Speed and direction of b and anti-b quark

[FilipLand]

Homework Statement

A Higgs particle of mass M is moving in the z-direction with speed $\Beta_H$c compared to the lab system.
It decays to a b-quark ans anti-b quark, each of mass m. What is the speed and direction of the b-quark compared to the lab-system, if the Higgs system,
(a) it is emitted in the forward direction.
(b) emitted in backwards direction.
(c) is emitted in y direction.

Homework Equations

Perhaps the lorentz transformation velocities.
$$V_y = \frac{v´_y}{\gamma (1-v´_x\frac{u}{c^2})}$$

Rapidity:
$$\eta (v)=artanh(\frac{v}{c}) $$
$$\eta(v´)= \eta (v´) + \eta(u)$$

The Attempt at a Solution

I tried to set $$ v_M * M(1-\frac{u^2}{c^2})^-1/2 = (v_b+b_anti-b)*m*(1-\frac{u^2}{c^2})^-1/2$$
which is equivalent to $$(v_b+b_anti-b)*m=v_M * M$$

Tha Higgs particle was coming in z-direction so $$v_M=v_z= \frac{v´_z}{\gamma(1+\frac{v'_x\beta_H}{c})}$$
And v_b and v_anti-b = v_y (in +- direction)

and
$$v_{\pm y}= \frac{v´_y}{\gamma(1\pm v´_x\beta_H/c}$$

Then I can set v´_x =0 for the higgs particle to simplify v_z to $$v_z=\frac{v´_z}{\gamma}$$

But this should be solved using rapidity, i.e velocity addition so I think I'm out on deep water..

Any ideas?

 

[Gene Naden]

Yes, thanks for asking. You should identify the second frame of reference. (the first frame of reference is the lab system). Then you can use the velocity addition formula very simply. Start by identifying the direction of the quark.

I am not familiar with rapidity but I don't think it is necessary in order to solve this problem.

Good luck!

 

[FilipLand]

Yes, thanks for asking. You should identify the second frame of reference. (the first frame of reference is the lab system). Then you can use the velocity addition formula very simply. Start by identifying the direction of the quark.

I am not familiar with rapidity but I don't think it is necessary in order to solve this problem.

Good luck!

Thanks! I think I got something but its pretty general :)