Relativistic Models for Neutrino-Nucleus Scatteringchimera.roma1.infn.it/OMAR/Elba_XI/barbaro.pdf · Relativistic Models for Neutrino-Nucleus Scattering Maria B. Barbaro Universita

Relativistic Models for Neutrino-Nucleus Scattering

Maria B. BarbaroUniversita' di Torino and INFN, Italy

Elba XI Workshop on Electron-Nucleus Scattering

June 21-25, 2010 , Marciana Marina, Isola d'Elba

Collaboration

J.E. Amaro (Granada) J.A. Caballero (Sevilla) T.W. Donnelly (M.I.T.) C. Maieron (Grenoble) A. Molinari (Turin) I. Sick (Basel) J.M. Udias (Madrid)

Motivation

Current and future experiments searching for neutrino oscillations use complex nuclei astargets and rely on the precise prediction of neutrino-nucleus cross sections for their analysis

Motivation

The Relativistic Fermi Gas underpredicts the MiniBooNEtotal cross section by about 20%

unless a very high axial massMA=1.35 GeV/c² is used

More realistic nuclear models are neededin order to understand this result

MiniBooNE data, Phys. Rev. D81, 092005 (2010)

Current and future experiments searching for neutrino oscillations use complex nuclei astargets and rely on the precise prediction of neutrino-nucleus cross sections for their analysis

Outline

Charged-Current neutrino-nucleus scattering formalism

- Quasielastic peak

- Delta-resonance region

Relativistic models:

1. Relativistic Fermi Gas

2. Phenomenological approach based on SuperScaling analysis of (e,e')

3. Relativistic Mean Field Model

4. Semi-relativistic Shell Model

Results: comparison with MiniBooNE quasielastic differential cross sections

Summary and Conclusions

(l,l') inclusive scattering

l

−

l−

W −

e , e '

e

e '

, l

d2

d ' d=Mott v L RLvT RT

d2

d ' d=0 V CC RCC2V CL RCLV L L RL LV T RT±2V T ' RT '

2 electromagnetic response functions

5 weak response functions

l= , e ,

CC neutrino reaction formalism

lA l−B

lA lB [ d2

ddk ' ]=0 F2

0=G cosc

2

22[k ' cos /2]2 tan /2=

∣Q2∣

4 '−∣Q2∣

F2=V CC RCC2V CL RCLV L L RL LV T RT

2V T ' RT '

=1 −1

charge-charge(00) charge-long.(03) long.-long.(33) transverse(11+22)

transverse-axial(12)

Weak current j= jV− j

AF2=X L

VVXC /LAA XT

VVAA XT 'VA

Axial-vectorVector

nuclear response function

elementary cross section

opposite sign for neutrino and antineutrino

V K kinematical factors

effective angleoutgoing lepton

Generalized Rosenbluth separation (different components of the hadronic tensor ) W

Quasi-elastic peak

Single nucleon current:

jV=u P 'F1

i

2 mN

F2Qu P

jA=uP' GA

1

2mN

GP Q5 u P

j= jV− jA

R LVV=2

[GE

1]2 , RT

VV=2[GM

1]2

R LLAA=2

2 RCC

AA=−

RCL

AA=2

[GA

1−GP

1]2

RTAA=2 1[GA

1]2 , RT 'VA=21GM

1GA1

The nuclear responses are purely isovector, typically transverse and have vector-vector (VV), axial-axial (AA) and vector-axial (VA) contributions

ln l−p , lp ln

=q /2mN , =/2mN , =2−2 dimensionless variables

L-T-T' separation MiniBooNE kinematics

,−

VV-AA-VA separation MiniBooNE kinematics

,−

Delta-resonance excitation

n− , p−

n− , p 0

j=T u

P ' , s ' u P ,s=V

C3V ,C 4

V ,C5V ,C6

V 5 A C3

A ,C4A ,C5

A ,C6A

isospin factorRarita-Schwinger spinor

vertex tensor

vector axial-vector

CVCC6V=0

PCACC6A=C5

A/

24 6 form factors :

C3V=2.051∣Q2

∣/0.54 GeV 2−2 , C4

V=

−mN

m

C 3V , C5

V=0

C3A=0 , C4

A=−0.3

1−1.21∣Q2∣/2 GeV 2

∣Q2∣

[1∣Q2∣/1.282 GeV 2

]2 , C5

A=−4 C4

A

Alvarez-Ruso et al., PRC59, 3386 (1999)

8 form factors

Relativistic Fermi Gas The nucleus is a collection of free nucleons described by Dirac spinors u(p,s) The only correlations between nucleons are the Pauli correlations Fully Lorentz covariant

2 parameters:

Response functions: RK ,=GK , f RFG

=

2 mN

,=q

2mN

,=2−2

dimensionless variables

single-nucleonfunctions “super-scaling” function

(universal)

,=1

F

−

11RFG scaling variable

F=T F /mN Fermi kinetic energy

kF Fermi momentum

Es energy shift typically~20 MeV : '=−Es

'= ' , '

f RFG=341−21−2

Super-Scaling Analysis of (e,e')

Plot the reduced cross section

as a function of the scaling variable (from RFG) or (from non-relativistic scaling analysis)

for different kinematics and nuclei

No q-dependence → first kind scaling No A-dependence → second kind scaling } super-scaling

F q , y =d/dd

Mott v LG LvT GT

y

y≃k F

[Day,McCarthy,Donnelly,Sick,Ann.Rev.Nucl.Part.Sci.40(1990); Donnelly & Sick, PRC60(1999),PRL82(1999)]

Scaling of first kind

SLAC data at different kinematics

Plot F(q,y) versus yfor different q

F q , y =d/dd

Mott v LG LvT GT

Reduced cross section:



F q , y F y for q∞

y-scaling

F q , y =d /dd

Mott v LG LvT GT


scaling function



F q , y =d/ddv LGLvT GT


F q , y F y for q∞

Note that at y>0 the scaling is not good, due to the presence of resonances, meson production, etc.


y-scaling

scaling function

QEP


Scaling of second kindDonnelly & Sick, PRC60, 065502 (1999): define a super-scaling function

f q ,=kF×F q ,

and plot it versus ψ for a variety of nuclei at the same kinematics

Ee=3.6GeV , e=160

In the scaling region (ψ'<0) a universal behavior is seen, with very little dependence on the nuclear species

Scaling of II kind

In the region above ψ'=0 where resonances, meson production and the start of DIS enter the 2nd-kind scaling is not as good

A

QEP

Scaling of 0-th kind (fL=fT)?

Is f L=f T ?

Although the amount of available data separated into longitudinal (L) and transverse (T) responses is small, one can attempt a scaling analysis with what does exist:

f Lq ,=kF×RLq ,

A eNL/Mott vL

f T q ,=kF×RT q ,

A eNT /Mott vT

L and T scaling functions:

Scaling of 0-th kind?

Is f L=f T ?

Donnelly & Sick, PRC60, 065502 (1999)

Although the amount of data separated into longitudinal (L) and transverse (T) responses is small, one can attempt a scaling analysis with what does exist:

f Lq ,=kF×RLq ,

A eNL/Mott vL

f T q ,=kF×RT q ,

A eNT /Mott vT


T

Inelastic contributions

Some violation below the QEP


Is f L=f T ?



f Lq ,=kF×RLq ,

A eNL/Mott vL

f T q ,=kF×RT q ,

A eNT /Mott vT


L T

In contrast, the L results show a universal behavior




f Lq ,=kF×RLq ,

A eNL/Mott vL

f T q ,=kF×RT q ,

A eNT /Mott vT

L T

However:q<570 MeV/c New Rosenbluth separated JLab data are needed!

Scaling in the QEP: summary Scaling of I kind is reasonably good below the QE peak (ψ<0) Scaling of II kind is excellent in the same region Violations of scaling, particularly of I kind, occur above the QEP and reside

mainly in the transverse response The longitudinal response super-scales

Scaling in the QEP: summary Scaling of I kind is reasonably good below the QE peak (ψ<0) Scaling of II kind is excellent in the same region Scaling violations, particularly of I kind, occur above the QEP and reside

mainly in the transverse response The longitudinal response super-scales

L

1. Asymmetric shape: long tail at high energy transfer

An experimental longitudinal super-scaling function has been extracted from (e,e') data [Jourdan, NPA603, 117 ('96)]

2. Only 4 parameters for all kinematics and all nuclei

3. Represents a strong constraint on nuclear models

Scaling in the Delta region

[ d2

dd ]' '

=[ d2

dd ]exp

−[ d2

dd ]QE

1) subtract the QE contribution obtained from Superscaling hypothesis form the total experimental cross section

2) divide by the elementary N →∆ c.s.

F ''=[ d2

dd ]

' '

M vL GLvT GT

3) multiply by the Fermi momentum

f ' '=kF F ' '

4) plot versus the appropriate scaling variable

=q ,

=11

4m

2 /mN2−1 inelasticity

Scaling in the Delta region

[ d2

dd ]' '

=[ d2

dd ]exp

−[ d2

dd ]QE

1) subtract the QE contribution obtained from Superscaling hypothesis form the total experimental cross section

2) divide by the elementary N →∆ c.s.

F ''=[ d2

dd ]

' '

M vL GLvT GT

3) multiply by the Fermi momentum

f ' '=kF F ' '

4) plot versus the appropriate scaling variable

=q ,

=11

4m

2 /mN2−1 inelasticity

E e=0.3−4GeV

=12−1450

12 C , 16 O

This approach can work only at Ψ∆<0,since at ΨΔ>0 other resonances and the tail of DIS contribute

Amaro, Barbaro, Caballero, Donnelly, Molinari, Sick, PRC71 (2005)

Test of superscaling

R Lq ,=GL q , f QE GL q , f

RT q ,=GT q , f QE GTq , f

d2

dd=Mott v L RLvT RT

f QE

f RFG

f

Test of superscaling

R Lq ,=GL q , f QE GL q , f

RT q ,=GT q , f QE GTq , f

d2

dd=Mott v L RLvT RT

12C E e=0.961GeV ,=37.5016O

Amaro et al., PRC71, 015501 (2005)

data from http:/faculty.virginia.edu/qes-archive

Application to neutrino scatteringJust as for the electron scattering reactions in the QE and Δ regions, we use the scaling functions determined above, but now multiply by the corresponding charged-current neutrino reaction cross sections for the Z protons and N neutrons in the nucleus -----► SuperScaling Approximation (SuSA)

2. CC neutrino reactions are isovector only, whereas electron scattering contains both isoscalar and isovector contributions (the transverse EM response is, in fact, predominantly isovector at high energy).Thus, in going from electron scattering, where the universal scaling function came from the L response (essentially 50% isoscalar and 50% isovector) to CC neutrino reactions we have had to invoke

Scaling of the 3rd Kind : f(T=0) = f(T=1)

where the isospin nature of the scaling functions is assumed to be universal.

Limits of the approach:

1. it can be applied only at high enough momentum transfers, where the scaling ideas make sense: at low q and ω collective effects (like giant resonances) become important

CC neutrino cross section

RFG

SuSA

ΔQEP

Amaro et al., PRC71, 015501 (2005)

12 C

Relativistic Impulse Approximation

In order to understand the microscopic origin of superscaling and test the validity of the superscaling approach to neutrino scattering we have explored other relativistic nuclear models:

1) Relativistic Mean Field Model - RMF

2) Semi-relativistic Shell Model (with FSI) - RSM

Both models are based on RIA:

J N , q =∫d pF pq J N

B p

Scattering off a nucleus ⇒ Incoherent sum of single-nucleon scattering processes

Nuclear current ⇒ One-body operator

1) RMF model: basic ingredients

B : bound nucleon w.f. ⇒ self-consistent Dirac-Hartree solutions derived within a RMF approach using a relativistiv Lagrangian with scalar and vector potentials [Horowitz&Serot, NPA368(1981),PLB86(1979); Serot&Walecka, Adv.Nucl.Phys.16 (1986)]

F: outgoing nucleon w.f. ⇒ different treatments of Final State Interactions (FSI):

1. RPWIA (no FSI)

2. RDWIA solutions of Dirac equation containing relativistic potentials:

a. ROP relativistic optical potential, successfully applied to exclusive (e,e'p) reactions [Udias et al., PRC48 (1993), PRC51 (1995), PRC64 (2001)], where the imaginary part represents the loss of flux into inelastic channels. For inclusive reactions, (e,e'), we simply use the real part of the optical potential (energy-dependent, A-indep. parameterizations EDAIC,EDAIO,EDAICa, Clark et al., PRC47 (1993) ) ⇒ rROP

b. distorted waves obtained with the same mean field as in the bound state ⇒ RMF

RMF Super-scaling Function

The RMF is the only model producing a non-symmetric scaling function, in reasonable agreement with the data

Only the description of FSI provided by RMF leads to an asymmetricscaling function in agreement with the behavior of the data

Test of SuSA in the RMF model

Shift at ψ'<0 and breakdown of I-kind scaling of ~ 25-30% at ψ'>0 (compatible with data) for q> 0.5 GeV/c

I kind II kind

Good scaling of II kind (but dependenton the prescription for the current, not so good with CC1)

Test of scaling of 0th kind

In the RMF model

Scaling of 0-th kind is violated.

f T f L

2) Semi-relativstic Shell Model

- Suited to describe closed shell nuclei- Initial state : Slater determinant with all shells occupied- Impulse approximation: final states are particle-hole excitations coupled to total angular momentum

- Single particle and hole wave functions are obtained by solving the Schrödinger equation with a Woods-Saxon potential including central, spin-orbit and Coulomb terms

∣i ⟩

∣f ⟩=∣ ph−1J ⟩

- Relativistic kinematics are taken into account by the substitution

as the eigenvalue of the Schrödinger equation for the outgoing nucleon

pp 1p /2mN

- Semi-relativistic currents involve an expansion of the relativistic nucleon currents in the bound nucleon momentum to first order J=up ' , s ' up ,s =p /mN

q , can be large

JV0=0i '0×⋅

JVT=1

Ti '1 ×

J A0= '0⋅ ' '0

T⋅

J Az= '3⋅

T

J AT= ' '1 ' '3

T⋅

0=2GEV

, '0=

2GMV −GE

V

1

1=2GEV

, '1=2GMV

Vector Axialetc., provide the required relativistic behavior

Semi-relativstic Shell Model (SRSM)The model superscales

and yields essentially the same results as the RFG (symmetric scaling function) but for extended tails outside the RFG response region.

' [MeV ]

RFG

SRSM-WS

' '

q=0.5,0 .7,1,1.3,1 .5GeV /c

12C ,16O ,40 Ca

,−

12C

Relativistic Shell Model with FSI

Improve the FSI:

1. Re-write the Dirac equation as a second order equation for the up component

2. Include the Darwin term

3. The function satisfies the Schrödinger equation

up

upr =K r , E r

r

[ −2mN

U DEB r , E ]r = E2−mN2

2mN

r

Dirac Equation Based potential (energy dependent)

Superscaling Functions in SRSM+FSI

SRSM-WS

SRSM+FSI

RMF

' '

The SRSM with FSI is essentially equivalent to the RMF in the longitudinal cahnnel,but it does not contain the L-T scaling violations

SRSM-WS

SRSM+FSI

,− 12C

Results for neutrino reactions

SuSA

E=1GeV

Integrated cross sections

[Amaro et al., Phys. Rev. Lett. 98, 242501 (2007)]

Relativstic Mean Field ≈ SuperScaling Approach

Relativstic Fermi Gas ≈ Semi-relativistic Shell Model (with WS potential)

Integrated cross sections

[Amaro et al., PRL 98, 242501 (2007)]

Comparison with MiniBooNE

flux

⟨E⟩=0.788 GeV

We average the double differentialcross section

over the MiniBooNE Neutrino flux:

d2

dcosdT

[PRD81,092005 (2010)]

Data are given in 0.1 GeV bins of Tμ and 0.1 bins of cosθμ

RFG

Comparison with MiniBooNE data at 0.9<cosθ<1

SuSARFG


SuSARFG

Pauli blocking is active in this region (low momentum transfers, q≤0.4 GeV/c): this explains the huge difference between the RFG (where PB is included by definition) and the SuSA (which has no PB) results.

At very low angles both RFG and SuSA are compatible with the data, except for the Pauli-blocked region, where super-scaling ideas are not applicable.



SuSARFG

Pauli blocking is active in this region (low momentum transfers, q≤0.4 GeV/c): this explains the huge difference between the RFG (where PB is included by definition) and the SuSA (which has no PB) results.

At very low angles both RFG and SuSA are compatible with the data, except for the Pauli-blocked region, where super-scaling ideas are not applicable.

MA=1.03 GeV/c²

SuSARFG

For θ~30° the RFG is still compatible with the data, the SuSA result is too low at the QEP.

MA=1.03 GeV/c²

Comparison with MiniBooNE data at 0.8<cosθ<0.9

[Data from PRD81,092005 (2010)]

SuSARFG

For θ~60° both RFG and SuSA underestimate the experimental data.

MA=1.03 GeV/c²


[Data from PRD81,092005 (2010)]

SuSARFG

For θ~60° both RFG and SuSA underestimate the experimental data.

MA=1.03 GeV/c²

RFG,MA=1.35 GeV/c²A RFG with high axial mass fits the data: what is the physics?


[Data from PRD81,092005 (2010)]

RMF versus SuSA (preliminary)

RMF RMF

SuSA SuSA

Beyond Impulse Approximation: two-body currents

At the kinematics where meson-exchange currents (MEC) play a significant role superscaling violations may occur in the transverse channel.

contact: pion-in-flight:

“correlation” diagrams:

N ,

MECMeson Exchange Currents contribute to the 1p-1h sector:

and to the 2p-2h one:(just some of the many diagrams)

1p-1h MEC

[Amaro et al., NPA723, 181 (2003)]

I kind scaling

II kind scaling

RFG

including Δ-MECat various q

including Δ-MECat various kF

The Δ-MEC diagrams give the dominant contribution

The response is calculated on the RFG basis and ismainly transverse (althoughrelativistically there is a smallL contribution)

Both kinds of scaling are violated

1p-1h MEC interfere distructivelywith the 1-body current, loweringthe cross section

2p-2h MEC

'0

'0

RFG

RFG

[De Pace et al., NPA741, 249 (2004)]

Scaling is broken both below and abovethe QEP

They contribute outside the RFG response region -1<ψ<1

They give a positive contribution of about10-15% at the QEP

However, the calculation is not completebecause it does not include the correlations required by gauge invariance

Very demanding calculation, even in RFG

Summary and Conclusions

Puzzle: the Relativistic Fermi Gas overestimates the (e,e') data and underestimates CC neutrino cross section

The SuSA model, based on super-scaling and designed to fit inclusive electron scattering data, gives even worse agreement with the experiment

The Relativistic Mean Field model is in qualitative agreement with the SuSA predictions and also fails to reproduce the MiniBooNE data at high scattering angles

The Relativistic Shell model, including Final State Interactions through a Dirac equation based potential, is in good agreement with the RMF results

The SuSA approach relies on scaling of 0th kind [fL=fT] and of 3rd kind [f(T=0)=f(T=1)]: a better understanding of scale-violating contributions, such as MEC, is needed in order to use (e,e') data to predict neutrino cross sections

Future (present) work:

1. Complete the calculation of MEC in the 2p-2h sector

2. Implement scaling violations (of 0, I and II kind) in the SuSA approach

3. Compare results with other models in the same conditions: Relativistic Green's Function (Pavia group), Coherent Density Fluctuation model (Sofia group), .... anybody is welcome!

Thank You

Relativistic Effects

Non-relativistic

Rel. kinematicsNon-rel. currents

Relativistic

DEB potential

Documents

Relativistic Models for Neutrino-Nucleus Scatteringchimera.roma1.infn.it/OMAR/Elba_XI/barbaro.pdf · Relativistic Models for Neutrino-Nucleus Scattering Maria B. Barbaro Universita