Early Universe Nucleosynthesis

[Mordred]

SU(2)

\[{\small\begin{array}{|c|c|c|c|c|c|c|c|c|c|}\hline Field & \ell_L& \ell_R &v_L&U_L&d_L&U_R &D_R&\phi^+&\phi^0\\\hline T_3&- \frac{1}{2}&0&\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}&0&0&\frac{1}{2}&-\frac{1}{2} \\\hline Y&-\frac{1}{2}&-1&-\frac{1}{2}&\frac{1}{6}&\frac{1}{6}& \frac{2}{3}&-\frac{1}{3}&\frac{1}{2}&\frac{1}{2}\\\hline Q&-1&-1&0&\frac{2}{3}&-\frac{1}{3}&\frac{2}{3}&-\frac{1}{3}&1&0\\\hline\end{array}}\]

\(\psi_L\) doublet

\[D_\mu\psi_L=[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^+\tau^-W_\mu^-)-i\frac{g}{2}\tau^3W^3_\mu+i\acute{g}YB_\mu]\psi_L=\]\[\partial_\mu-i\frac{g}{\sqrt{2}}(\tau^+W_\mu^-)+ieQA_\mu-i\frac{g}{cos\theta_W}(\frac{t_3}{2}-Qsin^2\theta_W)Z_\mu]\psi_L\]

\(\psi_R\) singlet

\[D_\mu\psi_R=[\partial\mu+i\acute{g}YB_\mu]\psi_R=\partial_\mu+ieQA_\mu+i\frac{g}{cos\theta_W}Qsin^2\theta_WZ_\mu]\psi_W\]

 with \[\tau\pm=i\frac{\tau_1\pm\tau_2}{2}\] and charge operator defined as

\[Q=\begin{pmatrix}\frac{1}{2}+Y&0\\0&-\frac{1}{2}+Y\end{pmatrix}\]

\[e=g.sin\theta_W=g.cos\theta_W\]

\[W_\mu\pm=\frac{W^1_\mu\pm iW_\mu^2}{\sqrt{2}}\]

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

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

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

mass euqenstates given by \(A_\mu\) an \(Z_\mu\)

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

\[B_\mu= Z_\mu sin\theta_W+A_\mu cos\theta_W\]

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

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

ghost field given by

\[\acute{\psi}=e^{iY\alpha_Y}\psi\]

\[\acute{B}_\mu=B_\mu-\frac{1}{\acute{g}}\partial_\mu\alpha Y\]

 

 [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

 

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 April 24 by Mordred

[Mordred]

inflationary gravity waves

Weak field limit transverse , traceless components with \(R_{\mu\nu}=0\)

\[h^\mu_\mu=0\]

\[\partial_\mu h^{\mu\nu}=\partial_\mu h^{\nu\mu}=0\]

\[R_{\mu\nu}=8\pi G_N(T_{\mu\nu}-\frac{1}{2}T^\rho_\rho g_{\mu\nu})\]

vacuum T=0 so \(\square h_{\mu\nu}=0\)

transverse traceless wave equation

\[\nabla^2h-\frac{\partial^2h}{c^2\partial t^2}=\frac{16\pi G_N}{c^4}T\]

inhomogeneous perturbations of the RW metric

\[ds^2=(1+2A)dt^2-2RB_idtdx^i-R^2[(1+2C)\delta_{ij}+\partial_i\partial_j E+h_{ij}]dx^idx^j\]

where A,B,E and C are scalar perturbations while \(h_{ij}\) are the transverse traceless tensor metric perturbations

each tensor mode with wave vector k has two transverse traceless polarizations.

\[h_{ij}(\vec{k})=h_\vec{k} \bar{q}_{ij}+h_\vec{k} \bar{q}_{ij}\]

*+x* polarizations

The linearized Einstein equations then yield the same evolution equation for the amplitude as that for a massless field in RW spacetime.

\[\ddot{h}_\vec{k}+3H\dot{h}_\vec{k}+\frac{k^2}{R^2}h_\vec{k}=0\]

https://pdg.lbl.gov/2018/reviews/rpp2018-rev-inflation.pdf

Edited April 26 by Mordred

[Mordred]

 

A possible antineutrino cross section calculation massless case

\[\vec{v}_e+p\longrightarrow n+e^+\]

Fermi constant=\(1.1663787(6)*10^{-4} GeV^{-2}\)

\[\frac{d\sigma}{d\Omega}=\frac{S|M|^2\acute{p}^2}{M_2|\vec{p_1}|2|\vec{p_1}|(E_1+m_2c^2)-|\vec{p_1}|\prime{E_1}cos\theta}\]

Fermi theory

\[|M|^2=E\acute{E}|M_0^2|=E\acute{E}(M_Pc^2)^2G^2_F\]

\[\frac{d\sigma}{d\Omega}=(\frac{h}{8\pi}^2)\frac{M_pc^4(\acute{E})^2G^3_F}{[(E+M_p^2)-Ecos\theta]}\]

\[\frac{d\sigma}{d\Omega}=(\frac{h}{8\pi}^2)\frac{M_pc^4(\acute{E})^2G^3_F}{M_pc^2}(1+\mathcal{O}(\frac{E}{M_oc^2})\]

\[\sigma=(\frac{\hbar cG_F\acute{E}^2}{8\pi})^2\simeq 10^{-45} cm^2\] \

Edited April 27 by Mordred

[Mordred]

\[\vec{v}_e+p\longrightarrow n+e^+\]

\[\array{ n_e \searrow&&\nearrow n \\&\leadsto &\\p \nearrow && \searrow e^2}\]

Edited April 27 by Mordred

[Orion1]

An electroweak nuclear interaction between an anti-electron neutrino and a proton, producing a positively charged electroweak boson and generating a neutron and a positron. 

[math]\vec{\nu}_e + p^+ \overset{W^+}{\longrightarrow} n^0 + e^+[/math]
[math]\;[/math]
[math]\array{\vec{\nu}_e \searrow && \nearrow n^0 \\ & \overset{W^+}{\leadsto} & \\ p^+ \nearrow && \searrow e^+}[/math]

Any discussions and/or peer reviews about this specific topic thread?

Reference:
Wikipedia - Electroweak Interaction:
https://en.wikipedia.org/wiki/Electroweak_interaction

 

Edited April 28 by Orion1

[Mordred]

nice I wish there was a way to show the propagators better in latex. As there  is two symbols for propagators in the Feymann rules. Wavy line being one the other dotted line (ghost propagators) the problem isn't the horizontal but the diagonals for triple and quartic interactions.

lecture_16.pdf (usp.br)

second link has the ghost propagators

https://arxiv.org/pdf/1209.6213

 

Edited April 28 by Mordred

[Mordred]

Accelerator physics 

Frenet-Serret Frame/coordinates Hamilton form reference

reference 1)

https://arxiv.org/pdf/1502.03238

reference 2) Particle accelerator Physics by Helmut Weidemann third edition

particle trajectory

 

r(z)=ro(z)+δr(z)

define 3 vectors as

ux(z) unit vector ⊥ to trajectory

uz(Z)=dro(z)dz unit vector || to beam trajectory

 

uy(z)=uz(z)+ux(z)

 "to form an orthogonal coordinate system moving along the trajectory with a reference particle at r0(z) . In beam dynamics we identify the plane defined by vectorsux and uz(z ) as the horizontal plane and the plane orthogonal to it as the vertical plane, parallel to uy . Change in vectors are determined by curvatures "

 

dUz(z)d(z)=kxUz(z)

 

dUy(z)dz=kyUz(z)

k_x and k_y are the curvatures in the horizontal and vertical plane.

gives particle trajectory as

\[r(x,y,z)=r_o(z)+x(z)U_x(z)+y(z)U_y(z)\]

"where\( r_0(z)\) is the location of the coordinate system’s origin (reference particle) and (x,y) are the deviations of a particular particle from \(r0(z)\). The derivative with respect to z is then

\[\frac{d}{dz}r(x,y,z)=\frac{dr_o}{dz}+xz\frac{dU_x(z)}{dz}+\frac{dU_y(z)}{dz}+\acute{x}(z)U_x(z)+\acute{y}(z)U_y(z)\]

\[dr=U_xdx+U_ydy+U_zhdz\]

\[h=1+k_{0x}x+k_{0y}y\]

curvilinear coordinate beam dynamic Langrangian

\[\mathcal{L}=-mc^2\sqrt{1-\frac{1}{c^2}(\dot{x}^2+\dot{y}^2+h^2\dot{z}^2)}+e(\dot{x}A_x+\dot{y}A_y+h\dot{z}A_z)=-e\phi\]

 

reference  2) 1.8O and 1.81

see floquet coordinates below

 

Edited April 29 by Mordred

[Mordred]

https://en.wikipedia.org/wiki/Floquet_theory

for A(x) aka Floquet coordinates

https://personal.math.ubc.ca/~ward/teaching/m605/every2_floquet1.pdf

https://www.cfm.brown.edu/people/dobrush/am34/Mathematica/ch2/floquet.html

 

Edited April 29 by Mordred

[Mordred]

Cosmological Principle implies

\[d\tau^2=g_{\mu\nu}dx^\mu dx^\nu=dt^2-a^2t{\frac{dr^2}{1-kr^2}+r^2d\theta^2+r^2\sin^2\theta d\varphi^2}\]

the Freidmann equations read

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

for \[\rho=\sum^i\rho_i=\rho_m+\rho_{rad}+\rho_\Lambda\]

\[2\frac{\ddot{a}}{a}+(\frac{\dot{a}}{a})^2+\frac{k}{a^2}=-8\pi Gp\]

for 

\[p=\sum^ip_i=P_{rad}+p_\Lambda\]

with conservation of the energy momentum stress tensor

\[T^{\mu\nu}_\nu=0\]

\[\dot{p}a^3=\frac{d}{dt}[a^3(\rho+p)]\Rightarrow \frac{d}{dt}(\rho a^3)=-p\frac{d}{dt}a^3\]

\[p=\omega\rho\]

given w=0 \(\rho\propto a^{-3}\) for matter, radiation P=1/3 \(\rho\propto{-3}\), Lambda w=-1.\(p=-\rho\) for k=0

\[H_o^2=\frac{k}a^2_O=\frac{8\pi G}{3}(\rho^0+\rho_{rad}^0\\rho_\Lambda)\]

dividing by \(H^2_0\) and \(P^0_{crit}=\frac{3H^2_0}{8\pi G}\)

gives

\[1=-\frac{k}{h_0^2a^2_0}+\Omega^o_m+\Omega^0_{rad}=\Omega^0_\Lambda\]

\[\Omega_k^0=-\frac{k}{h^2_0a^2_0}\Rightarrow 1=\Omega_k^0+\Omega^0_{rad}+\Omega^0_\Lambda\]

densities can be written as

\[\rho_{rad}=\rho^0_{rad}(\frac{a_o}{a})^4=\frac{3}{8\pi G}H_0^2\Omega^0_{rad}(\frac{a_o}{a})^4\]

\[\Omega_m=\rho^0_m(\frac{a_o}{a})^3=\frac{3}{8\pi G}H_0^2\Omega^0_{rad}(\frac{a_o}{a})^3\]

\[\rho_\Lambda=\rho_\Lambda^0=\frac{3}{8\pi G}H_o^2\Omega^0_\Lambda\]

\[-\frac{k}{a^2}=\overbrace{-\frac{k}{a^2_0H_o^2}}^{\Omega^0_k}H^2_0(\frac{a_o}{a})^2\]

with \(1+z=\frac{a_0}{a}\) densities according to scale factor as functions of redshift.

\[\rho_{rad}=\frac{3}{8\pi g}H^2_o\Omega^0_{rad}(\frac{a_o}{a}^4=\frac{3}{8\pi G}H^2_0\Omega^0_{rad}(1+z)^4\]

\[\rho_m=\frac{3}{8\pi g}H^2_o\Omega^0_m(\frac{a_o}{a}^3=\frac{3}{8\pi G}H^2_0\Omega^0_m(1+z)^3\]

\[\rho_\Lambda=\frac{3}{8\pi G}H_0^2\Omega^0_\Lambda\]

\[H^2=H_o^2[\Omega^2_{rad}(1+z)^4+\Omega_m^0(1+z)^3+\Omega_k^0(1+z)^2+\Omega_\Lambda^0]\]

the Hubble parameter can be written as 

\[H=\frac{d}{dt}ln(\frac{a(t)}{a_0}=\frac{d}{dt}ln(\frac{1}{1+z})=\frac{-1}{1+z}\frac{dz}{dt}\]

look back time given as

\[t=\int^{t(a)}_0\frac{d\acute{a}}{\acute{\dot{a}}}\]

\[\frac{dt}{dz}=H_0^{-1}\frac{-1}{1+z}\frac{1}{[\Omega_{rad}(1+z^4)+\Omega^0_m(1=z0^3+\Omega^0_k(1+z)^2+\Omega_\Lambda^0]^{1/2}}\]

\[t_0-t=h_1\int^z_0\frac{\acute{dz}}{(1+\acute{z})[\Omega^0_{rad}(1+\acute{z})^4+\Omega^0_m(1+\acute{z})^3=\Omega^0_k(1+\acute{z})^2+\Omega^0_\Lambda]^{1/2}}\]

 

 

 

 

 

Edited May 18 by Mordred

[Mordred]

https://arxiv.org/abs/2004.11140v1

Cl(6) preons handy reference 

Edited July 23 by Mordred

[Mordred]

Self adjoint ODEs

\[\acute{p_0}(x)=p_1(x)\]

\[\mathcal{L}=\frac{d}{dx}[p_o{x}\frac{d}{dx}]+p_2(x)\]

\[\mathcal{L}u=\acute{(p_o \acute{u})}+p_2u\]

integration by parts

\[\int^b_a=v^\ast(x)\mathcal{L}u(x)dx=\int^b_a[v^\ast\acute{(p_o\acute{u})}+v^\ast p_2 u]dx\]

\[=v\ast p_o\acute{u}]^b_a+\int^b_a[-(v^\ast)p_o\acute{u}+v^\ast p_2 u]dx\]

second integration to be continued.

self reminder of goal 

Dirichlet and Neumann boundary conditions

applicability to Chebyshev differential equation first and second order polynomials. (via Sturm-Luiville)

 

 

Edited August 1 by Mordred

[Mordred]

Christoffels for the FLRW metric in spherical coordinates.

\[ds^2=-c(dt^2)+\frac{a(t)}{1-kr^2}dr^2+a^2(t)r^2 d\theta^2+a^2(t)r^2sin^2d\phi\]

\[g_{\mu\nu}=\begin{pmatrix}-1&0&0&0\\0&\frac{a^2}{1-kr^2}&0&0\\0&0&a^2 r^2&0\\0&0&0&a^2r^2sin^2\theta \end{pmatrix}\]

\[\Gamma^0_{\mu\nu}=\begin{pmatrix}0&0&0&0\\0&\frac{a}{1-(kr^2)}&0&0\\0&0&a^2r^2&0\\0&0&0&a^2r^2sin^2\theta \end{pmatrix}\]

\[\Gamma^1_{\mu\nu}=\begin{pmatrix}0&\frac{\dot{a}}{ca}&0&0\\\frac{\dot{a}}{ca}&\frac{a\dot{a}}{c(1-kr^2)}&0&0\\0&0&\frac{1}{c}a\dot{a}r^2&0\\0&0&0&\frac{1}{c}a\dot{a}sin^2\theta \end{pmatrix}\]

\[\Gamma^2_{\mu\nu}=\begin{pmatrix}0&0&\frac{\dot{a}}{ca}&0\\0&0&\frac{1}{r}&0\\\frac{\dot{a}}{ca}&\frac{1}{r}&0&0\\0&0&0&-sin\theta cos\theta \end{pmatrix}\]

\[\Gamma^3_{\mu\nu}=\begin{pmatrix}0&0&0&\frac{\dot{a}}{ca}\\0&0&0&\frac{1}{r}\\0&0&0&cot\theta\\\frac{\dot{a}}{c}&\frac{1}{r}&cot\theta&0\end{pmatrix}\]

\(\dot{a}\) is the velocity of the scale factor if you see two dots its acceleration in time derivatives. K=curvature term

Newton limit geodesic

\[\frac{d^r}{dt^2}=-c^2\Gamma^1_{00}\]

Christoffel Newton limit

\[\Gamma^1_{00}=\frac{GM}{c^2r^2}\]

Covariant derivative of a vector \(A^\lambda\)

\[\nabla_\mu A^\lambda=\partial_\mu A^\lambda+\Gamma_{\mu\nu}^\lambda A^\nu\]

 

Using above to break down 

Local maximally Symmetric subspace (local Euclid)

from reference

https://www.sissa.it/app/phdsection/OnlineResources/104/Adv.GR-Lect.Notes.pdf

Killing equation

\[\nabla_\mu x_\mu=\nabla_nu x_\mu=0\]

where x is the killing vector and defines an isometry

\[\mathbb{M}=\mathcal{R}\times\sum^3\]

due to symmetries and Corpernicus Principle we can reduce to 2 dimensions for curvature terms Possible curvatures flat, spherical, hyperbolic

\[ds^2=-dt^2+R(t)d\sigma^2\]

\(d\sigma^2\) =space  independent scale factor (a)

\[d\sigma^2=\gamma_{ij}dx^idx^j\]

 

\[d\sigma^2=\frac{d\bar{r}^2}{1-k\bar{r}}^2+\bar{r}^2\Omega^2\]

Using Raychaudhuri equations reference below

setting shear and twist to zero and Raychaudhuri expansion is

\[V=\frac{4}{3}\pi R^3\]

\[R=R(t)=a(t)R_0\]

\[\theta=\lim\limits_{\delta V\rightarrow 0}\frac{1}{V}\frac{\delta V}{\delta\tau}=\frac{1}{\frac{4}{3}\pi 3 R^3}\frac{4}{3}\pi 3 R^2\dot{R}=3\frac{\dot{a}}{a}=3H\]

Raychaudhuri for expansion becomes 

\[\dot{\theta}=-\frac{\theta^2}{3}-R_{\mu\nu}u_\mu u^\nu\]

where \(u^\mu\) is purely time-like

\[3\dot{H}=-3 H^2-R_{00}\Longrightarrow 3\frac{\ddot{a}}{a}=R_{00}\]

\[R_{00}=8\pi G_N(T_{00}-\frac{1}{2}T g_{00})\]

with relations in article below (missing in above reference) and employing last equation becomes

\[\frac{\ddot{a}}{a}=-\frac{4\pi G_N}{3}(\rho+3p)\]

 

https://amslaurea.unibo.it/18755/1/Raychaudhuri.pdf

Edited August 4 by Mordred

[Mordred]

Breit Wigner references

https://arxiv.org/pdf/1608.06485

https://arxiv.org/pdf/1608.06485

Cross section for specific processes

https://pdg.lbl.gov/2010/reviews/rpp2010-rev-cross-section-formulae.pdf

https://citeseerx.ist.psu.edu/document?

repid=rep1&type=pdf&doi=1a8f4d739a9c49f16f562bd2751d6b7d5339e3e4

Edited October 21 by Mordred

[Mordred]

https://inis.iaea.org/collection/NCLCollectionStore/_Public/25/026/25026515.pdf

\[d_L\rightarrow U^d_L d_L\]

\[d_R\rightarrow U_R^d d_R\]

\[u_L\rightarrow U_L^uu_L\]

\[u_R\rightarrow U_R^uu_R\]

\[\mathcal{L}=\frac{q_2}{\sqrt{2}}[W^+_\mu\bar{u}^i_L\gamma^\mu(V)^{ij}d^j_L+W^-_\mu\bar{d}^i_L\gamma^\mu(V^\dagger)^{ij}\mu^j_L\]

 

\[v_{ckm}=\begin{pmatrix}1&0&0\\0&c_{23}&s_{23}\\0&-s_{23}&c_{23}\end{pmatrix}\begin{pmatrix}c_{13}&0&s_{13}^{1\delta}\\0&1&0\\-s_{13}^{i\delta}&0&c_{13}\end{pmatrix}\begin{pmatrix}c_{12}&s_{12}&0\\-s_{12}c_{12}&0\\0&0&1\end{pmatrix}\]

\[\begin{pmatrix}c_{12}c_{13}&c_{12}c_{13}&s_{13}^{-i\delta}\\-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta}&s_{23}c_{13}\\s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta}&-c_{12}s_{23}-s_{12}c_{23}-s_{12}c_{23}s_{12}e^{i\delta}&c_{23}c_{12}\end{pmatrix}\]

\[s_{ij}=sin\theta_{ij}\]

\[c_{ij}=cos\theta_{ij}\]

\[ic=[Y_\mu y^\dagger_\mu ,Y_d Y^\dagger_d]=[U_\mu M^\dagger_\mu,U_d M^2_d U^\dagger_d]=U_\mu[M^2_\mu,VM^2_dV^\dagger]U^\dagger_\mu\]

Vandermond formula needed for above for next step...

Commutator of the Quark Mass Matrices in the Standard Electroweak Model and a Measure of Maximal CP Nonconservation

C.Jarlskog

 

https://kernel-cdn.niconi.org/2021-10-19/1634657105-497108-physrevlett551039.pdf

https://pages.cs.wisc.edu/~sifakis/courses/cs412-s13/lecture_notes/CS412_12_Feb_2013.pdf

Valndermonde polynomial interpolation for Langrange to reduce computations for curve fitting.

 

https://orionquest.github.io/Numacom/lectures/interpolation.pdf

 

Edited October 29 by Mordred

[Mordred]

Langrange polynomial interpolation programming steps for Vandermonde

https://people.clas.ufl.edu/kees/files/LagrangePolynomials.pdf