IIFT-UAM/CSIC-20-110, HUPD-2003, FTUAM-20-13
Meson spectrum in the large N limit. Margarita Garc´ıa P´erez, a Antonio Gonz´alez-Arroyo, a,b
Masanori Okawa c a Instituto de F´ısica Te´orica UAM-CSIC, Nicol´as Cabrera 13-15, Universidad Aut´onoma de Madrid,Cantoblanco, E-28049 Madrid, Spain b Departamento de F´ısica Te´orica, M´odulo 15, Universidad Aut´onoma de Madrid, Cantoblanco,E-28049 Madrid, Spain c Graduate School of Advanced Science and Engineering, Hiroshima University, Higashi-Hiroshima,Hiroshima 739-8526, Japan
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We present the result of our computation of the lowest lying meson massesfor SU(N) gauge theory in the large N limit (with N f /N −→ N (169, 289 and361) to monitor the N-dependence of the most sensitive quantities. Our methodology isbased upon a first principles approach (lattice gauge theory) combined with large N volumeindependence. We employed both Wilson fermions and twisted mass fermions with maximaltwist. In addition to masses in the pseudoscalar, vector, scalar and axial vector channels, wealso give results on the pseudoscalar decay constant and various remormalization factors. a r X i v : . [ h e p - l a t ] N ov ontents A.1 Wilson fermions 37A.2 Twisted mass fermions 43
Large N gauge theories [1] sit at the crux of different approaches to quantum field theory.This is one of the motivations for studying these theories. For the most basic case of a pureYang-Mills theory we face the difficulties of a strong non-perturbative dynamics without thehelp of supersymmetry. Nonetheless, the theory exhibits many simplifications with respectto theories at finite N . This comes without paying the price of loosing some of the mainnon-perturbative phenomena, such as confinement or existence of a mass gap, which remaina challenge for a rigorous proof. From a physicist standpoint it is important to extract thevalues of the main physical parameters of the theory. These values will remain a testingground for new ideas and methodologies aiming at an analytic or numerical approach tothe theory. In particular, as mentioned earlier, the large N theory seems more easilyaddressable from studies originating from string theory, as the AdS/CFT approach [2–4]. The main first principles approach to non-perturbative dynamics is given by Wilson’slattice gauge theories [5]. Thus, it is extremely desirable to attack the problem using this– 1 –ethodology. The importance of this goal has been realized from long time ago by a smallfraction of the community, pioneered by the work of Mike Teper and collaborators [6, 7] (SeeRef. [8] for a review of results). The intrinsic difficulty of this goal comes from the fact thatthe large N results follow from extrapolation of those obtained at several small values of N .Fortunately for this program, it seems that many of the physical quantities have a relativelymild dependence on N . On the negative side it turns out that the large N gauge theoriespossess simplifications that are not present in theories at finite N , making the study of thelatter harder. To name one, which is of particular relevance for this work, we have the rolepaid by quark degrees of freedom in the fundamental representation. In the large N limitthese quarks are non-dynamical, so that the so-called quenched approximation becomesexact. On the other hand, at finite N fundamental quarks are dynamical. Generatingthe configurations becomes much more costly, and if the quenched approximation is usedone has to be careful in treating the chiral behaviour of the theory. There is certainlyan interplay between the small quark mass and large N limits that one has to be carefulabout [9, 10].The present work follows a completely different approach. The idea is to exploit one ofthe simplifications that the large N limit produces. In particular, we will employ the prop-erty of volume independence, originating from an observation by Eguchi and Kawai [11].In essence, the statement says that finite volume corrections go to zero in the large N limit.Although the original formulation was proven incorrect in the continuum limit [12], severalways to enforce the property have been found [12–15]. Here we will make use of an ideaoriginating soon after the work of Eguchi and Kawai by two of the present authors [16, 17].The main point is to use appropriately chosen twisted boundary conditions [18–20]. Inthis way one can preserve the necessary symmetries to guarantee that the original proofof Eguchi and Kawai holds. As a matter of fact one can take volume independence tothe extreme and reduce the lattice volume to single point. This gives a matrix modelof d = 4 matrices known as the Twisted Eguchi-Kawai model (TEK). The advantage ofthis formulation is that because of the reduction in the number of space-time degrees offreedom, one can go to much larger values of N . Thus large N extrapolations becomeunnecessary and some corrections like those associated to quark loops become negligiblysmall. There are, however, finite N corrections which turn out to be of a different naturethan those of the ordinary theory. These corrections are of two types. One takes the formof ordinary finite volume lattice corrections on a lattice of size ( √ N ) . Hence, a fixed valueof N determines a scaling window in which the effective size of the box in physical unitsremains large enough. Going to smaller lattice spacings one would enter the femtoworldlike in standard lattice gauge theories. To enlarge the window one has to increase the valueof N . The other type of correction is related to an incomplete cancellation of non-planarcontributions. The nature of these corrections are related to effects in non-commutativefield theory [21–23]. Fortunately, the formulation of TEK has a free integer flux parameterthat can be tuned to minimize these corrections and used to quantify their effect [24].In summary, our formulation of the large N lattice theory (the TEK model) has thecapacity of producing estimates of the physical observables of the theory with all statisticaland systematic errors under control: measurable and improvable. In the last few years we– 2 –ave been running a number of tests [25] to ensure the capacity of our method to producephysical results within the standard computational resources available at present. Thefirst continuum observable that was studied was the string tension. Indeed, the earlyestimates [26, 27] using our method preceded in many years any other large N latticeestimate. More recently [28], a measurement to a few percent precision has been obtained,which is at least of the level of precision as other recent determinations [29, 30]The present work aims at producing a calculation of the low-lying meson spectrumin the large N limit with all statistical and systematic errors under control. The work isthe result of several years of study. The methodology was developed in Ref. [31]. Sev-eral tests and technical improvements, as well as partial results have been presented inother works [32–34]. The meson spectrum has also been investigated by other authorsusing different techniques. Indeed, this was pioneered by extrapolations from quenchedstudies [35, 36]. There are also studies which employed the idea of volume independenceto give determinations [37]. More recent determinations based on the extrapolation byother authors [38–40] give results with similar precision to ours. The exact compatibilitydemands that all estimates have well quantified errors. This is our purpose here. It mustbe said that the existence of different methodologies is very welcome since there are obvi-ously pros and cons in each of them. We have already mentioned some advantages of ourmethod in relation with the quenched approximation and the chiral limit. However, on thenegative side our method does not allow the computation of 1 /N corrections, which arealso important phenomenologically. Combining our results with finite N estimates couldbe very interesting.The structure of the paper is as follows. In the next section we review the philosophyand main formulas involved in our methodology, referring to the original references forexplicit derivations. Section 3 is of the most technical nature. Readers mostly interestedin the results might skip it. However, for us this section is crucial since it lists and ana-lyzes all the steps involved in giving final numbers and the possible sources of errors theymight introduce. A lot of effort has been put into it so that our final mass values havetrustworthy statistical and systematic errors. Section 4 contains the presentation of ourresults starting with those associated with the pseudoscalar channel. In this channel weuse both Wilson and maximally twisted mass fermions. This turns out to be crucial toobtain a determination of the pseudoscalar decay constant. The vector, scalar, and axialvector meson masses are computed as well. In Section 5 we use the previous information topresent our final table of values of the meson masses in the continuum limit. Values are pre-sented with statistical errors and systematic errors mainly arising from the extrapolationto vanishing lattice spacing. Readers interested in the results should address directly tothis section. We also compare our results with other determinations and predictions of themeson masses and the pion decay constant. Possible future improvements are discussed.At the end of the paper we give a long list of tables containing the explicit lattice results ofour simulations. This might be useful for other researchers who might want to use our bareresults for comparison or display. We have ourselves profited from other authors doing thesame. – 3 – Meson masses from reduced models
As explained in the introduction, the goal of this paper is to compute the masses of mesonswith small number of flavours in the large N limit of d = 4 Yang-Mills theory. Ourmethod is based on reduced models. In particular we will be using the twisted Eguchi-Kawai model, which is a model involving d SU(N) matrices without any space-time label.Despite its conceptual simplicity this matrix model has observables whose expectationvalue in the large N limit coincides with that of ordinary lattice gauge theory at infinitevolume and infinite N . In particular this refers to Wilson loops. This statement can beproven non-perturbatively under certain assumptions by Schwinger-Dyson equations[11,16], perturbatively to all orders [17, 41] and also tested up to 5 decimal places by directevaluation of Wilson loops in the ordinary and reduced model [25].Let us briefly revise the probability distribution and action density of the TEK model.The former is given in terms of the latter by means of the partition function Z T EK = d − (cid:89) µ =0 (cid:18)(cid:90) dU µ (cid:19) e − S TEK (2.1)where the integration on the SU(N) U µ matrices uses the invariant Haar measure on thegroup. The action S T EK follows by contracting to a point the action of ordinary latticegauge theories in a finite box with twisted boundary conditions. Many different types oflattice actions can be used, but most of the numerical results have been obtained using thesimple Wilson action S T EK = − bN (cid:88) µ (cid:54) = ν z νµ Tr (cid:16) U µ U ν U † µ U † ν (cid:17) (2.2)where 1 /b is the lattice equivalent of ‘t Hooft coupling λ , and the factors z νµ = z ∗ µν =exp { πin νµ /N } are Nth roots of unity. The integer-valued antisymmetric twist tensor n νµ is irrelevant in the large N limit, provided it is chosen in the appropriate range. Abad choice can yield very large finite N corrections and even a possible breaking of thesymmetry conditions for reduction to hold [42–44]. In practice our choice has been the so-called symmetric twist which demands N to be the square of an integer N = ˆ L . Then onetakes n νµ = k ˆ L for µ > ν , where k is an integer coprime with ˆ L . The choice is irrelevantprovided one satisfies the criteria given in Ref. [41].Reduction implies that the expectation value of a Wilson loop W ( C ) for an SU(N)gauge theory at infinite volume and infinite N can be obtained as follows W ( C ) = lim N −→∞ z ( C ) N (cid:104) Tr( U ( C )) (cid:105) T EK (2.3)where the right-hand side is an expectation value in the TEK model of the trace of theproduct of link variables following the sequence of directions specified by C (no space-timelabels for TEK links). The factor z ( C ) is a product of the z νµ factor for a collection ofplaquettes with boundary on C .The previous formula (2.3) can be proven non-perturbatively from Schwinger-Dysonequations assuming center symmetry [11, 16], shown to be valid in perturbation theory to– 4 –ll orders [17], and checked directly to very high precision numerically [25]. Furthermore,these results give information about how big has N to be to acquire a given precision.Indeed, part of the finite N corrections assume the form of finite size corrections for alattice of size ( √ N ) = ˆ L . Hence, when computing extended observables it is necessarythat they all fit inside such an effective box. Typical loop sizes should then be smallerthan ˆ L . Lattice masses should also satisfy M ˆ L (cid:29)
1. In practice, due to scaling, thisaffects the range of values of b to be covered. All these limitations are pretty standard inall lattice gauge theory studies. Here we simply have to look at ˆ L = √ N as the effectivebox size. The whole procedure has been tested quite successfully in computing the stringtension [28] in the large N limit and other observables [45]. In those studies we went asfar as N = 1369, corresponding to a box size of 37 , which is quite satisfactory for latticestandards. Unfortunately, in the present study we are unable to reach those values becauseof the computational effort involved in computing the quark propagator. Our work ismostly based on results obtained for N = 289 corresponding to a box size of 17 . Thislimits the range of our b values, but still seems sufficient to obtain good estimates of thecorresponding masses. We have also obtained some results at ˆ L = 13 and 19, allowing usto test the effect of finite N .The methodology for computing meson masses has been explained in previous pub-lications of two of the present authors [31]. We will assume that the number of flavoursof quarks in the fundamental stays finite in the large N limit. Hence, quarks are non-dynamical and the configurations are generated with the pure gauge TEK model. Quarksdo propagate in the background of these gauge fields. One can allow the quark fields topropagate in a lattice of any size (with restrictions) including infinite size. In practice, sincethe gauge fields live in an effective box of size ˆ L , we can restrict the quarks to live in abox of the same size, although we will double it in the time direction for ease in computingcorrelators.The meson masses are obtained by measuring the exponential decay in time of cor-relators among quark bilinear operators projected to vanishing spatial momentum. Forexample, one can consider objects of the form O ( t ) = ¯Ψ( t, (cid:126)x ) O ( (cid:126)x, (cid:126)y )Ψ( t, (cid:126)y ) (2.4)where O is also a matrix in spinor and colour indices. Correlators of these bilinears,after integration over the fermion fields, become expectation values in the probabilitydistribution provided by the TEK action: C ( t ) = (cid:104) O (0) O (cid:48) ( t ) (cid:105) = (cid:104)− Tr( O ( (cid:126)x, (cid:126)y ) D − (0 , (cid:126)y ; t, (cid:126)z ) O (cid:48) ( (cid:126)z, (cid:126)u ) D − ( t, (cid:126)u ; 0 , (cid:126)x )) (cid:105) (2.5)where D − ( x, y ) stands for the lattice quark propagator between two space-time points.The lattice propagator is the inverse of the Dirac operator. On the lattice there are severaloptions. Most of our results are obtained for the Wilson-Dirac operator, although we willalso use the twisted mass [46–48] operator. Other possibilities are feasible but have notbeen used in this work. The set of meson operators used for our work includes both localand non-local ones and will be explained later. All of the operators are gauge-invariant and– 5 –rojected to vanishing spatial momentum. It is important to use various operators withthe same quantum numbers, since the corresponding masses should coincide. This helps inobtaining more precise determination of the masses. Another comment is that in Eq. (2.5)we have omitted disconnected pieces, which are subleading at large N . This is an importantsimplification, which erases the difference between flavour singlet and non-singlet channels.The description done in the previous paragraph looks completely standard. The maindifference in our case is that quarks propagate in the background of volume independentgauge fields (up to a twist). This simplifies considerably the structure of the propagatorsand the meson correlators. An analogy can be drawn with the situation of electronspropagating in a crystal solid. The background field in that case is the potential createdby the ions of the solid, which is periodic with a period of the order of the lattice spacing.The motion of the electron in an infinite or much bigger solid translates into a dependenceof the propagator on the Bloch momentum. Our case is not exactly identical since thegauge field is only periodic up to a twist. The gauge field only repeats itself exactly after √ N steps in any direction. However, we only consider situations in which the correlationlengths are much smaller than this scale. For that purpose, obviously N should be largeenough.Following the derivation given in Ref. [31] we arrive at the following formula for themeson correlator in momentum spaceˆ C ( p ) = (cid:88) q (cid:104) Tr( OD − ( q , (cid:126)q ) O (cid:48) D − ( q + p , (cid:126)q )) (cid:105) (2.6)where the trace runs over colour and spinorial indices. In a first approximation the opera-tors O and O (cid:48) are just different matrices of the Dirac Clifford algebra defining the quantumnumbers of the meson to be studied. Finally, D − ( q , (cid:126)q ) is the inverse of the Dirac operatorin our particular setting. We will first of all focus on Wilson fermions, on which most ofour results are based. As explained in Ref. [31], after some algebra one can write the Diracoperator as D WD ( p ) = I − κ d − (cid:88) µ =0 (cid:16) ( I + γ µ ) ⊗ W µ ( p ) + ( I − γ µ ) ⊗ W † µ ( p ) (cid:17) (2.7)where W µ ( p ) is the following N × N matrix W µ ( p ) = e ip µ a U µ ⊗ Γ ∗ µ (2.8)where U µ are the SU(N) matrices generated by the standard Monte Carlo method for theTEK model and Γ µ are fixed SU(N) matrices satisfyingΓ µ Γ ν = e πin µν /N Γ ν Γ µ (2.9)where n µν is the same twist tensor used in the TEK action. With our choice of twisttensor the solution of the previous equation is unique up to similarity transformationsand multiplication by elements of the center. The first ambiguity is irrelevant since our– 6 –bservables are traces. The phase ambiguity can be reabsorbed into U µ . In practice, wecan pick any particular solution and keep it fixed.A final comment affects the range of values of momenta. This is determined by thesize of the lattice in which the quark fields live. If quarks propagate in an infinite lattice,then p µ is an angle. However, since the gluon fields are equivalent to those living in abox of size √ N , it is reasonable to confine the fermions to live in a similar box. In thatcase, momenta range over integer multiples of 2 π/ ( a √ N ). This choice has a bonus, sincethen the momentum factors can be reabsorbed in U µ , and one can omit the sum over q in Eq. (2.6). For the total temporal momentum we chose p = πn / ( a √ N ) with n theinteger ranging from 0 to 2 √ N −
1, allowing propagators to extend longer in the timedirection. The correlator in configuration space is then given by C ( t ) = 12 √ N √ N − (cid:88) n =0 e − iπtn / ( a √ N ) ˆ C ( p ) (2.10)It is this observable that has been used to extract masses of Wilson fermions. The analysis to be presented is based on several years of work accumulating configura-tions generated with the TEK probability density based on Wilson action as shown inEqs. (2.1)-(2.2). The main parameters defining each simulation are the value of ˆ L = √ N ,the inverse ‘t Hooft coupling b , and the value of the twist flux integer k . The total numberof configurations accumulated for each set of parameters are given on Table 1. For eachvalue a number of thermalization steps was performed initially. We do not appreciate anyMonte Carlo time dependence of our results. The configurations used for the analysis weregenerated using the overrelaxation method explained in Ref. [49], which gives shorter auto-correlation times that the more traditional modified heath-bath ´a la Fabricius-Hahn [50].The number of sweeps performed from one configuration to the next N s , also shown inTable 1, is chosen large to ensure that the configurations are largely independent.The choice of parameters explained in the previous paragraph was dictated by thestandard lattice requirement of having small lattice spacings a ( b ) in physical units. Thesevalues can be extracted from our previous analysis of the string tension [28], which usedmuch larger values of N and many more values of b . Having 4 different values of b in our casewill allow us to test the scaling behaviour of our data. Scale-setting is an important stepin providing physical values in the continuum limit. The idea is to express all dimensionfulmagnitudes in terms of an appropriate physical unit. Scaling dictates that choosing oneunit or other is irrelevant as we approach the critical point (in our case b = ∞ ). Apartfrom the string tension one can choose other physical units. The constancy of the ratio ofunits as we approach the continuum limit is by itself a check of scaling. Many proposalsappear in the literature. A good unit must be easily computable and less affected bycorrections such as lattice artifacts or finite volume corrections. One possible choice is– 7 – k b N confs N s √ σa ( b ) a ( b ) / ¯ r a ( b ) / √ t l ( b, N ) √ σ
169 5 0.355 1600 1000 0.2410(31) 0.2389(20) 0.2271(1) 3.133(40)169 5 0.360 1600 1000 0.2058(25) 0.2015(12) 0.1933(1) 2.675(33)289 5 0.355 800 1000 0.2410(31) 0.2389(20) 0.2271(1) 4.097(53)289 5 0.360 800 1000 0.2058(25) 0.2015(12) 0.1933(1) 3.499(43)289 5 0.365 800 1000 0.1784(17) 0.1722(11) 0.1661(1) 3.032(29)289 5 0.370 800 1000 0.1573(18) 0.1501(10) 0.1434(1) 2.674(30)361 7 0.355 800 3000 0.2410(31) 0.2389(20) 0.2271(1) 4.579(59)361 7 0.360 800 3000 0.2058(25) 0.2015(12) 0.1933(1) 3.910(48)361 7 0.365 800 3000 0.1784(17) 0.1722(11) 0.1661(1) 3.389(32)361 7 0.370 800 3000 0.1573(18) 0.1501(10) 0.1434(1) 2.989(34)
Table 1 : Our data sample: For each value of N , k and b , we list the number of configu-rations N confs , number of sweeps N s between configurations, the lattice spacing in variousunits ( √ σa ( b ), a ( b ) / ¯ r , a ( b ) / √ t ) and the effective box size l ( b, N ) in 1 / √ σ units.given by the scale ¯ r introduced in Ref. [28]. Essentially, it is similar in spirit to Sommerscale [51] but defined in terms of square Creutz ratios. Our determination, mainly basedin our extensive analysis done for our aforementioned string tension paper, is included inthe table. A very popular scale lately is the quantity √ t defined in terms of the Wilsonflow [52]. We have analyzed this quantity for the TEK model and various values of b and N . Using this information we were able to extrapolate the value of √ t to N = ∞ .The results are given in Table 1. Errors come from a simultaneous fit to various N andare presumably underestimated. Remarkably a simultaneous fit to the ratio √ t / ¯ r and b ≥ .
36 gives 1 . χ smaller than 1. This is a good check of scaling given thatboth scales involved are based on quite different observables and procedures. Using ourdata for b ≥ .
36 we can give determinations √ t σ = 1 . r √ σ = 1 . . b = 0 . (cid:46)
2% in the continuum limit.The standard lattice condition of having sufficiently large physical volumes now trans-lates into large enough values of N . We recall that √ N plays the role of effective size inlattice units. Using the lattice spacing values described in the previous paragraph one cancompute the lattice effective linear size of our data l ( b, N ) ≡ √ N a ( b ). This is given intable 1 in string tension units. As explained earlier, our main observables are the meson correlation function in channel γ A and γ B at time distance tC ( t ; γ A , γ B ) = 12 √ N √ N − (cid:88) n =0 e − iπtn / ( a √ N ) Tr (cid:2) γ A D − (0) γ B D − ( p ) (cid:3) (3.1)– 8 –eson π ρ a a b γ A γ γ i γ γ i (cid:15) ijk γ j γ k J P C − + −− ++ ++ + − Table 2 : Quantum numbers of the meson channels analyzed in this work and spin-parityassignment.For the case of Wilson fermions, the Wilson-Dirac matrix D WD ( p ) depends on the valueof the hopping parameter κ which is a function of the bare quark mass. D WD ( p ) acts oncolor ( U µ ), spatial (Γ µ ) and Dirac ( γ µ ) spaces and its explicit form is D WD ( p ) = 1 − κ d − (cid:88) µ =0 (cid:104) (1 − γ µ ) ˜ U µ Γ ∗ µ + (1 + γ µ ) ˜ U † µ Γ tµ (cid:105) (3.2)with ˜ U µ =0 = e ip a U µ =0 = e iπn / √ N U µ =0 , ˜ U µ =1 , , = U µ =1 , , . (3.3) γ A and γ B assign meson quantum numbers in the continuum limit. The choice of elementsand their continuum spin parity correspondence is given in Table 2.Eq. (3.1) corresponds to correlators of ultralocal operators. In order to have a rangeof operators having the same quantum numbers we make use of the smearing method.Smearing can be implemented by replacing γ A in Eq. (3.1) by the operator [38]: γ A → γ (cid:96)A ≡ D (cid:96)s γ A , D s ≡
11 + 6 c (cid:34) c (cid:88) i =1 (cid:16) ¯ U i Γ ∗ i + ¯ U † i Γ ti (cid:17)(cid:35) . (3.4)Here, (cid:96) is the smearing level and ¯ U i is the APE-smeared spatial link variable obtained afteriterating n ape times the following transformation U (cid:48) i = Proj SU(N) (1 − f ) U i + f (cid:88) j (cid:54) = i ( U j U i U † j + U † j U i U j ) , (3.5)with c and f free smearing parameters. Here Proj SU(N) means the projection onto theSU(N) matrix. In this paper, we took n ape = 10, c = 0 . f = 0 . C ( t ; γ (cid:96)A , γ (cid:96) (cid:48) B ) = 12 √ N √ N − (cid:88) n =0 e − iπtn / ( a √ N ) Tr (cid:104) D (cid:96)s γ A D − (0) D (cid:96) (cid:48) s γ B D − ( p ) (cid:105) . (3.6)The computation of the traces and the inversion of the Dirac operator is performed bymeans of a stochastic evaluation based on the use of Z random sources [53, 54]. Let z ( α, β, γ ) be the source vector having color ( U µ ) index 1 ≤ α ≤ N , spatial (Γ µ ) index1 ≤ β ≤ N and Dirac ( γ µ ) index 1 ≤ γ ≤
4. The source vectors take values z ( α, β, γ ) =– 9 – √ ( ± ± i ) and satisfy (cid:104) z ∗ ( α (cid:48) , β (cid:48) , γ (cid:48) ) z ( α, β, γ ) (cid:105) = δ α (cid:48) α δ β (cid:48) β δ γ (cid:48) γ , when averaged over randomsources. Now, we can replace the trace in Eq. (3.6) by the following expression:Tr (cid:104) [ D (cid:96)s γ A D − (0) D (cid:96) (cid:48) s γ B D − ( p ) (cid:105) (3.7)= (cid:68) z † D (cid:96)s γ A D − (0) D (cid:96) (cid:48) s γ B D − ( p ) z (cid:69) (3.8)= (cid:68) z † D (cid:96)s γ A γ Q − (0) D (cid:96) (cid:48) s γ B γ Q − ( p ) z (cid:69) (3.9)where Q ( p ) = D ( p ) γ . To account for the inversion of the Dirac operator, we solve thefollowing 2 √ N equations for y ( p ) Q ( p ) y ( p ) = z, p = πn a √ N with 0 ≤ n ≤ √ N − x A(cid:96) Q (0) x A(cid:96) = γ γ † A D (cid:96)s z (3.11)Averaging over random sources, and taking into account that D s is a Hermitian operator,we have (cid:68) ( x A(cid:96) ) † D (cid:96) (cid:48) s γ B γ y ( p ) (cid:69) = (cid:68) z † D (cid:96)s γ A γ Q − (0) D (cid:96) (cid:48) s γ B γ Q − ( p ) z (cid:69) . (3.12)In the actual average, we use 5 random sources. We are using the Conjugate Gradientinversion, so Q − actually means Q − = QQ − .For each value of b , many different values of κ have been used (the list will be givenlater). By fitting the dependence of the PCAC quark mass (defined later) on κ , one candetermine κ c ( b, N ), which is the chiral limit at which this mass vanishes. The list of valuesobtained in our simulation are collected in Table 3.As explained in the previous section, we have also studied twisted mass fermions withmaximal twist [46]. This corresponds to a modified Dirac operator of the form D ± TM = D WD (cid:12)(cid:12) κ = κ c ± i κ c µγ (3.13)where the Wilson-Dirac operator is computed at the value κ c ( b, N ) determined earlierwith Wilson fermions. After a chiral rotation of the fermion fields this can be written asthe Dirac operator of a fermion field of mass µ plus an irrelevant modified Wilson term.The interesting bilinear operators from which we compute correlators involve two differentfields having opposite values of the µ -term. It is equivalent to having non-singlet bilinearoperators involving two flavours. This modified Dirac operator has many advantages.For example, the constant µ is proportional to the quark mass and monitors the explicitviolation of chiral symmetry. It has also important advantages from the point of viewof lattice artefacts [47] and most importantly for our purposes allows the determinationof the pseudoscalar decay constant without unknown renormalization factors [55] (for areview of properties and results, plus a list of other relevant references we refer the readerto Ref. [48]).As shown in the last section, the original formula for the correlator in momentum spaceEq. (2.6) involves a momentum sum. In principle choosing different ranges only produces– 10 –nite N corrections. A priori it seems that the formula involving the momentum sum isharder to obtain since one would need to invert the propagator for all values of q . However,there is a trick that can be used that costs no extra effort. The idea is for each gauge fieldconfiguration to generate a random value of q within the right range and with a uniformdistribution. In particular if we consider that the quark field lives in an infinite lattice, forevery gauge field configuration we can generate a random angle α µ and replace the link inthe Dirac operator as follows: U µ −→ e iα µ U µ (3.14)and then apply the same source and inversion procedure as explained earlier. The methodproduces results at no extra computational cost and we verified that for the free theorythe method gives the right propagator. Since the modified links are in U(N) rather thanSU(N), we did not impose the unit determinant condition in the Ape smearing procedure.For Wilson fermions we used both methods giving compatible results within errors. Forconsistency, all the data and masses presented here for the Wilson fermions were obtainedwith the simple version with no momentum sum. However, our twisted mass results areobtained with the random angle method. Comparison of the results obtained with bothmethods provides an explicit test that finite N corrections are under control.In relation with the previous comment, it is clear that stronger finite N corrections areexpected when we approach the chiral limit, since the effective lattice size becomes smallin units of the inverse pion mass. To avoid this problem, all our results have been obtainedfor large enough values of m π l . The results are then extrapolated to the chiral limit. Weshould point out that because of our large values of N this has less problems that for smallones, since chiral logs are strongly suppressed. This is one advantage of using our method.Finally, as explained earlier, once we obtain the expectation values of the correlators asa function of momenta we Fourier transform them to a time dependent function. For largeenough time-separations these correlation functions would decay exponentially in time, thecoefficient of which determines the minimal mass in that channel in lattice units. This isin short the procedure. In practice this becomes much more complicated and requires anefficient treatment of all data. This will be explained in the following subsection. As just mentioned, meson masses are extracted from the exponential decay in time ofappropriate correlation functions. Although such correlators receive contribution from atower of states, the lowest mass in the corresponding channel dominates at large times andcan be determined from the coefficient of the late single-exponential decay. Nonetheless,this is not as straightforward as it may seem; the determination of the large time decay inan actual simulation faces two problems: the fact that the time extent of the lattice is finiteand the deterioration of the signal to noise ratio at large time separation. Therefore, for aprecise determination of the ground-state mass it is advantageous to work with operatorsfor which the projection onto the ground state is maximized and the single-exponentialdecay sets in early on. Smearing is one step in that direction; it increases the overlap withthe ground state wave function by extending the operator range at distances comparable– 11 –ith the inverse wavelength of the particle. Beyond that, further improvement can beobtained by using a variational approach [56, 57]. The idea is very simple and is basedon constructing a linear combination of operators with the same quantum numbers, withcoefficients tuned to optimize the ground state projection. We will describe below theparticular implementation of the variational procedure used in this work.To start with, consider a basis of N op operators with the right quantum numbers, interms of which a N op × N op correlator matrix is built: C ab ( t ) = (cid:104) O † a ( t ) O b (0) (cid:105) . (3.15)The variational strategy looks for solutions to the generalized eigenvalue problem (GEVP): C ab ( t ) v ( i ) b ( t , t ) = λ ( i ) ( t , t ) C ab ( t ) v ( i ) b ( t , t ) , (3.16)for given t and t , with t > t . This is obviously equivalent to solving the standardeigenvalue problem for the matrix: C − ( t ) C ( t ) C − ( t ) , (3.17)with eigenvalues λ ( i ) ( t , t ) and corresponding eigenvectors C / ( t ) v ( i ) ( t , t ). The corre-lation matrix given in eq. (3.15) can be rotated to the v ( i ) basis, leading to the matrix:˜ C ij ( t, t , t ) = v ( i ) a ( t , t ) C ab ( t ) v ( j ) b ( t , t ) , (3.18)which is diagonal at t = t with eigenvalues given by λ ( i ) ( t , t ). With a complete basis ofoperators and for infinite time extent, one would obtain λ ( i ) ( t , t ) = exp( − m i ( t − t )).The objective of the procedure is to select the set of operators and the choice of t and t to optimize the projection onto the lowest states, in particular the ground state. Thestandard GEVP proceeds by repeating the diagonalization at all values of t = t , looking fora plateaux in the effective masses extracted from λ ( i ) ( t , t ). Given the limited time extentof our lattice, we have preferred instead to fix the value of t and work with the correlatormatrix defined by ˜ C ( t, t , t ). To determine the ground state mass in each channel, we usethe diagonal matrix element related to the largest eigenvalue λ max ( t , t ) = max i λ ( i ) ( t , t ).Let us denote by v max ( t , t ) the corresponding eigenvector in terms of which we introducean optimal , within the given choice of operators and selected values of t and t , correlationfunction: C opt ( t, t , t ) = v max ∗ a ( t , t ) C ab ( t ) v max b ( t , t ) . (3.19)The ground-state mass is extracted in the usual way from the exponential decay at largetimes of this correlator.For the specific implementation in this work, we have taken t = a and t = 2 a , with a the lattice spacing. The choice obeys to a compromise to maximize the projection ontothe ground state without affecting the signal to noise ratio which becomes worse for largervalues of t i . We have tested that other choices in the range of values of t i below 4 a givecompatible results. The operators in the basis are constructed by using different smearinglevels of the quark bilinear operator as described in section 3.2. We have considered up– 12 – C ( t ) t √ σ m π = 1.56(2), χ =0.38 m π * = 4.0(2), χ =0.36 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 M ( t ) / √ σ t √ σ m π = 1.56(2) N=289, b=0.355 κ =0.1600 0.0001 0.001 0.01 0.1 1 10 0 0.5 1 1.5 2 2.5 3 3.5 4 C ( t ) t √ σ m ρ = 2.15(5), χ =0.76 , m ρ * = 4.7(3), χ =0.65 0 1 2 3 4 5 0 0.5 1 1.5 2 2.5 3 3.5 4 M ( t ) / √ σ t √ σ m ρ = 2.15(5) N=289, b=0.355 κ =0.1600 Figure 1 : Plots on the left show the time dependence of the optimal correlator for pseu-doscalar (top) and vector (bottom) mesons. Plots on the right show the time dependenceof the effective masses. The red band indicates the mass of the ground-state determinedfrom the fit to the correlator with the length of the band indicating the fitting range andthe width indicating the jacknife error. – 13 – C ( t ) t (cid:125)(cid:125)√ σ m a =3.09(8), χ =0.18 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 M ( t ) / (cid:125)(cid:125) √ σ t (cid:125)(cid:125)√ σ m a = 3.09(8) N=289, b=0.355 κ =0.1600 C ( t ) t (cid:125)(cid:125)√ σ m a =3.46(9), χ =0.41 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 M ( t ) / (cid:125)(cid:125) √ σ t (cid:125)(cid:125)√ σ m a = 3.46(9) N=289, b=0.355 κ =0.1600 C ( t ) t (cid:125)(cid:125)√ σ m b =3.59(11), χ =0.60 0 1 2 3 4 5 6 7 0 0.5 1 1.5 2 M ( t ) / (cid:125)(cid:125) √ σ t (cid:125)(cid:125)√ σ m b = 3.59(11) N=289, b=0.355 κ =0.1600 Figure 2 : As in fig. 1 for, from top to bottom, a , a and b mesons.– 14 –o 10 operators corresponding to fermion smearing levels: 0, 1, 2, 3, 4, 5, 10, 20, 50, 100.Ground-state meson masses in each channel are extracted by fitting the optimal correlatorto a hyperbolic-cosine. The optimization procedure involves, first, a choice of the operatorsin the basis and, second, the choice of fitting range [ t min , t max ]. These choices are the mainsources of systematic errors in the mass determination. We will discuss the selection criteriain sec. 3.4. Let us just mention here that they respond to two general considerations: thereduction of excited state contamination and the limitation of finite size effects and signalreduction at large times. While smearing improves the projection onto the ground state,at the same time it leads to larger finite size effects and larger statistical errors at largetime, for this reason we vary the selection of operators in the basis to limit the maximalamount of smearing. In general we have used the full basis in the pseudoscalar channeland up to 50 smearing levels for the other meson channels.In order to illustrate the quality of the fits that can be attained with this procedure,we present in figs. 1, 2 one example in each meson channel. The figure corresponds to alattice with N = 289 at b = 0 .
355 and κ = 0 .
16. For each channel, we display the timedependence of the optimal correlator (left) and the effective mass (right). The red bandin the plot shows, for illustration, how the determination extracted from the hyperboliccosine fit compares with the effective mass plateaux, with the length of the band indicatingin each case the fitting range [ t min , t max ].In addition to the ground state mass, we have also attempted to estimate the massof the first excited state in each meson channel. For that purpose, we have performeddouble exponential fits keeping the ground state mass, m , fixed to the one extracted fromthe optimal correlator. To be specific, we look at correlators derived including in thevariational basis the 4 lowest smearing levels. This correlator is fitted, in the range t ∈ [(2 √ σ ) − , t max + a ], with a combination of two hyperbolic cosine functions with one massparameter free and the other fixed to the ground-state mass. This allows to determinethe first excited state mass m ∗ . The correlator fits for the pseudoscalar and vector mesonsshown in fig. 1 include these two contributions with m and m ∗ fixed to the values determinedin this way. The goal of this paper is to provide values for the mesons masses and decay constants forthe large N gauge theory based on first principles that can be used as a test for otherestimates based on other types of ideas and approximations. For that purpose the finalnumbers have to be supplied with trustworthy errors. Some of these errors are statistical.These are indeed the simplest conceptually. In this paper we have employed the well-knownjacknife method, based on splitting the sample into several groups of equal size, computingthe quantity in question by averaging over all but one of these groups, and equating theerror to the dispersion of these values.A much more difficult task is the estimate of systematic errors. These have differentsources and are not necessarily symmetric around the mean value. Most of the errorsthat we will be concerned with are typical of all lattice calculations, but our methodology– 15 –nhances or reduces some of those. A very important characteristic of the lattice approachis that all the errors can be estimated. Here we gather together the errors that are characteristic of the numerical procedure. Weare fairly confident that the generation of configurations is safe. We have previously carriedthe same procedure for much larger values of N (up to 1369) and much larger values of b (up to 0.385) and found no relevant systematic effects. Thus, this should even be moreso in the more restricted range of parameters that we are exploring here. Furthermore wealso played with different thermalization steps and updates per measurement and foundno deviations.Concerning the methodological aspects entering in the evaluation of the observables,we made several tests to quantify the errors committed. In particular we used the sameprograms and methods for the free field theory (coupling b −→ ∞ ). The advantage isthat the propagators and correlators can be computed analytically in this case, so that thecomparison with data gives an idea of the numerical errors involved. In this respect wetested the effect of taking a given number of sources in the calculation of the meson corre-lators. With 200 sources we found complete agreement between the analytic and numericalcomputation within errors. With 5 sources some systematic deviations are observed forsmall values of the quark mass. However, the main effect turns out to be in the mesoncorrelators for p = 0 which produces the addition of a small constant to the correlator. Inconfiguration space the effect only shows up at large correlation times. Thus, in our massextraction algorithms we have set the upper limit of the fitting range to avoid this region,which is also the most affected by other statistical and systematic errors.In estimating the final meson masses and decay constants systematic errors are ex-pected to arise from finite lattice spacing and finite size effects. In our particular construc-tion, finite size is traded by finite N . There is an antagonistic interplay among these twoerror sources. For discretization errors to be small, a ( b ) should be small, while for finitesize errors to be small √ N a ( b ) should be large. This is pretty tight given that most of ourdata are obtained for ˆ L = √ N = 17.The sensitivity to finite size effects is larger when there are large correlation lengthsinvolved. Thus, in our measurements we have stayed away from the worse region, keepingthe pion mass times the effective length large enough. Anyhow, to monitor and estimatethe size of these errors we have studied the most sensitive quantities with two other valuesof N (in addition to our four values of b ).It is important to realize that the finite-size effects in the meson correlators havetwo sources. On one side, the gauge field configurations generated with the TEK modelcorrespond to a finite volume of size ˆ L , as Wilson loop expectation values show. Onthe other hand the quarks can be made to propagate in a box of the same size or larger.These two options for the quark box size can be easily implemented, as explained in theprevious section. Thus, the effect can be monitored. We tested this first for the case offree fermion fields ( b −→ ∞ ) in which the meson propagator can be computed analytically– 16 –by a momentum sum) as well. Indeed, the effects of using different quark box sizes aresizably smaller for the twisted mass Dirac operator than for the Wilson-Dirac one.For our dynamical configurations at finite b we have also monitored the effect of thefermion propagation volume. The results obtained with the simplest formula, which in-volves no explicit sum over spatial momenta, corresponds to a fermion box of size ˆ L .The results corresponding to an infinite quark box were obtained, as explained before, bygenerating momenta p randomly with a uniform distribution for each configuration. Theprocedure does not involve any additional cost in computer time and is seen to give thecorrect results in our free fermion case. We did not find significant differences within errorsbetween the random momenta results and the others.Finally, we come to the systematic errors which arise from the procedure to extractthe masses from the time-dependent meson correlators. This is by far the largest source ofsystematic errors. In our procedure masses are obtained from a fit to an exponential decayof the time-correlation function of a properly selected combination of mesonic operatorswith the appropriate quantum numbers. The results then depend on the selection of theoperator, the region of fitting and the functional form chosen for the fit. As discussedin sec. 3.3 , the mesonic operator in each channel is extracted from a variational analysisaimed at improving the projection onto the ground state. The basis for this variationalanalysis is constructed in terms of smeared operators with smearing level 0, 1, 2, 3, 4, 5,10, 20, 50 and 100. The results that will be presented correspond to a particular choicethat we consider optimal and includes all operators up to smearing level 20, 50 or 100,depending on the state under study and the physical volume of the lattice. The selectionof fitting range, [ t min , t max ] responds to two general considerations: t min is tuned to reducecontamination from excited states, and t max to limit finite size effects and the poor signalto noise ratio at large times. Concrete choices for these quantities depend not only onthe channel considered but also on the amount of smearing of the operators in the basis;including operators with larger smearing level improves the projection onto the groundstate and allows to reduce the value of t min , but, at the same time, the signal to noiseratio deteriorates at large times, limiting the value of t max . As a rule of thumb, given ameson state and operator choice, t min is fixed in physical units, around the value where theeffective mass plateaux sets in, and t max is set to a fraction of the maximal time separation( a √ N ). In order to estimate the systematic effects in the selection of smearing and fittingranges, we have analyzed the spectrum obtained if only operators with smearing levelbelow 10 are included in the variational basis; as an example we display in fig. 3 the fitsto the pseudoscalar correlator, corresponding to the case presented previously in fig. 1.The masses obtained are in good agreement giving m π / √ σ = 1 . t min , t max ], however, inthe case of the heaviest states ( a , a and b ) we also included a constant contributionthat improved the χ of the fits at large times. In most cases the constant turned out tobe small and the resulting masses compatible within errors with those obtained with no– 17 –onstant added. C ( t ) t √ σ m π = 1.585(13), χ =0.48 m π * = 4.3(2), χ =0.44 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 M ( t ) / √ σ t √ σ m π = 1.585(13) N=289, b=0.355 κ =0.1600 Figure 3 : The plot on the left shows the time dependence of the correlator in the pseu-doscalar channel obtained from the variational analysis including operators up to smearinglevel 10. The plot on the right shows the time dependence of the effective masses. The redband indicates the mass of the ground-state pion determined from the fit to the correla-tor with the length of the band indicating the fitting range and the width indicating thejacknife error.
We start with the most important channel: that involving correlators among pseudoscalarmeson operators. Because of γ -hermiticity the correlation function is positive definite. Ourresults show a very neat exponential fall-off of the propagator for large enough separations.Using the optimal operator, as explained in the previous section, the exponential behaviourextends to rather low separations. Fig. 1 shows some characteristic correlators illustratingthe typical error size. From these correlators a fit allows us to determine the mass ofthe lowest energy state, which we will call the pion. We show in Fig. 4 an example ofthe dependence of the square of this mass in units of the string tension as a function of– 18 – M π κ ) N=269, b=0.355N=269, b=0.360N=269, b=0.365N=269, b=0.370 0 0.02 0.04 0.06 0.08 0.1 0.12 3.05 3.1 3.15 3.2 3.25 3.3 M P C A C κ ) N=289, b=0.355N=289, b=0.360N=289, b=0.365N=289, b=0.370
Figure 4 : Dependence of the pion mass square (left) and the PCAC mass (right) in unitsof the string tension as a function of 1 / (2 κ ).1 / (2 κ ). We see that the dependence is linear. Indeed, this is what we expect from thepseudo-Goldstone boson nature of the pion. The quantity 1 / (2 κ ) is a linear function ofthe quark mass. The vanishing of the pion mass occurs when the quark mass vanishes andwe recover the explicit chiral symmetry, which remains spontaneously broken. In Table 3we list κ πc , the value of κ at which our linear fit predicts the vanishing of the pion mass.We also give the slope and the square root of the Chi-square per degree of freedom of thelinear fit ( ¯ χ ≡ (cid:112) χ / dof ). The slope is scaled by a ( b ) in string tension units. This scalingis the expected one if both the lattice pion mass M π = m π a ( b ) and the bare quark mass M q ≡ κ − κ c = Z S m q a ( b ), are scaled with the lattice spacing as shown. N b
Slope / √ σ κ πc ¯ χ Z P / ( Z A Z S ) κ c ¯ χ
169 0.355 10.4(1.4) 0.16247 (38) 0.20 0.894 (47) 0.16248 (15) 0.08169 0.360 11.2(2.5) 0.15983 (47) 0.26 0.0.917 (37) 0.159827(91) 0.50289 0.355 11.4(3) 0.162689(96) 0.19 0.8681(62) 0.162815(31) 0.44289 0.360 11.8(7) 0.16015 (20) 0.16 0.878 (15) 0.160129(57) 0.11289 0.365 10.9(4) 0.15854 (11) 0.66 0.872 (14) 0.158014(45) 0.22289 0.370 11.6(6) 0.15666 (16) 0.66 0.908 (15) 0.156283(43) 0.18361 0.355 11.2(7) 0.16287(17) 0.06 0.840 (22) 0.162978(82) 0.07361 0.360 11.1(6) 0.16022(15) 0.15 0.873 (17) 0.160243(49) 0.01361 0.365 12.0(1.2) 0.15815(22) 0.19 0.917 (44) 0.158008(99) 0.28361 0.370 11.1(9) 0.15681(18) 0.10 0.884 (26) 0.156380(49) 0.13
Table 3 : Determination of the critical κ from the linear dependence of the pion masssquare ( κ πc ) or the PCAC mass ( κ c ) on 1 / (2 κ ). We also provide the slopes of the linear fits.In the case of the PCAC mass, the slope determines the ratio of renormalization constants Z P / ( Z A Z S ). In the case of the pion mass square we express the slope in string tensionunits. The quantity ¯ χ denotes the square root of the reduced Chi-square ( ¯ χ ≡ (cid:112) χ / dof ).– 19 – M π µ N=289, b=0.355N=289, b=0.360N=289, b=0.365N=289, b=0.370
Figure 5 : Dependence of the pion mass square in lattice units as a function of the twistedmass parameter µ for N = 289.An alternative way to determine the onset of chiral symmetry is to look directly atthe Ward identity and the PCAC relation. This allows the definition of a quark mass,called PCAC mass M PCAC , whose vanishing signals the point where the explicit breakingvanishes. This involves expectation values of the axial vector and pseudoscalar operatorbetween the vacuum and the pion at rest. Hence, the PCAC mass is affected by operatorrenormalization and depends on the operator that we use. On the lattice we consider theultralocal version of the axial vector current and of the pseudoscalar operator. To computeit we follow a similar procedure as in Ref. [38] by computing the correlation functions ofthe two operators in question with the optimized version of the pseudoscalar operator usedfor the extraction of the pion mass. Hence, the values of the M PCAC masses are determinedby fitting to a constant the ratio M PCAC ( t ) = C ( t + a ; γ γ , γ opt5 ) − C ( t − a ; γ γ , γ opt5 )4 C ( t ; γ , γ opt5 ) (4.1)This new mass is proportional to the quark mass M q with proportionality constant given bythe ratio of renormalization factors Z P / ( Z A Z S ). Therefore, from the vanishing of M PCAC one can extract an alternative derivation of the value of the critical κ , that should coincidein the continuum, infinite volume limit, with the one determined from the vanishing of thepion mass. The values of κ c extracted in this way, as well as the slopes and values of ¯ χ are also given in Table 3 – an example of the quality of the fits is shown in Fig. 4. Thisdetermination of κ c is more precise and will be taken as our best estimate of the chirallimit. – 20 –ll the previous results concerned Wilson fermions. However, we have also studiedmaximally twisted mass fermions [46]. For that one has to start with the Dirac operatorfor massless Wilson fermions, hence at κ = κ c . Then, as explained in Eq. (3.13), one addsa mass term ± i κ c µγ , where the opposite sign is used for the two propagators involved ina meson correlator (see Eq. (3.13) for the lattice twisted mass Dirac operator). This, aftera chiral rotation, is equivalent to having two flavours of equal mass µ and computing thecorrelation function of non-singlet meson operators. The unfamiliar reader is invited toconsult the literature [46–48, 55]. By construction, the pseudoscalar meson mass vanishes(approximately) at µ = 0. Furthermore, since µ is proportional to the quark mass, the pionmass square should depend linearly on µ . Our results agree with this linear dependencewith intercept equal to zero. The fitted slopes are given in Table 4, again scaled with thelattice spacing in string tension units. As an example, fig. 5 displays the data and thelinear (one-parameter) fit for N = 289. N b
Slope / √ σ ¯ χ
169 0.355 19.54(25) 0.76169 0.360 18.79(41) 1.07289 0.355 19.38(32) 2.10289 0.360 19.61(33) 0.61289 0.365 19.32(33) 0.94289 0.370 20.71(54) 0.64361 0.355 19.07(45) 2.87361 0.360 18.98(31) 1.71361 0.365 18.82(21) 1.10361 0.370 20.06(36) 1.07
Table 4 : Slope, in units of the string tension, of the linear dependence of the pion masssquare on the twisted mass parameter µ .Having determined the mass of the pion we can also look at the masses of the excitedstates with the same quantum numbers. The results of the GEVP for the first excited statesturn out not to be precise enough and we have opted for estimating the mass of the excitedstates by performing double exponential fits to the correlators, keeping the ground statemass fixed. The results for Wilson and twisted mass fermions are presented in tables 12and 16. An extrapolation to the chiral limit using a linear dependence in the PCAC mass,at given b and N , works well and gives the intercepts presented in table 5, which haverather large statistical errors. For each value of b we have 4 determinations dependingon the methodology (Wilson or twisted mass) and the value of N , which are displayed asfunction of the lattice spacing in fig. 6. This can give an idea of the systematic errors.Given that they are quite large, we cannot perform a reliable continuum extrapolation ofthe results, a constant fit to all the data gives for instance m (0) π ∗ / √ σ = 4 . χ = 1 .
6. Further work would be required to improve these results.– 21 – b m (0) π ∗ / √ σ (W) ¯ χ m (0) π ∗ / √ σ (Tw) ¯ χ
289 0.355 3.42(26) 0.22 4.45(56) 0.12289 0.360 4.00(42) 0.16 4.16(51) 0.10289 0.365 4.70(18) 0.28 4.67(27) 0.10289 0.370 4.48(20) 0.28 4.45(38) 0.10361 0.355 3.40(46) 0.13 4.17(69) 0.18361 0.360 3.60(35) 0.10 4.62(30) 0.10361 0.365 4.32(38) 0.39 4.36(16) 0.13361 0.370 5.02(31) 0.10 5.09(33) 0.13
Table 5 : Masses of first excited state in the pseudoscalar channel extrapolated to thechiral limit for Wilson (W) and twisted mass (Tw) fermions. m ρ * / √ σ Wilson, N=289Wilson, N=361 m π * / √ σ a √ σ Wilson, N=289Wilson, N=361Tw mass, N=289Tw mass, N=361
Figure 6 : Lattice spacing dependence of the first excited states in the pseudoscalar andvector channels, extrapolated to the chiral limit.
Another very interesting quantity to study is the pion decay constant f π . This is given interms of the matrix element of the axial current between the vacuum and the pion states.– 22 –n contrast with what happens with masses, this quantity is sensitive to normalizationand renormalization of the operator involved. The naive extension of the standard QCDdefinition diverges in the large N limit as √ N . Thus, as previous authors, we define thelarge N decay constant as follows f π = √ F π = (cid:115) N m π (cid:104) | A (0) | π ; (cid:126)p = 0 (cid:105) (4.2)where A is the temporal component of the axial vector current. The definition is cooked tocoincide with the standard one for N = 3. As it stands, the definition is symbolic if we donot specify how the | π ; (cid:126)p = 0 (cid:105) state is normalized. The formula assumes the relativisticallyinvariant normalization (cid:104) p | q (cid:105) = (2 π ) (2 p ) δ ( (cid:126)p − (cid:126)q ).In practice the matrix element appearing in Eq. (4.2) can be extracted from the cor-relation function of the temporal component of the axial current with any operator thatacts as a pion interpolating field. Dividing out by the square root of the correlation func-tion of the interpolating field with itself, we eliminate the dependence on the choice ofinterpolating field. The mechanism is true both on the continuum as on the lattice. As alattice interpolating field we can use the optimal operator selected by the GEVP method,reducing the possible contamination with excited states. In any case, this can be moni-tored by verifying that the correlator decreases in time as exp( − tm π ). This is essentiallythe same method used by other authors as for example Ref. [38]. A final word of warningcomes from the aforementioned necessity of normalizing and renormalizing the lattice Ax-ial current operator to match with the continuum definition. The normalization is ratherstandard. With our choice of the Wilson Dirac operator the lattice quark field is givenby Ψ L = (cid:112) a / (2 κ )Ψ. In addition the Axial current is a composite operator and getsa renormalization factor Z A . This depends on the choice of lattice discretized operator.Here we restrict ourselves to the ultralocal version A L ( x ) = ¯Ψ L ( x ) γ γ Ψ L ( x ). Finally,we can express the lattice spacing and the pion mass in string tension units to determine F π / ( √ σZ A ) for each of our datasets. The results are collected in the Table 15.Do our data scale? Examining the results for N=289, one can see that if we comparedata points having the same value of the pion mass, the tendency is that F π / ( √ σZ A )tends to decrease with increasing b . For example the data at b = 0 . κ = 0 . b = 0 .
365 and κ = 0 . F π / ( √ σZ A )is 10% higher. However, one cannot deduce from this a sizable violation of scaling, sincethe renormalization factor Z A introduces a explicit b -dependence of the result which goesin the same direction. Hence, to verify this one would need to know the value of Z A foreach b .Driven by this problem we decided to try a different method for determining the valueof F π . This is actually the main reason why we decided to include twisted mass fermions.One of the advantages of this method is that it allows a determination of F π without arenormalization factor [55]. Furthermore, the procedure is somewhat different and servesas an additional test of the robustness of our results. The value of f π is extracted from the We thank our colleagues Carlos Pena and Fernando Romero-L´opez for discussions about this point – 23 –xpression f π = 2 µ √ m π √ N (cid:104) | ¯Ψ( x ) γ Ψ( x ) | π ; (cid:126)p = 0 (cid:105) (4.3)Deriving the appropriate reduced lattice formula poses no problem. Again, the matrixelement is extracted from the correlation function of the ultralocal pseudoscalar operatorwith the optimal one. The propagator is now the inverse of the twisted mass one, givenin the previous section. This includes the Wilson-Dirac operator set at the value of κ c determined earlier for Wilson fermions. Namely the value at which the extrapolated PCACmass vanishes. The results then depend on the additional parameter µ , for which we choose4 values for each data set. As mentioned earlier, the pion mass square obtained in eachcase vanishes linearly in µ . The corresponding results for F π are given in Table 17.We still observe that some of the results having similar values of m π have non-compatible values of F π . However, after analyzing the results in detail, we discoveredthat this phenomenon is due to a sizable dependence on the effective volume. In Fig. 7 wedisplay the values of F π as a function of the pion mass square arranged into sets havingsimilar values of the effective box size a √ N = l √ σ . The data scale well within errors.However, the values of F π tend to decrease sizably for small volumes. Having results forthree different values of N has been crucial in discovering the origin of this disagreement.In order to get maximal information from our data we have tried to parameterizethe volume dependence of the results by fitting the data to an exponentially decreasingfunction of the effective size: F π ( m π ) + B ( m π ) exp {− C √ σl } . The coefficient C turns outto be 1.36(11) which explains why our results obtained for relatively small volumes are stillsizably dependent on the volume. Curiously we did not find any significant dependencein the determination of the masses. The fitted function F π ( m π ) gives our estimate ofthe pion decay constant at infinite volume and infinite N . In the region covered by ourdata it can be described well by a second degree polynomial in m π . The same goes forthe coefficient B ( m π ) of the exponentially decreasing term. Fig. 8 shows all our datafor N = 289 and N = 361 together with the solid lines giving the result of the fit. Animportant quantity is the value of F π in the chiral limit. We have done various fits withdifferent parameterizations and in all cases it seems that a value of F π (0) = 0 . F π from the twisted mass data we can come back to analyzethe results obtained for Wilson fermions. Knowing the value of F π the Wilson data allowsa non-perturbative determination of Z A . Unfortunately, this cannot be done point by pointsince the twisted mass and Wilson data do not correspond to exactly the same values of m π . However, it is highly non-trivial that all the Wilson fermion data can be well describedby the same functional dependence on m π up to a multiplicative factor 1 /Z A . Remarkablyit can, as shown in Fig. 8, where the twisted mass data are displayed together with theWilson Dirac data after a suitable rescaling.As a spin-off we have obtained a non-perturbative determination of Z A . The corre-sponding values are listed in Table 6. This can be compared with the value computed in– 24 – l √ σ ~ 3 b=0.355, N=169b=0.365, N=289b=0.370, N=361 l √ σ ~ 3.5 F π / √ σ b=0.360, N=289b=0.365, N=361 l √ σ > 3.9 m π / σ b=0.355, N=289b=0.355, N=361b=0.360, N=361 Figure 7 : Dependence of the pion decay constant on the pion mass square at approxi-mately fixed physical volume for twisted mass fermions.perturbation theory to two loop order in Ref. [58] giving Z A = 1 − . λ − . λ (4.4)It is well-known that truncated perturbative results on the lattice give poor estimateswhen computed in terms of the bare coupling constant λ = 1 /b . Much better results followwhen using improved couplings λ I of which there are various examples. In Table 6 we usevarious customary definitions to re-express the perturbative formula Eq. 4.4 in terms ofthose improved couplings. In general, the perturbative estimates tend to give higher valuesthan our non-perturbative result. Remarkably the same phenomenon takes place for SU(3)as commented in Ref. [38]. We have also analyzed the lowest lying spectrum in the vector channel using in this caseWilson fermions. The analysis, following the strategy presented in sec. 3.3, has allowed adetermination of the ρ mass, as well as the mass of the first excited state in this channel, ρ ∗ . The final results for all our lattices are compiled in table 13 in Appendix A.1.– 25 – N=289 F π / √ σ Tw mass, b=0.355 b=0.360 b=0.365 b=0.370Wilson , b=0.355 b=0.360 b=0.365 b=0.370
N=361 F π / √ σ m π / σ Tw mass, b=0.355 b=0.360 b=0.365 b=0.370Wilson , b=0.355 b=0.360 b=0.365 b=0.370
Figure 8 : Dependence of the pion decay constant on the pion mass square. The linescorrespond to the function F ( m π ) + B ( m π ) exp {− C √ σl } , with F ( m π ) = 0 . . m π /σ − . m π /σ , B ( m π ) = − . .
5) + 2 . m π /σ − . m π /σ , and C = 1 . N = 289 and N = 361, the value of ¯ χ for the fit is 0.85. Non-perturb. Non-perturb. Perturb. λ I = Perturb. λ I = Perturb. λ I =b N=289 N=361 1 / ( bP ( b )) − P ( b )) 8(1 − P ( b ))0.355 0.807(7) 0.806(4) 0.9161 0.8960 0.87660.360 0.827(10) 0.812(6) 0.9154 0.8971 0.87970.365 0.854(8) 0.801(10) 0.9149 0.8983 0.88250.370 0.85(2) 0.81(1) 0.9147 0.8994 0.8849 Table 6 : The first and second columns contain our non-perturbative determination of Z A from the N = 289 and N = 361 data. The remaining columns contain two-loopperturbative predictions using various definitions of the improved coupling λ I in terms ofthe plaquette value P ( b ). – 26 – m ρ / √ σ m PCAC / √ σ N=289, b=0.355 b=0.360 b=0.365 b=0.370N=361, b=0.355 b=0.360 b=0.365 b=0.370
Figure 9 : Dependence of the vector meson mass as a function of the PCAC mass, all inunits of the string tension. m ρ ( ) / √ σ l(b,N) √ σ m ρ (0) / √ σ = 1.676 (33)N=169N=289N=361 Figure 10 : Finite size dependence of the chirally extrapolated vector meson mass, m (0) ρ / √ σ . The purple band in the plot is the chiral extrapolation displayed in fig. 9,determined by a joint fit of the PCAC mass dependence of the data with l ( b, n ) ≥ . b Slope m (0) ρ / √ σ ¯ χ
169 0.355 1.2(1.6) 1.76(27) 0.44169 0.360 3.2(1.5) 1.29(26) 0.08289 0.355 2.27(20) 1.713(63) 0.05289 0.360 2.44(27) 1.673(84) 0.12289 0.365 2.22(27) 1.792(91) 0.16289 0.370 2.81(25) 1.557(90) 0.13361 0.355 2.44(59) 1.68(12) 0.001361 0.360 2.67(44) 1.59(10) 0.03361 0.365 2.81(67) 1.61(15) 0.20361 0.370 2.87(67) 1.69(14) 0.21 l ( b, N ) ≥ . Table 7 : Slope and intercept of the chiral extrapolation of the vector meson mass. Thelast row is determined by a joint fit of all the data with l ( b, n ) ≥ . N b m (0) ρ ∗ / √ σ (W) ¯ χ
289 0.355 4.55(31) 0.22289 0.360 4.91(28) 0.16289 0.365 5.29(21) 0.28289 0.370 5.14(14) 0.28361 0.355 4.44(43) 0.1361 0.360 4.38(37) 0.1361 0.365 5.19(39) 0.18361 0.370 5.17(32) 0.15
Table 8 : Intercept of the chiral extrapolation of the first excited state in the vector channel.The value of the ρ mass shows a linear dependence in the PCAC quark mass, asexpected at leading order in chiral perturbation theory in the large N limit [59]. Thisdependence is displayed in fig. 9 for all our lattices. By performing a simultaneous fitto all the points with physical size l ( b, n ) ≥ .
4, we obtain an intercept in the chirallimit given by m (0) ρ / √ σ = 1 . m (0) ρ / √ σ = 1 . χ from 0.44 to 1.0. We have alsoperformed independent fits to each data set, keeping b and N fixed. The correspondingintercepts and slopes are given in table 7. In fig. 10, the values of the intercepts aredisplayed as a function of the physical volume, l ( b, N ); with the exception of the smallestlattice volume, there is no significant finite volume effect in the results. The scaling withthe lattice spacing is very good and we see no clear lattice spacing dependence within theestimated errors.A common way of displaying the dependence of the ρ mass on the quark mass is to use– 28 – m ρ / m ρ ( ) m π / (m ρ (0) ) π / (m ρ (0) ) N=289, b=0.355N=289, b=0.360N=289, b=0.365N=361, b=0.355N=361, b=0.360N=361, b=0.365N=361, b=0.370
Figure 11 : Dependence of the ρ mass on the pion mass square, both normalized in unitsof the value of the ρ mass in the chiral limit m (0) ρ .the value of m (0) ρ to set the scale and monitor the dependence of m ρ /m (0) ρ on ( m π /m (0) ρ ) ,this eliminates renormalization ambiguities in the definition of the PCAC quark mass andmay get rid of some of the systematic errors in the determination of the scale. All ourresults for N = 289 and N = 361, with l ( b, N ) √ σ > .
9, are plotted in this way on fig. 11.The dependence on the pion mass square is linear with a slope given by 0.307(5) and with¯ χ = 0 .
45 .We have as well determined the masses of the first excited state in the vector channel,following the same procedure discussed for the pseudoscalar. The results are presented intable 13. The intercepts of the linear extrapolation to the chiral limit are given in table 13and displayed as a function of the lattice spacing in fig. 6. Assuming that the lattice spacingdependence is within errors, and fitting all the results to a constant gives a mass for theexcited state of 5.0(2) in units of the string tension with a value of ¯ χ = 1 . The ground-state masses in the a , a and b meson channels, for N = 289 and variouslattice spacings, are presented in table 14 in Appendix A.1. They have been obtained usingWilson fermions. As mentioned in section 3.3, we extract the masses from a fit to the optimal correlator, including in this case operators with smearing up to 50 in the basis.This allows to determine the ground state masses with good accuracy, although our resultsare not precise enough to extract the masses of the excited states in each channel.As in the case of the ρ meson, the extracted masses depend linearly on the PCACmass. Figures 12 - 14 exhibit this linear dependence for the three different channels. A– 29 – m a / √ σ m PCAC / √ σ N=289, b=0.355N=289, b=0.360N=289, b=0.365N=289, b=0.370
Figure 12 : Dependence of the a meson mass as a function of the PCAC mass, all inunits of the string tension. The yellow band corresponds to a joint fit of the b = 0 .
360 and b = 0 .
365 data sets. m a / √ σ m PCAC / √ σ N=289, b=0.355N=289, b=0.360N=289, b=0.365N=289, b=0.370
Figure 13 : Dependence of the a meson mass as a function of the PCAC mass, all inunits of the string tension. The yellow band corresponds to a joint fit of the b = 0 .
360 and b = 0 .
365 data sets. – 30 – m b / √ σ m PCAC / √ σ b=0.355b=0.360b=0.365b=0.370 Figure 14 : Dependence of the b meson mass as a function of the PCAC mass, all inunits of the string tension. The yellow band corresponds to a joint fit of the b = 0 .
360 and b = 0 .
365 data sets.
N b m (0) a / √ σ Slope m (0) a / √ σ Slope m (0) b / √ σ Slope ¯ χ
289 0.355 2.49(10) 3.01(32) 3.04(11) 2.13(36) 3.23(14) 1.83(47) 0.5/0.1/0.1289 0.360 2.25(7) 3.44(24) 2.92(9) 2.43(33) 3.02(27) 2.31(93) 0.9/0.1/0.1289 0.365 2.23(7) 3.64(24) 3.00(9) 2.52(28) 2.99(21) 2.23(68) 0.5/0.1/0.2289 0.370 2.15(6) 3.31(20) 2.92(10) 2.37(31) 3.26(20) 1.84(58) 0.4/0.1/0.2289 2.24(5) 3.54(17) 2.95(6) 2.53(21) 3.01(17) 2.23(55) 0.7/0.6/0.2
Table 9 : Slope and intercept of the chiral extrapolation of the masses in the a , a , and b meson channels. The last row is determined by a joint fit of the data with b = 0 . b = 0 .
360 and 0 .
365 gives as interceptsin the chiral limit: m (0) a = 2 . √ σ , m (0) a = 2 . √ σ , and m (0) b = 3 . √ σ . Thecorresponding slopes are given in table 9, where we also collect the parameters resultingfrom fits done at a fixed value of the inverse lattice coupling b . As shown in fig. 15, theresults show good scaling towards the continuum limit. The dependence on the latticespacing is negligible in the case of the a and b states, while the scalar meson a showsa tendency to decrease towards the continuum limit. The bands in the plot are obtainedfrom a constant fit to all the results, which works well in all cases except for the a mass.This issue will be further discussed in section 5, where we will present our final estimatesfor the continuum extrapolated masses. – 31 – m b (0) / √ σ = 3.16 (10)b m ( ) / √ σ m a (0) / √ σ = 2.97 (5)a √ σ m a (0) / √ σ = 2.24 (7)a Figure 15 : Lattice spacing dependence of the chirally extrapolated a , a , and b mesonmasses for N = 289. The bands correspond to a simultaneous fit of all the results to aunique value, independent of the lattice spacing. In this section we will summarize our results and analyze them in relation with other esti-mates and expectations of meson masses at large N . We have provided a lot of informationin tables which could help other researchers to draw their own conclusions. Our method-ology has the advantage of having negligible standard large N corrections. However, othertype of large N corrections arise, taking the form of finite volume corrections. This canbe understood and avoided by working at large enough effective volumes in physical units.Indeed, after monitoring for this effect we do not see appreciable dependence in the masses.On the contrary, the effect is quite sizable in the pion decay constant f π , and extrapola-tion is needed to obtain sensible results. Fortunately, in this case we have two independentmethods (Wilson and twisted mass fermions) to provide additional support to our analysis.To obtain the physical masses from lattice computations one has to extrapolate to thecontinuum limit. We have expressed the masses obtained at all lattice spacing values inunits of the string tension, so that in the continuum limit they should approach a constantvalue, which is the final continuum result. All the meson masses, with the exception of– 32 –he pion mass, depend linearly on the quark mass. On the other hand spontaneous chiralsymmetry breaking implies that the pion mass square also depends linearly on the quarkmass. For a more physical comparison we preferred to combine the two previous resultsand express the meson masses at each lattice spacing as follows m X √ σ = m (0) X √ σ + Y X m π σ (5.1)where X specifies the meson in question. Indeed, this is precisely what we see. However,the value of the masses in the chiral limit m (0) X √ σ at each lattice spacing value, were obtainedby extrapolating to vanishing PCAC mass, since that quantity is more precise and lessaffected by finite effective volume corrections than the pion mass itself. Once that is fixed,one can determine the slope coefficient Y X from the remaining data. For the case of F π our results can be well fitted with the following simple parameterization: F π σ = ( F (0) π ) σ + Y F m π σ (5.2)In this case a large N extrapolation is also needed as explained in the previous section.The final stage is the extrapolation of the chiral limit meson masses and slopes Y X tothe continuum limit. Our lattice results, obtained for four different values of the latticespacing, have to be extrapolated to vanishing spacing. In general, to the leading approxi-mation, the continuum value is approached linearly or quadratically in the lattice spacingdepending on the quantity, the system under study and particular discretization employed.For Wilson fermions we expect a linear approach. However, the majority of our physicalobservables are consistent with a negligible small slope compared to the statistical errors.This can be seen from Figs. 10, 15. Indeed, fitting the values obtained for masses andslopes for all lattice spacings to a unique value gives values of ¯ χ which are well below 1,signalling a good fit. In accordance with that, we have given as our best estimates of thecontinuum values the ones obtained by this simultaneous fit. These values are collected inTable 10. The described situation applies for all the slopes Y X as well as the masses of the ρ , a and b mesons in the chiral limit. The only exception is the value of the a mass. Inthis case the fit is not statistically favourable giving a value of ¯ χ = 1 .
7. More importantly,the data shows a systematic trend towards a decrease with the lattice spacing. Thus, ourbest estimate for the mass of a will follow from the procedure explained below.In any case, it is not possible to exclude a possible linear or quadratic dependenceon the lattice spacing even for the cases in which the data is consistent with a constantvalue. The extrapolation of the data using linear or quadratic dependencies on the latticespacing also depends on which observable ( √ σ , ¯ r or t ) is used to fix the scale, despitethe good agreement seen in table 1. For the case of the ρ , a and b meson masses, thedifferent extrapolations do not ameliorate the ¯ χ of the constant fit. However, we made useof the span defined by all the different extrapolations to give an estimate of the systematicerrors, since they exceed those coming from other sources. This shows the typical probleminvolved in extrapolations. The boundaries of the range covered by the different continuumlimit extrapolations relative to the best statistical average define a positive and a negative– 33 – ρ a b a F πm (0) X √ σ . (cid:32) +9 − (cid:33) . (cid:32) +8 − (cid:33) . (cid:32) +11 − (cid:33) . (cid:32) +41 − (cid:33) . Y X Table 10 : Masses of mesons in the chiral and continuum limit and slopes corresponding toEq. (5.1). The central value is our best statistically significant estimate, corresponding toa constant fit for the ρ , a and b mesons and a linear fit in a ( b ) for the a mass (see text).The column vector give the maximum positive and negative shifts of our best estimateneeded to cover the range of all different continuum limit extrapolations. They can beinterpreted as the systematic errors of our continuum limit results (see text).value, shifting up or down the best value. These shifts are shown in Table 10 as a columnvector of values to be added or subtracted to the last significant digits of the best value.The situation for the a mass is quite different. In this case, all the extrapolationsbased on the different choices of scale-setting predict a decrease with the lattice spacing a ,accompanied with a better statistical significance, signalled by values of ¯ χ of order 0.75-0.9.The continuum limit values obtained by fitting a linear dependence in a/ ¯ r and a/ √ t arequite consistent with each other. After converting them back to string tension units using¯ r √ σ = 1 .
035 and √ t σ = 1 .
078 we get m (0) a / √ σ = 1 .
829 and 1 .
816 respectively. Hence,we take the weighted average as our best estimate shown on Table 10. The systematicerrors are depicted in the same fashion ranging from the constant fit to the linear fit using a √ σ .In the case of the slopes Y X (shown in Table 10) the systematic errors associated tothe continuum limit extrapolations lie well within the range of statistical errors. Our bestestimates for the pion decay constant and its corresponding slope are given, followed bythe statistical and systematic error. The latter also depends on the large N extrapolationpresented in the previous section.Now let us compare our results with other predictions and calculations. The firstcomparison can be made with QCD and its meson spectrum. Setting the string tension to √ σ = 440 M eV , the values of the physical I=1 meson masses in string tension units aregiven by 1.76, 2.82, 2.82 and 2.23 for ρ , a , b and a respectively, which are not too farfrom the 1.71, 2.99, 3.175 and 1.855 of our large N best estimates for the physical value of m π /σ ∼ .
1. Hence, it seems that, in what respects to meson masses, large N provides afairly good approximation to the real world.Our next comparison is with other lattice determinations of the spectrum. As men-tioned in the introduction, results using the quenched approximation extrapolated to large N started some years ago [35, 36, 60]. Results based on ideas of volume independencesimilar to ours appeared even earlier [37, 61]. Very interestingly, some results have alsobeen obtained with two flavours of dynamical quarks, with values reasonably close [39].The most complete published lattice work is that of Ref. [38]. It involves a very detailedand thorough work producing results on large N spectroscopy by extrapolation of the re-– 34 –ults obtained at various N , including some at N = 17. These results are obtained at aunique value of the lattice spacing, so that an analysis of the lattice spacing dependenceis not possible. However, since the authors use Wilson fermions and a Wilson action witha value of the coupling which seems to correspond to b=0.36 at large N , we can comparetheir results with what we obtain at that same coupling. The chiral limit masses for a and b and F π are perfectly consistent within errors. Our result for the ρ in the chiral limit(1.63(8)) is slightly larger than their result (1.54(1)), while for the a it is the other wayround (2.25(7) versus 2.40(3)). In any case, there is a very good qualitative agreement,which is remarkable given the very different methodology used by the two determinations.Incidentally our results on the rho mass and F π do not agree with those of Refs. [37, 61],despite they being closer methodologically to ours.An effort to obtain results in the continuum limit was done by some members of thesame collaboration. In particular a detailed study with 4 lattice spacing (like ours) forSU(7) has appeared in proceedings of conferences [62]. A more detailed chart includingan estimate of the continuum extrapolation of meson masses at large N was presented inthe thesis dissertation of Luca Castagnini [63] . The study covers 4 values of the latticespacing with the finer lattices similar to ours and one point coarser than our data. Thefinal results involve a double extrapolation to N = ∞ and a = 0, which is achieved by a4 parameter fit to the data of each observable. Hence, the results have to be taken cumgranum salis, as recognized by the author. Nonetheless, the final table is strikingly similarto our best values giving chiral limit masses of 1.687(24), 2.93(11), 2.97(13) for ρ , a and b respectively and F π = 0 . a obtained by the fit for a and b are small,in numerical agreement with our results which are consistent with vanishing slopes withinerrors. There is certain disagreement in their predicted slope in a for the ρ . Mysteriouslywe end up having the same estimate for the rho mass in the chiral limit and large N .The case for the scalar meson a deserves special attention. Both our results as well asthose of Refs. [62, 63] point towards a sizable positive slope with a √ σ . Our best estimatefor the a mass in the chiral limit 1.83(15) is remarkably close to the value 1.81(17) givenin Ref. [63]. This agreement gives a stronger evidence that our observed a-dependence isnot a statistical fluctuation but rather a genuine effect. It seems that the scalar mesons arerather special, also having a larger quark mass dependence Y a than the remaining mesons.Indeed, the same happens in QCD and a long lived discussion has been centered aboutthem (see Ref. [64] for a recent review). Actually, it has been proposed that the study ofthe behaviour of the masses at large N could help in settling some points [65]. A gooddeal of the controversy has to do with the isoscalar f (500), which in the large N limitis degenerate with the a . Hence, studying the leading 1 /N corrections is presumablycrucial. Furthermore there are certain predictions of quite different nature [66, 67] thatsuggest that the large N a mass might coincide with that of the rho meson, a possibilitywhich is not inconsistent with our results.Altogether, it is fair to say that our continuum results are consistent with the onesby Castagnini and, as emphasized earlier, this is more remarkable given the differences in We thank Marco Bocchicchio and Gunnar Bali for pointing us to this reference – 35 –ethodology of our approaches. Even the scale-setting is different, since we use our ownmeasurements of the string tension and two other methods to fix the scale.Our next goal is to comment on other works and results concerning the meson spectrumat large number of colours obtained with other methods. A new paradigm has arisen fromthe AdS/CFT correspondence [2–4], mapping field theory problems into others involvingstring theory, supergravity or just gravity. This is, no doubt, an interesting breakthroughintroducing a new perspective in describing some field theoretical phenomena and allowingthe computation of some observables. The large N limit seems to be crucial in making thesenew methods feasible. The original ideas concern theories which are very different fromQCD, being conformal, supersymmetric, and having fermions and scalars in the adjointrepresentation. As we move away from this situation the level of rigour in the connectiondecreases. Nonetheless, theories with quarks in the fundamental [68], with reduced or no-supersymmetry and with running couplings, have been studied. Interesting calculationsincluding meson masses have been performed [69–72]. We address the reader to the reviewin Ref. [73] for a very nice account and a more complete list of references. It is worthmentioning that even in theories which are not exactly QCD some observables becomerather close to our results. For example, the slope Y ρ obtained in Ref. [70] is indeed consis-tent with our result within errors. We should also emphasize that more phenomenologicalmethods inspired by holography have been proposed [74–76]. As a general rule, some ofthese papers can use our results to fix some of the parameters of their models, but theyhave the potential of predicting higher excited states which are more difficult to obtain bylattice methods. Along these lines it also worth mentioning the modified string proposalof Ref. [77] which gives meson spectrum results which in some cases are quite compatiblewith our results.A final comment concerns possible future improvement of our work. Given the effortinvolved, the large uncertainties in taking the continuum limit are somewhat discouraging.To obtain a better control it is sometimes good to go to coarser lattices where the effectis more pronounced. However, for coarser lattices using the simplest parameterizationbecomes more doubtful and adding more parameters spoils the advantage. From our pointof view it is better to go to finer lattices and to reduce the errors. For that the mostimportant limitation is the value of N , whose square root translates into an effective latticesize. The present limit is only computational and enters in the cpu resources neededto compute the propagator. Reaching larger values allows a longer time extent of thecorrelators and hence longer plateaus. Furthermore, one could reach larger values of b and, hence, smaller values of the lattice spacing without running into finite effective sizeproblems. In relation to this, it should be mentioned that there is no need to take volumereduction to the extreme and simulate the 1 point lattice TEK model. The results couldbe achieved by running in a small lattice of size L × L . However, we advise researcherstrying to follow this road to use appropriately chosen twisted boundary conditions. Inparticular, using symmetric twist the effective lattice size would become L √ N and goodresults could be obtained with much smaller values of N at very small volumes.– 36 – cknowledgments We acknowledge interesting conversations about various aspects with G. Bali, M. Bocchic-chio, B. Lucini, C. Pena and F. Romero-L´opez. M.G.P. and A.G-A acknowledge financialsupport from the MINECO/FEDER grant FPA2015-68541-P and PGC2018- 094857-B-I00and the MINECO Centro de Excelencia Severo Ochoa Program SEV-2016-0597. M. O. issupported by JSPS KAKENHI Grant Number 17K05417. This publication is supportedby the European project H2020-MSCAITN-2018-813942 (EuroPLEx) and the EU Horizon2020 research and innovation programme, STRONG-2020 project, under grant agreementNo 824093. This research used computational resources of the SX-ACE system providedby Osaka University through the HPCI System Research Project (Project ID: hp170003and hp180002). We acknowledge the use of the Hydra cluster at IFT.
A Raw data
A.1 Wilson fermions – 37 – b κ am pcac ¯ χ
169 0.355 0.1592 0.05672 (84) 1.33169 0.355 0.1600 0.04260 (99) 1.25169 0.355 0.1607 0.0306 (11) 1.14169 0.360 0.1570 0.05151 (76) 0.68169 0.360 0.1577 0.03899 (70) 0.76169 0.360 0.1585 0.02385 (82) 0.95289 0.355 0.1565 0.10781 (45) 1.64289 0.355 0.1575 0.08986 (43) 1.50289 0.355 0.1585 0.07238 (43) 1.33289 0.355 0.1592 0.06041 (43) 1.21289 0.355 0.1600 0.04695 (42) 1.12289 0.355 0.1607 0.03525 (42) 1.12289 0.360 0.1550 0.09070 (95) 0.59289 0.360 0.1560 0.07251 (82) 0.43289 0.360 0.1570 0.05467 (72) 0.25289 0.360 0.1577 0.04231 (67) 0.17289 0.360 0.1585 0.02808 (75) 0.24289 0.365 0.1535 0.0814 (11) 0.67289 0.365 0.1540 0.0720 (10) 0.68289 0.365 0.1545 0.06269 (99) 0.70289 0.365 0.1550 0.05350 (95) 0.74289 0.365 0.1555 0.04440 (90) 0.80289 0.365 0.1562 0.03187 (83) 0.96289 0.365 0.1570 0.01799 (66) 1.19289 0.370 0.1520 0.0821 (13) 1.29289 0.370 0.1525 0.0722 (12) 1.37289 0.370 0.1530 0.0624 (11) 1.44289 0.370 0.1535 0.0526 (10) 1.50289 0.370 0.1540 0.04292 (94) 1.53289 0.370 0.1547 0.02958 (84) 1.49289 0.370 0.1555 0.01481 (73) 1.30361 0.355 0.1592 0.06118 (50) 0.37361 0.355 0.1600 0.04794 (46) 0.47361 0.355 0.1607 0.03655 (43) 0.54361 0.360 0.1570 0.05631 (39) 0.87361 0.360 0.1577 0.04396 (36) 0.88361 0.360 0.1585 0.02998 (33) 0.88361 0.365 0.1555 0.04669 (77) 0.30361 0.365 0.1562 0.03374 (76) 0.26361 0.365 0.1570 0.0185 (11) 0.23361 0.370 0.1540 0.04372 (69) 0.27361 0.370 0.1547 0.03063 (60) 0.27361 0.370 0.1555 0.01601 (49) 0.27
Table 11 : PCAC mass for Wilson fermions.– 38 – b κ m π / √ σ ¯ χ m π ∗ / √ σ ¯ χ
169 0.355 0.1592 1.658 (38) 0.59 3.40 (08) 0.59169 0.355 0.1600 1.428 (44) 0.46 3.21 (07) 0.46169 0.355 0.1607 1.217 (53) 0.46 3.02 (11) 0.45169 0.360 0.1570 1.745 (80) 0.39 3.85 (16) 0.44169 0.360 0.1577 1.536 (80) 0.43 3.65 (15) 0.43169 0.360 0.1585 1.188 (99) 0.30 3.28 (12) 0.27289 0.355 0.1565 2.406 (26) 0.64 4.63 (30) 0.54289 0.355 0.1575 2.189 (24) 0.54 4.47 (23) 0.47289 0.355 0.1585 1.957 (23) 0.47 4.31 (20) 0.42289 0.355 0.1592 1.782 (22) 0.42 4.17 (20) 0.38289 0.355 0.1600 1.563 (21) 0.38 3.99 (20) 0.36289 0.355 0.1607 1.345 (21) 0.36 3.79 (19) 0.33289 0.360 0.1550 2.445 (48) 0.39 5.07 (31) 0.47289 0.360 0.1560 2.181 (43) 0.31 4.88 (28) 0.43289 0.360 0.1570 1.893 (40) 0.22 4.69 (25) 0.36289 0.360 0.1577 1.670 (41) 0.16 4.54 (26) 0.29289 0.360 0.1585 1.370 (65) 0.10 4.24 (43) 0.18289 0.365 0.1535 2.540 (39) 0.21 5.38 (14) 0.19289 0.365 0.1540 2.392 (36) 0.21 5.29 (14) 0.19289 0.365 0.1545 2.239 (34) 0.21 5.20 (14) 0.18289 0.365 0.1550 2.081 (33) 0.20 5.11 (14) 0.18289 0.365 0.1555 1.917 (32) 0.20 5.03 (14) 0.17289 0.365 0.1562 1.680 (31) 0.21 4.93 (15) 0.17289 0.365 0.1570 1.400 (34) 0.26 4.97 (25) 0.18289 0.370 0.1520 2.725 (56) 0.41 5.50 (16) 0.29289 0.370 0.1525 2.544 (55) 0.38 5.36 (15) 0.28289 0.370 0.1530 2.359 (55) 0.34 5.22 (16) 0.28289 0.370 0.1535 2.169 (57) 0.30 5.09 (17) 0.27289 0.370 0.1540 1.975 (59) 0.25 4.96 (18) 0.25289 0.370 0.1547 1.700 (62) 0.18 4.81 (20) 0.21289 0.370 0.1555 1.382 (73) 0.13 4.79 (26) 0.16361 0.355 0.1592 1.808 (19) 0.22 4.28 (17) 0.45361 0.355 0.1600 1.593 (18) 0.19 4.11 (16) 0.43361 0.355 0.1607 1.382 (17) 0.17 3.92 (16) 0.40361 0.360 0.1570 1.860 (22) 0.27 4.41 (14) 0.30361 0.360 0.1577 1.638 (21) 0.21 4.25 (14) 0.26361 0.360 0.1585 1.352 (22) 0.18 4.03 (15) 0.21361 0.365 0.1555 1.901 (29) 0.15 5.03 (11) 0.32361 0.365 0.1562 1.634 (34) 0.12 4.88 (13) 0.23361 0.365 0.1570 1.242 (70) 0.10 4.53 (29) 0.09361 0.370 0.1540 2.029 (33) 0.16 5.29 (12) 0.23361 0.370 0.1547 1.750 (35) 0.08 5.20 (14) 0.16361 0.370 0.1555 1.381 (47) 0.03 5.13 (21) 0.13
Table 12 : Wilson fermion masses in the pseudoscalar channel, extracted from a fit of thetime dependence of the optimal correlator. The excited mass is extracted from a doubleexponential fit with the ground-state mass fixed, as indicated in the text.– 39 – b κ m ρ / √ σ ¯ χ m ρ ∗ / √ σ ¯ χ
169 0.355 0.1592 2.08 (17) 0.98 4.13 (0.21) 0.92169 0.355 0.1600 1.945 (64) 0.98 4.12 (0.13) 0.89169 0.355 0.1607 1.922 (91) 0.76 4.30 (0.22) 0.65169 0.360 0.1570 2.08 (40) 0.52 4.67 (0.28) 0.36169 0.360 0.1577 1.910 (74) 0.49 4.59 (0.12) 0.35169 0.360 0.1585 1.668 (94) 0.43 4.46 (0.11) 0.32289 0.355 0.1565 2.727 (44) 1.04 4.93 (0.22) 0.88289 0.355 0.1575 2.561 (45) 0.98 4.85 (0.22) 0.84289 0.355 0.1585 2.395 (46) 0.91 4.78 (0.23) 0.78289 0.355 0.1592 2.280 (48) 0.85 4.74 (0.23) 0.73289 0.355 0.1600 2.152 (53) 0.76 4.71 (0.25) 0.65289 0.355 0.1607 2.048 (59) 0.68 4.71 (0.29) 0.60289 0.360 0.1550 2.742 (61) 0.50 5.40 (0.26) 0.48289 0.360 0.1560 2.538 (58) 0.51 5.31 (0.24) 0.51289 0.360 0.1570 2.328 (58) 0.51 5.21 (0.22) 0.52289 0.360 0.1577 2.176 (60) 0.50 5.14 (0.22) 0.50289 0.360 0.1585 1.998 (73) 0.45 5.07 (0.21) 0.41289 0.365 0.1535 2.814 (68) 0.45 5.76 (0.16) 0.35289 0.365 0.1540 2.691 (66) 0.47 5.69 (0.16) 0.36289 0.365 0.1545 2.569 (65) 0.47 5.62 (0.16) 0.36289 0.365 0.1550 2.449 (65) 0.47 5.56 (0.16) 0.37289 0.365 0.1555 2.333 (68) 0.46 5.50 (0.17) 0.36289 0.365 0.1562 2.191 (81) 0.40 5.47 (0.20) 0.32289 0.365 0.1570 2.04 (11) 0.31 5.45 (0.25) 0.25289 0.370 0.1520 3.013 (80) 0.34 5.99 (0.18) 0.26289 0.370 0.1525 2.846 (78) 0.35 5.87 (0.17) 0.28289 0.370 0.1530 2.678 (77) 0.37 5.75 (0.16) 0.31289 0.370 0.1535 2.506 (77) 0.38 5.64 (0.15) 0.32289 0.370 0.1540 2.332 (77) 0.38 5.55 (0.14) 0.34289 0.370 0.1547 2.086 (84) 0.36 5.43 (0.14) 0.34289 0.370 0.1555 1.80 (11) 0.31 5.32 (0.15) 0.30361 0.355 0.1592 2.300 (40) 0.42 4.68 (0.14) 0.53361 0.355 0.1600 2.166 (42) 0.47 4.63 (0.15) 0.56361 0.355 0.1607 2.051 (46) 0.48 4.59 (0.16) 0.58361 0.360 0.1570 2.322 (37) 0.55 4.86 (0.12) 0.58361 0.360 0.1577 2.164 (38) 0.62 4.76 (0.11) 0.63361 0.360 0.1585 1.981 (41) 0.72 4.64 (0.17) 0.70361 0.365 0.1555 2.346 (52) 0.25 5.64 (0.20) 0.29361 0.365 0.1562 2.154 (56) 0.26 5.54 (0.18) 0.24361 0.365 0.1570 1.893 (96) 0.36 5.36 (0.23) 0.26361 0.370 0.1540 2.486 (78) 0.16 5.75 (0.25) 0.37361 0.370 0.1547 2.268 (79) 0.12 5.62 (0.23) 0.35361 0.370 0.1555 1.979 (87) 0.15 5.38 (0.18) 0.43
Table 13 : Masses in the vector channel for Wilson fermions.– 40 – b κ m b / √ σ ¯ χ m a / √ σ ¯ χ m a / √ σ ¯ χ
289 0.355 0.1565 4.06 (12) 0.94 3.990 (88) 0.75 3.807 (77) 0.16289 0.355 0.1575 3.92 (11) 0.87 3.842 (87) 0.65 3.622 (77) 0.14289 0.355 0.1585 3.78 (11) 0.79 3.693 (87) 0.55 3.425 (69) 0.14289 0.355 0.1592 3.69 (11) 0.72 3.587 (88) 0.48 3.277 (72) 0.15289 0.355 0.1600 3.59 (11) 0.60 3.461 (90) 0.41 3.093 (80) 0.18289 0.355 0.1607 3.52 (13) 0.46 3.343 (99) 0.34 2.863 (92) 0.17289 0.360 0.1550 4.04 (23) 0.28 3.988 (87) 0.33 3.728 (55) 0.31289 0.360 0.1560 3.83 (25) 0.37 3.782 (80) 0.41 3.479 (50) 0.31289 0.360 0.1570 3.65 (23) 0.48 3.578 (75) 0.47 3.206 (47) 0.29289 0.360 0.1577 3.52 (22) 0.57 3.432 (72) 0.53 2.984 (50) 0.25289 0.360 0.1585 3.32 (21) 0.69 3.246 (71) 0.66 2.656 (60) 0.24289 0.365 0.1535 4.03 (21) 0.41 4.152 (90) 0.42 3.841 (78) 0.30289 0.365 0.1540 3.90 (20) 0.39 4.019 (86) 0.41 3.688 (75) 0.27289 0.365 0.1545 3.76 (20) 0.36 3.885 (84) 0.40 3.528 (72) 0.26289 0.365 0.1550 3.63 (19) 0.34 3.753 (82) 0.38 3.360 (69) 0.24289 0.365 0.1555 3.51 (19) 0.32 3.622 (81) 0.37 3.178 (67) 0.23289 0.365 0.1562 3.36 (19) 0.30 3.447 (82) 0.34 2.889 (69) 0.20289 0.365 0.1570 3.27 (22) 0.35 3.259 (86) 0.33 2.553 (70) 0.13289 0.370 0.1520 4.25 (23) 0.45 4.17 (13) 0.37 3.835 (92) 0.27289 0.370 0.1525 4.12 (22) 0.45 4.02 (137) 0.38 3.656 (87) 0.26289 0.370 0.1530 3.99 (21) 0.45 3.86 (11) 0.38 3.471 (81) 0.25289 0.370 0.1535 3.86 (20) 0.46 3.71 (11) 0.39 3.278 (76) 0.24289 0.370 0.1540 3.74 (19) 0.46 3.56 (10) 0.39 3.078 (71) 0.24289 0.370 0.1547 3.59 (19) 0.46 3.35 (10) 0.41 2.788 (62) 0.22289 0.370 0.1555 3.48 (21) 0.39 3.17 (11) 0.41 2.438 (55) 0.23
Table 14 : Masses in the a , a and b channels for Wilson fermions.– 41 – b κ Z − A F π / √ σ ¯ χ
169 0.355 0.1592 0.322 (11 ) 0.72169 0.355 0.1600 0.285 (13 ) 0.82169 0.355 0.1607 0.248 (13 ) 0.76169 0.360 0.1570 0.301 (19 ) 0.51169 0.360 0.1577 0.261 (17 ) 0.63169 0.360 0.1585 0.198 (19 ) 0.80289 0.355 0.1565 0.4389 (69) 0.95289 0.355 0.1575 0.4194 (59) 0.82289 0.355 0.1585 0.3972 (50) 0.69289 0.355 0.1592 0.3796 (44) 0.63289 0.355 0.1600 0.3567 (39) 0.60289 0.355 0.1607 0.3325 (39) 0.66289 0.360 0.1550 0.422 (10 ) 0.46289 0.360 0.1560 0.3982 (82) 0.39289 0.360 0.1570 0.3688 (67) 0.30289 0.360 0.1577 0.3431 (65) 0.24289 0.360 0.1585 0.302 (10) 0.23289 0.365 0.1535 0.4184 (89) 0.36289 0.365 0.1540 0.4041 (87) 0.29289 0.365 0.1545 0.3878 (87) 0.24289 0.365 0.1550 0.3691 (89) 0.19289 0.365 0.1555 0.3470 (93) 0.17289 0.365 0.1562 0.3081 (98) 0.19289 0.365 0.1570 0.2423 (97) 0.27289 0.370 0.1520 0.4228 (120) 0.59289 0.370 0.1525 0.4039 (111) 0.58289 0.370 0.1530 0.3823 (104) 0.56289 0.370 0.1535 0.3568 (100) 0.54289 0.370 0.1540 0.3259 (95) 0.53289 0.370 0.1547 0.2697 (86) 0.51289 0.370 0.1555 0.1780 (91) 0.44361 0.355 0.1592 0.3830 (46) 0.39361 0.355 0.1600 0.3612 (40) 0.45361 0.355 0.1607 0.3396 (36) 0.53361 0.360 0.1570 0.3793 (59) 0.56361 0.360 0.1577 0.3566 (53) 0.52361 0.360 0.1585 0.3242 (52) 0.48361 0.365 0.1555 0.3701 (66) 0.23361 0.365 0.1562 0.3345 (73) 0.17361 0.365 0.1570 0.2678 (135) 0.31361 0.370 0.1540 0.3702 (61) 0.40361 0.370 0.1547 0.3268 (63) 0.38361 0.370 0.1555 0.2483 (77) 0.28
Table 15 : Pion decay constant for Wilson fermions.– 42 – .2 Twisted mass fermions
N b κ c µ m π / √ σ ¯ χ m π ∗ / √ σ ¯ χ
169 0.355 0.02477 2.500 (32) 0.21 4.62 (24) 0.74169 0.355 0.01736 2.067 (25) 0.29 4.36 (21) 0.76169 0.355 0.01344 1.818 (23) 0.47 4.20 (18) 0.80169 0.355 0.00852 1.477 (21) 0.69 3.94 (15) 0.86169 0.360 0.02081 2.431 (42) 0.56 4.67 (13) 0.65169 0.360 0.01570 2.092 (40) 0.51 4.45 (13) 0.63169 0.360 0.01217 1.851 (39) 0.44 4.29 (13) 0.60169 0.360 0.00716 1.497 (38) 0.31 4.05 (14) 0.51289 0.355 0.02477 2.523 (18) 0.73 5.24 (39) 0.70289 0.355 0.01736 2.068 (15) 0.87 5.03 (32) 0.83289 0.355 0.01344 1.802 (14) 0.84 4.92 (33) 0.79289 0.355 0.00852 1.424 (14) 0.59 4.69 (41) 0.56289 0.360 0.02081 2.521 (38) 0.68 5.14 (44) 0.89289 0.360 0.01570 2.159 (34) 0.57 4.93 (31) 0.72289 0.360 0.01217 1.886 (32) 0.49 4.78 (37) 0.57289 0.360 0.00716 1.448 (30) 0.46 4.49 (31) 0.40289 0.365 0.01814 2.487 (37) 0.50 5.33 (17) 0.40289 0.365 0.01293 2.082 (34) 0.47 5.13 (17) 0.39289 0.365 0.00933 1.782 (32) 0.50 5.00 (19) 0.44289 0.365 0.00615 1.497 (31) 0.54 4.90 (22) 0.52289 0.370 0.01552 2.551 (59) 0.53 5.57 (31) 0.41289 0.370 0.01194 2.221 (52) 0.52 5.31 (26) 0.42289 0.370 0.00827 1.861 (50) 0.49 5.05 (24) 0.43289 0.370 0.00534 1.555 (54) 0.44 4.84 (26) 0.42361 0.355 0.02477 2.548 (21) 0.54 3.17 (95) 0.76361 0.355 0.01736 2.082 (16) 0.45 5.15 (29) 0.57361 0.355 0.01344 1.806 (14) 0.37 4.96 (29) 0.48361 0.355 0.00852 1.411 (13) 0.28 4.64 (33) 0.35361 0.360 0.02081 2.520 (25) 0.37 5.43 (20) 0.73361 0.360 0.01570 2.150 (20) 0.43 5.21 (19) 0.72361 0.360 0.01217 1.871 (18) 0.44 5.07 (18) 0.68361 0.360 0.00716 1.421 (14) 0.36 4.91 (20) 0.49361 0.365 0.01814 2.503 (24) 0.83 5.30 (15) 0.61361 0.365 0.01293 2.076 (19) 0.74 5.02 (12) 0.55361 0.365 0.00933 1.758 (17) 0.60 4.83 (10) 0.45361 0.365 0.00615 1.456 (18) 0.41 4.69 (10) 0.33361 0.370 0.01552 2.500 (40) 0.41 5.56 (22) 0.41361 0.370 0.01194 2.183 (33) 0.31 5.41 (21) 0.34361 0.370 0.00827 1.835 (29) 0.24 5.31 (21) 0.26361 0.370 0.00534 1.530 (32) 0.24 5.26 (25) 0.21
Table 16 : Masses in the pseudoscalar channel for twisted mass fermions.– 43 – b κ c µ F π / √ σ ¯ χ
169 0.355 0.02477 0.3644 (32) 0.78169 0.355 0.01736 0.3171 (19) 0.77169 0.355 0.01344 0.2870 (17) 0.78169 0.355 0.00852 0.2376 (19) 0.81169 0.360 0.02081 0.3425 (25) 0.99169 0.360 0.01570 0.3058 (19) 0.85169 0.360 0.01217 0.2744 (23) 0.72169 0.360 0.00716 0.2116 (33) 0.52289 0.355 0.02477 0.3775 (11) 0.91289 0.355 0.01736 0.3338 (13) 0.92289 0.355 0.01344 0.3080 (14) 0.82289 0.355 0.00852 0.2695 (19) 0.58289 0.360 0.02081 0.3652 (27) 0.62289 0.360 0.01570 0.3305 (20) 0.54289 0.360 0.01217 0.3032 (20) 0.49289 0.360 0.00716 0.2542 (32) 0.45289 0.365 0.01814 0.3537 (24) 0.75289 0.365 0.01293 0.3127 (20) 0.70289 0.365 0.00933 0.2761 (24) 0.64289 0.365 0.00615 0.2311 (32) 0.57289 0.370 0.01552 0.3387 (40) 0.57289 0.370 0.01194 0.3030 (30) 0.54289 0.370 0.00827 0.2573 (39) 0.48289 0.370 0.00534 0.2068 (55) 0.40361 0.355 0.02477 0.3812 (14) 0.46361 0.355 0.01736 0.3363 (12) 0.44361 0.355 0.01344 0.3106 (14) 0.41361 0.355 0.00852 0.2748 (19) 0.38361 0.360 0.02081 0.3708 (20) 0.30361 0.360 0.01570 0.3372 (14) 0.31361 0.360 0.01217 0.3116 (15) 0.31361 0.360 0.00716 0.2681 (21) 0.24361 0.365 0.01814 0.3558 (25) 1.03361 0.365 0.01293 0.3161 (19) 0.91361 0.365 0.00933 0.2825 (23) 0.71361 0.365 0.00615 0.2424 (33) 0.45361 0.370 0.01552 0.3518 (20) 0.46361 0.370 0.01194 0.3184 (16) 0.39361 0.370 0.00827 0.2752 (24) 0.27361 0.370 0.00534 0.2277 (37) 0.20
Table 17 : Pion decay constant for twisted mass fermions.
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