Is Large-N (? 6)3 theory still mathematically inconsistent?

Matsubara, Yoshimi; Suzuki, Tsuneo; Yotsuyanagi, Ichiro

doi:10.1007/bf01436516

Cited by 5 publications

(4 citation statements)

References 11 publications

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“…How can the presence of a singular FP at the end of the BMB line be explained from the large N limit? As for the first question, it is widely believed that the BMB FP has no counterpart at finite N [6,7,9,[15][16][17] which would imply that the BMB phenomenon is, at best, a curiosity of the N ¼ ∞ case. As for the second question, to the best of our knowledge, no progress has been made these past decades [6,7,9,[15][16][17].…”

Section: Introductionmentioning

confidence: 99%

Finite N origin of the Bardeen-Moshe-Bander phenomenon and its extension at N=∞ by singular fixed points

2020

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We study the OðNÞ model in dimension three (3d) at large and infinite N and show that the line of fixed points found at N ¼ ∞-the Bardeen-Moshe-Bander (BMB) line-has an intriguing origin at finite N. The large N limit that allows us to find the BMB line must be taken on particular trajectories in the ðd; NÞ plane: d ¼ 3 − α=N and not at fixed dimension d ¼ 3. Our study also reveals that the known BMB line is only half of the true line of fixed points, the second half being made of singular fixed points. The potentials of these singular fixed points show a cusp for a finite value of the field and their finite N counterparts a boundary layer.

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Section: Introductionmentioning

confidence: 99%

Finite N origin of the Bardeen-Moshe-Bander phenomenon and its extension at N=∞ by singular fixed points

2020

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“…(3 ·19) Note that it is unnecessary to evaluate the O(l/N) terms in G and K, since they do not contribute to the effective potential up to 0(1). Substituting (3 ·16) ~ (3 ·18) into (3 -12), we find VN is equal to Win (2·7) plus Mf(¢c·¢c)/2+N-1~0(¢c-<f;c)(¢c-<f;J with Mf and Mb satisfying (2·8) and (3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13)(14)(15)(16)(17)(18)(19), respectively.…”

Section: (2·4)mentioning

confidence: 98%

“…The LN effective potential is obtained from Win (2)(3)(4)(5)(6)(7) when Mb and Mf satisfy the gap equations (2)(3)(4)(5)(6)(7)(8) and (2)(3)(4)(5)(6)(7)(8)(9). Hence (2)(3)(4)(5)(6)(7)(8)(9)(10)(11)(12)(13) means the effective potential is complex. The vacuum is not stable in the gM f < 0 case including the phase with the spontaneous breaking of scale invariance.…”

Section: (2·4)mentioning

confidence: 99%

A Nontrivial Ultraviolet Fixed Point and Stability of Three-Dimensional O(N) Model with Fermions and Bosons

Suzuki

Yamamoto

1986

Progress of Theoretical Physics

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A three-dimensional O(N) model with both fermions and scalar bosons are studied extensively up to the second order in the 1/ N expansion. Fermionic interactions work towards stabilizing the vacuum as seen from the Gaussian variational method. Vacuum stability requires a constraint among dimensionless coupling constants. The effective potential is calculated up to the second order. Renormalizing the effective potential, we calculate the renormalization-group functions. There are some fixed points. The renormalization-group flow diagram shows that there exists an ultraviolet stable fixed point compatible with the vacuum stability constraint. The fixed point corresponding to that in the supersymmetric limit is seen to be unstable unless there is an exact supersymmetry from the beginning. § 1. IntroductionIt is an important problem in relativistic quantum field theory what are criteria for choosing non-trivial and mathematically consistent theories. It has been claimed recently I) that the simplest (rjJ4)d theory is trivial for d>4 (and possibly also for d=4 2 )) in the limit of infinite ultraviolet cutoff. Are all non-asymptotically-free theories trivial?The l}(rjJ6)3 theory is also renormalizable and non-asymptotically-free. The l/N analyses of its O(N) symmetric version have revealed existence of the nontrivial ultraviolet fixed point (UVFP) at l} = 192, when the coupling is defined by (6N2)-Il) ((P) 3.3)_5)It was observed, however, by a variational method that the theory becomes unstable for l} >(47r)2 and l}(47r)2, it was argued in Ref. 6) that a new UVFP exists at l} = (47r)2 with which a new phase is associated. But it has no stable vacuum, either.7),8)When an N -component fermi field is introduced in addition to the N -component scalar field, there appear other types of renormalizable and 1/ N -calculable interactions in three dimensions. 9 )-II)As is well-known in supersymmetric (SUSY) theories, such fermionic interactions are expected to improve the ultraviolet behavior of the system. It is the aim of this work to study effects of such new interactions on the stability problem and to discuss the ultraviolet behavior of the model up to the next-to-leading 1/ N (NLN) order.We first analyse the system using the Gaussian variational method in the next section. We find that fermionic interactions drastically improve stability of the vacuum. Stability requires a constraint among coupling constants. In § 3, we calculate the effective potential up to the NLN order following Cornwall-lackiw-Tomboulis. 12 ) Renormalization of divergences of the effective potential is performed in § 4. Renormalization-group flow and fixed-point structure are studied in § 5. A stable UVFP is found to exist in a domain compatible with the constraint. The final section is devoted to conclusions.

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“…How can the presence of a singular FP at the end of the BMB line be explained from the large N limit? As for the first question, it is widely believed that the BMB FP has no counterpart at finite N [6,7,9,[15][16][17] which would imply that the BMB phenomenon is, at best, a curiosity of the N = ∞ case. As for the second question, to the best of our knowledge, no progress has been made these last decades [6,7,9,[15][16][17].…”

mentioning

confidence: 99%

The finite $N$ origin of the Bardeen-Moshe-Bander phenomenon and its extension at $N=\infty$ by singular fixed points

Fleming,

Delamotte,

Yabunaka

2020

Preprint

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We study the O(N ) model in dimension three (3d) at large and infinite N and show that the line of fixed points found at N = ∞ -the Bardeen-Moshe-Bander (BMB) line-has an intriguing origin at finite N . The large N limit that allows us to find the BMB line must be taken on particular trajectories in the (d, N )-plane: d = 3 − α/N and not at fixed dimension d = 3. Our study also reveals that the known BMB line is only half of the true line of fixed points, the second half being made of singular fixed points. The potentials of these singular fixed points show a cusp for a finite value of the field and their finite N counterparts a boundary layer.

show abstract

Is Large-N (? 6)3 theory still mathematically inconsistent?

Cited by 5 publications

References 11 publications

Finite N origin of the Bardeen-Moshe-Bander phenomenon and its extension at N=∞ by singular fixed points

Finite N origin of the Bardeen-Moshe-Bander phenomenon and its extension at N=∞ by singular fixed points

A Nontrivial Ultraviolet Fixed Point and Stability of Three-Dimensional O(N) Model with Fermions and Bosons

The finite $N$ origin of the Bardeen-Moshe-Bander phenomenon and its extension at $N=\infty$ by singular fixed points

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