R. Browera b, A. Hasenfratz€, C. Rebbia>h* E. Weinberg", O. Witzelh**

" Department of Physics, Boston University, Boston, Massachusetts 02215, USA b Center for Computational Science, Boston University, Boston, Massachusetts 02215, USA c Department of Physics, University of Colorado, Boulder, CO 80309, USA

Received October 15, 2014

We investigate the transition between spontaneous chiral symmetry breaking and conformal behavior in the SU{3) theory with multiple fermion flavors. We propose a new strategy for studying this transition. Instead of changing the number of flavors, we lift the mass of a subset of the fermions, keeping the rest of the fermions near the massless chiral limit in order to probe the transition.

Contribution for the JETP special issue in honor of V. A. Rubakov's 60th birthday

DOI: 10.7868/S004445101503009X


The discovery of the Higgs meson has put in place the final piece of the electroweak sector of the Standard Model [1,2]. But, contrary to a theory like QCD, where asymptotic freedom guarantees that the ultraviolet cutoff can be removed, the theory of a self-interacting scalar needs an ultraviolet completion. The theoretically potentially viable options, compatible with present experimental limits, fall into two main categories: supersymmetric and composite Higgs models. Composite Higgs models are based on some new strongly coupled but chirally broken gauge-fermion sector where the Higgs particle is a fermionic 0++ bound state. The idea of a composite model was first introduced as "Technicolor" [3,4], which was later generalized to "Extended Technicolor" [5,6] to accommodate a mechanism for generating fermion masses. Rather soon it became clear that simple "scaled-up QCD" models cannot satisfy elect roweak phenomenological constraints. "Walking Technicolor", a model where the underlying coupling constant evolves very slowly, was proposed to remedy the situation [7,8]. The notion

E-mail: rebbi'fflbu.edu

** Present address: School of Physics & Astronomy, The University of Edinburgh, EH9 3FD, UK

of walking technicolor was particularly important because it brought to light the role that the proximity of an infrared fixed point could play for strongly coupled theories of elect roweak symmetry breaking.

A general feature of theories of electroweak symmetry breaking based on strong dynamics is that they predict a variety of new composite states. However, to date, no additional states have been found at the LHC other than the Higgs particle. Thus, any strong dynamics model of electroweak symmetry breaking faces the challenge of predicting the existence of a bound scalar state with the IIlclSS and properties of the Higgs boson, while the other states of the system must be significantly heavier. In passing, it is worth noting that even the occurrence of a Higgs-like scalar was not given in the original formulation of such theories. Indeed one of the motivation for their introduction was to provide electroweak symmetry breaking in the absence of any Higgs particle in the low-energy spectrum.

We note that in QCD, the 0++ <r(550) resonance is very close to other hadronic excitations. A phe-nomenologically viable model must exhibit some new phenomena that separate the a from the rest of the non-Goldstone spectrum. One possibility which is currently getting a lot of attention is a theory which exhibits near-conformal symmetry. The presence of a softly broken conformal symmetry might give origin to a low -rilciSS scalar, possibly as a pseudo-dilaton state,

Fig. 1. Two-loop perturbative 3 function for SU{3) with different numbers of flavors, indicating asymptotically free, fixed point, and IR-free behavior

while all other composite states would appear at much higher rilctSS. This could happen if the theory is close to an infrared fixed point as suggested in walking Technicolor scenarios.

In the generalization of QCD to SU(ATC) theories with ATj fundamental flavors, the two-loop 3 function is given by

/%) = - /,;/' + 0(g7


3o =

ßi =

11 ~3






— !\f2 3 r






When the number of flavors is small, both 3o and 3i are negative, indicative of QCD-like asymptotic freedom. As Eqs. (1) and (2) show, as the number of flavors increases, first 3i changes sign at N^ and the 3 function develops a nontrivial zero at some g = g„ value, and then at N'f > N^ the value of 3q also changes sign and

asymptotic freedom is lost. Figure 1 illustrates this behavior for Nc = 3. For Arjc) < Nf < N'p the two-loop 3 function suggests the presence of an infrared fixed point at some value g„ of the coupling constant. If g„ is small, a perturbative study of the infrared behavior of the theory might be warranted, but for larger values of g„ the investigation must proceed through nonper-t urbat ive t echniques.

In the specific case of SU(3), lattice studies indicate that the 12 fundamental flavor theory exhibits an

infrared fixed point [9 16]. Moreover, recent lattice investigations of the SU(3) theory with 8 fundamental and 2 sextet flavors have given tantalizing evidence for the existence of a low -rilclSS scalar [IT, 18], as would be needed for a model that describes the Higgs scalar.

While the above considerations may offer some hope that the explanation of electroweak symmetry breaking could be found in a strongly interacting gauge theory, an enormous amount of work remains to be done to support or invalidate such a conjecture. Indeed, beyond the intrinsic difficulty of calculating observables of a non-Abelian gauge theory in the intermediate coupling domain, a task that can presently be tackled only by-lattice simulation techniques, one also needs to contend with the fact that, contrary to QCD, the parameters of the underlying theory (number of colors, number of flavors, fermion representation) are not a priori known, and hence one needs to explore a range of possible models. With this paper, we wish to make a contribution to the methodology that can be used in the search for a successful theory.


We consider a system with Nf fermions such that Nf€< Nf < Nj\ hence, the theory, while still asymptotic free, has an infrared fixed point. One would like (1) to find the value N^ at which the fixed point appears. A caveat in this question is that the number of flavors is not a continuous variable, but instead an integer. Further, there is no guarantee that the nearest integer below Nj€^ is close enough to the infrared fixed point

to show the phenomenologically desired properties.

We suggest here a different strategy, namely, to consider a theory with Ar£ light and N^ heavy flavors. The Nt light flavors are kept near the chiral limit m£ « 0, while the jVfe heavy flavors have a variable, heavier mass in/, > in(. Specifically, we study an SU(3) system with Nt = 4 and Nh = 8 such that the light flavors on their own form a chirally broken system, whereas in the iii/, —¥ ni( limit the Ar£ +Ar/, = 12 flavors describe a mass- deformed conformal system with an infrared fixed point.

To understand the expected behavior as m/, changes, we consider the renormalization group (RG) flow of the Nt + N't, theory in the limit m£ = 0, sketched in Fig. 2. The heavy fermion mass m/, is a relevant parameter, and for nonzero values it traces out an RG trajectory leading away from the infrared fixed point (IRFP) of the 12-flavor theory to the trivial fixed point of the 4-flavor theory. For large values

Fig. 2. Illustration of the expected renormalization-group flow lines for the Ay + A'/, flavor theory. The thick line shows the RG trajectory connecting the con-formal infared fixed point (IRFP) at ;rt/, = rnf = 0 (12 flavors) and the trivial fixed point of the 4-flavor theory at m-h = oc. The thin lines are RG flow lines that first approach, then run along the RG trajectory. As m-h —0, the flow lines spend increasingly more time around the IRFP, creating a "walking" scenario, while as m-h increases, the heavy flavors decouple and the RG flows resemble the running of the 4-flavor system

of ini,. the heavy flavors decouple and the model is essentially the A^-flavor chirally broken theory. In the other limit, when n%h = mi = 0, the system is conformal and the RG flow runs into the IRFP. Finite in/, ^ 0 breaks conformal symmetry, but for small m/, the RG flow approaches the IRFP and stays around it for a while before eventually running toward the trivial infrared fixed point. This is the desired walking behavior, where the length of walking can be controlled by tuning /rift. Thus our theory interpolates between the running behavior of the 4-flavor chirally broken theory and the walking behavior of the 12-flavor mass- deformed conformal theory.

Phenomenologically, it would be more interesting to set ATt = 2 as opposed to 4 to ensure that the system has only three massless Goldstone bosons as needed for electroweak symmetry breaking, but due to our lattice fermion formulation, it is simpler in this pilot study-to work with four light flavors. As mentioned above, there is increasing evidence that the 12-flavor system is infrared conformal with a relatively small anomalous dimension 7m « 0.24 [13,15,19], which may be too small to satisfy phenomenological walking constraints. A system closer to the conformal window with a larger anomalous dimension might have been more realistic, but for technical reasons we kept the number of flavors as a multiple of four.


We have performed simulations at various values of n>( with the intention that the light fermions should be taken in the chiral limit, or as close to it as possible, while the IIlclSS of the heavier fermions is varied with in/, > in(. Our calculations are still in progress, the results we present here are preliminary.

The simulations have been carried out using nHYP smeared staggered

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