# Hadronic bound states in SU(2) from Dyson–Schwinger equations

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## Abstract

By using the Dyson–Schwinger/Bethe–Salpeter formalism in Euclidean spacetime, we calculate the ground state spectrum of \(J\le 1\) hadrons in an SU(2) gauge theory with two fundamental fermions. We show that the rainbow-ladder truncation, commonly employed in QCD studies, is unsuitable for a description of an SU(2) theory. This we remedy by truncating at the level of the quark–gluon vertex Dyson–Schwinger equation in a diagrammatic expansion. Results obtained within this novel approach show good agreement with lattice studies. These findings emphasize the need to use techniques more sophisticated than rainbow-ladder when investigating generic strongly interacting gauge theories.

## Keywords

Gauge Theory Higgs Boson Chiral Limit Gluon Propagator Landau Gauge## 1 Introduction

Quantum chromodynamics (QCD) is a strongly interacting gauge theory whose study has proven to be one of the most formidable challenges of modern theoretical physics. While the high-energy regime of QCD is by now relatively well explored in terms of perturbation theory, the arguably more interesting (and intrinsically non-perturbative) phenomena such as dynamical chiral symmetry breaking and confinement are yet to be fully understood.

One of the strategies which might lead to our better understanding of QCD is to investigate theories which are QCD-like, but have certain properties that make them technically less challenging than QCD itself. A prime example is provided by studies of SU(2) gauge theories with an even number of fermion flavors. Lattice simulations of these theories at non-zero chemical potential do not suffer from the sign problem, and such models thus provide ideal conditions to study the phase diagram of strongly interacting matter [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11].

Here we wish to concentrate on the situation with two fundamentally charged Dirac fermions [1, 8, 10, 11]. Such a theory may also be interesting in the context of a unified description of cold asymmetric Dark Matter (DM) and dynamical electroweak (EW) symmetry breaking [12, 13, 14], wherein the ground state hadronic spectrum at \(T=0\), \(\mu = 0\) is of great importance. It is exactly this hadronic spectrum that will be the central focus of our study.

In this paper we use the non-perturbative, continuous and covariant formalism of Dyson–Schwinger (DSE) and Bethe–Salpeter (BSE) equations in Euclidean spacetime [15, 16, 17, 18]. When applied to QCD, the most common truncation one can make is that of rainbow-ladder (RL), wherein the quark–antiquark interaction kernel is replaced by a dressed one gluon exchange. It is the simplest approximation scheme that respects the axial-vector Ward–Takahashi identity (axWTI), thus preserving the chiral properties of the theory and the (pseudo)-Goldstone boson nature of light pseudoscalar mesons. With a judicious choice of model dressing functions, the RL truncation has been applied relatively successfully to QCD phenomenology for both mesons [19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30] and baryons [31, 32, 33, 34, 35].

However, as we will show in this paper, the RL truncation performs unsatisfactorily when adapted to an SU(2) theory with two fundamental flavors, even though the theory is expected to have QCD-like dynamics. We discuss possible reasons for this in more detail in Sect. 2. Here we only comment that we strongly believe that (most) of the inadequacy of RL method comes from its weak connection to the underlying gauge sector. Remedying this requires the use of beyond rainbow-ladder (BRL) techniques, with our preference toward those based on the diagrammatic expansion of quark–gluon vertex DSE [36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46]. While there are other BRL methods available [47, 48, 49, 50, 51, 52, 53], we choose the diagrammatic approach as it makes it easier to study the influence of the gauge sector on hadronic observables. Our aim in this paper is thus not only to provide a continuum calculation complementary to the lattice investigations of [12, 13], but also to explicitly demonstrate the importance of using BRL methods when studying generic strongly interacting theories.

This manuscript is organized as follows. In Sect. 2 we discuss the DSEs relevant for our calculation, and also describe in some detail the approximations and model inputs we employ. In Sect. 3 we describe the extrapolation procedures used to obtain hadron masses, and provide estimates for errors coming from extrapolation. The results are discussed and compared to relevant lattice data in Sect. 4. We conclude in Sect. 5.

## 2 Framework

### 2.1 Rainbow-ladder

In the RL approach, the model dressing function \(\lambda (k^2)\) of Eq. (5) is often combined with the dressing of the gluon propagator \(D_{\mu \nu }(k^2)\) into a single model function, constructed to reproduce correctly some hadronic observables, usually \(m_\pi \) and \(f_\pi \). While this method has shown considerable success in QCD phenomenology (see e.g. [55, 56] for some of the limitations of the model), in an SU(2) theory the approach seems rather unsuitable, especially in the \(1^{+\,\!+}\) channel: see Table 2 for details.

There are two primary reasons why RL will not perform satisfactorily in a generic strongly interacting theory. One reason is with regards to its very limited interaction structure (\(\gamma ^\nu \times \gamma ^\mu \)) which offers no variation in interaction strength across different meson channels. The second is that the connection to the underlying gauge dynamics is typically lost in the construction of an effective quark–gluon interaction; this prevents adequate rescaling of parameters such as \(g^2N_c\) that cannot be translated into a re-parameterization of an effective model.

### 2.2 Beyond rainbow-ladder

We emphasize here that this model is, in a sense, highly constrained. Namely, once the input for the ghost and gluon propagators (which we will discuss shortly) and the truncation of the quark–gluon vertex DSE have been chosen, all other parts of the calculation are fixed. The BSE kernel follows from the axWTI, and the model dressing \(\lambda (k^2)\) of Eq. (5) follows from the tree-level projection of the full quark–gluon vertex. We will re-iterate this point in Sect. 4.1, when we provide an estimation of the model dependence.

## 3 Bound states from space-like \(P^2\)

One of the consequences of working with Euclidean spacetime is that access to time-like quantities, such as masses of bound states, requires an analytic continuation of the component Green functions to complex momenta.

While this is only a minor technicality thanks to many well-established techniques in the literature [22, 49, 83, 84, 85], there are situations in which existing methods do not apply, or which are simply too complicated to implement. In this case, indirect methods can be employed that enable access to a limited number of time-like quantities [86, 87].

In the next two sections we describe two techniques that have been widely used, and compare their performance in cases where direct analytic continuation is possible. This provides an estimate of the methods applicability to the study at hand.

### 3.1 Eigenvalue extrapolation

Since \(\lambda (P^2)\) is a continuous function of \(P^2\), one can conceive that its continuation from space-like \(P^2>0\) to time-like \(P^2<0\) may be obtained through appropriate function fitting and extrapolation. The transformation of the eigenvalue \(g\left( \lambda \right) = 1 - 1/\lambda \), see Ref. [26], removes a considerable degree of intrinsic curvature in the region close to the pole, rendering simple linear extrapolation viable provided the extrapolation is not *far*.

In the top panel of Fig. 7 we show the eigenvalue extrapolation of \(\lambda (P^2)\) for various \(J^{PC}\) states. The data is first transformed via \(g\left( \lambda \right) \), before a linear fit \(f(P^2) = a + b x\) is performed. Finally, we plot the inverse function of \(g\), \(\lambda _{\mathrm {fit}}=g^{-1}(f(P^2))\) as solid lines. Exact results, obtained via calculation in the complex plane, are included as labeled points.

### 3.2 Inverse vertex function extrapolation

Results for vertex pole extrapolation for QCD rainbow-ladder in the chiral limit, compared with the result computed through direct analytic continuation. All units are in MeV. The points \(P^2\) are taken from the region \((0,L)\); the errors on extrapolated results come from the fitting procedure

\(J^{PC}\) | Calc | \(R^{(2,2)}\) (\(L=0.5\)) | \(R^{(2,2)}\) (\(L=1.0\)) |
---|---|---|---|

\(0^{{-}{+}}\) | 0 | \(1\) | \(1\) |

\(0^{{+}{+}}\) | 658 | \(657\,(23)\) | \(656\,(23)\) |

\(1^{{-}{-}}\) | 738 | \(731\,(27)\) | \(728\,(27)\) |

\(1^{{+}{+}}\) | 900 | \(899\,(33)\) | \(899\,(33)\) |

A second source of errors has to do with the applicability of the extrapolation procedure, as one would expect the whole method to become less reliable as one probes deeper into the \(P^2 <0\) region (i.e. the technique is less reliable for heavier mesons). Although it is very hard to quantify this, a comparison with exact results suggests that these effects are quite small for the inverse vertex approximation. In light of other systematic errors, present in both the continuum and lattice investigations of the SU(2) gauge theory, we will ignore this uncertainty in Sect. 4. As an additional check on the extrapolation method, we performed calculations with different fit ranges for \(P^2\), with the total momentum sampled in the region \((0,L)\) (in GeV\(^2\)), and with \(L\) given in the table. In the next section we employ the method with \(L=0.5\), which appears empirically to have the best performance.

## 4 Results

### 4.1 Estimation of model dependence

As already highlighted, the majority of model dependence stems from the truncation of the quark–gluon vertex DSE. Other parts of the calculation are constrained either by the underlying gauge dynamics (i.e. the ghost and gluon propagator which are taken from appropriate lattice or continuum calculations) or by chiral symmetry (in the process of truncating the BSE kernel). Thus, we can test the sensitivity of the truncation by varying the solution of the quark–gluon vertex within the constraints imposed by chiral symmetry breaking and the axWTI.

Chiral limit results for meson masses in rainbow-ladder (RL) and beyond rainbow-ladder (BRL) truncations, compared with lattice data for an SU(2) theory. All units are in TeV. Errors of the BRL results come from the extrapolation procedure. For the \(0^{+\,\!+}\) state, our continuum result is for an isoscalar; lattice results are forthcoming

### 4.2 Discussion

Comparison of our results with the lattice [12, 13] requires the scale to be set by equating the electroweak (EW) scale with the pseudoscalar meson (‘pion’) decay constant, i.e. \(v_{\mathrm {EW}} = f_{\pi } =246\) GeV. This puts the theory under investigation in the context of dynamical EW symmetry breaking, otherwise known as Technicolor (TC) [93, 94].

The drawbacks that the SU(2) model discussed here (and any other model with QCD-like dynamics) faces as a Technicolor template are by now well known. These include the problems with precision tests on flavor-changing neutral currents [95], and the composite ‘Higgs boson’ which is expected to be very heavy. This latter problem is seen here, whereupon we do not find an isoscalar scalar (‘sigma’) meson (a TC version of the Higgs boson) below \(1.33\) TeV. This situation, however, might change drastically if one considers explicitly the couplings to Standard Model particles [96], or more general EW embeddings [97]. Another promising approach to Technicolor phenomenology is to use nearly conformal theories as Technicolor templates [98, 99, 100, 101]. As we are presently concerned with the QCD-like aspects of the model under investigation, we will not comment on its possible Technicolor applications further.

For the discussion of results it would be useful to have an estimate on the mass of an isoscalar scalar meson, calculated in a method different from our DSE/BSE approach. Since the lattice results for this particle are yet to come, we will use the values obtained by means of group theory scaling, which for the model under investigation gives \(m_{\sigma }\in \) \(\left[ 1,1.5\right] \) TeV [96]. Taking this into consideration, it seems that the RL method fares well for the sigma meson, and to a lesser extent, the rho meson. In the \(1^{+\,\!+}\) channel, this truncation performs inadequately, with a result which deviates by about 30 % from the central lattice value. It is arguable whether or not one can modify the RL method so that it is better suited for an SU(2) theory, thus performing reasonably well for all considered mesons. Based on our current results, and given the limitations of the RL framework, we are skeptical toward this prospect.

On the other hand, the results of the BRL approach compare well with lattice, especially when employing the dressed three-gluon vertex. Since there are considerable error margins present in both the continuum and the lattice investigations, stronger statements about the agreement of our methods will have to wait for more refined calculations.

Regarding the continuum calculation, dressing of the three-gluon vertex seems to lead to a better agreement with the discretized approach, but the overall impact of this modification is relatively mild, and all meson masses are rather robust in this respect. This leaves open the possibility that more elaborate modifications of our model (i.e. inclusion of additional diagrams and higher \(n\)-point Green functions in the quark–gluon vertex DSE) might not change the results appreciably. However, note that dynamical contributions that can collectively be termed ‘pion cloud’ effects are known to be important, and they are the focus of present and future investigations.

As a final remark, we note that the calculation of the baryonic spectrum in this theory does not require any additional effort. An SU(2) gauge theory possesses an enlarged (Pauli–Gürsey) flavor symmetry, which implies that chiral multiplets will contain both mesons and baryons (diquarks). In other words, a meson with \(J^P\) quantum numbers will be degenerate with a \(J^{-P}\) diquark. This degeneracy (which breaks down if the chemical potential is raised above some critical value \(\mu _c\)) has been confirmed in numerous lattice investigations [2, 4, 5, 12, 13].

## 5 Conclusions and outlook

We presented a Dyson–Schwinger/Bethe–Salpeter calculation of ground state hadron masses in a theory with two colors and two fundamentally charged Dirac fermions. We employed a novel beyond rainbow-ladder method and obtained good agreement with lattice results for spin one mesons: however, improved calculations will be needed to reduce uncertainties in both lattice and continuum approaches.

For \(J=0\) mesons, we demonstrated that chiral dynamics is satisfied (i.e. the GMOR relation holds) and obtained the mass of the sigma meson to be in good agreement with the analysis based on group theory scaling. Additionally, we showed that the rainbow-ladder method performs unsatisfactorily in this strongly interacting template. This underlines the need to use more sophisticated techniques when studying generic non-Abelian gauge theories.

Besides masses, the beyond rainbow-ladder approach we outlined here can also be used to study hadronic decays and form factors. A first step toward accessing these quantities is to extend the calculation to complex Euclidean momenta. However, the technical complications which arise are considerable and are subject to future investigation.

## Notes

### Acknowledgments

We would like to thank R. Alkofer, C. S. Fischer, A. Maas, H. Sanchis-Alepuz, and F. Sannino for useful discussions and a critical reading of this manuscript. This work was supported by the Helmholtz International Center for FAIR within the LOEWE program of the State of Hesse, a Lise-Meitner fellowship M1333–N16 from the Austrian Science Fund (FWF), and from the Doktoratskolleg “Hadrons in Vacuum, Nuclei and Stars” (FWF) DK W1203-N16. Further support by the European Union (Hadron Physics 3 project “Exciting Physics of Strong Interactions”) is acknowledged.

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