More on the flavor dependence of $m_\varrho / f_\pi$

In previous work, arXiv:1905.01909, we have calculated the $m_\varrho / f_\pi$ ratio in the chiral and continuum limit for $SU(3)$ gauge theory coupled to $N_f = 2,3,4,5,6$ fermions in the fundamental representation. The main result was that this ratio displays no statistically significant $N_f$-dependence. In the present work we continue the study of the $N_f$-dependence by extending the simulations to $N_f = 7, 8, 9, 10$. Along the way we also study in detail the $N_f$-dependence of finite volume effects on low energy observables and a particular translational symmetry breaking unphysical, lattice artefact phase specific to staggered fermions.


Introduction and summary
We study the flavor number dependence of the ratio of the vector meson mass and the pseudoscalar decay constant in SU (3) gauge theory. The ratio is significant for a large class of beyond Standard Model theories envisioning a strongly interacting Higgs sector and a composite Higgs boson [1]. The elementary fermion ingredients of the composite Higgs boson may form other bound states, such as a vector meson, which would be one of the new, so far undetected, particles the theory predicts. The pseudoscalar decay constant sets the scale, in many theories it is simply identified with v = 246. 22 GeV, the symmetry breaking scale of the Standard Model. Having non-perturbative results for m /f π then determines the vector meson mass m in physical units. This beyond Standard Model scenario, and variants thereof, attracted enormous interest in the lattice community in the past decade . For a recent reviews of the available lattice results see [35] and references therein.
Apart from the phenomenological motivation the N f -dependence of our ratio is an interesting QFT question on its own. Once both f π and m are understood to be defined at finite fermion mass m and the chiral limit is only taken for the ratio, m /f π is a meaningful quantity both inside and outside the conformal window. Outside the conformal window both the denominator and nominator are finite in the chiral limit with an obviously finite ratio. Inside the conformal window both m and f π behave as O(m α ) for small m with the same exponent α, again leading to a finite ratio in the chiral limit. Hence the ratio is meaninful and well-defined on the full range 0 ≤ N f ≤ 16, including the quenched case N f = 0 and the last integer flavor number N f = 16 before asymptotic freedom is lost at N f = 33/2. Formally, N f = 33/2 corresponds to a free theory [36] and as such m = 2m and f π = √ 12m, leading to m /f π = 1/ √ 3. This is an order of magnitude smaller than ∼ 8 found for 2 ≤ N f ≤ 6. Hence on the range 7 ≤ N f ≤ 16 the ratio will drop an order of magnitude and it is not a priori known whether the drop will be gradual or rapid, nor is it known if the onset of the conformal window somewhere around 10 ≤ N f ≤ 13 is connected to it in any way. Motivated by both the phenomenological implications and the purely QFT aspects we continue the investigation with 7 ≤ N f ≤ 10 in the present work. Even though we would like to know the behavior for 11 ≤ N f ≤ 16 as well, finite volume effects are growing as a function of N f so rapidly that unfortunately we must postpone these flavor numbers to future work.
The organization of the paper is as follows. In section 2 we first study the N fdependence of an unphysical lattice phase specific to staggered fermions. The reason for doing so is that as N f grows the size of the unphysical phase in the (β, m) plane grows and one must avoid it in order to perform the physically relevant chiral-continuum limit. Section 3 details our study of the finite volume effects, the upshot of which is that as N f is growing so do finite volume effects. In fact the growth is rather rapid and is the main reason N f = 10 is the highest flavor number we can reliably simulate at the moment. The chiral-continuum limit is investigated in section 4 once the bare parameters are chosen such that unphysical phases are avoided and finite volume effects are suppressed sufficiently. We end with conclusions and possible outlook to future work in section 5.

Discretization and unphysical phases with staggered fermions
The lattice discretization in the present work follows exactly [37]; 4 steps of stout smearing [38,39] is applied to naive staggered fermions with smearing parameter = 0.12. A combination of the HMC and RHMC algorithms [40,41] with or without rooting are used to have the desired continuum flavor number N f . Both in [37] and the present work simulations are run at particular points of the (β, m) phase diagram at given N f . It is important that the bare parameters are all in the region of phase space which is continuously connected to the physical β → ∞ region, especially because unphysical phases do exist with staggered fermions.
The possibility that in the (β, m) bare parameter space an unphysical Aoki-like phase might exist with staggered fermions was first pointed out in [42,43]. Using staggered -2 - chiral perturbation theory it was shown that decreasing the mass on coarse lattices can lead to condensation of taste split meson states which in turn means that the vacuum becomes unstable. The new vacuum has different symmetries from the one expected in the continuum and in particular the so-called staggered shift symmetry, which is a translation by a single site accompanied by a phase factor for fermion fields, is broken. Briefly, taste split meson masses M 2 in staggered chiral perturbation theory receive a continuum-like contribution from the fermion mass, O(m), but also a contribution from taste splitting operators, O(a 2 ). If the latter is negative and large in absolute value compared to the former, M 2 may turn negative, leading to the aforementioned instability. Convincing numerical evidence for this scenario was provided in [44] for N f = 8, 12 and the relationship between the staggered perturbation theory picture and the actual numerical results were further clarified in [45].
In this section we study the unphysical, shift symmetry broken phase with our particular discretization and the full range of flavor numbers 2 ≤ N f ≤ 10 contained in both [37] and the present work. The main conclusion will be that even though unphysical phases do exist for N f > 2 and we do map them out, our simulation points are all in the physical phase, justifying our chiral-continuum extrapolations.
The single site shift symmetry in question is [46] where χ(x) is the staggered field at integer site x,μ is the unit vector on the lattice in direction µ and ξ µ (x) = (−1) ν>µ xν . In this convention the staggered signs in the Dirac operator are η µ (x) = (−1) ν<µ xν . The staggered action is clearly invariant under this set of transformations.
-3 -As discussed in [44] a suitable order parameter for the study of the potential spontaneous breaking of (2.1) is the difference of plaquettes on neighboring sites. More precisely, in terms of the plaquette P (x), where the sum over the lattice involves only even x µ coordinates. Clearly, if the sum would be over the entire lattice ∆ µ P would always be zero. In this way ∆ µ P measures if translational invariance in direction µ holds for the plaquette or not. In the physical phase, where translational invariance for gluonic observables is present, ∆ µ P = 0 for all µ. The unphysical phase will be signaled by ∆ µ P = 0 for at least one direction µ.
It is a straightforward exercise to map the observable ∆ µ P as a function of (β, m) for the various flavor numbers. A useful quantity to monitor is the square ∆ µ P ∆ µ P involving a sum over µ. Two typical results are shown for ∆ µ P ∆ µ P at fixed β as a function of m and at fixed m as a function of β in figure 1 with N f = 7 on 18 4 lattices. It is not our goal to obtain very precise values for (β c , m c ) corresponding to the spontaneous breaking of translational invariance, for our purposes an estimate will suffice which can be read off from results of the type shown in figure 1. A detailed finite size scaling study would be required for anything more precise. As we will see our simulation points are so far away from the (β c , m c ) phase boundaries that a rough estimate is indeed sufficient.
Performing the scans on 12 4 and 18 4 lattices shows that volume dependence is negligible on our level of precision. The summary of our results for the phase boundaries are shown in figure 2 for all flavor numbers where the thickness of the boundaries include the uncertainty related to our crude reading off of (β c , m c ) on fixed 18 4 lattice volumes.
For each N f > 2 a triangle shaped region corresponds to the spontaneously broken shift symmetry phase at finite (β, m). This triangle presumably extends down to m = 0 at two particular β values. The bare mass, above which translational symmetry is unbroken for all β is a growing function of N f as can be seen in figure 2. Not surprisingly, the particular β above which translational symmetry is unbroken for all masses is a decreasing function of N f . At N f = 3 we could not resolve the triangle shape because the broken phase only occures for very small masses, but nevertheless could find a transition. Interestingly, we could not detect any translational symmetry broken phase for N f = 2, perhaps because no such phase exists or perhaps because it occurs at extremely small masses.
The (β, m) values for N f = 2, 3, 4, 5, 6 which were used in the chiral-continuum extrapolations in [37] were listed in tables 3 and 4 of said work while the same parameters are listed in table 3 for the present work with N f = 7, 8,9,10. Clearly, all parameters used for the chiral-continuum extrapolations are in the physical phase and far from the (β c , m c ) phase boundaries.  Table 1. Volume dependence of m π and f π and fixed lattice spacing and fermion mass, together with the infinite volume extrapolated results using (3.1) and (3.2). The χ 2 /dof of the extrapolations are also shown.
enough in order to suppress finite volume distortions of the ratio m /f π especially because as the volume is increasing m and f π are moving in the opposite direction. The finite volume effects are thus enhancing each other in the ratio and too small volumes will lead to an overestimation of m /f π . An upper bound on the size of finite volume effects sets a lower bound on m π L for each N f . The results in [37] have shown that this lower bound is heavily N f -dependent in the range 2 ≤ N f ≤ 6. In the present work these finite volume investigations are extended to 7 ≤ N f ≤ 10.
The main low energy quantities m π and f π are measured at fixed lattice spacing and mass m for various lattice volumes, since these observables are expected to be the most sensitive to the finite volume. The m π L dependence of these quantities are given by with some m π∞ , f π∞ , C m and C f parameters. The details follow the procedure explained in [37], in particular we have in terms of the Bessel function K 1 . The sum is over integers (n 1 , n 2 , n 3 , n 4 ) with n 2 = n 2 1 + n 2 2 + n 2 3 + 4n 2 4 = 0 where µ = 4 corresponds to the time direction. The function g(x) describing the finite volume effects represents the lightest particle, the pion, going around the finite volume in all 3 space and the time direction any number of times. The leading contribution for our geometry, where the lattice is largest in the time direction comes from the pion going around each spatial direction once, corresponding to n = (±1, 0, 0, 0), (0, ±1, 0, 0), (0, 0, ±1, 0). If only these terms are kept we are led to the familiar finite volume effects given by a single exponential, We have repeated the finite volume fits with the above leading order single exponential expression as well and the results did not change within statistical uncertainties hence the data can not distinguish between the two sets of extrapolations. Note that the finite volume extrapolations (3.1) with either (3.2) or (3.3) do not depend on chiral perturbation theory at all, they hold for any massive QFT with m π taking the place of the lightest mass. In particular even if N f is inside the conformal window but a finite mass is introduced leading to finite masses for physical excitations, finite volume effects are still described by (3.1) and (3.2) or approximately (3.3).
The results of our fits of the type (3.1) with (3.2) are shown in figure 3. The main conclusion from the finite volume volume study is that as N f is increasing the minimal m π L required for at most 1% finite volume effects needs to grow. On the full range 2 ≤ N f ≤ 10 including the results from [37] the bounds can be interpolated by the simple expression, For instance at N f = 10 we have m π L > 7.66, about twice as large as the corresponding bound at N f = 2. Apart from the exponential finite volume effects discussed above for f π and m π , there might be further finite volume effects influencing m because of its possible decay to 2 pions. For our simulation points is however stable.
The conclusion from this section is that our simulation results suffer from at most 1% finite volume effects for N f = 7, 8, 9 and at most 1.5% for N f = 10, resulting in at most 3% distortion in the ratio m /f π , well below our statistical uncertainties.

Chiral-continuum extrapolation
Apart from the observables m π , f π and m the gradient flow scale t 0 was also measured to set the scale in the chiral-continuum extrapolations. The right hand side in the definition of t 0 [47],  Once t 0 is measured along with our low energy quantities of interest the chiralcontinuum extrapolation is performed via where X = f π or m . The continuum mass dependence is given by C 0 + C 1 m 2 π t 0 and the two terms C 2 and C 3 parametrize cut-off effects in both the chiral limit value C 0 and the slope C 1 .
The measured data for m π , f π , m , t 0 are shown in figure 3. The lattice geometry was always L 3 × 2L, the collected number of thermalized configurations O(1000) and every 10 th was used for measurements. For each flavor number, simulations are performed at 3 lattice spacings with 4 masses at each. Hence the chiral-continuum extrapolations (4.2) correspond to dof = 8 in each case.  . The χ 2 /dof of the extrapolation is also shown. The solid black line corresponds to the resulting continuum mass dependence C 0 + C 1 m 2 π t 0 , i.e. dropping C 2 and C 3 which are responsible for the cut-off effects. The deviations from the data at given bare coupling β shown by different colors, and the straight line are indicative of said cut-off effects. The absolute scale on the axis can not be directly compared between different flavor numbers because the definition of t 0 was N f -dependent, see (4.1).
-8 -  Table 2. Continuum results for each N f in the chiral limit.  Figure 5. The N f -dependence of m /f π in the chiral-continuum limit. The results with 2 ≤ N f ≤ 6 are from [37] and 7 ≤ N f ≤ 10 corresponds to this work. The result of a constant fit as a function of N f is also shown.
The chiral-continuum extrapolations are shown in figure 4 and the results are tabulated in table 2. The full N f -dependence of m /f π in the chiral-continuum limit for 2 ≤ N f ≤ 10 using also the results from [37] is shown in figure 5.
It was observed in [37] that there is no statistically significant N f -dependence in the ratio for 2 ≤ N f ≤ 6, at least on the level of precision available there. A statistically good constant fit gave m /f π = 7.95 (15). We can now repeat the constant fit on the new range 7 ≤ N f ≤ 10 and the result is m /f π = 7.01(40) with χ 2 /dof = 0.97, which represents a slight 2σ-decrease. Nevertheless combining all results on the full range 2 ≤ N f ≤ 10 we obtain m /f π = 7.85 (14) with χ 2 /dof = 1.10 which is our final result. 1 Apparently, the free value m /f π = 1/ √ 3 at N f = 33/2 is still about an order of magnitude away.

Conclusion
In this work we continued our study of the ratio m /f π in the chiral-continuum limit. Constant fits as a function of N f on the two ranges 2 ≤ N f ≤ 6 and 7 ≤ N f ≤ 10 show -9 -a decrease on the 2σ-level but a constant fit on the full range 2 ≤ N f ≤ 10 is still a statistically acceptable result and leads to m /f π = 7.85 (14). The main conclusion is the reinforcement of the picture arising from [37], namely that m /f π is a robust quantity once the gauge group is fixed and does not depend much, if at all, on the fermion content. Applied to composite Higgs models inspired by strong dynamics, this would mean that a potential measurement of a new so far unobserved vector resonance inherent in these types of models, would not select the flavor number. The measured vector mass would rather place constraints on the gauge group [48].
Our ratio has a well-defined meaning in the chiral limit both inside and outside the conformal window. If the free value m /f π = 1/ √ 3 = 0.577 is to be reached at N f = 16.5, an order of magnitude drop ought to take place beyond N f = 10. If the trends of finite volume effects follow (3.4) in any sense, the N f > 10 simulations will be very challenging. It would be most interesting to work out the perturbative corrections to 1/ √ 3 close to the upper end of the conformal window, i.e. not much below N f = 16.5 where perturbation theory is reliable. Hopefully the full range 2 ≤ N f ≤ 16 can then be covered by a combination of non-perturbative simulations and perturbative results. The onset of the conformal window would probably leave some sort of imprint on the flavor dependence of the ratio, a subject we leave for future work.