Absence of chiral symmetry breaking in multi-flavor strongly coupled lattice gauge theories

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Ph. de Forcrand, S. Kim and W. Unger, arXiv:1208.2148 [hep-lat]

Original

Cited in the paper.

The expansion amounts to a loop expansion with a modified gauge boson propagator which includes the fermion bubble contribution to the self-energy, and modified n n -point gauge boson vertices which include the contribution of the fermion loop with n n external bosons; and all gauge boson interaction legs carrying 1 / N F 1/\sqrt{N_{F}} factors. The expansion correctly indicates that the theory should become more “semiclassical” for large number of fermion flavors, but clearly cannot deal with the strong coupling limit

Cited in the paper.

More generally, the global symmetry is G × G G\times G where G G is the subgroup of U ⁡ ( N F ) U(N_{F}) which commutes with all the matrices { γ ⁡ [ b ] } \{\gamma[b]\} . For staggered fermions G = U ⁡ ( N F ) G=U(N_{F})

Cited in the paper.

Closed paths not so coupled give of course disconnected graphs canceling against the denominator

Cited in the paper.

Indeed, the rules of Grassmann integration automatically take into account the exclusion principle summing over all fermionic path integral configurations so as to give the appropriate cancellations between configurations contributing to connected and disconnected graphs. As a result, after the factoring of disconnected graphs, one is left with unrestricted sums for the remaining connected graphs [ 1 ]

Cited in the paper.

Note that the tilings must always conform to the rule of equal number of U U ’s and U † U^{\dagger} ’s on each bond in order to obtain a non-vanishing result upon integration over the gauge field

Cited in the paper.

Equivalently, viewing the plaquettes as vertices, a reduced graph is a tree

Cited in the paper.

The solution for any m m gives [ 3 ] , [ 11 ] : G ⁡ ( m ) = [ d ​ ( m 2 + 2 ​ d − 1 ) 1 / 2 − m ⁡ ( d − 1 ) ] ( d 2 + m 2 ) ​ 1 𝐂 ​ 𝟏 𝐅 . G(m)={[d(m^{2}+2d-1)^{1/2}-m(d-1)]\over(d^{2}+m^{2})}\;{\bf 1_{\scriptscriptstyle C}}{\bf 1_{\scriptscriptstyle F}}\;

Cited in the paper.

Some interesting proposals based on simulation results have been made in [ 5 ] . For earlier, well-known discussions, which, however, assume chiral symmetry breaking at sufficiently strong coupling for any number of flavors, see [ 16 ] . In this connection also note that a first order phase transition at fixed N F N_{F} , within the conformal window, to a chirally broken phase as β \beta is decreased has been seen in [ 17 ] . In the present context this might be interpreted as crossing the phase boundary ( N F / N C ) c ​ ( β ) (N_{F}/N_{C})_{c}(\beta) extending into the ( β , ( N F / N C ) ) (\beta,(N_{F}/N_{C})) plane from the point β = 0 \beta=0

Cited in the paper.

For various ideas along these lines see P. A. Lee, N. Nagaosa and X-G Wen, Rev. Mod. Phys 78

Original

2006

Later among the works it cites.

E. T. Tomboulis, Phys. Rev. D 85

Original

2012

Closest in time.

P. H. Damgaard, U. M. Heller, A. Krasnitz and P. Olesen, Phys. Lett. B 400

Original

2012

Closest in time.