Warped Passages - Lisa Randall [121]
In most unified theories, the only way to make one particle heavy and the other one light is to introduce a huge fudge factor. No physical principle predicts that the masses should be so different; a very carefully chosen number is the only way to make things work. That number has to have thirteen digits of accuracy, otherwise wither the proton would decay or the weak gauge boson masses would be too large.
Particle physicists call the necessary fudge fine-tuning. A fine-tune is when you adjust the parameter to get exactly the value you want. The word “tuning” is used because it is like tweaking a piano string to get precisely the right note. But if you wanted to get a frequency of a few hundred hertz correct to thirteen-digit accuracy, you would have to listen to it for ten billion seconds—320 years—to check that it was right. Thirteen-digit accuracy is hard to come by.
I could make other fine-tuning analogies, but I promise you they’ll all sound contrived. For example, consider a huge corporation where one person is in charge of expenditure and another is in charge of receipts. Suppose that they never communicate with each other, but at the end of the year the corporation is supposed to have spent almost precisely the amount it took in, with less than a dollar remaining, or else the corporation with fail. Yep, that’s a contrived example. And there’s a good reason for that. No sensible situations depend on fine-tuning, no one wants their fate (or the fate of their business) to hang on an unlikely coincidence. Yet almost any Grand Unification Theory with a light Higgs particle has such a dependency problem. A theory in which the physical predictions depend so sensitively on a parameter is very unlikely to be the whole story.
But the only way to get a small enough Higgs particle mass in the simplest GUT is to fudge the theory. The GUT model offers no good alternative. This is a serious problem for most models that unify in four dimensions, and many physicists, including myself, are uncertain about unification of forces because of it.
And the hierarchy problem gets even worse. Even if you were willing to simply assume, without any underlying explanation, that one particle is light and the other extremely heavy, you would still run into problems with an effect called quantum mechanical contributions, or just quantum contributions. These quantum contributions must be added to the classical mass to determine the true, physical mass that the Higgs particle would have in the real world. And those contributions are generally far larger than the few hundred GeV mass that the Higgs particle requires.
Let me warn you that the discussion in the next section about quantum contributions, based as it is on virtual particles and quantum mechanics, is not going to be intuitive. Don’t try to imagine a classical analog; what we are about to consider is a purely quantum mechanical effect.
Quantum Contributions to the Higgs Particle’s Mass
The previous chapter explained how a particle generally will not travel through space unchallenged. Virtual particles can appear and disappear, and thereby influence the path of the original particle. Quantum mechanics tells us that we always have to add up the contributions to any physical quantity from all such possible paths.
We have already seen that such virtual particles make the strength of forces depend on distance in a way that has been measured and agrees quite well with predictions. The same types of quantum contribution that give energy dependence to the forces also influence the size of masses. But in the case of the mass of the Higgs particle—unlike the strengths of forces—the consequences of virtual particles don’t look as if they’ll coincide with what experiments