7 hours ago, Mordred said:

Feymann Integrals 

set   c=ℏ =1, (D-1), 

momentum o particle p as D dimensional vector D(00) =E/c with D-1 remainder spatial components

Minkowskii scalar product of pa,pb

pa⋅pb=pμagμνpμb

set propogator of a scalar particle momentum p and mass m

1p2+m2 consider graph G with Next external edges, Nint internal edges and L=loop number for connected graphs

page 16 forwards on Feymann graph rules

Feymann Integrals by Stefan Weinzierl

https://arxiv.org/abs/2201.03593

 

project goal examination of the gauge group langrangians with above reference and applying the QFT creation and annihilation operators and QFT propogators

Electroweak Lanqrangian

L−14WαμνWμνα−14Wμνμν+Ψ¯¯¯¯iγμDμΨ

W1,2,3μ[ and Bμ are the 4 spin 1 fields 

Covariant derivative

D2=∂μ+igWτ2−ig´2Bμ

W+ andW− bosons are expressed as

W±μ=12−−√W1μ∓iW2μ)

γ and Z as

Aμ=Bμ cosθw+W3μ sinθW

Zμ=−Bμ sinθw+W3μ cosθW

 

The Cabibbo-Kobayashi-Maskawa matrix

 

 

Why did all the math symbols and structures dissappear? Will have to redo this from scratch 

2 hours ago, Mordred said:

Why did all the math symbols and structures dissappear? Will have to redo this from scratch 

Hey Mordred, it's good to see you back.

Hope you made your fortune during you long(ish) abscence.

 

I can still see you maths, they haven't disappeared.

But the forum layout has been updated so rolls up a longer post with  "expand" at the bottom.

 

Unfortunately it dropped the math structure in the fractions and dropped the superscript to subscripts etc. I will simply redo it. At least the Electroweak section and just reference the statements for the Feymann Integrals I want to explore. Maybe a few days though. The reference is one I just recently found and is extremely informative.

Electroweak Langrangian

L=−14WαμνWμνα−14BμνBμν+Ψ¯¯¯¯iγμDμΨ

W1,2,3μ and Bμ

Covariant Derivative

Dμ=∂μ+igWμτ2−ig´2Bμ

mass eigenstates observed in experiments are linear combinations of the electroweak eigenstates. Hence  W− and  W+  

W±μ=12−−√(W1μW2μ)

γ and Z are\

Aμ=BmucosθW+W3μsinθW

Zμ=BmusinθW+W3μcosθW

Cabibbo-Kobayashi-Maskawa matrix

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

 

symmtric massless

SU(2)L⊗U(1)γ

−LW=g2√UIL¯¯¯¯¯¯γμ1DiLW+μ+hc

UIL and DIL is a vector in generation space of the up/down quark interaction-eigenstates.

while 1 is a unit-matrix in generation space.

Symmetry break (weak)

LY−UIL¯¯¯¯¯¯FUIRH0∗+DILGDIRH0+hc

VeV  〈H0〉=v/2–√ F and G are Yukawa matrices. 

quark mass terms

MU=Fν2–√::MD=Gν2–√

Gauge Interaction

LW=gsqrt2UL¯¯¯¯¯¯γV†LDLW+μ+hc

where V†L  is the mixing matrix for quarks giving n generations (n*n unitary matrix) with

n2 parameters, in which n(n-1)/2 can be chosen as real angles and n(n+1)/2 are phases

subsequent transformation

v=PUV†LP∗D

v=⎛⎝⎜VudVcdVtdVusVcsVtsVubVcbVtb⎞⎠⎟

 

for 3 generations of quarks

 

 

THE CKM QUARK-MIXING MATRIX 

1) https://escholarship.org/content/qt1jt6c151/qt1jt6c151_noSplash_4cb0f722c2c328730fbac6c5d0971db9.pdf

Feymann Integrals by Stefan Weinzierl

2) https://arxiv.org/abs/2201.03593

PMNS mixing matrix

https://pdg.lbl.gov/2020/reviews/rpp2020-rev-neutrino-mixing.pdf

 

 

Edited December 5, 20223 yr by Mordred

I give up this is the 4th time this got screwed up the last time was when i tried inserting the url for an image