A Prime-Index Estimate of Neutrino- Nucleon Cross SectionsνN Cross Section Calculations • The calculations also follow these production classifications for ν Energy. • The only

A Prime-Index Estimate of Neutrino-Nucleon Cross Sections

Ali R. Fazely

Ali R. Fazely, Miami Conference, December 16 -22, 2015

Outline

What is a Neutrino?

How to Calculate 𝝂-N Cross Section Using Q.M.?

Prime-Index Estimate

Conclusions and Open Questions


What is a Neutrino?• An elementary particle, not part of

ordinary matter

• No electric charge, ½ integer spin, very little mass

• First postulated by Pauli (1930) to explain beta-decay

• Lepton Family:

– e-, Electon Neutrino 𝝂e

– µ, Muon Neutrino 𝝂µ

– τ, Tau Neutrino 𝝂τ

• Possesses only weak interaction and gravity

• There are lots of them!

Beta Decay

of a Neutron

n


Neutrinos within the Standard Model

https://simple.wikipedia.org/wiki/Elementary_particle


Where do Neutrinos come from?

Source of ’s

Nuclear Reactors

few MeV

Human Body

= 3.4 x 108/day

Earth’s Radioactivity

6 x 106/cm2s

Accelerators

E ~ 0.01 – 350

GeV

The Sun

e

Earth = 6 x 1010 / cm2s

E 0.1 – 17 MeV

The Big Bang, relic

neutrinos

= 330 / cm3

E = 0.0002 eV

Super Nova

SN1987

E 1-80 MeV

Atmospheric ’s

e,,e,1 /cm2s

E0.1– 10000GeV

- -

Spallation

Neutron SourceAli R. Fazely, Miami Conference, December 16 -22, 2015

http://www.lanl.gov/

http://www.lanl.gov/

6

Neutrinos as Cosmic Messengers

Protons: deflected by

magnetic fields. p

Photons: easily

absorbed by CMB

backgrounds.

γ

Neutrinos: not

deflected by magnetic

fields. Low interaction

cross-section.

ν


Supernova 1987A


Type II SupernovaOn average, every 30 years or so in our galaxy

a massive star with M > 8M explodes

Gravitational instability due to C, O, and Si fusion

into a Fe – Ni core

Gravity overcomes the electron pressure and collapse

begins; nuclear densities are reached with a core

radius of R ~ 10 km, with E = GM2/R ~ 1059 MeV

Neutrinos are trapped in the neutrino sphere and

materials bounce, cooling of the neutron star by

neutrino emission, shock wave and explosion

Ekin ~ 0.01 E.

99% of the energy is carried off by neutrinos!

Approximately ~ 1058 Neutrinos are Produced!


SN87A EventsIMB & Kamioka

IMB

KAM


“I think I can safely say that

nobody understands Quantum

Mechanics” –

- Richard Feynman

The only tool to calculate 𝝂-N cross section

is Quantum Mechanics


Neutrino Production Sources

ν Energy

10 MeV 80 MeV 350 GeV 20 TeV To GZK (EeV)

Solar & Reactor < 10 MeV

Supernova < 80 MeV

10 MeV < Accelerator < 350 GeV

100 MeV < Atmospheric < 20 TeV

Astrophysical > TeV


νN Cross Section Calculations

• The calculations also follow these production classifications

for ν Energy.

• The only free nucleon we have is the proton.

• All neutrino interactions with neutrons must occur through a

reaction with a nucleus.

Shell Model, RPA, EFT for Eν < 100 MeV

Impulse Approximation (Fermi Gas Model, Spectral

Functions) for Eν < ~ 1 GeV

Quark Parton Model for Eν > 1 GeV





In the cross section equation, the most challenging part to

calculate is the matrix element part.

At low neutrino energies, i.e. reactors, and to some extent

Supernova, |M|2 = |MF|2 + |MGT|2 ⋍ 1 + 3 = 4, from neutron

β-decay.

However, things become more complicated as neutrino

energy increases and nuclear and nucleon structure effects

come into play!



The nucleon weak current is;

Q-dependence of vector form factors (F1 and F2) can be measured

from e-scattering. The pseudo-scalar is small and the axial form

factor (FA) can only be measured by neutrino scattering from

nucleons.





Cross section measurements

Particle data group http://pdg.lbl.govAli R. Fazely, Miami Conference, December 16 -22, 2015

Structure Function Method

Formaggio and Zeller, Rev. Mod. Phys., Vol. 84, No. 3, July–September 2012


Parameterized Cross SectionsEν < 60 MeV

Vogel & Beacom,Phys. Rev.

D60, 053003 (1999)

Eν < 300 MeV

Strumia and Vissani, Phys.

Lett. B564 42-54, (2003)

Eν > 1.0 PeV, Eν in GeV

R. Gandhi, et al., Astro. Part.

Phys. 52 81, (1996)


There is an easier way!

Estimate the total neutrino

nucleon cross sections by using

the relation between prime

numbers and their indices.


Prime Numbers• Prime numbers are numbers that are divisible by themselves

and “one” only. By this definition, 1 is not considered a prime

• It is a formidable mathematical challenge to determine if a number is prime. Many algorithms have been developed to find new prime numbers.

• No method exists linking primes to their positional rank. For example, the 1143rd prime can only be realized by indexing every prime from the first one, 2, to the 1143rd, which is 9221.

• Understanding Primes would lead to appreciating the complexity of composites, twin primes, lonely primes, factorization of semi primes and other number theory problems.

22Ali R. Fazely, Miami Conference, December 16 -22, 2015

PrimesAre there infinitely many primes?The answer is yes!

Euclid (325-265 BC) proved by contradiction that there are indeed infinitely many primes!

Assume that you know the largest prime P. We then form a composite C such that,C = 2 × 3 × 5 × 7 … × P. Then,C + 1 is either a prime or not. If it is a prime then it is larger than P. If it is not a prime, then it cannot be divisible by any of the primes smaller than P. Again the prime has to be larger than P.


Prime - Index Relation


Prime NumbersIndex Prime

1 2

2 3

3 5

“ “

7 17

8 19

9 23

11 31

“ “

19 67

“ “

114 619

“ “

1143 9221

“ “

1187 961925Ali R. Fazely, Miami Conference, December 16 -22, 2015

Prime-Index Method

The method takes the index of the prime as the energy and the

prime itself then provides the cross section for that energy.

Because there are no units available, we assign the index to be

in units of MeV. Note, the choice of MeV is made because it is

of the order of the neutron-proton mass difference or of the

nuclear binding energy. The cross sections are then expressed in

units of (0.70) of 10-42 cm2 or in units of 0.70 ab (ab stands for

atto - barn = 10-18 barn).

For example, at a neutrino energy of 19 MeV which is the index

of the prime number 67, the cross section is 0.70 × 67 or ~ 47

ab, i.e. 47 × 10-42 cm2.


νN Cross Sections

27

Energy (MeV σ (p × 0.7)(ab)

2 2.1

. .

5 7.7

“ “

7 11.9

8 13.3

. .

11 21.7

. .

19 47

. .

500 2499.7

. .

1100 5831.7

. .

3500 22826.3Ali R. Fazely, Miami Conference, December 16 -22, 2015

Prime-Index Prediction of 𝝂-N Cross

Sections

Pearson Correlation Coefficients are 0.98 and 0.99 for 𝝂 and 𝝂, respectively

Prime × 0.70

Prime × 0.26

𝝂-N

𝝂-N


Ultra High Energies


Astrophysical Muon NeutrinosNorthern Sky The distribution

of reconstructed

muon energy

proxy for events

sample,

compared to the

expected

distributions for

an E-2 flux.

Significance =

3.7σ330 TeV<E<1.4 PeVPRL 115, 081102

(2015)


Observation of Highest Energy Neutrinos2.6 ± 0.3 PeV, observed June 11, 2014

(ICRC 2015, July 30 to August 6, 2015, The Hague, The Netherlands.)


Prime-Index Prediction

Prime-Index Prediction

Next Generation, IceCube-Gen2


A simulated 60-PeV horizontal muon

Ultra High Energies

• Here we use Gauss’s prime number theorem to

generate ν/ν -N cross sections.

• σ (ab) = 0.70 E(MeV)lnE(MeV) (ν-N);

• σ (ab) = 0.26 E(MeV)lnE(MeV) (ν-N)

• Gravity effects and higher dimensions are

proposed that increase ν-N cross sections by

several orders of magnitude.


Effects of Gravity and Extra DimensionsSolid Black Line is the Prime-Index Relation Prediction!

Tao Han and Dan Hooper, New Journal of Physics 6 (2004) 150


Conclusions

A simple method is proposed to estimate and predict 𝝂-

N Cross Sections over six decades of neutrino energy

and is compared to experimental data.

The method provides a quick, reliable estimate of

neutrino-nucleon/nucleus cross sections essential to rate

calculations in various neutrino experiments.

The Pearson Correlation Coefficient was calculated to

be 0.98 and 0.99 for neutrinos and antineutrinos,

respectively.

The method predicts rise in cross sections as

suggested by p-brane and extra dimensions.

Question: Why so good?


Documents

A Prime-Index Estimate of Neutrino- Nucleon Cross SectionsνN Cross Section Calculations • The calculations also follow these production classifications for ν Energy. • The only