About the determination of the bottom quark's -1/3 charge?

[jeebs]

Hi,
I have been trying to find out about the upsilon meson (the bottom-antibottom quark pairing) for a piece of work I'm preparing. Since it has a neutral charge, I've been trying to find out how it was decided that what came to be known as the bottom quark had a charge of -1/3 rather than +2/3. I have a couple of things that I don't understand though.

I came across a paper, "Observation of a narrow resonance formed in e+e- annihilation at 9.46 GeV" (1978), which talks about measuring the mass of the particles produced via electron-positron collisions at the PLUTO detector. The upsilon production cross section apparently allows determination of the "electronic partial width", $$\Gamma _e_e$$, which as I understand it is synonymous with the branching ratio, ie. the ratio of a occurence of specific decay mode to the total decays.
(Since this experiment starts with colliding electrons & positrons, does this $$\Gamma _e_e$$ refers to a collision that produces no different particles, or does it refer to some new particle being made in the collision that has then decayed into an electron-positron pair again? Or is it something else entirely?)

Anyway, the paper says that "models for quark binding in nonrelativistic potentials relate $$\Gamma _e_e$$ to the charge of the constituent quarks". However, none of the references included in the paper give any explanation of this so-called model, and textbooks/googling around have not turned up anything useful, apart from one thing I spotted, a quark-antiquark interaction potential:
$$V(r) = -\frac{4}{3}\alpha _s\frac{1}{r} + \frac{r}{a^2}$$.
However, I have a problem with this - looking at the units/dimensions, this equation seems to make no sense. The paper I found this in says that "the length a is assumed to be a universal constant characterizing the quark confinement interaction", and "the Coulmobic interaction has a strength $$\alpha _s(m_Q)$$ whose m_Q (presumably quark mass) dependence is given by the well known renormalization group formula from color gauge theory".
This means that we have potential energy (or just a potential) on the left hand side, and two quantities of different dimensions on the right hand side, so this has confused me.
I'm guessing that the relationship between quark mass, $$\Gamma _e_e$$ and the quark charge is contained in this $$\alpha _s(m_Q)$$ thing, but I'm still none the wiser really.

Can anyone clear any of this stuff up for me?
Thanks.

 

[fzero]

(Since this experiment starts with colliding electrons & positrons, does this $$\Gamma _e_e$$ refers to a collision that produces no different particles, or does it refer to some new particle being made in the collision that has then decayed into an electron-positron pair again? Or is it something else entirely?)

It refers to an intermediate state, in this case the quarkonium particle, and a final state that is an ep pair with total energy equal to the quarkonium energy. The primary channel here is through a virtual photon.

Anyway, the paper says that "models for quark binding in nonrelativistic potentials relate $$\Gamma _e_e$$ to the charge of the constituent quarks". However, none of the references included in the paper give any explanation of this so-called model, and textbooks/googling around have not turned up anything useful, apart from one thing I spotted, a quark-antiquark interaction potential:
$$V(r) = -\frac{4}{3}\alpha _s\frac{1}{r} + \frac{r}{a^2}$$.
However, I have a problem with this - looking at the units/dimensions, this equation seems to make no sense. The paper I found this in says that "the length a is assumed to be a universal constant characterizing the quark confinement interaction", and "the Coulmobic interaction has a strength $$\alpha _s(m_Q)$$ whose m_Q (presumably quark mass) dependence is given by the well known renormalization group formula from color gauge theory".
This means that we have potential energy (or just a potential) on the left hand side, and two quantities of different dimensions on the right hand side, so this has confused me.

The units work out correctly because the formula is written in units where $$\hbar=c=1$$. In these units, length has dimension inverse to energy. $$\alpha_s$$ is the strong force analogue of the fine structure constant and is dimensionless.

I'm guessing that the relationship between quark mass, $$\Gamma _e_e$$ and the quark charge is contained in this $$\alpha _s(m_Q)$$ thing, but I'm still none the wiser really.

Not quite. $$\alpha _s(m_Q)$$ measures the strength of the interaction due to the strong force between the quark and anti-quark at the energy scale $$m_Q$$. It's not related directly to the $$\Gamma_{ee}$$ branching ratio, since that primarily depends on the electromagnetic coupling to the virtual photon.

The first-order branching ratio one obtains from the potential model you mention is

$$\Gamma_{ee}^0 = \frac{4\alpha^2 q_Q^2 }{M^2} | \psi(0)|^2.$$

I'm bound to get some things wrong if I actually try to derive this, but I can explain some factors. I believe this formula appears in Barger and Philips, "Collider Physics," so hopefully you could go there for more discussion.

First, the factor $$| \psi(0)|^2$$ is the probability density for finding a zero distance between the quark and anti-quark. This is a required condition for annihilation to take place. $$M^2$$ sets the energy scale for the process, since we're assuming that the kinetic energy of the quarks is small compared to the binding energy (also called the nonrelativistic quark model).

If we try to compute the amplitude for the process $$q\bar{q}\rightarrow ep$$ via a photon, we can reproduce the charge factors. First we find one factor of $$q_Q e$$, the charge of the quark (in units of the electron charge) from the coupling of $$q\bar{q}$$ to the virtual photon. There is also a factor of the electron charge $$e$$ coming from the coupling of ep to the photon at the other end of the Feynman diagram. Together this gives a factor of $$q_Q \alpha$$, where $$\alpha$$ is the fine structure constant. Not what appears in the branching ratio is the square of the amplitude, so we know that we'll have a factor $$\alpha^2 q_Q^2$$.

 

[jeebs]

Thanks that's been helpful.

It refers to an intermediate state, in this case the quarkonium particle, and a final state that is an ep pair with total energy equal to the quarkonium energy. The primary channel here is through a virtual photon.

am I right in thinking that if $$\Gamma _t_o_t_a_l = \Gamma_e_e + \Gamma _?$$, then $$\Gamma_?$$ will be the partial width of upsilon production, in other words, the remaining portion of upsilons that have not decayed into an electron positron pair?

 

[Vanadium 50]

At the time, the charge assignment of the bottom quark (or if you prefer, the identification of the E-288 discovery as the bottom quark) was determined by the time-honored principle of "theoretical prejudice". The K-M paper and the Harari paper all assumed (correctly, as it happened) that the t-b mass hierarchy followed the c-s mass hierarchy.

There is also evidence from neutral meson mixing that the heaviest quark has charge +2/3, which would indicate that E-288 found the -1/3 member of the weak isodoublet, although this argument was probably not very popular at the time.

The key is the branching fraction, not so much the partial widths. The branching fraction of a 10 GeV quarkonium state with the quantum numbers of the upsilon is about 2% if the quark has charge -1/3 and around 8% if it has charge +2/3. Once this was measured at CLEO and CUSB, there was no longer any ambiguity.