Investigating BSM Models with Large Scale Separation

Mass-split systems based on a conformal infrared fixed point provide a low-energy effective description of beyond the standard model systems with large scale separation. We report results of exploratory investigations with four light and eight heavy flavors using staggered fermions, and up to five different values for the light flavor mass, five different heavy flavor masses, and two values of the bare gauge coupling.


Introduction
We focus on "beyond the standard model" (BSM) theories based on strong dynamics. These theories assume the existence of a strongly coupled system of gauge fields and fermions with a broken chiral symmetry: three Goldstone bosons provide the longitudinal components of the W's and Z, while the Higgs emerges either as a pseudo-Goldstone boson or as a dilaton. Experiment demands that a viable theory accommodates a large separation between the scale of electroweak symmetry breaking and the scale of excitations in the new strongly coupled sector. This is achieved by formulating the strongly coupled theory in such a way that it lies within, or close to, the conformal window, i.e. the range of parameters for which a theory has an infrared fixed point (IRFP) but is still asymptotically free (see e.g. [1][2][3][4][5]). Now, while with some gauge groups a theory can be inside the conformal window with few fermions, in other cases, and most notably with the SU(3) gauge group, a large number of fermions is required to get a model with an IRFP. In this latter case only a small subset of the fermions can have vanishing mass: the other fermions must be given a mass to avoid the occurrence of way too many Goldstone bosons. Thus the theory becomes a "mass broken conformal theory", that is a theory which would have an IRFP with all fermions massless, but which is kept away from the IRFP by giving mass to a subset of its fermions. Purpose of this paper is to study the properties of such "mass broken conformal theories". We will proceed by presenting first some unproven, but plausible theoretical considerations based on non-perturbative Wilsonian renormalization group arguments, following them with the illustration of numerical results which appear to validate our assumptions. The conclusion will be that "mass broken conformal theories" have remarkable properties of universality, not unlike QCD where with massless quarks the theory is independent of any coupling constant.

Theoretical Considerations
In this section we will formulate some theoretical expectations. We consider an SU(3) lattice gauge theory with N f identical fermions. We take the lattice to be of infinite extent, we denote by β, in the standard manner, a parameter proportional to the inverse gauge coupling squared, denote by am the mass of the fermions in lattice units 1 , and assume that the fermions have been discretized in a manner which preserves chiral symmetry for am = 0 and avoids unwanted fermionic copies. When we mention an observable, we take this to be the actual value taken by the observable, which is in principle theoretically well defined.
For our discussion, let us take first N f = 4. The fermion mass am can be taken to zero. The theory exhibits chiral symmetry breaking. Observables such as the pseudoscalar decay constant aF π , masses aM etc. have well defined finite values in lattice units which depend only on β. The continuum limit is recovered by letting β → ∞: in this limit dimensionless ratios of observables approach their continuum values, any non-vanishing observable can be used to set the scale and all other observables are fully determined in a manner independent of any free parameter (dimensional transmutation.) We may take one or two masses, am 1 , am 2 , different from zero. As β → ∞ to recover the continuum limit we must also send am 1 , am 2 → 0 in a controlled manner. The spectrum of masses and other observables will depend now on the way am 1 , am 2 go to zero as β → ∞ and the ratios of observables to F π will tend to finite limits. We recover QCD with massless up and down quarks and non-vanishing strange and charm masses. Nothing new here: we described a QCD-like theory with massless up and down quarks and non-vanishing strange and charm quark masses.
Let us consider then the SU(3) theory with 12 fermions of mass am = 0. We assume that the theory is conformal. The correlations functions will exhibit non-trivial power law behavior at large distances. If we rescale lengths appropriately, the long range behavior with any two values of β will be the same, because under renormalization group transformations the theory runs to to an IRFP, in the space of infinite couplings. (The location of the IRFP will depend on the specifics of the renormalization.) The continuum behavior of the correlation functions is recovered by going to very large lattice distances. If we are close enough to the IRFP, the lattice will approximate well the continuum also at moderate and small distances: then the short distance behavior of the correlation functions will be trivial because the theory is asymptotically free in the UV.
Assume now that we keep 4 fermions massless and give a mass am to the other 8 fermions [6][7][8]. The observables (in lattice units) will now depend on β and am. As in the case with N f = 4, at large distances the correlation functions will exhibit exponential behavior and we can measure a non-vanishing aF π as well as non-vanishing masses and other observables. In a renormalization transformation, in the space of infinite couplings, the system will move away from the IRFP with am = 0 because am is a relevant parameter. To recover the continuum limit with aF π → 0 we must let am approach its fixed-point value am = 0, but, and here is the crucial point, contrary to the case of the theory with N f = 4, now we do not need to let β → ∞, since, as am → 0, the theory will flow closer and closer to the IRFP no matter what the original β is. If we take am sufficiently small, the ratios of observables, whether built out of the massless fermions, massive fermions or both, and in particular the ratios of observables to F π , will be independent of am and β. If we give two different masses, am 1 , am 2 , to the massive fermions, or if we give a small mass am to the massless fermions, hyperscaling arguments [8,9] show that the ratios of observables to F π or among themselves only depend on the ratios of the bare Lagrangian masses.
In conclusion, we can take F π to set the scale, and then the continuum theory is fully defined by the ratio of Lagrangian masses. In particular, if we only have a common mass parameter am for the 8 massive fermions, while the 4 light fermions are kept at zero mass, then the theory is fully defined without any free parameter. In this case am plays a role similar to β for massless QCD.
• We also simulated a system with β In Figure 1 we present the value of the pseudoscalar decay constant, aF π in lattice units. aF π is seen to decrease as am h decreases, following the expectation that it should go to zero as am h approaches the fixed-point value am h = 0.
Our light spectrum results for β = 4.0 are summarized in the four panels of Fig 2. (In this figure, as well as in the following figures, we use the symbols π, ρ, . . . as a short-hand for pseudoscalar, vector etc., referring to the particle state with the same quantum numbers as in QCD.) We plot ratios of hadron masses M H over F π , and as functions of m /m h , in order to remove the scale dependence. We note that M π /F π decreases towards zero for m /m h → 0, giving evidence of chiral symmetry breaking. We also observe that the 0 ++ is light, with M 0 ++ < M ρ , tracking the pion.  Ratios of the masses of π, ρ, a 0 , a 1 , n, and of the 0 ++ over F π . The left panel shows results for QCD taken from the PDG [14], the right panel averages for a (mass deformed) 12 flavor theory [11,[15][16][17].
In Figure 3 we superimpose the data obtained with different values of m h . Hyperscaling arguments [8,9] indicate that in the vicinity of the IRFP all scale independent quantities will be only a function of m /m h , with no separate dependence on m h . The way in which the data appear to line-up in Fig. 3 provides strong evidence for hyperscaling.  We also calculated the spectrum of masses for states made of heavy fermions. The spectra of light-light and heavy-heavy meson composites is illustrated in the the three panels of Fig. 4. The heavy-heavy spectrum also gives clear evidence of hyperscaling.
The upward bend in the heavy-heavy spectrum is in part due to the fact that light-light F π shows a marked decrease for m /m h → 0.  . Ratios of the masses of π, ρ, a 1 over F π for light-light and heavy-heavy composites. The left panels shows result for QCD [14] with the (unphysical) η s taken from [18], the right panels for a (mass deformed) 12 flavor theory [11,[15][16][17].
In Figure 5 we show the ratio of the heavy-heavy π and a 1 masses over the mass of light-light and heavy-heavy ρ. Once again the upward bend in the left panel can be attributed to the decrease of the light-light M ρ in the denominator. Ratios against heavy-heavy M ρ show little variation. Figure 6 presents more detailed evidence for the fact that chiral symmetry is broken. The left panel shows that as the ratio m /m h goes to zero, M π tends to zero while F π tends to a finite limit, a clear indication of chiral symmetry breaking. In order to display physical values for F π , as opposed to aF π (in lattice units), we multiply F π by a scale, denoted by a , which represents the lattice spacing for β = 4.0, am = 0.003, am h = 0.080 as determined through the Wilson flow. The values in lattice units, aM π , aF π at all other values of am , am h are converted to the values in Fig. 6 multiplying them by a /a as determined by the Wilson flow. The sharp increase of M ρ /M π for m /m h → 0, which is seen in the right panel of Fig. 6, is more evidence of chiral symmetry breaking. In Section 2 we stated that in the neighborhood of the IRFP the coupling constant β is an irrelevant parameter. In order to verify that this is indeed the case we performed additional simulations with β = 4.4, am h = 0.070. If β is an irrelevant parameter, then the ratios of physical quantities should be largely independent of its value. In Figure 7 we plot the results for M π , M ρ , M a 1 , in units of F π , obtained from all our simulations. We already observed how the data corresponding to different m h at β

Conclusions and Outlook
Our results give evidence for hyperscaling in the heavy fermion masses and for the irrelevance of the gauge coupling: for sufficiently small m h , ratios of physical quantities will depend only on m /m h and will be independent of β. The continuum limit would be reached for m , m h → 0 with m /m h kept fixed. In particular, in the chiral limit m = 0, m h would only serve to set the scale.
There is some analogy between m h in our model and the bare coupling constant g in QCD: near the fixed-point the physical values of masses and other observables do not depend on them. In principle, in the chiral limit, the theory built in the neighborhood of the IR fixed-point would be a self-consistent, parameter free theory, very much like QCD with massless quarks.
However, even if a theory built on the IR fixed-point were self-consistent, like QCD it would have to be embedded into a more general BSM framework. Then, on various phenomenological grounds (scaling dimensions of the condensate, of baryonic operators etc.), it can be argued that the theory should be strongly coupled: thus, still within the conformal window, but as close as possible to its beginning.
There is some evidence that an SU(3) theory with 10 massless flavors may be conformal [19,20]. This makes it quite valuable to perform an investigation, similar to the one presented in these proceedings, but with 4 massless and 6 massive fermions. Because of computational and, to some extent, theoretical difficulties [21], however, a simulation with 10 flavors could not avail itself of the simplifications brought about by staggered fermions. Moreover the need of preserving a clearly defined chiral limit practically forces one to using the much more computationally demanding domain-wall discretization. This is currently planned by the Lattice Strong Dynamics collaboration (http://lsd.physics.yale.edu/). Hopefully, the efforts of the LSD collaboration and/or some other group endowed with sufficient computational resources will clarify, in a not too distant future, the properties of the SU(3) four-plus-six fermion system.