Mass anomalous dimension in SU(2) with two adjoint fermions
Francis Bursa, Luigi Del Debbio, Liam Keegan, Claudio Pica, Thomas Pickup
aa r X i v : . [ h e p - ph ] D ec DAMTP-2009-65
Mass anomalous dimension in SU(2) with two adjointfermions.
Francis Bursa
Jesus College, Cambridge, CB5 8BL, United Kingdom
Luigi Del Debbio, Liam Keegan, Claudio Pica
SUPA, School of Physics and Astronomy,University of EdinburghEdinburgh EH9 3JZ, United Kingdom
Thomas Pickup
Rudolf Peierls Centre for Theoretical Physics,University of Oxford, Oxford OX1 3NP, United Kingdom
Abstract
We study SU (2) lattice gauge theory with two flavours of Dirac fermions in theadjoint representation. We measure the running of the coupling in the Schr¨odingerFunctional (SF) scheme and find it is consistent with existing results. We discusshow systematic errors affect the evidence for an infrared fixed point (IRFP). Wepresent the first measurement of the running of the mass in the SF scheme. Theanomalous dimension of the chiral condensate, which is relevant for phenomenolog-ical applications, can be easily extracted from the running of the mass, under theassumption that the theory has an IRFP. At the current level of accuracy, we canestimate 0 . < γ < .
56 at the IRFP.
Introduction
Experiments at the LHC are about to probe nature at the TeV scale, where new physicsbeyond the Standard Model (BSM) is expected to be found. The existence of a newstrongly–interacting sector that is responsible for electroweak symmetry breaking is aninteresting possibility. Technicolor was originally proposed thirty years ago, and strongly–interacting BSM has been revisited in many instances since then. Recent reviews can befound in Refs. [1, 2].In order to be phenomenologically viable, technicolor theories need to obey the con-straints from precision measurements at LEP [3, 4]. Moreover the symmetry breakingneeds to be communicated to the Standard Model, so that the usual low–energy physicsis recovered. This is usually achieved in the so-called Extended Technicolor (ETC) mod-els by invoking some further interaction at higher energies that couples the technicolorsector to the Standard Model. At the TeV scale the remnants of this coupling are higher–dimensional operators in the effective Hamiltonian, which are suppressed by powers of thehigh energy scale, M , that characterises the extended model. Amongst these operatorsare a mass term for the Standard Model quarks, and four–fermion interactions that wouldcontribute to flavour–changing neutral currents (FCNC). Thus there is a tension on thepossible values of M : on the one hand M needs to be large so that FCNC interactionsare suppressed, on the other hand M needs to be small enough to generate the heavierquark masses. In particular, the effective operator for the Standard Model quark massesis: L m = 1 M h Φ i ¯ ψψ , (1.1)where ψ indicates the quark field, and Φ is the field in the technicolor theory which is re-sponsible for electroweak symmetry breaking. In the traditional technicolor models, whichare realised as SU( N ) gauge theories, Φ = ¯ΨΨ is the chiral condensate of techniquarks.Let us emphasise that quark masses are defined in a given renormalisation scheme and ata given scale. For instance the data reported in the Particle Data Group summaries [5]usually refer to the quark mass in the MS scheme at 2 GeV. The coefficient that appearsin Eq. (1.1) is the chiral condensate at the scale M : h ¯ΨΨ i (cid:12)(cid:12) M = h ¯ΨΨ i (cid:12)(cid:12) Λ exp (cid:20)Z M Λ dµµ γ ( µ ) (cid:21) , (1.2)where γ is the anomalous dimension of the scalar density, and Λ is the typical scale ofthe technicolor theory, Λ ≈ h ¯ΨΨ i ∼ Λ , and therefore the naive expectation for the quark masses is m ∼ Λ /M .Eq. (1.2) suggests a possible way to resolve the tension due to the large quark masses.If the technicolor theory is such that γ is approximately constant (and large) over a suffi-ciently long range in energies, then the running above will generate a power enhancement1f the condensate. This scenario has been known for a long time under the name of walkingtechnicolor [6, 7, 8]. Gauge theories with a large number of fermions have been traditionalcandidates for walking theories; the fermions slow down the running of the coupling andcan potentially lead to the required power–enhancement. More recent incarnations havebeen proposed that are constructed as SU(N) gauge theories with fermions in higher–dimensional representations of the colour group [9, 10, 11]. These theories could have agenuine IR fixed point (IRFP), or simply lie in its vicinity. The existence of an IRFPis a difficult problem to address since it requires to perform quantitative computationsin a strongly–interacting theory. Lattice simulations can provide first–principle resultsthat can help in determining the phenomenological viability of these models; numericalsimulations of models of dynamical electroweak symmetry breaking have attracted grow-ing attention in recent years [12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26,27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. A number of theories have beenstudied: SU(3) with 8, 10, 12 flavours of fermions in the fundamental representation,SU(3) with fermions in the sextet representation, and SU(2) with fermions in the adjointrepresentation. These studies have focused either on the spectrum of the theories, or onthe running of the coupling computed in the Schr¨odinger functional (SF) scheme, findingsome tantalising numerical evidence for IR behaviours different from what is known fromQCD.Existing simulations of the Schr¨odinger functional have identified a possible fixed pointin many of the above–mentioned theories by noticing a flat behaviour of the runningcoupling in this scheme over a given range of energy scales.In this work we consider the SU(2) theory with two flavours of adjoint fermions, andcompute the running coupling in the SF scheme. We confirm the results obtained inRef. [32], and present a more refined analysis of the lattice data. We focus on the runningof the mass in the SF scheme, from which we can extract the mass anomalous dimensionthat appears in Eq. (1.2). Current simulations are still plagued by systematic errors,which we examine in detail both for the coupling and the mass. These errors are thelargest limitation to drawing strong conclusions from the lattice data. These limitationsare common to all the studies performed so far, more extensive work is required in orderto reach robust conclusions. Our results for the anomalous dimension of the mass providecrucial input for these studies that aim at exact results for non-supersymmetric gaugetheories in the non-perturbative regime. 2 SF formulation
We define the running coupling g non-perturbatively using the Schr¨odinger Functionalscheme [41, 42]. This is defined on a hypercubic lattice of size L , with boundary conditionschosen to impose a background chromoelectric field on the system. The renormalisedcoupling is defined as a measure of the response of the system to a small change in thebackground chromoelectric field. Specifically, the spatial link matrices at t = 0 and t = L are set respectively to: U ( x, k ) | t =0 = exp [ ητ a/iL ] , (2.1) U ( x, k ) | t = L = exp [( π − η ) τ a/iL ] , (2.2)with η = π/ P + ψ = 0 , ψP − = 0 at t = 0 , (2.3) P − ψ = 0 , ψP + = 0 at t = L , (2.4)where the projectors are defined as P ± = (1 ± γ ) /
2. The fermion fields also satisfyperiodic spatial boundary conditions [44]. As we mentioned above, one can readily verifyin perturbation theory that these boundary conditions impose a constant chromoelectricfield.We use the Wilson plaquette gauge action, and Wilson fermions in the adjoint repre-sentation, as implemented in Ref. [17]. Note that we have not improved the action, andtherefore our results are going to be affected by O ( a ) lattice artefacts. The same approachhas been used so far for the preliminary studies of this theory in Ref. [32].The coupling constant is defined as g = k (cid:28) ∂S∂η (cid:29) − (2.5)with k = − L /a sin( a /L ( π − η )) chosen such that g = g to leading order in per-turbation theory. This gives a non–perturbative definition of the coupling which dependson only one scale, the size of the system L .To measure the running of the quark mass, we calculate the pseudoscalar densityrenormalisation constant Z P . Following Ref. [45], Z P is defined by: Z P ( L ) = p f /f P ( L/ , (2.6)3here f and f P are the correlation functions involving the boundary fermion fields ζ and ζ : f = − / L Z d u d v d y d z h ζ ′ ( u ) γ τ a ζ ′ ( v ) ζ ( y ) γ τ a ζ ( z ) i , (2.7) f P ( x ) = − / Z d y d z h ψ ( x ) γ τ a ψ ( x ) ζ ( y ) γ τ a ζ ( z ) i . (2.8)These correlators are calculated on lattices of size L , with the spatial link matrices at t = 0 and t = L set to unity.The Schr¨odinger Functional boundary conditions remove the zero modes that arenormally an obstacle to simulating at zero quark mass [46]. This means we can rundirectly at κ c . We determine κ c through the PCAC mass in units of the inverse latticespacing am ( L/ am ( x ) = ( ∂ + ∂ ∗ ) f A ( x )2 f P ( x ) (2.9)and f A ( x ) = − / Z d yd z h ψ ( x ) γ γ τ a ψ ( x ) ζ ( y ) γ τ a ζ ( z ) i . (2.10)Here the lattice derivatives ∂ and ∂ ∗ are defined by ∂ f ( x ) = f ( x + 1) − f ( x ) and ∂ ∗ f ( x ) = f ( x ) − f ( x − L , withthe spatial link matrices at t = 0 and t = L set to unity.We define κ c by the point where am vanishes. We measure am for 5 values of κ in theregion − . < am < . κ to find an estimate of κ c . Theerror on κ c is estimated by the bootstrap method.In practice we achieve | am | . . am ∼ .
02, for which we found no shift in g or Z P within thestatistical uncertainty of the measured values, so the effect of our quark mass can safelybe neglected. We have performed two sets of simulations in order to determine the running couplingand Z P . The parameters of the runs are summarised respectively in Tab. 1, and 2. Thevalues of κ c are obtained from the PCAC relation as described above.4 L =6 L =8 L =12 L =162.00 0.190834 - - -2.10 0.186174 - - -2.20 0.182120 0.181447 0.1805 -2.25 0.180514 0.179679 - -2.30 0.178805 0.178045 - -2.40 0.175480 0.174887 - -2.50 0.172830 0.172305 0.17172 0.171722.60 0.170162 0.169756 - -2.70 0.167706 - - - β L =6 L =8 L =12 L =162.80 0.165932 0.165550 0.16505 -3.00 0.162320 0.162020 0.161636 0.1616363.25 0.158505 - 0.1580 -3.50 0.155571 0.155361 0.155132 0.1551323.75 0.152803 - - -4.00 0.150822 0.150655 - -4.50 0.147250 0.14720 0.14712 0.147128.00 0.136500 0.13645 0.136415 - Table 1: Values of β , L , κ used for the determination of g . The entries in the table arethe values of κ c used for each combination of β and L .Note that Z P is determined from a different set of runs at similar values of β , L , κ . β L =6 L =8 L =12 L =162.00 0.190834 - - -2.05 0.188504 - 0.18625 -2.10 0.186174 - - -2.20 0.182120 0.181447 0.1805 -2.25 0.180514 0.179679 - -2.30 0.178805 0.178045 - -2.40 - 0.174887 - -2.50 0.172830 0.172305 0.17172 0.171722.60 0.170162 0.169756 - -2.70 0.167706 - - - β L =6 L =8 L =12 L =162.80 0.165932 0.165550 0.16505 -3.00 0.162320 0.162020 0.161636 0.1616363.25 0.158505 - 0.1580 -3.50 0.155571 0.155361 0.155132 0.1551323.75 0.152803 - - -4.00 0.150822 0.150655 0.15051 -4.50 0.14725 0.14720 0.14712 0.147128.00 0.13650 0.13645 0.136415 0.13641516.0 0.1302 0.1302 0.1302 0.130375 Table 2: Values of β , L , κ used for the determination of Z P . The entries in the table arethe values of κ c used for each combination of β and L . Recent studies have focused on the running of the SF gauge coupling, and have highlighteda slow running in the lattice data for this quantity [13, 15, 31, 32]. This is clearly differentfrom the behaviour observed in QCD–like theories [43, 47]. These results are certainlyencouraging, but have to be interpreted with care. Lattice data can single out at best arange of energies over which no running is observed. However it is not possible to concludefrom lattice data only that the plateau in the running coupling does extend to arbitrarilylarge distances, as one would expect in the presence of a genuine IRFP. On the otherhand, if the plateau has a finite extent, i.e. if the theory seems to walk only over a finiterange of energies, then the behaviour of the running coupling in the absence of a genuine5xed point depends on the choice of the scheme, and therefore the conclusions becomeless compelling.Let us discuss the scheme dependence of the running coupling in more detail. Thequantities we are interested in are the beta function and the mass anomalous dimension: µ ddµ g ( µ ) = β ( g ) , (3.1) µ ddµ m ( µ ) = − γ ( g ) m ( µ ) , (3.2)where g, m are the running coupling and mass in a given (mass-independent) renormali-sation scheme. Note that γ in Eq. (3.2) is the anomalous dimension of the scalar density,which appears also in Eq. (1.2); γ differs from the usual mass anomalous dimension byan overall sign. Both β and γ can be computed in perturbation theory for small valuesof the coupling constant: β ( g ) = − g (cid:2) β + β g + β g + O ( g ) (cid:3) , (3.3) γ ( g ) = g (cid:2) d + d g + O ( g ) (cid:3) . (3.4)The coefficient β , β , d are scheme–independent; expressions for β , β for fermions inarbitrary representations of the gauge group have been given in Ref. [14], while for thefirst coefficient of the anomalous dimension, we have: d = 6 C ( R )(4 π ) , (3.5)where C ( R ) is the quadratic Casimir of the fermions’ colour representation. In the specificcase we are studying in this work d = 3 / (4 π ).Different schemes are related by finite renormalisations; the running of the couplings ingoing from one scheme to the other is readily obtained by computing the scale dependencewith the aid of the chain rule. Let us consider a change of scheme:¯ g ′ = φ (¯ g, m/µ ) , (3.6)¯ m ′ = ¯ m F (¯ g, ¯ m/µ ) . (3.7)We impose two conditions on φ : it must be invertible, and should reduce to φ (¯ g ) =¯ g + O (¯ g ) for small values of ¯ g . Eq. (3.7) encodes the fact that a massless theory remainsmassless in any scheme. The picture simplifies considerably if one considers only mass–independent renormalisation schemes; the functions φ and F only depend on the coupling¯ g , and one finds: β ′ (¯ g ′ ) = β (¯ g ) ∂∂ ¯ g φ (¯ g ) (3.8) γ ′ (¯ g ′ ) = γ (¯ g ) + β (¯ g ) ∂∂ ¯ g log F (¯ g ) . (3.9)6he scheme–independence of the coefficients β , β , d can be obtained by expanding thefunctions that describe the mapping between the two schemes, φ and F , in powers of g . Eqs. (3.8), (3.9) summarise the main features that we want to highlight here. Theconditions we imposed on φ imply that ∂∂ ¯ g φ (¯ g ) > i.e. asymptotic freedom cannotbe undone by a change of scheme. The existence of a fixed point is clearly scheme–independent: if β (¯ g ∗ ) = 0 for some value ¯ g ∗ of the coupling, then β ′ has also a zero. Notethat the value of the critical coupling changes from one scheme to the other, ¯ g ′∗ = φ (¯ g ∗ ),however the existence of the fixed point is invariant. Similarly, the anomalous dimensionis scheme–independent at a fixed point, since the second term in Eq. (3.9) vanishes there.Moreover, if the change of scheme only involves a redefinition of the coupling, but leavesthe mass unchanged, then the anomalous dimension does not vary.Unfortunately none of these conclusions holds in the absence of a fixed point. Inparticular, a flat behaviour of the running coupling over a finite range of energies can beobtained in any theory by a suitably–chosen change of scheme.It is worth stressing here another important point concerning the numerical studiesof running couplings. There are instances where the beta function of an asymptoticallyfree theory remains numerically small. This is the case of the theory considered in thiswork, namely SU(2) with 2 flavours of adjoint Dirac fermions, in the perturbative regime.In this case the running of the coupling is very slow from the very beginning, and thisis independent of the possible existence of an IRFP at larger values of the coupling. Asa consequence high numerical accuracy is needed in order to resolve a “slow” running;therefore numerical studies of potential IRFP need high statistics, and a robust controlof systematics. In particular it is important to extrapolate the step-scaling functionscomputed on the lattice to the continuum limit, in order to eliminate lattice artefactswhich could bias the analysis of the dependence of the running coupling on the scale.This is particularly relevant for the studies of potential IRFP, since lattice artefacts couldmore easily obscure the small running that we are trying to resolve. Some of thesedifficulties were already noted in Ref. [32]; current results, including the ones presentedin this work, are affected by these systematics.More extensive simulations are therefore needed in order to remove the lattice artefactsby performing a controlled extrapolation of the lattice step scaling functions defined belowin Sect. 4. The scale L at which the coupling is computed and the lattice spacing a mustbe well separated. This last step is a crucial ingredient in the SF scheme, since it decouplesthe details of the lattice discretization from the running of the couplings at the scale L that we want to determine. Asymptotically free theories are effectively described by aperturbative expansion at small distances. In this regime, the degrees of freedom are theelementary fermions and the gauge bosons, renormalized couplings can be computed inperturbation theory, and different schemes can be related by perturbative calculations.7he evolution of the running coupling can be followed starting from this high–energyregime and moving towards larger distances. If the theory has an IRFP, the value ofthe running coupling approaches some finite limit ¯ g ∗ as L is increased, i.e. the runningcoupling must lie in the interval [0 , ¯ g ∗ ]. Its running can be traced from the UV regime upto the limiting value, which is approached from below. Larger values of ¯ g can be obtainedin a lattice simulation; however the interpretation of these points is less transparent. Onepossibility is that the lattice theory in some region of bare parameter space lies in the basinof attraction of some non-trivial UV fixed point where a different continuum theory canbe defined. The running coupling would then approach the IRFP value from above. Thenon-trivial UV fixed point is clearly difficult to identify, thereby making the extrapolationto the continuum limit rather tricky in this case.A more pragmatic approach could be to ignore the issue of the existence of a non-trivial UV fixed point, and simply explore the limit L/a ≫
1, assuming that the startingpoint is the lattice theory with a cutoff, and that we are only interested in the regimewhere distances are large compared to the cutoff. This interpretation is prone to sys-tematic errors due to potential O (Λ a ) term, where Λ is some physical mass scale in thetheory. These terms are not necessarily small, even if the limit a/L → g > ¯ g ∗ could be affected by non-universallattice artefacts.Studies of the running couplings in the SF scheme are a useful tool to expose thepossible existence of theories that show a conformal behaviour at large distances. However,the results of numerical simulations have to be interpreted with care; they are unlikelyto provide conclusive evidence about the existence of a fixed point by themselves, butthey can be used to check the consistency of scenarios where the long–range dynamicsis dictated by an IRFP. A more convincing picture can emerge when these analyses arecombined with spectral studies [17, 18, 30, 40], or MCRG methods [37]. We have measured the coupling g ( β, L ) for a range of β, L . Our results are reported inTab. 3, and plotted in Fig. 1: it is clear that the coupling is very similar for different L/a at a given value of β , and hence that it runs slowly.In Fig. 2 we compare our results to those obtained in Ref. [32]. Our results are directlycomparable since we use the same action and definition of the running coupling, and itis reassuring to see that they agree within statistical errors. The numbers reported inthe figure have been obtained using completely independent codes; they constitute animportant sanity check at these early stages of simulating theories beyond QCD.8he running of the coupling is encoded in the step scaling function σ ( u, s ) asΣ( u, s, a/L ) = g ( g , sL/a ) (cid:12)(cid:12) g ( g ,L/a )= u , (4.1) σ ( u, s ) = lim a/L → Σ( u, s, a/L ) , (4.2)as described in Ref. [42]. The function σ ( u, s ) is the continuum extrapolation of Σ( u, s, a/L )which is calculated at various a/L, according to the following procedure. Actual simula-tions have been performed at the values of β and L reported in Tab. 1. β g L/a=16L/a=12L/a=10 2/3L/a=9L/a=8L/a=6
Figure 1: Data for the running coupling as computed from lattice simulations of theSchr¨odinger functional. Numerical simulations are performed at several values of thebare coupling β , and for several lattice resolutions L/a . The points at
L/a = 9 , areinterpolated. 9 L =6 L =8 L =12 L =162.00 4.237(58) - - -2.10 3.682(39) - - -2.20 3.262(31) 3.457(59) - -2.25 3.125(19) 3.394(54) - -2.30 3.000(25) 3.090(46) - -2.40 2.813(21) 2.887(44) - -2.50 2.590(20) 2.682(35) 2.751(68) 3.201(324)2.60 2.428(16) 2.460(29) - -2.70 2.268(14) - - - β L =6 L =8 L =12 L =162.80 2.141(12) 2.218(22) 2.309(40) -3.00 1.922(10) 1.975(25) 1.958(32) 2.025(157)3.25 1.694(5) - 1.830(90) -3.50 1.522(4) 1.585(11) 1.626(30) 1.603(76)3.75 1.397(3) - - -4.00 1.275(3) 1.320(7) - -4.50 1.101(3) 1.128(5) 1.152(10) 1.106(64)8.00 0.558(1) 0.567(2) 0.574(3) - Table 3: Measured values of g on different volumes as a function of the bare coupling β . g ( β , L / a ) Figure 2: The results of our numerical simulations are compared to recent results obtainedin Ref. [32]. Different symbols correspond to different values of the lattice bare coupling β , corresponding respectively to β = 2 . , . , . , . , .
0. Empty symbols correspond tothe data obtained in this work. Full symbols correspond to the data in Ref. [32]. Symbolshave been shifted horizontally for easier reading of the plot.Starting from the actual data, we interpolate quadratically in a/L to find values of g ( β, L ) at L = 9 , , so that we obtain data for four steps of size s = 4 / L → sL : L = 6 , , , sL = 8 , , ,
16. Then for each L we perform an interpolation in β using10he same functional form as Ref. [31]:1 g ( β, L/a ) = β N " n X i =0 c i (cid:18) Nβ (cid:19) i (4.3)We choose to truncate the series with the number of parameters that minimises the χ per degree of freedom.All the subsequent analysis is based on these interpolating functions, and does notmake further use of the original data. Using the fitted function in Eq. (4.3), we computeΣ( u, / , a/L ) at a number of points in the range u ∈ [0 . , . a/L using these points to give a single estimate of σ ( u ) ≡ σ ( u, / u are shown in Fig. 3. The L = 6 data werefound to have large O ( a ) artifacts, and are not used in the continuum extrapolation. The L = 16 data have a large statistical error, which limits their current impact on the con-tinuum extrapolation. The sources of systematic uncertainty in our final results for σ ( u )are due to the interpolation in L and β and to the extrapolation to the continuum limit.Full details of the statistical and systematic error analysis are provided in Appendix A.The resulting values for σ ( u ) with statistical errors only can be seen as the blackcircles in Fig. 4. The red error bars in Fig. 4 also include systematic errors, but usingonly a constant continuum extrapolation. This is equivalent to the assumption that latticeartefacts are negligible in our data. A similar assumption has been used in Ref. [32], wherethe data at finite a/L were used directly to constrain the parameters that appear in the β function of the theory. The study of the lattice step scaling function, and its continuumextrapolation, that we employ for this work, will ultimately allow us to obtain a fullcontrol over the systematic errors.The step scaling function encodes the same information as the β function. The relationbetween the two functions for a generic rescaling of lengths by a factor s is given by: − s = Z σ ( u,s ) u dx √ xβ ( √ x ) . (4.4)The step scaling function can be computed at a given order in perturbation theory byusing the analytic expression for the perturbative β function, and solving Eq. (4.4) for σ ( u, s ). On the other hand, it can be seen directly from the definition of σ ( u, s ) in Eq. (4.2)that an IRFP corresponds to σ ( u, s ) = u .Our current values for the step scaling function are consistent with a fixed point in theregion g ∼ . − .
2, as reported in Ref. [32]. Further simulation at higher g is limitedby the bulk transition observed in Ref. [18, 30] at β ≃ . σ ( u ), which is dominated by systematic errors.11 .00 0.05 0.10 0.15 0.20a/L1.01.52.02.53.03.5 Σ ( / , u , a / L ) u=1.0u=2.0u=3.0 Figure 3: Results for the lattice step–scaling function Σ(4 / , u, a/L ). The dashed linesrepresent the initial value of u . The point at x = 0 yields the value of σ ( u ), i.e. theextrapolation of Σ to the continuum limit. The error bar shows the difference betweenconstant and linear extrapolation functions, and gives an estimate of the systematic errorin the extrapolation as discussed in the text.12 .0 1.0 2.0 3.0u1.0001.0101.0201.030 σ ( u ) / u Statistical
Figure 4: The relative step–scaling function σ ( u ) /u obtained after extrapolating the latticedata to the continuum limit. The black circles have a statistical error only. The red errorbars also include systematic errors, but using only a constant continuum extrapolation(i.e. ignoring lattice artifacts). Note that a fixed point is identified by the condition σ ( u ) /u = 1. 13 .0 1.0 2.0 3.0u0.981.001.021.041.061.081.10 σ ( u ) / u Statistical
Figure 5: The relative step–scaling function σ ( u ) /u obtained after extrapolating the latticedata to the continuum limit. The black circles have a statistical error only, the red errorbars include systematic errors but using only a constant continuum extrapolation, andthe grey error bars give an idea of the total error by including both constant and linearcontinuum extrapolations. The running of the fermion mass is determined by the scale–dependence of the renormal-isation constant for the pseudoscalar fermion bilinear Z P defined in Eq. (2.6). Note that Z P is both scheme and scale dependent. The same step scaling technique described forthe gauge coupling can be used to follow the nonperturbative evolution of the fermionmass in the SF scheme. In this work, we follow closely the procedure outlined in Ref. [48].We have measured the pseudoscalar density renormalisation constant Z P ( β, L ) for arange of β, L . Our results are reported in Tab. 4, and plotted in Fig. 6, where we see thatthere is a clear trend in Z P as a function of L at all values of β .The lattice step scaling function for the mass is defined as:Σ P ( u, s, a/L ) = Z P ( g , sL/a ) Z P ( g , L/a ) (cid:12)(cid:12)(cid:12)(cid:12) g ( L )= u ; (5.1)14he mass step scaling function in the continuum limit, σ P ( u, s ), is given by: σ P ( u, s ) = lim a → Σ P ( u, s, a/L ) . (5.2) β Z P L/a=6L/a=8L/a=9L/a=10 2/3L/a=12L/a=16
Figure 6: Data for the renormalisation constant Z P as computed from lattice simulationsof the Schr¨odinger functional. Numerical simulations are performed at several values ofthe bare coupling β , and for several lattice resolutions L/a . The points at
L/a = 9 , are interpolated. β L =6 L =8 L =12 L =162.00 0.3016(6) - - -2.05 0.3265(11) - 0.2466(6) -2.10 0.3469(6) - - -2.20 0.3845(6) 0.3550(7) 0.3087(6) -2.25 0.4028(6) 0.3707(7) - -2.30 0.4203(6) 0.3841(7) - -2.40 - 0.4134(7) - -2.50 0.4762(6) 0.4406(9) 0.3970(7) 0.3763(39)2.60 0.5012(7) 0.4624(7) - -2.70 0.5228(6) - - - β L =6 L =8 L =12 L =162.80 0.5424(7) 0.5025(6) 0.4639(6) -3.00 0.5770(7) 0.5381(7) 0.5008(8) 0.4647(55)3.25 0.6120(6) - 0.5342(30) 0.5063(44)3.50 0.6385(7) 0.6030(7) 0.5580(10) 0.5523(43)3.75 0.6654(6) - - -4.00 0.6830(6) 0.6501(6) 0.6197(14) -4.50 0.7173(7) 0.6859(6) 0.6547(4) 0.6341(27)8.00 0.8261(3) 0.8114(3) 0.7956(2) 0.7827(11)16.0 0.9146(4) 0.9082(2) 0.9005(5) 0.8887(15) Table 4: Measured values of Z P on different volumes as a function of the bare coupling β . 15he method for calculating σ P ( u ) ≡ σ P ( u, /
3) is similar to that outlined in Sec. 4 forcalculating σ ( u ). Interpolation in β is accomplished using a function of the form: Z P ( β, L/a ) = n X i =0 c i (cid:18) β (cid:19) i (5.3)Full details of the procedure are given in Appendix B. Again the errors are dominatedby systematics, in particular the choice of continuum extrapolation function. In Fig. 7we see that, unlike g , Z P has a significant variation with a/L that is fit well by a linearcontinuum extrapolation. The constant extrapolation is only used to quantify the errorsin extrapolation. Σ P ( / , u , a / L ) u=0.90u=2.70 Figure 7: Results for the lattice step–scaling function Σ P (4 / , u, a/L ). The point at x = 0yields the value of σ P ( u ), i.e. the extrapolation of Σ P to the continuum limit. The errorbar shows the difference between constant and linear extrapolation functions, and givesan estimate of the systematic error in the extrapolation as discussed in the text.16 σ P ( / , u ) Statistical Error
Figure 8: The step-scaling function for the running mass σ P ( u ), using a linear continuumextrapolation. The black circles have a statistical error only, the red error bars includesystematic errors using a linear continuum extrapolation. The grey error bars come fromalso including a constant extrapolation of the two points closest to the continuum, andgive an idea of the systematic error in the continuum extrapolation.Using the fact that σ P ( u, s ) = m ( µ ) /m ( µ/s ) for µ = 1 /L , we can perform an iterativestep scaling of the coupling and the mass to determine the running of the mass with scale.However, since we observe no running of the coupling within errors this is not particularlyinteresting.The mass step scaling function is related to the mass anomalous dimension (see e.g.Ref. [48]): σ P ( u ) = (cid:18) uσ ( u ) (cid:19) ( d / (2 β )) exp "Z √ σ ( u ) √ u dx (cid:18) γ ( x ) β ( x ) − d β x (cid:19) . (5.4)We find good agreement with the 1-loop perturbative prediction, as shown in Fig. 8.In the vicinity of an IRFP the relation between σ P and γ simplifies. Denoting by γ ∗ the value of the anomalous dimension at the IRFP, we obtain: Z m ( µ/s ) m ( µ ) dmm = − γ ∗ Z µ/sµ dqq , (5.5)17nd hence: log | σ P ( s, u ) | = − γ ∗ log s . (5.6)We can therefore define an estimatorˆ γ ( u ) = − log | σ P ( u, s ) | log | s | , (5.7)which yields the value of the anomalous dimension at the fixed point. Away from thefixed point ˆ γ will deviate from the anomalous dimension, with the discrepancy becominglarger as the anomalous dimension develops a sizeable dependence on the energy scale.We plot the estimator ˆ γ in Fig. 9. Again the error bars come from evaluating the aboveexpression using the extremal values of σ P ( u ) at each u . We see that the actual valueof ˆ γ is rather small over the range of interest. In particular at g = 2 .
2, the benchmarkvalue for the IRFP tentatively found in Ref. [32], we have ˆ γ = 0 . +43 − using just thelinear continuum extrapolation, and ˆ γ = 0 . +76 − if we include the constant continuumextrapolation as well. In the presence of an IRFP ˆ γ yields the value of the anomalousdimension, and therefore the values above can be used to bound the possible values of γ ∗ .The results of Ref. [32] suggest the IRFP is in the range g = 2 . − .
2; at the extremes ofthis range we find γ ∗ = 0 . +85 − and 0 . +15 − using just the linear continuum extrapolation,and γ ∗ = 0 . +105 − and 0 . +15 − including the constant continuum extrapolation. Overthe entire range of couplings consistent with an IRFP, γ ∗ is constrained to lie in therange 0 . < γ ∗ < .
56, even with our more conservative assessment of the continuumextrapolation errors. 18 .0 0.5 1.0 1.5 2.0 2.5 3.0 3.5u0.00.10.20.30.40.5 ^ γ ( u ) Statistical Error
Figure 9: The mass anomalous dimension estimator ˆ γ ( u ). The dashed line shows the1-loop perturbative result, the black circles have a statistical error only, and the red errorbars include systematic errors using a linear continuum extrapolation. The grey errorbars also include a constant extrapolation of the two points closest to the continuum,giving an idea of the systematic error involved in the continuum extrapolation. In this paper we have presented results for the running of the Schr¨odinger Functionalcoupling g and the mass anomalous dimension γ .Turning first to the running of the coupling, our results are completely consistent withthose of Ref. [32]. Our statistical errors are larger; however, we have carried out ouranalysis in a way that aims at disentangling clearly the scale dependence from the latticeartefacts. Our analysis can be systematically improved as more extensive studies areperformed, and will ultimately allow us to take the continuum limit with full control overthe resulting systematic errors. Our results appear to show a slowing in the running of thecoupling above g = 2 or so, and are consistent with the presence of a fixed point wherethe running stops at somewhat higher g . This is consistent with the analysis of Ref. [32].However, once we include the systematic errors from the continuum extrapolation we find19hat our results no longer give any evidence for a fixed point. The fundamental reason forthis is that the running of the coupling is very slow in this theory and so great accuracyis needed, in particular near a possible fixed point.By contrast, we find that the behaviour of the anomalous dimension γ is much easier toestablish. The systematic errors from the continuum extrapolation are much smaller thanthe signal, and we find a moderate anomalous dimension, close to the 1-loop perturbativeprediction, throughout the range of β explored. In particular, in the range g = 2 . − . . < γ < .
56. These values aremuch smaller than those required for phenomenology, which are typically of order 1-2.Such large values of γ are clearly inconsistent with our results. The anomalous dimensionat the fixed point can be computed analytically using the all–order beta function proposedin Ref. [49]. The result can be expressed as a function of group–theoretical factors only.Using the conventions described in the Appendix of Ref. [14] for these group–theoreticalfactors, the result in Ref. [49] yields γ = 3 /
4, which is not too far from the bound wequote above. Given the uncertainty in the exact value of the SF coupling at the fixedpoint, γ = 3 / Z P at L = 8and L = 12. By contrast for the running of the coupling we must measure the differenceof two quantities that are almost the same, since the beta function is expected to be small.Furthermore, the anomalous dimension is crucial for phenomenology; if it is not large thenthe presence or absence of walking behaviour becomes academic. Hence the implicationsof our measurement of γ for the phenomenology of minimal walking technicolor call for amore precise study.Our conclusion that γ is not large is unlikely to be affected by using larger lattices.One can see this by considering the continuum extrapolations in Fig. 7. For γ to reach,say, 1 in the continuum limit, we would need Σ P to be 3 / .
75 at a/L = 0. Howeverwe see that the dependence on a/L is much too small for this to be possible, and indeedis in the wrong direction. Only a very unlikely conspiracy of lattice artifacts would makeit possible for Σ P to be as small as 0.75 in the continuum limit. On the other hand thevalue of ¯ g corresponding to the IRFP is currently not known with sufficient accuracy.The results presented here are the first computation of the anomalous dimension ata putative fixed point; the systematic errors need to be reduced to make our conclusionsmore robust. In particular, using larger lattices would give results at smaller a/L andhence make the continuum extrapolations more accurate. It may also be necessary touse an improved action in the long term to achieve the precision required to show the20xistence of an IRFP or of walking behaviour. However, as described above, this is veryunlikely to affect our phenomenologically most important result, namely that γ is notlarge. Recent results in Ref. [50] suggest that the anomalous dimension can be computedusing finite–size scaling techniques. A comparison of different techniques will improve thedetermination of the anomalous dimension. Acknowledgments
A Coupling error analysis
We directly measure the Schr¨odinger Functional coupling g and perform multiple stagesof interpolation and extrapolation to extract the continuum step scaling function σ ( u ) ≡ σ ( u, / g L/a c . ± .
057 0 . ± .
050 1 . ± .
001 0 . ± .
003 0 . ± . c − . ± . − . ± . − . ± . − . ± . − . ± . c . ± . − . ± . − . ± . − . ± . c . ± .
125 0 . ± . c − . ± . − . ± . c − . ± . c . ± . c − . ± . χ dof .
85 2 .
42 1 .
73 3 .
45 3 . dof Table 5: Interpolation best fit parameters for g .21n order to estimate our errors for each of these stages we perform multiple bootstrapsof the data. The full procedure to get a single estimate of σ ( u ) can be summarised asfollows: • Generate N b × N a bootstrapped ensembles of the data and extract mean and errorfor each. • For each bootstrap, interpolate in a/L to find values at L = 9 , . • From each set of N a of these find the mean and standard deviation, to give N b interpolated data points with error bars. • For each of the N b bootstraps do a non-linear least squares fit for g ( β, L ) interpo-lation functions in β , an example is shown in Fig. 10. • Use these functions to find N b estimates of Σ( a/L, u ) for L = 8 ,
9, and from thisextract a mean and error for each a/L . • Perform a single weighted continuum extrapolation in a/L using these points to give σ ( u ).This process is repeated N m times, bringing the total number of bootstrap replicas ofthe data to N a × N b × N m . This gives N m estimates of σ ( u ), from which a mean and1-sigma confidence interval is extracted. 22 β m ea s u r e d g / b e s t -f it i n t e r po l a ti on Figure 10: Example of an interpolation function for L = 8, with a ± σ confidence interval,compared with measured g data points.However, the systematic errors that result from varying the number of parametersin the interpolation functions or the continuum extrapolation functions are significantlylarger than the statistical errors for the optimal set of parameters.In order to quantify this, we repeated the entire bootstrapped process of calculating σ ( u ) with a range of different interpolation and extrapolation functions, each of whichgives an estimates for σ ( u ), with a statistical error.Specifically, we included two different choices for the number of parameters in theinterpolating functions at each L . We kept the best fit, outlined in Tab. 5 and added thefunction with the second lowest χ per degree of freedom as shown in Tab. 6. The errorin the continuum extrapolation was estimated by including both constant and linearextrapolation functions. All possible combinations of these functions gave us a set of2 = 32 values for σ ( u ), each with a statistical error, which spanned the range of thesystematic variation.For each value of u the resulting extremal values of σ ( u ) were used as upper and lowerbounds on the central value. 23 L/a c . ± .
057 0 . ± .
050 1 . ± .
001 0 . ± .
003 0 . ± . c − . ± . − . ± . − . ± . − . ± . − . ± . c . ± . − . ± . − . ± . − . ± . c . ± .
125 0 . ± . c − . ± . − . ± . c − . ± . c . ± . c − . ± . χ dof .
85 2 .
42 1 .
73 3 .
45 3 . dof Table 6: Interpolation next-best fit parameters for g . B Mass error analysis
The mass error analysis follows the same procedure as outlined in Appendix A with g replaced by Z P . The function used to interpolate Z P in β is given in Eq. 5.3, and anexample fit is shown in Fig. 11. The c i giving the smallest reduced χ are given in Tab. 7and those with the second smallest in Tab. 8.In addition, Z P converges faster than g and we have better 16 data so we can use3 points in our continuum extrapolations. Again the L = 6 data were found to havelarge O ( a ) artifacts so are not used in the continuum extrapolation, and for the constantextrapolation only the two points closest to the continuum limit are used. The fits forboth g and Z P are required to determine σ P ( u ), so independently varying the choice ofthe number of parameters for these now gives 2 = 1024 values for σ P ( u ), each with astatistical error. Z P L/a
12 16 c . ± .
30 0 . ± .
09 1 . ± .
01 1 . ± .
01 1 . ± .
01 1 . ± . c . ± . − . ± . − . ± . − . ± . − . ± . − . ± . c − . ± . − . ± .
64 4 . ± .
54 1 . ± .
05 2 . ± .
31 1 . ± . c . ± .
14 36 . ± . − . ± . − . ± . − . ± . − . ± . c − . ± . − . ± .
04 7 . ± . c . ± .
32 57 . ± . c − . ± . χ dof .
42 1 .
66 2 .
24 4 .
82 6 .
68 6 . dof
11 8 5 6 6 3
Table 7: Interpolation best fit parameters for Z P .24 P L/a
12 16 c . ± .
07 1 . ± .
46 0 . ± .
02 1 . ± .
01 0 . ± .
03 0 . ± . c − . ± . − . ± .
46 0 . ± . − . ± . − . ± . − . ± . c . ± .
46 34 . ± . − . ± .
87 1 . ± . − . ± .
60 0 . ± . c − . ± . − . ± .
42 58 . ± . − . ± .
97 5 . ± . c . ± .
72 405 . ± . − . ± .
92 0 . ± . − . ± . c − . ± . − . ± .
59 71 . ± . c . ± . χ dof .
46 1 .
75 2 .
32 5 .
97 7 .
47 8 . dof
12 7 4 5 5 4
Table 8: Interpolation next-best fit parameters for Z P . β m ea s u r e d Z _ P / b e s t -f it i n t e r po l a ti on Figure 11: Example of an interpolation function for L = 8, with a ± σ confidence interval,compared with measured Z P data points. 25 eferences [1] Christopher T. Hill and Elizabeth H. Simmons. Strong dynamics and electroweaksymmetry breaking. Phys. Rept. , 381:235–402, 2003. [arXiv:hep-ph/0203079].[2] Francesco Sannino. Dynamical Stabilization of the Fermi Scale: Phase Diagram ofStrongly Coupled Theories for (Minimal) Walking Technicolor and Unparticles. 2008.arXiv:0804.0182 [hep-ph].[3] Michael Edward Peskin and Tatsu Takeuchi. A New constraint on a strongly inter-acting Higgs sector.
Phys. Rev. Lett. , 65:964–967, 1990.[4] Guido Altarelli and Riccardo Barbieri. Vacuum polarization effects of new physicson electroweak processes.
Phys. Lett. , B253:161–167, 1991.[5] C. Amsler and others (Particle Data Group).
Phys. Lett. , B667:1, 2008. and 2009partial update for the 2010 edition.[6] Bob Holdom. Techniodor.
Phys. Lett. , B150:301, 1985.[7] Bob Holdom. Flavor changing suppression in technicolor.
Phys. Lett. , B143:227,1984.[8] Koichi Yamawaki, Masako Bando, and Ken-iti Matumoto. Scale Invariant Techni-color Model and a Technidilaton.
Phys. Rev. Lett. , 56:1335, 1986.[9] Dennis D. Dietrich and Francesco Sannino. Walking in the SU(N).
Phys. Rev. ,D75:085018, 2007. [arXiv:hep-ph/0611341].[10] Roshan Foadi, Mads T. Frandsen, Thomas A. Ryttov, and Francesco Sannino. Min-imal walking technicolor: Set up for collider physics.
Phys. Rev. , D76:055005, 2007.[arXiv:0706.1696 [hep-ph]].[11] Roshan Foadi, Mads T. Frandsen, and Francesco Sannino. Constraining Walking andCustodial Technicolor.
Phys. Rev. , D77:097702, 2008. [arXiv:0712.1948 [hep-ph]].[12] Simon Catterall and Francesco Sannino. Minimal walking on the lattice.
Phys. Rev. ,D76:034504, 2007. [arXiv:0705.1664 [hep-lat]].[13] Thomas Appelquist, George T. Fleming, and Ethan T. Neil. Lattice Study ofthe Conformal Window in QCD-like Theories.
Phys. Rev. Lett. , 100:171607, 2008.[arXiv:0712.0609 [hep-ph]]. 2614] Luigi Del Debbio, Mads T. Frandsen, Haralambos Panagopoulos, and Francesco San-nino. Higher representations on the lattice: perturbative studies.
JHEP , 06:007, 2008.[arXiv:0802.0891 [hep-lat]].[15] Yigal Shamir, Benjamin Svetitsky, and Thomas DeGrand. Zero of the discretebeta function in SU(3) lattice gauge theory with color sextet fermions.
Phys. Rev. ,D78:031502, 2008. [arXiv:0803.1707 [hep-lat]].[16] Albert Deuzeman, Maria Paola Lombardo, and Elisabetta Pallante. The physics ofeight flavours.
Phys. Lett. , B670:41–48, 2008. [arXiv:0804.2905 [hep-lat]].[17] Luigi Del Debbio, Agostino Patella, and Claudio Pica. Higher representations on thelattice: numerical simulations. SU(2) with adjoint fermions. 2008. arXiv:0805.2058[hep-lat].[18] Simon Catterall, Joel Giedt, Francesco Sannino, and Joe Schneible. Phase dia-gram of SU(2) with 2 flavors of dynamical adjoint quarks.
JHEP , 11:009, 2008.[arXiv:0807.0792 [hep-lat]].[19] Benjamin Svetitsky, Yigal Shamir, and Thomas DeGrand. Nonperturbative infraredfixed point in sextet QCD.
PoS , LATTICE2008:062, 2008. arXiv:0809.2885 [hep-lat].[20] Thomas DeGrand, Yigal Shamir, and Benjamin Svetitsky. Exploring the phase dia-gram of sextet QCD.
PoS , LATTICE2008:063, 2008. arXiv:0809.2953 [hep-lat].[21] Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, and Chris Schroeder.Nearly conformal electroweak sector with chiral fermions.
PoS , LATTICE2008:058,2008. arXiv:0809.4888 [hep-lat].[22] Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, and Chris Schroeder.Probing technicolor theories with staggered fermions.
PoS , LATTICE2008:066, 2008.arXiv:0809.4890 [hep-lat].[23] Albert Deuzeman, Maria Paola Lombardo, and Elisabetta Pallante. The physics ofeight flavours.
PoS , LATTICE2008:060, 2008. [arXiv:0810.1719 [hep-lat]].[24] Albert Deuzeman, Elisabetta Pallante, Maria Paola Lombardo, and E. Pallante.Hunting for the Conformal Window.
PoS , LATTICE2008:056, 2008. arXiv:0810.3117[hep-lat].[25] Ari Hietanen, Jarno Rantaharju, Kari Rummukainen, and Kimmo Tuominen. Spec-trum of SU(2) gauge theory with two fermions in the adjoint representation.
PoS ,LATTICE2008:065, 2008. [arXiv:0810.3722 [hep-lat]].2726] Xiao-Yong Jin and Robert D. Mawhinney. Lattice QCD with Eight Degenerate QuarkFlavors.
PoS , LATTICE2008:059, 2008. [arXiv:0812.0413 [hep-lat]].[27] Luigi Del Debbio, Agostino Patella, and Claudio Pica. Fermions in higher represen-tations. Some results about SU(2) with adjoint fermions.
PoS , LATTICE2008:064,2008. arXiv:0812.0570 [hep-lat].[28] Thomas DeGrand, Yigal Shamir, and Benjamin Svetitsky. Phase structure of SU(3)gauge theory with two flavors of symmetric-representation fermions.
Phys. Rev. ,D79:034501, 2009. [arXiv:0812.1427 [hep-lat]].[29] George T. Fleming. Strong Interactions for the LHC.
PoS , LATTICE2008:021, 2008.[arXiv:0812.2035 [hep-lat]].[30] Ari J. Hietanen, Jarno Rantaharju, Kari Rummukainen, and Kimmo Tuominen.Spectrum of SU(2) lattice gauge theory with two adjoint Dirac flavours.
JHEP ,05:025, 2009. [arXiv:0812.1467 [hep-lat]].[31] Thomas Appelquist, George T. Fleming, and Ethan T. Neil. Lattice Study ofConformal Behavior in SU(3) Yang-Mills Theories.
Phys. Rev. , D79:076010, 2009.[arXiv:0901.3766 [hep-ph]].[32] Ari J. Hietanen, Kari Rummukainen, and Kimmo Tuominen. Evolution of thecoupling constant in SU(2) lattice gauge theory with two adjoint fermions. 2009.arXiv:0904.0864 [hep-lat].[33] A. Deuzeman, M. P. Lombardo, and E. Pallante. Evidence for a conformal phase inSU(N) gauge theories. 2009. arXiv:0904.4662 [hep-ph].[34] Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, and Chris Schroeder.Topology and higher dimensional representations.
JHEP , 0908:084, 2009.[arXiv:0905.3586 [hep-lat]].[35] Thomas DeGrand and Anna Hasenfratz. Comments on lattice gauge theories withinfrared- attractive fixed points.
Phys. Rev. , D80:034506, 2009. [arXiv:0906.1976[hep-lat]].[36] Thomas DeGrand. Volume scaling of Dirac eigenvalues in SU(3) lattice gauge theorywith color sextet fermions. 2009. arXiv:0906.4543 [hep-lat].[37] Anna Hasenfratz. Investigating the critical properties of beyond-QCD theories us-ing Monte Carlo Renormalization Group matching.
Phys. Rev. , D80:034505, 2009.[arXiv:0907.0919 [hep-lat]]. 2838] L. Del Debbio, B. Lucini, A. Patella, C. Pica, and A. Rago. Conformal vs confiningscenario in SU(2) with adjoint fermions. 2009. arXiv:0907.3896 [hep-lat].[39] Zoltan Fodor, Kieran Holland, Julius Kuti, Daniel Nogradi, and Chris Schroeder.Nearly conformal gauge theories in finite volume. 2009. arXiv:0907.4562 [hep-lat].[40] C. Pica, L. Del Debbio, B. Lucini, A. Patella, and A. Rago. Technicolor on theLattice. 2009. arXiv:0909.3178 [hep-lat].[41] Martin Luscher, Peter Weisz, and Ulli Wolff. A Numerical method to compute therunning coupling in asymptotically free theories.
Nucl. Phys. , B359:221–243, 1991.[42] Martin Luscher, Rajamani Narayanan, Peter Weisz, and Ulli Wolff. The Schrodingerfunctional: A Renormalizable probe for nonAbelian gauge theories.
Nucl. Phys. ,B384:168–228, 1992. [arXiv:hep-lat/9207009].[43] Martin Luscher, Rainer Sommer, Ulli Wolff, and Peter Weisz. Computation of therunning coupling in the SU(2) Yang- Mills theory.
Nucl. Phys. , B389:247–264, 1993.[arXiv:hep-lat/9207010].[44] Stefan Sint and Rainer Sommer. The Running coupling from the QCDSchrodinger functional: A One loop analysis.
Nucl. Phys. , B465:71–98, 1996.[arXiv:hep-lat/9508012].[45] Stefano Capitani, Martin Luscher, Rainer Sommer, and Hartmut Wittig. Non-perturbative quark mass renormalization in quenched lattice QCD.
Nucl. Phys. ,B544:669–698, 1999. [arXiv:hep-lat/9810063].[46] Stefan Sint. On the Schrodinger functional in QCD.
Nucl. Phys. , B421:135–158,1994. [arXiv:hep-lat/9312079].[47] Michele Della Morte et al. Computation of the strong coupling in QCD with twodynamical flavours.
Nucl. Phys. , B713:378–406, 2005. [arXiv:hep-lat/0411025].[48] Michele Della Morte et al. Non-perturbative quark mass renormalization in two-flavorQCD.
Nucl. Phys. , B729:117–134, 2005. [arXiv:hep-lat/0507035].[49] Thomas A. Ryttov and Francesco Sannino. Supersymmetry Inspired QCD BetaFunction.