Infrared conformality and bulk critical points: SU(2) with heavy adjoint quarks
Biagio Lucini, Agostino Patella, Antonio Rago, Enrico Rinaldi
CCERN-PH-TH/2013-215, Edinburgh 2013/12
Infrared conformality and bulk critical points: SU(2)with heavy adjoint quarks
Biagio Lucini a Agostino Patella b,c
Antonio Rago c Enrico Rinaldi d a College of Science, Swansea University, Singleton Park, Swansea SA2 8PP, UK b PH-TH, CERN, CH-1211 Geneva 23, Switzerland c School of Computing and Mathematics & Centre for Mathematical Science, Plymouth University,Plymouth PL4 8AA, UK d Higgs Centre for Theoretical Physics, SUPA, School of Physics and Astronomy, University ofEdinburgh, Edinburgh EH9 3JZ, UK
E-mail: [email protected], [email protected],[email protected], [email protected]
Abstract:
The lattice phase structure of a gauge theory can be a serious obstruction toMonte Carlo studies of its continuum behaviour. This issue is particularly delicate whennumerical studies are performed to determine whether a theory is in a (near-)conformalphase. In this work we investigate the heavy mass limit of the SU(2) gauge theory with N f = 2 adjoint fermions and its lattice phase diagram, showing the presence of a criticalpoint ending a line of first order bulk phase transition. The relevant gauge observables andthe low–lying spectrum are monitored in the vicinity of the critical point with very goodcontrol over different systematic effects. The scaling properties of masses and susceptibil-ities open the possibility that the effective theory at criticality is a scalar theory in theuniversality class of the four–dimensional Gaussian model. This behaviour is clearly dif-ferent from what is observed for SU(2) gauge theory with two dynamical adjoint fermions,whose (near-)conformal numerical signature is hence free from strong–coupling bulk effects. a r X i v : . [ h e p - l a t ] S e p ontents Despite the recent identification of a light Higgs boson at the LHC [1, 2], unveiling themechanism of electroweak symmetry breaking is still an open problem in theoretical par-ticle physics. Among the possibilities still on the table, the suggestion that a novel stronginteraction displaying confinement [3, 4] and an anomalous dimension of the chiral conden-sate of order one [5–7] can be tested quantitatively with lattice simulations. In the questfor a theory that could realise concretely this scenario, several gauge theory models, withmatter in the fundamental or in a two–index representation and various flavour and colourcontent have been studied with Monte Carlo methods (see [8–10] for recent reviews), withthe space of parameters narrowed down using analytical input [11, 12].A large anomalous dimension is expected to arise near the onset of the conformalwindow. Hence, to unambiguously ascertain conformality for a gauge theory, one needsto be able to robustly determine whether the theory has or is near to an infrared (IR)fixed point. Recent Monte Carlo studies have shown that the identification of IR fixedpoints on the lattice is not straightforward, since in numerical simulations the system has afinite size (while conformality would require infinite distances to be explored) and fermionshave a finite mass (the conformal limit being for massless matter fields). In addition,lattice simulations are performed at fixed cutoff and for particular choices of the discretisedaction. Because of these unavoidable complications, there is wide consensus that, in latticesimulations aimed at ascertaining (near-)conformality of a gauge theory, evidences basedon different approaches and different techniques need to be collected before one can excludespurious lattice signatures being mistaken for genuine IR fixed points in the continuum. Inthe last few years, thanks to the joint effort of various groups, this programme has beencarried out for SU(2) gauge theory with two adjoint Dirac fermions, for which the scalingof the spectrum [13–17] and the behaviour of the coupling constant under RG flow [18–21] strongly suggest IR conformality (see also [22–24] for earlier numerical investigations).– 1 –ore recent lattice studies of the theory are focused on controlling systematic effects [25–28]and on precise measurements of the anomalous dimension [29, 30]. However the persistenceof the IR fixed point in the continuum limit is still under investigation.Lattice simulations in the (near-)conformal regime are made more difficult by thelack of experimental guidance. For this reason, all possible sources of uncertainties andambiguities need to be carefully analysed. In this paper, we study the possibility thatnumerical indications of conformality in SU(2) gauge theory with two Dirac flavours in theadjoint representation are in fact due to the presence of a second order transition pointin a system related to this theory. Namely we refer to a SU(2) gauge theory with mixedfundamental–adjoint action [31], which the aforementioned theory with dynamical fermionsreduces to at leading non–trivial order in the hopping parameter expansion. This is apotential effect that has not been considered before in the literature. For its investigation,we will compare the scaling of the spectrum in the gluonic sector found for SU(2) with twoadjoint Dirac fermions with the scaling of the pure gauge spectrum of the mixed actionsystem near its quantum critical point. In doing so, we shed some light on the nature ofthis point, solving some controversies in the earlier literature [32–37].The rest of the paper is organised as follows. After introducing the system we haveinvestigated and elucidating its relationship with SU(2) with two adjoint Dirac fermions(Sect. 2), in Sect. 3 we present our results on the location of the critical point in the barecoupling plane. These are obtained from the study of plaquette differences in the twocoexisting vacua along the first order phase transition line ending in it. We also showresults for the susceptibility of the plaquette. Sect. 4 reports on our data for the spectrum,whose scaling properties are investigated in Sect. 5. Our findings are then summarised inSect. 6. Preliminary results of our investigation have been reported in [38].
In the Wilson discretisation of fermions, the lattice Dirac operator for a single fermionspecies of mass am (in lattice units) transforming in the representation R of the gaugegroup is given by M αβ ( ij ) = ( am + 4 r ) δ ij δ αβ − (cid:104) ( r − γ µ ) αβ U R µ ( i ) δ i,j − ˆ µ + ( r + γ µ ) αβ (cid:0) U R µ ( i ) (cid:1) † δ i,j +ˆ µ (cid:105) , (2.1)where i and j are lattice site indices, α and β are Dirac indices, µ is an Euclidean directionand the γ matrices are formulated in Euclidean space. U R µ ( i ) is the link variable in therepresentation R of the gauge group SU( N c ). The path integral of a theory with N f flavourstransforming in the representation R is then given by Z = (cid:90) ( D U µ ( i )) (det M ( U µ )) N f e − S F , (2.2)where S F is the gauge action, which for simplicity will be taken as the Wilson plaquetteaction: S F = β fund (cid:88) i,µ>ν (cid:18) − N c Re Tr F ( U µν ( i )) (cid:19) . (2.3)– 2 –ere, U µν ( i ) the plaquette in the ( µ, ν )–plane from point i and β fund = 2 N c /g , with g thebare coupling. The sum over all the points i is done over the four–dimensional lattice L × T .Tr F is the trace operator defined in the fundamental representation of the SU( N c ) gaugegroup. For reasons that will be clear below, we call this the fundamental action (wherefundamental refers to the fact that the plaquette is in the fundamental representation).For large bare quark mass, det M can be expanded in powers of the hopping parameter κ = [2( am + 4 r )] − . At the leading non–trivial order, this gives Z = (cid:90) ( D U µ ( i )) e − S eff , (2.4)with S eff = S F + S R (2.5)and (up to irrelevant constants) S R = ˜ β R (cid:88) i,µ>ν (cid:18) − d R Re Tr R ( U µν ( i )) (cid:19) , (2.6)where Tr R is the trace in the representation R, d R the dimension of that representationand ˜ β R = 8 κ d R (cid:0) r − r (cid:1) . (2.7)Eqs. (2.5-2.7) show that at high bare mass the dynamical system is approximated by agauge system with a mixed action, i.e. with an action that, in addition to the fundamentalWilson term, has a coupling to the plaquette in the representation R governed by themass of the fermions (assumed to be large). These variant actions are known to have anon-trivial phase structure in the plane of the couplings (see e.g. [39]).If we specialise our derivation to the adjoint representation, Eq. (2.5) becomes a par-ticular case of the mixed fundamental–adjoint gauge action S = β fund (cid:88) i,µ>ν (cid:18) − N c Re Tr F ( U µν ( i )) (cid:19) + β adj (cid:88) i,µ>ν (cid:18) − N c − A ( U µν ( i )) (cid:19) , (2.8)where Tr A is the trace in the adjoint representation (whose dimension is d A = N c − F by Tr A ( U ) = | Tr F ( U ) | − N c = 2, simulations with the mixed fundamental–adjoint action were alreadycarried out in the early days of lattice gauge theories [31] and more recently in Ref. [32,33, 35, 36]. A mixed–action study of the finite temperature transition for SU(3) wasalso considered in Ref. [40]. These studies suggest the existence of a second order phasetransition point in the bare coupling plane, which for SU(2) is attained at β adj ≈ . β fund ≈ .
22. However, due to growing autocorrelation times, which makes it difficult– 3 –o study the system in a neighbourhood of this critical point, this evidence has not beenconsidered conclusive [37].Due to the relationship between SU(2) with mixed fundamental–adjoint action andSU(2) with two Dirac flavours in the adjoint representation and to hints for near–conformalityin the latter, it is important to analyse carefully the physics of the pure gauge system in itsbare parameter space. In particular we explore the region around ( β fund = 1 . , β adj = 1 . end pointtriple point Z SO(3) β β β fundadj adj (cid:22) = Figure 1 . Sketch of the phase diagram for the lattice system defined by Eq. (2.8). The lines offirst order phase transitions are shown and are explained in Sect. 3.
From previous works on small lattices [31] and from analytical arguments [39], it wasknown that the lattice system described by Eq. (2.8) had an interesting non–trivial phasediagram. A sketch of the phase diagram is presented in Fig. 1. The bare parameter spaceof the couplings ( β fund , β adj ) contains two first order phase transition lines that belong totheories in different limits of the system: the Z gauge theory at β adj = ∞ and the SO(3)gauge theory at β fund = 0. These two lines merge in a triple point at finite values of thecouplings and continue as a single line toward the fundamental Wilson axis β adj = 0. Thissingle line is thought to be a line of bulk first order transition points that ends around β adj ≈ .
25. Due to the possibility of this end–point being of second order, it is necessaryto carefully investigate the nature of the transition on large volume lattices and at highstatistics, which is required by the large autocorrelation times of the system. For our Monte– 4 – .199 1.1995 1.2 1.2005 1.201 1.2015 β fund < P l aq F > L=16 cold startL=16 hot startL=12 cold startL=12 hot start
Figure 2 . Fundamental plaquette expectation values for several β fund couplings at β adj = 1 .
275 ontwo different volumes L = 12 ,
16. The hysteresis cycle, clearly visible on the larger volume, is hardto identify on the smaller one. A similar consistent picture holds for the expectation values of themodulus of Polyakov loops.
Carlo simulations we employ a biased Metropolis algorithm that has been proven to havean heatbath–like efficiency [41, 42]. This algorithm helped us reduce autocorrelations withrespect to the standard multi–hit Metropolis on which earlier results are based .We first checked for the presence of the bulk transition line on hypercubic lattices aslarge as L = T = 40 by simulating at β adj > .
25. We monitored local observables suchas the fundamental and adjoint plaquettes and the Polyakov loops in the four directions,together with their normalised susceptibilities. The bulk transition manifests itself with ajump in the expectation value of the plaquettes as the couplings are varied in the (pseudo-)critical region. Moreover, in the same region a clear hysteresis cycle in the plaquettesappears when Markov chains are started from random (hot) or unit (cold) gauge configu-rations. The presence of metastable states characterised by different values of the plaquetteallows us to follow the bulk transition line and estimate the location of its end–point, wherethe plaquette gap disappears in the thermodynamic limit.The fundamental plaquette is shown in Fig. 2 for several β fund values in the pseudo–critical range at fixed β adj = 1 . β adj approches 1 .
25 from above. However, we notedthat the transition to the asymptotic regime is sharp, with values of relevant observablesapproximately independent of L and T as soon as the small volume regime is exited. On An alternative algorithm with an efficiency similar to the one used in this work has been proposedin [43]. – 5 –he largest volume used for this part of the study, for which L = T = 40, and with morethan 400000 measurements, we find that the plaquette gap between the two vacua is non–zero at β adj = 1 .
25, in contrast with the results at smaller L of Ref. [35]. A careful studyof the plaquette gap, defined as∆ p fund = (cid:10) Plaq F, (cid:11) − (cid:10) Plaq F, (cid:11) , (3.1)where the subscripts 1 and 2 refer to the distinct vacua at couplings centered in the hys-teresis loop, shows a definite trend towards zero. The trend in ∆ p fund is plotted in Fig. 3,where, for each estimate, we used the smallest volume where the first order nature of thetransition was manifest. Numerical values are summarised in Tab. 1. We note that theautocorrelation time dramatically increases as one approaches the critical point (see the τ f and τ a columns of Tab. 2). We were unable to give an estimate of the autocorrelationtime at β adj = 1 .
25, where a large L = 40 lattice was required. Hence, on this latter pointthe systematic error is not fully under control. For this reason the point at β adj = 1 .
25 willnot be considered in our subsequent analyses.By interpolating both ∆ p fund and the equivalent quantity ∆ p adj for the adjoint pla-quette using polynomials, we identify a region 1 . < β adj < .
25 where the plaquette gapvanishes. This is our first estimate for the location of the end–point. This estimate willbe refined in the following using scaling analysis of the susceptibilities and of the spectralobservables. β adj ∆ p f und Figure 3 . ∆ p fund as defined in Eq. (3.1). Also shown as a grey shaded area is the approximateposition of the critical β adj value at which ∆ p fund is expected to vanish: 1 . < β adj < .
25. Aconsistent result is found using the adjoint plaquette (∆ p adj ). The plotted points are reported inTab. 1. In the region below the approximate location of the end–point, we have checked that thetransition becomes a crossover, signalled by the lack of volume scaling in the fundamental– 6 –nd adjoint plaquette susceptibilities χ Pf , χ Pa : the height and the location of peaks ofthe susceptibility (determined with a scan in β fund at fixed β adj ) are consistent across thedifferent volumes. An example is reported in Fig. 4. The location of the peak can befollowed in the ( β fund , β adj ) plane and separates a strong coupling region at small β fund from a region closer to the weak coupling limit ( β fund → ∞ ). We summarise the maximumvalues for χ Pf and χ Pa along this crossover line in Tab. 2. In the same table we alsoreport an estimate of the integrated autocorrelation times for the fundamental and adjointplaquettes.The peak of the plaquette susceptibilities can be used to give a new estimate of theend–point location. In a similar analysis, the conventional way of proceeding is to usereweighting to locate the maximum of susceptibilities. We have attempted this procedure,but reweighting proves to be unviable due to the small overlap of the plaquette distributionsat neighbour β fund for lattices of the required size. This situation is depicted in Fig. 5,where the small overlap is visible for relative variations in β fund that are less than onepart in a thousand. Since carrying out a reweighting programme in the critical regionwill be computationally proibitive, we reverted to an estimate of the maximum involvinga comparison of values at neighbour simulated β fund .Our results show that by approaching β adj ≈ .
25 from below, the maximum of thesusceptibility increases, at fixed L volume. Its scaling form can be fitted by χ (max)Pf = A ( β (crit)adj − β adj ) − γ , (3.2)with A = 0 . β (crit)adj = 1 . γ = 1 . χ / dof = 0 .
67) for the data β adj = 1 . .
22, at L = 20. Our numerical data for χ (max)Pf and their best fit are shown inFig. 6. χ (max)Pf is the integrated plaquette–plaquette correlation function. The compatibilitywithin 1.2 standard deviations of the fitted value γ = 1 . γ = 1 (predictedby the mean–field theory) gives a first hint that the model could be in the universalityclass of the 4d Gaussian model, whose critical properties are described by the mean–fieldapproximation. The fitted β (crit)adj provides another estimate of the end–point location, whichis compatible with the value obtained from ∆ p fund . The stability of the fit is checked bychanging the number of points included and the fitted parameters are summarised in Tab. 3.One can also study the scaling as a function of β fund . Since our calculations weredesigned for the scaling in terms of β adj , our resolution for this analysis is not optimal,because β fund is measured rather than inputed. This affects most the β fund values far fromthe critical point, and in particular β fund = 1 . − .
42. Hence, we use this analysis onlyas a consistency check of general scaling behaviour. In terms of β fund , the scaling form of χ (max)Pf is given by χ (max)Pf = A ( β fund − β (crit)fund ) − γ . (3.3)Our best fit (displayed with the numerical data in Fig. 7) gives A = 0 . β (crit)fund =1 . γ = 1 . χ / dof = 1 .
41) for the data at β fund = 1 . .
37 and L = 20.Tab. 4, which reports the values of the fitted parameters for various choices of the fit range,suggests that the extracted value of the critical coupling and of the critical exponent remain– 7 –table across the different fits. Finally, despite using the same notation, we remark that theexponents γ in Eqs. (3.2) and (3.3) need not to be the same. The fact that the measuredvalues are consistent within errors is a likely indication that both the β adj and the β fund directions have a non–zero projection along the dominant direction of the renormalisationgroup flow at the relevant infrared fixed point. β fund χ P f L=12L=16L=20
Figure 4 . Lack of volume scaling in the fundamental plaquette susceptibility at β adj = 1 .
10. Thepeak location does not move up to volumes L × T = 20 . The properties of the spectrum and the scaling of different masses as the critical surface ofa fixed point is approached give important information on the low–energy dynamics of thetheory. In the following we explain our analysis of the low–lying spectrum in the crossoverregion. Since this is the first study of this kind, we focused on controlling possible sourcesof systematic errors such as autocorrelations and finite–size effects. The aim is to extractthe light glueball spectrum in the thermodynamic limit and to study the scaling propertiesof ratios of masses while approaching the end–point in a controlled manner, e.g. along aspecified trajectory in the bare parameter space.We simulate the SU(2) Yang–Mills theory at six different adjoint couplings β adj =1 . , . , . , . , . , .
20. For each of them we simulate a range of fundamentalcouplings β fund such that both the strong coupling and the weak coupling limits are inves-tigated. The location of the simulated points in the two–dimensional space of the couplingsis plotted in Fig. 8, together with the location of the bulk phase transition and its estimatedend–point (cfr. Sect. 3). We use several lattice volumes ranging from 6 ×
12 to 48 × .605 0.61 0.615 0.62 0.625 0.63Plaq F β fund =1.328 β fund =1.329 β fund =1.330 Figure 5 . Fundamental plaquette distributions at β adj = 1 .
10 for L × T = 20 . The overlap ofdifferent distributions is not enough to allow for a stable multi–histogram reweighting analysis ofthe susceptibility. Note that the distance between the points is much finer than the one in Fig. 4. β adj χ P f Figure 6 . Scaling of the fundamental plaquette susceptibility maximum values for different β adj couplings toward the location of the bulk transition end–point. The estimated β (crit)adj from a fit withEq. (3.2) is highlighted by the grey shaded region on the right. and we noted a drastic increase of them for 1 . ≤ β adj ≤ .
20 (cfr. Tab. 2). Therefore,different measurements are separated by N τ gauge updates to reduce autocorrelations, with– 9 – .25 1.3 1.35 1.4 β fund χ P f Figure 7 . Scaling of the fundamental plaquette susceptibility maximum values for different β fund couplings toward the location of the bulk transition end–point. The estimated β (crit)fund from a fit withEq. (3.3) is highlighted by the grey shaded region on the left. β fund β ad j First order transitionExpected location of the end-point β adj =1.20 β adj =1.18 β adj =1.16 β adj =1.10 β adj =1.05 β adj =1.00Crossover line Figure 8 . Location of the bulk phase transition (grey area delimited by thin dashed black lines)and of the points where we measured the spectrum of the theory (red squares). The estimatedlocation of the bulk transition end–point is indicated by concentric circles. The thick dashed blueline joins the points where χ Pf reaches its maximum and defines a crossover region. – 10 – τ = 250 at β adj = 1 .
16 up to N τ = 800 at β adj = 1 .
20. Large statistics ensembles with N meas ≥ a m t o r Figure 9 . Dependence of the torelon mass am tor on its length L at fixed β adj = 1 . β fund = 1 . a √ σ = 0 . am tor ( L ) at L = 16. Torelon masseslarger than the cutoff are not considered as reliable estimates. First of all, we extracted the string tension a √ σ , which was then used to set the overallscale. This measure of the dynamical mass gap is extracted from long spatial Polyakovloop correlators. The asymptotic large–time behaviour of these correlators is governed bythe lightest torelon state. Assuming that a confining flux tube with massless modes bindsa static quark-antiquark pair, the mass of the lightest torelon am tor can be used to obtainthe string tension according to the ansatz am tor ( L ) = a σL − π L − π L a σ . (4.1)The validity of the above equation is checked a posteriori by comparing the extracted stringtension at various sizes L and by evaluating the size of the subleading finite L correction c = π / (18 L a σ ) with respect to the leading one c = − π/ (3 L ). This procedure isillustrated on a typical set of data in Fig. 9, where the string tension is not fitted, butrather extracted using Eq. (4.1) and the data point am tor ( L ) at L = 16. We observe that,when the loop is too short (i.e. such that am tor ( L ) < . . ≤ am tor ( L ) ≤ .
0, that are correctly– 11 –
10 15 20 25 30L0.20.220.240.260.280.3 a σ / β fund =1.256 β fund =1.257 β fund =1.258 β fund =1.259 Figure 10 . Dependence of the string tension a √ σ on the length of the spatial torelons L at fixed β adj = 1 .
20 and for different fundamental couplings. The filled points identify the results used toestimate the infinite volume limit of a √ σ . β fund a σ / L = 6L = 8L = 10L = 16L = 24L = 32
Figure 11 . String tension measured from several different volumes L = 6-32 at β adj = 1 .
05. Atweaker coupling β fund > .
39 larger volumes are needed to keep finite–volume effects under control.The dashed vertical line indicates the approximate position of the maximum in χ Pf . described by Eq. (4.1). For the numerical value at the highest simulated L in the regimeof validity of Eq. (4.1) (which in this case is the result at L = 16) one gets c /c (cid:39) . a √ σ as a function of L and β fund at fixed β adj ). Our results show that significant finite–size effects are absent when La √ σ >
3, which we satisfied in our simulations using largespatial volumes for the smallest values of a √ σ . Indeed, in the explored range of couplings,the string tension can change by a factor of 5 and, whereas small L ∼ −
10 volumesare sufficient at stronger couplings, larger ones are needed towards weak coupling. Thesituation is shown in Fig. 11 and it is representative of all the simulated β adj points.We can estimate the infinite–volume limit of this observable in the following way: whenthe two largest simulated volumes at each point give consistent results within two standarddeviations, we take the largest volume result as an estimate of the thermodynamic limit(provided am tor is below the cutoff). When the aforementioned criterion is not fulfilled, wedo not give an infinite–volume estimate. However, if a single volume simulation is available,we still report it in plots that show results on various volumes. We use the same approachfor estimating the large volume limit also for the other spectral observables studied in thissection.Another important point that we mentioned in Sect. 3 is the increase in the numberof measurements that are needed closer to β adj ≈ .
25 due to large autocorrelation times.For example, in Fig. 12 we show that almost a tenfold increase in statistics is needed toreduce the systematic error in the identification of the effective mass plateaux for am tor at a m e ff
500 measures; 100 bins1000 measures; 200 bins2000 measures; 400 bins4000 measures; 800 bins
Figure 12 . Effective am tor at β adj = 1 .
20 and β fund = 1 .
256 for L = 10, T = 20. The statisticsis doubled until a plateaux is clearly identified. The dashed green lines indicate the 1- σ contour ofthe fitted effective mass between t = 3 and t = 7. Data points are horizontally shifted for clarity. Our second spectral observable is the mass of the scalar glueball state am ++ . Inorder to measure this mass, the vacuum subtracted correlators of smeared Wilson loops,– 13 – a m e ff L=24 β fund =1.31L=32 β fund =1.31L=32 β fund =1.32L=32 β fund =1.33 Figure 13 . Scalar glueball effective mass at β adj = 1 .
16 and for three values of β fund . A comparisonbetween L = 24 and L = 32 results for β fund = 1 .
31 is also shown. The 1- σ contour of the fittedmasses is plotted on top of the respective fitted points. Data points are horizontally shifted forclarity. a m e ff Full variational basis L=24Glueballs only L=24Bitorelons only L=24Full variational basis L=32Glueballs only L=32Bitorelons only L=32
Figure 14 . Effective mass of the scalar ground state obtained using different variational basis. Twovolumes are compared at β fund = 1 . β adj = 1 .
20. No significant difference is present. Points areshifted for clarity. symmetrised to have 0 ++ quantum numbers, have been inserted in a variational basis for– 14 – .254 1.256 1.258 1.26 1.262 1.264 β fund m i x T ( % ) L = 10L = 16L = 24L = 32
Figure 15 . Relative contribution of the bi–torelon operators to the scalar ground state for severalfundamental couplings at β adj = 1 .
20. On the largest volume, this contribution drops significantly.Values at different volumes are shifted horizontally for clarity. a generalised eigenvalue problem. In addition, in order to identify finite–size artefacts, adifferent type of scalar operators made by symmetrised Polyakov loops winding in oppo-site directions around the spatial torus has been used in the same variational calculation.This second operator set couples mainly to bi–torelon excitations, which are suppressedin the large volume limit and can be used to identify these spurious contaminations ofthe spectrum in the scalar channel. For further technical details, we refer the reader toRef. [44].The scalar glueball mass is reliably estimated thanks to the large variational operatorbasis used in our calculation, which allows us to obtain long and robust effective massplateaux. Fig. 13 provides an example of effective masses for a large 32 lattice at fixed β adj for three β fund values, with a comparison with results from a smaller 24 ×
32 forone value of β fund . For larger β fund values, we used bigger volumes in order to checkexplicitly for finite–size effects. Moreover, the contributions of spurious states has beeninvestigated by looking at the extracted spectrum using the different kind of operatorsdescribed above. Surprisingly, for some values of the couplings we have noticed a large O (50%) contribution of the bi–torelon operators to the ground state; a variational analysiscontaining only Wilson loop operators or, separately, only Polyakov loop operators, turnedout to give the same results for the ground state masses. An example of the effectivemass plot obtained from such different variational operator bases is shown in Fig. 14. Thisconfirms that bi–torelon operators can give sizeable contributions to correlators used toextract the scalar ground state mass. However, by using larger lattices we could confirmthat the contribution of these operators dropped down to less than 10%, as expected. This– 15 –s clearly depicted in Fig. 15, where the relative bi–torelon operators contribution to theground state is shown at β adj = 1 .
20 and for several volumes. In our computation, carehas been taken in choosing the lattice size in such a way that bi–torelon contamination inthe scalar spectrum is negligible. Results for the infinite volume limit of a √ σ and am ++ at various fundamental and adjoint couplings are included in Tab. 5 to 10.The strategy applied for the extraction of the tensor glueball mass am ++ is similarto that used for am ++ . Here, the lattice operators are symmetrised to project only ontothe E irreducible representation of the cubic group which is subduced from the spin 2 ofthe full continuum rotational symmetry. We remark that for some of the β adj couplingswe could not reliably estimate the thermodynamic limit of am ++ due to somewhat largerfinite–size effects. It is known that this channel is more difficult to extract due to theheavier mass of its ground state. Results in the thermodynamic limit are summarised inTab. 11 for this observable. β fund a m e ff L = 10L = 16L = 24L = 32
Figure 16 . Scalar glueball mass at β adj = 1 .
16 for a wide range of fundamental couplings. Contraryto the monotonicity of a √ σ in the same range of β fund , am ++ develops a dip in the crossover region,before raising and decreasing again in the weak coupling limit. The dashed vertical line indicatesthe approximate position of the maximum in χ Pf . Let us move to the description of the features of the extracted spectrum, when the relavantsources of finite–size effects are taken into account. The most interesting feature of thescalar channel spectrum is the presence of a slight dip for a certain region of β fund couplingsaround the crossover region, where χ Pf reaches its maximum value. This dip becomes morepronounced and at its bottom the mass value gets lighter as we increase β adj towards thetransition end–point. At β adj = 1 .
00 the dip is still only a mild plateaux that am ++ – 16 – .34 1.36 1.38 1.4 1.42 β fund a m σ ++ ++ Figure 17 . The measured low–lying spectrum of a √ σ , am ++ and am ++ at β adj = 1 .
05 in theinfinite–volume limit. The dashed vertical line indicates the approximate position of the maximumin χ Pf . reaches before starting decreasing again towards the weak coupling region. However, at β adj = 1 . am ++ drops dramatically, to form the dip shown in Fig. 16. A similar situationhas been found at β adj = 1 .
18 and 1 .
20. It is important to recall that such a behaviour isabsent in both a √ σ and am ++ , which smoothly decrease as functions of β fund . A situationwhere the infinite–volume limit has been estimated is shown in Fig. 17.In a neighbourhood of a second order phase transition point the light lattice degreesof freedom that become massless at the critical point define an effective long–distancecontinuum theory. As the critical point is approached, their mass goes to zero accordingto some scaling exponents that characterise the dynamics at large distances. For oursystem, only am ++ seems to become light at the end–point of the first order line. Inorder to investigate its approach to the end–point, we fit the measured am ++ using theparameterisation am ++ = A ( β (crit)adj − β adj ) P , (5.1)which is inspired by the scaling of the correlation length ξ near a critical point: ξ = ξ | T /T c − | − ν . The critical exponent ν is 0 . A =1 . β (crit)adj = 1 . P = 0 . χ / dof = 1 .
07. We also fitted thedata keeping a fixed P = 0 . A = 1 . β (crit)adj = 1 . χ / dof = 1 .
97. Both fits are compared to the data inFig. 18. Taken at face value, our results suggest that the mean–field scaling is not ruledout. Indeed if we exclude the point at β adj = 1 .
00, the value of P gets closer to the mean–– 17 –eld prediction (Tab. 12). As shown in Tab. 13, a similar analysis of the scaling of am ++ in terms of β fund gives compatible results. To resolve the issue of whether the system isdescribed by the mean–field theory, one would need to go closer to the end–point, which iscurrently computationally proibitive for the resources at our disposal. The m ++ / √ σ ratiois shown in Fig. 19. β adj a m ++ fit: P=0.42(3)fit: P=0.5 (fixed) Figure 18 . am ++ for different values of β adj and on the trajectory defined by the maxima of χ Pf .The fitting function from Eq. (5.1) is used to represent the data. The shaded grey area comprisesthe values of the critical point coming from the two different fits in the plot. Motivated by the need to better understand possible roles of lattice artefacts in investi-gations of gauge theories in the (near-)conformal regime, we have studied a SU(2) puregauge theory with a modified lattice action with couplings to both the fundamental andthe adjoint plaquettes. This theory, which is related to SU(2) gauge theory with two Diracflavours in the adjoint representation in the limit of large bare fermion mass, is known tohave a bulk phase transition with an end–point relatively close to the fundamental couplingaxis. The controversial nature of this end–point is resolved and our estimates for its loca-tion are summarised in Tab. 14. Using our improved gluon spectroscopy techniques [44],we measured the string tension, the scalar glueball mass and the tensor glueball mass. Westudied their scaling properties when the end–point is approached along a controlled trajec-tory that follows the peaks of the fundamental plaquette susceptibility. To our knowledge,this is the first systematic study of the gluonic spectrum in this model. For this reason, wecarefully checked that we are reasonably free from finite–size effects and (mostly thanks tothe simulation algorithm used in this work) the autocorrelation time of our observables waskept under control. The spectrum extrapolated to infinite volume shows a non–constant– 18 – β adj m ++ / σ / Figure 19 . The ratio between am ++ and a √ σ for different values of β adj and on the trajectorydefined by the maxima of χ Pf . The ratio decreases below one when the bulk transition end–pointis approached (∆ p fund = 0). The shaded grey area indicates the estimated location of the criticalpoint obtained by the scaling analysis (Eq. (5.1)). m ++ / √ σ ratio when approaching the end–point in a controlled manner (see Fig. 19). Inparticular, since the 0 ++ state is the only light degree of freedom near the end–point andthe scaling is marginally compatible with being described by the critical exponents of the4d Gaussian model, it is conceivable that the corresponding effective theory be a scalartheory described by the mean-field approximation. This behaviour is in contrast with theinfrared dynamics of SU(2) gauge theory with two adjoint Dirac fermions, where such aratio is driven to a constant by a conformal fixed point and is consistent with the contin-uum SU(2) Yang–Mills value m ++ / √ σ ∼ .
7. Therefore, we can reasonably conclude thatthe observed spectral signals of near-conformality in SU(2) gauge theory with two adjointDirac fermions are not affected by the second order phase transition point of the relatedgauge system with mixed fundamental-adjoint action. It would be instructive to performa similar analysis for gauge theories with fermions in the symmetric or antisymmetric rep-resentation and N c ≥
4, for which the stability of fluxes of higher N -lity in pure gauge (seee.g. [45]) could create a more complicated phase structure in the effective theory at largemass. Acknowledgments
We would like to thank Philippe de Forcrand, Claudio Bonati and Guido Cossu for fruitfulcomments and discussions. We are indebted with Urs Heller for guidance on the algorithmused in our simulations. B.L. acknowledges financial support from the Royal Society (grantUF09003) and STFC (grant ST/G000506/1) and the hospitality of the Aspen Center forPhysics during the final stage of this work, which allowed him to discuss various aspects– 19 –f the project with the participants to the workshop
Lattice Gauge Theory in the LHCEra , and in particular with R. Brower, A. Hasenfratz, Y. Meurice and T. Tomboulis. E.R.was funded by a SUPA Prize Studentship and a FY2012 JSPS Postdoctoral Fellowshipfor Foreign Researchers (short-term). The simulations discussed in this work have beenperformed on a cluster partially funded by STFC and by the Royal Society, on systems madeavailable to us by HPC Wales and on the HPCC Plymouth cluster facilities at PlymouthUniversity.
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B501 (2001) 128–133, [ hep-lat/0012025 ]. – 22 – β adj β fund ∆ p fund ∗ ∗ Table 1 . The estimated location of the hysteresis centre at different β adj values. Each value ∆ p fund is measured on a volume L , which is the minimum one needed to discern the two metastable statesof the first order transition. The errors on ∆ p fund are estimated by comparing its value on all thesimulated points in the hysteresis region. However, the starred point comes from a single simulation,and the error comes from the difficulty of estimating the expectation values (cid:10) Plaq F, (cid:11) and (cid:10) Plaq F, (cid:11) in the presence of long autocorrelation times. β adj β fund χ (max)Pf τ f χ (max)Pa τ a ∗ Table 2 . The maximum of the fundamental and adjoint plaquette susceptibilities χ (max)Pf , χ (max)Pa and the integrated autocorrelation time of the fundamental and adjoint plaquettes τ f , τ a for differentadjoint couplings β adj and at fixed volume L = 20. The statistical errors come from a jackknifeprocedure with bins of 3 τ measures, while the error on β fund is estimated by the distance betweenneighbouring simulated points. The two lines at β adj = 1 .
00 are due to the fact that both β fund =1 .
40 and β fund = 1 .
41 give compatible values for the measured maximum of the susceptibility, whichindicates that the real maximum falls in between the two simulated β fund . The starred point at β adj = 1 .
00 was used for the calculation of the spectrum at β adj = 1 .
00, while the other point atthe same value of β adj was used for the susceptibility analyses. The fact that at fixed β adj different(close) values of β fund are suitable for a scaling analysis of different observables is due to the mildnessof the crossover at the boundary of the critical region. – 23 –ange β adj β (crit)adj A γ χ / dof1.00-1.20 1.2453(55) 0.078(5) 1.05(6) 0.661.00-1.22 1.2459(28) 0.077(3) 1.06(3) 0.501.05-1.20 1.2451(87) 0.078(10) 1.05(10) 1.00 ∗ Table 3 . Fit results for χ (max)Pf according to the formula χ (max)Pf = A ( β (crit)adj − β adj ) − γ . The starredvalue uses the starred point in Tab. 2. Boldfaced values are used in the text. range β fund β (crit)fund A γ χ / dof1.241-1.40 1.2206(34) 0.053(5) 1.09(6) 1.69 ∗ Table 4 . Fit results for χ (max)Pf according to the formula χ (max)Pf = A ( β fund − β (crit)fund ) − γ . Thestarred value uses the starred point in Tab. 2. Boldfaced values are used in the text. β adj = 1 . β fund L am ++ t i - t f L a √ σ t i - t f Table 5 . Values for am ++ and a √ σ on the lattices used as an estimate of the infinite volume limitfor different β fund at β adj = 1 .
00. The time interval t i - t f indicates the fitting window used for thereported values. – 24 – adj = 1 . β fund L am ++ t i - t f L a √ σ t i - t f ∗ Table 6 . Same as Tab. 5 for β adj = 1 .
05. The stars ∗ indicate quantities for which only a singlevolume simulation is available. β adj = 1 . β fund L am ++ t i - t f L a √ σ t i - t f ∗ ∗ ∗ Table 7 . Same as Tab. 5 for β adj = 1 .
10. The stars ∗ indicate quantities for which only a singlevolume simulation is available. β adj = 1 . β fund L am ++ t i - t f L a √ σ t i - t f ∗ Table 8 . Same as Tab. 5 for β adj = 1 .
16. The stars ∗ indicate quantities for which only a singlevolume simulation is available. – 25 – adj = 1 . β fund L am ++ t i - t f L a √ σ t i - t f ∗ ∗ ∗ ∗ Table 9 . Same as Tab. 5 for β adj = 1 .
18. The stars ∗ indicate quantities for which only a singlevolume simulation is available. β adj = 1 . β fund L am ++ t i - t f L a √ σ t i - t f Table 10 . Same as Tab. 5 for β adj = 1 . – 26 – adj = 1 . β adj = 1 . β fund L am ++ t i - t f β fund L am ++ t i - t f ∗ ∗ Table 11 . Values for am ++ on the lattices used as an estimate of the infinite volume limitfor different β fund at β adj = 1 .
00 and β adj = 1 .
05. The time interval t i - t f indicates the fittingwindow used for the reported values. The stars ∗ indicate quantities for which only a single volumesimulation is available. range β adj β (crit)adj A P χ / dof1.00-1.20 1.2308(59) 1.19(5) 0.42(3) 1.071.05-1.20 1.2330(96) 1.23(13) 0.44(7) 1.481.00-1.20 1.2455(26) 1.31(2) 0.5 1.971.05-1.20 1.2420(28) 1.36(3) 0.5 1.37 Table 12 . Fit results for am ++ according to the formula am ++ = A ( β (crit)adj − β adj ) P . In thelast two lines, P is kept fixed to 0.5. range β fund β (crit)fund A P χ / dof1.256-1.41 1.2346(47) 1.30(8) 0.40(4) 1.641.256-1.37 1.2326(77) 1.37(20) 0.43(7) 2.231.256-1.41 1.2210(25) 1.50(3) 0.5 3.141.256-1.37 1.2246(26) 1.57(5) 0.5 2.10 Table 13 . Fit results for for am ++ according to the formula am ++ = A ( β fund − β (crit)fund ) P . Inthe last two lines, P is kept fixed to 0.5. method β (crit)adj β (crit)fund latent heat (Eq. (3.1)) 1.22 - 1.25 – χ (max)Pf scaling (Eq. (3.2)) 1.2460(38) 1.2229(31) am ++ scaling (Eq. (5.1)) 1.2308(59) 1.2346(47)with fixed exponent ( P = 0 .
5) 1.2455(26) 1.2210(25)
Table 14 . Summary of the different estimates of the critical ( β fund , β adj ) values described in thetext.) values described in thetext.