Early Universe Nucleosynthesis

[Mordred]

This thread will take me a considerable amount of time as I will be examining various treatments of BB nucleosynthesis and development of an eventual article of processes involved. for the initial stages I will simply be gathering the relevant formulas.

Prior to symmetry  Break Relevant equations

 The FLRW metric of the LCDM universe is used by the LCDM model of the Big bang to describe the evolution history of our Observable universe. The model starts at 10^{-43} seconds forward from a low entropy, hot dense state. One plausible explanation of how our universe began prior to that include quantum fluctuations.  The model only describes our Observable portion as we do not know what occurs beyond the Cosmological event horizon. The FLRW metric is given as follows

\[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\]

\[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\]

where k is the curvature term, a is the scale factor both being dimensionless quantities. The contributions of each particle species via their corresponding equations of state is determines how our universe expands. The evolution history can be determines as a function of Cosmological redshift via the following equation

\[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

where the standard model may be represented by the covariant derivative form of the Langrangian 

\[\mathcal{L}=\underbrace{\mathbb{R}}_{GR}-\overbrace{\underbrace{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}_{Yang-Mills}}^{Maxwell}+\underbrace{i\overline{\psi}\gamma^\mu D_\mu \psi}_{Dirac}+\underbrace{|D_\mu h|^2-V(|h|)}_{Higgs}+\underbrace{h\overline{\psi}\psi}_{Yukawa}\]

 

\[V_{ckm}=V^\dagger_{\mu L} V_{dL}\]

The gauge group of electroweak interactions is 

\[SU(2)_L\otimes U(1)_Y\] where left handed quarks are in doublets of \[ SU(2)_L\] while right handed quarks are in singlets

the electroweak interaction is given by the Langrangian

\[\mathcal{L}=-\frac{1}{4}W^a_{\mu\nu}W^{\mu\nu}_a-\frac{1}{4}B_{\mu\nu}B^{\mu\nu}+\overline{\Psi}i\gamma_\mu D^\mu \Psi\]

where \[W^{1,2,3},B_\mu\] are the four spin 1 boson fields associated to the generators of the gauge transformation \[\Psi\]

The 3 generators of the \[SU(2)_L\] transformation are the three isospin operator components \[t^a=\frac{1}{2} \tau^a \] with \[\tau^a \] being the Pauli matrix and the generator of \[U(1)_\gamma\] being the weak hypercharge operator. The weak isospin "I" and hyper charge \[\gamma\] are related to the electric charge Q and given as

\[Q+I^3+\frac{\gamma}{2}\]

with quarks and lepton fields organized in left-handed doublets and right-handed singlets: 

the covariant derivative is given as

\[D^\mu=\partial_\mu+igW_\mu\frac{\tau}{2}-\frac{i\acute{g}}{2}B_\mu\]

\[\begin{pmatrix}V_\ell\\\ell\end{pmatrix}_L,\ell_R,\begin{pmatrix}u\\d\end{pmatrix}_,u_R,d_R\]

The mass eugenstates given by the Weinberg angles are

\[W\pm_\mu=\sqrt{\frac{1}{2}}(W^1_\mu\mp i W_\mu^2)\]

with the photon and Z boson given as

\[A_\mu=B\mu cos\theta_W+W^3_\mu sin\theta_W\]

\[Z_\mu=B\mu sin\theta_W+W^3_\mu cos\theta_W\]

the mass mixings are given by the CKM matrix below

\[\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}\]

Bose Einstein Statistics

\[n_i = \frac {g_i} {e^{(\varepsilon_i-\mu)/kT} - 1}\]

Fermi-Dirac statistics

\[ n_i = \frac{g_i}{e^{(\epsilon_i-\mu) / k T} + 1}\]

Maxwell Boltzmann

\[\frac{N_i}{N} = \frac {g_i} {e^{(\epsilon_i-\mu)/kT}} = \frac{g_i e^{-\epsilon_i/kT}}{Z}\]

Saha Boltzmann equation (calculate hydrogen decoupling

\[\frac{n_i+n_e}{n_i}=\frac{2}{\omega^3}\frac{g_i+1}{g_i}exp[-\frac{(\epsilon_i+1-\epsilon_i)}{k_BT}\]

Edited March 23 by Mordred
corrections

[Sensei]

For clarity, write a description of all variables and their dimensions/units.

 

Edited December 13, 2022 by Sensei

[Mordred]

1 minute ago, Sensei said:

For the clearance, write a description of the all variables and their dimensions/units.

 

good plan will do as much as possible

[MigL]

What is the intended purpose ?

I simply assume that, as the universe cools, particles of the appropriate mass/energy are created at the appropriate temperatures.
It is the annihilation of these particles with their anti-particles that poses the problem of preponderance of matter WRT anti-matter.

[Mordred]

6 minutes ago, MigL said:

What is the intended purpose ?

I simply assume that, as the universe cools, particles of the appropriate mass/energy are created at the appropriate temperatures.
It is the annihilation of these particles with their anti-particles that poses the problem of preponderance of matter WRT anti-matter.

The intended purpose will be to eventually migrate this into a full article using current cosmological parameters. Its a project I've been building towards for several years now and has always been a primary focus of my studies. Now I'd like to formalize it however evidently there is issues going on with lengthy edits as I just lost all the latex work above yet again lmao

[MigL]

Never been a fan of LaTex either.

[Mordred]

Just now, MigL said:

Never been a fan of LaTex either.

seems to be a bit of a glitch that the latex structures tend to drop ah well those equations are fairly straightforward to fix up.

[Mordred]

\[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Particle& Spin & g & Q &B&L_e &L_\mu&M (Mev)&\tau\\\hline \gamma&1&2&0&0&0&0&<3*10^{-33}&stable\\\hline e^-&1/2&2&-1&0&1&0&0.511&>2*10^{22}yrs\\\hline e+&1/2&2&1&0&-1&0&0.511&>2*10^{22}yrs\\\hline v_e&1/2&1&0&0&1&0&,5*10^{-5}&stable\\\hline \overline{v}_e&1/2&1&0&0&-1&0&<5*10^{-5}&stable\\\hline \mu^-&1/2&2&-1&0&0&1&105.7&2.2*10^{-6}sec\\\hline \mu^+&1/2&2&1&0&0&-1&105.7&2.2*10^{-5}sec\\\hline v_\mu&1/2&1&0&0&0&1&<0.25&>10^{32}yrs\\\hline \overline{v}_\mu&1/2&1&0&0&0&-1&<0.25&>10^{32}yrs \\\hline p&1/2&2&1&1&0&0&938.3&.10^{32}yrs\\\hline \overline{p}&1/2&2&-1&-1&0&0&938.3&>10^{32}yrs\\\hline n&1/2&2&0&1&0&0&939.6&898 sec\\\hline \overline{n}&1/2&2&&*-1&0&0&939.6&898 sec\\\hline \pi^0+&0&1&1&0&0&0&139.6&1.39*10^{-8}sec\\\hline \pi^-&0&1&0&0&0&0&135.0&8.7*10^{-17}sec\\\hline \pi^+&0&1&-1&0&0&0&139.6&2.6*10^{-8}sec\\\hline\end{array}}\]

 

will have to go through and update these entries table is as follows g is degrees of freedom, electric charge Q, Baryon number B,

(note need to add tau and tau neutrino for the lepton family, gauge bosons W,Z,g and Higgs as well as quarks)

electron lepton number\[ L_e\]

muon lepton number\[L_\mu\]

Edited December 14, 2022 by Mordred

[Mordred]

support document list

https://physics.nist.gov/cuu/Constants/codata.pdf

Fundamental constants 2018

https://physics.nist.gov/cuu/pdf/wall_2018.pdf

Cosmic inventory (2004)

https://arxiv.org/abs/astro-ph/0406095

2018 Planck datasets

https://www.aanda.org/articles/aa/full_html/2020/09/aa33910-18/aa33910-18.html

Planck cosmological parameters 2018

https://www.aanda.org/articles/aa/pdf/2020/09/aa33910-18.pdf

Equilibrium temperature of Hydrogen

https://iopscience.iop.org/article/10.3847/1538-4357/ab2d2f/pdf

 

 

 

Edited December 15, 2022 by Mordred

[Mordred]

FLRW Metric equations

\[d{s^2}=-{c^2}d{t^2}+a({t^2})[d{r^2}+{S,k}{(r)^2}d\Omega^2]\]

\[S\kappa(r)= \begin{cases} R sin(r/R &(k=+1)\\ r &(k=0)\\ R sin(r/R) &(k=-1) \end {cases}\]

\[\rho_{crit} = \frac{3c^2H^2}{8\pi G}\]

\[H^2=(\frac{\dot{a}}{a})^2=\frac{8 \pi G}{3}\rho+\frac{\Lambda}{3}-\frac{k}{a^2}\]

setting \[T^{\mu\nu}_\nu=0\] gives the energy stress mometum tensor as 

\[T^{\mu\nu}=pg^{\mu\nu}+(p=\rho)U^\mu U^\nu)\]

\[T^{\mu\nu}_\nu\sim\frac{d}{dt}(\rho a^3)+p(\frac{d}{dt}(a^3)=0\]

which describes the conservation of energy of a perfect fluid in commoving coordinates describes by the scale factor a with curvature term K=0.

the related GR solution the the above will be the Newton approximation.

\[G_{\mu\nu}=\eta_{\mu\nu}+H_{\mu\nu}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}\]

Thermodynamics

Tds=DU+pDV Adiabatic and isentropic fluid (closed system)

equation of state

\[w=\frac{\rho}{p}\sim p=\omega\rho\]

\[\frac{d}{d}(\rho a^3)=-p\frac{d}{dt}(a^3)=-3H\omega(\rho a^3)\]

as radiation equation of state is

\[p_R=\rho_R/3\equiv \omega=1/3 \]

radiation density in thermal equilibrium is therefore

\[\rho_R=\frac{\pi^2}{30}{g_{*S}=\sum_{i=bosons}gi(\frac{T_i}{T})^3+\frac{7}{8}\sum_{i=fermions}gi(\frac{T_i}{T})}^3 \]

\[S=\frac{2\pi^2}{45}g_{*s}(at)^3=constant\]

temperature scales inversely to the scale factor giving

\[T=T_O(1+z)\]

with the density evolution of radiation, matter and Lambda given as a function of z

\[H_z=H_o\sqrt{\Omega_m(1+z)^3+\Omega_{rad}(1+z)^4+\Omega_{\Lambda}}\]

 

 

 

 

Edited December 16, 2022 by Mordred

[Mordred]

Higgs Inflation Single scalar field Modelling.

\[S=\int d^4x\sqrt{-g}\mathcal{L}(\Phi^i\nabla_\mu \Phi^i)\]

g is determinant

Einstein Hilbert action in the absence of matter.

\[S_H=\frac{M_{pl}^2}{2}\int d^4 x\sqrt{-g\mathbb{R}}\]

set spin zero inflaton as

\[\varphi\]

minimally coupled Langrangian as per General Covariance in canonical form. (kinetic term)

\[\mathcal{L_\varphi}=-\frac{1}{2}g^{\mu\nu}\nabla_\mu \varphi \nabla_\nu \varphi-V(\varphi)\]

where \[V(\varphi)\] is the potential term

integrate the two actions of the previous two equations for minimal scalar field gravitational couplings

\[S=\int d^4 x\sqrt{-g}[\frac{M_{pl}^2}{2}\mathbb{R}-\frac{1}{2}g^{\mu\nu}\nabla_\mu\varphi \nabla_\nu \varphi-V(\varphi)]\]

variations yield the Euler_Langrene

\[\frac{\partial \mathcal{L}}{\partial \Phi^i}-\nabla_\mu(\frac{\partial \mathcal{L}}{\partial[\nabla_\mu \Phi^i]})=0\]

using Euclidean commoving metric

\[ds^2-dt^2+a^2(t)(dx^2+dy^2=dz^2)\]

this becomes

\[\ddot{\varphi}+3\dot{\varphi}+V_\varphi=0\]

\[S=\int d^4 x\sqrt{-g}[\frac{M_{pl}^2}{2}\mathbb{R}-\frac{1}{2}g^{\mu\nu}\nabla_\mu\varphi \nabla_\nu \varphi-V(\varphi)]\]

and 

\[G_{\mu\nu}-\frac{1}{M_{pl}}T_{\mu\nu}\]

with flat commoving geometry of a perfect fluid gives the energy momentum for inflation as 

\[T^\mu_\nu=g^{\mu\lambda}\varphi_\lambda \varphi_\nu -\delta^\mu_\nu \frac{1}{2}g^{\rho \sigma} \varphi_\rho \varphi_\sigma V(\varphi)]\]

 

\[\rho=T^0_0=\frac{1}{2}\dot{\varphi}^2+V\]

\[p=T^i_i (diag)=\frac{1}{2}\dot{\varphi}^2-V\]

\[w=\frac{p}{\rho}\]

\[w=\frac{1-2 V/\dot{\varphi^2}}{1+2V/\dot{\varphi^2}}\]

***method by Fernando A. Bracho Blok Thesis paper.***

https://helda.helsinki.fi/bitstream/handle/10138/322422/Brachoblok_fernando_thesis_2020.pdf?sequence=2&isAllowed=y

now to examine it to other Higgs single scalar field field methodologies.

in particular

https://arxiv.org/abs/1402.3738

equation 16 of the above article matches 2.38 and 2.39 of the Brachoblok paper with two different methodologies. (cool need to further study both methods)

\[\rho=T^0_0=\frac{1}{2}\dot{\varphi}^2+V\]

\[p=T^i_i (diag)=\frac{1}{2}\dot{\varphi}^2-V\]

https://arxiv.org/abs/1303.3787 (for this I will need to research Jordon frame) in particular page 23 (single scalar Higgs)

 goals check list.

(single scalar field (Higgs prior to electroweak symmetry breakings. For symmetry break (Higgs and Yukawa couplings of the CkMS unity triangle.). Follow through with each particle species including generations via Higgs). (details and preliminary work aforementioned). Hydrogen, lithium and deuterium dropout. (Saha equations) for particle species Maxwell Boltzmann, Bose-Einstein and Fermi Dirac. statistics). Apply Principle of General Covariance throughout.

 

 

 

 

 

Edited December 23, 2022 by Mordred

[Mordred]

Phase space equations applications to Bolztmann.

\[q_i=ap_i\]

commoving coordinate of particle as r^i. 

proper momenta as p_i

\[P_i=\frac{m_a dx_i}{\sqrt{-ds^s}=(1-\psi}q_i\]

particle density in canonical phase space distribution

\[(f^a,P_j,\tau)\]

\[dNa=f_a(r_i,P_j \tau)d^3 r_id^3P_j\]

for every particle species and their polarizations (a) the energy momentum is given in the Newtonian gauge by the expression (first order) by

\[T_{a\nu}^{\mu}=\int d^3 p_i \frac{p^\mu p_\nu}{p^0}f_a\]

with \[p^0=-p^0=\sqrt{(q/a)^2+m_a^2}\]

\[p^i=p_i=q_i/a\]

obeys Boltzmann equation of the form

\[\dot{f}+\dot{r}^i\frac{\partial f}{\partial r^i}+\dot{q}\frac{\partial f}{\partial q}+\frac{\partial f}{\partial \tau}_c\]

Relativistic Wigner Function Approach to Neutrino Propagation in Matter.

 

https://arxiv.org/pdf/hep-ph/9810347.pdf

Signatures of Relativistic Neutrinos in CMB Anisotropy and Matter Clustering

https://arxiv.org/pdf/astro-ph/0310198.pdf

 

 

 

Edited December 29, 2022 by Mordred

[Mordred]

Fermi's Golden Rule

\[\Gamma=\frac{2\pi}{\hbar}|V_{fi}|^2\frac{dN}{DE_f}\]

density of states

\[\langle x|\psi\rangle\propto exp(ik\cdot x)\]

with periodic boundary condition as "a"\[k_x=2\pi n/a\]

number of momentum states

\[dN=\frac{d^3p}{(2\pi)^2}V\]

decay rate

\[\Gamma\]

Hamilton coupling matrix element between initial and final state

\[V_{fi}\]

density of final state

\[\frac{dN}{dE_f}\]

number of particles remaining at time t (decay law)

\[\frac{dN}{dt}=-\Gamma N\]

average proper lifetime probability

\[p(t)\delta t=-\frac{1}{N}\frac{dN}{dt}\delta t=\Gamma\exp-(\Gamma t)\delta t\]

mean lifetime \[\tau=<t>=\frac{\int_0^\infty tp (t) dt}{\int_0^\infty p (t) dt}=\frac{1}{\Gamma}\]

relativistic decay rate set 

\[L_o=\beta\gamma c\tau\] average number after some distance x

\[N=N_0\exp(-x/l_0)\]

 

Edited January 1, 2023 by Mordred

[Mordred]

Early Universe Cross section list

Breit Wigner cross section

\[\sigma(E)=\frac{2J+1}{2s_1+1)(2S_2+1)}\frac{4\pi}{k^2}[\frac{\Gamma^2/4}{(E-E_0)^2+\Gamma/4)}]B_{in}B_{out}\]

E=c.m energy, J is spin of resonance, (2S_1+1)(2s_2+1) is the #of polarization states of the two incident particles, the c.m., initial momentum k E_0 is the energy c.m. at resonance, \Gamma is full width at half max amplitude, B_[in} B_{out] are the initial and final state for narrow resonance the [] can be replaced by

\[\pi\Gamma\delta(E-E_0)^2/2\]

The production of point-like, spin-1/2 fermions in e+e− annihilation through a virtual photon at c.m.

\[e^+,e^-\longrightarrow\gamma^\ast\longrightarrow f\bar{f}\]

\[\frac{d\sigma}{d\Omega}=N_c{\alpha^2}{4S}\beta[1+\cos^2\theta+(1-\beta^2)\sin^2\theta]Q^2_f\]

where

\[\beta=v/c\]

c/m frame scattering angle

\[\theta\] 

fermion charge

\[Q_f\]

if factor [N_c=1=charged leptons if N_c=3 for quarks.

if v=c then (ultrarelativistic particles)

\[\sigma=N_cQ^2_f\frac{4\pi\alpha^2}{3s}=N_cQ^2_f\frac{86.8 nb}{s (GeV^2)}\]

2 pair quark to 2 pair quark

\[\frac{d\sigma}{d\Omega}(q\bar{q}\rightarrow \acute{q}\acute{\bar{q}})=\frac{\alpha^2_s}{9s}\frac{t^2+u^2}{s^2}\]

cross pair symmetry gives

\[\frac{d\sigma}{d\Omega}(q\bar{q}\rightarrow \acute{q}\acute{\bar{q}})=\frac{\alpha^2_s}{9s}\frac{t^2+u^2}{t^2}\]

 

 

 

 

 

Edited March 31 by Mordred

[Mordred]

Higgs cross sections partial width's

\[\Gamma(H\rightarrow f\bar{f})=\frac{G_Fm_f^2m_HN_c}{4\pi \sqrt{2}}(1-4m^2_f/m^2_H)^{3/2}\]

\[\Gamma(H\rightarrow W^+ W^-)=\frac{GF M^3_H\beta_W}{32\pi\sqrt{2}}(4-4a_w+3a_W^2)\]

\[\Gamma(H\rightarrow ZZ)=\frac{GF M^3_H\beta_z}{64\pi\sqrt{2}}(4-4a_Z+3a_Z^2)\]

Edited May 21, 2023 by Mordred

[Mordred]

Bump, still examining this still trying to figure out thermal equilibrium dropout of several other particles and relevant atoms via Saha equations

[Mordred]

Deuterium BBN reference

https://www.astro.uvic.ca/~jwillis/teaching/astr405/astr405_lecture4.pdf

equation 18

\[\frac{\mathcal{n}_D}{\mathcal{n}_p\mathcal{n}_n}=6(\frac{m_nk\tau}{\pi\hbar^2})^{-3/2}exp(\frac{Q_D}{k\tau}\]

Edited April 10, 2023 by Mordred

[Mordred]

just setting reminder equations that I find handy, in this case the Langrene  that correlates the action of the various particle interations  ( close to a unification....lol also reminds me how to do some interesting latex techniques...

[latex] \mathcal{L}=\underbrace{\mathbb{R}}_{GR}-\overbrace{\underbrace{\frac{1}{4}F_{\mu\nu}F^{\mu\nu}}_{Yang-Mills}}^{Maxwell}+\underbrace{i\overline{\psi}\gamma^\mu D_\mu \psi}_{Dirac}+\underbrace{|D_\mu h|^2-V(|h|)}_{Higgs}+\underbrace{h\overline{\psi}\psi}_{Yukawa}[/latex]

 [latex]D_\mu[/latex] minimally coupled gauge covariant derivative. h Higg's bosonic field [latex] \chi[/latex] is the Goldstone boson (not shown above) Goldstone no longer applies after spontaneous symmetry breaking [latex]\overline{\psi}[/latex] is the adjoint spinor

[latex]\mathcal{L}_h=|D\mu|^2-\lambda(|h|^2-\frac{v^2}{2})^2[/latex]

[latex]D_\mu=\partial_\mu-ie A_\mu[/latex] where [latex] A_\mu[/latex] is the electromagnetic four potential 

QCD gauge covariant derivative

[latex] D_\mu=\partial_\mu \pm ig_s t_a \mathcal{A}^a_\mu[/latex] matrix A represents each scalar gluon field

 

 

Single Dirac Field

[latex]\mathcal{L}=\overline{\psi}I\gamma^\mu\partial_\mu-m)\psi[/latex]

under U(1) EM fermion field equates to 

[latex]\psi\rightarrow\acute{\psi}=e^{I\alpha(x)Q}\psi[/latex]

due to invariance requirement of the Langrene above and with the last equation leads to the gauge field [latex]A_\mu[/latex]

[latex] \partial_\mu[/latex] is replaced by the covariant derivitave

[latex]\partial_\mu\rightarrow D_\mu=\partial_\mu+ieQA_\mu[/latex]

where [latex]A_\mu[/latex] transforms as [latex]A_\mu+\frac{1}{e}\partial_\mu\alpha[/latex]

Single Gauge field U(1)

[latex]\mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}[/latex]

[latex]F_{\mu\nu}=\partial_\nu A_\mu-\partial_\mu A_\nu[/latex]

add mass which violates local gauge invariance above

[latex]\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{2}m^2A_\mu A^\mu[/latex] guage invariance demands photon be massless to repair gauge invariance add a single complex scalar field

[latex]\phi=\frac{1}{\sqrt{2}}(\phi_1+i\phi_2[/latex]

Langrene becomes

[latex] \mathcal{L}=\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+|D_\mu \phi|^2-V_\phi[/latex]

where [latex]D_\mu=\partial_\mu-ieA_\mu[/latex]

[latex]V_\phi=\mu^2|\phi^2|+\lambda(|\phi^2|)^2[/latex]

[latex]\overline{\psi}=\psi^\dagger \gamma^0[/latex] where [latex]\psi^\dagger[/latex] is the hermitean adjoint and [latex]\gamma^0 [/latex] is the timelike gamma matrix

the four contravariant matrix are as follows

[latex]\gamma^0=\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}[/latex]

[latex]\gamma^1=\begin{pmatrix}0&0&0&1\\0&0&1&0\\0&0&-1&0\\-1&0&0&0\end{pmatrix}[/latex]

[latex]\gamma^2=\begin{pmatrix}0&0&0&-i\\0&0&i&0\\0&i&0&0\\-i&0&0&0\end{pmatrix}[/latex]

[latex]\gamma^3=\begin{pmatrix}0&0&1&0\\0&0&0&-1\\-1&0&0&0\\0&1&0&0\end{pmatrix}[/latex]

where [latex] \gamma^0[/latex] is timelike rest are spacelike

V denotes the CKM matrix usage

[latex]\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}[/latex] 

[latex]V_{ckm}=V^\dagger_{\mu L} V_{dL}[/latex]

the CKM mixing angles correlates the cross section between the mass eigenstates and the weak interaction eigenstates. Involves CP violations and chirality relations.

Dirac 4 component spinor fields

[latex]\gamma^5=i\gamma_0,\gamma_1,\gamma_2,\gamma_3[/latex]

4 component Minkowskii with above 4 component Dirac Spinor and 4 component Dirac gamma matrixes are defined as

[latex] {\gamma^\mu\gamma^\nu}=2g^{\mu\nu}\mathbb{I}[/latex] where [latex]\mathbb{I}[/latex] is the identity matrix. (required under MSSM electroweak symmetry break}

in Chiral basis [latex]\gamma^5[/latex] is diagonal in [latex]2\otimes 2[/latex] the gamma matrixes are

[latex]\begin{pmatrix}0&\sigma^\mu_{\alpha\beta}\\\overline{\sigma^{\mu\dot{\alpha}\beta}}&0\end{pmatrix}[/latex]

[latex]\gamma^5=i{\gamma_0,\gamma_1,\gamma_2,\gamma_3}=\begin{pmatrix}-\delta_\alpha^\beta&0\\0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}[/latex]

[latex]\mathbb{I}=\begin{pmatrix}\delta_\alpha^\beta&0\\0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}[/latex]

Lorentz group identifiers in [latex](\frac{1}{2},0)\otimes(0,\frac{1}{2})[/latex]

[latex]\sigma\frac{I}{4}=(\gamma^\mu\gamma^\nu)=\begin{pmatrix}\sigma^{\mu\nu\beta}_{\alpha}&0\\0&-\sigma^{\mu\nu\dot{\alpha}}_{\dot{\beta}}\end{pmatrix}[/latex]

[latex]\sigma^{\mu\nu}[/latex] duality satisfies [latex]\gamma_5\sigma^{\mu\nu}=\frac{1}{2}I\epsilon^{\mu\nu\rho\tau}\sigma_{\rho\tau}[/latex]

a 4 component Spinor Dirac field is made up of two mass degenerate Dirac spinor fields U(1) helicity 

[latex](\chi_\alpha(x)),(\eta_\beta(x))[/latex]

 

[latex]\psi(x)=\begin{pmatrix}\chi^{\alpha\beta}(x)\\ \eta^{\dagger \dot{\alpha}}(x)\end{pmatrix}[/latex]

the [latex](\alpha\beta)=(\frac{1}{2},0)[/latex] while the [latex](\dot{\alpha}\dot{\beta})=(0,\frac{1}{2})[/latex]

this section relates the SO(4) double cover of the SU(2) gauge requiring the chiral projection operator next.

chiral projections operator

[latex]P_L=\frac{1}{2}(\mathbb{I}-\gamma_5=\begin{pmatrix}\delta_\alpha^\beta&0\\0&0\end{pmatrix}[/latex]

[latex]P_R=\frac{1}{2}(\mathbb{I}+\gamma_5=\begin{pmatrix}0&0\\ 0&\delta^\dot{\alpha}_\dot{\beta}\end{pmatrix}[/latex]

 

Weyl spinors

[latex]\psi_L(x)=P_L\psi(x)=\begin{pmatrix}\chi_\alpha(x)\\0\end{pmatrix}[/latex]

[latex]\psi_R(x)=P_R\psi(x)=\begin{pmatrix}0\\ \eta^{\dagger\dot{a}}(x)\end{pmatrix}[/latex]

 

 

also requires Yukawa couplings...SU(2) matrixes given by

[latex]diag(Y_{u1},Y_{u2},Y_{u3})=diag(Y_u,Y_c,Y_t)=diag(L^t_u,\mathbb{Y}_u,R_u)[/latex]

[latex]diag(Y_{d1},Y_{d2},Y_{d3})=diag(Y_d,Y_s,Y_b)=diag(L^t_d,\mathbb{Y}_d,R_d[/latex]

[latex]diag(Y_{\ell 1},Y_{\ell 2},Y_{\ell3})=diag(Y_e,Y_\mu,Y_\tau)=diag(L^T_\ell,\mathbb{Y}_\ell,R_\ell)[/latex]

the fermion masses

[latex]Y_{ui}=m_{ui}/V_u[/latex]

[latex]Y_{di}=m_{di}/V_d[/latex]

[latex]Y_{\ell i}=m_{\ell i}/V_\ell[/latex]

Reminder notes: Dirac is massive 1/2 fermions, Weyl the massless. Majorona  fermion has its own antiparticle pair while Dirac and Weyl do not.  The RH neutrino would be more massive than the LH neutrino, same for the corresponding LH antineutrino and RH Neutrino via seesaw mechanism which is used with the seesaw mechanism under MSM. Under MSSM with different Higgs/higglets can be numerous seesaws.  The Majorona method has conservation violations also these fermions must be electric charge neutral. (must be antiparticles of themselves) the CKM and PMNS are different mixing angels in distinction from on another. However they operate much the same way. CKM is more commonly used as its better tested to higher precision levels atm.

Quark family is Dirac fermions due to electric charge cannot be its own antiparticle. Same applies to the charged lepton family. Neutrinos are members of the charge neutral lepton family

CKM is also a different parametrisation than the Wolfenstein Parametrization in what way (next study)

Lorentz group

Lorentz transformations list spherical coordinates (rotation along the z axis through an angle ) \[\theta\]

\[(x^0,x^1,x^2,x^3)=(ct,r,\theta\phi)\]

\[(x_0,x_1,x_2,x_3)=(-ct,r,r^2,\theta,[r^2\sin^2\theta]\phi)\]

 

\[\acute{x}=x\cos\theta+y\sin\theta,,,\acute{y}=-x\sin\theta+y \cos\theta\]

\[\Lambda^\mu_\nu=\begin{pmatrix}1&0&0&0\\0&\cos\theta&\sin\theta&0\\0&\sin\theta&\cos\theta&0\\0&0&0&1\end{pmatrix}\]

generator along z axis

\[k_z=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}\]

generator of boost along x axis::

\[k_x=\frac{1\partial\phi}{i\partial\phi}|_{\phi=0}=-i\begin{pmatrix}0&1&0&0\\1&0&0&0\\0&0&0&0\\0&0&0&0 \end{pmatrix}\]

boost along y axis\

\[k_y=-i\begin{pmatrix}0&0&1&0\\0&0&0&0\\1&0&0&0\\0&0&0&0 \end{pmatrix}\]

generator of boost along z direction

\[k_z=-i\begin{pmatrix}0&0&0&1\\0&0&0&0\\0&0&0&0\\1&0&0&0 \end{pmatrix}\]

the above is the generator of boosts below is the generator of rotations.

\[J_z=\frac{1\partial\Lambda}{i\partial\theta}|_{\theta=0}\]

\[J_x=-i\begin{pmatrix}0&0&0&0\\0&0&0&0\\0&0&0&1\\0&0&-1&0 \end{pmatrix}\]

\[J_y=-i\begin{pmatrix}0&0&0&0\\0&0&0&-1\\0&0&1&0\\0&0&0&0 \end{pmatrix}\]

\[J_z=-i\begin{pmatrix}0&0&0&0\\0&0&1&0\\0&-1&0&0\\0&0&0&0 \end{pmatrix}\]

there is the boosts and rotations we will need

and they obey commutations

\[[A,B]=AB-BA\]

SO(3) Rotations list

set x,y,z rotation as

\[\varphi,\Phi\phi\]

\[R_x(\varphi)=\begin{pmatrix}1&0&0\\0&\cos\varphi&\sin\varphi\\o&-sin\varphi&cos\varphi \end{pmatrix}\]

\[R_y(\phi)=\begin{pmatrix}cos\Phi&0&\sin\Phi\\0&1&0\\-sin\Phi&0&cos\Phi\end{pmatrix}\]

\[R_z(\phi)=\begin{pmatrix}cos\theta&sin\theta&0\\-sin\theta&\cos\theta&o\\o&0&1 \end{pmatrix}\]

Generators for each non commutative group.

\[J_x=-i\frac{dR_x}{d\varphi}|_{\varphi=0}=\begin{pmatrix}0&0&0\\0&0&-i\\o&i&0\end{pmatrix}\]

\[J_y=-i\frac{dR_y}{d\Phi}|_{\Phi=0}=\begin{pmatrix}0&0&-i\\0&0&0\\i&i&0\end{pmatrix}\]

\[J_z=-i\frac{dR_z}{d\phi}|_{\phi=0}=\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}\]

with angular momentum operator

\[{J_i,J_J}=i\epsilon_{ijk}J_k\]

with Levi-Civita

 

\[\varepsilon_{123}=\varepsilon_{312}=\varepsilon_{231}=+1\]

\[\varepsilon_{123}=\varepsilon_{321}=\varepsilon_{213}=-1\]

SU(3) generators Gell Mann matrix's

\[\lambda_1=\begin{pmatrix}0&-1&0\\1&0&0\\0&0&0\end{pmatrix}\]

\[\lambda_2=\begin{pmatrix}0&-i&0\\i&0&0\\0&0&0\end{pmatrix}\]

\[\lambda_3=\begin{pmatrix}1&0&0\\0&-1&0\\0&0&0\end{pmatrix}\]

\[\lambda_4=\begin{pmatrix}0&0&1\\0&0&0\\1&0&0\end{pmatrix}\]

\[\lambda_5=\begin{pmatrix}0&0&-i\\0&0&0\\i&0&0\end{pmatrix}\]

\[\lambda_6=\begin{pmatrix}0&0&0\\0&0&1\\0&1&0\end{pmatrix}\]

\[\lambda_7=\begin{pmatrix}0&0&0\\0&0&-i\\0&i&0\end{pmatrix}\]

\[\lambda_8=\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\0&1&0\\0&0&-2\end{pmatrix}\]

commutation relations

\[[\lambda_i\lambda_j]=2i\sum^8_{k=1}f_{ijk}\lambda_k\]

with algebraic structure

\[f_{123}=1,f_{147}=f_{165}=f_{246}=f_{246}=f_{257}=f_{345}=f_{376}=\frac{1}{2},f_{458}=f_{678}=\frac{3}{2}\]

with Casimer Operator

\[\vec{J}^2=J_x^2+J_y^2+j_z^2\]

 

 

 

 

 

Edited May 17, 2023 by Mordred

[Mordred]

right hand neutrino 

details to examine in particular 3 LH neutrinos with 4 

https://arxiv.org/pdf/1911.05092.pdf

https://arxiv.org/pdf/1901.00151.pdf

https://arxiv.org/pdf/2109.00767v2.pdf

question to examine how many seesaw mechanism would 3 doublet 4 singlet Higgs entail and would this lead to Pati-Salam solutions pertaining to SO(10 MSSM).

needs further examination Mikheyev–Smirnov–Wolfenstein (MSW) potential

3.5 KeV xray anomoly

https://arxiv.org/abs/1402.2301

requirements sterile neutrino mass terms must be in the KeV range to satisfy sterile neutrinos as a DM candidate

 

[Genady]

22 hours ago, Mordred said:

just setting reminder equations that I find handy, in this case the Langrene  that correlates the action of the various particle interations  ( close to a unification....lol also reminds me how to do some interesting latex techniques...

L=RGR−14FμνFμνYang−MillsMaxwell+iψ¯¯¯γμDμψDirac+|Dμh|2−V(|h|)Higgs+hψ¯¯¯ψYukawa

 Dμ minimally coupled gauge covariant derivative. h Higg's bosonic field χ is the Goldstone boson (not shown above) Goldstone no longer applies after spontaneous symmetry breaking ψ¯¯¯ is the adjoint spinor

Lh=|Dμ|2−λ(|h|2−v22)2

Dμ=∂μ−ieAμ where Aμ is the electromagnetic four potential 

QCD gauge covariant derivative

Dμ=∂μ±igstaAaμ matrix A represents each scalar gluon field

 

 

Single Dirac Field

L=ψ¯¯¯Iγμ∂μ−m)ψ

under U(1) EM fermion field equates to 

ψ→ψ´=eIα(x)Qψ

due to invariance requirement of the Langrene above and with the last equation leads to the gauge field Aμ

∂μ is replaced by the covariant derivitave

∂μ→Dμ=∂μ+ieQAμ

where Aμ transforms as Aμ+1e∂μα

Single Gauge field U(1)

L=14FμνFμν

Fμν=∂νAμ−∂μAν

add mass which violates local gauge invariance above

L=−14FμνFμν+12m2AμAμ guage invariance demands photon be massless to repair gauge invariance add a single complex scalar field

ϕ=12√(ϕ1+iϕ2

Langrene becomes

L=14FμνFμν+|Dμϕ|2−Vϕ

where Dμ=∂μ−ieAμ

Vϕ=μ2|ϕ2|+λ(|ϕ2|)2

ψ¯¯¯=ψ†γ0 where ψ† is the hermitean adjoint and γ0 is the timelike gamma matrix

the four contravariant matrix are as follows

γ0=⎛⎝⎜⎜⎜1000010000−10000−1⎞⎠⎟⎟⎟

γ1=⎛⎝⎜⎜⎜000−1000001−101000⎞⎠⎟⎟⎟

γ2=⎛⎝⎜⎜⎜000−i00i00i00−i000⎞⎠⎟⎟⎟

γ3=⎛⎝⎜⎜⎜00−10000110000−100⎞⎠⎟⎟⎟

where γ0 is timelike rest are spacelike

V denotes the CKM matrix usage

⎛⎝⎜⎜d´s´b´⎞⎠⎟⎟⎛⎝⎜VudVcdVtdVusVcsVtsVubVcbVtb⎞⎠⎟⎛⎝⎜dsb⎞⎠⎟  

Vckm=V†μLVdL

the CKM mixing angles correlates the cross section between the mass eigenstates and the weak interaction eigenstates. Involves CP violations and chirality relations.

Dirac 4 component spinor fields

γ5=iγ0,γ1,γ2,γ3

4 component Minkowskii with above 4 component Dirac Spinor and 4 component Dirac gamma matrixes are defined as

γμγν=2gμνI where I is the identity matrix. (required under MSSM electroweak symmetry break}

in Chiral basis γ5 is diagonal in 2⊗2 the gamma matrixes are

(0σμα˙β¯¯¯¯¯¯¯¯¯¯σμαβ0)

γ5=iγ0,γ1,γ2,γ3=(−δβα00δα˙β˙)

I=(δβα00δα˙β˙)

Lorentz group identifiers in (12,0)⊗(0,12)

σI4=(γμγν)=⎛⎝σμνβα00−σμνα˙β˙⎞⎠

σμν duality satisfies γ5σμν=12Iϵμνρτσρτ

a 4 component Spinor Dirac field is made up of two mass degenerate Dirac spinor fields U(1) helicity 

(χα(x)),(ηβ(x))

 

ψ(x)=(χαβ(x)η†α˙(x))

the (αβ)=(12,0) while the (α˙β˙)=(0,12)

this section relates the SO(4) double cover of the SU(2) gauge requiring the chiral projection operator next.

chiral projections operator

PL=12(I−γ5=(δβα000)

PR=12(I+γ5=(000δα˙β˙)

 

Weyl spinors

ψL(x)=PLψ(x)=(χα(x)0)

ψR(x)=PRψ(x)=(0η†a˙(x))

 

 

also requires Yukawa couplings...SU(2) matrixes given by

diag(Yu1,Yu2,Yu3)=diag(Yu,Yc,Yt)=diag(Ltu,Yu,Ru)

diag(Yd1,Yd2,Yd3)=diag(Yd,Ys,Yb)=diag(Ltd,Yd,Rd

diag(Yℓ1,Yℓ2,Yℓ3)=diag(Ye,Yμ,Yτ)=diag(LTℓ,Yℓ,Rℓ)

the fermion masses

Yui=mui/Vu

Ydi=mdi/Vd

Yℓi=mℓi/Vℓ

Reminder notes: Dirac is massive 1/2 fermions, Weyl the massless. Majorona  fermion has its own antiparticle pair while Dirac and Weyl do not.  The RH neutrino would be more massive than the LH neutrino, same for the corresponding LH antineutrino and RH Neutrino via seesaw mechanism which is used with the seesaw mechanism under MSM. Under MSSM with different Higgs/higglets can be numerous seesaws.  The Majorona method has conservation violations also these fermions must be electric charge neutral. (must be antiparticles of themselves) the CKM and PMNS are different mixing angels in distinction from on another. However they operate much the same way. CKM is more commonly used as its better tested to higher precision levels atm.

Quark family is Dirac fermions due to electric charge cannot be its own antiparticle. Same applies to the charged lepton family. Neutrinos are members of the charge neutral lepton family

CKM is also a different parametrisation than the Wolfenstein Parametrization in what way (next study)

Lorentz group

Lorentz transformations list spherical coordinates (rotation along the z axis through an angle )

θ

 

 

(x0,x1,x2,x3)=(ct,r,θϕ)

 

 

(x0,x1,x2,x3)=(−ct,r,r2,θ,[r2sin2θ]ϕ)

 

 

 

x´=xcosθ+ysinθ,,,y´=−xsinθ+ycosθ

 

 

Λμν=⎛⎝⎜⎜⎜10000cosθsinθ00sinθcosθ00001⎞⎠⎟⎟⎟

 

generator along z axis

 

kz=1∂ϕi∂ϕ|ϕ=0

 

generator of boost along x axis::

 

kx=1∂ϕi∂ϕ|ϕ=0=−i⎛⎝⎜⎜⎜0100100000000000⎞⎠⎟⎟⎟

 

boost along y axis\

 

ky=−i⎛⎝⎜⎜⎜0010000010000000⎞⎠⎟⎟⎟

 

generator of boost along z direction

 

kz=−i⎛⎝⎜⎜⎜0001000000001000⎞⎠⎟⎟⎟

 

the above is the generator of boosts below is the generator of rotations.

 

Jz=1∂Λi∂θ|θ=0

 

 

Jx=−i⎛⎝⎜⎜⎜00000000000−10010⎞⎠⎟⎟⎟

 

 

Jy=−i⎛⎝⎜⎜⎜0000000000100−100⎞⎠⎟⎟⎟

 

 

Jz=−i⎛⎝⎜⎜⎜000000−1001000000⎞⎠⎟⎟⎟

 

there is the boosts and rotations we will need

and they obey commutations

 

[A,B]=AB−BA

 

SO(3) Rotations list

set x,y,z rotation as

 

φ,Φϕ

 

 

Rx(φ)=⎛⎝⎜10o0cosφ−sinφ0sinφcosφ⎞⎠⎟

 

 

Ry(ϕ)=⎛⎝⎜cosΦ0−sinΦ010sinΦ0cosΦ⎞⎠⎟

 

 

Rz(ϕ)=⎛⎝⎜cosθ−sinθosinθcosθ00o1⎞⎠⎟

 

Generators for each non commutative group.

 

Jx=−idRxdφ|φ=0=⎛⎝⎜00o00i0−i0⎞⎠⎟

 

 

Jy=−idRydΦ|Φ=0=⎛⎝⎜00i00i−i00⎞⎠⎟

 

 

Jz=−idRzdϕ|ϕ=0=⎛⎝⎜0i0−i00000⎞⎠⎟

 

with angular momentum operator

 

Ji,JJ=iϵijkJk

 

with Levi-Civita

 

 

ε123=ε312=ε231=+1

 

 

ε123=ε321=ε213=−1

 

SU(3) generators Gell Mann matrix's

 

λ1=⎛⎝⎜010−100000⎞⎠⎟

 

 

λ2=⎛⎝⎜0i0−i00000⎞⎠⎟

 

 

λ3=⎛⎝⎜1000−10000⎞⎠⎟

 

 

λ4=⎛⎝⎜001000100⎞⎠⎟

 

 

λ5=⎛⎝⎜00i000−i00⎞⎠⎟

 

 

λ6=⎛⎝⎜000001010⎞⎠⎟

 

 

λ7=⎛⎝⎜00000i0−i0⎞⎠⎟

 

 

λ8=13–√⎛⎝⎜10001000−2⎞⎠⎟

 

commutation relations

 

[λiλj]=2i∑k=18fijkλk

 

with algebraic structure

 

f123=1,f147=f165=f246=f246=f257=f345=f376=12,f458=f678=32

 

with Casimer Operator

 

J⃗ 2=J2x+J2y+j2z

 

 

 

 

 

 

I've noticed that the expressions for single Dirac field, chiral projection operators, and single complex scalar field have typos.

[Mordred]

Thanks For pointing that out. I will make the corrections once I get a chance though I may just change that section to a more standardized notation.

+1 for catching that appreciate it.

edit: Yeah I see what you mean I am going to change it to a more standardized format. Thanks again for the catch. I had pulled it from some old note I had put together a few years back. Likely an older format for the Majorana basis there is better and clearer methods. It was from my older notes when I was studying Majorana.

yeah figured out what is the issue is I couldn't recall why I needed the identity matrix

[latex]\mathbb{I}[/latex]

the format pertains to MSSM where the identity matrix is a requirement. I won't be using this format so will change it to the MSM format with the modern tilde to denote Majorona fields. Its from back when I was studying Majorona under Pati-Salam. Its required for the supersymmetric partner identities. Completely forgot about that lmao

 

Edited May 18, 2023 by Mordred

[Mordred]

Higgs cross sections partial width's

\[\Gamma(H\rightarrow f\bar{f})=\frac{G_Fm_f^2m_HN_c}{4\pi \sqrt{2}}(1-4m^2_f/m^2_H)^{3/2}\]

\[\Gamma(H\rightarrow W^+ W^-)=\frac{GF M^3_H\beta_W}{32\pi\sqrt{2}}(4-4a_w+3a_W^2)\]

\[\Gamma(H\rightarrow ZZ)=\frac{GF M^3_H\beta_z}{64\pi\sqrt{2}}(4-4a_Z+3a_Z^2)\]

\[N_c=3\] for quarks 1 for leptons

\[a_w=1-\beta^2_W=\frac{4m^2_w}{m^2_H}\]

\[a_Z=1-\beta^2_Z=\frac{4m^2_Z}{m^2_H}\]

explicitely

\[\Gamma(H\longrightarrow gg)=\frac{\alpha_s^2G_FM^3_H}{36\pi^3\sqrt{2}}|\sum_q I(\frac{m^2_q}{m^2_H}|^2\]

Higgsstralung with k in c.m momentum of Higgs boson

\[\sigma(g_i\overline{q}_j)\rightarrow=\frac{\pi \alpha^2 |V_{ij}|^2}{36sin^4\theta_W}\frac{2k}{\sqrt{s}}\frac{k^2+3m^2_W}{(s-m^2_W)^2}\]

\[\sigma(f\acute{f}\rightarrow ZH)=\frac{2\pi\alpha^2|v_{ij}|^2(\ell^2_f+r^2_f)}{48n_csin^4\theta_Wcos_W^2}\frac{2k}{\sqrt{s}}\frac{k^2+3m_Z^2}{(s-m^2_Z)^2}\]

note last equation shows all quarks contribute to ZZ fusion process

Edited May 21, 2023 by Mordred

[Mordred]

SO(3,1) universal cover SL(2C)  spin1/2

Lie group Pauli matrices

\[SL(2\mathbb{C})={M\in Mat(2\mathbb{C});det(M)=1}\]

\[(X= 2*2) Hermitian-matrices \begin{pmatrix}x^2+x^3&x^1-ix^2\\x^1+ix^2&x^0-x^3\end{pmatrix}\]

\[\sigma_0=\begin{pmatrix}1&0\\0&1\end{pmatrix}\]

\[\sigma_1=\begin{pmatrix}0&1\\1&0\end{pmatrix}\]

\[\sigma_2=\begin{pmatrix}0&i\\-i&0\end{pmatrix}\]

\[\sigma_3=\begin{pmatrix}1&0\\0&-1\end{pmatrix}\]

\[det(x)=x_0^2-x_1^2-x_2^2-x_3^2\]

 

\[\Psi=\begin{pmatrix}\Psi+\\\Psi-\end{pmatrix}\in\mathbb{C}^2\]

\[(M,\Psi)\rightarrow M\cdot\Psi\]

where Dirac spinors consist of 2 Weyl spinors

 

Edited May 21, 2023 by Mordred

[Mordred]

Higgsstralung with k in c.m momentum of Higgs boson

\[\sigma(g_i\overline{q}_j)\rightarrow=\frac{\pi \alpha^2 |V_{ij}|^2}{36sin^4\theta_W}\frac{2k}{\sqrt{s}}\frac{k^2+3m^2_W}{(s-m^2_W)^2}\]

\[\sigma(f\acute{f}\rightarrow ZH)=\frac{2\pi\alpha^2|v_{ij}|^2(\ell^2_f+r^2_f)}{48n_csin^4\theta_Wcos_W^2}\frac{2k}{\sqrt{s}}\frac{k^2+3m_Z^2}{(s-m^2_Z)^2}\]

note last equation shows all quarks contribute to ZZ fusion process.

V denotes the CKM matrix usage

[latex]\begin{pmatrix}\acute{d}\\\acute{s}\\\acute{b}\end{pmatrix}\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\V_{cd}&V_{cs}&V_{cb}\\V_{td}&V_{ts}&V_{tb}\end{pmatrix}\begin{pmatrix}d\\s\\b\end{pmatrix}[/latex] 

[latex]V_{ckm}=V^\dagger_{\mu L} V_{dL}[/latex]

the CKM mixing angles correlates the cross section between the mass eigenstates and the weak interaction eigenstates. Involves CP violations and chirality relations.

Kk cool the first 2 equations show how the cross section correlates to the CKMS with the Higgs already factored in on the partial widths. The partial widths correlate to the detector channels. 

 

@GenadyI'm going to need the MSSM chiral operators. Simply as I have the supersymmetric cross sections and would like to examine them further. 

Edited May 21, 2023 by Mordred

[Genady]

1 hour ago, Mordred said:

GenadyI'm going to need the MSSM chiral operators. Simply as I have the supersymmetric cross sections and would like to examine them further. 

Sorry @Mordred, I can't participate. I'm not trained on this level.