# Dilaton EFT Framework For Lattice Data

###### Abstract

We develop an effective-field-theory (EFT) framework to analyze the spectra emerging from lattice simulations of a large class of confining gauge theories. Simulations of these theories, for which the light-fermion count is not far below the critical value for transition to infrared conformal behavior, have indicated the presence of a remarkably light singlet scalar particle. We incorporate this particle by including a scalar field in the EFT along with the Nambu-Goldstone bosons (NGB’s), and discuss the application of this EFT to lattice data. We highlight the feature that data on the NGB’s alone can tightly restrict the form of the scalar interactions. As an example, we apply the framework to lattice data for an SU(3) gauge theory with eight fermion flavors, concluding that the EFT can describe the data well.

## 1 Introduction

Lattice simulations of strongly interacting gauge theories indicate that infrared conformal behavior sets in with a sufficiently large number of light fermions latticeCFT ; latticeCFT2 ; latticeCFT3 ; latticeCFT4 ; latticeCFT5 ; latticeCFT6 ; latticeCONF1 ; latticeCONF2 ; latticeCONF3 ; latticeUNC1 ; latticeUNC2 ; latticeUNC3 ; latticeUNC4 ; latticeUNC5 . In addition, a remarkably light singlet scalar particle appears in the spectrum of recent simulations as this number is increased toward the critical value for the transition to conformal behavior (the “bottom of the conformal window”) LSDresult ; latticeSCALAR1 ; latticeSCALAR2 ; latticeSCALAR3 ; latticeSCALAR4 . This has led to the suggestion that the light scalar should be interpreted as a dilaton, and that this interpretation could become even more accurate as the fermion number is taken closer still to the transition value.

Lattice simulations for gauge theories of this type have been carried out for a set of small fermion masses , with extrapolation to typically discussed by fitting the results to continuum chiral perturbation theory. This is equivalently an interpretation in terms of a chiral-Lagrangian EFT consisting of (pseudo) Nambu-Goldstone bosons (NGB’s) with a small mass . The expansion parameter is proportional to where is the NGB decay constant, and can indeed be small for the range of values employed in the simulations.

A more general approach is to employ an EFT consisting of the NGB’s together with a description of a light singlet scalar consistent with its interpretation as a dilaton. Several authors have begun this program EFTDilaton1 ; EFTDilaton3 ; EFTDilaton4 ; EFTDilaton5 ; BLL . In this paper, we develop such a framework for comparison with existing lattice results as well as future simulations. Lattice results have so far been obtained for values such that the NGB mass is of the same order as the scalar mass LSDresult ; latticeSCALAR1 ; latticeSCALAR2 ; latticeSCALAR3 ; latticeSCALAR4 . These, in turn, are relatively small compared to the masses of other composite states, so that the use of an EFT consisting of only these degrees of freedom should provide a good first approximation. If and when simulations can be done at even smaller values of , such that the NGB mass drops clearly below the scalar mass, which in turn remains well below the other physical scales, the framework will remain reliable.

The EFT we employ involves a decay constant for the scalar as well as . In the EFT, enters as the order parameter for scale symmetry breaking. Both constants descend from the underlying, confining gauge theory with , and we expect them to have values set by the confinement scale. A small scalar mass parameter also descends from the underlying, theory. With both and small compared to and , the quantum loop corrections can be small, and are neglected in this paper. We instead provide a fit to existing lattice results, and a framework for future simulations, employing the EFT at only the classical level. A notable feature of the framework is that lattice data on only the NGB’s (their mass and decay constant), which are currently measured most precisely, are sufficient to determine a key parameter of the EFT and tightly restrict the form of the scalar potential.

In Section 2, we describe the EFT, employing a general form for the scalar potential. In Section 3, we discuss the application of the EFT to lattice data and then describe a fit to current data from the LSD collaboration, drawing some conclusions about the form of the EFT for that case. In Section 4, we develop simple, linearized expressions to fit future lattice data over a small range of fermion masses . We summarize and conclude in Section 5.

## 2 The EFT

The low-energy EFT is built from the scalar field and a set of NGB fields . The latter arise from the spontaneous breaking of chiral symmetry, and the former, to the extent that it can be interpreted as a dilaton, arises from the spontaneous breaking of conformal symmetry. The purely scalar part of the EFT consists of a kinetic term along with a potential arising from the explicit breaking of conformal symmetry in the underlying theory, which we take to be small:

(1) |

We assume that the potential has a minimum at some value , and that it is comparatively shallow, so that the mass of the fluctuations around the minimum satisfies .

A specific choice of the potential amounts to supplementing the EFT with partial information from the underlying dynamics. Two examples from the literature are

(2) | |||||

(3) |

normalized such that in each case is the scalar mass. The first is a weakly-coupled potential such as the one appearing in the standard model. It can arise from the deformation of an underlying conformal field theory (CFT) by relevant operators. The second has been proposed in Ref. GGS as a way to model the behavior of a CFT deformed by a nearly marginal operator (see also Ref. EFTDilaton1 ). We will allow the lattice data to determine the form of the potential.

The NGB’s arising from the spontaneous breaking of chiral symmetry are described in terms of a field transforming as , with and the matrices of and transformations. (This approach can be adapted to other symmetry groups and breaking patterns). The field satisfies the nonlinear constraint . We hence write:

(4) |

where the coupling to the dilaton field (introduced here as a compensator field to maintain the scale invariance of this term in the Lagrangian) is dictated by the fact that the kinetic term has scaling dimension . The field can be parametrized through where and are the generators of normalized as . In contrast with the linear-sigma-model description of chiral symmetry breaking, more generally , as the underlying strong dynamics may involve many condensates besides the chiral-symmetry-breaking one.

In lattice calculations of particle masses and decay constants in the underlying gauge theory, chiral symmetry (as well as conformal symmetry) must be explicitly broken by the introduction of a fermion mass term of the form . For the study of confining dynamics, is taken to be small compared to the confinement scale. The small explicit breaking is implemented in the EFT through the term

(5) |

where , with determined by the chiral condensate of the underlying theory (). The product is RG-scale independent, with each factor typically defined at the UV cutoff (the lattice spacing). The parameter is argued to be the scaling dimension of in the underlying theory BLL . This scaling dimension is an RG-scale dependent quantity, which could vary from at UV scales where the theory is perturbative to smaller values near the confinement scale. Analyses of near-conformal theories have suggested a scaling dimension at this scale Georgi . We keep as a free parameter to be fitted to the lattice data.

Expanding around gives

(6) |

generating a negative contribution to the scalar potential as well as an NGB mass term. The new contribution to the scalar potential shifts both the VEV and the mass of the scalar field . The shifted VEV will, through Eq. (4), re-scale the NGB kinetic term, and hence the NGB decay constant.

## 3 Comparison To Lattice Data

### 3.1 General Discussion

Lattice simulations are currently carried out for gauge theories with fairly small (). For these cases, cannot be too large if the theory is to be in the confining phase. Our program is to use the full EFT to describe current and future lattice results for the NGB’s and the light scalar, the latter having already been observed for example in the simulations. The parameters , , , and the scalar potential , have no dependence on the fermion mass and are held fixed as the parameter is varied. In this paper, this will be done using the EFT at only the classical level, neglecting corrections of order and . We discuss this approximation further in Section 5.

A set of physical, -dependent quantities , , and emerge from the EFT. The quantity is the -value that minimizes the full potential

(7) |

can be measured, for example, in NGB scattering. It is finite assuming only that is stable and increases at large more rapidly than . In the case of the potential given by Eq. (2), is determined by the equation

(8) |

whereas for the potential in Eq. (3), it is determined by

(9) |

In general, depends on the interplay between the two parts of the potential . The physical scalar mass is determined by the curvature of the full potential at its minimum. The remaining two quantities, and , can be identified after properly normalizing the NGB kinetic term. They are given in general by simple scaling formulae:

(10) | |||||

(11) |

(see also Ref. EFTDilaton1 ). For a given value of , the dependence of each of the four quantities , , , and on is described by the four parameters , , , , and whatever additional parameters enter the scalar potential . One immediate prediction is that and have the same functional dependence on .

The comparison to lattice data will focus first on the quantities and , which are currently known most precisely. For this purpose, it is helpful to note that the two scaling relations, Eqs. (10) and (11) give

(12) |

where . Fitting lattice data to Eq. (12) can allow an accurate determination of .

Another key question is to what extent the form of the scalar potential can be determined by a fit to lattice data. With the small amount of data available so far, only limited progress can be made on this “inverse-scattering” problem. We will find it helpful, even with the current data, to consider the slope of the scalar potential at the value of () that minimizes the full potential . From Eqs. (7) and (10),

(13) |

Since , a plot of the data for versus provides a measure of the slope of at versus itself. This slope vanishes in the chiral limit , corresponding to , since then (the minimal point of itself). As is increased, the slope of increases through positive values. We will use Eq. (13) to analyze data from the LSD collaboration in the next sub-section.

This procedure can be taken to the next stage by bringing the lattice data on into the analysis. From Eqs. (10), (11) and (13), together with the definition , one can derive an expression for the second derivative of at :

(14) |

Thus data for could be used in the analysis alongside the and data, to allow a fit that can better constrain both the scalar potential, and the other free parameters of the Lagrangian.

### 3.2 Application to the LSD Data

We next apply our EFT framework to the LSD collaboration data for the gauge theory with . These data, which cover the smallest fermion mass range studied as of yet for this theory, are currently limited to , , and . They are shown in Figs. 1a and 1b. A list of the numerical values and errors has been provided to us by the LSD collaboration. We first note that the lattice data for and are remarkably linear throughout the range of values. The data for are compatible with linearity but the errors are large. The data are consistent with an expected intercept of . A finite intercept is expected in the case of the data.

The linearity of the data combined with the substantial variation of with leads, through the scaling relation Eq. (12), to a determination of . Since the data for is itself near-linear in and is varying substantially, must be close to . We fit the data assuming that an additional, conservative systematic uncertainty should be assigned to it, for both and . This is consistent with the estimate of finite-volume and lattice-discretization artifacts reported in Ref. LSDresult . In addition, there are systematic uncertainties associated with the EFT we employ. We discuss these briefly in Section 5, but do not include them in our fit. The result of our fit to Eq. (12), treating both and as free parameters, is

(15) |

with uncertainty and (where ). This result is not inconsistent with and therefore with (the zeroth-order chiral perturbation theory formula for )^{1}^{1}1If the lattice data for were not so linear in , they could still be consistent with the scaling relation Eq. (12), but with .. By contrast, the substantial variation of with looks nothing like
zeroth-order chiral perturbation theory. In our EFT, its variation with is naturally
accommodated at the classical level.

The near-linearity with of the data provides more detailed information. Through Eq. (10), it implies that must also be near linear in . This suggests a relation similar to that of Eq. (8) which arises from the potential, together with , but it doesn’t rule out other forms for the potential. To proceed, we use Eq. (13) relating the slope of at to the product . In Fig. 2, we plot the LSD data for against . Error bars representing the 2% systematic uncertainty are shown. Since each point on the vertical axis is proportional to the slope of at and each point on the horizontal axis is proportional to , the points in the figure are displaying the shape of the scalar potential for the theory.

The data indicate clearly that increases with for a range of beyond its minimum at a rate much faster than , confirming that the scalar sector of the EFT is self-interacting. The data are in fact consistent with the large- behavior as in Eq. (2). For this potential, Eqs. (10) and (13) give

(16) |

where . The data can be fit to this form, with and treated as independent parameters. The best fit is represented by the red line in Fig. 2. The fit parameters are

A fit deriving from other forms of the scalar potential qualitatively similar to is also possible. An example is , for which

(17) |

This can also lead to a good fit, but only with a very small value of , indicating a parameter hierarchy in the EFT that could be unnatural. Here, we don’t show this fit or others based on alternative forms of the potential. While potentials qualitatively unlike can be ruled out, the limited amount of data available does not yet allow us to distinguish between a variety of similar forms. As more data points become available, spread over a larger range of fermion masses, the above method can be used to determine the functional form of the scalar potential with increasing precision, over a larger range of field values. We note again that the NGB lattice data alone ( and ) can provide this information.

To take this analysis further and to isolate and determine the EFT parameters beyond and , the lattice data for , shown in Fig. 1a, can also be included in the fits. Because of the large statistical errors currently associated with these points, they don’t yet add precision to the analysis of the form of the scalar potential, but they are sufficient to provide some approximate information about the parameter and the associated physical quantity . It can be seen that for , and for any potential with the character of which fits the Fig. 2 data well, the relation must hold. Using the fact that throughout the fermion mass range, along with Eq. (10), we have the rough prediction throughout the range. In the future, more information about the potential can also be found by including data. Eq. (14) then provides a measure of the second derivative of at the minimum of the full potential .

## 4 Small Mass-Shift Approximation - A Side Note

Looking to the future, lattice data for the gauge theory with could extend to smaller values as well as include more densely spaced points in the range of Figs. 1a and 1b. There will also be data for as a function of . Simulations of other theories could produce additional interesting data for each of the masses and decay constants. These results could appear linear as a function of or exhibit nonlinear behavior. For future analysis of such data sets using our EFT and allowing for a general form of the scalar potential , it could be helpful to linearize the physical quantities about a reference value . In this section, we briefly describe this approach.

With restricted to a small enough neighborhood of , the quantities of interest will be sensitive to the shape of the full potential only in the neighborhood of its minimum with . The full potential can therefore be approximated as

(18) |

with

(19) |

where is the minimum of the scalar potential for the reference value and . is the scalar mass at the reference value and is a free parameter controlling the strength of the scalar cubic self-interaction. We expect it to be .

We make the replacements

where . The quantities , , , and then have the following dependence on :

(20) | ||||

(21) | ||||

(22) | ||||

(23) |

where

(24) |

One can fit lattice data as a function of using these formulae and their extensions to higher order. The expansion is reliable providing . The four parameters , , , and are simply the values of , , , and at the reference point. The additional two parameters and can be determined by the slope of the curves at the reference point. At higher orders, additional parameters describing the shape of the potential will enter.

The parameter is itself sensitive to the shape of the potential. In the absence of the chiral-symmetry-breaking second term in Eq. (7), we have for and for . Away from the chiral limit, at some reference value , the contribution of the second term must be taken into account. The value of will depend on the shape of , the value of and the other parameters, and the choice of the reference mass . For the case , it will remain the case that if .

## 5 Summary and Conclusion

We have developed a simple EFT framework for the interpretation of lattice results for confining gauge theories, in which the light-fermion count is near to but below the critical value for transition to conformal behavior. The lattice studies indicate that a remarkably light scalar appears in the spectrum along with the NGB’s and higher-mass states. Interpreting the scalar as a dilaton, we have included only it and the NGB’s in the EFT, and allowed a general form for the dilaton potential.

The presence of a small fermion mass in the underlying gauge theory, necessary for lattice simulations, leads to a chiral-symmetry-breaking term in the EFT. The coupling of this term to the scalar field is described by an unknown parameter , to be fit to lattice data. We provided expressions for the masses and decay constants of the scalar particle and NGB’s appropriate for comparison to lattice data, noting that the data can be used to determine as well as the shape of the scalar potential.

We applied this framework to the current LSD collaboration data for an gauge theory with , which covers the smallest fermion-mass range studied for this theory. Even with the limited data available so far, we concluded that and that the scalar potential grows approximately like beyond its stable minimum. Among the parameters of our EFT in the chiral limit, we have so far provided only an estimate of and the product , for the case of the potential (see Eq. (16)). Nevertheless, the fact that our EFT, used at the classical level, can accommodate the lattice data as well as it does supports our starting assumption that the scalar particle is weakly self-interacting, that is . As more data points become available, our method can be used to determine the functional form of the scalar potential with increasing precision, over a larger range of field values.

For purposes of analyzing future lattice data, we developed expressions for the masses and decay constants of the scalar and NGB’s, linearized in about a reference value . This framework is well suited to analyze future data that are dense in the neighborhood of a reference value. The scalar potential is obtained from data as a Taylor series, making it possible to exclude potentials that are inconsistent with data in a systematic way.

The EFT we have employed neglects the effects of heavier states such as the vector and axial-vector bound states produced by the underlying gauge theory. In the case of the LSD data for the gauge theory with , the masses of these states have been measured for each of the values in Fig. 1. Throughout this range, dropping to for the lowest value. The data for , with their larger statistical errors, also satisfy a similar bound. The axial state is still heavier leading to corrections that are even more suppressed.

Our framework has also neglected higher order corrections in perturbation theory arising from loops of NGB’s and the scalar. By inspection of the lattice data in Fig. 1, one can see that for all but one of the points, . It is also the case for most of our fits, although not described in detail here, that throughout the range, . Thus, from the data, one can see that loop-expansion quantities such as and are expected to be relatively small. The loop expansion also has counting factors that can grow with , as well as chiral logarithms, and these have to be included in a full analysis of these corrections. This is beyond the scope of the present paper. We note here only that the order of magnitude of these corrections varies very little throughout the mass range of Figs. 1a and 1b, so that their systematic effect should be possible to control.

More generally, our EFT framework can be applied to lattice data from any strongly coupled gauge theory with a light-fermion count below the bottom of the conformal window, but close enough to exhibit a light scalar in the spectrum. A current example could be the gauge theory with a doublet of fermions in the symmetric-tensor representation. Our framework and analysis can be refined further as the amount and quality of lattice data increases, with the ultimate goal of a full “inverse-scattering” reconstruction of the scalar potential from the data.

###### Acknowledgements.

We thank Enrico Rinaldi for his assistance in providing the latest data and plots from the LSD collaboration. We also thank Biagio Lucini, Pavlos Vranas, George Fleming and Andrew Gasbarro for helpful discussions. The work of TA and JI is supported by the U.S. Department of Energy under the contract DE-FG02-92ER-40704. The work of MP is supported in part by the STFC Consolidated Grant ST/L000369/1.## References

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