Large-N reduction of SU(N) Yang-Mills theory with massive adjoint overlap fermions
LLarge-N reduction of SU( N ) Yang-Mills theory withmassive adjoint overlap fermions A. Hietanen a,b , R. Narayanan a a Department of Physics, Florida International University, Miami, FL 33199, USA b School of Physical Sciences, Swansea University Singleton Park, Swansea SA2 8PP,UK
Abstract
We study four dimensional large-N SU( N ) Yang-Mills theory coupled toadjoint overlap fermions on a single site lattice. Lattice simulations alongwith perturbation theory show that the bare quark mass has to be takento zero as one takes the continuum limit in order to be in the physicallyrelevant center-symmetric phase. But, it seems that it is possible to take thecontinuum limit with any renormalized quark mass and still be in the center-symmetric physics. We have also conducted a study of the correlationsbetween Polyakov loop operators in different directions and obtained therange for the Wilson mass parameter that enters the overlap Dirac operator. Keywords: N limit of gauge theories has many intriguing properties. Oneof these is continuum reduction [1]. It states that one obtains correct infi-nite volume zero temperature results by working on a finite volume latticeas long as the center symmetry is intact. In [2], it was proposed, that fora Yang-Mills theory with massless adjoint fermions with periodic boundaryconditions, the volume can be reduced down to a single site as opposed tothe pure gauge case [3], where weak coupling analysis shows all the cen-ter symmetries to be broken [4]. This has been confirmed both by latticetechniques and by perturbation theory [5, 6, 7, 8, 9, 10, 11, 12, 13].The question we want to address in this paper is what occurs at the large N continuum limit when fermions have a mass. The large N continuumlimit is taken by first extrapolating N → ∞ and then b → ∞ , where b is theinverse ’t Hooft coupling, g N . It has been argued in [14] that for any finitemass, a center symmetry unbroken phase exists at sufficiently small volume.Lattice studies using Wilson fermions has shown a large range of masses at Email addresses: [email protected] (A. Hietanen), [email protected] (R. Narayanan)
Preprint submitted to Physics Letters B November 2, 2018 a r X i v : . [ h e p - l a t ] N ov xed lattice spacing where the center symmetry remains intact [10].In this paper, we address the question of center symmetry both in thelattice and in the continuum using massive adjoint overlap fermions [16,17]. We show that the critical bare quark mass µ c , above which the centersymmetry is broken, is zero at the continuum limit. However, on a latticewith a finite lattice spacing, µ c >
0. Values of masses, which are accessibleto lattice simulations, depend on how µ c scales as a function of the latticespacing.We study the problem with one Weyl fermion, f = 0 .
5, both by pertur-bation theory and lattice simulations. Using perturbation theory we showthat center symmetry is broken even when quarks are given an arbitrarilysmall mass. We have performed lattice simulations with different b and N .The lattice results confirm with perturbation theory and we find a µ c ( b )that decreases as b increases. We do not see any evidence of scaling of µ c ( b )versus b . Our numerical results indicate that we can obtain the continuumlimit with arbitrary physical mass for the adjoint quarks.All details pertaining to the single site lattice model with adjoint overlapfermions are described in [9]. To study the continuum limit starting fromthe single site action, we use the weak coupling expansion and write the linkmatrices as U µ = e ia µ D µ e − ia µ ; D ijµ = e iθ iµ δ ij , (1)and perform an expansion in a µ . The θ iµ are the eigenvalues of the Polyakovloop operator and they have to uniformly distributed in the range [ − π, π ]and uncorrelated in all four directions in order to correctly reproduce infinitevolume continuum perturbation theory. The leading order result is S = (cid:88) i (cid:54) = j (cid:40) ln (cid:34)(cid:88) µ sin (cid:0) θ iµ − θ jµ (cid:1)(cid:35) − f ln µ − µ (cid:80) µ sin θ iµ − θ jµ − m w (cid:115)(cid:18) (cid:80) µ sin θ iµ − θ jµ − m w (cid:19) + (cid:80) µ sin ( θ iµ − θ jµ ) , (2)where the first line of RHS is the contribution from gauge fields [4] and thesecond line is the contribution from f flavors of Dirac fermions [9]. The barequark mass is m w µ √ − µ with µ ∈ [0 ,
1] and m w is the Wilson mass parameter.The gauge action has its minimum, −∞ , when all the angles θ iµ are equal.With one massless Weyl fermion ( f = 0 .
5) the fermionic part cancels out theinfinity and renders the action finite. In [9] we used Monte Carlo techniques2o find out the actual minimum. Namely, we consider the Hamiltonian H = 12 (cid:88) µ,i (cid:0) π iµ (cid:1) + βS. (3)For large β , the Boltzmann measure e − H is dominated by the minimum.Hence, this minimum can be found by performing a HMC update for the π, θ system.To reduce rounding errors in equations of motions, we introduce a regu-lator ∆ to the gauge field action S g → (cid:88) i (cid:54) = j ln (cid:34)(cid:88) µ sin (cid:0) θ iµ − θ jµ (cid:1) + ∆ (cid:35) . (4)In the computations we choose ∆ = 10 − , which is much smaller than theaverage difference between angles 2 π/N when N < Z N symmetriesis [4] P µ = 12 (cid:18) − N | Tr U µ | (cid:19) = 1 N (cid:88) i,j sin (cid:0) θ iµ − θ jµ (cid:1) . (5)If P µ = , then the Z N symmetry in that direction is unbroken.In Fig. 1 we reproduce the results of [9] with a large N scaling. To betterobserve the center symmetry breaking, the measurements P i are ordered foreach configuration s.t. P < P < P < P . This indicates that centersymmetry is probably restored when Wilson mass is in the range 3 . 0. The range of m w with broken symmetry does not depend on N .It is possible that center symmetry is broken in a subtle manner inthe range 3 . < m w < . 0. For example, the eigenvalues of the individ-ual Polyakov loop operator might be uniformly distributed but they mightshow correlations in different directions . In order to have a correct sumover all momenta as one would have in a infinite lattice, we need to ensurethat the traces of Polyakov loops vanish and there are no correlations be-tween different directions on the single site model. Let us assume that theeigenvalues are uniformly distributed and choose πjN , j = 1 , · · · , N as the N eigenvalues. Let π j , j = 1 · · · , N denote a permutation of j = 1 · · · , N .We compute the correlated action, S c , with θ jµ = πjN and compare it to theuncorrelated action, S u , with θ jµ = ππ µj N where π µ are different permutationsfor different µ .Fig. 2 shows the difference, S u − S c , as a function of Wilson mass m w withdifferent N . The value for S u is obtained by averaging over several different This is the problem with quenched Equchi-Kawai model [15]. w P µ N = 35, µ = 1 N = 35, µ = 2 N = 35, µ = 3 N = 35, µ = 4 N = 87, µ = 1 N = 87, µ = 2 N = 87, µ = 3 N = 87, µ = 4 N = 137, µ = 1 N = 137, µ = 2Ν = 137, µ = 3Ν = 137, µ = 4 Figure 1: Plot of P µ as a function of Wilson mass. w -0.200.20.40.60.81 S u - S c N = 15N = 57N = 137 Figure 2: The difference between actions with correlated S c and uncorrelated eigenvaluesto each direction. w P N = 18N = 57N = 87 Figure 3: Plot of P µ as a function of m w with massive quarks. random permutations but the fluctuations get smaller as N increases andit is sufficient to consider just one random permutation as N → ∞ . Theuncorrelated minimum is preferred when 1 < m w < 5. There is againvirtually no dependence on N . This combined with the restriction of centersymmetry restoration gives 3 < m w < m w does not include zero.In a typical free field analysis of overlap fermions [18, 19], one shows that itcorrectly represents a single Dirac flavor in a region around zero momentumas long as 0 < m w < 2. Momentum in our case is replaced by ( θ iµ − θ jµ )and we want to cover the whole range of allowed momenta. If this doesnot occur, we will not have proper reduction or a correct realization of thecenter symmetric phase. Because the range of allowed momenta (volume ofthe Brillouin zone) in the conventional free field analysis increases as m w increases, we see why m w close to zero is not appropriate. Since we donot have a concept of doublers on a single site lattice, we do not require m w < π and have propersampling of all momenta as per the infinite lattice, we find a range of allowed m w than includes m w > 2. One can also understand why m w cannot be5 µ P b=1b=3b=5 Figure 4: Plot of P as a function of mass for three different b with N = 15 arbitrarily large since we would be approaching the limit of na¨ıve fermionswhich does not have a center symmetric phase on a single site lattice [9].Once fermions have a non-zero mass, the fermionic contribution to (2)is always finite. Then the minimum of the perturbative action is dominatedby the pure gauge part and occurs when all the eigenvalues are the same.The effect of finite N is is demonstrated in Fig. 3 for µ = 0 . 1. We have onlyplotted the component P , since it determines the center symmetry breakingpoint. The symmetry breaking is evident as N → ∞ .For the actual lattice simulation, we used HMC-algorithm describedin [9]. All the simulations were performed with f = 0 . 5, Wilson mass m w = 5, and they consist of about 100 independent measurements. Ther-malization is fast and requires only about ten iterations. Most of the simu-lations were performed with N = 15, but to study 1 /N effects we did alsosimulations with N = 11 and N = 18. The purpose of the simulations areto find out the critical mass µ c for center symmetry breaking as a functionof N and b .In Fig. 4 we have plotted P as a function of mass with N = 15 for This value is slightly high, since it is on the high end of (6). This is because theargument presented with regard to Fig. 2 was realized after we obtained the numericalresults presented in this section. µ P N=11N=15N=18 Figure 5: Plot of P as a function of mass for three different N with b = 5 b = 1 , 3, and 5. The data shows that µ c ( b ) does decrease with increasing b but the decrease is clearly slower than scaling would dictate. The range ofcenter symmetry breaking is between 0 . . b in the range [1 , µe π b fixed as we take b → ∞ in orderto take the continuum limit at a fixed physical mass. 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