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I read that Quantum Chromodynamics is a theory with a mass gap. What is a mass gap in laymen terms? Why some theories have it? Which theories does not have it?

Note: I searched for mass gap before asking, all topics about it assume that the reader knows what it is, or define it in an advanced and technical way. So I'm looking for a simple answer if possible.

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related: physics.stackexchange.com/q/4930 –  user1631 Jun 22 '12 at 0:50
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2 Answers

up vote 15 down vote accepted

A mass-gap means that aside from the vacuum (totally empty space), the next higher energy state has an energy which is bigger than zero by a finite amount, not by an arbitrarily small amount. This usually means no massless particles, since massless particles can have arbitrarily low energy.

Another way of saying mass-gap which is somewhat more mathematical is that all correlation functions (the statistical versions of quantum fields) are exponentially decaying, so that the field values in imaginary time are independent when you go out far enough away. This is to contrast with a theory with no mass gap, where the correlations go to zero slowly, as a power of the distance between the points.

The mathematical definition is that there exists a positive constant A such that the energy of any state obeys:

$\langle\psi|H|\psi\rangle \ge \langle 0| H |0\rangle + A$

Where $|\psi\rangle$ is any state and $|0\rangle$ is the vacuum state.

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I believe he is asking for a soft answer –  Argus Jun 22 '12 at 3:25
    
@Argus: The first paragraph is soft, but one should mention the other stuff (which is not that much harder). –  Ron Maimon Jun 22 '12 at 7:25
    
I like that one. It is "as soft as possible, but not softer." –  Kostya Jun 22 '12 at 21:23
    
@RonMaimon: The answer would be perfect, if you would have added an extra condition. It's not true for "any state" $|\psi\rangle$, but for any $|\psi\rangle$ orthogonal to the vacuum state $| 0 \rangle$. Would you change it, in case somebody (like me) would read it much later. Cheers, Z –  Zoltan Zimboras Jan 25 at 23:20
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One says that a quantum field theory has a mass gap when zero is an isolated point in the mass spectrum of the theory. This is the case if and only if the mass of all asymptotic states (scattering products) is bounded below by a positive number.

QED has no mass gap, as observable photons are massless states.

In QCD, the observed bound states are mesons and baryons (and perhaps heavier objects such as glueballs or pentaquarks, etc.). Thus there is a mass gap. Note that asymptotic states must be uncolored, which rules out the massless gluons as candidates.

See also the entry ''What is the mass gap?'' in Chapter B6 ''Bound states and applications'' of my theoretical physics FAQ at http://www.mat.univie.ac.at/~neum/physfaq/physics-faq.html

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Many thanks for your answer. I know see why gluons are not part of the definition. I hope I can vote up, but I don't have enough points! –  user09876 Jun 22 '12 at 15:13
    
gluons are confined, so they aren't really massless. –  Ron Maimon Jun 22 '12 at 17:39
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