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General issues with the implementation of theory models in - - PowerPoint PPT Presentation

Sep 11, 2022 139 likes •671 views

General issues with the implementation of theory models in generators A. Nikolakopoulos , N. Jachowicz, K. Niewczas, J. T. Sobczyk , R. Gonzalez-Jimenez, J.M. Udias 1 NuSTEC workshop, Pittsburgh USA A. Nikolakopoulos Outline I. Nucleon

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  • A. Nikolakopoulos

NuSTEC workshop, Pittsburgh USA

General issues with the implementation

  • f theory models in generators
  • A. Nikolakopoulos , N. Jachowicz, K. Niewczas, J. T. Sobczyk , R. Gonzalez-Jimenez, J.M. Udias
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Outline

  • A. Nikolakopoulos

NuSTEC workshop, Pittsburgh USA

  • I. Nucleon complexity
  • II. Nuclear complexity
  • III. Final state interaction

Underlying message: More exclusive signals higher dimensional problems →

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v+N → π + N + l : counting variables

  • A. Nikolakopoulos

5 Four vectors = 5x4 = 20 variables

  • 4 : on mass shell relations
  • 4 : initial nucleon known (at rest)
  • 4 : Energy-momentum conservation
  • 3 : Freedom to choose reference frame

And invariance along q (known direction of one four vector) = 5 independent variables

Ev , cosθl , El , Ωπ* or Ev, Q2,W, Ωπ*

NuSTEC workshop, Pittsburgh USA

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v+N → π + N + l : Born approximation

  • A. Nikolakopoulos

Leptonic part ( PW approximation ) known → Hadronic part modelling effort → Exploit these facts:

  • Lepton tensor is known
  • Hadronic part is invariant under

rotation along q and is the product of Hadronic current with its conjugate → Separate the φ* dependence

NuSTEC workshop, Pittsburgh USA

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Separating the variables

  • A. Nikolakopoulos

Example for the A structure function: Here the Hadron tensor depends on 3 variables:

W, Q2 , cosθπ

* and φπ *= 0

And in total one needs 15 elements of the hadron tensor

For inclusive:

Only A survives integration over pion angles:

And responses depend on Q2 and W

NuSTEC workshop, Pittsburgh USA

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What we know from electro- and photoproduction What we know from electro- and photoproduction

  • A. Nikolakopoulos

Many approaches in the literature:

  • MAID07 -DCC ( e.g. Sato and Lee) -Effective Lagrangian approaches,ChpT , ...

Ingredients:

  • Nucleon resonances
  • Background terms : Born term, Vector meson exchanges
  • cross channel resonances
  • Final state interactions
  • …
  • Many parameters fitted to > 20000 datapoints:

NuSTEC workshop, Pittsburgh USA

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What we know from electro- and photoproduction What we know from electro- and photoproduction

  • A. Nikolakopoulos

NuFA FACT19, Daegu Korea

Many approaches in the literature:

  • MAID07 -DCC ( e.g. Sato and Lee) -Effective Lagrangian approaches, ...

Ingredients:

  • Nucleon resonances
  • Background terms : Born term, Vector meson exchanges
  • Final state interactions
  • …
  • Many parameters fitted to > 20000 datapoints:
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What we know from electro- and photoproduction What we know from electro- and photoproduction

  • A. Nikolakopoulos

NuFA FACT19, Daegu Korea

Many approaches in the literature:

  • MAID07 -DCC ( e.g. Sato and Lee) -Effective Lagrangian approaches, ...

Ingredients:

  • Nucleon resonances
  • Background terms : Born term, Vector meson exchanges
  • Final state interactions
  • …
  • Many parameters fitted to > 20000 datapoints:

For neutrinos no such dataset is available

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Electroproduction data

  • A. Nikolakopoulos

Write lepton tensor for polarized electron explicitly

NuSTEC workshop, Pittsburgh USA

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Electroproduction data: e+p n + → π+

  • A. Nikolakopoulos

LEM from R. Gonzalez-Jimenez et al.

  • Phys. Rev. D 95, 113007 (2017)

Based on HNV model Data from E89-038 CLAS experiment, 1999, V. Burket, R. Minehart MAID07 : Drechsel, D., Kamalov, S.S. & Tiator, L. Eur. Phys. J. A (2007) 34: 69

NuSTEC workshop, Pittsburgh USA

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Electroproduction data: e+p n + → π+

  • A. Nikolakopoulos

LEM from R. Gonzalez-Jimenez et al.

  • Phys. Rev. D 95, 113007 (2017)

Based on HNV model Data from E89-038 CLAS experiment, 1999, V. Burket, R. Minehart MAID07 : Drechsel, D., Kamalov, S.S. & Tiator, L. Eur. Phys. J. A (2007) 34: 69

NuSTEC workshop, Pittsburgh USA

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Electroproduction data: e+p n + → π+

  • A. Nikolakopoulos

LEM from R. Gonzalez-Jimenez et al.

  • Phys. Rev. D 95, 113007 (2017)

Based on HNV model Data from E89-038 CLAS experiment, 1999, V. Burket, R. Minehart MAID07 : Drechsel, D., Kamalov, S.S. & Tiator, L. Eur. Phys. J. A (2007) 34: 69

NuSTEC workshop, Pittsburgh USA

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Structure functions for neutrinos

  • A. Nikolakopoulos

NuSTEC workshop, Pittsburgh USA

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Angular distributions for neutrinos

  • A. Nikolakopoulos

HNV, DCC and LEM vary in structure functions, still more or less agree on angular cross

  • section. (Around Delta peak)

Could this influence neutrino oscillation analysis ?

NuSTEC workshop, Pittsburgh USA

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Angular distributions for neutrinos

  • A. Nikolakopoulos

In (most) event generators: Isotropic distribution in CMS. → Computationally easy

What is the difficulty ?

✗ Time to compute cross section

→ Actually rather fast The problem is efficiency in Sampling the phase space

NuSTEC workshop, Pittsburgh USA

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How to introduce the fivefold CS ?

  • A. Nikolakopoulos

Sample inclusive cross section in the traditional way:

Tabulate or Calculate in situ inclusive structure functions for the interaction Functions only of Q2 and W, very fast interpolation in 2D.

This gives an event with Q2 and W

NuSTEC workshop, Pittsburgh USA

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How to introduce the fivefold CS ?

  • A. Nikolakopoulos

given a Q2 and W, distribution of cosθ* is determined by A

A is a smooth function and can usually be interpolated by a polynomial of degree 2

NuSTEC workshop, Pittsburgh USA

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How to introduce the fivefold CS ?

  • A. Nikolakopoulos

given a Q2 and W, distribution of cosθ* is determined by A

A is a smooth function and can usually be interpolated by a polynomial of degree 2

NuSTEC workshop, Pittsburgh USA

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How to introduce the fivefold CS ?

  • A. Nikolakopoulos

given a Q2 and W, distribution of cosθ* is determined by A

A is a smooth function and can usually be interpolated by a polynomial of degree 2 Calculation of A(cos) for fixed Q2 and W is very cheap Interpolation with degree 2 polynomial means: Cumulative distribution function Is a monotonic degree 3 polynomial → Can be inverted analytically → Inversion sampling

NuSTEC workshop, Pittsburgh USA

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How to introduce the fivefold CS ?

  • A. Nikolakopoulos

given a Q2 and W, distribution of cosθ* is determined by A

A is a smooth function and can usually be interpolated by a polynomial of degree 2 Calculation of A(cos) for fixed Q2 and W is very cheap Interpolation with degree 2 polynomial means: Cumulative distribution function Is a monotonic degree 3 polynomial → Can be inverted analytically → Inversion sampling

NuSTEC workshop, Pittsburgh USA

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How to introduce the fivefold CS ?

  • A. Nikolakopoulos

given a Q2 , W, and cos θ* distribution of φ* is

By calculation of A at 3 points one gets a cosine according to the theoretical distribution With efficiency 100%

Again we determine the CDF algebraically. → The CDF can be inverted numerically to give φ*

NuSTEC workshop, Pittsburgh USA

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How to introduce the fivefold CS ?

  • A. Nikolakopoulos

NuStec workshop, Pittsburgh

given a Q2 , W, and cos θ* distribution of φ* is

By calculation of A at 3 points one gets a cosine according to the theoretical distribution With efficiency 100%

Again we determine the CDF algebraically. → The CDF can be inverted numerically to give φ*

First results, sampling in the full phase space, still some issues to be checked and algorithms to be explored

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v+A → π + N + X+l : counting variables

  • A. Nikolakopoulos

6 Four vectors = 6x4 = 24 variables

  • 4 : on mass shell relations
  • 4 : initial nucleus known (at rest)
  • 4 : Energy-momentum conservation
  • 3 : Freedom to choose reference frame

And invariance along q (known direction of one four vector) = 9 independent variables

Ev , cosθl , El , Ωπ , ΩN, kπ

  • 1 : Final nucleus left in a hole state

(i.e. integrate over final nucleus energy) = 8 independent variables We go from a 2 3 process to a 2 4 process → → But there are no additional constraints because residual nucleus can be in any state. So from 5 9 variables (one can also interpret the extra 4 variables as four-vector of → initial bound nucleon) interaction

NuSTEC workshop, Pittsburgh USA

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How to introduce the fivefold CS ?

  • A. Nikolakopoulos

given a Q2 , W, and cos θ* distribution of φ* is

By calculation of A at 3 points one gets a cosine according to the theoretical distribution With efficiency 100%

Again we determine the CDF algebraically. → The CDF can be inverted numerically to give φ*

NuSTEC workshop, Pittsburgh USA

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v+A → π + N + X+l : Born approximation

  • A. Nikolakopoulos

interaction Nuclear modeling = finding a good approximation for the wavefunctions

Ψi and Ψf contain the whole initial and final state

NuSTEC workshop, Pittsburgh USA

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Impulse approximation

  • A. Nikolakopoulos
  • I. Interaction with only one particle of complex system
  • II. The incident particle (Q) is unaffected by the system (in BA)

Reduces the problem to finding single particle states in nuclear medium:

NuSTEC workshop, Pittsburgh USA

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Impulse approximation

  • A. Nikolakopoulos
  • I. Interaction with only one particle of complex system
  • II. The incident particle (Q) is unaffected by the system (in BA)

Reduces the problem to finding single particle states in nuclear medium: With p = pm = q – p’N -k’π This is a six dimensional integral with a lot of matrix multiplication…

NuSTEC workshop, Pittsburgh USA

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Factorization

  • A. Nikolakopoulos

Replace these by asymptotic momenta

NuSTEC workshop, Pittsburgh USA

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Relativistic Plane wave Impulse approximation

  • A. Nikolakopoulos

p’N = pN k’π = kπ

NuSTEC workshop, Pittsburgh USA

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Plane wave Impulse approximation

  • A. Nikolakopoulos

Projection onto positive energy states

Matrix element becomes proportional to initial momentum distribution Combination of off-shell plane wave spinor expression And probability of finding momentum p in nucleus

NuSTEC workshop, Pittsburgh USA

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Plane wave Impulse approximation

  • A. Nikolakopoulos

Projection onto positive energy states

Matrix element becomes proportional to initial momentum distribution Combination of off-shell plane wave spinor expression And probability of finding momentum p in nucleus

NuSTEC workshop, Pittsburgh USA

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Plane wave Impulse approximation

  • A. Nikolakopoulos

Projection onto positive energy states

Matrix element becomes proportional to initial momentum distribution Combination of off-shell plane wave spinor expression And probability of finding momentum p in nucleus

Side note: Difference between RPWIA and PWIA was explored in:

Nucl.Phys. A632 (1998) 323-362 No big difference for inclusive responses in CC2 operator Larger effect for more ‘off-shell’ operators, and for transverse-longitudinal interference

NuSTEC workshop, Pittsburgh USA

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Plane wave Impulse approximation

  • A. Nikolakopoulos

Projection onto positive energy states

Matrix element becomes proportional to initial momentum distribution Combination of off-shell plane wave spinor expression And probability of finding momentum p in nucleus

NuSTEC workshop, Pittsburgh USA

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Plane wave Impulse approximation

  • A. Nikolakopoulos

Matrix element becomes proportional to initial momentum distributions (some examples):

  • RFG : plane waves up to kF
  • LFG : plane waves up to kF but kF depends on nuclear density

→ possible to introduce additional density dependence

  • IPSM : e.g. from mean field (HF/RMF/harmonic oscillator)

different shells have different momentum distribution and → separation energies

  • IPSM + correlations : account for high momentum components

in nuclear momentum distribution

NuSTEC workshop, Pittsburgh USA

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Plane wave Impulse approximation

  • A. Nikolakopoulos

Matrix element becomes proportional to initial momentum distributions (some examples):

  • RFG : plane waves up to kF
  • LFG : plane waves up to kF but kF depends on nuclear density

→ possible to introduce additional density dependence

  • IPSM : e.g. from mean field (HF/RMF/harmonic oscillator)

different shells have different momentum distribution and → separation energies

  • IPSM + correlations : account for high momentum components

in nuclear momentum distribution Comparison of these different spectral functions For exclusive nucleon knockout (RFG, LDA, RMF, Rome model)

NuSTEC workshop, Pittsburgh USA

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Factorization, with FSI

  • A. Nikolakopoulos

Transition matrix:

In general, dependence on q, pN and kπ ( 7 variables ) Contrast with RPWIA : depends only on pm = pN + kπ – q

Spreading of the energy momentum relation in a potential Particles have fixed energy and are only on shell asymptotically → Probing of multiple initial momentum states

NuSTEC workshop, Pittsburgh USA

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Kinematic dependence

  • A. Nikolakopoulos

In general, dependence on q, pN and kπ ( 7 variables )

Contrast with RPWIA : depends only on pm = pN + kπ – q

Dependence on q and kN Becomes less important for high momenta

NuSTEC workshop, Pittsburgh USA

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Kinematic dependence

  • A. Nikolakopoulos

In general, dependence on q, pN and kπ ( 7 variables )

Contrast with RPWIA : depends only on pm = pN + kπ – q

Dependence on q and kN Becomes less important for high momenta Energy dependent potentials

NuSTEC workshop, Pittsburgh USA

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Final state interactions

  • A. Nikolakopoulos

NuSTEC workshop, Pittsburgh 2019

Distinction between:

  • I. HARD FSI

Secondary interactions

(e.g. Absorption, charge exchange, ...)

Treated in Cascade model

  • II. SOFT FSI

Influence of nuclear medium on energy-momentum of particle Not included in Cascade

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Final state interactions

  • A. Nikolakopoulos

NuSTEC workshop, Pittsburgh 2019

  • I. HARD FSI

Secondary interactions

(e.g. Absorption, charge exchange, ...)

Treated in Cascade model

  • II. SOFT FSI

Influence of nuclear medium on energy-momentum of particle Not included in Cascade

In principle: coupled channels In practice : Optical potentials Imaginary part removes inelasticities from the final state

Inclusive Exclusive ↔

Look at one channel Flux is lost in inelasticities

Don’t look at the final state All inelastic channels contribute

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Final state interactions

  • A. Nikolakopoulos

NuSTEC workshop, Pittsburgh 2019

Inclusive Exclusive ↔

Look at one channel Flux is lost in inelasticities

Don’t look at the final state All inelastic channels contribute

Potentials are energy dependent because Inelasticity grows as more channels open

RGF (A. Meucci, C. Giusti, et al. ) : recover flux lost in inelastic channels RROP: Use real part of optical potential to conserve flux ED-RMF: Phenomenological reduction of real RMF potential

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Distortion of the outgoing nucleon

  • A. Nikolakopoulos

NuSTEC workshop, Pittsburgh USA

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Distortion of the outgoing nucleon

  • A. Nikolakopoulos

RMF results for (e,e’)

  • R. Gonzalez-Jimenez, A. Nikolakopoulos, N. Jachowicz and J.M.

Udias, arXiv:1904.10696 Pion production calculation also with full RMF model (including nucleon FSI) MEC from G. Megias et al. PRD 91, 073004 (2015)

NuSTEC workshop, Pittsburgh USA

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Distortion of the outgoing nucleon

  • A. Nikolakopoulos

Carbon

Titanium Argon Arxiv:1909.07497

NuSTEC workshop, Pittsburgh USA

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(e,e’ p) and Final-State Interactions

  • A. Nikolakopoulos

NuSTEC workshop, Pittsburgh USA

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(e,e’ p) and Final-State Interactions

  • A. Nikolakopoulos

Observation/Assumption: The effect of the optical potential accounts almost only for ‘hard’ rescattering events. So the MC can take care of this but the model should take into account the real part of the potential to give A good inclusive cross section

NuSTEC workshop, Pittsburgh USA

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Random Phase Approximation

  • A. Nikolakopoulos

Take into account long range nuclear excitations Every possible combination of excited SP states in nuclear medium

Mean field propagator

NuSTEC workshop, Pittsburgh USA

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Random Phase Approximation

  • A. Nikolakopoulos
  • Start from (basically) free initial and final states

➔ Large effect of RPA is needed to introduce interactions

  • MF initial and final states

➔ Effect of RPA is smaller

NuSTEC workshop, Pittsburgh USA

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Random Phase Approximation

  • A. Nikolakopoulos
  • Start from (basically) free initial and final states

➔ Large effect of RPA is needed to introduce interactions

  • MF initial and final states

➔ Effect of RPA is smaller

LaLargest reduction for low w and q

→ in QE scattering this corresponds to low Nucleon momenta → This is the region where FSI is most important Orthogonality + Spreading of wavefunction

NuSTEC workshop, Pittsburgh USA

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Nucleon FSI and Q2 distributions

  • A. Nikolakopoulos

Reduction at low Q2 Compared to RPWIA Pion potential is still Missing, one expects A reduction in the same kinematic region

NuSTEC workshop, Pittsburgh USA

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Nucleon FSI and Q2 distributions

  • A. Nikolakopoulos

Does a deficit also show up in other distributions ?

Nucleon FSI leads to an

  • verall reduction in pion

angle Slightly stronger forward reduction

Pion angle E = 2032 P3p proton in carbon

NuSTEC workshop, Pittsburgh USA

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Nucleon FSI and Q2 distributions

  • A. Nikolakopoulos

Does a deficit also show up in other distributions ?

In lepton angle mostly Forward lepton reduction

lepton angle E = 2032 P3p proton in carbon

NuSTEC workshop, Pittsburgh USA

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  • A. Nikolakopoulos

Conclusions

  • I. Nucleon complexity

→ Angular distributions require higher dimensional sampling

  • II. Nuclear complexity

→ Nuclear degrees of freedom require higher dimensional sampling

  • III. Final state interaction

→ Consistently describing inclusive and exclusive signals is complicated → Nuclear effects depend on kinematics of outgoing hadrons → higher dimensional problems

NuSTEC workshop, Pittsburgh USA