LLarge N reduction in deformed Yang-Mills theories
Helvio Vairinhos ∗ Centro de Física do Porto, Universidade do Porto, Porto, PortugalE-mail: [email protected]
We explore, at the nonperturbative level, the large N equivalence between ordinary SU(N) Yang-Mills theory on R and on R × S with double-trace deformations. In particular, we compare thevalues of the 0 ++ glueball mass obtained in both sides of the equivalence. ∗ Speaker. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlike Licence. http://pos.sissa.it/ a r X i v : . [ h e p - l a t ] N ov arge N reduction in dYM theories Helvio Vairinhos
1. Introduction
In their seminal work [1], Eguchi and Kawai noted that the physics of Euclidian d = Z N symmetry of the theory ispreserved in the continuum limit. However, the center symmetry in bosonic lattice gauge theoriesis often spontaneously broken at small physical volumes, thus invalidating large N reduction insuch cases [2].One way to avoid the spontaneous breaking of the center symmetry is to reduce the theorydown to a finite physical volume that is larger than a critical value (below which the center symme-try naturally breaks) [3]. Alternatively, a clever choice of twisted boundary conditions allows largeN reduction down to zero volume [4].Recently, Ünsal and Yaffe proposed [5] a double-trace deformation of SU(N) Yang-Mills the-ory living on R × S that allows full reduction of the volume of the compact direction withoutspoiling the centre symmetry. The lattice action of this deformed Yang-Mills (dYM) theory is S dYM = − N b ∑ p ReTr U p + L z (cid:98) N / (cid:99) ∑ n = c n ∑ (cid:126) x ∈ R | Tr Ω n (cid:126) x | (1.1)where b = β / N = λ − is the inverse lattice ’t Hooft coupling, U p are the plaquettes, Ω (cid:126) x are thePolyakov loops wrapping S , c n are the deformation coefficients, and L z is the size of the compactdirection in lattice units.We test the validity of the large N equivalence between conventional Yang-Mills on R anddYM on R × S with a fully reduced compact direction. We do so by comparing the 0 ++ glueballmass estimated numerically from Monte Carlo simulations of their lattice-regularized theories.If such an equivalence holds at large N, then the dYM theory Eq.1.1 could be used to study thelarge N limit of pure Yang-Mills theory at a smaller computational cost, due to the full reductionof one direction. For the same reason, it could also be used to study the large N meson spectrum.The preliminary results from our simulations of the dYM theory show a large N glueball massthat is different from the value estimated for Yang-Mills theory [6], which is likely due to poorstatistics. Further numerical studies of dYM will solve this issue.
2. Numerical simulations
We simulate the dYM theory of Eq.1.1 with gauge groups N = , ,
5, which have at most twoindependent double-trace deformations.In order to test the validity of large N reduction we simulate the dYM theory directly on L × L = , , , , , Deformed Yang-Mills theories with more than one deformed direction are also possible to define, but they are noteasier to simulate. The action of dYM theory on R d − k × S k includes O ( N k ) deformation terms, which makes numericalsimulations rather expensive for k >
1. While it is possible that only a subset of those deformations might be necessaryto keep the center symmetry intact, it is also possible that an excessive number of deformation terms may lead to anon-trivial large N limit for S dYM different from the large N limit of conventional Yang-Mills, at strong coupling. arge N reduction in dYM theories Helvio Vairinhos R S Ω x L t Figure 1:
Polyakov loop wrapping the compact direction of R × S . In conventional Yang-Mills, such lattices would be in the deconfined phase, which is char-acterized by a non-zero expectation value of the Polyakov loop wrapping the smallest direction(Fig.1). In dYM, however, the centre symmetry is intact for sufficiently large values of the defor-mation coefficients c n , and a physical confining phase is expected to exist for sufficiently small b .In our simulations we use the conservative values c = . c = . Since the lattice action Eq.1.1 is nonlinear with respect to the link variables (due to the double-trace terms), the Cabibbo-Marinari pseudo-heatbath algorithm [8] cannot be implemented directly(it requires that the lattice action is linear with respect to each link variable).However, it is possible to linearize Eq.1.1 with respect to the link variables by adding a set ofauxiliary Gaussian degrees of freedom to the action and a set of Hubbard-Stratonovich transforma-tions that remove the nonlinear terms [9]. In that situation, the Cabibbo-Marinari pseudo-heatbathalgorithm can then be applied directly. We use such an algorithm in our simulations, which is fasterand decorrelates quicker than the Metropolis alternative.Our preliminary calculations involved the generation of O ( ) thermalized configurations foreach lattice spacing and each N . ++ glueball masses We estimate the mass of the 0 ++ glueball in lattice units from the two-point function of appro-priately smeared and blocked spatial plaquette operators, ¯ u p , whose quantum numbers match thoseof the 0 ++ glueball state: (cid:104) ¯ u p ( ) † ¯ u p ( τ ) (cid:105) ∝ e − am ++ τ + · · · (2.1)To set the scale, we estimate the string tension in lattice units, a √ σ , from the mass of thetorelon states, am l . The torelon masses are extracted from two-point functions of appropriatelysmeared and blocked (cid:126) p = l wrapping non-compact directions: (cid:104) ¯ l ( ) † ¯ l ( τ ) (cid:105) ∝ e − am l τ + · · · (2.2)The string tension in units of the lattice spacing is then estimated assuming a Lüscher correction: La σ = am l + π L + · · · (2.3)In our simulations we choose values of the lattice coupling that correspond to sizes of thenon-compact directions larger than 1 fm, so that the system is in the confined phase.The dimensionless ratios m ++ / √ σ for various lattice sizes and gauge groups are given inFig.2. The continuum limit extrapolations were performed after removing the outliers.3 arge N reduction in dYM theories Helvio Vairinhos m ++ / √ σ a σ SU (3) SU (4) SU (5) Figure 2:
Continuum limit of the ++ glueball masses for several gauge groups. After extracting the continuum values of the 0 ++ glueball masses, we take their large N limitassuming a leading 1/N correction (Fig.3), m ++ √ σ (cid:12)(cid:12)(cid:12)(cid:12) N ≈ . ( ) + . ( ) N (2.4)which we compare with the result obtained from simulations of Ytheory [6], m ++ √ σ (cid:12)(cid:12)(cid:12)(cid:12) N ≈ . ( ) + . ( ) N (2.5)There is a discrepancy in both the large N limit of the glueball mass and, more pronouncedly,in the leading-order 1/N correction.
3. Conclusion
We observe a discrepancy between the large N values of the 0 ++ glueball mass calculatedin dYM and in conventional Yang-Mills [6]. This is possibly due to deficient statistics in ourestimations (which produced a number of outlier points that we disregarded in the extrapolations).A more precise study will be presented elsewhere, in order to improve the accuracy of our results,and to determine whether large N equivalence between Yang-Mills and dYM holds.We observe an even larger discrepancy between the values of the leading 1/N correction inYang-Mills and dYM. Such a discrepancy is natural, since the theories are very different at finiteN, and there is no reason to expect their 1/N corrections to be related. However, such a large value4 arge N reduction in dYM theories Helvio Vairinhos m ++ / √ σ /N Figure 3:
Large N limit of the ++ glueball masses. for the leading 1/N correction in dYM means that this theory is not as close to its large N limitthan conventional Yang-Mills is. Precise estimations of large N observables from dYM might alsorequire simulations for larger values of N than usual.We plan to increase the statistics of the present study and analyse the nonperturbative spectrumof dYM in more detail, in order to quantify unambiguously the validity of large N reduction in thepresence of double-trace deformations.
4. Acknowledgements
We would like to thank to Masanori Hanada, Marco Panero, Mithat Ünsal, Mike Teper andJoão Penedones for stimulating interactions during the development of this work. We would alsolike to thank the warm hospitality at the Yukawa Institute for Theoretical Physics, where part ofthis work was developed. Our lattice calculations were carried out on the Avalanche cluster atFEUP/University of Porto.HV is supported by FCT (Portugal) under the grant SFRH/BPD/37949/2007 and the projectCERN/FP/123599/2011.
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