Spectroscopy of SU(4) composite Higgs theory with two distinct fermion representations
Venkitesh Ayyar, Thomas DeGrand, Maarten Golterman, Daniel C. Hackett, William I. Jay, Ethan T. Neil, Yigal Shamir, Benjamin Svetitsky
SSpectroscopy of SU (4) composite Higgs theory with two distinctfermion representations
Venkitesh Ayyar, Thomas DeGrand, Maarten Golterman, Daniel C. Hackett, William I. Jay, Ethan T. Neil,
1, 3, ∗ Yigal Shamir, and Benjamin Svetitsky Department of Physics, University of Colorado, Boulder, Colorado 80309, USA Department of Physics and Astronomy, San Francisco State University,San Francisco, CA 94132, USA RIKEN-BNL Research Center, Brookhaven National Laboratory,Upton, New York 11973, USA Raymond and Beverly Sackler School of Physics and Astronomy,Tel Aviv University, 69978 Tel Aviv, Israel (Dated: May 9, 2018)
Abstract
We have simulated the SU(4) lattice gauge theory coupled to dynamical fermions in the funda-mental and two-index antisymmetric (sextet) representations simultaneously. Such theories arisenaturally in the context of composite Higgs models that include a partially composite top quark.We describe the low-lying meson spectrum of the theory and fit the pseudoscalar masses and de-cay constants to chiral perturbation theory. We infer as well the mass and decay constant of theGoldstone boson corresponding to the non-anomalous U(1) symmetry of the model. Our resultsare broadly consistent with large- N c scaling and vector-meson dominance. PACS numbers: 11.15.Ha, 12.39.Fe, 12.60.Rc, ∗ [email protected] a r X i v : . [ h e p - l a t ] M a y . INTRODUCTION Gauge theories coupled simultaneously to more than one fermion representation—“multirep” theories—open a new dimension in the study of gauge dynamics. Apart fromthe influence of each fermion species on the gauge field and vice versa, phase transitionsand symmetry breaking in each species can affect the others dramatically. Of course, QCDalready contains light quarks, strange quarks, and heavy quarks, and the influence of eachspecies on the others is an old and continuing object of QCD calculations. The differenceis that QCD’s quarks are all equivalent, in that a tuning of the masses can change one intoanother. Fermions in inequivalent representations, on the other hand, enter the dynamicswith different strengths irrespective of their masses.As usual, symmetries offer the clearest perspective on the physics of inequivalent fermions.Each species has its maximal flavor symmetry, while no symmetries mix the different species.If all the fermions are made massless, the chiral symmetries of the species remain distinct.One symmetry could break spontaneously while others do not. This is a generalization ofthe old issue of scale separation, which was originally seen as a possible separation of a chiralscale from the confinement scale of the gauge theory [1–3]. It is possible that inequivalentrepresentations, simultaneously coupled to the gauge field, define independent chiral scales.This might find expression in the finite-temperature physics of the theory, in the form ofdistinct phase transitions for each fermion species as well as for the confinement physicsof the gauge field. Alternatively, one phase transition, possibly governed by the largestquadratic Casimir of the fermion representations, might trigger all the others to occur atthe same scale.We present here the first results of our work on the SU(4) gauge theory with N f = 2Dirac fermions in each of two distinct representations, the fundamental and two-indexantisymmetric (a real representation). We have chosen this model because it is close to amodel proposed by Ferretti for a hypercolor theory that yields a composite Higgs boson [4, 5]and a partially composite top quark [6]. Ferretti’s model [7] contains 5 Majorana fermions inthe sextet representation and 3 Diracs fermions in the fundamental; simulating this fermioncontent requires the costly rational hybrid Monte Carlo (RHMC) algorithm, and so, instead,we study the theory with 4 Majoranas (equivalent to 2 Dirac fermions) in the sextet and2 Diracs in the fundamental. In Ferretti’s model, the massless sextet Majorana fermionsΨ condense to break their chiral symmetry according to SU(5) → SO(5), whereupon theStandard Model’s Higgs multiplet appears as Nambu–Goldstone (NG) bosons; our symmetrybreaking scheme is SU(4) → SO(4). The fundamental fermions ψ ( ) in Ferretti’s modelare brought in so that the theory will possess fermionic baryons constructed as ψ ( ) ψ ( ) Ψ“chimera” bound states, to be used as top partners; they condense (again, in the chirallimit) according to SU(3) L × SU(3) R → SU(3) V . In our model, the corresponding symmetry-breaking scheme is SU(2) L × SU(2) R → SU(2) V . We believe that our model contains all thequalitative physics of Ferretti’s model while offering a laboratory for developing quantitativetechniques.Multirep theories of physical significance are not easy to come by. Apart from the phe-nomenological requirements, Ferretti’s choice of model is constrained [8] by the simple factthat higher-representation matter fields push gauge theories into the conformal window un-less the number of fermions is quite small. It is essential that the gauge theory of hypercolor This scheme is not directly useful for model building since the SU(4) / SO(4) coset does not accommodatethe Higgs field. Thoseresults fit nicely into the body of work on QCD and its generalizations to larger values of N c . The analysis there, similar to QCD studies, related the gauge coupling β and hoppingparameter κ to a physical scale r (the Sommer scale) and the quark mass m q , and used thelatter as an abscissa for plotting particle masses and decay constants. Here, of course, thespace of bare couplings consists of the gauge coupling β and two hopping parameters κ and κ for the two fermion species. We translate these into the scale parameter t , derived fromthe Yang–Mills gradient flow, and the two quark masses m and m .Our main tool for understanding the meson spectrum is a recent generalization of chiralperturbation theory ( χ PT) to the low-energy sector of a two-representation theory [12].This form of χ PT provides formulas for masses, decay constants, and chiral condensates atnext-to-leading order, with m and m as independent variables. These formulas contain animportant qualitatively new piece of physics compared to QCD—communication betweenthe different species. They describe, for instance, the dependence of the masses of the NGbosons of all the broken chiral symmetries on both fermion masses.Another new feature of the two-representation theory is the existence of a non-anomaloussinglet axial current, and a corresponding singlet NG boson that must be included in thelow-energy chiral theory. This particle is denoted ζ in Ref. [12] and is of significant phe-nomenological interest for composite Higgs models [8, 13, 14]. In this work we do not probethis singlet pseudoscalar state directly. Nevertheless we extract information about it indi-rectly, via its virtual contributions to the properties of the flavored NG bosons associatedwith chiral symmetry breaking of the individual representations. In particular, its decayconstant in the chiral limit is a parameter in the chiral Lagrangian and thus appears as afit parameter, allowing us to infer its mass using the leading-order formula.Besides the pseudoscalar channel, we calculate masses and matrix elements of the lightestvector bosons. The vector is the lightest narrow resonance in QCD, and its properties areclosely related to those of the pseudoscalars within the framework of vector meson dominance(VMD). We explore the evidence for VMD in our theory and its consequences for the decaywidth of the vector. This is of particular phenomenological interest, since in composite Higgsmodels, the vector resonance is often one of the first signatures expected in collider searches.The paper is organized as follows. In Sec. II we describe the lattice theory, the observableswe use, and ensembles we generated. In Sec. III we describe our application of χ PT, includingthe discretization effects of Wilson fermions, and our scale setting method which is basedon t . In Sec. IV we present our results for the pseudoscalar spectrum and decay constants,including the flavor singlet ζ . We present the vector particles in Sec. V and use VMD toestimate decay widths. In Sec. VI we discuss our results from the point of view of large- N c predictions, and present our overall conclusions.The tables containing the various measured quantities have been collected together inAppendix A. In Appendix B we explain technical aspects of our analysis of lattice data. InAppendix C we review the definition of the U(1) axial current and of the mass parameter inWilson χ PT. Finally, Appendix D contains a calculation of perturbative Z -factors for thenHYP lattice action with dislocation-suppression. A preliminary exploration of the chimera states—using configurations generated with only fundamentaldynamical fermions—was presented in [11]. I. THE LATTICE THEORYA. Symmetries
The chiral symmetry of the fundamental fermions and its expected breaking are the sameas in two-flavor QCD. The specifics of chiral symmetry breaking for the sextet representationare less well-known, so we will discuss them briefly; a more detailed explanation is given in[10, 12].The sextet representation of SU(4) is a real representation. Our model has two Diracfermions charged under this representation, ψ ( ) i , ¯ ψ ( ) i , i = 1 , , which are equivalent to fourMajorana fermions Ψ I , I = 1 , . . . ,
4. The global symmetry of the continuum theory isthus also SU(4). Using the language of Majorana fermions, the bilinear condensate (cid:104) Ψ I Ψ J (cid:105) is symmetric in its Majorana-flavor indices. Hence, after spontaneous symmetry breakingone expects the unbroken symmetry to be SO(4) [15]. One consequence of the enlargedsymmetry is that ¯ ψ ( ) ψ ( ) mesons and ψ ( ) ψ ( ) diquarks (both gauge-singlet objects) aremembers of a degenerate multiplet of the unbroken group.As usual, the chiral symmetries of the theory are explicitly broken by the Wilson termin the lattice action. The lattice theory thus has the same flavor symmetry as expectedin the continuum theory after spontaneous symmetry breaking: SU(2) V × U(1) B for thefundamental representation and SO(4) for the sextet. Our use of Wilson fermions thusassumes that the spontaneous breaking of chiral symmetries is as would be forced by abilinear condensate, and all measured correlation functions reflect this.A special feature of the two-representation theory is the existence of a conserved U(1)axial current. While the individual U(1) currents J ( )5 µ and J ( )5 µ are anomalous, one can forma linear combination J µ of these currents that decouples from F ˜ F . Condensation of eitherfermion species then spontaneously breaks the non-anomalous axial symmetry, giving riseto a singlet NG boson that we denote ζ . We review the normalization of the U(1) currentin Appendix C 1. B. Lattice action and parameters
Our lattice action contains gauge-field terms and two fermion actions, one for each rep-resentation: S = S gauge + S ( ) F + S ( ) F . (2.1)Each fermion action is a Wilson–clover action built of gauge links constructed by nHYPsmearing [16, 17]. In S ( ) F the smeared links are promoted to the sextet representation [10].There are two hopping parameters, κ and κ . We set both clover coefficients equal to unity, c SW = 1, a choice known to work well with nHYP smearing in QCD [18] and with fermionsin higher representations [19].The gauge-field action takes the form S gauge = βS plaq + γS NDS . (2.2)The first term is the usual plaquette action, while the second is an nHYP dislocation-suppression (NDS) term [20], constructed from the nHYP-smeared links. The NDS term isdesigned to avoid singularities in the nHYP smearing. For the present study, we hold theratio γ/β fixed at 1/125 and use β as a free bare parameter.4oncurrent with the work described here, we are also studying the finite-temperaturephase structure of the theory [21, 22]. Comparison of the sextet-only limit of this theory toearlier published results [10] shows that the use of the NDS action removes the previously-observed bulk transition from the interesting region of parameter space (see also Ref. [23]).In the multirep theory, we see no evidence for a bulk transition anywhere near the rangeof bare parameters at which we run, indicating that all of our ensembles correspond to theconfined continuum phase with broken chiral symmetry.We extract masses and decay constants in the usual way from two-point correlationfunctions. We denote pseudoscalar masses and decay constants in the representation r by M P r and F P r , respectively. The corresponding quantities in the vector channel are denotedby M V r and F V r .We define the fermion masses m and m by imposing the axial Ward identity (AWI), ∂ µ (cid:104) | A ( r ) µa ( x ) O r (0) | (cid:105) = 2 m r (cid:104) | P ( r ) a ( x ) O r (0) | (cid:105) , (2.3)where x (cid:54) = 0, and a is an isospin index. We use the local unimproved axial current A ( r ) µa andpseudoscalar density P ( r ) a in each representation r . For the determination of the AWI mass,we do not renormalize these currents because the mass itself is not a physical observables;based on our perturbative renormalization of these currents described in Appendix D (usedfor calculation of decay constants), the effect of including the renormalization would be smallanyway, amounting to a few-percent shift of the masses. For O r we take a pseudoscalarsource. When the distinction between representations is irrelevant, we will refer to thefermion mass defined by Eq. (2.3) as m AWI . Further information about our conventions andmethods for spectroscopy is given in Appendix B.
C. Scale setting
We set the scale in our simulations using the flow scale, t , introduced by L¨uscher [24].The flow scale is defined by the implicit equation t (cid:104) E ( t ) (cid:105)| t = C, (2.4)where E ( t ) = G aµν G aµν ( t ) is constructed from the clover form of the field strength G aµν atflow time t . Here C is a dimensionless number, conventionally [24] taken to be 0.3 in QCD.With this choice, √ t corresponds to a length scale of 0 .
14 fm ( i.e., an energy scale of 1.4GeV) in QCD simulations [25, 26].For an arbitrary gauge theory, any value for C is a priori as good as any other. However,for comparison to existing studies with different gauge groups, it is useful to let C vary with N c . Arguments from large- N c QCD, supported by lattice data [26, 27], suggest that t ∼ N c at leading order. For the SU(4) simulations of this work we therefore use t (cid:104) E ( t ) (cid:105)| t = 0 . ×
43 = 0 . . (2.5)Lattice calculations give masses as dimensionless numbers M a and gradient-flow scalesas t /a . Dimensionless products like ˆ M ≡ M √ t eliminate the lattice spacing a , and ourtables and figures will display such quantities. To aid the intuition, one can mentally convert M √ t to M/ (1 . χ PT in Section III below.5 . Ensembles
The ensembles used in this study are listed in Tables II–IV in Appendix A. They fallinto three groups. The short runs with the smallest lattices, of size V = n s × n t = 16 × β, κ , κ ). Themost important observables for this step were √ t , the scale defined by gradient flow (seeTables V–VII), and the masses M P r of the pseudoscalars constructed respectively fromfermions in the r = and representations (see Tables VIII–X).The goal of this orientation was to find couplings that give t /a = O (1) along withpseudoscalar masses that are reasonably light, for subsequent comparison to χ PT. It turnedout that these short runs yielded results that are in themselves usable for the chiral fits tobe presented below, and hence we include them in our analysis.As can be seen in the tables, some ensembles differ in small changes to their κ r values.Our orientation runs found that t /a and aM P are often sensitive to these small changes.We demanded that our ensembles satisfy the criterion M P r
L > L = n s a is the spatial size of the lattice. This is the familiar rule of thumb from QCD,based on the fact that leading-order finite-volume corrections are proportional to e − M π L ; amore detailed study of finite-volume effects in our data is given in Appendix B 4. Weconsidered cutting data above a maximum value of t /a beyond which finite-volume effectsseverely contaminate determination of the flow scale; such a cut was found to be unnecessaryfollowing the cuts on M P L . We did eliminate ensembles with t /a < .
94 because in thesecases the flow did not enter a linear regime. These correspond to a large lattice spacing—inQCD language, 1 /a < . V = 16 ×
32. Finally, we have four extended runs on lattices with V = 24 × t /a and small ˆ M P , so that the constraint M P L >
L/a .The pseudoscalar masses for all the ensembles are given in Tables VIII–X. To show ourcoverage of M P values, we map them in the ( M P , M P ) plane in Figs. 1 and 2. The firstshows the pseudoscalar masses obtained for 0 . < √ t /a < .
41, which translates to acutoff of 1 . < /a < √ t /a = 1 .
05, or 1 /a = 1 .
45 GeV). The second plot represents ensembles in the range1 . < √ t /a < .
64, or 2 GeV < /a < . III. CHIRAL PERTURBATION THEORY
The standard framework for analyzing the light pseudoscalar sector is χ PT. The gener-alization of χ PT to a theory with fermions in two different representations was developedin Ref. [12], and the next-to-leading-order (NLO) results of this work provide the basis forour fits for the pseudoscalar masses and decay constants. We will also need Wilson chiralperturbation theory (W χ PT), the extension of chiral perturbation theory to include thediscretization errors of Wilson fermions [28–33].6 .2 0.3 0.4 0.5 0.6 0.7 M P t M P t / × × FIG. 1. Map of our ensembles in the plane of pseudoscalar masses M P r . These are coarse lattices,with 0 . < √ t /a < .
41. We define arbitrarily √ t = (1 . − for comparison with QCD.For most of these ensembles 1 /a (cid:39) .
45 GeV by this measure.
A. Using a yardstick
We need a yardstick with which to measure dimensionful quantities as the fermion massesare varied. In this paper, we use √ t for the characteristic length scale of every ensemble.To measure an observable in units of t simply means to multiply it by the power of t thatrenders it dimensionless. Since t itself admits a chiral expansion [34], the resulting dimen-sionless quantity admits a chiral expansion whenever the original dimensionful observabledoes.To see how this works, consider a gauge theory with mass-degenerate fermions of mass m , all in the same representation. In continuum χ PT, the NLO expression for the decayconstant is F NLO = F (cid:18) c Bm π F log (cid:0) Bm/µ (cid:1) + L BmF (cid:19) . (3.1) B and F are the familiar parameters of the LO lagrangian. [Our normalization conventionfor the pseudoscalar decay constant is larger by √ χ PTliterature. See Eq. (4.1) below.] We recall that the LO pseudoscalar mass is( M ) LO = 2 Bm . (3.2)The remaining parameters in Eq. (3.1) are µ , the renormalization scale, and L , which is a(dimensionless) linear combination of the NLO low-energy constants (LECs), whose valuedepends on the choice of µ . The coefficient c of the logarithmic term is a calculable numberthat depends only on the fermion representation and on the number of flavors [35].7 .2 0.3 0.4 0.5 0.6 0.7 M P t M P t / × 18 × × FIG. 2. Same as Fig. 1, but here we plot ensembles on fine lattices, √ t /a > .
41. If we fix √ t = (1 . − then this means 1 /a > /a (cid:39) . The NLO result for t is t NLO0 = t , ch (cid:18) k BmF (cid:19) , (3.3)where t , ch is the value of t in the chiral limit, and ˜ k is a new LEC. Notice that this expres-sion depends analytically on the fermion mass m . As was shown in Ref. [34], logarithmiccorrections to t occur for the first time at the next-to-next-to-leading order (N LO).Combining Eqs. (3.1) and (3.3) we obtain the NLO result for the dimensionless productˆ F ≡ F √ t , ˆ F NLO = F (cid:112) t , ch (cid:18) c Bm π F log (2 Bmt ) + ( L + ˜ k /
2) 2
BmF (cid:19) . (3.4)Here we have chosen the renormalization scale to be µ = t − / , ch . LECs are independent of thefermion mass, and to preserve this feature we rescale them with t , ch , for example defining˚ F = F √ t , ch . Equation (3.4) can then be written asˆ F NLO = ˚ F (cid:32) c B ˚ m π ˚ F log (cid:0) B ˚ m (cid:1) + ( L + ˜ k /
2) 2 ˚ B ˚ m ˚ F (cid:33) . (3.5)The expansion parameter is now ˚ m , which is the fermion mass m measured in units of t , ch .8quation (3.5) is inconvenient because ˚ m is not known for a given ensemble until t , ch is known. Finding t , ch (in units of t of the given ensemble) requires a complicated fittingprocedure that we wish to avoid. Instead, we opt for rescaling all observables of a givenensemble, including the fermion mass, with t of the same ensemble. Introducing ˆ m ≡ m √ t we now use Eq. (3.3) to relate the rescaled masses,˚ m = ˆ m (cid:18) − ˜ k BmF (cid:19) , (3.6)which allows us rewrite Eq. (3.5) asˆ F NLO = ˚ F (cid:32) c B ˆ m π ˚ F log (cid:0) B ˆ m (cid:1) + ( L + ˜ k /
2) 2 ˚ B ˆ m ˚ F (cid:33) . (3.7)The transition from ˚ m to ˆ m left no trace, because the difference is a higher-order correction.More generally, at NLO the transition from ˚ m to ˆ m can always be absorbed into a redefi-nition of the LECs. (A case where the redefinition is non-trivial is the NLO result for thepseudoscalar mass.)An appealing feature of Eq. (3.7) is that it looks the same as Eq. (3.1). In particular, thecoefficient of the logarithmic term is unchanged. The only minor change is that the coefficientof the NLO analytic term is now L + ˜ k / L . (At N LO things would becometechnically more complicated, because N LO logarithmic corrections for t would have to beincorporated as well.) It can be checked that this nice feature generalizes to an arbitraryfermion content. In the NLO fit formulae that we present below, all the logarithmic termswill thus have the same coefficients as in the usual continuum NLO expressions [12, 35]. B. Wilson chiral perturbation theory
The extension of continuum chiral perturbation theory to include the discretization errorsof Wilson fermions goes under the name of Wilson chiral perturbation theory, or W χ PT.In the light pseudoscalar sector, W χ PT allows us to extrapolate both to the chiral limit, m →
0, and to the continuum limit, a →
0. W χ PT comes in two variants, depending onthe choice of a power counting scheme. In this paper we follow the “generic small mass,” orGSM, power counting, defined by p ∼ m ∼ a , (3.8)where p is an external momentum, m is the fermion mass, and a is the lattice spacing,all measured in terms of a typical hadronic scale. The alternative power counting scheme,known as the “large cutoff effects,” or LCE, power counting, is defined by p ∼ m ∼ a . (3.9)The GSM scheme is appropriate when the fermion mass is not too small, and O ( a ) effectsmay be considered as subleading corrections. (We must, of course, remain within the chiralregime, meaning that ˆ m = m √ t is small.) In particular, our determination of the criticalsurface κ cr , where the mass of fermions in representation r vanishes, is done via extrapolationfrom the GSM regime. As a result, we do not probe the possible existence of an Aoki phase.For more details, see Appendices B 3 and C 2.9he fermion mass appearing in the LO lagrangian of W χ PT is the so-called shifted mass ,defined by m shifted = m ctm + aW /B, (3.10)where m ctm is the fermion mass of continuum χ PT, and W is a new LEC from W χ PT. Thedifference between the shifted and continuum masses vanishes in the continuum limit. Forthis lattice study, we need to know how the shifted mass m shifted compares to the fermionmass m AWI measured in our simulations via the axial Ward identity Eq. (2.3). As was shownin Ref. [36], m shifted = m AWI , up to corrections that are higher order in either of the abovepower counting schemes. In view of the important role that this result plays in our analysis,we briefly summarize the derivation of Ref. [36] in Appendix C 2. For our chiral fits we thusdefine ˆ m = m AWI √ t . (3.11)The last ingredient we need for our fits is the lattice spacing. Since we are measuring alldimensionful quantities in units of t , it is natural to adopt a mass-dependent prescription,and to measure also the lattice spacing in units of t . We thus introduceˆ a ≡ a/ √ t . (3.12)The Wilson discretization effects of any hatted (dimensionless) observable will be accountedfor by an expansion in ˆ a .In QCD, it is common to choose a mass-independent scale-setting prescription, wherebythe lattice spacing is a function of the bare coupling β , but is independent of the barefermion masses (see for example Refs. [37, 38]). In brief, for every constant- β plane, thisprocedure requires finding the point where certain dimensionless quantities (such as M π /F π and M K /F K ) attain their real-world values. The value in lattice units of some dimensionfulobservable at the reference point is then used to determine the lattice spacing in physicalunits.Here we have opted for mass-dependent scale setting because of several important differ-ences. First, the BSM context does not provide us with any experimental results that couldbe used to define a reference point. This problem might be circumvented by invoking thechiral limit as a reference point on each constant- β plane. This, however, has the undesirablefeature that the scale setting procedure would necessarily involve an extrapolation.Second, in our model, as in many other models that have been studied in the BSMcontext, we observe a rapid change of t /a with the fermion mass, especially when thelatter becomes light. Moreover, this phenomenon is quite general, and is seen for virtuallyany quantity that might be used to set the scale; its proper interpretation is thus that thelattice spacing itself is changing rapidly. The underlying reason is that, in comparison withQCD, BSM theories tend to have a large number of fermionic degrees of freedom, whichhave a strong screening effect on the bare coupling. When we consider the dependence ofa hatted quantity, such as ˆ M P , on the hatted mass parameter, ˆ m , we expect to see somedeviations from the continuum values, but such scaling violations should be small whenthe bare coupling is small enough. By contrast, as explained above, the lattice spacing ˆ a itself can vary rapidly with the fermion mass(es). By using the mass-dependent scale-settingprescription of Eq. (3.12) we can incorporate this effect into our analysis. As we will see,the remaining scaling violations in the hatted quantities are small and amenable to W χ PT.10 . Summary of χ PT formulae
Our central fits below will include terms through NLO in the GSM power counting.These formulae depend exclusively on the dimensionless quantities we have introduced inthe previous subsections. The NLO expressions for the pseudoscalar masses of the tworepresentations are( ˆ M P ) NLO = 2 ˆ m ˚ B (cid:18) L M ˆ m + L M ˆ m + 12 ∆ −
45 ∆ ζ (cid:19) (3.13)+ ˚ W M ˆ a ˆ m + ˚ W M ˆ a ˆ m + ˚ W M ˆ a , ( ˆ M P ) NLO = 2 ˆ m ˚ B (cid:18) L M ˆ m + L M ˆ m −
14 ∆ −
15 ∆ ζ (cid:19) (3.14)+ ˚ W M ˆ a ˆ m + ˚ W M ˆ a ˆ m + ˚ W M ˆ a , while the expressions for the decay constants are( ˚ F P ) NLO = ˚ F (cid:0) L F ˆ m + L F ˆ m − ∆ (cid:1) + ˚ W F ˆ a , (3.15)( ˚ F P ) NLO = ˚ F (cid:0) L F ˆ m + L F ˆ m − (cid:1) + ˚ W F ˆ a . (3.16)The one-loop chiral logarithms enter as∆ = 2 ˆ m ˚ B π ˚ F log (cid:16) m ˚ B (cid:17) , (3.17)∆ = 2 ˆ m ˚ B π ˚ F log (cid:16) m ˚ B (cid:17) , ∆ ζ = ˆ M ζ π ˚ F ζ log (cid:16) ˆ M ζ (cid:17) , where the dimensionless mass-squared of the singlet NG boson is defined byˆ M ζ = 85 (cid:32) F ˆ m ˚ B + ˚ F ˆ m ˚ B ˚ F ζ (cid:33) . (3.18)This corresponds to the LO result of Ref. [12], rescaled by t . Further technical detailsrelated to the ζ and our conventions for the conserved axial current appear in Appendix C 1.The most important parameters in the expressions above are the LO LECS of the con-tinuum two-representation theory (rescaled by √ t , ch ): ˚ B , ˚ B , ˚ F , ˚ F , and ˚ F ζ . The dimen-sionless parameters L Mrs and L Frs , r = 4 , , are linear combinations of the continuum NLOLECs and of similar NLO LECs originating from the chiral expansion of the flow scale [cf.Eq. (3.3)]. The general form of the analytic NLO continuum terms was discussed in [12].Because we do not have enough independent quantities to distinguish the individual NLOLECs, we instead consider L Mrs and L Frs as the parameters for the fit. Finally, the various˚ W parameters account for the NLO analytic terms of W χ PT in the GSM power-countingscheme. Overall, these formulae contain 21 undetermined parameters, which we will fitbelow using 172 correlated points of data: four data points for each of our 43 ensembles.11e have not presented NLO fit formulae for the mass and decay constant of the singletNG boson ζ . We do not make use of these formulae in this work because we have notcalculated fermion-disconnected diagrams, which is technically challenging, and so we donot have direct access to the singlet sector. Nevertheless, through their dependence on ∆ ζ ,virtual ζ loops contribute to the masses and decay constants of the other NG bosons atNLO. In the next section we will explore what can be learned about the singlet sector fromthis effect.Another interesting quantity is the chiral condensate in each representation. At lowest or-der in χ PT (equivalently, in the corresponding chiral limit, ˆ m r → r = − ˚ B r ˚ F r . (3.19)Instead of measuring the condensates directly—a formidable task with Wilson fermions—we will make use of Eq. (3.19) to extract their values in the (double) chiral limit from ouranalysis of the pseudoscalar masses and decay constants. IV. PSEUDOSCALAR MESONSA. Masses and decay constants
We begin with the pseudoscalar mesons, which become NG bosons in the chiral limit.For a first look, we plot in Fig. 3 the squared masses ˆ M P r . The sextet mass ˆ M P is plottedagainst the AWI mass ˆ m of the sextet fermion, ignoring the dependence on the fundamentalfermion mass ˆ m , and likewise for ˆ M P , plotted against ˆ m . As expected from leading-orderchiral perturbation theory, the overall behavior of each squared mass is approximately linear.One supposes that the scatter around the straight lines is due to the hidden dependence onthe other fermion mass, as well as corrections from NLO and from lattice artifacts. We willexamine this hypothesis shortly.The pseudoscalar decay constants are defined by (cid:10) (cid:12)(cid:12) A ( r )4 a (cid:12)(cid:12) P ( r ) b (cid:11) = δ ab M P r F P r , (4.1)at zero spatial momentum, which is the convention that gives F π (cid:39)
130 MeV in QCD. Wecalculate F P r with the procedure described in Appendix B 2, renormalizing according to theanalogue of Eq. (B5). We plot the (rescaled) decay constants ˆ F P r in Fig. 4. The data showa steady rise with ˆ m r . The same qualitative behavior is seen in QCD, where the pion decayconstant is an increasing function of the quark mass.We have presented in Sec. III C the predictions of χ PT in NLO for pseudoscalar observ-ables. We conduct a joint fit of the four observables ˆ M P r and ˆ F P r to the NLO formulae ofEqs. (3.13)–(3.16). On each ensemble, we use single-elimination jackknife to construct the6 × χ that is minimized for the fit. Wedo not include correlations with the flow scale t , which has negligible error compared tothe other quantities we extract.The full NLO fit to 21 parameters and 172 −
21 = 151 degrees of freedom gives χ /DOF= 0.48. Table I contains the resulting values for the LECs and demonstrates the presenceof important lattice artifacts in our data. For the masses, the most significant terms are the O ( ˆ m r ˆ a ) artifacts, in the same representation. For the decay constants, the O (ˆ a ) artifacts12 .00 0.02 0.04 0.06 0.08 0.10 0.12 m r p t M P r t FundamentalSextet
FIG. 3. Squared mass of the two pseudoscalar species, each plotted against the AWI mass of thecorresponding fermion species, all in units of the flow scale t . are also significant. From an empirical perspective, these four NLO Wilson terms form anecessary minimal set of artifact terms for modeling the data.Figures 5 and 6 illustrate the sizes of the Wilson artifacts (red) emerging from this fit. Inthese figures, the “corrected” data (dark blue) result from subtracting the lattice artifactsfrom the data (light blue), allowing us to extrapolate to the continuum limit, ˆ a → σ in the fit parameters.) In order to display a smooth curvefor the continuum NLO result for the decay constants, we have included only the same-representation terms when drawing the green band (indicated by “continuum NLO SREP”in the figure). The remaining scatter and deviation in the subtracted data (dark blue) isevidence of coupling between the representations.Table I demonstrates that all five leading-order LECs are well-determined by the NLOfit. We note that the singlet decay constant ˚ F ζ is larger than ˚ F and similar in size to ˚ F .Because measurement of chiral logarithms is known to be a difficult task in QCD studies,we return to the question of the stability of this result below.Turning our attention to the NLO LECs, we examine the communication between therepresentations. The ratios L M /L M and L F /L F quantify the relative influence of the sex-tet fermions on ˆ M P and ˆ F P , respectively, in the continuum theory. Similarly, the ratio˚ W M / ˚ W M measures the relative influence of the sextet artifact term compared to the fun-damental artifact term in ˆ M P . Taking into account correlations, the following ratios are13 .00 0.02 0.04 0.06 0.08 0.10 0.12 m r p t F P r p t FundamentalSextet
FIG. 4. Decay constant of each pseudoscalar species plotted against the mass of the correspondingfermion species, in units of the flow scale t . different from zero at the 2 σ level, L F /L F = +0 . , (4.2)˚ W M / ˚ W M = +0 . . (4.3)The converse influence of the fundamentals upon the sextets follows from exchanging(4 (cid:11) L M /L M , L F /L F , and ˚ W M / ˚ W M are all consistent with zero. De-spite the large uncertainties, this suggests that the sextets influence the fundamentalssignificantly, while the converse is not true. The same qualitative conclusion is also evident,for instance, in the NLO continuum behavior of the decay constants. Figure 6 shows thatsubtracted data (dark blue) are, to good approximation, a smooth function of ˆ m only.In contrast, the corresponding fundamental result in Fig. 5 (also in dark blue) exhibits aconspicuous jaggedness, indicating important dependence on the sextet fermion mass. B. Stability of the NLO fit
In this subsection we explore the stability of the NLO fit. First, since we are using priorsto ensure convergence of the non-linear fitting procedure, it is important to verify that ourresults were not biased by them. To this end, we have redone the fit using the results ofthe first fit as initial guess, while multiplying the width of all priors by 10. Figure 7 showsthe results of both fits for the 5 LO LECs in the two lines at the bottom. The results areindistinguishable, indicating that the LO LECs were not influenced by the priors. (Thesame is also true for the NLO LECs.) 14
O: ˚ B . B . F . F . F ζ . L F . L F . L F . L F . L M . L M . (1) L M . L M . W F − . W F − . W M . W M − . W M − . W M . W M . W M − . χ PT formulae.
The chiral fit provides a-posteriori justification for the use of the GSM power-countingscheme, where O ( a ) terms are not part of the LO lagrangian. Both fermion masses in ourensembles lie roughly in the range 0 . (cid:46) ˆ m r (cid:46) . . (4.4)The range of lattice spacings we explore is0 . (cid:46) ˆ a (cid:46) . . (4.5)[Recall that our scale setting implies ˆ a = a /t by definition, see Eq. (3.12).] The O ( m r )contribution to the pseudoscalar masses is 2 ˚ B r ˆ m r , while the O ( a ) contribution is ˚ W Mr ˆ a .For our fermion masses and lattice spacings, these contributions lie approximately withinthe following ranges 2 ˚ B r ˆ m r : 5 × [0 . , . ≈ [0 . , . , (4.6)˚ W Mr ˆ a : 0 . × [0 . , . ≈ [0 . , . . (4.7)We see that the O ( m r ) terms are at least an order of magnitude larger than the O ( a ) terms,showing that the GSM power counting is the appropriate one (as long as this picture is notupset by large N LO corrections, see below).15 .40.20.00.20.40.60.8 M P t Lattice dataArtifactsData - ArtifactsLO continuum0.00 0.02 0.04 0.06 0.08 0.10 m p t F P p t Lattice dataArtifactsData - ArtifactsNLO continuum: single rep
FIG. 5. Breakdown of the contribution of lattice artifacts in the joint fit to χ PT for the fundamentalmasses and decay constants. .40.20.00.20.40.60.8 M P t Lattice dataArtifactsData - ArtifactsLO continuum0.00 0.02 0.04 0.06 0.08 0.10 m p t F P p t Lattice dataArtifactsData - ArtifactsNLO continuum: single rep
FIG. 6. Breakdown of the contribution of lattice artifacts in the joint fit to χ PT for the sextetmasses and decay constants. χ PT are ξ r ≡ B r ˆ m r / π ˚ F r . With the central-fit values for the LECs, these fermionmasses correspond to the following ranges for the expansion parameters,0 . (cid:46) ξ (cid:46) . , (4.8)0 . (cid:46) ξ (cid:46) . . (4.9)We see that the the maximum of the sextet expansion parameter ξ is smaller by a factorof 2.5 than the fundamental expansion parameter ξ . The main reason is that ˚ F is sig-nificantly larger than ˚ F , as might be expected based on the relative dimension of the tworepresentations (see Sec. VI A). . . . . . Central fit10x all priors ˆ m < 0.09 ˆ m < 0.07 ˆ m < 0.05 M P L > 4.5 M P L > 5.0Exclude N t = 18 ◦ B . . . . ◦ B .
10 0 .
12 0 . ◦ F .
14 0 .
16 0 .
18 0 . ◦ F . . . . ◦ F ζ FIG. 7. Exploring the stability of leading-order LECs in chiral fits. We take the NLO result todefine our central values, which appear at the bottom of each column. The variations are describedin the text.
It is quite plausible that ξ is sufficiently small that the expansion in m converges wellover our entire ensemble set. The same may not be true for ξ , whose value can be as largeas 0.5. In the next three lines of Fig. 7 we study the influence on the LO LECs of droppingensembles at the high end of the ˆ m range: ˆ m > . > .
07, and finally > .
05. We seethat truncating our data set has only a modest effect on the ˚ F r and ˚ B r parameters. Onthe other hand, since we only obtain ˚ F ζ through NLO logarithms, it is not surprising thatthe increase in the error bar of ˚ F ζ is much more pronounced. Indeed, when we restrict toˆ m < .
05, ˚ F ζ is only 2 σ away from zero. 18he next two lines in Fig. 7 investigate the possible influence of finite-volume effects onour central analysis; further discussion appears in Appendix B 4. The minimum cutoff on M P L in data used in the central fit is varied from its initial value of 4.0 in our main analysisto 4.5 and 5.0, excluding more data that may be expected to have the largest finite-volumecontamination. Finally, in the top line we repeat our fit with all V = 16 ×
18 ensemblesexcluded from the analysis, in order to test for systematic effects in our correlator analysisdue to the smaller time extent. No significant change to our results is seen in any case.The main systematic uncertainty about this non-QCD system is the neglect of N LOcorrections. We do not really know how high can we go in ξ and ξ if we want thesecorrections to remain below a certain level. While our stability tests give us some insight,we do not have enough data for a quantitative study of N LO. Nevertheless, we take thesmallness of ξ and our stability tests on ˆ m as evidence that the data are in the regime whereNLO ChPT applies, even if we do not have enough information to quantify the correspondingsystematic error. C. The singlet Goldstone boson ζ As explained in Sec. III C, the chiral fit in the fundamental and sextet sectors allows usto probe the ζ meson as well. We examine its mass in the chiral-sextet limit, ˆ m → M ζ , constructed using Eq. (3.18) and the parameters of the central fit, inthe continuum (ˆ a →
0) limit, as a function of the mass ˆ m of the fundamental fermions.The figure shows that the singlet boson is consistently lighter than the pseudoscalar of thefundamental sector in this limit.We can make a conservative prediction regarding the ζ mass as follows. As we have justexplained, we do not know how large ˆ m can be while keeping the N LO corrections below,say, 10% or 20%. Lowering the maximal value of ˆ m raises the uncertainty in ˚ F ζ , as seen inFig. 7. Still, even if we lower the maximal value of ˆ m so as to, say, double the uncertaintyof ˚ F ζ , we would still find that M ζ < M at the 1 σ level.The chiral-sextet limit is interesting for composite-Higgs models. In many models, in-cluding those proposed by Ferretti and Karateev [8], the symmetries of the Standard Modelare embedded into the unbroken global symmetries, so that neither the fundamental nor thesextet fermions are required to be strictly massless. Nonetheless, successful models are likelyto have very light sextet fermions, because a large sextet mass would prevent the Higgs fieldfrom condensing even after the generation of a potential from the coupling of the Higgs toStandard Model fields. V. VECTOR MESONSA. Masses and decay constants
We now turn to our results for vector masses and decay constants. Vector-meson decayconstants appear in the literature with a variety of conventions. We define F V r to have unitsof energy, (cid:10) (cid:12)(cid:12) V ( r ) ia (cid:12)(cid:12) V ( r ) jb (cid:11) = δ ab δ ij F V r M V r , (5.1)where the vector meson is at rest. The indices are i, j = 1 , , , for the spatial directions, andas usual, a, b = 1 , , , for isospin. This definition is frequently used in the phenomenology19 .00 0.02 0.04 0.06 0.08 0.10 0.12 m p t M t FundamentalU(1) Goldstone
FIG. 8. Mass squared ˆ M ζ of the non-anomalous NG boson in the combined continuum (ˆ a → m →
0) limits, as extracted using Eq. (3.18) and the central fit’s parameters,plotted against ˆ m . The pseudoscalar mass ˆ M in the fundamental sector in the same limit isshown for comparison. literature on precision electroweak observables, for example Ref. [39].Figures 9 and 10 show results for ˆ M V r and ˆ F V r , respectively, each plotted against thefermion mass ˆ m r in the same representation. As before, we measure all quantities in unitsof t . The data for these plots are listed in Tables XI–XIII. Both quantities shows a modest,plausibly linear rise against the fermion mass, albeit with a large spread.Figure 11 shows the ratio of the pseudoscalar and vector masses, M P r /M V r , again plottedagainst the fermion mass ˆ m r in the same representation. This ratio is greater than or equalto a half for all but the smallest masses. Because the decay V → P P is p -wave, the vectoris stable if M P /M V > . (cid:112) − k /M V , where k min = 2 π/L is the minimum nonzeromomentum. This condition is satisfied for both representations on all of our ensembles, sothe vectors are indeed stable.We model ˆ M V r and ˆ F V r as linear functions of the fermion mass in the same representationand of the lattice spacing, for example,ˆ M V = c + c ˆ m + c ˆ a . (5.2)For this analysis, we restrict ourselves to the 30 ensembles for which we were able to measurethe vector decay constants (see Tables XI—XIII). The individual correlated fits are success-ful, with typical χ /DOF (cid:46) . − .00 0.02 0.04 0.06 0.08 0.10 m r p t M V r p t FundamentalSextet
FIG. 9. Vector masses vs fermion masses in units of the flow scale t . B. Vector meson dominance and the KSRF relations
The pseudoscalar and vector decay constants are related through the hypothesis of vectormeson dominance (VMD). Kawarabayashi, Suzuki, Riazuddin, and Fayyazuddin (KSRF)showed long ago [40–42] that VMD leads to the prediction F V = √ F P , (5.3)independent of representation. Figure 14 shows the ratio F V r /F P r in each representation,after subtracting lattice artifacts. The KSRF prediction is qualitatively successful. (In QCD,the experimental value is roughly 1.66.)Another result of KSRF is that the on-shell coupling constant g VPP mediating the decayof a vector into two pseudoscalars is given by g V P P = M V F P . (5.4)We plot this ratio in Fig. 15. As already noted, in our ensembles the vector meson is stable.Nevertheless, we may use the KSRF result as a phenomenological estimate for the behaviorclose to the chiral limit. Using the tree-level formula for the V → P P decay width in thelimit where M P r (cid:28) M V r , Γ V → P P (cid:39) g VPP M V π , (5.5)we can estimate the the mass-to-width ratio for each vector resonance,Γ V → P P M V (cid:39) M V πF P . (5.6)21 .00 0.02 0.04 0.06 0.08 0.10 m r p t F V r p t FundamentalSextet
FIG. 10. Vector decay constants vs fermion masses in units of the flow scale t . From Fig. 15 we thus obtain Fig. 16. For the physical ρ meson, this ratio has a value ofroughly 0.23. (The experimental value is 0.19.) VI. DISCUSSIONA. Large- N c counting We want to put our results in context with comparisons to QCD. Large- N c counting (for N c colors) is a way to do that. Any quantity Q is expected to scale across N c as Q ( N c ) = N pc (cid:18) Q + Q N c + Q N c + · · · (cid:19) , (6.1)where p is some characteristic exponent determined by large- N c considerations, and the Q i are a set of expansion coefficients. Before we get to the (limited) comparisons between dif-ferent N c ’s we can make, our theory gives us a unique opportunity to compare the expansioncoefficients for different representations. More specifically, if we neglect all the subleadingcorrections, our data allow us to compare the leading expansion coefficient Q between thefundamental and two-index antisymmetric representations, for various obesrvables.We start with meson masses, which are predicted to be independent of N c [ p = 0 inEq. (6.1)]. Figs. 3 and 9 reveal near-independence of representation of the pseudoscalarand vector masses when plotted against the corresponding fermion mass. This is furthersupported by the near equality of ˚ B and ˚ B in Table I. We conclude that Q is roughlyindependent of representation for the pseudoscalar and vector meson masses.22 .00 0.02 0.04 0.06 0.08 0.10 0.12 m r p t M P r / M V r FundamentalSextet
FIG. 11. The mass ratio M P r /M V r in a fixed representation.
Decay constants scale as √ N c for single-index fermions and as N c for two-index fermions.As for the leading expansion coefficient, a possible guess might be that the product N pc Q follows the leading large- N c behavior of (dim r ) / . This would imply that there exists aconstant c such that Q ≈ c for mesons made of fundamental-representation fermions, while Q ≈ c/ √ N c = 4 we thus expect F /F ≈ (cid:112) N c / √
2. The ratio F /F is plotted in Fig. 17for the pseudoscalar and vector mesons, showing good (perhaps too good) agreement withthis expectation. Another consequence is that the ratio F V r /F P r is expected to be roughlyindependent of representation r , in agreement with Fig. 14 and the KSRF relation.In this context, we can also compare the fermion condensates in the two representations.We can use the results of the chiral fit to calculate these (per flavor), each in its correspondingchiral limit. Using Eq. (3.19) we find ˆΣ / ˆΣ = 2 . B / ˚ B (cid:39) F / ˚ F (cid:39) √ M V r /F P r to the SU(3) case. We expect thatthis ratio for either SU(4) representation will be smaller than its value in QCD, which is770 MeV /
130 MeV = 5 .
9. This is borne out by Fig. 15. At a more quantitative level, large- N c scaling predicts this value to be 5 . × (cid:112) / ≈ . N c prediction of 5 . × / ≈ . ρ meson. 23 .40.20.00.20.40.60.81.01.2 M V p t Lattice dataArtifactsData - ArtifactsNLO continuum: single rep0.00 0.02 0.04 0.06 0.08 0.10 0.12 m p t F V p t Lattice dataArtifactsData - ArtifactsNLO continuum: single rep
FIG. 12. Breakdown of the contribution of lattice artifacts in the empirical models for the vectormasses and decay constants in the fundamental representation. .40.20.00.20.40.60.81.01.21.4 M V p t Lattice dataArtifactsData - ArtifactsNLO continuum: single rep0.00 0.02 0.04 0.06 0.08 0.10 m p t F V p t Lattice dataArtifactsData - ArtifactsNLO continuum: single rep
FIG. 13. Breakdown of the contribution of lattice artifacts in the empirical models for the vectormasses and decay constants in the sextet representation. .00 0.02 0.04 0.06 0.08 0.10 0.12 m r p t F V r / F P r FundamentalSextet
FIG. 14. The ratio of the vector and pseudoscalar decay constants in each representation. TheKSRF prediction is a constant value of √ B. Conclusions
In this paper we have described the low-lying mesonic spectrum of SU(4) gauge theorycoupled to dynamical fermions in the fundamental and sextet representations. These multi-rep simulations are the first of their kind. Our choice of this theory was inspired by its closesimilarity to a composite-Higgs model first studied by Ferretti [7].Our analysis focused on the masses of the pseudoscalar and vector states and their as-sociated decay constants. Using the extension of χ PT that accounts for the discretizationerrors of Wilson fermions, we carried out a global fit of the pseudoscalar masses and de-cay constants of the two representations, to NLO in the GSM power-counting scheme. Wefound significant lattice artifacts, which we were able to subtract, obtaining predictions forcontinuum-limit values. Our chiral fit provides mild evidence for coupling between the twofermion representations, a novel feature of multirep theories.Through both the mass terms and the Wilson terms, our lattice setup incorporates theexpected symmetry breaking patterns: SU(2) L × SU(2) R → SU(2) V in the fundamentalsector, and SU(4) → SO(4) in the sextet sector. We did not carry out a dedicated study ofalternative symmetry breaking patterns. Still, the success of the chiral fits provides someconfirmation that the above symmetry breaking patterns are the right ones.The main theoretical uncertainty of our chiral fits concerns the size of N LO effects.Thanks to a large decay constant, the chiral expansion converges quickly in the sextet sector,supporting the hypothesis that N LO effects are small in this sector. In the fundamentalsector the chiral expansion converges more slowly. Hence, keeping N LO effects below acertain comfortable level might require the exclusion of ensembles where ˆ m is on the highside. More quantitative statements cannot be made given our data.26 .00 0.02 0.04 0.06 0.08 0.10 0.12 m r p t M V r / F P r FundamentalSextet
FIG. 15. The ratio of the vector mass and pseudoscalar decay constant in a fixed representation.KSRF identify this quantity with the coupling g VPP . In QCD, this ratio is roughly 5.9.
The correlation functions that we calculated probe directly the pseudoscalar states madepurely of fundamental or purely of sextet fermions. This is reflected in the stability of theLO parameters ˚ F r and ˚ B r if we constrain the maximal value of ˆ m to successively smallervalues. The last LO parameter, the decay constant ˚ F ζ of the axion-like singlet NG boson,is not well-determined because we have not calculated propagators in the ζ channel. The ζ meson does contribute through virtual loops to the correlation functions we have studied.Accordingly, our fits depend on ˚ F ζ , but only through NLO logarithmic terms. ˚ F ζ is moresensitive to the upper limit on ˆ m ; as a result, so is our prediction for the mass of the ζ boson. Nonetheless, we have argued that the ζ is lighter than the fundamental-sector NGbosons, M ζ < M , in the sextet-chiral limit ˆ m →
0, a limit which is interesting for thephenomenology of Ferretti’s model. In a full composite-Higgs model, however, the massesof all pseudoscalar states can receive important corrections from the couplings to StandardModel fields.In modeling our results for the vector mesons, we found that the ratio of pseudoscalarto vector decay constants agrees well with the KSRF result based on vector meson domi-nance. As discussed in Sec. VI A, comparing the KSRF prediction for the decay rate of thevector meson in the chiral limit to the QCD case shows reasonable agreement with large- N c counting.Although our estimates for Γ V /M V depend on the well-motivated but non-rigorous as-sumption of vector meson dominance, the resulting narrowness is almost certainly generic.In large- N c , the widths of mesons made of fundamental-representation fermions scales as1 / √ N c and thus they become narrower as N c increases. Insofar as large- N c provides thecleanest explanation for the narrowness and existence of mesons in QCD, the vector mesonsshould become narrower in theories with more colors. We proposed that in multirep theories,27 .00 0.02 0.04 0.06 0.08 0.10 0.12 m r p t Γ V / M V FundamentalSextet
FIG. 16. Tree-level estimates for the width-to-mass ratio of the vector mesons according to KSRF.The KSRF estimate for this ratio is roughly 0.23 in QCD. the generalization of √ N c is (dim r ) / , a hypothesis supported by our data. This result isgood news for phenomenologists looking to constrain models like the Ferretti model, sincenarrower states typically provide clearer signals in collider data.As we have mentioned, we are also exploring the phase diagram of this multirep theory[21]. We have been looking for—and not finding—scale separation between the representa-tions in the confinement and chiral transitions. We are also studying the baryon spectrum,a particularly interesting sector of the theory given its connection to top-quark physics andpartial compositeness in the Ferretti model.Other interesting avenues for the future work in this model (or multirep composite Higgstheories more generally) include quantities related to the Higgs potential. The contributionof the Standard Model’s gauge fields to the Higgs potential, Π LR , is conceptually identicalto the physics of electromagnetic splittings between pions in QCD and has been the subjectof a recent pilot study on the lattice [43]. The top-quark contribution to the Higgs potentialis considerably more challenging [44, 45]. ACKNOWLEDGMENTS
Computations for this work were carried out with resources provided by the USQCDCollaboration, which is funded by the Office of Science of the U.S. Department of Energy;with the Summit supercomputer, a joint effort of the University of Colorado Boulder andColorado State University, which is supported by the National Science Foundation (awardsACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado StateUniversity; by the Cy-Tera Project, which is co-funded by the European Regional Develop-ment Fund and the Republic of Cyprus through the Research Promotion Foundation; andthe DECI resource ARCHER based in the United Kingdom at the University of Edinburgh28 .00 0.02 0.04 0.06 0.08 0.10 0.12 m p t F / F Pseudoscalar F /F Vector F /F FIG. 17. The ratio of the decay constants in the sextet and fundamental representations in boththe pseudoscalar (dark green) and vector (light green) channels. with support from PRACE.This work was supported in part by the U.S. Department of Energy under grants de-sc0010005 (Colorado) and de-sc0013682 (M. G. ), and by the Israel Science Foundationunder grant no. 449/13 (Tel Aviv). Brookhaven National Laboratory is supported by theU. S. Department of Energy under contract de-sc0012704.29 ppendix A: Data tables
Ensemble β κ κ Configurations1 7.2 0.13173 0.13423 672 7.2 0.1318 0.1341 293 7.2 0.132 0.134 424 7.3 0.1314 0.1333 175 7.3 0.1315 0.1333 176 7.308 0.1304 0.1339 297 7.31 0.1305 0.1339 178 7.32 0.13 0.134 179 7.33 0.1314 0.1332 1710 7.33 0.1314 0.1333 1711 7.33 0.1315 0.1335 1712 7.4 0.1307 0.133 1713 7.4 0.131 0.133 2914 7.5 0.13 0.132 1715 7.5 0.13 0.1325 2916 7.5 0.13 0.1327 2917 7.5 0.13 0.1328 2918 7.5 0.1305 0.1327 2919 7.75 0.129 0.131 2920 7.75 0.129 0.1315 29TABLE II. List of ensembles with V = 16 ×
18 generated for this study. Configurations areseparated by 4 Monte Carlo trajectories. nsemble β κ κ Configurations21 7.25 0.13095 0.13418 6122 7.25 0.13147 0.13395 7123 7.276 0.13157 0.13364 9624 7.3 0.13117 0.13363 6125 7.3 0.13118 0.13361 9626 7.3 0.13162 0.1334 7127 7.308 0.1304 0.13393 9628 7.33 0.1314 0.1332 9629 7.4 0.1307 0.133 9630 7.55 0.129 0.1325 8431 7.55 0.13 0.1325 8432 7.65 0.128 0.131 4933 7.65 0.129 0.1308 4934 7.65 0.13 0.131 8435 7.65 0.13 0.132 8436 7.75 0.128 0.131 8437 7.75 0.129 0.1308 5438 7.75 0.1295 0.1315 3439 7.85 0.129 0.1308 44TABLE III. List of ensembles with V = 16 ×
32. Configurations are separated by 10 Monte Carlotrajectories.Ensemble β κ κ Configurations40 7.51 0.1307 0.1328 13341 7.55 0.13 0.1327 8042 7.55 0.1305 0.1325 9143 7.55 0.1307 0.13234 80TABLE IV. List of ensembles with V = 24 ×
48. Configurations are separated by 10 Monte Carlotrajectories. nsemble t /a ˆ m ˆ m t and fermion masses ˆ m r = m r √ t in the ensembleswith volume V = 16 × nsemble t /a ˆ m ˆ m
21 1.093(9) 0.0422(7) 0.0203(10)22 1.135(9) 0.0279(11) 0.0251(12)23 1.128(24) 0.0243(7) 0.0326(7)24 1.132(12) 0.0345(8) 0.0323(14)25 1.100(10) 0.0331(5) 0.0325(5)26 1.111(9) 0.0228(6) 0.0381(8)27 1.174(10) 0.0556(7) 0.0220(9)28 1.095(12) 0.0282(7) 0.0427(7)29 1.226(10) 0.0416(8) 0.0403(8)30 1.418(12) 0.0865(11) 0.0414(15)31 1.845(18) 0.0495(11) 0.0340(13)32 0.916(5) 0.1068(8) 0.0858(15)33 1.067(5) 0.0816(10) 0.0896(8)34 1.463(15) 0.0459(18) 0.0801(22)35 2.294(22) 0.0382(13) 0.0357(21)36 1.556(12) 0.1077(12) 0.0708(10)37 1.754(15) 0.0730(19) 0.0771(16)38 2.621(20) 0.0465(13) 0.0402(14)39 2.670(22) 0.0602(14) 0.0599(12)TABLE VI. Same as Table V, but in the ensembles with volume V = 16 × t /a ˆ m ˆ m
40 2.260(16) 0.0196(4) 0.0194(9)41 2.166(11) 0.0468(5) 0.0205(4)42 2.182(12) 0.0264(5) 0.0293(6)43 2.118(6) 0.0189(5) 0.0360(7)TABLE VII. Same as Table V, but in the ensembles with volume V = 24 × nsemble ˆ M P ˆ M P ˆ F P ˆ F P M P r = M P r √ t and decay constants ˆ F P r = F P r √ t in the ensembles with volume V = 16 × nsemble ˆ M P ˆ M P ˆ F P ˆ F P
21 0.366(9) 0.263(10) 0.119(6) 0.142(9)22 0.305(9) 0.303(8) 0.105(4) 0.151(5)23 0.275(6) 0.341(7) 0.108(4) 0.162(5)24 0.340(5) 0.340(9) 0.119(4) 0.168(9)25 0.339(3) 0.344(6) 0.107(4) 0.148(13)26 0.279(7) 0.368(11) 0.103(4) 0.167(13)27 0.423(4) 0.279(5) 0.127(4) 0.159(7)28 0.300(8) 0.391(8) 0.115(6) 0.173(6)29 0.372(6) 0.391(4) 0.126(3) 0.173(8)30 0.559(8) 0.408(12) 0.156(6) 0.187(7)31 0.429(10) 0.375(9) 0.140(9) 0.189(10)32 0.597(8) 0.554(5) 0.159(9) 0.208(13)33 0.514(8) 0.576(9) 0.154(8) 0.219(11)34 0.412(9) 0.565(9) 0.141(7) 0.224(8)35 0.400(9) 0.408(10) 0.132(6) 0.192(17)36 0.636(7) 0.538(8) 0.166(6) 0.210(8)37 0.530(5) 0.571(7) 0.154(4) 0.223(12)38 0.443(14) 0.428(15) 0.135(9) 0.188(13)39 0.505(13) 0.529(17) 0.148(8) 0.216(8)TABLE IX. Same as Table VIII, but in the ensembles with volume V = 16 × M P ˆ M P ˆ F P ˆ F P
40 0.278(4) 0.291(10) 0.114(4) 0.167(7)41 0.418(5) 0.295(7) 0.139(4) 0.169(4)42 0.317(6) 0.355(8) 0.125(4) 0.182(8)43 0.267(9) 0.394(8) 0.114(4) 0.184(5)TABLE X. Same as Table VIII, but in the ensembles with volume V = 24 × nsemble ˆ M V ˆ M V ˆ F V ˆ F V M V r = M V r √ t and decay constants ˆ F V r = F V r √ t in theensembles with volume V = 16 ×
18. Some ensembles did not yield reliable measurements of F V r because of insufficient statistics. The figures and tables omit data from such ensembles. nsemble ˆ M V ˆ M V ˆ F V ˆ F V
21 0.56(2) 0.55(3) 0.19(2) 0.275(1)22 0.51(2) 0.58(3) 0.17(3) 0.265(1)23 0.53(3) 0.61(1) 0.18(1) 0.263(1)24 0.56(2) 0.61(2) 0.19(1) 0.265(2)25 0.52(3) 0.59(2) 0.19(1) 0.265(1)26 0.50(3) 0.62(2) – –27 0.59(2) 0.57(3) 0.20(1) 0.250(2)28 0.55(3) 0.65(2) 0.19(1) 0.290(3)29 0.59(1) 0.66(2) 0.20(1) 0.287(1)30 0.73(2) 0.71(2) 0.24(3) 0.308(2)31 0.65(2) 0.70(5) 0.21(1) 0.291(2)32 0.74(1) 0.78(1) 0.24(2) 0.316(2)33 0.70(1) 0.79(1) 0.22(1) 0.319(1)34 0.66(3) 0.82(3) 0.22(3) 0.339(2)35 0.68(5) 0.77(5) 0.20(5) 0.310(3)36 0.81(1) 0.80(3) 0.25(1) 0.326(2)37 0.74(2) 0.82(2) 0.23(4) 0.322(4)38 0.69(4) 0.76(5) 0.24(2) 0.334(3)39 0.75(2) 0.83(2) 0.24(2) 0.350(4)TABLE XII. Same as Table XI, but in the ensembles with volume V = 16 × M V ˆ M V ˆ F V ˆ F V
40 0.57(6) 0.61(2) – –41 0.64(2) 0.60(4) 0.17(1) 0.29(2)42 0.59(3) 0.66(5) – –43 0.57(3) 0.70(2) 0.20(2) 0.32(1)TABLE XIII. Same as Table XI, but in the ensembles with volume V = 24 × ppendix B: Technical matters—lattice1. Correlator fitting In calculating correlation functions, we use a smeared source operator on the t = 0 timeslice while using both point and smeared operators at the sink, projected onto zero spatialmomentum. Smearing is done after fixing to the Coulomb gauge, always with smearingradius r = 6 a . On the large lattices we use antiperiodic boundary conditions in timefor the fermion propagators. For the V = 16 ×
18 ensembles, on the other hand, wesuperimpose propagators computed with periodic and anti-periodic boundary conditions,which effectively doubles the temporal size of the lattice—a technique sometimes called the“P+A trick” (see Ref. [46] and references therein).After restricting each correlator to a range [ t min , t max ] we find acceptable fits with single-exponential models, that is, without inclusion of excited states. In each representation, weextract the fermion mass m r from the axial Ward identity (2.3) via joint fits to the (cid:104) AP (cid:105) and (cid:104) P P (cid:105) correlators with a point sink.We use the publicly available Python packages lsqfit [47] and gvar [48] for nonlinearfitting and classical error propagation. When computing ratios of quantities derived fromdifferent fits, we use single-elimination jackknife to propagate errors including correlations.For each correlator, our fitting procedure is as follows. First, we vary the initial and finaltimes [ t min , t max ] used in the fits, amounting to a grid search over possible range fits. Thebest fit is chosen automatically using a criterion from the QCD literature with a preferencefor small χ / dof, large fit ranges, and well-determined fit parameters [49]. We maximize Q ≡ p × N dof (cid:80) n σ p n , (B1)where p is the unconstrained p -value, N dof denotes the number of degrees of freedom in thefit, and σ p n denotes the statistical error in the n th fit parameter. Although this criterionis ultimately arbitrary, it coincides with intuition about which fits ought to be consideredgood and removes subjective bias. We confirm that masses emerging from this procedureare consistent with expectations from effective mass plots; a representative comparison isshown in Fig. 18. We have also experimented with a “two-state” double-exponential ansatz,observing no significant changes in the ground-state masses within the combined statisticaland systematic errors estimated using this procedure.For an estimate of the systematic uncertainty associated with our fit-choice procedure, wecompute the spread in the model parameters emerging from all nominally good fits satisfying Q ≥ .
1. We then combine the statistical and systematic uncertainties conservatively using σ tot = σ stat + σ syst . (B2)The systematic error assigned by this procedure is often comparable to the statistical error,and is occasionally significantly larger. The error estimates for the fermion masses m r inTables V–XIII include this fit-range systematic.
2. Decay constants and operator renormalization
The lattice operators appearing in the correlation functions are subject to finite renor-malization in order to obtain the continuum-normalized operators that are required for38 t/a E ff e c t i v e m a ss m a ( β, , , N s , N t ) = (7 . , . , . , , t/a E ff e c t i v e m a ss m a ( β, , , N s , N t ) = (7 . , . , . , , FIG. 18. Representative plot showing the effective mass ma extracted from a smeared-source,point-sink pseudoscalar correlator on a typical 16 ×
18 ensemble (top) and 16 ×
32 ensemble(bottom) used in this study. The black lines indicate the mass and error (including range-fitsystematic uncertainty) extracted from a full nonlinear fit of the correlator. The horizontal widthof the black lines indicates the starting and ending times ( t i , t f ) used in the best range fit asdetermined by the MILC criterion. Note that in the lower panel, the small difference between thecentral value from the best fit and the effective mass at larger t/a is covered by the statistical +systematic error band, showing that our range-fit uncertainty is working as expected. determination of decay constants. We carry out this procedure in lattice perturbation the-ory including tadpole improvement; the procedure is described in Appendix D. The explicitrelationship between lattice and continuum operators is given by Eq. (D28).Simultaneous fits to the smeared-source, point-sink ( s, p ) and smeared-source, smeared-sink ( s, s ) correlation functions allow us to extract the mass, decay constant, and smearedamplitude. For example, the (cid:104) V V (cid:105) correlators in representation r give us the vector decay39onstant F V r defined in Eq. (5.1). The fit functions are C ( s,p ) V r ( t ) = A sV r A pV r M V r (cid:0) e − M V r t + e − M V r ( T − t ) (cid:1) , (B3) C ( s,s ) V r ( t ) = A sV r M V r (cid:0) e − M V r t + e − M V r ( T − t ) (cid:1) , (B4)giving the vector mass M V r and the point and smeared amplitudes A pV r and A sV r , respec-tively. In order to obtain decay constants with continuum normalizations, we apply therenormalization factors of Eq. (D28). The result is F V r = Z V r (cid:18) − κ r κ cr (cid:19) A pV r M V r . (B5)
3. Fermion mass determination and κ cr To determine the critical values κ cr , which enter into the field normalization for decayconstants defined in Appendix B 2, we perform a global fit to the AWI fermion masses inunits of the flow scale t as given in Tables V–VII. We use the model function √ t m = c + c β + κ ( d + d β ) + κ ( d (cid:48) + d (cid:48) β ) (B6)and similarly for √ t m (with a separate set of coefficients). We find that these terms,which are a subset of all possible combinations of the bare parameters { β, κ , κ } throughquadratic order, are sufficient to provide reasonable fit quality.Since we are interested in the regions where m r →
0, we use only those ensembles thathave √ t m r < .
08, a value determined empirically by inspecting our data for deviationsfrom the simple analytic behavior of Eq. (B6). Our fits give χ / dof of 16/24 and 21/24 forfitting √ t m and √ t m , respectively. The resulting κ c curves at two β values are shownin Fig. 19.As noted above, because κ c is determined by extrapolation, we do not probe the existenceof a possible Aoki phase.
4. Study of finite-volume corrections
All of our ensembles satisfy the criterion M P r
L > M ζ L > Q on our ensembles using the Wilson flow to smear the gauge fields outto t/a = 5 . Q in all cases.First, to obtain a theoretical estimate of the expected size of finite-volume effects, we con-sider the size of leading-order finite-volume correction to tadpole diagrams in chiral pertur-bation theory [9, 33, 52]. The dimensionless figure of merit for this effect is 2 I ( M P r , L ) /F P r , .09 0.10 0.11 0.12 0.13 0.14 0.15 β = 7 . β = 7 . FIG. 19. κ c curves with uncertainties for both representations at two values of the bare coupling β , based on our fits to √ t m r as described in the text. The red (horizontal) curve is where m vanishes, defining the function κ c ( β, κ ). The blue (vertical) curve is where m vanishes, defining κ c ( β, κ ). where I ( M, L ) = 6 (cid:18) M π (cid:19) (cid:115) π ( M L ) e − ML . (B7)This quantity gauges the effect of mesons interacting with their finite-volume image points.In Fig. 20 we plot 2 I ( M P r , L ) /F P r for all of our ensembles, with each representation r plotted against the corresponding ˆ m r . We therefore expect that finite volume correctionsdo not exceed a few percent in the ensembles of this study. This formula assumes theapplicability of chiral perturbation theory, which requires that F P r L (cid:38)
1; over all of ourensembles we find that F P L (cid:38) . F P L (cid:38) . M P r , F P r , and t on ensembles with severalspatial volumes at two sets of bare parameters, ( β, κ , κ ) = (7 . , . , . . , . , . V = 16 ×
32 ensembles36 and 37 as listed in Table III. Four ensembles hold N t = 32 fixed and vary the spatialvolume as N s = 12 , , ,
18; the fifth ensemble at each point has N s = 24 and N t = 48.Results of this test are shown in Figs. 21-23 below. For both sets of bare parameters, allobservables down to the smallest N s = 12 are seen to be within ±
5% of the central valueobtained on N s = 24, and within 2-3% for N s = 16 which is the smallest spatial volumeused in our central analysis.Finally, we have included explicit variation of the finite-volume cut on pseudoscalar mesonmasses (i.e. minimum cut on M P r L ) in the stability analysis of our central chiral fit, aspresented in Sec. IV B and Fig. 7. We also consider the effects of the finite temporal directionby cutting the N t = 18 ensembles out of the analysis. All of the fit results are seen to bestable at the one-sigma level as we vary the finite-volume cut. We conclude that finite-volumeeffects are not significant in our results at the level of precision we obtain.41 .00 0.02 0.04 0.06 0.08 0.10 0.12 m r p t -6 -5 -4 -3 -2 I ( M P r , n s ) / F P r FundamentalSextet
FIG. 20. Quantifying the size of leading-order finite volume corrections, Eq. (B7).
Appendix C: Technical matters—the axial current and W χ PT1. Conserved
U(1)
Axial Current
The conserved axial current is given by J µ = (cid:88) r = , q r J ( r )5 µ . (C1)As usual, the normalization of U(1) currents is arbitrary. We normalize the individual axialcurrents J ( r )5 µ such that all right-handed fields have unit charge. The ratio q /q is then fixedby the group traces of the two representations. For the normalization of J µ we adopt thesame prescription as in Ref. [12]. The resulting charges are q = 2 √ , q = − √ . (C2)These charges were used in the χ PT formulae of Sec. III C.Tracing Eqs. (3.13)–(3.16) back to the general NLO expressions of Ref. [12], one can checkthat they only depend on the ratios q r /F ζ . These ratios are independent of the choice ofnormalization for the axial current, because a rescaling of J µ by a factor λ implies a rescalingby the same factor of both the charges q and q and of the singlet’s decay constant F ζ . Allother LECs in Eqs. (3.13)–(3.16) are independent of this rescaling (for more details, seeRef. [12]). Of course, once the normalization of the charges is fixed according to Eq. (C2),the normalization of F ζ is fixed as well. 42 .460.480.500.520.54 a M P a M P
12 14 16 18 24 N s a F P
12 14 16 18 24 N s a F P ( , , ) = (7.75, 0.128, 0.131) FIG. 21. Explicit test of the dependence of the pseudoscalar masses and decay constants on spatialvolume at bare parameters ( β, κ , κ ) = (7 . , . , . ±
5% with respect to the mean value of the rightmost N s = 24 point.
2. AWI mass and Wilson chiral perturbation theory
In this appendix we review the proof of Ref. [36] that the mass defined by imposing theaxial Ward identity, m AWI , is equal to the shifted mass m shifted , which is the mass parameteroccurring in the LO lagrangian of W χ PT, up to higher-order corrections. For simplicity, wewill consider the GSM power counting used throughout this paper.A nice feature of the GSM scheme is that the LO lagrangian takes the same form as in thecontinuum. The reason is that the only new operator at O ( a ) is the Pauli term a ¯ ψσ µν F µν ψ ,which has the same chiral transformation properties as the fermion mass term. The Pauliterm enters the Symanzik action with a coefficient that depends linearly on the parameter c SW of the clover term in the Wilson action. After the transition to the chiral effectivetheory, the non-derivative terms in the LO lagrangian take the form L m = − F χ † Σ + Σ † χ ) − F A † Σ + Σ † ˆ A ) , (C3)where the mass term has been mapped to the first term on the right-hand side, and the Pauliterm to the second. χ is the usual spurion of continuum χ PT, and ˆ A is a new spurion withthe same chiral transformation properties as χ . The “expectation values” of the spurionsare [53] χ = 2 Bm ctm , ˆ A = 2 W a, (C4)43 .360.380.400.42 a M P a M P
12 14 16 18 24 N s a F P
12 14 16 18 24 N s a F P ( , , ) = (7.75, 0.129, 0.1308) FIG. 22. Explicit test of the dependence of the pseudoscalar masses and decay constants onspatial volume at bare parameters ( β, κ , κ ) = (7 . , . , . ±
5% with respect to the mean value of the rightmost N s = 24 point.
12 14 16 18 24 N s t / a ( , , ) = (7.75, 0.128, 0.131)
12 14 16 18 24 N s ( , , ) = (7.75, 0.129, 0.1308) FIG. 23. Explicit test of the dependence of the Wilson flow scale t /a on spatial volume at twosets of bare parameters, as described in the text. The dashed lines indicate variations of ±
5% withrespect to the mean value of the rightmost N s = 24 point. where B and W are low-energy constants (LECs). Substituting back into Eq. (C3) gives L m = − F Bm ctm + 2 W a ) tr(Σ + Σ † )= − F Bm shifted tr(Σ + Σ † ) , (C5)44here the shifted mass is defined by Eq. (3.10). For brevity, in the rest of this appendix wedenote the shifted mass as m .As explained in Sec. II B, in our numerical simulations we define the fermion mass m AWI by imposing the axial Ward identity, Eq. (2.3). Before we can use W χ PT, we need to knowthe relation between m AWI and the shifted mass m of Eq. (3.10), in terms of which the chiralexpansion is done.This relation was analyzed carefully in Ref. [36] for the case of two-flavor QCD. Thepseudoscalar density that was considered there, and which is also used in our work, is theusual local density, P a loc = ¯ ψγ T a ψ . (C6)For the axial current, Ref. [36] considered A aµ = A aµ, loc + ac A ∂ µ P a loc , (C7)where ∂ µ stands for a lattice derivative; the local axial current is A aµ, loc = ¯ ψγ γ µ T a ψ . (C8)For the purpose of this discussion, we may consider c A in Eq. (C7) as a free parameter. Inour numerical simulations we use the naive axial current, which corresponds to c A = 0.To first order in the pion field, the lattice operators are mapped to the effective theoryaccording to P a eff = √ F Bπ a (cid:0) O ( a ) (cid:1) , (C9) A aµ, eff = √ F ∂ µ π a (cid:0) O ( a ) (cid:1) , (C10)where π a is the effective pion field. The precise form of the O ( a ) corrections may be found inRef. [36]. In both equations, they depend linearly on the clover parameter c SW . In addition,the O ( a ) correction in Eq. (C10) depends linearly on c A . Plugging this into Eq. (2.3) andusing the LO pion mass, given by M = 2 Bm , we find m AWI = m (cid:0) O ( a ) (cid:1) + O ( a ) . (C11)To the order we are working, in general one expects also an O ( m ) correction. This correctionvanishes, however, because the continuum theory satisfies m AWI = m identically. While thederivation of Ref. [36] was given for a complex representation, a similar argument applies toreal (or pseudoreal) representations.Equation (C11) is robust in that changing the clover coefficient c SW or changing theparameter c A in Eq. (C7) will change the O ( am ) corrections, but will not affect the simul-taneous vanishing of m AWI and the shifted mass m . This feature is disrupted only by O ( a )effects, which is as it should be. Indeed, as shown in Ref. [28], depending on the sign of aparticular O ( a ) LEC, in the region where m ∼ a one either encounters the Aoki phase, ora first-order discontinuity line at which the pion mass reaches a non-zero minimum.As a corollary of Eq. (C11), we may use the value of m AWI , taken from the numericalsimulations, for the shifted mass m . At the order we are working, the differences betweenthe two are absorbed into a redefinition of NLO parameters of the chiral expansion. In this subsection we disregard renormalization factors. ppendix D: Gluon propagator and perturbative calculations for nHYP links To perform one loop perturbation theory for the NDS action [20], we need the gluonpropagator. This appendix describes its construction and gives perturbative results forcurrent renormalization factors.Normalized hypercubic links V x,µ are constructed from the dynamical gauge field U x,µ viathree successive smearing steps [16, 17]. Each step uses a weighted sum over staples, whichis then reunitarized. Explicitly,Ω x,ρ ; ξ = (1 − α ) U x,ρ + α (cid:16) U x,ξ U x +ˆ ξ,ρ U † x +ˆ ρ,ξ + U † x − ˆ ξ,ξ U x − ˆ ξ,ρ U x − ˆ ξ +ˆ ρ,ξ (cid:17) (D1a) V x,ρ ; ξ = P (Ω ξ,ρ ; ξ )Ω x,µ ; ν = (1 − α ) U x,µ + α (cid:88) ρ (cid:54) = µ,νξ (cid:54) = µ,ν,ρ (cid:16) V x,ρ ; ξ V x +ˆ ρ,µ ; ξ V † x +ˆ µ,ρ ; ξ + V † x − ˆ ρ,ρ ; ξ V x − ˆ ρ,µ ; ξ V x − ˆ ρ +ˆ µ,ρ ; ξ (cid:17) (D1b)˜ V x,µ ; ν = P (cid:16) Ω x,µ ; ν (cid:17) ˜Ω x,µ = (1 − α ) U x,µ + α (cid:88) ν (cid:54) = µ (cid:16) ˜ V x,ν ; µ ˜ V x +ˆ ν,µ ; ν ˜ V † x +ˆ µ,ν ; µ + ˜ V † x − ˆ ν,ν ; µ ˜ V x − ˆ ν,µ ; ν ˜ V x − ˆ ν +ˆ µ,ν ; µ (cid:17) (D1c) V x,µ = P (cid:16) ˜Ω x,µ (cid:17) . The reunitarization operator P is defined as V = P (Ω) ≡ Ω Q − / , (D2)where Q = Ω † Ω . (D3)The best way to understand the smearing is to go in reverse order. The staple sumextends into a different direction at each smearing step, such that each fat link V x,µ dependson a particular thin link U y,ν , if and only if there exists a hypercube to which both V x,µ and U y,ν belong.The dislocation-suppressing action adds a new term to the pure-gauge action S g S g = S plaq + S NDS , (D4)where the new term is S NDS = 12 N c (cid:88) x tr (cid:32) γ (cid:88) µ ˜ Q − x,µ + γ (cid:88) µ (cid:54) = ν Q − x,µ ; ν + γ (cid:88) ρ (cid:54) = ξ Q − x,ρ ; ξ (cid:33) . (D5)In practice we fix the α i ’s and take γ = γ = γ = γ = zβ where z is held constant. Theweak-coupling expansion to be sketched below gives the bare coupling g as1 g = β N c + 1 N c (cid:16) γ α γ α + γ α (cid:17) . (D6)To construct the gluon propagator we need to compute the gauge action in quadraticorder. This is pretty standard; a good reference is Ref. [54]. We take the lattice action, S = a (cid:88) x L ( ψ, ψ, U ) , (D7)46nd replace the link field by an expansion in terms of gauge fields U µ ( x ) = exp[ igaA µ ( x )] = 1 + igaA µ ( x ) − g a A µ ( x ) + · · · , (D8)where A µ ( x ) = (cid:80) a ( λ a / A aµ gives the decomposition into color components. There is anidentical expansion for the fat link, which we write as V µ ( x ) = exp[ igaB µ ( x )].The action has an expansion in powers of A . In terms of the integral over the four-dimensional Brillouin zone, (cid:90) q ≡ (cid:90) π/a − π/a d q (2 π ) , (D9)and the vector potential, A µ ( x ) = (cid:90) q A µ ( q ) e iq ( x + a ˆ µ/ , (D10)the free gauge boson action is S G = − (cid:90) pp (cid:48) (2 π ) δ ( p + p (cid:48) )[ A aµ ( p (cid:48) ) D abµν ( p ) A bν ( p )] . (D11)For the gauge boson, D abµν = δ ab D µν . Just as in a continuum theory, the gauge boson actioncannot be inverted to give the propagator without fixing the gauge. A conventional choicefor a gauge fixing term is [introducing ˆ k µ = 2 /a sin( ak µ / S gf = − (cid:88) µν (cid:90) k Tr 1 ξ ˆ k µ ˆ k ν A aµ ( − k ) A aν ( k ) . (D12)Then the gauge boson propagator is found by solving the field equation (cid:88) ν (cid:20) ξ ˆ k µ ˆ k ν + D µν ( k ) (cid:21) G ντ ( k ) = δ µτ . (D13)We simply do this numerically, inverting the four by four matrix for each k value.To perform the perturbative expansion of the NDS action, we look at each term in turn.Consider Q x,ρ ; ξ = Ω † x,ρ ; ξ Ω x,ρ ; ξ . (D14)Multiplying this out, we find that Q is a sum of loops of perimeter four and perimeter six,labeled P and E respectively. The planar E loops, E µν , are 1 × ± ν direction from site x . Hence, Q = 1 + α (1 − α ) (cid:88) j P j + α (cid:88) k E k . (D15)Expanding each term out in terms of A ’s, we discover that Q − = 1 − S Q where now theexpressions are quadratic functions of the gauge fields. Slightly abusing notation, we write P and E for the objects made of A µ ’s, so that S Q = α (1 − α ) (cid:88) j P j + α (cid:88) k E k (D16)47early identical results obtain for the other two Q ’s, with several small exceptions. First,the analogs of the plaquettes are built of one thin link A µ ( x ) while the other three links arefattened. Second, the perimeter-6 links are built entirely of fat links. Finally, in addition tothe 1 × E µν plaquettes, there are “chair” plaquettes C µνρ which are “folded” about the µ axis. They extend in four directions ( ± ν , ± ρ ).The analog of the fat to thin relation for the links ( U → V ) is a fat to thin gauge fieldrelation, B λ ( q ) = (cid:88) µ A λ ( q ) h µλ ( q ) (D17)This means that all the perimeter-six contributions can be easily computed beginning withthe fat links, whose gauge-unfixed action is D µν ( q ). The thin link action is D µν ( k ) = (cid:88) λ λ h µλ ( − q ) D λ λ ( q ) h νλ ( q ) . (D18)Finally, by convention the action D µν ( q ) ∼ δ µν k − k µ k ν so we have to rescale the actionto remove explicit factors of the coupling constants, giving S = 1 N (cid:16) P + z ˜ Q − + zQ − + zQ − (cid:17) , (D19)where N = 2 N c / ( βg ) = [1 + 2 z ( α / α + α )], basically Eq. (D6).The fattened terms in the action are awkward to compute. The plaquette is an example.It is probably best to record F µρ ( q ) = i ˆ q µ B ρ ( q ) + A µ ( q ) exp( − iq ρ / − B µ ( q ) , exp( iq ρ /
2) (D20)where B µ is the fat link gauge potential, and then to substitute in Eq. (D17); to write(schematically) F µρ ( q ) = (cid:88) λ A λ ( q ) C λµν ( q ) ; (D21)and then D λ λ ( q ) = (cid:88) µν C λ µν ( − q ) C λ µν ( q ) . (D22)We also need perturbative expressions for the partially fattened links. They are A µ,ρη ( q ) = A µ ( q ) + α (cid:2) ˆ q µ ˆ q ω A ω ( q ) − ˆ q ω A µ ( q ) (cid:3) , (D23)where ω (cid:54) = µ, ν, ρ , and˜ A µ,ρ ( q ) = A µ ( q ) + α (cid:88) η (cid:54) = µ,ρ ˆ q η (cid:16) α − ˆ q ω α (cid:17) [ A η ( q )ˆ q µ − A µ ( q )ˆ q η ] , (D24)where ω (cid:54) = µ, η, ρ .Our simulations are done with z = 1 / z , the perturbative propertiesof the NDS gauge action are almost indistinguishable from those of a pure Wilson action.Here are three examples. 48irst, the plaquette has an expansion Tr U plaq /N c = 1 − g pC F where C F is the quadraticCasimir for fundamentals and p is a constant. For the Wilson action, p = 0 .
5. Thisexpression is often replaced by − ln (cid:28) N c Tr U plaq (cid:29) = g pC F . (D25)This defines a coupling g in the so-called α V scheme, because the potential is written as V ( q ) = − C F πα V ( q ) q (D26)The scale of the coupling is set by the Lepage–Mackenzie [55] prescription,log q ∗ = (cid:82) d q log qI ( q ) (cid:82) d q I ( q ) . (D27)Results for p and q ∗ for several values of z are given in Table XIV. z p q ∗ z V q ∗ z A q ∗ z P q ∗ z S q ∗ . . − .
38 1 . − .
30 1 . − .
12 2 .
28 0 .
04 2 . .
008 0 .
504 3 . − .
37 1 . − .
30 1 . − .
11 2 .
42 0 .
05 2 . .
02 0 .
510 3 . − .
38 1 . − .
29 1 . − .
12 2 .
48 0 .
04 2 . .
05 0 .
525 3 . − .
40 1 . − .
31 1 . − .
15 2 .
40 0 .
02 2 . .
10 0 .
548 3 . − .
42 1 . − .
34 1 . − .
19 2 . − .
02 2 . z . Uncertainties are ± p . The quantities z V , z A , z P and z S are the renormalization factors for the local vector, axial vector, pseudoscalar, andscalar currents. With the gluon propagator in hand we can immediately compute the static Coulombpotential. With our conventions, the continuum potential is V ( r ) = 1 / (4 πr ), and so plottingthe rescaled lattice potential 4 πrV ( r ) immediately exposes the lattice artifacts of a particularaction. We show results for this quantity in Fig. 24 for z = 1 / Q with engineering dimension D [we have in mind finding the M S (modified min-imal subtraction) value at scale µ ] is related to the lattice value by¯ Q ( µ = 1 /a ) = Q ( a ) (cid:18) − κ κ c (cid:19) Z Q , (D28)with Z Q = 1 + α C F π z Q , (D29)49 IG. 24. Comparison of the potential for the NDS action, with γ/β = 1 /
125 (diamonds), withthat of the usual Wilson action (crosses). where α = g / (4 π ), C F is the quadratic Casimir for the fermion, and z Q is a scheme matchingnumber. Results for nHYP clover fermions without the NDS term are tabulated in Ref. [56]and allow us to check the z = 0 limit. For more discussion of the calculation of the z Q ’s, seeRef. [57]. Table XIV gives selected values of z Q for the vector, axial vector, pseudoscalar,and scalar currents.To evaluate the final Z -factors, we run α V ( q ) obtained at aq (cid:63) = 3 .
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