Thursday, 13 September 2018

standard model - Fermion mass Higgs mechanism


How does a fermion, like an electron, get its mass through the Higgs-mechanism? Can someone explain me this with formulas (Lagrangian)?


I know that the Yukawa interaction has something to do with this, is that right?


Maybe when I'm right, there is a term:


$$g \bar{\Psi} \Phi \Psi?$$



Answer



It is about the "the 5-th force."


As you said the Yukawa term introducing the interaction between scalar field $\Phi$ and fermion $\Psi$ field: $$g \bar{\Psi} \Phi \Psi$$


The Higgs mechanism causes the $\Phi$ field condense at a classical expectation value (v.e.v: vacuum expectation value), due to the Higgs potential $U(\Phi)$, so $\Phi$ tend to find a classical minimum, which causes:



$$\Phi(x,t) \to \langle \Phi \rangle=m$$


as a fix value $m$. You can imagine this process as originally $\Phi(x,t)$ is a field variable free to have any real/complex values at any spacetime $(x,t)$ point due to quantum fluctuation. However, the Higgs mechanism causes $\langle \Phi \rangle=m$ finding a (local) classical stable minimum value of the potential $U(\Phi)$.


The remarkable result is that $\Phi(x,t)$ semi-classically now have to take the fix value at $m$ at any spacetime point! (This is the remarkable fact of the 5-th force: Higgs field introduces mass to fermions i.e. quarks, leptons, in the Standard Model. Some people coin the name the 5-th force - a different mechanism from the 4 fundamental forces.)


Add: Some people like to think about (fermions,W$^{\pm}$,Z$^{0}$ bosons) particles moving in the ocean of Higgs fields, thus (fermion,W,Z) particles become massive due to the buoyancy force effects in the Higgs ocean.


The mass $M$ of fermion fields now can be read as


$$g \bar{\Psi} \Phi \Psi \to (g\cdot m) \bar{\Psi} \Psi=M \bar{\Psi} \Psi $$ with fermion mass $M=g\langle \Phi \rangle=g\cdot m$.


Note that now Fermion mass takes the fixed value at $g \langle \Phi \rangle$, BUT there is quantum fluctuation around the v.e.v. ($\langle \Phi \rangle+\delta \Phi $) to cause fermion field interacting with the Higgs fluctuation $\delta \Phi $. You can draw a Feynman diagram to compute its effect.


No comments:

Post a Comment

Understanding Stagnation point in pitot fluid

What is stagnation point in fluid mechanics. At the open end of the pitot tube the velocity of the fluid becomes zero.But that should result...