Explorations beyond dilaton chiral perturbation theory in the eight-flavor SU(3) gauge theory
EExplorations beyond dilaton chiral perturbation theoryin the eight-flavor SU(3) gauge theory
Maarten Golterman a and Yigal Shamir ba Department of Physics and Astronomy, San Francisco State University,San Francisco, CA 94132, USA b Raymond and Beverly Sackler School of Physics and Astronomy,Tel Aviv University, 69978, Tel Aviv, IsraelWe continue our study of spectroscopy data for the SU(3) gauge theory witheight fundamental fermions, motivated by the effective field theory framework ofdilaton chiral perturbation theory (dChPT). At leading order dChPT predicts aconstant mass anomalous dimension γ m , consistent with the assumed proximityof an infrared fixed point. For the relatively large fermion masses simulatedby the LatKMI collaboration, the influence of the infrared fixed point dimin-ishes, and our fits suggest that γ m starts running. Since a complete higher-orderanalysis is not feasible with presently available data, we adopt a more phe-nomenological approach. We propose a partial extension to higher orders, whichincorporates the running of γ m into the tree-level lagrangian. We find that thisextension successfully describes the full fermion-mass range of the LatKMI data,including the pion taste splittings which arise from using staggered fermions inthe lattice simulations. We also investigate a more general class of dilaton po-tentials proposed in the literature, using both the LSD and LatKMI data sets,concluding that these data favor the form predicted by dChPT.1 a r X i v : . [ h e p - l a t ] D ec . INTRODUCTION Lattice simulations of the SU(3) gauge theory with eight Dirac fermions in the funda-mental representation have revealed the existence of a flavor-singlet scalar particle, which,at the fermion masses explored in these simulations, is approximately degenerate with thepions—the Nambu–Goldstone bosons associated with chiral symmetry breaking [1–4]. Asimilar light scalar has been found also in the SU(3) gauge theory with two sextet fermions[5–9], or with four light, and six [10] or eight [11] heavy fundamental fermions. The existence of a light flavor-singlet scalar particle roughly degenerate with the pionsmeans that, besides the pions, any effective field theory (EFT) description of the low-energybehavior has to include a field that represents this scalar particle. Here, our starting pointis dilaton chiral perturbation theory (dChPT), an EFT in which the lightness of the scalarparticle is assumed to arise from approximate scale invariance of the underlying theory inthe infrared [17–21]. Increasing the number of (massless) fermionic degrees of freedom willeventually take the theory into the conformal window, where the non-abelian gauge theoryis still asymptotically free, but develops an infrared fixed point (IRFP). The idea is that,with eight flavors, the SU(3) gauge theory is still outside the conformal window, but closeenough to the conformal sill—the number of flavors where the IRFP first develops—thatthe breaking of scale invariance in the infrared is governed by the proximity of the IRFP.The key assumption is then that the distance to the conformal sill can be treated as a smallparameter, in which a systematic power counting can be developed. The scalar particle,which we will refer to as the dilaton, is interpreted as a pseudo Nambu–Goldstone boson(pNGB) for the approximate scale symmetry [17]. The mass of the dilaton is controlled bythis small parameter, just as the fermion mass leads to a parametrically small pion mass.Since the fermion mass breaks scale invariance too, the dilaton mass will also depend on thefermion mass.In a previous paper [33] we applied leading-order (LO) dChPT to numerical data for theeight-flavor SU(3) gauge theory produced in lattice simulations by the LSD collaboration [3].We showed that, over the fermion mass range in these simulations, LO dChPT successfullydescribes the pNGB sector of the theory, including the dilaton. In Ref. [3] staggered fermionswere used, which exhibit taste splittings—a lattice artifact mass splitting of the pion mul-tiplet caused by a partial breaking of the flavor symmetry group in the staggered fermionformulation. We showed that dChPT explains the pattern of taste splittings in the pionsector observed in Ref. [3] as a function of the fermion mass. The vacuum expectation valueof the dilaton field depends on the fermion mass already in LO, leading to a fermion-massdependence of pNGB decay constants and masses that is qualitatively different from QCD.This includes the taste splittings, which are also qualitatively different from the pattern seenin QCD with staggered fermions.Given this success, our goal in this paper is to investigate whether dChPT can alsobe applied to the other major lattice study of the eight-flavor SU(3) gauge theory, by theLatKMI collaboration [4]. This study also used staggered fermions, and presented extensivespectroscopy data for the pNGB sector, including taste splittings. The KMI simulationswere done at larger fermion masses than those of LSD. Even if dChPT is the correct EFT,the question arises whether one can fit the KMI data using LO dChPT, or, alternatively, For reviews of lattice work, see Refs. [12–16]. For early work, and for other low-energy approaches, see Refs. [22–32]. For reviews, see Refs. [34, 35]. We will often shorten “LatKMI” to just “KMI.” γ m . In dChPT, at leading order, the mass anomalous dimension is constant, γ m = γ ∗ , where γ ∗ is the mass anomalous dimension at the nearby IRFP. dChPT allows fora non-constant γ m , but the power counting underlying dChPT accommodates corrections toa constant γ m only through higher orders. In order to systematically compare dChPT withthe KMI data, we would thus have to consider dChPT to next-to-leading order (NLO) orbeyond. However, the relatively large number of additional parameters that would be neededalready at NLO, and limitations of the presently available lattice data, to be discussed below,prevent us from attempting a complete NLO fit.Instead, we will take a more phenomenological approach, based on the following observa-tion. The salient difference between the KMI and LSD data appears to be that a constant γ m cannot account for the full range of (larger) fermion masses explored in the KMI data. Wewill thus extend LO dChPT by only including higher-order effects that are directly relatedto γ m ; we will refer to this extension as γ -dChPT. This makes our approach not systematic,since most NLO and higher-order effects are left out. Strictly speaking, γ -dChPT shouldthus be viewed as a model approach.In order for LO dChPT to accommodate a varying γ m , we will modify the mass-dependentpart of the potential, as described in detail in Sec. II. This raises the question of what happensif one also considers a generalization of the dilaton part of the potential. A class of potentialsdepending on a new parameter ∆, generalizing the dilaton potential of dChPT, has beenproposed before [24, 25, 28, 32], and we will refer to this different extension of LO dChPT as∆-dChPT. One recovers LO dChPT, including its dilaton potential, by taking ∆ →
4. It isinteresting to also confront ∆-dChPT with the data. We will revisit the analysis of the LSDdata using ∆-dChPT by Ref. [32], and extend this investigation to the KMI data. Despiteclaims in the literature [32], ∆-dChPT takes us outside the systematic power counting ofdChPT, and should thus be considered as a more phenomenological approach to the low-energy behavior of the N f = 8 theory.This paper is organized as follows. In Sec. II we introduce γ -dChPT, in which LO dChPTis extended to accommodate a varying γ m . In Sec. III we first present our evidence that γ m , as well as other LO parameters, are changing over the KMI mass range in a fit to LOdChPT. We then apply γ -dChPT to the pNGB sector of the KMI data. We find that arather simple model for a varying γ m provides good fits of the KMI data, including tastesplittings. In Sec. IV we consider the generalized class of dilaton potentials, reviewing theapplication of ∆-dChPT to the LSD data, and applying it to the KMI data. Combiningthese results provides some evidence that the dilaton potential of LO dChPT is preferredby the data, i.e. , that the preferred value in ∆-dChPT is close to ∆ = 4. Finally, Sec. Vcontains our conclusions. In App. A we elaborate on the choice of a mass-independent scalesetting prescription. In App. B we investigate the claim of Ref. [32] that ∆-dChPT admitsa systematic power counting for any value of ∆, and show that this claim is incorrect. For an early study of γ m in the N f = 8 theory, see Ref. [36]. I. DILATON ChPT AND γ m In Sec. II A, we begin with a summary of LO dChPT. This is the EFT that was appliedto the LSD data in Ref. [33]. In Sec. II B we revisit the physics of hyperscaling, and itsmanifestation in LO dChPT. This leads us in Sec. II C to introduce γ -dChPT, where wegeneralize the low-energy lagrangian to accommodate a non-constant mass anomalous di-mension. We emphasize that this extension takes us outside the strict EFT framework. InSec. II D we present the hadronic quantities to be fit to the KMI data of Ref. [4] in the restof this paper. A. dChPT at lowest order
The euclidean LO lagrangian for dChPT is given by L = 12 f τ e τ ∂ µ τ ∂ µ τ + 14 f π e τ tr ( ∂ µ Σ † ∂ µ Σ) + L m ( τ, Σ) + L d ( τ ) . (2.1)The potential terms are L d ( τ ) = f τ B τ e τ ( c + c τ ) , (2.2a) L m ( τ, Σ) = − f π B π me (3 − γ ∗ ) τ tr (Σ + Σ † ) . (2.2b)Here Σ is the usual non-linear field describing the pion multiplet, while τ is the dilatoneffective field. L depends on the low-energy constants (LECs) f τ , f π , B π , B τ , γ ∗ , c and c .We define the theory in the Veneziano limit [37], in which N ≡ N c ∝ N f is taken to infinitykeeping the ratio n f = N f /N c fixed, with N f the number of fundamental-representationflavors and N c the number of colors. The power counting is [17] p ∼ m ∼ n f − n ∗ f ∼ /N . (2.3)The relation p ∼ m defines the power counting of ordinary ChPT. The small parametercontrolling the hard breaking of scale invariance is n f − n ∗ f , where n ∗ f is the limiting value of n f for the theory at the conformal sill: the boundary between the regime where the masslesstheory undergoes chiral symmetry breaking, and the regime where this theory is conformalin the infrared, i.e. , where the gauge coupling g runs into an infrared fixed point g ∗ .Invoking the proximity of the sill of the conformal window, we assume that the β functionis small at the chiral symmetry breaking scale, and that the corresponding value of g is closeto g ∗ . We can then expand the mass anomalous dimension γ ( g ) in powers of n f − n ∗ f around γ ∗ = γ ( g ∗ ), the mass anomalous dimension at the infrared fixed point at the conformal sill.For a detailed discussion of the construction of the LO lagrangian, and the underlying powercounting, see Refs. [17, 20].In the dilaton potential (2.2a), c is O (1), while c is proportional to the small expansionparameter n f − n ∗ f . For m = 0, we shift the τ field to τ + v , with v = (cid:104) τ (cid:105) (cid:12)(cid:12) m =0 (beforethe shift). After the shift, the dilaton expectation value v ( m ) = (cid:104) τ (cid:105) vanishes in the massless The dimensionful quantities, p and m , are measured in units of the dynamically generated infrared scaleof the massless theory. For a few more details about the power counting, see App. B. f π,τ = e v f π,τ , (2.4a)ˆ B τ = e v B τ , (2.4b)ˆ B π = e (1 − γ ∗ ) v B π , (2.4c)the lagrangian becomes L = 12 ˆ f τ e τ ∂ µ τ ∂ µ τ + 14 ˆ f π e τ tr ( ∂ µ Σ † ∂ µ Σ) + L m ( τ, Σ) + L d ( τ ) , (2.5)with L d ( τ ) = ˆ f τ ˆ B τ e τ V d ( τ ) , (2.6a) V d ( τ ) = c (cid:18) τ − (cid:19) , (2.6b) L m ( τ, Σ) = −
12 ˆ f π ˆ B π me (3 − γ ∗ ) τ tr (Σ + Σ † ) . (2.6c)The shift sets c = − c /
4, and now the whole LO lagrangian is O ( p ) in the power count-ing (2.3). We will assume c >
0, so that the potential L d + L m is bounded from below.Assuming m ≥
0, the potential is minimized by Σ = 1. The dilaton expectation value v = v ( m ) solves the saddle-point equation(3 − γ ∗ ) m c M = v e (1+ γ ∗ ) v , M = ˆ f τ ˆ B τ ˆ f π ˆ B π N f . (2.7)The solution is positive, and monotonically increasing with m . The spectroscopy data weconsidered in Ref. [33] can then be expressed as functions of m , M π F π = 1 d v ( m ) ≡ h ( m ) , (2.8a) F π = ˆ f π e v ( m ) (2.8b)= (cid:18) d mh ( m ) (cid:19) γ ∗ , (2.8c) M τ F π = d (1 + (1 + γ ∗ ) d h ( m )) . (2.8d)Explicitly, h ( m ) = 1(1 + γ ∗ ) d W (cid:18) (1 + γ ∗ ) d d m (cid:19) , (2.9)where W is the Lambert W -function. The parameters d , , , are defined in terms of theLECs of the tree-level lagrangian, d = 2 ˆ B π ˆ f − γ ∗ π , d = (3 − γ ∗ ) ˆ f π B π c M , d = ˆ f π B π , d = 4 c ˆ B τ ˆ f π . (2.10)In Ref. [33] we applied LO dChPT, as summarized above, to the LSD data [3]. Thekey assumptions underlying this analysis were: (a) the N f = 8, N c = 3 theory undergoeschiral symmetry breaking; (b) for the LSD mass range, the β function is small enough thatthe dChPT power counting is applicable. The results of our analysis corroborated theseassumptions. 5 . Hyperscaling Consider momentarily a mass-deformed infrared conformal theory. We can probe thetheory over a range of scales where g is so close to the infrared fixed-point g ∗ that all effectsof its running can be neglected. The breaking of scale invariance is then driven entirely bythe input bare fermion mass m . Under these circumstances, any hadronic mass M followsa simple hyperscaling law, M Λ UV ∼ (cid:18) m Λ UV (cid:19) γ ∗ . (2.11)Here Λ UV is an ultraviolet scale for which the approximation γ m ( µ ) = γ ∗ is valid for any µ ≤ Λ UV , and m = m (Λ UV ), where m ( µ ) is the running renormalized mass. Hyperscalingis based on the following simple observations:1. The renormalized mass, m = m ( µ ), runs as dictated by its anomalous dimension.By contrast, the renormalized coupling has attained its fixed-point value g ∗ (up tonegligible corrections), hence also the mass anomalous dimension has a fixed value γ ∗ = γ m ( g ∗ ).2. No physical scale is generated dynamically in the massless theory. When the fermionmass is non-zero, the induced physical scale M is set by the condition M ∼ m ( M ).Indeed, starting from the solution for m ( µ ) for a constant mass anomalous dimension, m ( µ ) m = (cid:18) µ Λ UV (cid:19) − γ ∗ , (2.12)the hyperscaling law (2.11) immediately follows by postulating that the typical hadron mass M satisfies M ∼ m ( µ ) for µ = M . For any γ ∗ >
0, the existence of the physical scale M isguaranteed if m (cid:28) Λ UV . Starting from m ( µ ) = m (cid:28) µ at µ = Λ UV , m ( µ ) keeps increasingas µ is decreased, until eventually the equality M = m ( M ) is reached.Returning to dChPT, in Ref. [33] we found that the LSD data is in the “large-mass”regime [20], where | n f − n ∗ f | ∼ c (cid:28) m M , (2.13)for all (bare) masses. As follows from the previous subsection, in LO dChPT, c encodes themagnitude of the β function at the chiral symmetry breaking scale. The large-mass regimeis thus an approximate hyperscaling regime, where the input fermion mass dominates thebreaking of scale invariance. Indeed, in Ref. [20] we showed that the leading mass dependencepredicted by LO dChPT in the large-mass regime is the hyperscaling relation (2.11), for allhadronic masses and decay constants. We also calculated corrections to this relation, whichare present in dChPT already at LO, because the β function at the chiral symmetry breakingscale, hence c , is (by assumption) parametrically small, but not vanishingly small as in amass-deformed infrared conformal theory. Moreover, we showed that as long as | n f − n ∗ f | log (cid:32) m | n f − n ∗ f |M (cid:33) (cid:28) , (2.14) See also App. B. m / M can be large. By Eq. (2.7), M is constructed from LECs which can be defined in the chiral limit. It is a strikingdifference between ordinary ChPT and dChPT that, because of the nearby IRFP, in dChPTa systematic low-energy expansion exists even if the fermion mass is not small relative tothe infrared scale of the massless theory, so long as inequality (2.14) holds.The fermion mass range explored in the KMI data is higher than in the LSD data. Thecomparison can be made, for example, in units of t , see Fig. 5 of Ref. [2]. We will return tothe comparison between the LSD and KMI data, and its limitations, in Sec. III E below. Asmentioned in the introduction, when we increase the input fermion mass the influence of theIRFP diminishes. Eventually, we will reach energy scales where the running of the couplingpicks up, and, as a result, so does the running of the mass anomalous dimension. In the nextsubsection, guided by this consideration, we will develop a generalized notion of hyperscaling,which is founded on the same principles as above, except that the assumption of a constantmass anomalous dimension is relaxed. This will lead to the framework of γ -dChPT, whereLO dChPT is extended to accommodate a varying mass anomalous dimension. We stressthat the power counting of dChPT allows for corrections to a constant γ m , but only viahigher-order terms in the expansion in n f − n ∗ f . In seeking an extension of LO dChPT thataccommodates a varying γ m we are thus asking for a partial resummation of these higher-order terms, under the assumption that these are the dominant higher-order corrections.We conclude this subsection with a technical comment. The hyperscaling law (2.11) canbe rewritten as m M ∼ (cid:18) m Λ UV (cid:19) γ ∗ γ ∗ . (2.15)It follows that the fermion mass m is always much smaller than any hadronic mass M (aslong as m (cid:28) Λ UV ), and the same is true for the decay constants F π and F τ . Moreover, inRef. [20] we showed that this conclusion extends to n f < n ∗ f , below the conformal window,and that it applies also to the masses of the pNGBs, M π and M τ . We will assume thatthe ratio m /M remains small also when the simple hyperscaling relations, Eqs. (2.11)and (2.15), are generalized to account for the running of γ m . Indeed, for the LSD data, m /M π ranges between 0 .
015 and 0 .
04, while for the KMI data it ranges between 0 .
07 and0 .
17. Since m /M π (cid:28)
1, this allows us to use a mass-independent renormalization scheme. As we will see below, this greatly simplifies our considerations.
C. Varying γ m and γ -dChPT We will now proceed to develop the extension of LO dChPT allowing for a scale-dependent γ m . The RG equation governing the dependence of the renormalized mass m on the renor-malization scale µ is closely related to the behavior of the renormalized mass under scaletransformations. In order to relate the two, we first review how a scale is introduced intothe bare theory; we will do this using dimensional regularization. For more details, we referto Ref. [19]. We regulate the action of the microscopic theory as S = (cid:90) d d x µ d − L ( x ) , (2.16) At extremely high energy scales perturbation theory will eventually take over, and the β function willtend to zero as dictated by asymptotic freedom. The β and γ functions in a mass-dependent scheme can be expanded in powers of m /M , and the firstterm in this expansion yields a mass-independent scheme that is a good approximation if m /M (cid:28) L is the bare lagrangian, and d is the number of dimensions. With the factor µ d − ,the bare action S is invariant under scale transformations if we promote the bare parameters µ and m to spurions. The scale transformation rules are m → λm , (2.17a) µ → λµ , (2.17b) A µ ( x ) → λ A µ ( λx ) , (2.17c) ψ ( x ) → λ / ψ ( λx ) , (2.17d)where A µ is the bare gauge field and ψ the bare fermion field.The function γ m , defined by the RG equation µm dmdµ = − γ m , (2.18)describes the response of the renormalized mass m to a change of the renormalization scale µ . In a mass-independent scheme, all renormalization factors depend on the scales µ and µ only through their ratio, µ/µ . Hence, γ m = γ m ( g ( µ/µ )) , (2.19)where g = g ( µ/µ ) is the running coupling. From now on, we will write γ m ( µ/µ ) for γ m ( g ( µ/µ )), with slight abuse of notation. We choose µ not to transform under scaletransformations: the transformation (2.17) describes a rescaling of all the dimensionful barequantities relative to a fixed renormalization scale.Once γ m is known we can express m ( µ ), the renormalized mass at an arbitrary renormal-ization scale µ , in terms of the bare mass, m = m ( µ ), by integrating Eq. (2.18) between µ and µ . Introducing the formal solutions E ± ( µ/µ ) = e ± (cid:82) log µ/µ dt γ m ( e t ) . (2.20)of the RG equations µ dE ± dµ = ± γ m ( µ/µ ) E ± , (2.21)one has m ( µ ) = E − ( µ/µ ) m . (2.22)Using Eq. (2.17) for the dependence of the bare parameters m and µ on the scale transfor-mation parameter λ , it follows that an infinitesimal scale transformation of the renormalizedmass is governed by the differential equation [19] ∂m ( λ ; µ ) ∂ log λ = (cid:18) γ m (cid:18) µλµ (cid:19)(cid:19) m ( λ ; µ ) , (2.23)which is solved by m ( λ ; µ ) = λ E − ( µ/ ( λµ )) m . (2.24)For constant γ m = γ ∗ , Eq. (2.20) simplifies to E ± ( µ/µ ) = (cid:18) µµ (cid:19) ± γ ∗ , (2.25)8ence m ( λ ; µ ) = λ γ ∗ m ( µ ) = λ γ ∗ (cid:18) µ µ (cid:19) γ ∗ m . (2.26)The second equation explains the origin of the factor λ γ ∗ . A factor λ comes from the trans-formation of m , Eq. (2.17a), while the remaining factor λ γ ∗ comes from the transformationof µ , Eq. (2.17b). With the transformation rules of the effective fields τ ( x ) → τ ( λx ) + log λ , (2.27a)Σ( x ) → Σ( λx ) , (2.27b)it follows that L m ( x ) in Eq. (2.2b) transforms into λ L m ( λx ), as required for the invarianceof the action.In order to accommodate a non-constant γ m , we replace L m of Eq. (2.2b) by L m = − f π e τ E − ( e τ f π /µ ) B π ( µ/µ ) m ( µ/µ ) tr (Σ + Σ † ) . (2.28)Let us derive the transformation properties of this lagrangian. The combination B π ( µ/µ ) m ( µ/µ )is by assumption RG invariant, and we can write B π ( µ/µ ) as B π ( µ/µ ) = B RG π E + ( µ/µ ) . (2.29)The new LEC, B RG π , is both RG invariant and scale invariant, also by assumption. Hence B π ( µ/µ ) m ( µ/µ ) = B RG π m , and using Eq. (2.17a) it follows that under a scale transfor-mation ∂∂ log λ B π ( µ/ ( λµ )) m ( λ ; µ ) = + B π ( µ/ ( λµ )) m ( λ ; µ ) . (2.30)The factor E − ( e τ f π /µ ) in Eq. (2.28) is invariant under a scale transformation by construc-tion, because the combination e τ f π /µ is. Noting that the scaling dimension of Σ is zero,and taking the contribution from the factor e τ into account, we obtain ∂∂ log λ L m (cid:12)(cid:12)(cid:12) λ =1 = 4 L m + x µ ∂∂x µ L m = ∂∂x µ ( x µ L m ) , (2.31)which establishes the invariance of the action. This conclusion is valid for any choice of thefunction γ m .The lagrangian for dChPT with a varying γ m function is given by Eq. (2.1), with now L m given by Eq. (2.28). The theory is invariant under the scale transformation of the effectivefields, Eq. (2.27), combined with the spurion transformation rules m ( µ ) → m ( λ ; µ ) , (2.32a) µ → λµ , (2.32b) c → c − log λ , (2.32c) c → c . (2.32d) Being µ independent, E − ( e τ f π /µ ) is trivially RG invariant. In Ref. [17] we introduced a space-time dependent spurion field χ ( x ) for the renormalized mass, but forour present purposes, a space-time independent spurion for m is sufficient. L d in Eq. (2.2a). As usual, once the spurions m , µ and c are set equal to their fixedvalues, this breaks the scale symmetry explicitly.We may again shift the τ field, as we did in Sec. II A, such that after the shift it has avanishing expectation value for m = 0. The LECs f π,τ and B τ are redefined as in Eq. (2.4),but now ˆ B π is defined asˆ B π ( µ/µ ) = ˆ B RG π E + ( µ/µ ) , ˆ B RG π = e v B RG π , (2.33)so that ˆ B π = e v B π . The lagrangian after the shift is again given by Eq. (2.5), but now with L m = −
12 ˆ f π e τ E − ( e τ ˆ f π /µ ) ˆ B π ( µ/µ ) m ( µ/µ ) tr (Σ + Σ † ) , (2.34)instead of Eq. (2.6c). Note that, instead of being a function of e τ f π /µ , now E − is a functionof e τ ˆ f π /µ .Let us now reconsider the trace anomaly. We first apply the scale transformation only tothe effective fields, setting the spurions equal to their fixed values. In this case, ∂∂ log λ = ∂τ∂ log λ ∂∂τ = ∂∂τ , (2.35)and we obtain the contribution of L m to ∂ µ S µ , the divergence of the dilatation current S µ (see App. D of Ref. [17]), (cid:18) ∂∂τ − (cid:19) L m = − (1 + γ m ( e τ ˆ f π /µ )) L m = − (1 + γ m ( e τ ˆ f π /µ )) mψψ (EFT) . (2.36)In the last step we identified L m with the EFT representation of mψψ in the underlyingtheory. This reproduces, in the EFT, the contributions from the fermions to the traceanomaly [38]. Recall that we have defined γ m to be a function of µ/µ , cf. Eq. (2.19).Replacing τ by v ( m ), its vacuum expectation value at non-vanishing m , we see that Eq. (2.36)effectively identifies the renormalization scale µ with F π = e v ( m ) ˆ f π , cf. Eq. (2.8b). Thisreveals a key feature of our construction of γ -dChPT: γ m is evaluated at a renormalizationscale equal to the physical scale F π , which, in turn, is a function of the input fermion mass.We comment that we chose the hadronic scale inside E − in Eq. (2.28) to be f π , but, toachieve the desired scaling behavior, we could equivalently choose f τ , or, more generally,any other hadronic scale m h that enters the dChPT lagrangian (or generalization thereof)via the combination e τ m h , such as, for example, the nucleon mass in the chiral limit.We now specialize to specific choices for the function γ m . First, for constant γ m = γ ∗ ,ˆ B π ( µ/µ ) E − ( e τ ˆ f π /µ ) = ˆ B RG π (cid:18) µµ (cid:19) γ ∗ e − γ ∗ τ (cid:18) µ ˆ f π (cid:19) γ ∗ = ˆ B π ( µ/ ˆ f π ) e − γ ∗ τ , (2.37)and the lagrangian L m in Eq. (2.34) reduces to Eq. (2.6c). This also implies ˆ B π ( µ/ ˆ f π ) = e (1 − γ ∗ ) v B π ( µ/f π ), consistent with Eq. (2.4c). The transformation rules of c and c get modified at higher orders. For a detailed discussion of L d , seeRefs. [17, 20]. We omit the contribution from the scale dependence of the space-time coordinates (compare Eq. (2.31)). In this special case, the dependence on µ drops out.
10e next introduce a new choice for γ m that we will be using for the actual fits to theKMI data. With t = τ + log( ˆ f π /µ ) we define E − ( e τ ˆ f π /µ ) = E − ( e t ) = e − ˜ F ( t ) , (2.38)where ˜ F ( t ) = ˜ γ t −
12 ˜ bt + 13 ˜ ct , (2.39)a cubic polynomial in t . The variable t is invariant under scale transformations, and, con-sistent with our general discussion, ˜ γ , ˜ b and ˜ c are LECs that do not depend on µ or µ .Re-expressing t in terms of τ , we write˜ F ( t ) = ˜ F (log( ˆ f π /µ )) + F ( τ ) , (2.40) F ( τ ) = γ τ − bτ + 13 cτ , (2.41)which defines the coefficients of the cubic polynomial F ( τ ) in terms of those of ˜ F ( t ), andlog( ˆ f π /µ ). Substituting into Eq. (2.34), and absorbing e − ˜ F (log( ˆ f π /µ )) into ˆ B π , the final formof the lagrangian becomes L m = −
12 ˆ f π ˆ B π m e τ − F ( τ ) tr (Σ + Σ † ) . (2.42)We will use the acronym γ -dChPT for the lagrangian defined by Eq. (2.1), with L d givenby Eq. (2.2a), and L m by Eq. (2.42) for some general function F ( τ ). Of course, for the caseof a linear F ( τ ), Eq. (2.42) reduces to Eq. (2.2b), and the lagrangian is just LO dChPT.As an EFT, dChPT is based on the power counting established in Refs. [17, 20] andreviewed above. As in ordinary ChPT, loop corrections in dChPT can be included system-atically; the power counting (2.3) dictates which terms occur at the next-to-leading order(NLO) [17], at the next-to next-to-leading order (NNLO), and so on. The same is true inthe large-mass regime, where the power counting is controlled by Eq. (2.14). This raises thequestion of how much γ -dChPT deviates from the strict EFT framework of dChPT itself. Ifwe rely on algebraic structure and symmetries only, this allows E − ( e τ ˆ f π /µ ) in Eq. (2.34),or, equivalently, F ( τ ) in Eq. (2.42), to depend on an infinite number of parameters, reflect-ing the model nature of γ -dChPT. But if, on the other hand, we assume that F ( τ ) takesthe form of Eq. (2.41), with γ ∼ ( n f − n ∗ f ) = 1 , b ∼ n f − n ∗ f , c ∼ ( n f − n ∗ f ) , (2.43)then the factor e − F ( τ ) may be obtained via partial resummation of terms from all orders inthe expansion in powers of n f − n ∗ f . It thus reflects a fairly modest departure from dChPT, inthat we will be taking into account some higher-order analytic terms, resummed into e − F ( τ ) ,while omitting other higher-order terms. In addition, we will not calculate any non-analytichigher-order corrections when fitting γ -dChPT to data. We will re-examine the scenario ofEq. (2.43) after presenting our fits to the KMI data in Sec. III. D. Hadronic quantities for varying γ m As in Sec. II A, we begin with the saddle-point equation. For m ≥ v = v ( m ) is the solution of (compare11q. (2.7)) (3 − γ m ) m c M = ve v + F ( v ) , (2.44)where now γ m = F (cid:48) ( v ) . (2.45)When F ( τ ) is linear in τ we reproduce the results of Sec. II A, whereas for F ( τ ) in Eq. (2.41)we have γ m = γ − bv + cv . (2.46)Equation (2.44) can be rewritten as m = d ˜ d − γ m v e v + F ( v ) , (2.47)with ˜ d = ˆ f π B π c M . (2.48)For a general function F , Eq. (2.47) cannot be explicitly inverted analytically. We will, ineffect, solve it numerically for m as a function of v , as described in Sec. III. In terms of v , F π is still given by Eq. (2.8b). The pion mass is now M π = 2 ˆ B π m e v − F ( v ) , (2.49)so that, using Eq. (2.47), the ratio M π /F π is given by M π F π = 1˜ d v − γ m . (2.50)The three equations (2.47), (2.8b) and (2.50) contain six parameters, ˜ d , d , ˆ f π and the threeparameters inside F : γ , b and c .We will not fit M τ to the KMI data, as the errors found in Ref. [4] are too large for such afit to have statistical relevance. We will, however, fit the staggered taste-splittings obtainedin Ref. [4]. With M Γ i the masses of the taste-split pions corresponding to the tastesΓ i ∈ { Γ , Γ µ , Γ µν , Γ µ , Γ I } , (2.51)we will fit the differences ∆(Γ i ) ≡ a ( M i − M π ) , (2.52)according to [39, 40]∆(Γ ) ≡ ∆ P = 0 , (2.53a)∆(Γ µ ) ≡ ∆ A = C E ( γ ) + 3 C E ( γ ) + C E ( γ ) + 3 C E ( γ ) , (2.53b)∆(Γ µν ) ≡ ∆ T = 2 C E ( γ ) + 2 C E ( γ ) + 4 C E ( γ ) , (2.53c)∆(Γ µ ) ≡ ∆ V = C E ( γ ) + C E ( γ ) + 3 C E ( γ ) + 3 C E ( γ ) , (2.53d)∆(Γ I ) ≡ ∆ S = 4 C E ( γ ) + 4 C E ( γ ) . (2.53e) We note that M Γ = M π is the mass of the Nambu–Goldstone pion. C , , , are LECs associated with the taste-breaking potential [40], and E ( γ i ) = e (4 − γ i ) v . (2.54)Equation (2.54) assumes that γ i , the anomalous dimensions of the taste-breaking four-fermion operators, are constant (see Ref. [33] for more details). A global fit of the dataincluding all the taste splittings has eight new parameters, coming from Eq. (2.53), in addi-tion to the six parameters of the basic fit. This is a large number of parameters, and, as wewill see, some of them are not sufficiently constrained by the available data. Thus, we willnot venture into an exploration of any scale dependence of the γ i .We end this section with a comment. While in LO dChPT the potential is boundedfrom below, in γ -dChPT with general F ( v ) the potential can be unbounded from below. Mathematically, this appears to be a problem, but we contend that it is physically irrelevant.Within the EFT framework, the potential can only be known for O (1) values of the fields.While the pion field is always O (1) because it is a compact field, this is not the case for τ .We thus need to restrict the EFT to O (1) values of τ “by hand.” In practice, this meansthat after fits to the data, we need to check that indeed values of v predicted by the fitsare O (1), and do not land in the large-field region. In all our fits with a varying γ m indeedunphysical regions of the potential occur at very large values of v , but they are separatedfrom the physical region by an exponentially large potential barrier. Consistently, our fitsnever explore the unphysical region of the potential. III. FITS TO THE LatKMI DATA
In this section, we will present our fits to data reported in Ref. [4], obtained by theLatKMI collaboration for the eight-flavor SU(3) gauge theory. We begin in Sec. III A witha discussion of these data and the policies we will follow when we use them. In Sec. III B,we present “window” fits. These are fits of M π /F π and aF π to the predictions of LOdChPT, for successive quintets of fermion masses, from the five lightest masses to the fiveheaviest ones. Altogether, ten different fermion masses were simulated in Ref. [4], making six(overlapping) windows. The window fits test the constancy of the LO dChPT parameters.We find a systematic trend of change for all fit parameters, by much more than their errorsallow, proving that the full KMI mass range cannot be fit to LO dChPT. Then, in Sec. III Cwe fit the data at all ten fermion masses simultaneously to γ -dChPT, the extension of LOdChPT with a varying γ m constructed in Sec. II C, with the special choice of γ m in Eq. (2.46).We find that this extension of dChPT successfully describes the KMI data set. Data fortaste-split pion masses is available for a more limited set of fermion masses, and we presentour fits including the taste splittings in Sec. III D. We end with a discussion of the scaledependence of γ m found in our fits in Sec. III E.The simulations of Ref. [4] were all performed at the same bare coupling. Invoking amass-independent scale setting prescription, this implies that all ensembles have a commonlattice spacing a . We elaborate on the choice of a scale setting prescription in App. A.We will be using lattice units in all our fits. This means taking µ = µ = 1 /a , and thus m ( µ ) = m ( µ ) = m . For polynomial F ( v ), a necessary and sufficient condition that the potential will be bounded from belowis that the highest power of v is even, and its coefficient is positive. . The LatKMI data The pion mass M π and decay constant F π were measured in Ref. [4] at ten bare-massvalues am ∈ { . , . , . , . , . , . , . , . , . , . } . (3.1)In Ref. [4] a great effort was made to also determine the dilaton mass M τ . It was foundthat indeed a dilaton exists, roughly degenerate with the pions. M τ was measured for only6 fermion masses, leaving out am = 0 .
05, 0 .
07, 0 .
08 and 0 .
1. More seriously, the statisticalerrors of M τ turn out to be too large to have any real impact on our fits. In the windowfits to LO dChPT (next subsection), we found that when we include a fit of M τ /F π toEq. (2.8d) in our global fit, d remains largely undetermined, while all other fit parametersdo not change. The only noticeable change is a higher p -value, as might be expected. Wethus omit the dilaton mass from the fits discussed in this paper.Other hadron masses were also determined, notably the vector meson mass aM ρ and thenucleon mass aM N . For these hadrons, the prediction from LO dChPT is that the ratios M ρ /F π and M N /F π should be independent of am [20]; this is also true if we extend LOdChPT to include a varying γ m . Excluding the two largest fermion masses, am = 0 .
08 and0.1, we found that we can fit M ρ /F π to a constant, with a p -value of 0.31. M N was measuredonly for a subset of the fermion masses, am ∈ { . , . , . , . , . , . , . } , (3.2)which leaves out am = 0 .
05, 0 .
07 and 0.1. Keeping only the 5 lightest masses, we foundthat a fit of M N /F π to a constant has a p -value of 0.07. This suggests that for larger fermionmasses, higher-orders corrections in dChPT (other than a varying γ m ) would be needed tofit these ratios. In addition, discretization effects could be playing a bigger role (see below).We will thus focus in this paper on the pion sector, considering M π /F π and aF π in Secs. III Band III C, and adding taste splittings in Sec. III D.Information on the systematic errors of aM π and aF π is incomplete. Mostly, they weremeasured on at least two different volumes, and we estimate the finite-volume error bytaking the difference between the results at the largest two volumes. For am = 0 .
012 onlyone volume is available. In this case we took the finite-volume errors to be the same as for am = 0 . am = 0 . am = 0 .
012 is the lightest fermion mass,this procedure may underestimate its finite-volume errors. A single volume was reportedalso for am = 0 .
08 and 0.1. For these fermion masses, the two largest ones, M π L is verylarge, and finite-volume corrections should be very small. We thus took the finite-volumeerrors for these two masses to vanish. We added the statistical error and the finite-volumeerror of aM π and aF π in quadrature. These errors were propagated to the ratio M π /F π , andcorrelations between this ratio and aF π were kept. As the simulations of Ref. [4] were done at a single bare coupling, no direct informationis available on the lattice spacing dependence, and it is not possible to take the continuumlimit. We are thus forced to ignore scaling violations in our fits, but it should be kept inmind that these affect our results in an unknown way. Generally speaking, M ρ and M N arelarger than M π , and are thus prone to larger discretization effects. Also, as an example, The pions are too heavy for the ρ to decay. Correlations between aM π and aF π on each ensemble are not available. We note that, in Ref. [33], wefound that these correlations are small in the LSD data. B C D E Frange 0.012–0.04 0.015–0.05 0.02–0.06 0.03–0.07 0.04–0.08 0.05–0.1 χ /dof 9.37/6 9.85/6 4.81/6 4.38/6 4.56/6 3.83/6 p -value 0.15 0.13 0.57 0.63 0.60 0.70 γ ∗ a ˆ f π d − log( ad ) 10.5(3) 10.0(2) 9.5(2) 9.2(4) 9.1(2) 9.0(2)TABLE 1. Fits of the KMI data to Eqs. (2.8a) and (2.8b), using selections of five successivefermion masses from the set (3.1). All parameter errors reported in this paper are hessian. for am = 0 .
08 Ref. [4] finds the central values aM π = 0 . aM ρ = 0 .
68 and aM N = 1 . t in Sec. III E. The only other information on lattice spacing effectscomes from pion taste splittings. The masses of taste-split pions, which were measured onlyon the seven ensembles with bare masses (3.2), will be considered in Sec. III D. B. Window fits
We begin with fitting M π /F π and aF π to the predictions of LO dChPT, Eqs. (2.8a)and (2.8b). We consider sets of five successive fermion masses, taking first the lightest fivemasses from the set (3.1), then the second to the sixth masses, etc. , for a total of six quintets.The results are shown in Table 1. All the fits are good. However, the parameter valueschange with the partial mass range, more than allowed by their errors. In particular, thelowest mass range (fit 1A) and the highest mass range (fit 1F) do not overlap, hence theirparameter errors are statistically independent. These fits are thus not consistent with eachother. A simultaneous fit of LO dChPT to all ten masses has a p -value of order 10 − .Clearly, the whole KMI mass range cannot be fit to LO dChPT.As dChPT admits a systematic expansion, the failure to describe a set of data at LOmeans that higher orders in the expansion are needed. However, already at LO, dChPTcontains more parameters than ordinary ChPT. Depending on the observables being fitted,many more would be needed for an NLO fit. We believe that much better data is requiredfor a meaningful NLO fit. As discussed in Sec. III A, the LSD and KMI data sets bothcontain only a single lattice spacing, leaving discretization errors as an uncontrolled sourceof systematic uncertainty. In addition, it may well be that more refined data, for additionalbare masses and/or with smaller statistical errors, would be needed to determine all theparameters in the NLO fit. We will label fits with a number for the table, and a letter for the fit in the table. For example, fit 1Arefers to fit A in Table 1, etc. . Fits with a varying γ m Being unable to carry out a full NLO fit at present, we are left with the option of partiallyextending LO dChPT by exploring different “directions” in “higher-order parameter space.”By its very nature, no such extension is fully systematic, and each extension should thusbe considered a model. Our assumption is that our model, γ -dChPT, captures the relevantphysics better than other extensions of LO dChPT.As we have discussed in Sec. II B, the physical mechanism that underlies the behaviorof the LSD data is hyperscaling. The KMI mass range is higher than the LSD one, whichmotivates us to consider a minimal modification of this physical picture. We assume thatthe KMI mass range is still governed by the same principles that produce hyperscaling inthe LSD mass range, except that, because of the diminishing influence of the IRFP, we nowhave to allow the mass anomalous dimension to vary. That consideration has led us to theframework of γ -dChPT, developed in Sec. II C.In this subsection, we will thus consider fits of the KMI data to γ -dChPT. Specifically,we consider fits of M π /F π and aF π to Eqs. (2.50) and (2.8b), where γ m is quadratic in v , cf. Eq. (2.46). We begin with a technical issue. The independent variable in these equations is v , which, in turn, can be determined in terms of am using Eq. (2.47). However, unlike inLO dChPT discussed in Sec. II A, Eq. (2.47) cannot be analytically inverted. Instead, inaddition to the parameters defining the γ -dChPT lagrangian, we introduce new parameters v i , one per ensemble. We fit the corresponding bare mass am ,i to Eq. (2.47), whilesimultaneously also fitting ( M π /F π ) i and ( aF π ) i , all as functions of the same parameter v i .Artificially introducing a tiny error for am ,i , the fit in effect solves Eq. (2.47) numericallyfor v i in terms of am ,i . Thus, for given values of the γ -dChPT parameters, v i is equal to v ( am ,i ) with numerical precision set by the “error” of the “data” am ,i . We have variedthe errors on am ,i between 10 − and 10 − , finding no discernible differences in the resultsof our fits. χ values remain equal to four decimal places, whether one includes the “ am part” in the computation of χ or not.As in Ref. [33], we can calculate ( a ˆ B π ) i on each ensemble using Eq. (2.49) and our fitresult for v i . In all cases studied in this paper the so-obtained values of ( a ˆ B π ) i are equalwithin error. This confirms the self-consistency of our assumption that the lattice spacing a is independent of the fermion mass.The results of our fits are shown in Table 2. Fit 2A includes all ten ensembles, fit 2Bleaves out the am = 0 . am = 0 . c = 0, i.e. , taking γ m in Eq. (2.46) to be a linear function of v . Fits with c = 0 including allten ensembles, or omitting the am = 0 . p -values, 0.001 and 0.01respectively. We do not show them in the table. However, if we omit both the am = 0 . .
08 ensembles, we obtain fit 2D, which is a good fit. The parameters a ˆ f π , ˜ d andlog( ad ) are relatively stable between the fits with c as a free parameter, and fit 2D, where c = 0. By contrast, the parameters defining the function γ m change substantially: Fit 2Dyields much smaller values for both γ and b than the other fits of Table 2.The results of fits 2B and 2D are shown in Fig. 1. The black points are data that wereincluded in the fits, whereas the magenta points were excluded. The lower left panel shows In principle, the formal inverse function m = m ( v ) may not be single valued. In practice, we found that v is monotonically increasing with m over the entire KMI mass range. The total number of parameters increases by the number of v i parameters, i.e. , by the number of ensemblesincluded in the fit. The number of data increases by the same amount (the am ,i ), leaving the number ofdegrees of freedom unchanged. B C Domitted — 0.1 0.1, 0.08 0.1, 0.08 χ /dof 20.7/14 11.5/12 10.0/10 14.8/11 p -value 0.11 0.48 0.44 0.19ˆ f π d − log( ad ) 10.1(2) 10.4(2) 10.6(3) 9.9(1) γ b c Fits of M π /F π and aF π to γ -dChPT, the extension of LO dChPT discussed in Sec. II C.The “omitted” row shows bare mass values from the set (3.1) which are not included in the fit, ifany. that if we simplify our ansatz for γ m to be linear in v , then the am = 0 .
08 and 0.1 ensemblesmust be excluded.We have proposed in Sec. II C that the exponential factor e − F ( v ) may originate from aresummation of the dominant contributions from all orders in the expansion in n f − n ∗ f .According to the hypothesis (2.43), b is an NLO parameter, while c is an NNLO parameter.One way to test this scenario is to examine the effect of truncating the Taylor expansion ofthe exponential factor. The range of values we find for v in the fits to the KMI data is 1 . ≤ v ≤ .
5. Considering first fit 2B, we can compare the numerical values of exp (cid:0) bv − cv (cid:1) ,and its version truncated at NNLO, namely 1 + bv + b v − cv . When we vary v from1.5 to 2.5, the exponential and its truncated version take values ranging from 3.4 to 15,respectively 3.4 to 13. The differences (taking the correlations into account) are − . b andits relative error allows for a more precise comparison. Varying again v from 1.5 to 2.5,exp (cid:0) bv (cid:1) varies from 1.15 to 1.48, while the expansion to NLO, 1 + bv , varies from 1.14to 1.39. The (correlated) differences are 0.010(4) and 0.09(3), respectively. Thus, while thebehavior of both forms is qualitatively similar, the differences are statistically significant.Fits with the truncated version give results consistent with fits 2B and 2D, but with lower p -values.Without more data it is difficult to decide which fit in Table 2 is the preferred one.Clearly, unless the two heaviest masses are dropped, c must be kept in the fit. Given its(conjectural) role as an NNLO parameter, it is to be expected that eventually c will beneeded to describe the data as the mass range is increased. Still, we cannot rule out thatthe main reason why fit 2D does not accommodate the two heaviest masses is large scalingviolations at those mass values.In all fits where the parameter c is present, it is always small compared to b , consistentwith the conjectured hierarchy (2.43). However, in the same fits, one cannot say that b issmall compared to γ . By contrast, in fit 2D, where c = 0, also b is clearly small compare to γ . The most appealing scenario thus appears to be the following. We exclude the two largest17 .00 0.02 0.04 0.06 0.08 0.101214161820 �� � � π � / � π � �� � � � π �� � � π � / � π � �� � � � π FIG. 1.
Upper panels: Fit results for M π /F π (left panel) and aF π (right panel) using fit 2B. Lowerpanels: similar, using fit 2D. Black points are fitted data, while magenta points were not includedin the fit. fermion mass values, because they require going to (at least) NNLO in the EFT expansion,and/or because they are afflicted by too large scaling violations. The remaining mass rangemay be amenable to an NLO dChPT fit, for which fit 2D is our closest substitute. D. Taste splittings
We now turn to fits which also include the taste splittings (2.52), i.e. , fits of M π /F π , aF π and ∆ A,T,V,S to γ -dChPT, augmented by Eq. (2.53). Our fits are limited to the smallerensemble set (3.2), where the taste-split pion masses were measured.We show five different fits in Table 3. Fit 3A includes all the parameters: the basic γ -dChPT parameters of Sec. III C, namely a ˆ f π , ˜ d , log( ad ), γ , b and c , as well as all eighttaste-splitting parameters of Eq. (2.53). Data from all seven ensembles in the set (3.2)are included in the fit. The p -value is very high. The results for the six basic γ -dChPTparameters are consistent with fit 2B. As for the taste-splitting parameters, most of them,namely, γ , , and log C , , , are not well determined by the fit. We conclude that fit 3Agives an excellent description of the data, but the data are not precise enough to determine With the caveats discussed in Sec. III B. Note that the ensemble set (3.2) does not include am = 0 . B C D Eomitted — — — — 0.08 χ /dof 17.5/28 38.2/34 29.5/29 50.1/35 22.9/29 p -value 0.94 0.28 0.44 0.05 0.78ˆ f π d − log( ad ) 10.3(2) 10.2(2) 9.68(8) 9.65(7) 9.98(13) γ b c − log C − γ − log C γ − log C γ − log C γ Fits of M π /F π , aF π and taste splittings to γ -dChPT. The “omitted” row shows baremass values from the set (3.2) which are not included in the fit, if any. For description see text. all parameters in the fit.We next consider fits omitting poorly determined parameters. Among the taste-splittingparameters, only log C and γ were determined with good precision. As for C , C and C ,if we take their errors seriously, using them as 1 σ bounds, these parameters are “allowed”to be very small relative to C (by factors ∼ × , ∼
10 and ∼ , respectively).Setting C = C = C = 0, we obtain fit 3B. This is a good fit, even though its p -value ismuch smaller than fit 3A, as one would expect. The results of fits 3A and 3B are in verygood agreement. The dominance of the taste splittings generated by the C E ( γ ) term isconsistent with the results we obtained for the LSD data [33], as well as with the familiartaste splittings found in QCD.In Sec. III C we saw that the parameter c can be omitted if the fermion masses am = 0 . am = 0 .
08 is present in the ensemble set (3.2),we also repeated fits 3A and 3B while setting c = 0, obtaining fits 3C and 3D, respectively.Finally, fit 3E is similar to fit 3D, except that the am = 0 .
08 ensemble is not included.Fit 3C, were we set c = 0 but keep all the taste-splitting parameters, is very good. Settingboth c = 0 and C = C = C = 0 leads to a relatively low p -value in fit 3D. After droppingthe am = 0 .
08 ensemble, in fit 3E the p -value is again very high.Our results for a ˆ f π , ˜ d , log( ad ) are fairly consistent in all the fits reported in Tables 2and 3. The values of the parameters defining the function γ m are consistent among the fitswhere c (cid:54) = 0: fits 2A, 2B, 2C, 3A and 3B. In the fits with c = 0 the values of γ and b are different, but again consistent across this group: fits 2D, 3C, 3D, and 3E. The values ofthe taste splitting parameters log C and γ are consistent in all the fits of Table 3, while19 .00 0.02 0.04 0.06 0.080.0000.0050.0100.015 �� � � � � � � � � ����� �� � FIG. 2.
Fit 3C of the taste splittings ∆ A,T,V,S of Eq. (2.53), as a function of am . From top tobottom: ∆ S , ∆ V , ∆ T , and ∆ A . the (poorly determined) values of the remaining taste splitting parameters are consistentbetween fits 3A and 3C.In fit 3C, LO dChPT has been minimally extended (within the framework of γ -dChPT)to include an NLO correction to the function γ m . This fit gives an excellent description ofthe ensemble set (3.2) with taste splittings included; the parameter c is not needed. We thusconsider fit 3C to be the preferred fit from Table 3. We plot the taste splittings of this fit inFig. 2. A caveat is that, even though all the taste-split pion masses were measured in Ref. [4],the data are not precise enough to determine all taste-splitting parameters. We recall thatthe QCD taste splittings are essentially independent of the fermion mass [34, 40]. Bycontrast, as for the LSD data [33], also in the KMI mass range the taste splittings varywith the fermion mass. This behavior can be successfully described in dChPT, where thescale dependence of the taste-breaking operators gives rise to mass dependent tree-level tastesplittings, through the factors E ( γ i ) in Eq. (2.53). E. Scale dependence of γ m The anomalous dimension function γ m obtained from two of the fits of Table 2 is shownin Fig. 3. The blue band represents fit 2B, where γ m = F (cid:48) ( v ) is quadratic in v (Eq. (2.46)),while the magenta band represents fit 2D, where γ m is linear in v . With Eq. (2.8b), we takethe argument of γ m to be v = log( aF π /a ˆ f π ), and then plot γ m as a function of aF π . The two γ m functions agree well in most of the interval containing the fitted data, 0 . ∼ < aF π ∼ < . By contrast, the LSD data, which we fitted in Ref. [33], contains only M µ and M µν [3]. Thanks to the dominance of C , the QCD taste splittings are also roughly equal to each other. .02 0.04 0.06 0.08 0.10 0.120.00.51.01.52.0 �� π γ � FIG. 3.
The running mass anomalous dimension γ m , obtained from fit 2B (blue band) and 2D(magenta band), plotted as a function of aF π (see text). The gray horizontal band is γ ∗ = 0 . ± . , from our fit to the LSD data [33]. The fitted KMI data have values of aF π between . and . . fit from Sec. III B as a set of horizontal bands (each stretching over its corresponding rangeof aF π ), these bands would be consistent with the blue and magenta bands in that interval. Figure 3 also shows the value γ ∗ = 0 . F π should also be lower than the corresponding KMI range, in physical units. Equivalently,the LSD values of aF π , properly converted to KMI lattice units, should lie to the left of theKMI range of aF π in Fig. 3.Since the LSD data is successfully described by a constant γ m = γ ∗ , we expect that alsoin the chiral limit γ m will remain constant, at a value consistent with γ ∗ . The continuityof γ m as a function of F π thus requires that, as F π is lowered from the KMI range into theLSD range, γ m will rise to a value consistent with γ ∗ , and then stay roughly constant allthe way to the chiral limit. It is intriguing that the strong dynamics of the N f = 8 systemmight induce this behavior of γ m . Fig. 3 shows that, when extrapolated below the KMIrange, the quadratic γ m of fit 2B overshoots γ ∗ , while the linear γ m of fit 2D undershootsit. The desired behavior of γ m over the combined KMI and LSD ranges cannot be describedby simple ansatzes such as the ones we have used. One cannot rule out, however, thatthe combined LSD and KMI mass ranges could be described by including higher orders indChPT systematically.Clearly, an investigation of the combined LSD and KMI mass ranges would be extremelyinteresting. However, this is just not possible with the existing data sets. We already We do not show window fits in Fig. 3 because the different bands become visually difficult to see. A γ m function that saturates to a constant value at strong coupling was observed in the SU(2) theorywith two adjoint Dirac fermions [41]. We have attempted a comparison of the LSD and KMI lattice scales, using t , ch , thechiral-limit value of the gradient-flow scale t [42], which we have determined for the LSDdata set in Ref. [33]. The comparison is deficient for several reasons. First, unlike in ordinaryChPT [43], dChPT does not predict the behavior of t as a function of the fermion mass[33], so the best we can do is a phenomenological fit. Second, usually the gradient flowscale (or its chiral limit) is used to compare the lattice spacings of ensembles generatedwith different bare couplings, but with the same lattice action. By contrast, here we arecomparing results obtained using two different lattice actions, hence the meaning of thecomparison is less clear. Finally, there are also scaling violations in the lattice observablesused to extract t , as well as in the gradient-flow equation. KMI used two lattice definitionsfor t which should agree in the continuum limit, but which consistently differ by some 15%over the entire KMI mass range; we do not have equivalent information about uncertaintiesassociated with the LSD data. With all these caveats in mind, our findings suggest thatthe ratio r = a (KMI) /a (LSD) is smaller than one. Using Eq. (3.1) together with Eq. (4.5)below, it follows that the KMI mass range is indeed higher than the LSD mass range, inagreement with the physical picture reflected in Fig. 5 of Ref. [2]. But, we are unable toturn this conclusion into a more quantitative statement.We close this section with a comment. As discussed above, our experimentation with t (and its chiral extrapolation) suggests that r <
1. Now, an alternative way to estimate r would be to take advantage of the fact that ˆ f π , the chiral-limit value of the pion decayconstant, is a physical observable. Expecting √ f π (LSD) ≈ ˆ f π (KMI) in physical units, itfollows that a ˆ f π (KMI) / ( √ a ˆ f π (LSD)) ≈ r . The reason why we only expect an approximateequality between √ f π (LSD) and ˆ f π (KMI), is the different scaling violations of the twolattice actions. In reality, using the value of a ˆ f π (LSD) from Ref. [2], and taking a ˆ f π (KMI) ∼ .
01, we find a ˆ f π (KMI) / ( √ a ˆ f π (LSD)) ∼
10, in stark conflict with the estimate r < per-se can accountfor this inconsistency. The problem must be related to the long extrapolation to the chirallimit inherent in the extraction of a ˆ f π . It does not necessarily imply that ( γ -)dChPT cannotbe trusted. The factor e v ( m ) = F π / ˆ f π is very sensitive to m , which makes a long extrapolationto the chiral limit much more difficult than in the case of QCD. For at least one of the datasets our fit result for a ˆ f π is likely to contain a large, and unaccounted for, source of systematicerror. A comparison of the values of ad obtained from the two data sets reveals a similar,and, in fact, more severe, problem, which presumably have a similar source, given that d = ˆ f π / (2 ˆ B π ). We comment that in order to compare a ˆ B π between the LSD and KMIlattice scales we have to apply an RG transformation, but once again, it is hard to see howsuch a transformation would suffice to match the values of d found in the two simulations. To make sure that the same physical mass range is covered, one can, for example, monitor the values ofsome observable, such as a hadron mass or a decay constant, in units of √ t . The factor of √ V. THE ∆ CLASS OF DILATON POTENTIALS
So far, we have considered a model modification of the LO dChPT form of L m , based onthe observation that the coupling of the underlying theory may start running at the physicalscale determined by a growing fermion mass, thereby inducing a varying mass anomalousdimension as well. In this section we turn to a class of modifications to the dilaton-potentialterm L d . Alternate forms of the dilaton potential were first applied to the LSD data inRef. [28]. In Ref. [32] a class of dilaton potentials L ∆ was proposed, defined by (compareEq. (2.6)) L ∆ ( τ ) = ˆ f τ ˆ B τ e τ V ∆ ( τ ) , (4.1a) V ∆ ( τ ) = c − ∆ (cid:18) − e (∆ − τ (cid:19) , (4.1b)where ∆ is a new free parameter. We have translated the notation of Ref. [32] to ournotation. In the limit ∆ →
4, the potential L d of Eq. (2.6a) is recovered. For ∆ = 2, L ∆ becomes the linear σ -model potential considered in Ref. [8]. We will refer to the low-energylagrangian with L d replaced by L ∆ as ∆-dChPT.Applying ∆-dChPT to the LSD data, Ref. [32] concluded that these data appear tofavor a value of ∆ around 3 .
5, with a large uncertainty. Correlations in these data werenot taken into account [32]. Moreover, correlations which occur because of the appearanceof F π in all three equations fitted in Ref. [32], as well as the appearance of M π in two ofthem, apparently were not taken into account either. In Sec. IV A we begin by collecting theexpressions needed to fit ∆-dChPT. In Sec. IV B we revisit the determination of ∆ using theLSD data, taking all correlations into account. This analysis departs from the framework ofLO dChPT (Sec. II A) only by replacing the dilaton potential L d by L ∆ . At this stage themass anomalous dimension is held fixed, cf. Eq. (2.2b). Then, in Sec. IV C, we explore fitsof the KMI data to the ∆ class of potentials. As in the previous section, we consider bothfixed- γ m fits to subsets of the KMI data, as well as fits with a varying γ m to the entire KMIdata set. We summarize our findings in Sec. IV D.Unlike the modification of L m to accommodate a running γ m , we are not aware of aconcrete physical motivation to replace L d by the more general form L ∆ . A closely re-lated question is whether or not ∆-dChPT is the leading order in a systematic low-energyexpansion for an arbitrary value of ∆.The potential L d , Eq. (2.6a), is based on the systematic power counting developed inRef. [17]. Since L d corresponds to the limit ∆ → i.e. , the low-energylagrangian consisting of Eq. (II A) with L d replaced by L ∆ , should thus be considered to bea model. A. Fitting data to L ∆ For the case of a constant γ m = γ ∗ , combining Eq. (4.1) with L m of Eq. (2.2b), one finds L ∆ ( τ ) is bounded from below for any −∞ < ∆ < ∞ . v to m , m = d d − e (∆ − v − ∆ e (1+ γ ∗ ) v . (4.2)It is then straightforward to derive the relations M π F π = 1 d − e (∆ − v − ∆ ≡ h ∆ ( m ) , (4.3a) F π = ˆ f π e v (4.3b)= (cid:18) d mh ∆ ( m ) (cid:19) γ ∗ , (4.3c) M τ F π = d (cid:16) γ ∗ − d h ∆ ( m ) (cid:17) , (4.3d)where we used the definitions (2.10).In the case of a varying γ m , Eq. (4.3b) is still applicable, while combining Eq. (4.1) with L m of Eq. (2.42), Eqs. (4.2) and (4.3a) generalize to m = d ˜ d − e (∆ − v − ∆ e v + F ( v ) − γ m , (4.4a) M π F π = 1˜ d (3 − γ m ) 1 − e (∆ − v − ∆ , (4.4b)where γ m is given in Eq. (2.45), and ˜ d is defined in Eq. (2.48).We now turn to fits of the LSD and KMI data, in order to explore to what extent theyconstrain the value of ∆. We emphasize again that this investigation is empirical, as nosystematic power counting is available for this model for arbitrary values of ∆. B. The LSD data
Data reported in Ref. [3] includes results at five different fermion masses, am i ∈ { . , . , . , . , . } . (4.5)All ensembles have the same bare coupling, and, in a mass-independent scheme, the samelattice spacing [33]. We fitted the LSD data to LO dChPT in Ref. [33]. Here, we repeatsome of those fits replacing L d by L ∆ , keeping ∆ as a free parameter. Our results are shownin Table 4. These fits correspond to four fits presented in Ref. [33]: Fits 4A and 4B are to becompared to the fits shown in Table 1 of Ref. [33], while fits 4C and 4D are to be comparedwith the third column of Table 3 and the second column of Table 4 in Ref. [33].As discussed in great detail in Ref. [33], it is not possible to fit all parameters in the taste-breaking sector with the available LSD data. Here we kept those taste-breaking parametersthat gave rise to the best fits of Ref. [33]. Furthermore, in Ref. [33] we argued that four-ensemble fits, which exclude the ensemble with the largest fermion mass, are better behaved.While the five-ensemble fits reported in Table 4 already have good p -values, again we findthat p -values for the four-ensemble fits are significantly better.24 B C Domitted — 0.00889 — 0.00889 χ /dof 8.72/9 2.50/6 15.18/13 5.52/8 p -value 0.56 0.87 0.30 0.70∆ 2.8(7) 3.5(7) 2.7(6) 3.5(7) γ ∗ d d − log( ad ) 11.6(9) 12.9(2.5) 11.3(7) 12.6(2.1) d − log C — — — — γ — — — — − log C — — 9.7(6) 10(2) γ — — 2.0(1) 2.4(7) − log C — — 8.3(7) 10(2) γ — — 1.96(6) 2.1(4) − log C — — 36(7) 17(11) γ — — − Fits of the LSD data to ∆ -dChPT. The fits to the right of the double vertical lineinclude taste breaking; those to the left do not. The “omitted” row shows bare mass values fromthe set (4.5) which are not included in the fit, if any. Parameter values for γ ∗ and log d are in good agreement with the corresponding fits inRef. [33]. The parameters d and d are very poorly determined by the fits; especially bythose with four ensembles. This is no surprise, as d and d relate directly to the dilatonpotential L ∆ , in which now a new parameter, ∆, has been introduced. The results for thetaste-breaking parameters are in reasonable agreement with Ref. [33] for the five-ensemblefit, and in good agreement for the four-ensemble fit. By holding ∆ fixed in the fit, we verifiedthat in the limit ∆ → . →
4, is the correct low-energy EFT. The linear σ -model value, ∆ = 2, isdisfavored. By contrast, the values found in the five-ensemble fits average to 2.8(7). This is1 . σ away from ∆ →
4, and, in fact, between the two options, it slightly favors the linear σ -model value. C. The KMI data
We next turn to fits of the KMI data, with L ∆ replacing L d . We first consider againwindow fits similar to those of Table 1, but now with ∆ an additional free parameter. Theresults are reported in Table 5. The fits are reasonably consistent with ∆ = 4, while the25 ange 0.012–0.04 0.015–0.05 0.02–0.06 0.03–0.07 0.04–0.08 0.05–0.1 χ /dof 9.16/5 8.11/5 4.81/5 3.42/5 3.69/5 3.82/5 p -value 0.10 0.15 0.44 0.64 0.59 0.57∆ 3.8(5) 4.4(1) 4.0(5) 3.2(8) 3.3(8) 4.0(6) γ ∗ a ˆ f π d − log( ad ) 9.7(1.3) 20(24) 9.5(1.7) 7.8(7) 7.7(8) 8.8(2.1)TABLE 5. Fits of the KMI data to ∆ -dChPT (with a constant γ m = γ ∗ ), with selections of fivesuccessive fermion masses in Eq. (3.1), shown in the top row. other parameters are generally consistent between Tables 5 and 1. As before, a constant γ m is not sufficient to describe the KMI data over the full mass range. However, while γ ∗ varieswith the mass range selected in the fit, ∆ does not. If we compare the values of ∆ betweentwo of the fits in Table 5, these values are always consistent within the smaller of the twoerrors (with the exception of the second fit, for which ∆ has an anomalously small error).The first and last values, 3 . . L m of Eq. (2.42), and a varying γ m as defined in Eq. (2.46). As before, this introduces twomore parameters ( b and c ) into the fits, for a total of seven parameters. We will refer to thisflavor of the low-energy lagrangian as γ ∆-dChPT.In Table 6 we show a scan in ∆: at each chosen value of ∆, we fit the other six parameters.The fit for ∆ = 3 . p -value rapidly decreases, dipping below 0 .
01 for ∆ < .
8. We verified thatthe p -value keeps decreasing down to ∆ = 2 (where the p -value is of order 10 − ). If weincrease ∆ above 4, the p -value increases until ∆ reaches 4.5, where the p -value appearsto start decreasing again. However, we found that fits with ∆ ≥ . .
5, essentiallyall of them are not determined by the fit. We have repeated the fits of Table 6 omitting the am = 0 . am = 0 . .
08 ensembles, and we have also redone suchfits setting c = 0 (as in fit 2D). The conclusions are always the same as for the fits shownin Table 6. The fit at ∆ = 3 . . . L ∆ potential, we have not attemptedto include taste splittings in the KMI case. D. Discussion
Taking the fits of the LSD and KMI data together, it is clear that no very precise state-ment about the value of ∆ can be made. The KMI data appear to exclude the σ -modelvalue ∆ = 2. dChPT, which corresponds to ∆ → γ m = γ ∗ , is consistent with26 χ p -value a ˆ f π ˜ d − log( ad ) γ b c ∗ − − Fits of M π /F π and aF π to γ ∆ -dChPT, for fixed values of ∆ . All fits have 14 degreesof freedom. The fit with the asterisk may not have fully converged, and its χ value is an upperbound to the true minimum. the fits shown in Tables 4 and 5. An exception is the second window fit, fit 5B, which yieldsa result with a rather small error, ∆ = 4 . L d replaced by L ∆ of Eq. (4.1). However, we do not wish to implythat attempts to understand data in terms of models are not interesting. Fits to models,including ∆-dChPT (with ∆ not constrained to be close to 4), can provide a valuable “stresstest” of dChPT. This is why we considered fits of the LSD and KMI data to ∆-dChPT;Ref. [32] can be seen as a similar exploration of only the LSD data.Fits of the LSD data, comparing in particular the values ∆ = 2 and ∆ →
4, wereconsidered also in Ref. [8]. There, it was found that both dChPT and ∆-dChPT with∆ = 2 provide good fits to data using all five of the LSD ensembles. This finding agreeswith our fits in Table 4: fits 4A and 4C are consistent with ∆ = 2, but are less than ∼ σ away from ∆ = 4.In summary, a precise determination of the favored value of ∆ is not possible withpresently available data. Taking the results based on fits to both the LSD and KMI datatogether, we arrive at an estimated range for ∆,3 . < ∆ < . . (4.6)Our lower bound is based on the four-ensemble fits to the LSD data, which favor a valuearound ∆ ∼ .
5, combined with the γ ∆-dChPT scan of Table 6, which strongly disfavorsvalues below 3.8. Any fit of the KMI data set must somehow account for the running of γ m . Including higher orders systematically is not an option here, because, as we prove inApp. B, the claim of Ref. [32] that ∆-dChPT admits a systematic expansion is incorrect.The model alternatives are to use a fixed value of γ m while limiting the mass range as inthe “window” fits, or else to use an explicitly varying γ m function. As for the window fits, See also Ref. [9] for related studies of the SU(3) theory with two sextet fermions, which also has a lightflavor-singlet scalar. We recall, however, that dChPT is strictly speaking not applicable to this theory, asthe Veneziano limit can be taken only for fermions in the fundamental representation. < .
5, the fits of Table 6 to the KMI data strongly disfavor ∆ < .
8. Based on allfits together, the σ -model value ∆ = 2 appears to be excluded. Once again, the caveatsdiscussed in the previous section regarding the LSD and KMI data sets, and, in particular,the lack of information about scaling violations, apply also to our conclusions in this section. V. CONCLUSION
Our main goal in this paper was to confront the EFT framework provided by dChPTwith the KMI data for the eight-flavor SU(3) gauge theory [4]. The KMI simulations wereperformed at larger fermion masses than the LSD ones [3], taking the theory further awayfrom conformality. Hence, even with the successful application of LO dChPT to the LSDdata, which we reported on in Ref. [33], there is no guarantee that LO dChPT can also beapplied to the KMI data.Indeed, we found that the full fermion-mass range of the KMI data cannot be fitted toLO dChPT. The natural next step would be to attempt an NLO fit in dChPT. However, aswe explained in Sec. III, this is not feasible with presently available data. First, the largenumber of parameters involved in any NLO dChPT fit requires extensive precision data fora successful fit. Moreover, the KMI data set (and, likewise, the LSD data set) has only asingle lattice spacing, making a continuum extrapolation impossible.Instead, we introduced γ -dChPT, a model extension of LO dChPT with a scale-dependentmass anomalous dimension, which can be interpreted as arising from partially resumminghigher orders in the EFT expansion. We found that γ -dChPT provides a successful descrip-tion of the KMI data over the entire mass range.Given the success in describing the LSD data using LO dChPT [33], and the KMI datausing γ -dChPT with a relatively simple ansatz for the γ m function, the question ariseswhether γ -dChPT can be used to fit the LSD and KMI data simultaneously. Over the KMImass range, γ m would then have to increase as the fermion mass is decreased, eventuallysaturating to a constant when reaching the lower LSD mass range (see Fig. 3). Once again,however, the inability to take the continuum limit makes it impossible to carry out thisprogram at this time. The lack of information on the lattice spacing dependence is evenmore severe when trying to consider the LSD and KMI data sets together, because theywere produced with different lattice actions, and thus, their scaling violations for any givenphysical observable are different functions of the corresponding lattice spacing.We also considered ∆-dChPT—another generalization of LO dChPT in which the dilatonpotential is replaced by a class of potentials depending on a new parameter ∆. We emphasizethat ∆-dChPT does not allow for a systematic power counting, and should thus be considereda model, except in the limit ∆ → γ m . We concluded that thepreferred range of our combined analysis of the LSD and KMI data is 3 . < ∆ < .
5. Thisis centered around ∆ = 4, where ∆-dChPT reduces to LO dChPT.Recently, LO dChPT has also been successfully applied to the light sector of the SU(3)28auge theory with four light and six heavy flavors [10]. dChPT provides for a systematictreatment of the pNGBs, the pions and the dilaton, of a near-conformal gauge theory, but itdoes rest on certain assumptions [17, 20]. These initial successes are thus encouraging. Wehope that, in the future, more extensive and refined data will become available, allowing forfurther and more stringent tests of dChPT.
Acknowledgments
We thank Julius Kuti for discussions, and for asking probing questions about the relationof γ m to dChPT. MG’s work is supported by the U.S. Department of Energy, Office of Sci-ence, Office of High Energy Physics, under Award Number DE-SC0013682. YS is supportedby the Israel Science Foundation under grant no. 491/17. Appendix A: Scale setting prescription
Any analysis of lattice data requires a scale setting prescription, and the basic choice isbetween mass-independent or mass-dependent prescriptions. In this paper, as in Ref. [33],we opted for a scale-independent prescription, and confirmed the self-consistency of thischoice by checking that the values of a ˆ B π on all ensembles agree within error (see Sec. III).Here we discuss the alternative of using a mass-dependent prescription. In QCD simu-lations it is common nowadays to use the gradient flow scale t for scale setting [42]. Inparticular, the ensemble value of t can be used for a mass-dependent prescription. Whatmakes t particularly convenient for setting the scale is that it can be determined with highprecision, and it admits a chiral expansion, with non-analytic terms in the quark mass enter-ing only at NNLO [43]. By contrast, as we showed in Ref. [33], in dChPT there is no (useful)chiral expansion for t . This implies that one cannot derive expansions for dimensionlessquantities such as √ t M π , √ t F π , etc. , using dChPT.If we are interested in dChPT fits, we are thus unable to use t for scale setting. Instead,we may consider using a physical quantity such as F π for a mass-dependent scale settingprescription. As explained in Sec. II D, with our mass-independent scale setting, the basic fithas six parameters, two of which, namely d and ˆ f π , have mass dimension equal to one. Oneach ensemble, we fit aF π and M π /F π to Eqs. (2.8b) and (2.50), respectively. In addition, wetreat the fermion mass am as a data point with a small fictitious error, in order to determinethe expectation value v of the dilaton field on each ensemble, via Eq. (2.47).If we use F π for mass-dependent scale setting, we may still fit M π /F π to Eq. (2.50) asbefore. In addition, combining Eqs. (2.8b) and (2.47) together, we may fit m/F π as mF π = d ˆ f π d (3 − γ m ) v e F ( v ) , (A1)and use it to determine v , now as a function of m/F π . This procedure gives us access only tothe ratio d / ˆ f π , instead of to ad and a ˆ f π separately, as in the fitting procedure of Sec. II D.We tried to repeat the fits from Table 2, using Eqs. (2.50) and (A1). The result wasthat almost all fit parameters remained completely undetermined. The nominal fit qualitywas always very high ( p -value ≥ . × − v i ’s are determined in terms of the am i ’s using Eq. (2.47), this does notchange the number of degrees of freedom. By contrast, within the mass-dependent fittingprocedure, we have in total only 20 relations to determine both the 5 fit parameters and the10 auxiliary v i ’s. This leaves us with just 5 degrees of freedom, which apparently is just notenough to resolve the fit parameters.Although we were unable to actually perform a fit with a mass-dependent prescriptionfor scale setting, we may consider the following “thought experiment.” Assume that thedata allowed for fits with a mass-dependent prescription, and that the results of those fitsare in agreement with Table 2. This would mean, in particular, that the value of the newdimensionless fit parameter, d / ˆ f π , obtained from fitting Eq. (A1) is consistent with theresults for a ˆ f π and ad reported in Table 2. Now, while we determine all parameters in thelarge-mass regime, which is where both the LSD and KMI data are, ˆ f π and d are LECsthat characterize the massless theory. As we discussed in Sec. III E, the values of a ˆ f π and ad extracted from the LSD and the KMI data sets appear to be in conflict, both with thechiral limit values of t (determined from a phenomenological fit), and with each other. It isinteresting to check what is the situation for the dimensionless ratio d / ˆ f π . Comparing thevalue of this ratio using the results in Table 2 to those from Ref. [33] reveals that there isstill a significant conflict, of roughly the same size as for a ˆ f π , though smaller than for ad .We conclude that restricting ourselves to a mass-dependent scale setting prescription wouldnot by itself alleviate the problem of the long extrapolation from the large-mass regime tothe chiral limit. Appendix B: Power counting
In Ref. [32] it was proposed that ∆-dChPT—in which the potential L d of Eq. (2.6a)is replaced by L ∆ of Eq. (4.1), with ∆ a new free parameter—admits a systematic powercounting. In this appendix, we show that the arguments given in Ref. [32] are not correct.The potential (4.1) was already considered in Refs. [24, 25]. In those papers it wasassumed that the lagrangian of the underlying theory contains an operator with scalingdimension ∆, with some unspecified value of ∆, and a coupling which may be small. Thisnaturally leads to the consideration of potentials such as Eq. (4.1) in the EFT describingthe same theory at low energy.By contrast, here the underlying theory is known: it is the asymptotically free SU(3)gauge theory with N f = 8 Dirac fermions in the fundamental representation. This theorydoes not fall into the class of theories considered in Refs. [24, 25].It is instructive to briefly recall how the breaking of scale invariance is introduced intothe (massless) quantum theory; and then, how this breaking translates to the EFT [17]. Ascan be seen in Eq. (2.16), regularizing the bare lagrangian requires the introduction of ascale factor, µ d − , with the limit d − → µ transform according to Eq. (2.17b), we may promote µ to a spurion, formally restoringscale invariance.In making the transition to the EFT, we will want to use the well-known fact that theEFT lagrangian must be analytic in the spurion fields, if the underlying lagrangian is analyticin the (same set of) spurions. Correlation functions can then be generated by differentiatingthe partition function of the EFT with respect to the spurion fields, and compared with30heir counterparts in the underlying theory by applying the same derivatives again. Thismatching procedure fixes the LECs of the EFT order by order, according to the powercounting.A technical obstacle is that the action (2.16) is non-analytic in the spurion µ . Toovercome this problem, we introduce a new spurion field σ ( x ), and replace µ ⇒ ˆ µ e σ ( x ) . (B1)The new scale transformation rules replacing Eq. (2.17b) are σ ( x ) → σ ( λx ) + log λ , (B2a)ˆ µ → ˆ µ . (B2b)Now ˆ µ is invariant under a scale transformation, which in turn is “carried” by the constantmode of the new spurion field. Writing σ ( x ) = σ + δσ ( x ), with the constraint (cid:82) d d x δσ ( x ) =0, it follows that σ → σ + log λ . (B3)With this replacement, the bare action (2.16) becomes S = ˆ µ d − (cid:90) d d x e ( d − σ ( x ) L ( x ) . (B4)Classically, the σ ( x ) dependence vanishes for d →
4, showing that any dependence of therenormalized theory on σ ( x ) represents quantum breaking of scale invariance [17, 19, 38].Since the underlying theory is now analytic in the spurion field σ ( x ), so must be the EFT[17]. Note that if, instead, one were to use µ as a scale spurion, there would be no reasonfor the EFT to be analytic in µ , for the simple reason that the underlying theory (2.16) isnon-analytic in µ . In Ref. [32], the starting point of the argument was to assume that the lagrangian of thelow-energy theory depends analytically on a spurion field ˜ µ ( x ), with the scale transformationrule ˜ µ ( x ) → λ − ∆ ˜ µ ( λx ) . (B5)It is clear from the previous discussion that the underlying gauge theory does not accom-modate such a spurion. Comparing transformation rules, one can, however, make the iden-tification ˜ µ ( x ) ≡ e (4 − ∆) σ ( x ) . (B6)As we have just shown, the correct EFT must be analytic in σ ( x ), but not in e σ ( x ) (nor inany power of e σ ( x ) ). It follows immediately that the EFT must be analytic in log ˜ µ ( x ), butnot in ˜ µ ( x ) itself. This proves that the arguments of Ref. [32] are not valid, because theincorrect assumption that the EFT is analytic in ˜ µ ( x ) served as their starting point.While this proves that the power counting claimed in Ref. [32] is unfounded, severalcomments are in order.First, we draw the reader’s attention that in Sec. II C we made use of the original spurion µ , instead of σ ( x ). The reason is that our goal in Sec. II C was to derive the extension The same statement applies if the constant spurion µ is promoted to a field. The spurion ˜ µ ( x ) is denoted as λ ( x ) in Ref. [32], see Eq. (A1) therein. We have reserved λ for the scaletransformations parameter, which in turn is denoted as e ρ in Ref. [32].
31f LO dChPT to the case of a running γ m . This requires mainly the consideration ofrenormalization-group and scale transformation properties, and, for this purpose, using µ as a (constant) scale spurion is sufficient. The γ -dChPT framework developed in Sec. II Cdoes deviate from the strict power counting of dChPT [17], though it can be viewed as aresummation of contributions from all orders under the assumption that these dominate. Asfor establishing the power counting itself, this necessitates the replacement of µ by ˆ µ e σ ( x ) , cf. Eq. (B1). Correspondingly, the transformation rules (B2) take over the transformationrule of µ in Eqs. (2.17) and (2.32). For the actual proof of the power counting, and adetailed discussion of the assumptions that it requires, we refer to Refs. [17, 20].A key step in constructing a power-counting scheme is the identification, in the underlyingtheory, of a small parameter in terms of which the EFT expansion is to be organized. Inordinary ChPT, the small parameter is the fermion mass m , which is also the “expectationvalue” of the chiral spurion, (cid:104) χ ( x ) (cid:105) = m . Chiral symmetry is restored for m →
0, which inturn allows to establish that the pion mass is parametrically small.By contrast, in Ref. [17], the small parameter controlling the hard breaking of scaleinvariance was identified as n f − n ∗ f , serving as a proxy for the β function at the chiralsymmetry breaking scale. More precisely, the hypothesis made in Ref. [17] is˜ β ∼ | n f − n ∗ f | η , n f (cid:37) n ∗ f , (B7)for some η >
0, where ˜ β = µ α ∂α∂µ , (B8)and α is the ’t Hooft coupling, α = g N/ (4 π ), evaluated at the chiral symmetry breakingscale. While we often assume η = 1 for simplicity, including earlier in this paper, thisassumption is not essential. The power counting is valid for any fixed η >
0; the η dependenceis restored trivially via the substitution | n f − n ∗ f | ⇒ | n f − n ∗ f | η .The small parameter | n f − n ∗ f | η does not appear explicitly in the underlying lagrangian,and, in particular, it is not identified with the expectation value of σ ( x ). Indeed, unlikechiral symmetry, which is restored for m = 0, there is no fixed value of σ ( x ) for whichscale invariance is not broken. Rather, the expansion of correlation functions in powers of σ ( x ) corresponds to an expansion in the number of insertions of the trace anomaly. In themassless limit, every such insertion is proportional to the β function at the chiral symmetrybreaking scale, hence to | n f − n ∗ f | η . For this argument to work, it is crucial to use the spurionfield σ ( x ), and not µ or ˜ µ ( x ). The role of σ ( x ), or of its constant mode σ , is analogousto that of the θ parameter in the large- N c limit of ChPT in which the U(1) A symmetry isrestored. For a detailed comparison, we refer to Ref. [17]. The upshot is that one cannotestablish a relation between the expectation value of µ or ˜ µ ( x ) and the β function of theunderlying theory. Hence, even if one were to allow the low-energy theory to depend onlyon integer powers of the ˜ µ ( x ) spurion, as was postulated in Ref. [32], there is no reasonto assume that its expectation value should tend to zero when the conformal window isapproached. Both assumptions of Ref. [32], analyticity in the spurion ˜ µ , and its smallnessin the conformal limit, are thus in conflict with the properties of the underlying theory, and,in general, not valid.This concludes our discussion of the claims made in Ref. [32] with regard to power count-ing. But, a little more can be said about the connection of the potential L ∆ with dChPT, In accordance with our general reasoning, in Eq. (2.28), L m indeed depends on log µ . →
4. According to the dChPT power counting developedin Ref. [17], the scale invariant dilaton potential e τ is multiplied by a potential ˜ V d ( τ ) thatbreaks scale invariance, ˜ V d ( τ ) = ∞ (cid:88) n =0 ˜ c n n ! τ n . (B9)The LECs ˜ c n scale as ˜ c n ∼ | n f − n ∗ f | nη , and with the power counting (compare Eq. (2.3)) p ∼ m ∼ | n f − n ∗ f | η ∼ /N , (B10)it follows that the term ˜ c n n ! τ n can only appear at N n − LO in dChPT. In particular, the tree-level potential V d ( τ ) of Eq. (2.6b) obtained after the τ shift corresponds to c = ˜ c = − c .When ∆ is close to 4, we may identify ˜ V d ( τ ) = V ∆ ( τ ), which, using Eq. (4.1b), impliesthat ˜ c = − c / ∆ and ˜ c n = (4 c / ∆)(∆ − n − , n ≥ . (B11)The first two terms in the expansion reproduce the LO potential V d ( τ ). It follows that,for any fixed value of ∆ such that | ∆ − | ∼ | n f − n ∗ f | η with any η >
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