Asymptotic freedom

In physics, asymptotic freedom is the property of some gauge theories in which the interaction between the particles, such as quarks, becomes arbitrarily weak at ever shorter distances, i.e. length scales that asymptotically converge to zero (or, equivalently, energy scales that become arbitrarily large).

Calculating asymptotic freedom

Asymptotic freedom can be derived by calculating the beta-function describing the variation of the theory's coupling constant under the renormalization group. For sufficiently short distances or large exchanges of momentum (which probe short-distance behavior, roughly because of the inverse relation between a quantum's momentum and wavelength), an asymptotically free theory is amenable to perturbation theory calculations using Feynman diagrams. Such situations are therefore more theoretically tractable than the long-distance, strong-coupling behavior also often present in such theories, which is thought to produce confinement.

Related Topics:
Beta-function - Coupling constant - Renormalization group - Momentum - Wavelength - Perturbation theory - Feynman diagram - Confinement

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Calculating the beta-function is a matter of evaluating Feynman diagrams contributing to the interaction of a quark emitting or absorbing a gluon. In non-abelian gauge theories such as QCD, the existence of asymptotic freedom depends on the gauge group and number of flavors of interacting particles. To lowest nontrivial order, the beta-function in an SU(N) gauge theory with n_f kinds of quark-like particle is

Related Topics:
Non-abelian - Gauge group - Flavor - SU(N)

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:eta_1(lpha) = { lpha^2 over pi} left( -{11N over 6} + {n_f over 3} ight)

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where lpha is the theory's equivalent of the fine-structure constant, g^2/(4 pi) in the units favored by particle physicists. If this function is negative, the theory is asymptotically free. For SU(3), the color charge gauge group of QCD, the theory is therefore asymptotically free if there are 16 or fewer flavors of quarks.

Related Topics:
Fine-structure constant - Color charge

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Question: What is N?

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Answer: For SU(3) N = 3,

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and eta_1 < 0 gives n_f < {33 over 2},

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See also asymptotic safety

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