are the structure functions, which depend on the scalars and . The terms depending on , and in Eq. (10) do not contribute to the cross section in Eq. (7) in the limit of lepton mass .

In terms of the Bjorken variables and y defined as

 x=Q22Mν,y=νEν,Q2=−q2,ν=p.qM, (11)

are expressed in terms of dimensionless structure functions

 Fν(¯ν)1(x,Q2) = MWν(¯ν)1(ν,Q2) Fν(¯ν)2(x,Q2) = νWν(¯ν)2(ν,Q2) Fν(¯ν)3(x,Q2) = νWν(¯ν)3(ν,Q2). (12)

The expression of the differential cross section, for deep inelastic scattering (DIS) of neutrino with a nucleon target induced by charged current reaction is now given by:

 d2σν(¯ν)dx dy = G2FMEνπ(1+Q2/M2W)2((y2x+m2ly2EνM)F1(x,Q2)+[(1−m2l4E2ν)−(1+Mx2Eν)y]F2(x,Q2) ± [xy(1−y2)−m2ly4EνM]F3(x,Q2))

In the above equation +sign(-sign) in the coefficient with is for neutrino(antineutrino). and are related by the Callan-Gross relation, leading to only two independent structure functions and . For l=e, we take and assume .

The nucleon structure functions are determined in terms of parton distribution functions for quarks and anti-quarks given by

 Fνp2 = 2x[d(x)+s(x)+¯u(x)+¯c(x)]    F¯νp2=2x[u(x)+c(x)+¯d(x)+¯s(x)] Fνn2 = 2x[u(x)+s(x)+¯d(x)+¯c(x)]    F¯νn2=2x[d(x)+c(x)+¯u(x)+¯s(x)] xFνp3 = 2x[d(x)+s(x)−¯u(x)−¯c(x)]    xFνn3=2x[u(x)+s(x)−¯d(x)−¯c(x)] xF¯νp3 = 2x[u(x)+c(x)−¯d(x)−¯s(x)]    xF¯νn3=2x[d(x)+c(x)−¯u(x)−¯s(x)] (14)

For the neutral current(NC) induced reaction

 νl(¯νl)(k)+N(p)→νl(k′)+X(p′), l= e, μ, τ (15)

the expression of the cross section(II.1) modifies by changing , the mass of boson and the corresponding NC structure functions are given by

 FNC2(νp,¯νp) = 2x((u2L+u2R)[u+c+¯u+¯c]+(d2L+d2R)[d+s+¯d+¯s]) (16) xFNC3(νp,¯νp) = 2x((u2L−u2R)[u+c−¯u−¯c]+(d2L−d2R)[d+s−¯d−¯s])

for the proton target and

 FNC2(νn,¯νn) = 2x((u2L+u2R)[d+c+¯d+¯c]+(d2L+d2R)[u+s+¯u+¯s]) (17) xFNC3(νn,¯νn) = 2x((u2L−u2R)[d+c−¯d−¯c]+(d2L−d2R)[u+s−¯u−¯s]),

for the neutron target. Here and .

### ii.2 Deep Inelastic Neutrino Nucleus Scattering

When the reaction given by Eq. (4) takes place in a nucleus, several nuclear effects have to be considered. One may categorize these medium effects into two parts, a kinematic effect which arises because the struck nucleon is not at rest but is moving with a Fermi momentum in the rest frame of the nucleus and the other one is a dynamic effect which arises due to the strong interaction of the initial nucleon in the nuclear medium. For details please see the discussion given in Refs. Marco:1995vb (); Sajjad ().

The expression for the differential scattering cross section for a nuclear target A is similar to Eq. (7) and is given by:

 dσν(¯ν)ACCdE′dΩ′=G2F(2π)2⋅∣∣→k′∣∣∣∣→k∣∣⋅(m2Wq2−m2W)2⋅Lαβν,¯νWν(¯ν)Aαβ (18)

where is given by Eq. (8) and , the nuclear hadronic tensor, is given by:

 Wν(¯ν)Aαβ = (qαqβq2−gαβ)Wν(¯ν)A1(PA,q) (19) + 1M2A(PAα−PA⋅qq2qα)(PAβ−PA⋅qq2qβ)Wν(¯ν)A2(PA,q) − ı2M2AϵαβρσPρAqσWν(¯ν)A3(PA,q)

where is the momentum of the nucleus A.

In the local density approximation, the nuclear hadronic tensor can be written as a convolution of the nucleonic hadronic tensor with the hole spectral function. For symmetric nuclear matter, this would be Sajjad2 ():

 Wν(¯ν)Aαβ=4∫d3r∫d3p(2π)3ME(p)∫μ−∞dp0Sh(p0,p,kF(→r)) Wν(¯ν)Nαβ, (20)

where is the Fermi momentum for symmetric nuclear matter which depends on the density of nucleons in the nucleus, i.e. . is the hole spectral function, is the chemical potential and have been taken from Ref. FernandezdeCordoba:1991wf (). is the hadronic tensor for the free nucleon target given by Eq. (10). M and are respectively the mass and energy of the nucleon.

The natural extension of the above expression for taking into account the non-symmetric nature of the target nucleus would be to consider separate distributions of Fermi seas for protons and neutrons. The expression for which is given by Sajjad3 ():

 Wν(¯ν)Aαβ = 2∫d3r∫d3p(2π)3ME(p)∫μp−∞dp0Sprotonh(p0,p,kF,p)⋅Wν(¯ν)pαβ (21) + 2∫d3r∫d3p(2π)3ME(p)∫μn−∞dp0Sneutronh(p0,p,kF,n)⋅Wν(¯ν)nαβ

where the factor in front of the integral accounts for the two degrees of freedom of the spin of the nucleons. In the above equation, and are the two different spectral functions and normalized respectively to the number of protons or neutrons in the nuclear target. () is the Fermi momentum of proton (neutron). For the proton and neutron densities in iron, we have used two-parameter Fermi density distribution and the density parameters are taken from Ref. Vries ().

### ii.3 Construction of the nuclear hadronic tensor for non-symmetric nuclear matter

The natural invariant quantities for deep inelastic scattering (DIS) of neutrinos with nuclei are:

 xA=Q22P⋅q;  νA=P⋅qMA;  yA=P⋅qP⋅k (22)

where is the Bjorken variable in the nucleus and ; is the inelasticity. These two variables are related to the nucleonic ones via:

 xA = xA;  yA=q0Eν=y (23)

where and are defined in Eq. (11). We can see that , although for the nuclear structure functions are negligible. The variable varies between the following limits:

 0≤yA≤11+MAxA2Eν≈11+Mx2Eν, (24)

so, for sufficient high neutrino energy we have .
If we express the differential cross section with respect to these variables (), we obtain the following expression in terms of the nuclear structure functions:

 d2σν(¯ν)ACCdxAdyA = G2FMAEνπ(m2WQ2+m2W)2(y2AxAFν(¯ν)A1 (25) + {1−yA−MAxAyA2Eν}Fν(¯ν)A2±xAyA(1−yA2)Fν(¯ν)A3)

For the neutral current induced neutrino interaction, the form of the differential cross section is the same as for the charged current induced process but with the following changes:

First we look at the denominator of the Paschos-Wolfenstein relationship, an expression similar to Eq. (2), but in terms of the differential scattering cross section, for which we subtract the charged current antineutrino-nucleus cross section from the charged current neutrino-nucleus cross section and obtain the expression as:

Since in the present study, we have neglected the W-boson propagator term.

We need to relate the nuclear structure functions to the nucleon ones via an integral with the spectral function. Therefore, we introduce the following notation to avoid writing every time the integration symbols. For example, we may rewrite Eq. (21) with the following notation:

 Wν(¯ν)Aαβ=⟨Wν(¯ν)pαβ⟩Sprotonh+⟨Wν(¯ν)nαβ⟩Sneutronh (28)

where stands for proton:

 ⟨Wν(¯ν)pαβ⟩Sprotonh=2∫d3r∫d3p(2π)3ME(p)∫μp−∞dp0Sprotonh(p0,p,kF,p)Wν(¯ν)pαβ (29)

and for the neutron the expression is the same when the indices for the proton are replaced by the neutron indices.

Taking the three momentum transfer along the Z-axis, i.e, , and writing the xx-component of the nuclear hadronic tensor (Eq. 19), we get it in terms of the nuclear structure function i.e.

 Wν(¯ν)Axx=Wν(¯ν)A1=Fν(¯ν)A1(xA)MA=Wν(¯ν)Ayy (30)

Similarly, if we take the xx-components of the nucleonic hadronic tensor given by Eq. (10) and remembering the fact that nucleons in the nucleus are not at rest, the xx-component of the nucleonic hadronic tensor is not related only to the nucleon structure function , but it is a mixture of and components like the expression written below:

 Wν(¯ν)Nxx=Wν(¯ν)N1+p2xM2Wν(¯ν)N2=Fν(¯ν)N1(xN)M+p2xM2Fν(¯ν)N2(xN)νN (31)

where ; and .

Using Eqs. (30) and (31) we may write:

 Fν(¯ν)A1(xA)MA=⟨Fν(¯ν)p1(xN)M+p2xMFν(¯ν)p2(xN)p⋅q⟩Sph+⟨Fν(¯ν)n1(xN)M+p2xMFν(¯ν)n2(xN)p⋅q⟩Snh (32)

The difference that appears in Eq. (27) may then be written as:

 FνA1(xA)MA−F¯νA1(xA)MA = ⟨1M(Fνp1(xN)−F¯νp1(xN))+p2xM(p⋅q)(Fνp2(xN)−F¯νp2(xN))⟩Sph (33) + ⟨1M(Fνn1(xN)−F¯νn1(xN))+p2xM(p⋅q)(Fνn2(xN)−F¯νn2(xN))⟩Snh

where

 Fνp1−F¯νp1 = dv−uv Fνp2−F¯νp2 = 2xN(dv−uv) Fνn1−F¯νn1 = −dv+uv=−(Fνp1−F¯νp1) Fνn2−F¯νn2 = 2xN(−dv+uv)=−(Fνp2−F¯νp2)

Here and are the valence PDFs and we are working in the so-called up and down quarks approximation, where we neglect strange and charm quarks contributions.

In the case of symmetric nuclear matter, we may relate the Fermi momentum with the baryon density via , where is the baryon density. For a non-symmetric nuclear matter, we have different densities for protons and neutrons and corresponding to those, we also have different Fermi momenta for protons and neutrons. These are related by

 k3F,p=3π2ρp(r);k3F,n=3π2ρn(r) (34)

Instead of discussing in terms of neutron number(N) and proton number(Z) as independent variables, we define two independent variables A=N+Z and their difference =N-Z such that:

 N = A2+δ2;  Z=A2−δ2 (35)

Dividing the above equations by the nuclear volume , we obtain the densities of neutrons and protons:

 ρn = ρ2+δ2V;ρp=ρ2−δ2V (36)

where and . Replacing the densities for neutrons and protons by their corresponding Fermi momenta one has in terms of , and V i.e.

 k3F,p = k3F−3π22δV;k3F,n=k3F+3π22δV. (37)

A non-zero value of would imply that we are looking for the deviations from the isoscalarity. For and , we are going to perform an expansion in powers of the parameter , and retaining the first order term only with the assumption that the higher orders would be negligible. For instance, the expansion for the Fermi momentum of the proton would be: