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| {{Refimprove|date=November 2010}}
| | Hi present. My name is Walter Talmadge nevertheless don't like when people use my full concept. Texas is where my house. To play hockey may be the thing Vehicles most. Administering databases is my job. Check out his website here: http://mobiletycoon.org/index.php?title=User:JulieXRGQqobpvr<br><br>Also visit my blog; [http://mobiletycoon.org/index.php?title=User:JulieXRGQqobpvr female artist] |
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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'''.
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| More specifically, suppose that we find that the beta function of a theory up to two loops has the form
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| : <math>\beta(g) = -b_0 g^3 + b_1 g^5 + \mathcal{O}(g^7) \, </math> | |
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| 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>:
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| : <math>g_\ast^2 = \frac{b_0}{b_1}.</math> | |
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| 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>.
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| 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
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| :<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>
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| 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),
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| :<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>.
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| ==See also==
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| *[[Beta function (physics)|Beta function]]
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| ==References==
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| *[http://link.aps.org/doi/10.1103/PhysRevLett.33.244 W. E. Caswell, Phys. Rev. Lett. 33 (1974) 244].
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| *T. Banks and A. Zaks, Nucl.Phys. B196, 189 (1982).
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| {{DEFAULTSORT:Banks-Zaks Fixed Point}}
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| [[Category:Quantum field theory]]
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| [[Category:Quantum chromodynamics]]
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| [[Category:Fixed points (mathematics)]]
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| [[Category:Renormalization group]]
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| [[Category:Conformal field theory]]
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Hi present. My name is Walter Talmadge nevertheless don't like when people use my full concept. Texas is where my house. To play hockey may be the thing Vehicles most. Administering databases is my job. Check out his website here: http://mobiletycoon.org/index.php?title=User:JulieXRGQqobpvr
Also visit my blog; female artist