Measurement of the mass anomalous dimension of near-conformal adjoint QCD with the gradient flow
Camilo Lopez, Georg Bergner, Istvan Montvay, Stefano Piemonte
MMeasurement of the mass anomalousdimension of near-conformal adjoint QCD with the gradient flow
Camilo Lopez ∗1,2 , Georg Bergner †1 , Istvan Montvay ‡3 , and StefanoPiemonte §41 University of Jena, Institute for Theoretical Physics,Max-Wien-Platz 1, D-07743 Jena, Germany Department of Physics, University of Colorado, Boulder, Colorado80309, United States Deutsches Elektronen-Synchrotron DESY, Notkestr. 85, D-22607Hamburg, Germany University of Regensburg, Institute for Theoretical Physics,Universit¨atsstr. 31, D-93040 Regensburg, Germany6th November 2020
The mass anomalous dimension is determined in SU(2) gauge theory coupledto N f fermions in the adjoint representation for N f = 2, 3 /
2, 1 and 1 /
2, wherehalf-integer flavor numbers correspond to Majorana fermions. The numericalmethod is based on gradient flow.The results show near-conformal behavior for N f = 2, 3 / N f = 2 which is relevant for a strongly interactingextension of the Standard Model and has been studied in several previousinvestigations. We check whether the method is able to resolve discrepan-cies in earlier results for this theory. Overall, the method based on gradientflow delivers reliable results in qualitative agreement with previously knownnumerical data. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - l a t ] N ov he landscape of strongly coupled gauge theories with different matter contents hasbeen the subject of many recent investigations. They are motivated by the search fora strongly interacting completion of the Standard Model and by an improvement ourgeneral understanding of strong interactions, see reviews [1, 2]. A QCD-like behavior ofthese theories implies a running of the coupling from asymptotic freedom correspond-ing to a Gaussian ultra violet (UV) fixed point towards confinement in the infrared(IR). Confinement is due to an exponential growth of the strong coupling implying non-perturbative phenomena like the existence of a mass gap even in the limit where all massparameters of the matter fields are zero. The remormalization group (RG) flow of QCDfrom the perturbative running towards the strongly coupled infrared has been a subjectof several studies.Increasing the matter content of the theory leads to a screening effect, which willchange this scenario. Eventually, asymptotic freedom is lost and the RG flow is changedtowards a more QED-like form. In between the QCD-like running and the theories withno asymptotic freedom we expect the existence of the so-called conformal window. Thematter fields in this regime screen the running in the IR, but asymptotic freedom isstill preserved in the UV. Consequently, the strong coupling increases until it reaches afixed point in the IR. The RG flow close to the fixed point is characterized by universalquantities like mass ratios and anomalous dimensions of certain operators.Once the lower boundary of the conformal window is reached, the slow running of thegauge coupling provides interesting scenarios for model building. A large scale separationappears while the theory is still at strong couplings. This could explain an enhancementof certain operators in the low energy effective theory if the anomalous dimensions arelarge enough. In addition, a light dilaton-like scalar field can exist. From the perspectiveof model building, a smaller number of matter fields is preferred. This has lead toconsider theories with fermions in higher representation which require a smaller numberof fields to reach a near-conformal behavior. One of the most prominent examples of suchtheories is the SU(2) gauge theory coupled to two adjoint Dirac fermions [3, 4, 5, 14, 7].The motivation for our investigation is the exploration of the general theory space forpossible realizations of strongly interacting gauge theories. We investigate SU(2) gaugetheories with different number of fermions in the adjoint representation (adjoint QCD)in order to characterize the lower boundary of the conformal window. We denote by N f the number of Dirac fermions. Half integer values of N f correspond to 2 N f Majoranafermions. The important question is not only whether a fixed point exists for thesetheories. Going into more detail, we focus here on the mass anomalous dimension asone of the most important characteristics of the infrared fixed point (IRFP). Our mainresults are related to the N f = 2 case, since the conformal behavior of this theory isindicated by several previous numerical lattice studies. Therefore, it is an ideal startingpoint to test new methods determining the RG flow close to the fixed point. In addition,our aim is to resolve the observed discrepancies in the previously obtained values for themass anomalous dimension.In earlier studies we have also considered N f = 1, N f = 3 /
2, as well as N f = 1 / N f = 1 / N = 1 supersymmetricYang-Mills theory and should be outside of the conformal window. These studies were2ntended to investigate how the mass anomalous dimension increases towards the lowerend of the conformal window until the fixed point disappears. In theories with goodindications for a conformal fixed point only quite small mass anomalous dimensionshave been observed. Therefore, more detailed studies could show if much larger valuescould be realized at all.The investigations of the IRFP on the lattice is hampered by several difficulties. TheRG flow is restricted in the ultraviolet by the finite lattice spacing and in the infraredby the finite mass parameters and by the finite volume. This restricts the range of scalesfor the investigations and leads to deformations of the flow which are hard to quantify.The methods we have applied in our earlier studies for the determination of the massanomalous dimension γ ∗ are based on the scaling of the mass spectrum [10] and themode number [11]. The bound state masses M are predicted to scale with the fermionmass m according to M ≈ m / (1+ γ ∗ ) . The mode number computed from the eigenvaluesof the Dirac operator is predicted to scale in a certain regime between the infrared,dominated by the mass and the volume corrections, and the ultraviolet, dominated bylattice artifacts and perturbative running. We have found that the anomalous dimensionvaries significantly with the bare gauge coupling β . Recently a method has been proposedto determine γ ∗ , which captures the detailed features of the RG running and seems tobe able to resolve the discrepancy among different β values [12]. The different bareparameters can eventually be extrapolated to a convergent flow towards a single fixedpoint value. In this work we test whether this method provides values consistent withour previous determinations and whether it might help to resolve the discrepancies. In this section we briefly review the method for the determination of the mass anomalousdimension first derived in [12]. The fixed point value γ ∗ of this quantity is extrapolatedfrom the RG flow. A RG step consists of a coarse-graining and a scale transformation(dilation). In Refs. [13, 12], it has been proposed that the gradient flow corresponds(asymptotically at large flow times) to the coarse-graining step. This relation arisesfrom the fact that, for a given flow time t , the gradient flow kernel smears the fields overa (Gaussian) radius √ t . In momentum space this is equivalent to imposing a smoothUV-cutoff. In the following we discuss how the effect of the dilation can be included, sothat we can explore the RG flow and its fixed points from the gradient flow.A RG step with parameter b > a → a = b a ,and changes the couplings as g → g , m → m . Generic fields φ with canonical dimension d φ are scaled according to their scaling dimension, which includes the anomalous η . Ageneric two-point function transforms as hO (0) O ( x ) i g,m = b − d O + γ O ) hO (0) O ( x /b ) i g ,m , (1)where the operator O ( x ) ≡ O [ φ ]( x ) is a lattice interpolator, which is in general amonomial of gauge or fermion fields of order n O . The scaling dimension ∆ O includes Eq. 1 holds as long as O are scaling operators, i.e. if they are eigenstates of the linearized RG equations. d O ) and anomalous dimension ( γ O ).Although the gradient flow doesn’t re-scale the smeared fields, we can include a renor-malization factor by relating t and b . At large t it is expected that b ∝ √ tφ b ( x /b ) = b ∆ φ φ t ( x ) = b d φ + η/ φ t ( x ) . Here the subscript b labels the blocked fields after the RG step . Assuming that x islarge enough to both neglect the re-scaling of the distances on the lattice and avoid theoverlapping of the operators as t increases, we get hO t (0) O t ( x ) ihO (0) O ( x ) i = t ∆ O − n O ∆ φ , (2)The index t indicates interpolators consisting of fields at flow time t . Note that it isassumed that the x dependence cancels between numerator and denominator for largeenough x . Moreover, the ratio doesn’t require us to know the exact relation b ∝ √ t , ifwe stay away from small t .In order to remove the dependence on the scaling dimension of the field φ , a ratio withoperators ( V ) of scaling dimension ∆ V = 0 is added. This is fulfilled if V is a conservedcurrent. For simplicity we assume that it has the same field content as O ( n O = n V and d O = d V ). This leads to hO t (0) O t ( x ) ihO (0) O ( x ) i hV (0) V ( x ) ihV t (0) V t ( x ) i = t γ O . (3)The computation of correlators with both interpolators consisting of fields at t requiresthe integration of the computationally expensive adjoint fermion flow equation. As atechnical simplification one considers the correlator with only the sink term at flow time t , R O ( t, x ) = hO (0) O t ( x ) ihO (0) O ( x ) i hV (0) V ( x ) ihV (0) V t ( x ) i = t γ O / . (4)The corrections are of order O ( a √ t/x ) and, following [12], they are neglected in theanalysis. The final formula for the scaling of γ O with the energy scale is (¯ t = ( t + t ) / γ O (¯ t ) = log( R O ( t ) / R O ( t ))log ( √ t / √ t ) . (5)It is assumed that R doesn’t depend for large enough x and hence R ( t, x ) shouldapproach a constant.In the measurements on the lattice several deformations of the running have to betaken into account. In a vicinity of the IRFP the relevant deformations are the finite We stress that this is only valid at large t , since as t → b → The Monte Carlo Renormalization Group method states that the right hand side of Eq.1 can be writtenas a correlator of blocked fields with respect to the UV action. µ is represented using µ = 1 / √ t . The continuum action of the SU(2) gauge theories with fermions in the adjoint repres-entation reads S = Z d x
14 ( F aµν F aµν ) + 12 N f X i =1 ¯ λ ia γ µ D abµ λ ib . (6)In this formulation, the 2 N f fermion fields λ i fulfill the Majorana condition, but forinteger N f it can be straightforwardly converted into a theory with N f Dirac fermions.The covariant derivative acts in the adjoint representation is defined as D abµ λ b = ∂ µ λ a + igA cµ ( T Ac ) ab λ b , (7)where T Ac are the Lie algebra generators as given by the structure constants. On the lat-tice, we employ a tree-level Symanzik improved gauge action and stout-smeared Wilsonfermions. The simulation parameters are summarized in Table 1. Note that a sign prob-lem can arise for odd number of fermions, but our simulations are performed in a rangeof fermion masses where it can be neglected.We measured the anomalous dimension of the pseudo-scalar operator, which is relatedto the mass anomalous dimension as γ m = − γ P S , in adjoint QCD with N f = 1 / , , / λγ λ .The natural choice of V in case of Wilson fermions is the vector current, as it is thesimplest conserved current on the lattice. It is however important to notice, that thelocal (continuum) vector current V aµ = ¯ λ τ a γ µ λ (8)is not exactly conserved at non-zero lattice spacing in contrast to the point-split latticevector current˜ V aµ = 12 (cid:20) ¯ λ ( x )( γ µ − U µ ( x ) τ a λ ( x + a ˆ µ ) + ¯ λ ( x + a ˆ µ )( γ µ + 1) U † µ ( x ) τ a λ ( x ) (cid:21) . (9)Here τ a are the Pauli matrices acting on the flavor indices of the two Dirac spinor fields λ . If the local current V aµ is used in the calculations, an additional dependence on therenormalization factor Z V = 1 remains. In particular, lattice artifacts are expected tointroduce a spurious running of Z V on the scale µ , and we must ensure numerically5hat we are exploring a scaling region where such effects are under control. We actuallyperform the measurements of γ P S with both currents for the N f = 2 case in order toensure that the running in Eq.(4) is dominated by the mass anomalous dimension andto cross check the resultsThe result of the computation is an effective anomalous dimension γ m as a functionof the flow scale ¯ t . For an IR conformal theory, it is expected that the running of γ m towards the IR stops at a certain energy scale and γ m converges to the fixed point value γ ∗ . In practice, however, it is impossible to consider arbitrary large flow times due tonon-zero size and finite mass effects in the IR. The value of γ ∗ has to be extrapolatedfrom intermediate flow times, which leads to the usual windowing problem. The mainresults are obtained at very small masses, like in [12], which means that the finite volumeeffects are the dominant deformations of the flow. We start our studies with the case of N f = 2 since in this case there are several previousstudies available for comparison. All studies indicate the existence of an IRFP, but theanomalous dimension varies significantly. Our aim is check whether the gradient flowmethod is able to resolve the discrepancies. We simulated four different β values (seeTable 1): β = 1 . β = 1 . β = 1 . β = 2 .
25. Two different volumes, V = 24 ×
64 and V = 32 ×
64, areconsidered to estimate finite volume effects. In all cases we have measured O (100) wellseparated configurations. The gradient flow-times are in the range 1 ≤ t/a ≤ t/a = 10. We use the followingscaling relation based on data at the volumes L and sL R O ( g, s t, s L ) = R O ( g, s t, sL ) + s − γ O ( R O ( g, t, sL ) − R O ( g, t, L )) + O ( g − g ) , (10)in order to obtain data in a third, even larger, effective volume V = s L ×
64 [12], forthe present parameters s = 32 /
24, i.e. L S ∼
42. This is done in order to reduce finitevolume effects.
The first part of the analysis is to determine the ratios R P S ( t, x ), which are easilyobtained from the vector and pseudo-scalar operators. In Fig. 2 we show the ratio R P S as a function of x at different flow-times.We compare the results from both the local and point-split vector currents. We seethat, for the local current, R P S ( x ) clearly converges to an almost constant plateau.For the point-split current, R P S has larger fluctuations. This general observation isconsistent for all the volumes and β values we considered. As a measure of the asymptoticvalue of R P S ( t ) at large distances in Eq. (5), we take the average of R over the interval x ∈ [15 , γ P S ( µ ) for our6hree larger β at lattice size L = 32, for the two kinds of vector currents. The resultsare compatible, but the uncertainties are larger in the point-split current case. Overallthe renormalization factor of V µ seems, as a first approximation, to be negligible for thefinal result. The local current is more stable but comes with an additional systematicuncertainty since it is not strictly conserved. The point-split vector current, on the otherhand, is conserved but leads to larger statistical errors. For N f = 2, we have employedboth vector currents in the analysis at β = 1 .
6, 1 . .
25. For N f = 3 / N f = 1,we have focused on the results obtained from the point-split current. x / a R ( x ) Local current, =1.7, L =32t=0t=1 t=2t=3 t=4t=5 Figure 1: R PS ( x ) from local vector current. x / a R ( x ) point-split current, =1.7, L =32t=0t=1 t=2t=3 t=4t=5 Figure 2: R PS ( x ) from point-split vector cur-rent. a P S local, =1.6point-split, =1.6local, =2.25 point-split, =2.25local, =1.7point-split, =1.7 Figure 3:
Comparison of γ PS from local andpoint-split vector currents. L = 32 a P S point-split =2.25, L =24=2.25, L =32=1.6, L =24 =1.6, L =32=1.7, L =24=1.7, L =32 Figure 4:
Volume dependence of γ PS computedwith the point-split vector current. The running of γ P S ( µ ) obtained with the local current for all β values and lattice sizes L = 24 ,
32 at N f = 2 can be seen in Fig. 4. For the final analysis mass and finitevolume volume corrections have to be taken into account. We assume that the masscorrections are under control since we have chosen runs with am P CAC < .
04. In ourinvestigations of the mass spectrum, we have seen that finite size effects are the mostimportant deformation at this parameter range. In order to reduce the finite size effects,7 f L S L T β κ am PCAC
Table 1: Summary of the simulation parameters considered in the analysis of adjointQCD with different number of fermions N f . The gauge coupling g is related to β = g and the bare fermion mass m to κ = m +4) . Wilson fermions implyadditive and multiplicative renormalization of the fermion mass, which meansa more relevant estimate for the mass parameter is obtained from the partiallyconserved axial current (PCAC) relation. Three levels of stout smearing havebeen done for most of the simulation, except for SYM, where only one level hasbeen considered.we utilize the volume scaling formula (10). We obtain the required input at t = s t byinterpolating the jackknife samples with an exponential function.After obtaining R P S ( s t ) for L = 42, we compute γ P S ( µ ). In Fig. 6 we see that the β = 1 . γ ∗ value in thedeep infrared limit, as they don’t overlap with the curves from the other three barecouplings. This is probably a result of additional systematic uncertainties due to thevicinity of the bulk transition and the larger finite mass corrections. Therefore, weconcentrate on the β = 1 .
6, 1 .
7, and 2 .
25 data. Fig. 6 shows that γ P S , although notbeing strictly constant, shows a very weak scale dependence. This is a hint for an atleast near conformal behavior. Moreover, the anomalous dimension of our larger β valuesoverlap at aµ ∼ .
08. This behavior is in fact what we expect to see in a near conformalsystem. Indeed, in the limit µ →
0, all bare couplings must yield the same universalvalue γ ∗ .We present the final extrapolations towards the infrared limit in Figs. 7 using the localcurrent, while the point-split current results follow in Fig. 8. To obtain the critical γ ∗ we performed a global (joint) polynomial fit, where the parameter at µ = 0, i.e. γ ∗ ,was common to all data sets. The remaining fit parameters are allowed to vary with β and the uncertainties are obtained from the extrapolation of the jackknife samples.We have cut off the β = 2 .
25 point-split data at aµ ∼ .
9, since volume correctionsstart to become very large as t increases. This fact is already observed in Fig.4, wherethe values of γ P S ( µ ) at L = 32 and L = 24 start to diverge from each other as the8nfrared is approached. We observe that the IR extrapolations from both local andpoint-split currents yield a compatible value of around γ ∗ ≈ .
2. It is smaller thanprevious estimations based on the mode number [11, 10], but it is consistent with theSchr¨odinger-Functional results in [4].At this point, a short comment on β = 1 . β value γ P S reaches a plateau at γ ∗ ≈ .
38, which is considerably larger than our extrapolatedvalue. In our previous investigations, we have seen that a significantly larger massanomalous dimension is obtained at this coupling. This value of the gauge coupling isjust above the bulk transition for our tree level Symanzik improved gauge action andcan in this sense be compared to the value of β = 2 .
25 for the plain Wilson action usedin [11]. Indeed, a similar value for the mass anomalous dimension is obtained with themode number in both cases [10]. However, the large difference to the other β value has tobe taken into account. This will discussed in more detail in the following section. Notethat we have also considered possible phase transitions between β = 1 . β = 1 . γ ∗ for N f = 2adjoint QCD in the literature. Table 3 summarizes our results and compare them toother studies found in the literature. From the Table it can be seen that our findingsare compatible with many studies performed in the last years. Moreover, since we get aunique γ ∗ value through the joint extrapolation, the gradient flow method might help toresolve the β dependence of γ ∗ in the deep infrared limit. Note that our previous datain [10], at β = 1 . . γ ∗ = 0 . γ ∗ = 0 . β = 1 .
6, as well as a larger β = 2 . We can gain more information on the existence of a strong interacting IRFP by measuringthe running of the renormalized gauge coupling. This can be computed through thegradient flow method, since the flowed gauge energy density is directly proportional tothe renormalized coupling [15]. There are two possible alternatives. The first possibilityis to consider the running of the gauge coupling simply as a function of the scale µ proportional to 1 / √ t g ( µ ) = 128 π N − t h E ( t ) i (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) t =1 / µ . (11)In this case, finite size and lattice spacing corrections must be addressed separatelyby extrapolating first the renormalized coupling to the infinite volume limit and then bysending the bare lattice gauge coupling to zero. This renormalized coupling is shown inFig. 5. The coupling shows a plateau at some intermediate scale, before it starts to runagain at smaller µ .The second alternative allows for a better control on finite volume effects. Here oneconsiders the running of the gauge coupling with respect to µ = 1 /L on the hypercubic9 .10 0.15 0.20 0.25 0.30 0.35 0.40 a g () = 1.5, L = 32= 1.5, L = 24= 1.6, L = 32 = 1.6, L = 24= 1.7, L = 32= 1.7, L = 24 = 2.25, L = 32= 2.25, L = 32 Figure 5: Gradient flow gauge coupling of N f = 2 adjoint QCD at different β andvolumes.torus of volume L at fixed c = √ t/L [16]. The (discrete) β -function can be readdirectly from the difference of the renormalized coupling measured after flowing thegauge fields for a time equal to t = p c/L for two lattices of size L and sLβ ( g ) = g ( sL ) − g ( L )log ( s ) , (12)where the step s is customary chosen to be 2 or 3 /
2. Here the constant c defines thescheme, and has to be chosen in such a way that lattice discretization errors are undercontrol. The continuum limit is reached in the limit L/a → ∞ extrapolated for fixedrenormalized coupling. When periodic boundary conditions are applied to gauge andfermion fields, as in our case, a finite volume correction has to be included in orderto compute a correctly normalized coupling [16]. Given the limited range of bare gaugecouplings of our simulations, we can compute the discrete β -function for only four points.Table 2 shows the results for the schemes c = 0 . , .
20 and 0 .
26 with s = 32 /
24. Itcan be seen that the β -function has a zero at β = 2 .
25, while it becomes negative for β = 1 . .
7. For larger c the β -function at β = 1 . β -function by considering more gauge couplings,what could serve to confirm that the theory is scale invariant in the infrared. By com-bining step-scaling together with the running of the mass anomalous dimension we couldbe able to extrapolate the value of γ ∗ more reliably.10 .07 0.08 0.09 0.10 0.11 0.12 0.13 a P S Ref. [10], = 1.5Ref. [10], = 1.7 = 1.5= 1.6 = 1.7= 2.25
Figure 6: N f = 2: γ P S ( µ ) at L = 42 for all available β values (local vector current). Forcomparison, the red and gray bands show the results obtained from the modenumber in Ref.[10]. a P S * = 0.23 ± 0.01 local = 1.6 = 1.7 = 2.25 Figure 7: N f = 2: Extrapolation of γ m to its critical value γ ∗ at µ → .00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 a P S * = 0.20 ± 0.04 point-split = 1.6= 1.7 = 2.25 Figure 8: N f = 2: Extrapolation of γ P S to its critical value γ ∗ at µ → a P S * = 0.36 ± 0.05 = 1.7, point-split Figure 9: N f = 3 /
2: Extrapolation of γ P S to its critical value γ ∗ for the point-splitvector current. 12 β g β ( g )0.14 2.25 1.9E-2 -8.0E-51.7 3.2E-2 -1.6E-31.6 3.5E-2 -2.4E-31.5 4.2E-2 -2.3E-30.20 2.25 1.9E-2 1.3E-41.7 3.1E-2 -1.1E-31.6 3.4E-2 -1.4E-31.5 4.1E-2 3.2E-40.26 2.25 1.9E-2 6.2E-41.7 3.1E-2 -1.3E-31.6 3.3E-2 -9.1E-41.5 4.2E-2 3.2E-3 Table 2: Discrete β -function of N f = 2. γ ∗ This study local: . point-split: . β = 1 . . β = 1 . . . . . . . . N f = 2. We performed the same measurements for all the others lattices summarized in Table 1.The analysis was done in the same way as in the N f = 2 case. The only difference isthat here we have notably less ensembles. In particular, for N f = 3 / β value. Although we employ the volume formula for this single β , we are of course notable to determine the β dependence of γ ∗ , as we did in the previous section. The resultsfor N f = 3 / µ → γ ∗ = 0 . γ ∗ is larger than in the two-flavourcase. Our result is in agreement with previous lattice investigations found in Refs. [3, 19].Especially, we get the same value as in Ref. [3], where the authors found γ ∗ ∼ .
38 basedon an analysis of the mode number.In the one-flavour case we only have one β and one volume. The results are shown in13ig. 10. Although for these parameters we are already able to see a (near-)conformalbehaviour, i.e. a very weak change in γ P S , the extrapolated value γ ∗ should be takenvery carefully. In particular, it is considerably smaller than the results in Ref. [9], whichestimated γ ∗ ∼ .
9. It is however a valuable piece of information to see that it is likelyfor the theory to lie in or very near the conformal window. The smaller values of γ indicate rather a conformal scenario.Finally, in SYM, as expected, we don’t see any freezing in the running of γ m . This canbe seen in Fig. 12. From Fig. 11 we see that the estimation of the plateau of R ( x ) isparticularly difficult. If one insists in averaging over some x interval, the resulting R ( t )yields a γ P S ( µ ) that shows a strong scale dependence in the whole considered rangeof µ . This result is indeed a signal for a system lying well below the lower edge ofthe conformal window, as predicted for SYM. In general, it is expected that the scaledependence becomes much stronger in a chiral broken theory, compared to the conformalcase. a P S * = 0.58 ± 0.18 non-local = 1.75, L = 24, point-split Figure 10: Extrapolation of γ P S to its critical value γ ∗ for N f = 1 using the point-splitcurrent. We have studied the mass anomalous dimension for different theories with fermions inthe adjoint representation using a recently proposed method based on the Gradient flow.We have started the analysis with the theory with N f = 2 Dirac flavors. Previous lat-tice investigations of this theory established strong indications for a conformal behavior.14 x / a R ( x ) SYM =1.7, L =32t=1t=2 0 5 10 15 20 25 30 x / a R ( x ) SYM =1.7, L =32 t=3t=4 Figure 11: R ( x ) in SU(2) SYM (local vector current). Large errors lead to an unreliablesignal at large x . a P S SYM, = 1.75, = 0.1495, L = 32 Figure 12: Scale dependence of γ P S in SU(2) SYM using the local vector current.15owever, the predictions for the mass anomalous dimension differ significantly. In par-ticular, different values of the coupling constant lead to different predictions. Our aimis to test the Gradient flow method and resolve these discrepancies.The results resemble some features of our investigations with other methods, in whichthe smallest β value favors a larger mass anomalous dimension. However, since themethod is able to resolve the scale dependence of the running of the renormalized massin a certain region of the renormalization group flow, we can extrapolate results fromdifferent gauge couplings to a prediction for the infrared fixed point. This leads to asmaller value of the mass anomalous dimension in the range of γ ∗ ≈ .
16 to 0 . γ ( µ ) dependence on the flow scale µ . As expected, nofixed point value can be extrapolated in the SYM theory ( N f = 1 / N f . We find that the extrapolated value increases forsmaller N f . The largest value is γ = 0 . N f = 1 which decreases to γ = 0 . N f = 3 /
2. These results give the first reasonable estimates from the gradient flowmethod, but further runs with different gauge couplings would be required for a moredetailed and well established picture.
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