Lattice simulations with eight flavors of domain wall fermions in SU(3) gauge theory

Lattice simulations with eight flavors of domain wall fermions in SU(3) gauge theory

T. Appelquist Department of Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06520, USA    R. C. Brower Department of Physics, Boston University, Boston, Massachusetts 02215, USA    G. T. Fleming Department of Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06520, USA    J. Kiskis Department of Physics, University of California, Davis, California 95616, USA    M. F. Lin Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, Illinois 60439, USA Computational Science Center, Brookhaven National Laboratory, Upton, New York 11973, USA    E. T. Neil Department of Physics, University of Colorado, Boulder, Colorado 80309, USA RIKEN–BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA    J. C. Osborn Argonne Leadership Computing Facility, Argonne National Laboratory, Argonne, Illinois 60439, USA    C. Rebbi Department of Physics, Boston University, Boston, Massachusetts 02215, USA    E. Rinaldi Lawrence Livermore National Laboratory, Livermore, California 94550, USA    D. Schaich Department of Physics, Syracuse University, Syracuse, New York 13244, USA    C. Schroeder Lawrence Livermore National Laboratory, Livermore, California 94550, USA    S. Syritsyn RIKEN–BNL Research Center, Brookhaven National Laboratory, Upton, New York 11973, USA    G. Voronov Department of Physics, Sloane Laboratory, Yale University, New Haven, Connecticut 06520, USA    P. Vranas Lawrence Livermore National Laboratory, Livermore, California 94550, USA    E. Weinberg Department of Physics, Boston University, Boston, Massachusetts 02215, USA    O. Witzel Center for Computational Science, Boston University, Boston, Massachusetts 02215, USA
24 October 2014

We study an SU(3) gauge theory with degenerate flavors of light fermions in the fundamental representation. Using the domain wall fermion formulation, we investigate the light hadron spectrum, chiral condensate and electroweak parameter. We consider a range of light fermion masses on two lattice volumes at a single gauge coupling chosen so that IR scales approximately match those from our previous studies of the two- and six-flavor systems. Our results for the spectrum suggest spontaneous chiral symmetry breaking, though fits to the fermion mass dependence of spectral quantities do not strongly disfavor the hypothesis of mass-deformed infrared conformality. Compared to we observe a significant enhancement of relative to the symmetry breaking scale , similar to the situation for . The reduction of the parameter, related to parity doubling in the vector and axial-vector channels, is also comparable to our six-flavor results.

11.10.Hi, 11.15.Ha, 11.25.Hf, 12.60.Nz, 11.30.Qc

Lattice Strong Dynamics (LSD) Collaboration

I Introduction

The discovery of a Higgs particle at the Large Hadron Collider Aad et al. (2012); Chatrchyan et al. (2012) was a major step towards the longstanding goal of determining the mechanism of electroweak symmetry breaking. The properties of this particle are so far consistent with the predictions of the standard model Chatrchyan et al. (2013); Aad et al. (2013), but could also result from new strong dynamics at or above the TeV scale. Walking technicolor theories, in which approximately conformal dynamics produce a slowly-running gauge coupling and a large mass anomalous dimension across a wide range of energy scales Holdom (1981); Yamawaki et al. (1986); Appelquist et al. (1986), are potential candidates to produce a light composite Higgs boson Appelquist et al. (2013a). Numerical lattice gauge theory calculations are a crucial non-perturbative tool to study such strongly-interacting gauge theories from first principles. In this paper we present results from lattice investigations of SU(3) gauge theory with fundamental fermions, a candidate walking theory.

In SU() gauge theories with massless fermions in the fundamental representation, chiral symmetry breaks spontaneously and the system confines if is sufficiently small. When reaches a certain value , with at which asymptotic freedom is lost, the theory flows to a chirally symmetric conformal fixed point in the infrared (IRFP) Caswell (1974); Banks and Zaks (1982). The region is called the conformal window for SU() with fundamental fermions. Around the upper end of the conformal window, , the IRFP is weakly coupled and can be investigated perturbatively. The fixed point moves to stronger coupling as decreases, motivating lattice studies of non-perturbative conformal or near-conformal dynamics for .

Many lattice calculations have been performed to search for precise values of , and more generally to explore the range of possible phenomena in these strongly-coupled gauge theories (cf. the recent review Kuti (2014) and references therein). For SU(3) gauge theories with fundamental fermions, these studies have focused on , 8, 10 and 12. Although the 6-flavor theory exhibits interesting dynamical differences compared to QCD Appelquist et al. (2010); Appelquist et al. (2011a, 2012a); Miura and Lombardo (2013), there is little doubt that it is chirally broken. Studies with larger are less conclusive. Continuum estimates that  Appelquist et al. (1996, 1999); Bashir et al. (2013) make these difficult investigations particularly interesting.

For , several lattice studies Appelquist et al. (2008, 2009); Deuzeman et al. (2008); Fodor et al. (2009); Hasenfratz (2010); Jin and Mawhinney (2010) concluded that the theory most likely undergoes spontaneous chiral symmetry breaking. More recently, Refs. Cheng et al. (2013a, b) reported that the 8-flavor system possesses a large effective mass anomalous dimension across a wide range of energy scales. The LatKMI Collaboration is investigating the light meson spectrum of the theory Aoki et al. (2013), arguing that at lighter fermion masses the spectrum may be described by chiral perturbation theory, while data at heavier appear to exhibit some remnant of IR conformality despite chiral symmetry breaking. They also find that the flavor-singlet scalar Higgs particle can be as light as the pseudoscalar meson (the would-be pion) throughout this range of  Aoki et al. (2014). Preliminary results from a large-scale USBSM project could not clearly confirm spontaneous chiral symmetry breaking with fermion masses as light as on a lattice volume Schaich (2013).

In contrast to the lattice studies summarized above, which all employ staggered fermions, we investigate using the domain wall fermion formulation that possesses improved continuum-like chiral and flavor symmetries. This work is the latest addition to our extensive investigations of SU(3) gauge theories with , 6, 8 and 10 flavors of degenerate domain wall fermions Appelquist et al. (2010); Appelquist et al. (2011a, 2012a, 2012b, 2013b). Domain wall fermions are more computationally expensive than staggered fermions, which limits the statistics we can obtain and is the reason we have not yet determined the fermion-line-disconnected contributions to flavor-singlet observables. In the next section we summarize our 8-flavor simulations, which generate ensembles of gauge configurations for a range of light fermion masses on two lattice volumes, at a single gauge coupling chosen so that IR scales approximately match for all .

We use these ensembles to investigate the light hadron spectrum, chiral condensate and electroweak parameter. Section III presents our spectrum analyses, first reviewing the determination of hadron masses and decay constants. Steady growth in the ratio of the vector meson mass compared to the pseudoscalar mass as we approach the chiral limit suggests that chiral symmetry breaks spontaneously for . Although we find the flavor non-singlet scalar meson to be heavy, , we have not yet determined the mass of the more interesting flavor-singlet Higgs particle. When we confront the fermion mass dependence of spectral quantities with expressions motivated by either spontaneous chiral symmetry breaking or mass-deformed IR conformality, we obtain comparable fit quality in each case.

In Section IV we explore the enhancement of relative to the symmetry breaking scale , which is of interest in the context of fermion mass generation. We find a significant enhancement of the ratio for compared to , similar to results we previously reported for  Appelquist et al. (2010); Fleming and Neil (2013). Finally, we study the electroweak parameter in Section V, also discussing the related issue of parity doubling in the vector () and axial-vector () channels. We follow the approach of Ref. Appelquist et al. (2011a) to calculate from the transverse vacuum polarization function. At the range of masses we can access on our lattice ensembles, we observe parity doubling and a reduction in that are also comparable to six-flavor results from Ref. Appelquist et al. (2011a). We summarize our conclusions and prospects for further progress in Section VI. The Appendix provides additional information about thermalization, auto-correlations and the topological charge.

Ii Simulation Details

ii.1 Parameters and algorithms

Our calculations are performed with the domain wall fermion (DWF) formulation Shamir (1993); Furman and Shamir (1995), where an auxiliary fifth dimension separates the left-handed and right-handed chiralities. We thereby obtain good chiral symmetry even at non-zero lattice spacing, with only a small chiral symmetry breaking effect quantified as the residual mass . With a non-zero input fermion mass , the effective fermion mass is in the DWF formulation.

Using the Iwasaki gauge action Iwasaki (1983), we tune the bare gauge coupling to to obtain in lattice units, where is the linear extrapolation of the vector meson mass to the chiral limit 111Such a linear chiral extrapolation of assumes that chiral symmetry is spontaneously broken, since in IR-conformal systems all masses vanish in the infinite-volume chiral limit. The assumption of spontaneous chiral symmetry breaking appears reasonable for . Because we tuned with a limited amount of initial data, our preliminary determination had large uncertainties, and is consistent with the somewhat smaller final result in Table 3.. This value of approximately matches those used in our 2- and 6-flavor investigations Appelquist et al. (2010) (cf. Table 3), and is equivalent to having a relatively large UV cutoff scale . If the theory is confined and chirally broken, then is related to the confinement scale. Having a large ratio helps to separate the IR physics from the UV physics, particularly in a theory where the gauge coupling may be running slowly.

We consider two lattice volumes, and , with parameters summarized in Table 1. The length of the fifth dimension is fixed to for both volumes. To check for possible thermalization or poor sampling effects Appelquist et al. (2012b), we generate two independent ensembles for each of the two lightest masses, and 0.015, one starting from a random (disordered) gauge configuration, the other from an ordered configuration. In all of our analyses we use a jackknife procedure with 50-trajectory blocks to reduce the effects of auto-correlations. The Appendix provides additional information about auto-correlations and the thermalization cuts listed in Table 1. As discussed in the Appendix, 50-trajectory jackknife blocks may not remove all auto-correlation effects, which could cause our statistical uncertainties to be underestimated. Since we do not have enough data to use larger blocks consistently, we increase our error estimates by 25% to account for this potential underestimation.

Start Traj. Therm. Blocks
0.010 dis 785 610 3
0.010 ord 2032 610 28
0.015 dis 1279 510 15
0.015 ord 1734 510 24
0.020 dis 1441 510 18
0.025 dis 1324 510 16
0.030 dis 1392 510 17
0.020 ord 6665 610 31
0.025 ord 2780 610 19
0.030 ord 2460 610 13
0.035 ord 2860 610 29
0.040 ord 2400 610 29
0.045 ord 2219 610 22
0.050 ord 4219 610 14
Table 1: Simulation parameters, starting configuration and total number of trajectories in each of our DWF ensembles. All ensembles use gauge coupling and in the fifth dimension. All our analyses use a jackknife procedure with 50-trajectory blocks, so the number of blocks used for each ensemble depends on the corresponding thermalization cut.

We generate gauge configurations using a Hybrid Monte Carlo algorithm with Hasenbuch mass preconditioning Hasenbusch and Jansen (2003), in a style similar to Ref. Aoki et al. (2011). For each fermion determinant representing two degenerate flavors, an intermediate mass is used to precondition the input fermion mass and the Pauli–Villars mass , resulting in a partition function of the form


for degenerate fermions. Here is the gauge action, is the hermitian two-flavor domain wall Dirac operator Blum et al. (2004), and represents the DWF Pauli–Villars field. We use intermediate mass for all input , and fix the trajectory length to be  molecular dynamics (MD) time unit. While the intermediate mass term introduces additional pseudofermion fields, it helps to reduce the overall fermion force in the MD steps, making it possible to use larger step sizes while maintaining a good acceptance rate Aoki et al. (2005). Another reduction in computational cost comes from the use of the chronological inverter Brower et al. (1997). To avoid the loss of reversibility, we set stringent stopping conditions for the conjugate gradient matrix inversion: for the MD evolution, and for the Metropolis step.

ii.2 Residual mass and renormalization constants

We calculate the residual mass following the standard procedure described in great detail by Refs. Blum et al. (2004); Allton et al. (2008). The results from our ensembles are recorded in Table 2 and plotted in Figure 1. Defining in the limit , we obtain from a simple linear extrapolation.

0.010 2.860(5) 0.70146(8)
0.015 2.939(4) 0.70173(5)
0.020 3.014(7) 0.70221(7)
0.025 3.104(10) 0.70273(5)
0.030 3.210(8) 0.70324(6)
Table 2: Residual mass and axial-vector current renormalization constant as functions of the input mass from our 8-flavor ensembles.
Figure 1: Residual mass as a function of the input mass from our 8-flavor ensembles, with linear extrapolation to determine . The line is solid throughout the fit range .

Table 2 also presents our results for the DWF axial-vector current renormalization constant , determined from the procedure described in Ref. Blum et al. (2004). Our analyses below involve both this renormalization constant as well as the corresponding for the vector current. Because we define these renormalization constants in the chiral limit , DWF obey the chiral symmetry relation up to lattice discretization effects of . From a linear chiral extrapolation we find , which we will use for both vector and axial-vector currents.

Table 3 summarizes our and results for all of , 6 and 8. Screening effects from the additional fermions require that we work at stronger gauge couplings (smaller ) as increases, in order to maintain comparable IR scales such as and the chirally-extrapolated baryon mass . As a result, increases by two orders of magnitude as we move from to , while moves farther from unity. If is made too small, zero-temperature lattice calculations encounter a strong-coupling bulk transition at which the system becomes dominated by discretization artifacts. In lattice QCD with domain wall fermions and the Iwasaki gauge action we use, this occurs around . As increases the bulk transition moves to slightly stronger coupling, for . Working with safely on the weak-coupling side of this transition results in an 8-flavor somewhat smaller than the values we obtained for and 6.

2 2.70 0.0263(1) 0.85042(8) 0.2166(27) 0.2984(64)
6 2.10 0.8278(22) 0.72615(7) 0.1991(33) 0.2425(95)
8 1.95  2.6836(72)  0.70011(10)  0.1710(36) 0.2441(54)
Table 3: dependence of the residual mass , the axial-vector current renormalization constant , and linear chiral extrapolations of the vector meson and baryon masses, and , respectively. Our results for and 6 were first presented in Ref. Appelquist et al. (2010).

We do not include the chirally-extrapolated pseudoscalar decay constant among the IR scales that we attempt to match between systems with different . This is because is more sensitive to the form of the extrapolation, and needs to be analyzed using next-to-leading-order chiral perturbation theory (NLO PT). While we presented such an analysis for in Ref. Appelquist et al. (2010), we find that our 6- and 8-flavor data are not within the radius of convergence of NLO PT.

Iii Light Hadron Spectrum

iii.1 Meson masses and decay constants

Using the lattice ensembles discussed in the previous section, we analyze the light hadron spectrum, focusing especially on the pseudoscalar meson (), the vector meson () and the axial-vector meson (). In chirally broken systems, the pseudoscalar meson is a pseudo-Nambu–Goldstone boson (PNGB), while the vector and axial-vector mesons may become more degenerate for theories near the conformal window. In addition we consider the connected (flavor non-singlet) scalar meson (), and in the next subsection we will study the lightest baryon ().

We measure meson and baryon two-point correlators every 10 MD trajectories with both point () and Coulomb-gauge-fixed wall () sources, as well as and sinks. Hence for each hadronic operator, we have four different types of two-point correlator, denoted as , , and , where the subscripts indicate the sink and source, respectively. To further increase statistics we also use two different source locations, and , where is the temporal extent of the lattice. In our analyses, we first average each correlator over the two source locations and block every 50 MD trajectories. We then perform a simultaneous jackknife fit of the four averaged meson correlators to the form


where is projected to zero spatial momentum and the trace is over flavor. (We normalize the flavor matrices , so that .)

For the pseudoscalar meson we can consider both and , while for the scalar, for the vector and for the axial-vector meson, with , 2, 3. While each of the four source–sink combinations has an independent amplitude , the meson mass is a common parameter in the simultaneous fit, in a way similar to Ref. Allton et al. (2008). Our limited statistics do not allow us to calculate a correlation matrix between the different source–sink combinations. Because the pseudoscalar meson couples to both the and channels, our fit provides a common mass and eight amplitudes. For the vector and axial-vector mesons, we first average over the three polarizations , 2, 3 to form a single correlator for each source–sink combination. Our fits then provide a common mass and four amplitudes for each of the vector and axial-vector states.

Since different source–sink combinations have different excited-state contaminations, it is important to permit an independent fit range for each correlator in the simultaneous fit. This allows us to make more efficient use of the available data while still avoiding excited-state effects. To set fit ranges, we inspect the effective masses of the individual correlators to identify the onset of plateaus. Since these bosonic correlators are symmetric around the middle timeslice in the lattice, we “fold” the correlators by averaging and for . Some representative effective mass plots from our ensembles with (combining ordered and disordered starts) are shown in Figure 2. To maintain some uniformity between different ensembles, we attempt to choose fairly conservative fit ranges that fall within plateaus for all . Table 4 lists the resulting fit ranges used in our meson spectrum analyses.

Figure 2: Representative effective masses for the pseudoscalar meson (left) and vector meson (right), from our ensembles with , combining ordered and disordered starts. The shaded bands show the fit ranges and the uncertainties of the fit results. The red and blue regions in each band indicate the fit ranges for and , respectively.
Operator 0.015 0.020 0.025 0.030 All
  [16, 32] [16, 32] [16, 32] [16, 32] [16, 32] [8, 16]
[16, 32] [16, 32] [16, 32] [16, 32] [16, 32] [8, 16]
[16, 32] [20, 32] [16, 32] [16, 32] [16, 32] [8, 16]
[16, 32] [16, 32] [16, 32] [16, 32] [16, 32] [8, 16]
[16, 32] [16, 32] [16, 32] [16, 32] [16, 32] [8, 16]
[16, 32] [16, 32] [16, 32] [16, 32] [16, 32] [8, 16]
[16, 32] [20, 32] [16, 32] [16, 32] [16, 32] [8, 16]
[16, 32] [16, 32] [16, 32] [16, 32] [16, 32] [8, 16]
[6, 32] [6, 32] [6, 32] [6, 22] [6, 22] [6, 16]
[6, 32] [6, 32] [6, 32] [12, 32] [12, 32] [6, 16]
[20, 32] [20, 32] [20, 32] [20, 32] [20, 32] [12, 16]
[6, 32] [6, 32] [6, 32] [6, 22] [6, 22] [6, 16]
[10, 16] [5, 16] [10, 16] [5, 16] [5, 16] [10, 16]
[10, 22] [5, 16] [10, 22] [5, 16] [5, 16] [10, 16]
[16, 24] [20, 28] [16, 24] [20, 28] [20, 28] [12, 16]
[6, 16] [6, 16] [6, 16] [6, 16] [6, 16] [6, 16]
[15, 20] [6, 16] [15, 20] [13, 20] [13, 25] [10, 16]
[15, 20] [6, 16] [15, 20] [13, 20] [13, 25] [10, 16]
[15, 25] [6, 16] [15, 25] [20, 25] [20, 25] [10, 16]
[15, 20] [6, 16] [15, 20] [13, 20] [13, 20] [10, 16]
Table 4: Fit ranges in used to determine meson masses.

Turning to the flavor non-singlet decay constants, we define them through


for the pseudoscalar, vector and axial-vector channels, respectively. Here with are polarization vectors, while and are the local (non-conserved) vector and axial-vector currents. As discussed in the previous section, we use for both the vector and axial-vector current renormalization constants and . The above definitions are consistent with the conventions in Ref. Peskin and Takeuchi (1992), with normalization such that the QCD pion decay constant is about 93 MeV.

The partially-conserved axial current (PCAC) relation also allows us to determine the pseudoscalar decay constant from the pseudoscalar matrix element Blum et al. (2004). For DWF, the PCAC relation is


where is the (partially-)conserved axial-vector current and is the local pseudoscalar current. This allows us to replace the axial-vector matrix element in Eq. (4) by the pseudoscalar matrix element, giving


The amplitudes we obtain from the simultaneous fits allow us to determine the decay constants in several different ways. For each ensemble, we take the jackknife average of these different determinations as our final result.

As discussed in Section II, for and 0.015 we generate separate ensembles using either ordered or disordered starting configurations, to check for possible bias from inadequate thermalization or from poor sampling of the topological sectors. Table 5 compares results for the meson masses and decay constants determined separately on these ordered- and disordered-start ensembles. For both and 0.015, the separate results agree well enough that we can combine the two Markov chains to perform our final analysis. The final fit results for the meson masses and decay constants are shown in Table 6 for the ensembles, and in Table 7 for the ensembles.

0.010 dis 0.1983(66) 0.2786(66) 0.3441(52) 0.0285(7) 0.0438(6) 0.0445(32)
0.010 ord 0.1844(21) 0.2659(20) 0.3403(65) 0.0299(8) 0.0456(9) 0.0422(15)
0.015 dis 0.2252(19) 0.3029(30) 0.4138(47) 0.0360(11) 0.0486(14) 0.0459(22)
 0.015  ord  0.2251(13)  0.2964(20)  0.3915(36)  0.0365(10)  0.0466(8)  0.0400(16)
Table 5: Comparison of meson masses and decay constants from ensembles with different starting configurations.

The reasonable agreement between results from ordered and disordered starts is in stark contrast to the 10-flavor case Appelquist et al. (2012b), where we observed significant disagreements that we attributed to frozen topological charges . While the topological charges sampled by our 8-flavor ensembles do not produce the desired gaussian distributions, tunnels frequently, as we show in the Appendix. This tunneling appears sufficient to eliminate the systematic discrepancy introduced when the topological charge is completely frozen.

0.010 0.1853(21) 0.2664(21) 0.3411(64) 0.296(12) 0.0297(7) 0.0453(7) 0.0425(15)
0.015 0.2252(10) 0.2990(18) 0.3999(37) 0.381(8) 0.0364(7) 0.0474(7) 0.0421(12)
0.020 0.2582(14) 0.3363(27) 0.4414(102) 0.471(34) 0.0395(7) 0.0495(9) 0.0425(21)
0.025 0.2949(11) 0.3759(38) 0.5052(49) 0.474(17) 0.0456(9) 0.0541(14) 0.0427(32)
 0.030  0.3280(12)  0.4131(35)  0.5502(38)  0.518(26)  0.0504(13)  0.0586(16)  0.0493(42)
Table 6: Results for meson masses and decay constants from ensembles.
0.020 0.3725(83) 0.468(13) 0.483(18) 0.383(6) 0.0305(6) 0.0722(29) 0.065(4)
0.025 0.3880(83) 0.485(4) 0.517(9) 0.423(8) 0.0378(12) 0.0745(16) 0.068(2)
0.030 0.3957(81) 0.486(13) 0.511(31) 0.439(23) 0.0459(14) 0.0775(29) 0.060(6)
0.035 0.3930(67) 0.501(6) 0.596(18) 0.497(15) 0.0536(15) 0.0779(17) 0.066(5)
0.040 0.4120(41) 0.524(7) 0.621(27) 0.559(31) 0.0590(13) 0.0801(16) 0.061(6)
0.045 0.4373(31) 0.550(6) 0.753(39) 0.678(65) 0.0641(12) 0.0852(22) 0.087(11)
 0.050  0.4544(52)  0.557(5)  0.716(95)  0.609(50)  0.0661(12)  0.0829(27)  0.068(29)
Table 7: Results for meson masses and decay constants from ensembles.

iii.2 Baryon mass

The zero-momentum projected two-point baryon correlator is


where the interpolating operator is


In the above equation are color indices and is the charge conjugation operator. The fermion fields have anti-periodic boundary conditions in the time direction. If we define the parity projection operators , then on a lattice with temporal extent , the large- behavior of can be written as


where and represent the ground states of the baryon and of its parity partner.

For each two-point correlator measurement, we average the positive- and negative-parity-projected correlators to define


When , the averaged correlator takes the simple exponential form


and we use single-exponential fits to determine the baryon mass . By considering , one would obtain the mass of the baryon’s parity partner . However, we could not reliably determine from our current data, and only report results for . We also present results only from our ensembles, since the temporal extent of the lattices is not large enough to provide reliable plateaus.

We consider only wall sources with point sinks to determine the baryon mass, as this source–sink combination provides the best signal-to-noise ratio. As in the analysis for the mesons, we average over the two source locations and block every 50 MD trajectories. Figure 3 presents two representative sets of effective mass results (for our ensembles with and 0.015, combining ordered and disordered starts), which show that the approach to a plateau varies significantly for different . We choose fit ranges in by requiring that the fit results do not change beyond statistical uncertainties upon dropping the first or last points. These fit ranges, and the corresponding baryon mass results, are tabulated in Table 8. As for our meson spectrum results, we combine ordered- and disordered-start ensembles to determine the final results for and 0.015. We find good agreement between computed separately on the ordered- and disordered-start ensembles.

Figure 3: Representative baryon effective masses from our ensembles with and 0.015, combining ordered and disordered starts while considering only wall sources and point sinks. The shaded bands indicate the fit ranges and the uncertainties of the fit results.
Fit range dis ord
 0.010    0.3905(33)  0.401(10)  0.390(3)
0.015 0.4446(39) 0.446(6) 0.444(5)
0.020 0.4962(41)
0.025 0.5590(50)
0.030 0.6189(41)
Table 8: Fit ranges and results for the baryon mass from ensembles.

In Figure 4 we plot our baryon mass results as functions of for all of our investigations with , 6 and 8. Via the Feynman–Hellmann theorem, the -dependence of is related to the baryon term Walker-Loud et al. (2009); Appelquist et al. (2014)


Estimating as the slope of vs. , we find that the 6- and 8-flavor slopes agree within uncertainties, and both are twice as large as the result for . In terms of the dimensionless , we find for , for and for , with in each case.

Figure 4: Baryon mass plotted against for , 6 and 8, each with on lattices. Lines connect points to guide the eye.

iii.3 Chiral symmetry breaking

We begin our discussion of the spectrum results by considering the Edinburgh-style plot in Figure 5, which presents the ratios vs.  for all of our investigations with , 6 and 8. For each we analyze lattice ensembles with the same five input fermion masses . These correspond to for , for and for . The ratios in Figure 5 are designed to exaggerate finite-volume effects, which we expect to increase the masses while decreasing , pushing the points up and to the right. Our results do not show this behavior for any . The 2-flavor ratios move steadily to the left, as we would expect from spontaneous chiral symmetry breaking: the pseudoscalar becomes a massless NGB in the chiral limit, while and remain non-zero. Although the 6-flavor results also move to the left, they do not move as much as do those for , and the 8-flavor points cluster in a small region of the plot. The lightest point for may hint at the onset of finite-volume effects, but such effects are not yet significant in these ratios. In addition to providing evidence that finite-volume effects are under control, Figure 5 illustrates some of the differences between the three systems with , 6 and 8.

Figure 5: Edinburgh-style plot of vs. , for , 6 and 8, each with on lattices. Lines connect points to guide the eye, and arrows indicate the direction of decreasing masses. Our results for and 6 were first presented in Ref. Appelquist et al. (2010).

The ratios and provide similar illustrations, which we consider in Figure 6. These quantities diverge in the chiral limit for chirally broken systems in which while and remain non-zero. Figure 6 again compares with our earlier results for and 6, considering the same on lattices but now plotting vs. . The 2-flavor results for increase rapidly, as we expect for a chirally broken theory with non-zero fermion mass. While this ratio does not grow so dramatically for and 8, both show similar monotonic increases as decreases, suggesting spontaneous chiral symmetry breaking. The 2-flavor results for the ratio are qualitatively similar, though the uncertainties are significantly larger. (We could not reliably determine for with , for which .) The 6- and 8-flavor results for are roughly constant within uncertainties. This behavior may be related to parity doubling, an issue we will discuss further in the context of the electroweak parameter in Section V.2.

Figure 6: (left) and (right) plotted against for , 6 and 8, each with on lattices. Lines connect points to guide the eye. Our and results for and 6 were first presented in Ref. Appelquist et al. (2010).

Although we find the mass to be larger than the TeV-scale in the absence of finite-volume effects, this has no direct implications for the mass of the flavor-singlet scalar Higgs particle. The flavor-singlet state is sensitive to fermion-line-disconnected contributions that are extremely expensive to compute, especially in the DWF formulation. These disconnected contributions appear crucial to the LatKMI Collaboration’s observation of a flavor-singlet scalar roughly degenerate with the pseudoscalar for  Aoki et al. (2014). As an alternative approach, we have explored gluonic operators that also couple to the scalar channel. Our preliminary results suggest that our ensembles do not possess sufficient statistics to permit robust glueball analyses, another consequence of working with expensive DWF. Because it is so important to determine whether the observed 125 GeV Higgs is consistent with new strong dynamics, in the future we plan to explore disconnected DWF calculations, to judge whether the available computational resources would provide reliable results.

If chiral symmetry does break spontaneously in the 8-flavor system, as Figure 6 suggests, then leading-order PT predicts that at small the PNGB mass squared is . The other masses and decay constants should be well modeled by a constant plus a term linear in . We are not working at light enough masses to expect to resolve chiral logarithms. We therefore further explore the hypothesis of spontaneous chiral symmetry breaking by considering fits to the following simple forms:


These fits are shown in Figure 7. Their quality varies significantly depending on the observable considered, as indicated by the values in the second column of Table 9.

The third, fourth and fifth columns of Table 9 explore the fit-range dependence of these chiral extrapolations, by omitting the lightest point , the heaviest point , and both of these points, respectively. Most quantities show relatively little sensitivity to the fit range, with the different intercepts agreeing within statistical uncertainties. There is also no significant systematic trend in the  values as data points are omitted from the fits. The significantly non-zero intercepts (and large ) we find for the pseudoscalar mass squared indicate that our results are not well described by leading-order PT, . This behavior is consistent with our expectation that our 8-flavor data are not within the radius of convergence of PT, which is also the case for in this range of  Appelquist et al. (2010); Neil et al. (2009).

Figure 7: Fits of our results for hadron masses (left) and decay constants (right) to the forms in Eq. (14) motivated by spontaneous chiral symmetry breaking. The lines are solid throughout the fit range .
0.014(1) 4.8 0.011(2) 3.1 0.016(2) 4.5 0.013(2) 4.3
0.171(4) 1.0 0.174(5) 1.2 0.164(5) 0.1 0.164(8) 0.1
0.216(7) 0.9 0.208(10) 0.8 0.221(9) 0.8 0.214(14) 1.1
0.170(21) 2.3 0.155(25) 2.8 0.216(29) 0.8 0.210(37) 1.5
0.244(5) 1.0 0.250(7) 0.7 0.235(9) 0.8 0.243(14) 1.2
0.017(1) 1.6 0.017(1) 2.4 0.019(2) 1.6 0.020(2) 2.8
0.037(1) 1.3 0.038(2) 0.8 0.035(2) 0.9 0.036(3) 1.1
0.040(3) 0.6 0.042(4) 0.04 0.037(5) 0.6 0.041(6) 0.002
Table 9: Intercepts and  for linear chiral extrapolations of the spectrum data in Tables 6 and 8, using the forms in Eq. (14) motivated by spontaneous chiral symmetry breaking and considering several different fit ranges. The large  and non-zero intercepts for the pseudoscalar mass squared indicate that our results are not well described by leading-order PT, .

iii.4 Conformal hypothesis

Even though Figure 6 suggests that the 8-flavor system exhibits spontaneous chiral symmetry breaking, it is still worthwhile to consider the alternate hypothesis that the theory is IR conformal in the infinite-volume chiral continuum limit. If this were the case, then the introduction of a non-zero fermion mass would produce bound states with non-zero masses governed by the mass anomalous dimension at the IR fixed point Miransky (1999); Lucini (2010); Del Debbio and Zwicky (2010). At leading order, all hadron masses (and decay constants Del Debbio and Zwicky (2011)) should scale as a power law with the same exponent,


Lattice calculations further break conformal symmetry through the introduction of a finite volume and finite UV cut-off (inverse lattice spacing) Appelquist et al. (2011b); Del Debbio and Zwicky (2014); Cheng et al. (2013c), leading to effects that we will not address in this work.

As in the previous subsection, we consider the simplest possible fits motivated by the IR-conformal hypothesis, to the power-law forms in Eq. (15). Figure 8 shows the results, which correspond to the and  listed in the second column of Table 10. Attempts to fit the axial-vector decay constant to a power-law do not succeed, and we omit this observable from the figure and table. The vector decay constant also leads to unphysically large ; the tension between the data and the power-law fit form can be seen in the right panel of Figure 8.

Figure 8: Fits of our results for hadron masses (left) and decay constants (right) to the power-law forms in Eq. (15) motivated by mass-deformed IR conformality. The lines are solid throughout the fit range .

In Table 10 we see that the quality of the other power-law fits is comparable to that of the corresponding linear fits considered in the previous subsection, with a similar range of . Not surprisingly, the observables with larger  from power-law fits tend to produce smaller  from linear fits. Omitting the lightest point from the power-law fits reduces  for some, but not all, of the observables.

0.64(2) 2.5 0.67(3) 2.4 0.63(2) 3.2 0.66(3) 4.4
1.14(5) 8.7 1.31(8) 6.8 0.91(6) 1.0 0.99(10) 0.9
0.94(6) 0.7 0.97(9) 1.0 0.91(7) 0.9 0.92(11) 1.7
0.74(16) 1.3 0.67(18) 1.7 1.00(28) 0.6 0.98(36) 1.3
1.04(4) 7.8 1.22(7) 4.2 0.83(6) 3.0 0.97(11) 2.8
0.86(10) 1.6 0.92(13) 2.1 0.88(16) 2.4 1.03(24) 3.8
3.20(49) 2.8 4.07(86) 1.7 2.12(44) 1.6 2.75(86) 1.5
Table 10: Mass anomalous dimension and  from power-law chiral extrapolations of the spectrum data in Tables 6 and 8, using the forms in Eq. (15) motivated by mass-deformed IR conformality and considering several different fit ranges.

More significantly, the predicted mass anomalous dimension varies over a wide range depending on the observable. While this would suggest that the conformal hypothesis breaks down when confronted with our data, we also see that for most observables shows significant sensitivity to the fit range. Taking into account the systematic uncertainties suggested by this sensitivity removes much of the tension between different observables, especially given our limited data and the other neglected systematic effects mentioned above. While these results do not support the hypothesis of mass-deformed infrared conformality, neither do they strongly disfavor it. The most reliable conclusion we can make is that if the 8-flavor theory were IR conformal, then it would possess a relatively large mass anomalous dimension .

We can attempt to test these power-law fit results by checking their consistency with finite-size scaling. Considering a mass-deformed IR-conformal theory in a finite spatial volume , finite-size scaling states that the hadron masses depend on the scaling variable , as


As above, we are neglecting several potential complications Del Debbio and Zwicky (2014); Cheng et al. (2013c), so we will not require that our data for different observables all scale with the same fixed . In Figure 9 we plot our and results for , and as functions of . Because there is little or no overlap between the data sets from our two different volumes, we cannot use standard finite-size scaling techniques Bhattacharjee and Seno (2001); Houdayer and Hartmann (2004) to obtain additional estimates for the mass anomalous dimension. Instead, we simply use the values from Table 10 as input, and observe that the points appear to form reasonably continuous extensions of the data that produced these predictions through the power-law fits discussed above.

Figure 9: Testing the consistency of the mass anomalous dimensions predicted by power-law fits to data (Table 10) with finite-size scaling. Filled points are from lattices, while empty points are from . The plot on the left retains all five points with , while that on the right includes only the four heavier points and uses the corresponding from Table 10.

Iv Chiral condensate enhancement

In composite Higgs models, the chiral condensate plays the role of the Higgs vacuum expectation value, generating masses for the standard model fermions through dimension-six interactions of the form . Similar dimension-six couplings generate flavor-changing neutral currents subject to stringent experimental constraints Chivukula and Simmons (2010). A generic way to satisfy these constraints is for the value of to be large compared to the symmetry breaking scale . Such condensate enhancement is conjectured to occur for chirally broken theories near the conformal window Holdom (1981); Yamawaki et al. (1986); Appelquist et al. (1986). In this section we investigate the dimensionless ratio for , comparing this system with our previous results for and 6 Appelquist et al. (2010); Fleming and Neil (2013).

From the leading-order PT expression


(the Gell-Mann–Oakes–Renner relation), we can identify three observables that reduce to in the chiral limit:


The subscript on indicates that this quantity (like and ) is evaluated at non-zero fermion mass , in contrast to the chiral-limit values and . We discussed and at length in the Section III, and Table 11 presents our results for , normalized per flavor (see also Figs. 14 and 16 in the Appendix). Because we measure directly, these data are dominated by a UV-divergent term . As a consequence, the three ratios in Eq. (19) have significantly different values in the range of we can access on lattices.

 0.010  0.015460(17)
0.015 0.021645(25)
0.020  0.027023(26) 0.027724(17)
0.025 0.033072(56) 0.033742(24)
0.030 0.039124(74) 0.039697(11)
0.035 0.045327(62)
0.040 0.051396(43)
0.045 0.057279(59)
0.050 0.063092(66)
Table 11: Results for direct measurements of the chiral condensate from and ensembles, normalized per flavor as in Refs. Appelquist et al. (2010); Fleming and Neil (2013).

Following Refs. Appelquist et al. (2010); Fleming and Neil (2013) we proceed by comparing each observable in Eq. (19) with the corresponding 2-flavor quantity, considering the ratios


where enumerates the three constructions , and . We take the ratio of 8- and 2-flavor results evaluated with the same input mass . However, these systems have significantly different residual masses (Table 3), which lead to different and in and . We plot all three ratios in the right panel of Figure 10, using the geometric mean on the horizontal axis. The left panel presents our previous results for from Ref. Fleming and Neil (2013).

Figure 10: Ratios of the three observables in Eq. (19) that reduce to in the chiral limit, for normalized by (left) and (right). The horizontal axis is the geometric mean .

The two plots in Figure 10 are remarkably similar, indicating that the chiral condensate enhancement we observe for is barely larger than what we found for . In particular, the most stable ratio , which does not involve our direct measurements, increases by only a few percent for compared to . Even so, is significantly enhanced for the 8-flavor theory compared to . Based on the 6-flavor joint chiral extrapolation inspired by NLO PT in Ref. Fleming and Neil (2013), we obtain the unrenormalized ratio in the chiral limit, with roughly 10% statistical uncertainty.

V Electroweak S-parameter

In this section we consider the electroweak parameter Peskin and Takeuchi (1992), following the approach of Ref. Appelquist et al. (2011a). The parameter remains one of the most important experimental constraints on electroweak symmetry breaking through new strong dynamics. is defined to vanish for the standard model, and its experimental value is consistent with zero Baak et al. (2012). Simply scaling up 2-flavor QCD data to the electroweak scale (and imposing the Higgs mass  GeV) would predict , providing strong evidence that QCD-like technicolor theories are ruled out.

In order to use a many-flavor gauge theory, such as the system under consideration, as the basis of a composite Higgs model, the parameter must be significantly reduced compared to scaled-up QCD. In Ref. Appelquist et al. (2011a) we observed such a reduction for compared to 2-flavor results, which decreases but does not eliminate the tension with experiment. Here we repeat this analysis for , finding similar results. We also explore the related issue of parity doubling in the vector and axial-vector channels.

v.1 Direct analysis of vacuum polarization

The parameter is given by


where is the number of fermion doublets to which we choose to give chiral electroweak couplings. Here we fix . is the transverse component of the difference between vector () and axial-vector () vacuum polarization tensors, as a function of euclidean . Since our domain wall fermion action ensures , it is straightforward to compute . These renormalization constants appear since we consider one conserved DWF current and one local current in each correlator Schaich (2011). As first reported by Ref. Boyle et al. (2010), the use of a single conserved DWF current suffices to ensure that lattice artifacts cancel in the difference. (Such cancellations appear to result from and lattice currents forming an exact multiplet under chiral rotations, which is also a feature of the local overlap currents used by Ref. Shintani et al. (2008).) Combining one conserved current with a local current reduces computational costs by roughly a factor of compared to using conserved DWF currents exclusively.

To determine , we fit our data for to a four-parameter Padé-type rational function of the form


The quadratic-in- denominator in this expression is motivated as a generalization of the single-pole-dominance approximation


Eq. (22) is the same fitting function we used in Ref. Appelquist et al. (2011a); subsequent studies Aubin et al. (2012); Golterman et al. (2013) have since provided more systematic support for using such rational functions to fit the -dependence of vacuum polarization functions.

Finally, the subtraction of in Eq. (21) removes from the spectrum the three NGBs eaten by the W and Z, and sets for the standard model with Higgs mass  GeV. Since we have not yet carried out the computationally demanding calculation of the (flavor-singlet scalar) Higgs mass in our lattice studies, we take


The first term in Eq. (24) would be appropriate if were comparable to the TeV-scale vector meson mass ; the second term corrects this for the physical  GeV 222Recently Ref. Foadi et al. (2013) argued that a 125-GeV composite Higgs may result from radiative corrections to a state with an intrinsic mass 600 GeV. This effect may decrease the second term of Eq. (24), moving our predictions closer to the experimental value for the parameter..

Computing for fixed from Eqs. 21 and 24, employing the thermalization cuts and jackknife blocks listed in Table 1, produces the 8-flavor results shown in Figure 11. This figure also includes the and 6 results previously published in Ref. Appelquist et al. (2011a), which we update to use  GeV rather than  GeV. As in previous sections, we plot vs.  in order to provide a more direct comparison between the three different theories.

The parameter is only well defined in the chiral limit . However, chiral symmetry breaking with light but massive flavors produces PNGBs. To obtain the phenomenological parameter, we must consider a chiral limit in which only three of these PNGBs become exactly massless NGBs to be identified with the longitudinal components of the W and Z. The other PNGBs must remain massive enough to have evaded experimental observation. (These PNGBs are all pseudoscalars, not to be identified with the 125 GeV Higgs, which comes from the flavor-singlet scalar spectrum that we have not yet investigated.)

Figure 11: Electroweak parameter with  GeV, for , 6 and 8 with fermion doublet assigned chiral electroweak couplings in Eq. (21). Our results for and 6 were previously published in Refs. Appelquist et al. (2011a); Schaich (2011).

For this requiremen