Yang-Mills Theories with Adjoint Fermions

PHYSICAL REVIEW D 97, 074510 (2018)

Running coupling from gluon and ghost propagators in the Landau gauge: Yang-Mills theories with adjoint fermions

† Georg Bergner1,* and Stefano Piemonte2, 1Friedrich-Schiller-University Jena, Institute of Theoretical Physics, Max-Wien-Platz 1, D-07743 Jena, Germany 2University of Regensburg, Institute for Theoretical Physics, Universitätsstr. 31, D-93040 Regensburg, Germany

(Received 14 December 2017; published 20 April 2018)

Non-Abelian gauge theories with fermions transforming in the adjoint representation of the gauge group (AdjQCD) are a fundamental ingredient of many models that describe the physics beyond the Standard Model. Two relevant examples are N ¼ 1 supersymmetric Yang-Mills (SYM) theory and minimal walking technicolor, which are gauge theories coupled to one adjoint Majorana and two adjoint Dirac fermions, respectively. While confinement is a property of N ¼ 1 SYM, minimal walking technicolor is expected to be infrared conformal. We study the propagators of ghost and gluon fields in the Landau gauge to compute the running coupling in the MiniMom scheme. We analyze several different ensembles of lattice Monte Carlo simulations for the SU(2) adjoint QCD with Nf ¼ 1=2; 1; 3=2, and 2 Dirac fermions. We show how the running of the coupling changes as the number of interacting fermions is increased towards the conformal window.

DOI: 10.1103/PhysRevD.97.074510

I. INTRODUCTION perturbation theory in a given scheme S. This provides the asymptotic expansion Strong interactions are responsible for the confinement ∞ of quarks and for the generation2 of the hadron masses in X g d S S S S S 3þ2i QCD. The coupling (g or α ≡ 4π) of the strong interactions μ g ðμÞ¼β ðg ðμÞÞ ¼ − b g ðμÞ : ð1Þ dμ i as a function of the energy scale μ decreases at high i¼0 energies and asymptotic freedom allows a description of the interactions in a perturbative framework. The coupling The first two coefficients b0 and b1 are scheme independent increases towards lower energies leading to color charge and asymptotic freedom requires b0 to be positive. A confinement, dynamical scale generation, and other non- truncation to the first few coefficients of the expansion perturbative effects. Asymptotic freedom is a general provides a good approximation at high enough μ, but it is feature of many non-Abelian gauge theories. The determi- not yet fully understood how the coupling of QCD runs in μ → 0 nation of the running as a function of the scale, the gauge the deep infrared regime . In particular, many group, and the number of fermions is crucial for a deeper questions have been raised concerning the physical con- S understanding of QCD and confinement. Moreover, it sequences of a singular behavior of g ðμÞ at small scales in determines the applicability of possible extensions of relation to confinement. Different possibilities have been Standard Model that solve for open issues related to discussed in literature, see for instance Ref. [1] for a review. unification of forces, the hierarchy problem, and dark The so-called infrared slavery scenario has been proved not matter. to be strictly required for a realization of confinement. The running of the coupling with the scale μ is expressed Supersymmetric Yang-Mills theories (SYM) are a by the β-function, that can be computed order by order in remarkable exception, since many nonperturbative features of these theories are known analytically. Instanton calculations lead to the conclusion that there exists a scheme *[email protected] β † where the -function can be computed exactly to all-orders [email protected] in perturbation theory. A well-known example is the NSVZ β Published by the American Physical Society under the terms of -function [2] the Creative Commons Attribution 4.0 International license. μ 3 3 Further distribution of this work must maintain attribution to β μ − gð Þ Nc ’ ðgð ÞÞ ¼ 2 μ 2 ; ð2Þ the author(s) and the published article s title, journal citation, 16π 1 − Ncgð Þ and DOI. Funded by SCOAP3. 8π2

2470-0010=2018=97(7)=074510(16) 074510-1 Published by the American Physical Society GEORG BERGNER and STEFANO PIEMONTE PHYS. REV. D 97, 074510 (2018) that provides the running of the strong coupling of N ¼ 1 The final purpose is to determine a scale Λ from the SYM. This theory is the minimal supersymmetric extension running of the coupling at higher energies separated from of the pure gauge sector of QCD describing the interactions the possible conformal scale invariant behavior in the between gluons and their supersymmetric partners, the infrared but also separated from the ultraviolet cutoff gluinos, both transforming in the adjoint representation of effects. This implies that we have to check whether we the gauge group SUðNcÞ. The running coupling of the can connect the infrared and ultraviolet fixed points, i.e. the NSVZ β-function includes all nonperturbative contribu- nearly conformal and the perturbative running. We also tions due to the nonrenormalization theorem [3]. have to consider the possibility of unphysical regions where The number of fermions Nf has a negative contribution a backward running of the coupling might be occur. β to the first two coefficients of the -function b0 and b1, and The third goal is an identification of the onset of the there exists a critical number of fermions where b0 is conformal window for theories with Nf fermions in adjoint positive while b1 is negative [4,5]. In this case the running representation. The behavior of the running coupling μ of the coupling freezes at some scale and the correspond- indicates the appearance of a fixed point and provides ing gauge theory is infrared conformal. It has been hence a signal for the onset of the conformal window. The suggested that a near infrared conformal theory might be comparison of the running for different N hence allows in responsible for the breaking of electroweak symmetry f principle to identify the critical N for a transition from a [6,7]. The Higgs boson would be a composite bound state, f interpreted as a dilaton of the corresponding low energy confining to a conformal behavior. effective theory. Many recent lattice calculations have been This work is organized as follows. In the first part, we performed to understand how the conformal window is start in the continuum and present the considered scheme for the running coupling in Sec. II and the considered approached as the number of fermions Nf is increased, exploring the phase space not only with fundamental theories in Sec. III. In Sec. IV. the procedure of our quarks, but also with fermions in the adjoint and the sextet numerical determination of the running coupling on the representation of the gauge group. lattice is explained. In the last sections, we present and In this contribution, we explore the nonperturbative discuss our nonperturbative results for the running coupling dynamics of gluons and ghosts in the Landau gauge by in the different theories, starting with the extreme cases means of numerical lattice simulations for an SU(2) gauge N ¼ 1 SYM (Sec. V) and MWT (Sec. VI) before we theory coupled to one, two, three and four Majorana consider the intermediate theories in Secs. VII and VIII. fermions, corresponding to Nf ¼ 1=2, 3=2, and 2 Dirac fermions. We provide results for the running coupling in II. THE RUNNING COUPLING IN THE MOM these theories defined in the Mini-MOM scheme. We SCHEMES AND THE SOLUTIONS OF investigate how the ghost and gluon propagators and the DYSON-SCHWINGER EQUATIONS running coupling change when the number of fermions is increased starting from a confining theory, N ¼ 1 SYM, to In the continuum and in the Landau gauge, the form of the point where the conformal window is reached at gluon and ghost propagators, denoted in the following Nf ¼ 2. respectively as Gab;μνðpÞ and DabðpÞ, is constrained by There are several long-term goals of our investigations. Lorentz invariance. In particular, only two scalar functions The first one is to measure the running coupling of N ¼ 1 ZðpÞ and JðpÞ, called gluon and ghost dressing functions, SYM. Like in QCD, the running coupling and the scale Λ is are required to describe the deviations due to quantum an important ingredient for further investigations, for interactions from the free noninteracting propagators. The example, of renormalized currents and condensates. In renormalized gluon and ghost propagators have therefore addition it might help to understand the implications of the the following structure: exact results like (2) for this theory. 2 The second important goal is a general comparison of the pμpν Zðp Þ G ;μνðpÞ¼δ δμν − ; ð3Þ running coupling in the MOM scheme for a QCD-like ab ab p2 p2 theory and a theory in the conformal window in order to see how the signs for an infrared fixed point appear in this 2 setup. MWT is a prime example for such a comparison δ Jðp Þ DabðpÞ¼ ab 2 : ð4Þ since the indications for a conformal behavior of this theory p have been found in a large number of numerical investigations. So far this comparison of the running coupling in In the momentum (MOM) scheme, the renormalization the MOM scheme has only been done using Dyson- condition for the ghost and the gluon dressing function is Schwinger equations Ref. [8,9] for the case of Nf fermions set such that the gluon and the ghost propagators are equal in the fundamental representation. We want to confront the to their tree-level counterpart at the reference renormaliza- scenario for the appearance of the infrared fixed point tion scale μˆ 2 ¼ p2. In other words, the renormalization presented in these studies with our numerical results. condition is

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Zðμˆ 2Þ¼Jðμˆ 2Þ¼1: ð5Þ As the number of interacting fermionic degrees of freedom increases, the antiscreening of gluons still wins at high There are various possible alternative prescriptions for energies but it is not strong enough to overcome the the renormalization of the coupling in the MOM schemes, screening provided by virtual quark loops at smaller which can be defined from the three or four gluon vertexes energies. Therefore the running of the coupling freezes, [10–14], or from the quark-quark-gluon or ghost-ghost- and gluon and ghost propagators both diverge following a gluon vertexes. In principle each definition of the coupling power law in the infrared limit. These conclusions arise leads to a different running. The Landau gauge has the from the solutions of coupled DSEs. In [8,9] the onset of peculiar property that the renormalized coupling α at scale the conformal window for fermions in the fundamental μ2 ¼ p2 determined from the ghost-ghost-gluon vertex can representation has been investigated applying this Dyson- be defined just in terms of the renormalized ghost and Schwinger approach. The result was a remarkably small 4 5 gluon dressing function as critical Nf ¼ . for the onset of the conformal window. It was found that the dressing functions follow to a good αðp2Þ¼αðμˆ 2ÞZðp2ÞJðp2Þ2: ð6Þ approximation a polynomial form of type (7) with ρ ¼ 0.15, which implies a decreasing Zðp2Þ and increasing 2 The scheme is based on the UV-finiteness of the ghost- Jðp Þ towards the infrared. This has to be confronted with ghost-gluon vertex in the Landau gauge, accordingly to the an alternative scenario in which both dressing functions so-called Taylor theorem [15]. The main advantage of this become constant at the appearance of the infrared fixed scheme is that it requires the knowledge of only two-point point. We want to compare these scenarios with our functions, that can be reliably measured on the lattice with numerical results. Oð20Þ gauge-fixed configurations. A four-loop perturbative calculation of the running of The “Taylor coupling” defined by (6) has been deeply αðμÞ in the MiniMOM scheme has been presented for investigated in QCD using nonperturbative lattice simu- fermions in various representations including the adjoint lations and the Dyson-Schwinger approach [14,16–19]. one in Ref. [27]. The perturbative calculation will be used The Landau gauge turns out to be advantageous for a to compute the Λ-parameter of the adjoint QCD (AdjQCD) numerical solution of the Dyson-Schwinger equations with different numbers of fermions in the MOM scheme (DSEs). There has been a long discussion on whether and to verify whether our simulations are in a region where the so-called “scaling solution” or “decoupling solution” of the lattice spacing is small enough such that the coupling DSEs are effectively realized in confining non-Abelian at high energies follows the asymptotic scaling. We can gauge theories. The “scaling solution” predicts that the disentangle the running of the bare lattice coupling glat as a gluon and ghost dressing functions exhibit a powerlike function of the lattice spacing a from the running of the behavior in the infrared regime of the form strong coupling of (6) as a function of the momentum, such that a clear separation of lattice cut-off effects from the 2 2ρ 2 −ρ “ ” ZðpÞ ∼ ðp Þ JðpÞ ∼ ðp Þ ; ð7Þ physical continuum properties of the running coupling is possible. Similar methods have been widely employed in implying that the coupling runs to an infrared fixed point in previous lattice calculations of the running coupling for the limit p → 0 [17].The“decoupling solution” implies that many BSM theories in the Schrödinger functional or the – the gluon propagator has the form of Yukawa massive Wilson flow schemes [28 34], however we will not attempt particles and pure Yang-Mills theory would be “infrared to use step scaling at directly zero fermion mass but we will trivial”. Lattice investigations have the tendency to prefer the compute the coupling from simulations at nonzero fermion “decoupling solution”, although the infinite volume limit and mass produced by our collaboration to study the spectrum the Gribov noise might be a limitation that prevents the of bound states. observation of the “scaling solution” [20,21]. Quenched lattice simulations on boxes larger than ð10 fmÞ4 and various III. THE CONTINUUM ACTION OF ADJOINT QCD strategies to fix the Landau gauge have been considered to We study on the lattice gauge theories with fermions in address the issue, finding an agreement with the “decoupling the adjoint representation of the gauge group SU(2), which solution” [22]. A similar behavior has been seen in action reads in the continuum unquenched lattice simulations [23],aswellasinstudies Z 2 of the Schwinger-Dyson equations with the PT-BFM scheme 1 1 XNf 4 a a ¯i μ ab i [24,25] and discussed analytically in the context of the S d x FμνFμν λ γ Dμ λ : ¼ 4 ð Þþ2 a b ð8Þ Gribov-Zwanziger formalism [26]. i¼1 The interactions of gauge fields with matter fields λ changes nontrivially the behavior of ghost and gluon The field fulfills the Majorana condition propagators [8,9]. There are two competing effects, namely λ¯ λT the antiscreening of gluons and the screening of quarks. a ¼ a C; ð9Þ

074510-3 GEORG BERGNER and STEFANO PIEMONTE PHYS. REV. D 97, 074510 (2018) and the covariant derivative acts in the adjoint representa- coupling using step-scaling and the Schrödinger tion as functional method has provided evidences for the existence of an infrared fixed point of the ab c A ab Dμ λb ¼ ∂μλa þ igAμðTc Þ λb; ð10Þ β-function [34,40]. (iii) Nf ¼ 3=2 and Nf ¼ 1 adjoint QCD A where Tc are the Lie algebra generators as given by the Very little is known about the properties of structure constants. In the following we use the Dirac Nf ¼ 1 AdjQCD. It can be considered as the limit counting of fermionic degrees of freedom, meaning that for of N ¼ 2 SYM in which the scalars get an infinite 2 instance Nf ¼ corresponds to AdjQCD with two Dirac or mass and decouple. N ¼ 2 SYM is known to be four Majorana fermions. A sign problem can arise for odd confining, however the confinement properties number of fermions, but our simulations are performed in a could be spoiled if SUSY is broken. Some prelimi- range of fermion masses where the sign problem can be nary studies of Nf ¼ 1 AdjQCD on the lattice have neglected. been presented in Ref. [41], observing a scaling of The general properties of adjoint QCD depend strongly the bound spectrum compatible with a conformal or on the number of fermions. There are three relevant near conformal theory with a large mass anomalous possibilities: dimension γ ≈ 1. The large mass anomalous di- 1 2 (i) Nf ¼ = adjoint QCD mension would be useful for particle phenomenol- N ¼ 1 supersymmetric Yang-Mills theory has ogy, however the breaking patter would not provide only a single Majorana fermion, the gluino, inter- the required Goldstone bosons for the breaking of acting with the gluon. The gluon and the gluino field electroweak symmetry, and further fermion flavors are related by a supersymmetriy transformation, in the fundamental representation must be included. which assumes on-shell the form To the best of our knowledge, there has been no previous lattice investigations of AdjQCD with → − 2 λγ¯ ϵ AμðxÞ Aμ i μ ; ð11Þ three Majorana fermions. The theory is predicted μν to be just close to the conformal window. The main λa → λa − σμνFa ϵ; ð12Þ interest for Nf ¼ 3=2 is therefore to understand how where σμν is proportional to the commutator ½γμ; γν the conformal properties of adjoint theories depends on the number of flavors. and Fμν is the field strength. The infinitesimal SUSY transformation is parametrized by a global Majorana spinor ϵ. Lattice calculations have shown that the IV. GAUGE FIXING AND CORRELATORS IN theory is confining, in the sense that there is a MOMENTUM SPACE ON THE LATTICE linearly rising of the potential between two static Our numerical simulations are done on a four- fundamental quarks which defines a string tension σ. dimensional lattice of ðLx;Ly;Lz;LtÞ points in each The bound state spectrum is organized in super- direction. We take the same lattice size Ls for the spacial multiplets of particles with equal masses, and directions and a larger temporal extend Lt. The bare gauge recently Monte Carlo simulations have been able coupling of the lattice theory is glat and the inverse bare 2 to show that supersymmetry is intact also at non- β Nc coupling is defined as lat ¼ g2 . The bare fermion mass perturbative level and restored on the lattice in the lat → 0 m is fixed by the Hopping parameter κ ¼ 1 . continuum limit a [35]. lat 4mlatþ8 (ii) Nf ¼ 2 adjoint QCD On the lattice, the Landau gauge is fixed by maximizing Nf ¼ 2 AdjQCD is also known as minimal the functional walking technicolor [36]. Predicted to be near the X † conformal window, the theory would be a good ΠðfUgÞ ¼ ReðTrðΩðxÞUμðxÞΩ ðx þ μÞÞÞ; ð13Þ candidate for electroweak symmetry breaking to the x;μ Standard Model without the Higgs field, while satisfying at the same time the experimental con- with respect to the gauge transformation ΩðxÞ. The gluon straint on the flavor-changing neutral currents. There fields AμðxÞ are related to the traceless-antihermitian (TA) have been several studies of the properties of the part of the gauge links UμðxÞ by the definition model, a conformal or near conformal behavior has been observed in particular from the scaling of the 1 – 0þþ AμðxÞ¼ UμðxÞ : ð14Þ masses of bound states [37 39]. The glueball iglat TA has been found to be the lightest bound state, however the observed mass anomalous dimension We employ a sequence of standard over-relaxation updates γ ≈ 0.3 would be too small for realistic particle to find the maximal of the functional ΠðfUgÞ [42], together phenomenology. The determination of the running with parallel tempering. We stop when the condition

074510-4 RUNNING COUPLING FROM GLUON AND GHOST … PHYS. REV. D 97, 074510 (2018) X 2 −14 ðΔμAμÞ < 10 ð15Þ symmetric momenta are chosen. There are several different x;μ criteria to cut the on-axis momenta. A possibility is to perform a cylindric cut to the momenta we measure, pffiffiffiffiffiffiffiffiffiffi is satisfied. Many different ΩðxÞ configurations will satisfy excluding directions where for instance ðp · d= p · pÞ < the above condition on the lattice and there has been several 0.95, with d ¼f1=2; 1=2; 1=2; 1=2g. We choose instead to studies to understand the impact of this uncertainty. Some constrain the maximal absolute value of the deviation from small effect induced by the Gribov noise has been dis- the diagonal direction, defined as covered in the deep infrared region of QCD for large volumes in Ref. [43], however the full interpretation of the X 1 X ˆ − ˆ 0 55 gauge fixing effects on the lattice has been debated, see pμ 4 pν < . : ð20Þ μ ν also Ref. [44]. Once the gauge-fixed links are known, the gluon Compared to the cylindric cut, the condition above allows propagator in the momentum space can be easily computed more off-diagonal directions in the low momentum region, from the Fourier transform of the gluon field in the real that are still interesting to understand the general behavior space. The computation of the ghost propagator requires an at low energy of gluon and ghost propagators. Field inversion of the Faddeev-Popov operator for each momen- propagators at low momenta are usually affected more tum of the ghost field. We solve the corresponding linear from finite volume effects rather than from lattice artifacts. system using the BiCGStab biconjugate gradient algorithm. 2 2 Finally, we exclude the lattice modes with a p larger than The definition of the Faddeev-Popov operator on the lattice four, to avoid ultraviolet regions affected by strong lattice is discussed in detail in Ref. [43]. artifacts. We compute the gluon and ghost propagators for each The running of the strong coupling is given by the ensemble on Oð20Þ gauge-fixed configurations well sep- β-function according to the relation arated by at least fifty molecular dynamics units. As described in [14,16,17], the running coupling can be b1 2 2 2 μ b0gðμÞ 2b 16π computed directly just in terms of the lattice bare ghost 0 ¼ 2 exp 2 Λ 16π 2b0gðμÞ and gluon dressing function in the MiniMOM scheme as Z μ 2 gð Þ 1 16π b1 2 − 0 2 g 2 2 2 × exp 0 þ 03 2 0 dg : ð21Þ α lat 0 βðg Þ b0g b g ðp Þ¼4π Z0ðp ÞJ0ðp Þ : ð16Þ 0 The parameter Λ appears as an integration constant of The running scale is defined by the lattice momentum and the differential of (1) and it depends on the scheme. The the renomalization scale is set at the inverse lattice spacing. nonperturbative estimation of 4πα μ g μ 2 allows to There are several possible equivalent definitions of the ð Þ¼ ð Þ determine the Λ-parameter, assuming a sufficiently high modulus squared of the lattice momentum pˆ energy where the running given by the perturbative β-function holds. The so-called “window problem” is n ny n n pˆ ¼ 2π x ; ; z ; t ;n∈ ð−L =2;L=2: ð17Þ related to the difficulties of finding in current lattice L L L L i i i x y z t calculations such an high energy region where μ is at the same time still far away from the cut-off 1=a where In the following we always plot scales of lattice momenta lattice artifacts are dominant. defined as The fit to perturbation theory is performed including the X3 2 functional form of the perturbative lattice correction 2 2 2 api [45,46] a p ¼ sin 2 : ð18Þ i¼0 α 2 X X ðapμ;p Þ k2 4 6 ≡ K 1 k1 pμ k3 pμ: Lorentz invariance is broken on the lattice and therefore pert 2 lat ¼ þ þ 2 þ α ðp Þ p μ μ results coming from momenta with the same a2μ2 but different lattice directions will in general differ. The leading ð22Þ terms that violate Lorentz invariance are proportional to X The correction Klat on the right hand side takes into account 4 ðapμÞ ; ð19Þ the breaking of Lorentz symmetry on the lattice. This 2 μ functional form stabilizes and reduces the χ =d:o:f: of the fit. The constants kn are determined from a global fit of and it can be easily proven that for a given μ2, the above the running coupling and are equal for all lattice momenta. expression is minimized if the momentum is equally The inclusion of more terms in the equation above does not distributed on all four components, i.e. if the most improve significantly the goodness of the fits. We check the

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TABLE I. Summary table of the analyzed ensembles. In the leaves the weak coupling regime. In addition, the inter- 1 2 case of Nf ¼ = , the PCAC mass is not measured and the pion action of gauge fields with Wilson fermions will introduce mass is defined in a partially quenched approach [48]. The lattice 2 lattice artifacts of the order of OðaÞ or OðglataÞ in the action consists of a tree-level Symanzik improved gauge action extrapolation to the continuum limit of the renormalized and Wilson fermions in the adjoint representation. The simu- gluon and ghost propagators. lations of Nf ¼ 1=2 are with one level, at Nf ¼ 2 and 3=2 with three level of stout smearing. The simulations of Nf ¼ 1 are done with tree-level clover improvement. V. N = 1 SUPERSYMMETRIC YANG-MILLS THEORY β κ 3 Nf lat Ls × Lt amPCAC amπ The study of N ¼ 1 SYM has been the subject of a long 1=2 323 64 1.75 0.1492 × 0.20275(74) project of the DESY-Münster Collaboration. The early 1=2 1.9 0.14387 323 × 64 0.21410(33) simulations at β ¼ 1.6 failed to observe the formation 1=2 1.9 0.14415 323 × 64 0.17520(22) lat of supermultiplets and the expected degeneracy in the 1=2 1.9 0.14435 323 × 64 0.14129(59) 3 particle spectrum due to a too coarse lattice spacing [49]. 1 1.75 0.1650 16 × 32 1 1.75 0.1650 243 × 48 0.062837(79) 0.4648(12) Restoration of supersymmetry on the lattice has been 1 1.75 0.1660 243 48 0.03567(11) 0.3313(11) recently reported in the continuum limit including results × β 1 75 β 1 9 3=2 1.5 0.1330 243 × 48 0.19515(20) 0.86625(73) at lat ¼ . and lat ¼ . [35,50,51]. Therefore we are 3=2 1.5 0.1340 243 × 48 0.15632(15) 0.74286(62) going to focus the analysis of the gluon and ghost 3=2 1.5 0.1351 243 × 48 0.10986(12) 0.58219(99) propagators only on these last two sets of ensembles, 3 β 1 6 3=2 1.6 0.1300 24 × 48 0.19161(26) 0.7966(15) excluding the lattices generated at lat ¼ . . 3=2 1.7 0.1285 323 × 64 0.173655(41) 0.69268(25) The bare gluon and the ghost dressing functions are 3 β 1 9 3=2 1.7 0.1300 32 × 64 0.129098(37) 0.55712(19) shown at fixed lat ¼ . in Figs. 1(a) and 1(b). There is no 3=2 1.7 0.1320 323 × 64 0.06635(12) 0.3312(20) dependence of JðapÞ and ZðapÞ on the gluino mass up to 2 1.5 0.1325 323 × 64 0.128840(55) 0.58848(98) our statistical accuracy. The renormalized gluon and ghost 2 1.5 0.1335 323 × 64 0.089619(74) 0.44212(28) dressing functions are shown in Figs. 1(c) and 1(d). The 2 1.5 0.1350 323 × 64 0.030414(45) 0.17063(65) renormalization condition (5) is set at μ ¼ 5.1 GeV in 2 1.7 0.1275 323 × 64 0.17697(22) 0.66093(22) QCD units, which means setting at that scale bare quantities 2 1.7 0.1285 323 × 64 0.147091(22) 0.57247(16) equal to their tree-level counterpart, i.e. equal to one in this 2 1.7 0.1300 323 64 0.100878(47) 0.42116(32) β 1 9 5 858 84 β × case. We use w0ð lat ¼ . Þ=a ¼ . ð Þ, w0ð lat ¼ 243 64 2 1.7 0.1300 × 0.101728(79) 0.4039(86) 1.75Þ=a ¼ 3.411ð18Þ and w0 ¼ 0.226 fm to set the scale 3 2 2.25 0.1250 32 × 64 [52]. The conversion of the momenta in physical units is 2 2.25 0.1275 323 × 64 β effectively a different rescaling of the x-axis for each lat, while the renormalization condition requires effectively a rescaling of the y-axis. Our results coming from the two consistency of the estimated Λ-parameter by performing different bare lattice couplings fall onto a unique curve, the fit to pertubation theory without lattice corrections, proving the multiplicative renormalization of ZðpÞ and of considering only the lattice momenta in strictly diagonal JðpÞ. In particular, ZðpÞ has a downward tendency at the μ β 1 75 directions. Alternatively, having measured gluon and ghost smallest accessible with our volumes at lat ¼ . . The propagators in more lattice directions, we could have ghost propagator appears instead to be divergent in the deep extrapolated the effects of Lorentz symmetry breaking infrared, and the ghost dressing function has an upward terms as in Ref. [47], however the computation cost tendency in the same regime. required to invert the Faddeev-Popov operator would have The running coupling defined in (16) is shown in Fig. 2. been more demanding. In addition, in the MiniMOM It decreases at high energy as predicted by asymptotic scheme, the renormalization constant of the ghost-ghost- freedom, but, as observed in QCD, at very small μ the gluon vertex is set to one, therefore we must also fit the upward tendency of the ghost dressing function is not absolute normalization of αðμÞ together with Λ. sufficient to compensate the downward tendency of the We employ in all our simulations for all theories the tree- gluon dressing function, so that the running coupling αðμÞ level Symanzik improved gauge action and stout-smeared peaks in the infrared at around ap ≃ 0.2, corresponding to μ ≈ 1 Wilson fermions, except for the Nf ¼ 1 AdjQCD where we GeV in QCD units. More extensive simulations at use tree-level clover improved Wilson fermions. The larger volumes are required to address the issue of the N 1 simulations of Nf ¼ 1=2 are with one level, at Nf ¼ 2 running coupling of the ¼ SYM in the deep infrared. and 3=2 with three level of stout smearing. The summary of According to the methods presented in Sec. IV we obtain the ensembles analyzed is presented in Table I. While free for the scale Λ the values gluon propagators have already an Oða4Þ improved form, 2 2 ΛMiniMOM β 1 75 0 17 3 a ð lat ¼ . Þ¼ . ð Þ; ð23Þ there are lattice artifacts of the order Oðglata Þ appear if one

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(a) (a)

(b)

FIG. 2. (a, b) The running of the coupling αðμÞ for the N ¼ 1 SYM theory. There is only a very weak dependence of αðμÞ on the gluino mass.

lightest particle. When extrapolated to the chiral limit the β lightest mass in lattice units is around 0.228(71) at lat ¼ (c) 1 7 β 1 9 . and around 0.174(14) at lat ¼ . . The dimensionless Λ-parameter in the MiniMOM scheme in w0 units,

ΛMiniMOM β 1 75 0 59 10 w0 ð lat ¼ . Þ¼ . ð Þ; ð25Þ

ΛMiniMOM β 1 9 0 70 9 w0 ð lat ¼ . Þ¼ . ð Þ; ð26Þ

are compatible within the errors for the two different bare gauge couplings we have analyzed. The knowledge of the Λ-parameter in the MS-scheme is a (d) crucial starting ingredient for the renormalization of energy- momentum tensor and of the supercurrents of N ¼ 1 FIG. 1. (a, b) Bare gluon and ghost dressing functions for N ¼ 1 SYM as a function of the lattice momentum ap at β 1.9. SYM using the Wilson Flow following the approach of lat ¼ – β 1 9 (c, d) Renormalized gluon and ghost dressing functions for Ref. [53 55]. Using the result at lat ¼ . and the con- N ¼ 1 SYM on a logarithmic scale of the physical momentum version factors coming from the perturbative expansion of p. The gluon dressing function tends to zero for p → 0. the ghost-ghost-gluon vertex of Ref. [14,16,56], we can also compute the Λ-parameter in the MS-scheme, which is ΛMiniMOM β 1 9 0 120 15 a ð lat ¼ . Þ¼ . ð Þ; ð24Þ ΛMS β 1 9 0 385 50 w0 ð lat ¼ . Þ¼ . ð Þ: ð27Þ where the quoted errors are purely systematic, coming from MS various choices of the fitting ranges (see Fig. 3). These In QCD units, we obtain Λ ¼ 336ð44Þ MeV, a value quite scales are of the same order of magnitude as the mass of the similar to the recent determinations of ΛMS in full QCD [57],

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β 1 5 β 1 7 lat ¼ . and lat ¼ . . An additional set of simulations at β 2 2 lat ¼ . is also analyzed, but these ensembles should be considered with special care since the Polyakov loop in spacial directions shows already indications for a transition to the deconfined phase at these parameters. The detailed analysis about the simulations, the properties of the bound spectrum and the scaling of the mode number of the Dirac operator has been published in Ref. [39]. A previous investigation of the behavior of gluon and ghost dressing function has been presented in Ref. [58], finding some (a) evidence of a possible conformal behavior of the Taylor coupling. We have now the possibility to extend the analysis to larger volumes and to ensembles closer to the chiral limit. The gluon and the ghost dressing functions have clearly a flatter behavior than in N ¼ 1 SYM, especially at large momentum, see Fig. 4. In particular, the gluon dressing function develops a plateau as the fermion mass is β 1 5 decreased. At lat ¼ . there is even a critical mass at which the gluon dressing function turns from a decreasing to an increasing function of the momenta, see Fig. 4(a). The ghost dressing function has instead always a decreas- (b) ing behavior. The physical consequence is that gluon and ghost propagators are becoming both divergent in the infrared limit as the fermion mass is decreased, at least up to the region of momenta explored in our simulations. This is consistent with the DSE data for a conformal theory [8,9]. The running coupling is presented for our three different bare lattice couplings in Fig. 5. In contrast to N ¼ 1 SYM, we observe a clear mass dependence of the running β 1 5 β 1 7 coupling at lat ¼ . and lat ¼ . . At the smallest fermion mass we have simulated (κ ¼ 0.1350 and β 1 5 lat ¼ . ), the running coupling develops a large plateau, (c) which is the result of a cancellation between the upward tendency of the gluon dressing function against the FIG. 3. (a) The running of the coupling αðμÞ for N ¼ 1 SYM. The running can be related to four-loop perturbation theory at decreasing behavior of the ghost dressing function. The high energy. (b) The ratio of the lattice results over the plateau extends to the full scaling region at our largest β 2 25 α μ μ perturbative prediction, including the lattice correction of lat ¼ . , where ð Þ has a very weak dependence on . Eq. (22), is constant (blue points). Consequently there is a fairly Besides this plateau, there is a considerable running of large window to where perturbation theory can be trusted. The fit coupling the far infrared region at β ¼ 1.5 and 1.7, which is is performed in the window delimited by the green vertical lines. decreased for smaller fermion masses. This far infrared part The orange points show the ratio of the lattice strong coupling is also subject to considerable finite volume effects at scales to the perturbative prediction from the same fit, but excluding smaller than aμ ∼ 0.4, see Fig. 5(d). from the ratio the fitted correction of Eq. (22). (c) Lattice We don’t see a significant running in the ultraviolet part. corrections to the perturbative runningP Klat including (blue) 6 Due to the flat behavior of the strong coupling, the fit to and excluding (orange) the term pμ from the fit. μ perturbation theory is very unstable and it is impossible to estimate the ultraviolet scale Λ. although the agreement depends on the scale chosen to relate The general behavior with a nearly zero running of the the two theories and could be accidental. coupling over a large energy scale is consistent with the expectations for an infrared fixed point. The fermion mass is a relevant direction at the fixed point and is responsible VI. Nf = 2 ADJOINT QCD for the running in a region that moves further to the infrared Our main ensembles for Nf ¼ 2 AdjQCD have been the smaller the fermion mass is. Only the gluon dressing generated at two different bare lattice gauge couplings function at the smallest fermion masses of β ¼ 1.5 shows a

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(a) (a)

(b) (b)

(d) (d) FIG. 5. Strong coupling constant for N 2 AdjQCD as a 2 f ¼ FIG. 4. Bare gluon and ghost dressing functions for Nf ¼ function of the renormalization scale aμ at different β . The β 1 5 lat AdjQCD as a function of the lattice momentum ap at lat ¼ . results show a clear fermion mass dependence and the strong β 1 7 and lat ¼ . . coupling effectively freezes toward the massless limit. decreasing behavior in the infrared as expected from the scaling form (7) with a positive ρ. From a fit we obtain a couplings above the fixed point value and hence in a region value of around ρ ∼ 0.08 which is considerably smaller not connected to the ultraviolet fixed point. than the value obtained in the DSEs approach for funda- It is worth to investigate in more detail the dependence of mental fermions. However, all of our other results are also the running coupling on the fermion mass and the devia- quite consistent with the alternative scenario where both, tions form α being a constant. The coupling reaches its ghost and gluon dressing functions become almost constant value at zero mass up to a correction that depends on the in the plateau region of the effective coupling. scale μ as can be clearly seen in Fig. 6. In Fig. 6(a), Unfortunately we are not able to connect the running to couplings at different scales collapse to a unique point up to ≃ 0 03 the perturbative one since up to the highest energies we are our statistical accuracy at amPCAC . , and even at 0 14 α μ able to explore the running seems to be dominated by the fermion masses as large as mPCAC ¼ . , many ð Þ are influence of the fixed point. There are two explanations, still compatible within the errors for sufficient large μ. either our energy scales are still too low, or we are at Hence up to our statistical accuracy the results are

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running of the coupling is expected. As shown in [61], this might arise from extrapolation of a forward running at larger fermion masses towards the massless limit. This implies that at a finite mass the extrapolations should intersect implying a zero running of the coupling. The general form of our results are indeed consistent with the (a) one obtained from a step-scaling methods in Ref. [61],but as pointed out in this reference the uncertainties of the extrapolations do not allow a discrimination of the backward running from a zero running in the massless limit. The alternative interpretation is that the running goes to zero only in the massless limit as dictated by hyperscaling. An infrared conformal theory does not possess any dimen- sionful scale related to long-range physics. In the chiral (b) limit, all masses of bound states, including decay constants FIG. 6. Mass dependence of the running coupling for N ¼ 2 and condensates, extrapolate to zero according to a uni- f versal scaling AdjQCD as a function of the bare amPCAC at different aμ. The data shown come from the ensembles at βlat ¼ 1.5 and βlat ¼ 1.7, 1 (there is a tiny horizontal offset for each μ to make the individual M ∝ m1þγ ; ð28Þ points visible). All the couplings collapse to unique point at ≃ 0 03 κ 0 1350 amPCAC . , corresponding to ¼ . [green points of Fig. 5(a)]. At the smallest κ ¼ 0.1325, the red and cyan points are where γ is the mass anomalous dimension of the theory at still compatible within the errors (ακ¼0.1325ð1.333Þ¼0.305ð13Þ the infrared fixed point [62]. Our previous investigations and ακ¼0.1325ð1.447Þ¼0.313ð12Þ), meaning that the (near-)zero confirmed the validity of the scaling relation of the mass β of the lat-function can survive even for nonzero fermion masses. spectrum for the same ensembles, see Ref. [39].Asa consequence, the momenta cannot be converted to physical units by choosing a common scale for all fermion consistent with a zero of the β-function in the infrared region masses in a confining theory. However, the almost linear at small enough fermion masses. These observations support behavior of α μ in Fig. 6 suggests that a possible the evidences from previous lattice investigations that the ð Þ universal function can be found for large enough momenta N ¼ 2 AdjQCD theory is infrared conformal [34,40]. f if the lattice scale aμ is measured in units of the fermion The fact that the fermion mass affects rather the far mass m as infrared part is quite expected and can be understood by PCAC comparison with perturbative QCD. The calculation of the μ first two coefficients of the β-function in the MOM scheme aμ → 1 ; ð29Þ 1 γ as a function of the quark mass is known from Ref. [59]. ðmPCACÞ þ The main result is that b0 and b1 converge quite rapidly μ2 2 γ γ ≈ 0 3 for >mf to their massless limit. Therefore, for small with equal to . [39]. This rescaling is equivalent enough fermion masses and higher energy scales, we to a conversion of the lattice spacing a in units of the expect the running of α to be as slow as the running at fermion mass with its appropriate dimension, employing zero fermion mass. A further study of massive schemes for a “mass-dependent” scale-setting scheme. The result of near-conformal theories has been presented in Ref. [60] in this momentum rescaling is presented in Fig. 7. All points the context of NSZV-inspired β-functions. collapse to a unique curve except those at very small There are two possible interpretations of the form of the momenta, affected by finite volume effects and by the fermion mass dependence, which are both consistent with typical peak structure of α in momentum schemes. The our data. The first one is to assume that there is a real and scaling observed for the running coupling as a function of exact zero of the beta function already at a finite fermion the fermion mass is also in agreement with the hypothesis β 1 5 2 mass. Our simulations at lat ¼ . have been done at a that Nf ¼ AdjQCD is an infrared conformal theory: in β rather strong coupling, since this value of lat is just above the infrared regime the coupling runs only for non- the bulk transition in our lattice setup. In the infrared vanishing values of the fermion mass, which break the conformal scenario, the strong coupling region above the conformal behavior explicitly, in a form that must scale as infrared fixed point is not connected to ultraviolet fixed (28) close to the infrared fixed point. The main con- point. Hence weak couplings below the IR fixed point are clusion of the investigations of MWT is that all our not reached when starting from the strong coupling side. ensembles are in a regime that is dominated by the Since the coupling has to run towards the lower value at the infrared fixed point with the mass as the only relevant infrared fixed point, depending on the scheme, a backward direction.

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(a)

(b)

FIG. 7. “Hyperscaling” of the running coupling for Nf ¼ 2 AdjQCD at βlat ¼ 1.5 and βlat ¼ 1.7 as a function of −1=ð1þγÞ −0.77 μðmPCACÞ ≈ μðmPCACÞ . The strong coupling at different fermion masses appears to collapse toward a universal μ function. Some deviation is observed for small at the smallest (c) fermion mass, probably due to finite volume effects.

FIG. 8. Bare gluon and ghost dressing functions for Nf ¼ 3=2 β 1 5 AdjQCD as a function of the lattice momentum ap at lat ¼ . . VII. Nf =3=2 ADJOINT QCD We have generated ensembles at three different β lat momentum region where the effects of the nonvanishing for N ¼ 3=2 AdjQCD. The target of our analysis is to f fermion masses can be neglected. check whether the running coupling is similar to N ¼ 2 f In contrast to the N 2 case, the running coupling can N 1 f ¼ AdjQCD or it is rather close to ¼ SYM, in order to be fitted to perturbation to check a possible relation to the understand how the transition from a confining to a continuum theory. The best fit of αðμÞ to perturbation conformal theory occurs when the number of interacting 2 theory with a reasonable χ and a wide range of μ comes fermions is increased. from our data at β ¼ 1.7 and κ ¼ 0.1320, due to the fact For the two ensembles simulated with the largest lat that the other ensembles have too large fermion masses that am at β 1.5 the gluon dressing function has a PCAC lat ¼ are not negligible even at quite large μ [see Figs. 10(a), downward tendency in the deep infrared region similar to 10(b), and 10(c)]. Following the same procedure as for the one observed in pure gauge SU(2), see Fig. 8(a). The N ¼ 1 SYM, we get dressing function ZðapÞ becomes monotonous in the whole considered range of momenta only at the smallest fermion ΛMiniMOM β 1 5 0 0051 8 mass, see Fig. 8(b). The ghost dressing function JðapÞ a ð lat ¼ . Þ¼ . ð Þ; ð30Þ shows instead a mild dependence both on ap and on the ΛMiniMOM β 1 6 0 0055 4 fermion mass, see Fig. 8(c). a ð lat ¼ . Þ¼ . ð Þ; ð31Þ The running coupling has a downward tendency in the ΛMiniMOM β 1 7 0 0054 5 deep infrared region for our smallest μ [in particular, at a ð lat ¼ . Þ¼ . ð Þ: ð32Þ β 1 5 β 1 6 lat ¼ . and lat ¼ . , see Figs. 9(a) and 9(b)], driven by the downward behavior of the gluon dressing function. All quoted errors are purely systematic corresponding to As for Nf ¼ 2 AdjQCD, we observe a clear dependence of variations coming from different choices of the fitting the running coupling on the fermion mass. The running of α intervals. The value of Λ in lattice units is significantly is slow but still incompatible with zero, at least in the high smaller than the masses of hadrons in lattice units at the same

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FIG. 9. The running of the running coupling αðμÞ on a logarithmic scale μ for Nf ¼ 3=2 AdjQCD. Many ensembles α μ β show that a rising tendency of αðμÞ that peaks in the infrared FIG. 10. Fit of ð Þ to perturbation theory at various lat with region. the largest κ available plotted on a linear scale μ. The fitting intervals are (a) (1.3,2.05), (b) (1.3,2.05), (c) (0.9,2.15).

þþ parameters, for instance the glueball 0 mass am0þþ ≃ VIII. Nf = 1 ADJOINT QCD 0.25ð3Þ. The contrast to the well-known confining gauge theories is quite significant, in QCD the value of the intrinsic Since moving from four to three Majorana fermions ultraviolet scale Λ is comparable to the hadron scales in the coupled to SU(2) gluodynamics has been sufficient for a various MOM schemes. This is in accordance with the significant change in the running of α, it is interesting to observation that the fermion mass sets the scale of the mass check what happens to Nf ¼ 1 AdjQCD. We have gen- β 1 75 spectrum of this theory approximately according to (29).It erated a set of ensembles at lat ¼ . and several values is, furthermore, remarkable how weak the dependence of aΛ of κ. The largest statistics has been collected from two on the bare lattice coupling is. Even though these observa- ensembles at κ ¼ 0.1650 and κ ¼ 0.1660 with a lattice size tions are not enough to provide a strong evidence for a of 243 × 48. The ghost and the gluon dressing functions 3 2 conformal behavior as in the Nf ¼ 2 case, they are clearly have a similar tendency as observed for Nf ¼ = AdjQCD much different from QCD-like theories. [see Figs. 11(a) and 11(b)], with a peak of ZðapÞ in the

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FIG. 12. (a) The running of the running coupling αðμÞ on a FIG. 11. Bare gluon and ghost dressing functions for Nf ¼ 1 logarithmic scale μ for the N ¼ 1 AdjQCD theory. (b) Com- AdjQCD as a function of the lattice momentum ap at βlat ¼ 1.75. f parison of finite volume effects at κ ¼ 0.1650. infrared and a mild dependence of the dressing functions on the fermion mass. The running coupling is shown in Fig. 12. Finite volume AdjQCD might eventually develop an infrared fixed point effects seems to be under control, see Fig. 12(b). There is a is very small and beyond the reach of our current volumes dependence of the strong coupling on the fermion mass and fermion masses. Further simulations at large volumes at small μ, while at larger energy scales the running and smaller fermion masses would be required to reach a approaches well the predicted perturbative β-function of full definitive conclusion about its existence. the massless theory. It is therefore interesting to compare the fits of α to two and four loop perturbation theory. As before, we fit for the four loop running the ultraviolet scale and absolute normalization of α to the data of the smallest fermion mass (κ ¼ 0.1660) in a region with aμ > 0.5 in order to ensure negligible fermion mass effects. The scale Λ in lattice units is

ΛMiniMOM β 1 7 0 20 3 a ð lat ¼ . Þ¼ . ð Þ; ð33Þ the quoted error is again systematic. The Λ-parameter has developed a quite large value with respect to the pion mass compared to the case of N ¼ 3=2.AsforN ¼ 1 SYM, it f FIG. 13. Comparison of the fit of αðμÞ to two and four loops is of the same order of magnitude as the mass gap. In perturbation theory plotted on a linear scale μ. The fitting interval addition, it is worth to note that also two-loop perturbation is (0.47,2.2). The fit to two-loop perturbation theory might appear theory might fit well our data at sufficiently large μ,see better, but the agreement at low aμ is most likely accidental, given Fig. 13. The main conclusion from these investigations is that for aμ < 0.3 the effects of the fermion mass on αðμÞ are not that the scale where the strong coupling of Nf ¼ 1 negligible, see Fig. 12(a).

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IX. CONCLUSIONS At our smallest PCAC mass, the gluon dressing functions of N 2 AdjQCD shows a similar functional form We have presented a lattice study of the gluodynamics f ¼ in the Landau gauge for AdjQCD and observed that the as the one expected from the DSE approach. Within the properties of gluon propagators depend crucially on the range of momenta that we can reliably explore in our number of adjoint fermions. We have determined the running simulations, the gluon propagator diverges in the infrared coupling in the MiniMOM scheme and investigated how it regime, consistent with the disappearance of the plateau changes when going from the QCD-like case to a theory in of the propagator determined from the DSEs. Hence the the conformal window. general picture appears to be consistent with the DSE results for a conformal theory with fermions in the For Nf ¼ 1=2,i.e.N ¼ 1 SYM, the gluon propagator has, similar to QCD, a nontrivial infrared behavior that does fundamental representation. not show a significant dependence on the fermion mass. The study of the running of α for an infrared conformal The gluon dressing function develops a stronger depend- theory requires special considerations concerning the tun- ence on the fermion mass the more fermions are coupled to ing of the lattice parameters, the continuum and chiral the gluons. The behavior of the ghost dressing functions extrapolations, and the fit to pertubation theory. Our results 2 appears to be less dependent on the number of fermions, for Nf ¼ AdjQCD do not show a significant dependence β although a stronger flattening is observed towards the on the bare gauge coupling lat. However, the investigated conformal window. range of bare couplings should be enlarged to understand The running coupling computed from the gluon and how the ultraviolet fixed point is approached and to clearly ghost dressing functions provides a clear evidence for the separate possible unphysical regions of bare lattice param- asymptotic freedom of adjoint QCD with Nf ¼ 1=2 and eters beyond the infrared fixed point, where a flat or even a Nf ¼ 1 fermions. For Nf ¼ 1 AdjQCD, there could only backward running could be expected [61]. be evidence for an infrared fixed point at very small scales compared to the one that we can reliably explore with ACKNOWLEDGMENTS our simulations. A very slow running of α is observed for Nf ¼ 3=2 AdjQCD. Consequently, this theory appears to We thank Gunnar S. Bali, Jacques C. R. Bloch, Biagio be close to or inside the conformal window. Lucini, Holger Gies, and Meinulf Göckeler for interesting We have estimated the Λ-parameter in the MiniMOM discussions. We thank Gernot Münster and Pietro Giudice and in the MS scheme for N ¼ 1 SYM. This result is for reading and commenting on the first version of the crucial for the renormalization of the supercurrents using manuscript. We thank Istvan Montvay for his support in the methods proposed in Ref. [53–55]. We have determined performing the numerical simulations. The authors grate- the Λ-parameter in the MiniMOM scheme also for Nf ¼ 1 fully acknowledge the Gauss Centre for Supercomputing e. V. (GCS) for providing computing time for a GCS Large- and Nf ¼ 3=2 AdjQCD, while the β-function of Nf ¼ 2 AdjQCD is strongly affected by the nonvanishing value of Scale Project on the GCS share of the supercomputer the fermion mass at the considered range of the scale μ. The JUQUEEN at Jülich Supercomputing Centre (JSC) and on the supercomputer SuperMUC at Leibniz Computing determined value of Λ for Nf ¼ 3=2 AdjQCD is several orders of magnitude smaller than the mass gap. Centre (LRZ). GCS is the alliance of the three national supercomputing centres HLRS (Universität Stuttgart), The results for Nf ¼ 2 AdjQCD, i.e. MWT, are com- pletely different from the QCD case. Consistent with the JSC (Forschungszentrum Jülich), and LRZ (Bayerische expectations for a conformal theory, one observes a plateau Akademie der Wissenschaften), funded by the German region with an almost constant αðμÞ and a strong fermion Federal Ministry of Education and Research (BMBF) mass dependence in the far infrared. At the considered and the German State Ministries for Research of Baden- energy range the fermion mass seems to be the only Württemberg (MWK), Bayern (StMWFK) and Nordrhein- relevant parameter that determines the running of αðμÞ. Westfalen (MIWF). Further computing time has been This is consistent with the fact that the mass is the only provided by the iDataCool cluster of the Institute for relevant direction at the fixed point. The specific form of Theoretical Physics at the University of Regensburg. the fermion mass dependence can be either interpreted as S. P. acknowledges support from the Deutsche Forschungs- an indication for backward running at zero fermion mass or gemeinschaft (DFG) Grant No. SFB/TRR 55, and G. B. in terms of hyperscaling. received support from Grant No. BE 5942/2-1.

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