# Mesons and Nucleons From Holographic Qcd

###### Abstract

We present in this talk a recent investigation on a unified approach within the framework of a hard-wall model of AdS/QCD. We first study a theoretical inconsistency in existing models. In order to remove this inconsistency, we propose a unified approach in which the mesons and the nucleons are treated on the same footing, the same infrared (IR) cutoff being employed in both fermionic and bosonic sectors. We also suggest a possible way of improving the model by introducing a five-dimensional anomalous dimension.

## 1 Introduction

The AdS/QCD model has been extensively used in describing a wide variety of phenomena in particle and nuclear physics and even in condensed matter physics. The model [1, 2, 3] is constructed as a 5D holographic dual of QCD based on the general wisdom of AdS/CFT [4, 5, 6], the 5D gauge coupling being identified by matching the two-point vector correlation functions in deep Euclidean region. Although the AdS/QCD model is based on an ad hoc ground, it reflects some of most important features of gauge/gravity duality. Moreover, it is rather successful in describing properties of mesons (see, for example, the following recent review [7]).

QCD is not a conformal theory, in particular, in the low-energy region, so one should also incorporate this property in constructing an effective AdS/QCD model. Consequently, different models have been developed. In Refs. [1, 2, 3], the size of the extra dimension (also known as the compactification scale) was fixed at the point that corresponds approximately to the QCD scale parameter , i.e., an infrared (IR) cutoff parameter was explicitly introduced. It is often considered as the confinement scale that also breaks sharply the conformal invariance. These AdS/QCD models are called the hard-wall model. On the contrary, there is an alternative approach called a soft-wall model in which the conformal invariance is broken smoothly by introducing the dilaton-like field in the 5D AdS space [8, 9, 10].

While both approaches describe meson properties relatively well, one confronts a serious problem in the baryonic sector. Since there is no theoretical reason that the confinement scale for the baryon should be the same as that for the meson, different values of the IR cutoff for the baryon have been introduced [11, 12, 13, 14, 15, 16]. Actually, the main reason for that is due to the fact that it is impossible to reproduce the meson and baryon properties, in particular, excited nucleon states [14], with the same value of the IR cutoff used. However, when one calculates the meson-nucleon coupling constants, a serious inconsistency arises [17]. In order to determine the coupling constants consistently, one must use the same IR cutoff. Otherwise, one cannot fully consider whole information on the meson and nucleon wavefunctions. This is the motivation of the present investigation and has been studied in our recent work [18]. In the present talk, we will briefly review how to resolve this inconsistency and will put forward possible methods to describe the meson and baryon on an equal footing.

## 2 The effective action of a hard-wall model with holographic mesons and nucleons

We start from a hard-wall model for mesons developed in Refs.[1, 2] and study its applications to nucleons[14, 18]. The model has a geometry of 5D AdS

(1) |

where stands for the 4D Minkowski metric: . The 5D AdS space is compactified by two different boundary conditions, i.e. the IR boundary at and the UV one at . Considering the global chiral symmetry of QCD, we need to introduce 5D local gauge fields and of which the values at play a role of external sources for and currents respectively. Since chiral symmetry is known to be broken to spontaneously as well as explicitly, we introduce a bi-fundamental field with respect to the local gauge symmetry , in order to realize the spontaneous and explicit breakings of chiral symmetry in the AdS side. Considering these two, we can construct the bi-fundamental 5D bulk scalar field in terms of the current quark mass and the quark condensate

(2) |

with isospin symmetry assumed.

The 5D gauge action in AdS space with the scalar bulk field and the vector field can be expressed as

(3) |

where covariant derivative and field strength tensors are defined as and . The 5D gauge coupling is fixed by matching the 5D vector correlation function to that from the operator product expansion (OPE): . The 5D mass of the bulk gauge field is determined by the relation [5, 6] where denotes the canonical dimension of the corresponding operator with spin and turns out to be because of gauge symmetry. The effective action (3) describes the mesonic sector[1, 2] completely apart from exotic mesons[19].

To consider baryons in the flavor-two () sector, one needs to introduce a bulk Dirac field corresponding to the nucleon at the boundary[14, 18]. This hard-wall model was applied to describe the neutron electric dipole moment[20] and holographic nuclear matter[21]. In this model, the nucleons are the massless chiral isospin doublets and which satisfy the ’t Hooft anomaly matching. Then the spontaneous breakdown of chiral symmetry induces a chirally symmetric mass term for nucleons

(4) |

where is the nonlinear pseudo-Goldstone boson field that transforms as under . The and represent the SU(2) Pauli matrices and the pion decay constant, respectively. Thus, we have to consider the following mass term in the AdS side

(5) |

where denotes the mass coupling (or Yukawa coupling) between and nucleon fields, which is usually fitted by reproducing the nucleon mass MeV. In this regard, we can introduce two 5D Dirac spinors and of which the Kaluza-Klein (KK) modes should include the excitations of the massless chiral nucleons and , respectively. By this requirement, one can fix the IR boundary conditions for and at .

Note that the 5D spinors do not have chirality. However, one can resolve this problem in such a way that the 4D chirality is encoded in the sign of the 5D Dirac mass term. For a positive 5D mass, only the right-handed component of the 5D spnior remains near the UV boundary , which plays the role of a source for the left-handed chiral operator in 4 dimension. It is vice versa for a negative 5D mass. The 5D mass for the bulk dimensional spinor is determined by the AdS/CFT expression

(6) |

and turns out to be . However, since QCD does not have conformal symmetry in the low-energy regime, the 5D mass might acquire an anomalous dimension due to a 5D renormalization flow. Though it is not known how to derive it, we will introduce some anomalous dimension to see its effects on the spectrum of the nucleon.

Summarizing all these facts, we are led to the 5D gauge action for the nucleons

(7) | |||||

(8) |

where

(9) |

The non-vanishing components of the spin connection are and are the Lorentz generators for spinors. The matrices are related to the ordinary matrices as .

The more details of the present approach can be found in Ref.[18].

## 3 Results and discussions

Before presenting the results of this work, we note the most of input parameters of the model such as , and are quite well fitted in the mesonic sector [1]. Hence, we have only one free parameter to reproduce the data in the baryonic sector. However, the IR cutoff in the baryonic sector, which is often interpreted as a scale of the confinement, takes different values from those in the mesonic sector. Actually, Ref. [14] performed two different fittings of these parameters. In the first fitting of Ref. [14], the and the were fixed in the mesonic sector, and the is fitted to the nucleon mass. In the second fitting, the and the were taken respectively to be and such that the masses of the nucleon and the Roper resonance were reproduced. Since there is no reason for a nucleon to have the same scale of the confinement as that for a meson, this might be an acceptable argument as far as one treats mesons and baryons separately. However, there is one caveat. When it comes to some observables such as the meson-baryon coupling constants, we need to treat the mesons and baryons on the same footing and require inevitably a common . Otherwise, we are not able to consider whole information on both mesons and baryons. Moreover, a model uncertainty brings on by the mass coupling . Thus, in the present section, we will carry out the numerical analysis very carefully, keeping in mind all these facts.

We first take different values of the from those in the mesonic sector and try to fit the data as was done in Ref. [14]. In this case, is defined as . In general, one can examine two different limits of the mass coupling (see Ref. [18]): In the limit of the small mass coupling, there are three free parameters , and . All other parameters can be related to . On the other hand, in the limit of the strong mass coupling, the can be fixed by some condition [18], which leaves only two free parameters. Obviously, the dependence on the current quark mass must be tiny because of its smallness, so we can simply neglect it. In this case, we have only one free parameter.

The results of the calculations are listed in table 3. In the upper part of the table, we present the results in the limit of the strong mass coupling. They are more or less the same as those obtained in Ref. [14]. For comparison, we list the results for the small mass coupling in the lower part of table 3 where the mass coupling is chosen to be . While the spectrum of the nucleon seems to be qualitatively well reproduced, that of the meson is fairly underestimated in comparison with the experimental data. In the case of the strong mass coupling, the situation becomes even worse. One can note however, as shown in Ref. [18], the ordering of the nucleon-parity states are correctly reproduced for .

The results listed in table 3 indicate that it is not
possible to reproduce the spectra of the meson and the
nucleon at the same time.^{1}^{1}1Note that in the present work we
do not aim at the fine-tuning of the parameters to reproduce the
experimental data. The output data in baryonic sector is quite
stable for changes in . As an attempt to improve the
above-presented results, we want to introduce an anomalous dimension
of the 5D nucleon mass while the 5D mass of the bulk vector field
does not acquire any anomalous dimension because of the gauge
symmetry.

The results are drawn in figures 1 and 2. One can see that, when anomalous dimension is set to zero (i.e. is fixed), the experimental data is badly reproduced. The inclusion of an additional parameter (i.e. considering as a free parameter) improves the output data well but leads to larger values of the possible anomalous dimension.

Note that here the nucleon mass is not used as an input. Varying the value of , we try to fit the spectrum of the nucleon. We present the results from two different parameter sets called model A and model B. In this analysis, we take the values of the and from Ref. [1]. Note that the meson mass is used as an input in model A, while model B corresponds to the global fitting done in Ref. [1]. The 5D nucleon mass is varied in the range of , its anomalous dimension being considered as mentioned before. As shown in figures 1 and 2, the best result is obtained with . Though the absolute value of the nucleon mass turn out to be overestimated in contrast to the previous analysis presented in table 3, qualitatively it is well reproduced within while meson mass is fitted around its experimental value.

We are now in a position to include meson-baryon coupling constants
in the present numerical analysis. We will consider here the and the coupling constants in addition to the
meson and the nucleon spectra. One has to keep in mind that in
order to calculate the meson-baryon coupling constants it is
essential to use the same for the mesonic and baryonic
sectors. Otherwise, it is not possible to keep whole information on
the wavefunctions. Thus, it is of utmost importance to compute all
observables with the same set of parameters. We perform a global
fitting procedure to obtain the results listed in
table 3. Note that we consider here the chiral limit
(), since its effects on the results are rather
tiny.^{2}^{2}2Note that in the chiral limit, the nucleon mass is
different from experiments, . For instance,
in the chiral limit [22].

We assume also that the 5D nucleon mass acquires a large anomalous dimension so that it may vanish, i.e., . The best fit is obtained with the parameters fitted as follows: , , and . The masses of the ground-state nucleon and the meson are in good agreement with the experimental data. Moreover, those of the excited states are qualitatively well reproduced within . However, the coupling constants are in general about underestimated. We mention that in Ref. [17] the dependence of the meson-baryon coupling constants on the was investigated without considering hadron spectra but the results for the coupling constants are more or less in the same level as in the present work.

## 4 Summary and outlook

We have investigated the mesons and the nucleons in a unified approach, based on a hard-wall model of AdS/QCD [1, 14]. In order to study the nucleon spectrum, we have developed an approximated method in which the effective potential can be expanded. The method of this approximation was shown to work very well. In particular, the correct ordering of the nucleon parity states was analytically as well as explicitly shown in this method. [18]

We then have performed several numerical analyzes, varying the model parameters such as the IR cutoff , the quark condensate , and the mass coupling (or Yukawa coupling) . In order to improve the results of the nucleon and the meson spectra on an equal footing, we have introduced an anomalous dimension of the 5D nucleon mass. We found that the zero 5D nucleon mass, , produces the best fit.

We have proceeded to compute the and coupling constants with the same IR cutoff used. This is crucial in order to keep whole information about the wavefunctions. We carried out the global fitting procedure in which we obtained the best fit with the values of the parameters: , , and . The mass spectra of the nucleon and the meson are in relatively good agreement with the experimental data within , whereas the and coupling constants underestimated by about .

Last but not least, we want to mention the following significant point. While the present AdS/QCD model for the baryon respects some important physics such as spontaneous chiral symmetry breaking, it still suffers from a serious flaw. Any construction of AdS/QCD models must satisfy the UV matching with QCD. However, the present framework of the AdS/QCD model for the baryon does not match the OPE results of QCD. It indicates that there is still a very important component missing in the present version of the model, that is, the correct surface terms in the 5D effective action are missing. The corresponding investigation is under way [25].

## Acknowledgments

U.T. Yakhshiev would like to thank the organizers of HNP09 for the possibility to give a talk and for the hospitality during his stay at Osaka University. The present work is supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (grant number: 2009-0073101). Y. Kim acknowledges the Max Planck Society(MPG), the Korea Ministry of Education, Science, Technology(MEST), Gyeongsangbuk-Do and Pohang City for the support of the Independent Junior Research Group at the Asia Pacific Center for Theoretical Physics(APCTP).

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