en>Brirush at 17:43, 12 January 2015

2015-01-12T17:43:28Z

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	In [[quantum chromodynamics]] (and also ''N'' = 1 [[Super QCD\|superquantum chromodynamics]]) with massless [[Flavour (particle physics)\|flavors]], if the number of flavors, ''N''<sub>f</sub>, is sufficiently small (i.e. small enough to guarantee [[asymptotic freedom]], depending on the number of [[color charge\|colors]]), the theory can flow to an interacting conformal [[Fixed point (mathematics)\|fixed point]] of the [[renormalization group]]. If the value of the coupling at that point is ~~less than one (~~''i.~~e.'' one can perform [[perturbation theory (quantum mechanics)\|perturbation theory]] in weak coupling), then the fixed point~~ is ~~called a '''Banks–Zaks fixed point'''~~. ~~The existence of~~ the ~~fixed point was first reported by William E~~. ~~Caswell in 1974, and later used by Banks and Zaks in their analysis of the phase structure of vector-like gauge theories with massless fermions~~. ~~For this reason one also justifiably finds references to a '''Caswell-Banks–Zaks fixed point'''.~~

	~~More specifically, suppose that we find that the beta function of a theory up to two loops has the form~~

	: ~~<math>\beta(g) = -b_0 g^3 + b_1 g^5 + \mathcal{O}(g^7) \, </math>~~

	~~where <math>b_0</math> and <math>b_1</math> are positive constants. Then there exists a value <math>g=g_\ast</math> such that <math>\beta(g_\ast) =0</math>:~~

	: ~~<math>g_\ast^2 = \frac{b_0}{b_1}.<~~/~~math>~~

	~~If we can arrange <math>b_0</math> to be smaller than <math>b_1</math>, then we have <math>g^2_\ast <1<~~/~~math>~~. ~~It follows that when the theory flows to the IR it is a conformal, weakly coupled theory with coupling <math>g_\ast<~~/~~math>~~.

	For the case of a [[Yang-Mills Theory\|non-Abelian gauge theory]] with gauge group [[special unitary group\|<math>SU(N_c)</math>]] and [[Dirac fermion]]s in the fundamental representation of the gauge group for the flavored particles we have

	~~:<math>b_0 = \frac{1}{16\pi^2}\frac{1}{3}(11N_c-2N_f) \;\;\;\; \text{ and }\;\;\;\; b_1~~ = ~~-\frac{1}{(16\pi^2)^2}\left(\frac{34}{3}N_c^2 - \frac{1}{2}N_f\left(2 \frac{N_c^2 -1}{N_c} + \frac{20}{3}N_c \right) \right)</math>~~

	where <math>N_c</math> is the number of colors and <math>N_f</math> the number of flavors. Then <math>N_f</math> should lie just below <math>\tfrac{11}{2}N_c</math> in order for the Banks–Zaks fixed point to appear. Note that this fixed point only occurs if, in addition to the previous requirement on <math>N_f</math> (which guarantees asymptotic freedom),
	:<~~math>\frac{11}{2}N_c~~>~~N_f>\frac{68N_c^2}{(16+20N_c)}~~<~~/math~~>
	where the lower bound comes from requiring <math>b_1>0</math>. This way <math>b_1</math> remains positive while <math>-b_0</math> is still negative (see first equation in article) and one can solve <math>\beta (g) = 0</math> with real solutions for <math>g</math>.

	~~==See also==~~
	*[[Beta function (physics)\|Beta function]]

	~~==References==~~
	*[http://~~link.aps~~.org/~~doi/10~~.~~1103/PhysRevLett.33.244 W. E. Caswell, Phys. Rev. Lett. 33 (1974) 244].~~
	*T. Banks and A. Zaks, Nucl.Phys. B196, 189 (1982).

	~~{{DEFAULTSORT:Banks-Zaks Fixed Point}}~~
	~~[[Category:Quantum field theory]]~~
	~~[[Category:Quantum chromodynamics]]~~
	~~[[Category:Fixed points (mathematics)]]~~
	~~[[Category:Renormalization group]]~~
	~~[[Category~~:~~Conformal field theory]~~]

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en>Brirush at 17:43, 12 January 2015

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