According to the textbooks and many online sources, the range of a boson is determined by its mass. For example, a photon has an unlimited range due to its lack of mass. The W bosons have mass and thus have a limited range. Then there's the gluon, a mass-less boson with a finite range. How is that possible? That's what we will explore in this post. First, let's define the variables we will be using:
Starting with the Klein-Gordon equation, we derive the range equation (see equation 5 below).
We can also derive the range equation if we start with Einstein's famous equation:
At equation 13 above, we plug in a gluon's zero mass and get an infinite range. Not good! So what's wrong with this equation? For one, it strongly resembles the Compton or De Broglie wavelength formula. Calculating a particle-wave's wavelength is not the same as calculating its range. The two concepts might overlap some of the time, but not all of time. For example, photons can have different wavelengths but all of them have the same range. What we really found at 13 is a particle wavelength with zero energy/c^2--it's an infinite wavelength. Coincidentally, it's a photon's range.
OK, so how to we fix this equation? We derive the missing variable alpha. Equation 18 below is Newton's force equation, which is derived from the range equation. Notice that the acceleration (c/t) is fixed. Variable c is constant, so is t, since t was pulled out of Planck's constant (h-bar). Acceleration should vary, so we add the alpha coefficient to get the proper Newton equation. From there we derive Hooke's equation at 23.
Hooke's equation sort of behaves like the strong force or a rubber band. As range x increases, so does the force. Now let's work backwards and re-derive the range equation (see 25 below). At equations 26, 27, 28, we test our new range equation with particles of varying masses and accelerations.
At 26 the gluon's range is limited if its mass is zero and its acceleration is infinite. At 27, the photon has unlimited range due to zero mass and zero acceleration. At 28, the bowling ball has infinite range in a friction-less environment if it has zero acceleration. Newton would be proud!
Now, re: the qluon, does it really have infinite acceleration? To explore this question, let's go back to Newton's equation and derive equation 30 below:
At equation 30 we have light speed (c) divided by time (t). Nothing can go faster than light, so how can there be infinite acceleration? We know that a constant tangential velocity going around a circle, or oscillating is a form of acceleration, since acceleration is change of direction as well as change of speed. At equation 31 below, acceleration (a) goes to infinity as radius x goes to zero. At 32 to 34 we convert x into time (t).
At 34 above acceleration (a) goes to infinity as time (t) goes to zero. So why would time go to zero? We know that a photon going light speed experiences no time. A gluon going light speed would also experience no time. Being confined to the nucleus, its displacement velocity could be less than c, but it's oscillation velocity is c if it is a mass-less particle. At 35 to 38 we set alpha equal to the Lorentz factor.
We plug c^2 into the Lorentz factor and get a value of infinity for alpha or 1/0. We plug this into equation 40, do a little math, and get infinite acceleration at 42.
Equation 42 states the following: Given a fixed amount of energy (E), zero mass, and infinite acceleration (due to zero time and strong interactions), the range (x) of the gluon is limited. Equations 43, 44 support this statement. Equation 45 shows that the range of a particle is simply its energy divided by the force (F).
The gluon's force is zero mass times infinite acceleration. If we multiply this force by time we get a momentum. Just for fun, let's set this momentum equal to mc (see 47). If we plug mc into the range equation (48), something cool happens.
After plugging in some numbers we get the mass of a nucleon (proton, neutron) at 51. We start with a mass-less gluon and end up with something with mass within the limited range of the nucleus.
Below, we go back a few steps and do the math for a photon. As expected, it has infinite range.
The key difference between the photon and the gluon is revealed at 53 above. In the photon's case, the zero Lorentz factor is coupled with the mass instead of the acceleration. This leads to zero mass times zero acceleration times an infinite distance equals a fixed amount of energy.
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