Aspects of Neutrino Interactions (Scattering at small Q 2 - Region)
DDO-TH 14/31
ASPECTS OF NEUTRINO INTERACTIONS(Scatterings at the small Q -Region) T. Hoinka , E. A. Paschos , L. Thomas Department of Physics, Technical University of Dortmund, D-44221, Dortmund, Germany (Presented by E. A. Paschos at the CETUP-Workshop on Neutrino Interactions, July 22-31, 2014at Lead/Dead Wood, South Dakota, USA)
Abstract:
The article begins with a description of chiral symmetry and its applica-tion to neutrino induced reactions. For small Q (forward direction) the process isdominated by the amplitute with helicity zero where the pion pole disappears whenmultiplied with the polarization vector. The remaining part of the amplitude is deter-mined by PCAC. For E ν > GeV the computed cross sections are in good agreementwith data. In coherent pion production we expect equal yields for neutrinos and an-tineutrinos a relation which for E ν > GeV is fulfilled. We discuss specific featuresof the data and suggest methods for improving them by presenting new estimates forthe incoherent background. Email Address: [email protected] Email Address: [email protected] Email Address: [email protected] a r X i v : . [ h e p - ph ] D ec . INTRODUCTION In neutrino reactions the small Q region provides the opportunity for estimating thevector and axial contributions accurately. We will concentrate in this region and describethe methods in some detail. We will discuss two processes:i. Coherent pion production on Nuclei, andii. The production of the Delta resonance and its subsequent decay into a pion and anucleon, where the propagation and development of the final state is influenced bynuclear corrections.There are two schools for pion production calculations at low Q : the traditional basedon PCAC (partially conserved axial current hypothesis) [1-5] and microscopic calculations[6-11]. The methods were reviewed in a recent article [12]. In estimates of PCAC there isthe question how large is the region of validity and will be addressed later in the article. W + µ – ν µ N Nπ ± k k'p p'p π θ ζ θ’α φ p π s i n θ p π s i n ζ q k ν p π Figure 1: Left: Feynman diagram for the charged current reaction. Right: Coordinatesystem used for angular dependance.In the region we consider the current from the leptonic vertex is expanded in terms offour polarization vectors. Three of them have helicity one and the scalar has helicity zero.The geometry of the process is defined in figure (1) where the weak current (virtual Wor Z) moves along the z-axis and together with the neutrino direction they define the x-z-plane. The pion can be produced outside this plane. The three vectors of momenta forma tetrahedron. The four-vector of the virtual current has the components q µ = ( q , , , q z ) and the polarization vectors with angular momentum one are (cid:15) µ ( λ = ±
1) = 12 ± i , (cid:15) µ ( λ = 0) = 1 (cid:112) Q | (cid:126)q | ν
2n addition there is the scalar polarization vector (cid:15) µ ( l ) = q µ (cid:112) Q . (1)For small Q there is a property of the weak interactions that simplifies the calculations.For ν (cid:29) Q the leptonic current for charged and neutral reactions is dominated by thepolarizations (cid:15) µ ( λ = 0) and (cid:15) µ ( l ) . In general, the amplitude for the axial hadronic currentis written as the sum of a pion pole and a smooth remainder − i (cid:10) π + N (cid:12)(cid:12) A + µ | N (cid:105) = √ f π q µ Q + m π T ( π + N → π + N ) − R µ , (2)where T ( π + N → π + N ) is π + N elastic amplitude. The divergence of the matrix elementon the left hand side of the equation is determined by PCAC. When this result is combinedwith the pion pole on the right-hand side the pion propagator cancels and the remaindergives a smooth function for q µ R µ q µ R µ = −√ f π T ( π + N → π + N ) (3)In reference [3] the transverse contribution to coherent scattering was estimated and wasfound that it is small relative to the contribution for zero helicity. The important polar-izations produce the cross section [3,4] d σ d Q d ν d t = G F | V ud | π νf π E ν Q (cid:40) ˜ L + 2 ˜ L l m π Q + m π + ˜ L ll (cid:18) m π Q + m π (cid:19) (cid:41) d σ π d t (4)The notation in this equation is standard with E ν , ν and Q defined to be the energy ofthe neutrino, the energy transfer between the leptons and q = − Q the square of the fourmomentum of the current, respectively. The pion decay coupling constant f π = 0 . GeVand L , L l and L ll are density matrix elements arising from the polarizations of theleptonic tensor [3,4]. We shall use equation (4) for coherent scattering and for the axialcontribution to the excitation of resonances.Coherent scattering is defined as scattering on a nucleus which remains in its groundstate. Thus there is no exchange of quantum numbers between the virtual current-pionsystem and the nucleus and only the axial current contributes, especially in the two po-larizations included in equation (4). For coherent scattering the cross section d σ ( πN ) / d t denotes the elastic scattering of a pion on the nucleus N. The size of the nucleus is largerelative to that of protons and the form factor obtained as a Fourier transform of thenuclear density is a fast falling function of t = ( q − p π ) . Data for elastic pion-Carbonscattering are available [13-15] and will be used in section 3. For the background wetake the incoherent sum of pion-proton and pion-neutron scattering and fold them withnuclear corrections (final state interactions). The amplitudes computed with the PCACrelation are propotional to the square of the pion mass and go to zero as m µ → . In ourcase the divergence of the hardonic matrix element has this property and it cancels thepion pole leaving a smooth function. In other words chiral symmetry determines a termwhich cancels the pion pole with the remainder being a smooth function. The remainderis transformed into a physical process for which we shall use experimental data. Usingpion-nucleus scattering data includes an implicit assumption that the amplitudes do notchange much when the pion is taken off the-mass-shell. This is correct for small values of Q and we must face the question how large can Q be in each process.3 . KINEMATICS For neutrino induced reactions the calculation must respect the physical boundaries dic-tated by the kinematics of the lepton vertex. In particular, the four momentum of thecurrent is space-like so that the variable t = ( q − p π ) can not reach the value t = 0 . . Anyintegration over | t | covers the range (cid:18) Q + m π ν (cid:19) < | t | < | t | (5)The upper limit of integration is the first diffractive minimum and can be extended toinfinity without any noticeable change in the numerical results. The lower limit for thevalue of | t | is important because d σ d t ( πA ) is a very fast falling function of | t | with thefollowing consequence: when one integrates to | t | → there is a sizable overestimate ofthe neutrino induced cross section (by almost a factor of two) [4].The calculation of the incoherent background is more complicated and has differentkinematics. It receives a contribution from the vector current and the vector ⊗ axial inter-ference term. For the kinematic variables we use the notation from figure (1). From thesquare of the vector p (cid:48) = ( q − p π ) + p and the definition of t we obtain ν = E π − t M , which in the approximation νE π ∼ | (cid:126)q || (cid:126)p π | gives ν ≈ M E π + (cid:2) Q − m π (cid:3) M − E π (1 − cos ζ ) (6)The mass M is the mass of the target, which for a nucleus is larger than the otherquantities . This gives the relation ν ∼ E π that we frequently use for coherent scattering.Balancing the three momenta of the muon and the current perpendicular to the neutrinodirection gives Q ≈ E ν E π E ν cos θ (cid:48) − E π (1 − cos θ (cid:48) ) (7)Two more relations follow from the geometry of the tertahedron in figure (1) sin α = sin θ sin ζ sin φ (8)and the addition theorem cos ζ = cos θ cos θ (cid:48) + sin θ sin θ (cid:48) cos φ (9)These relations are useful when we wish to change the variables referring to the current tothose referring to the neutrino direction. 4 . THE CROSS SECTIONS The most detailed evidence for coherent scattering is the observation of the characteristicsharp peak in the t distribution of the events. This was the evidence in the early ex-periments. In the analysis of the Minerva data the background is estimated by studyingthe t -distribution of pions for . < | t | < . Gev and then extrapolating the functionalform to smaller values, where the signal for coherent scattering peaks at | t | near zero.Distributions on the variable | t | are not available but integrals over | t | for various neutrinoenergies were presented at this meeting [16]. Instead of calculating the cross section again,we take the figures presented some time ago in reference [4] and plot in the figures the newexperimental points.The elastic pion-Carbon cross section was parametrized as follows: d σ ( πN )d t = a exp (cid:2) − b | t | (cid:3) (10)with values for the paramaters given in table (1) of [4] . The limits of integration weredescribed in the previous section. The values of a ( ν ) and b ( ν ) are given in the table and goup . GeV. Beyond that value the data were extrapolated to have constant values equalto the last entry in the table, i.e. a = 3 . (barn/GeV ) and b = 53 . (1/GeV ). As wementioned in the introduction, only the axial current contributes to coherent scattering,expecting neutrino and antineutrino induced reactions to be equal. The data is consistentwith this property and we shall use the same curves for both reactions. In article [4] thecross section was calculated up to E ν = 10 GeV. A similar calculation appeared in reference[5] for a smaller range of E ν up to . GeV; up to this energy the results of the two groupsagree.In figure (2) we show the neutrino data. We include the older data from K2K [17],SciBoone [18] and SKAT [19] and to them we added the results from Minerva reported atthe Workshop [16, Higuera]. A point from ArgNeut has a very large error and lies outsidethe figure. In figure (3) we present the same curves and added the experimental points forantineutrinos from ArgoNeut [20] and Minerva. For E ν > . GeV the agreement betweentheory and experiment is very good, however the errors for antineutrinos are larger. Inthese energies neutrino and antineutrino cross sections are consistent to being equal. Onlyat the first point with an average energy between . and . GeV the values of the twocross sections are different. These points do not seem to follow the trend of the higherpoints or of the curves and we discuss them below.The PCAC approach is valid for low values of Q . In the comparison with the curves, theexperimental points for energies up GeV agree with the curves with the Q having smallvalues, below . Gev . For the higher energies the points agree with curves whose Q are closer to . GeV. This is a consequence of the fact that we used for . < ν < GeVconstant values for a and b occuring in equation (8). If we use for b values smaller than . (1/GeV ), then similar curves are obtained for smaller values of Q . In addition,at this low energy the background is dominated by the production of the ∆ -resonancewhose angular distribution is (1 + 3 cos θ ∗ ) where θ ∗ and all other starred quantities referto center of mass. The contribution of the axial current to the cross section in the cmssystem is d σ d Q d ν d cos θ ∗ = G F | V ud | π (cid:26) νf π E ν Q ˜ L (cid:27) σ max M R Γ ( W − M R ) + M R Γ (cid:0) θ ∗ (cid:1) (11)5 K2K SKAT SciBooNE E υ GeV σ CC c m MINVERvA charged current (neutrino)
Figure 2: Integrated charged current cross section wit Q = 0 . , . , . and . GeV (bottom to top). The curves are from [4] and the data from [16,17,18,19]. SKAT0 2 4 6 8 10020406080100 E υ GeV σ CC c m charged current (anti-neutrino) MINVERvAArgoNeut
Figure 3: Integrated charged current cross section. The curves are the same as in figure 2and the data for antineutrinos from ArgoNeut [20] and MINERvA [16] and SKAT [19]6ith σ max = 199 · − cm and the other quantities referring to the resonance. Thisequation is analogous to equation (4) with the difference that now the variables are deter-mined by two body kinematics. The process does not exhibit a diffractive peak. For thedependence on the t variable we must substitute in equation (11) cos θ ∗ = 1 + t p ∗ . (12)The terms from the vector current squared and the interference have the same angulardependence so that the complete contribution from the ∆ -resonance can be analyzed inthe manner described. This way we obtain the t dependence for the background. Anothercriterion for separating the coherent from the background is provided by the opening angleof the pion relative to the neutrino direction. Starting with the triple differential crosssection in equation (11), we transformed it to the laboratory variables by computing theappropriate Jacobian. The results for a C target with an averaging over protons andneutrons are shown in figure (4). We selected several values for E ν , E π = 0 . GeV andset the azimuthal angle φ equal to zero. This figure can be compared with the plots forcoherent scattering in figure (2) of ref. [21] and figure (5) in this article. The incoherentcross section is relatively large and extends to larger values of θ π . It peaks at θ π ≈ ◦ incontrast to coherent scattering which peaks at θ π ≈ ◦ − ◦ and vanishes for θ π ≈ ◦ .A pure coherent signal appears only for θ π < ◦ .
0 20 40 60 80 100 120 140 E ν = 1 GeVE ν = 3 GeVE ν = 5 GeVE ν = 10 GeV Figure 4: Distributions on the polar angle for a C target and for various values of E ν ,for E π = 0 . GeV and φ = 0 The PCAC approach we described is valid for small values of Q . When we comparethe theoretical curves with experimental points we observe that for E ν < GeV the fourcurves are close together. Thus small values of Q are acceptable for the comparisonwith the data. For energies E ν > GeV the experimental points agree with the theoreticalcurves where Q is closer to . GeV. This may be a consequence of the fact that for ν > . GeV we used constant values for a ( ν ) and b ( ν ) . We did this because we couldnot find data at these energies. If we select smaller values for b ( ν ) then the coherent crosssection will be larger also for smaller values of Q . This observation suggest that we should7 Figure 5: Triple differential cross section for coherent scattering for E ν = 1 . GeV, E π =0 . GeV and φ = 0 try a fit by restricting Q = 0 . or . GeV and then searching for values of a ( ν ) and b ( ν ) which reproduce the coherent data .
4. SUMMARY
In the first chapter of the article we explained how chiral symmetry is applied to neutrinoand antineutrino reactions. It was emphasized that the dominant contribution does notcome from the pion pole, but from the cloud of mesons that surround the target [3,22].In fact for low values of the neutrino energy the relevant range of Q is small, so thatthe introduction of a form factor or of a propagator from axial mesons is not necessary.The new data indicate that for E ν (cid:38) GeV, higher values of Q may be necessary. Forcomparisons with new data we took the theoretical curves from earlier calculations [4] and[5] and used the same curves for neutrinos and antineutrinos. For coherent scattering theintegrated cross sections are expected to be equal and the data cofirm this expectation.In general, the agreement between theory and experiment is good, but the statisticaluncertainties are still large.Only at the first point with E ν ≈ E ¯ ν ≈ . GeV there is a difference between neutri-nos and antineutrions We point out that the background at this energy is not diffractiveand special care is necessary in order to determine the background from the ∆ -resonancewhich has a different dependence on the variable t . It may also be helpful to define vari-ables relative to the neutrino direction. In the econd chapter we summarized kinematicrelations written in terms of variables relative to the neutrino direction. In figure (4)we show the angular distributions of pions relative to the neutrino direction originatingfrom the production of the ∆ -resonance. We computed them for several neutrino ener-gies and for C target by taking the sum of scatterings on protons plus neutrons andthen multiplied the sum by six. The angular deependence of the baclground is broaderthan the distribution for coherent scattering shown in figure 5. These distributions were8omputed for Q < . GeV . A recent calculation [23] presented angular distributionswithout any restrictions on the range of Q . We hope that new estimates will be helpfulfor understanding the background and improving the accuracy of coherent scattering.
4. ACKNOWLEDGEMENT
This is an expanded version of a talk presented at the CETUP-Workshop in the summerof 2014. One of us (Emmanuel A. Paschos) wishes to thank the organizers of the workshopfor providing a stimulating atmosphere that lead to the improvements discussed in thisarticle. 9