Scalar mesons within a dynamical holographic QCD model
We show that the infrared dynamics of string modes dual to
states within a Dynamical AdS/QCD model of coupled dilaton-gravity
background gives the Regge-like spectrum of ’s scalars and
higher spin mesons consistent with experimental data. The pion mass
and its trajectory were also described with a scale deformation of
the metric and a rescaled string mass. The available experimental
decay widths of the decays provided a complementary check
of the proposed classification scheme for , ,
and as radial excitations of . For
we estimated a mixing angle of with other
keywords:Gauge/string duality, Meson Masses
The light scalar mesons are still challenging our imagination and models for quark spectroscopy . Nowadays, tetraquarks configurations are considered dominant for the structure of and scalar resonances . Also, the decay of and into pions (2 and 4) suggest a structure of , while the decay of into suggest a structure of . However those interpretations are not in agreement with colisions leading to  and . It leads to a possibility of interpret this resonances as a scalar glueballs, or a mixing between those structures. The real nature of scalars is a fundamental and controversial open question that needs more experimental data to help to solve this issue.
In this work, using the Regge trajectories and scalar ( family) decay widths into 2 () as guidelines, we find that these states can still be interpreted as states in the framework of a Dynamical AdS/QCD model . It has been applied with success to the description of the Regge trajectories of light mesons with nonzero spin. The model consists in the solution of five-dimensional gravity (5d) coupled to an active dilaton field, where the metric comes from a IR deformed anti-de Sitter space in order to have confinement by the area law behavior of the Wilson loop. We expect that the phenomenological 5d holographic QCD viewpoint ([7, 8, 9, 10, 11, 12, 6]), consistent with experimental data, can give some hints to construct a 10 dimensional (10d) dual model of QCD-like theories.
The AdS/CFT conjecture relates a 10d Type IIB superstring field theory in with correlators of an N=4 super Yang-Mills theory in a four dimensional flat space-time. The low energy limit of massless fields of a Type IIB superstring theory is described by a Type IIB supergravity. Therefore it is natural that the search for a QCD-like dual model starts from a 10d solution of the Type IIB supergravity theory. In this respect, an important step in this program is the investigation of solutions of Type IIB supergravity that are pure N=1 super Yang Mills in the IR. These solutions lead to dual models with some QCD properties as confinement and chiral symmetry breaking. There are some examples of supergravity solutions as Klebanov-Strassler , Klebanov-Tseytlin  and Maldacena-Nunez  which have some characteristics of an N=1 theory. On the other hand, the spectrum of those models are still far from the phenomenology. An interesting result of  is to show how to construct an effective 5d action with several scalars starting with a 10d consistent truncation of type IIB supergravity described by Papadopoulos-Tseytlin (PT) ansatz. In the 5d effective action, each scalar has a different sigma model metric. It suggests that it may be conceivable a possibility that phenomenological 5d effective models of QCD could play a role in describing mesons, and more generally hadrons.
Here we focus on the light-scalar and pseudoscalar sector of QCD. Our results show that an appropriate choice of the scaling factor in the 5d metric model gives the experimental Regge behavior (scalars and pseudoscalars) and partial decay width of ’s into two pions. The partial widths are obtained without introducing any free parameter beyond those implicit contained in the description of the scalar and pion Regge trajectories. We made the assumption that the holographic coordinate in the metric of the Dynamical AdS/QCD model for high spin mesons  can be rescaled in order to provide the effective potential for the string modes dual to the scalar and pseudoscalar mesons. The masses of the scalars are then obtained by a single rescale of the holographic coordinate in the metric form. We also describe the light pseudoscalar states. The scale of the holographic coordinate is contracted for the pseudoscalars in respect to the scalar case and a rescaled string mass is also introduced to allow a description of the pion mass.
The outcome from our work is that , , , , , , and are excited states of , and they belong to a Regge trajectory with a slope of about 0.5 half of the corresponding one for the -meson trajectory.
Additionally to support our interpretation of the nature of the light ’s scalars, we obtain the wave functions of the string modes dual to the scalar meson states and using the pion string amplitude we calculate the decay width of the scalars into two pions. The experimental observation of the systematic decrease found for for higher excitations is easily understood within the holographic view: the overlap between the radial excitations and the two pion state amplitudes are depleted by increasing the number of nodes of the scalar wave function. From that the decreasing pattern for the decay width appears.
The two-pion partial decay width for , , and , for which experimental information exists, are qualitatively consistent with the model results. No further parameter is required besides the coupling between the scalar field with the pions that is determined from the analysis of the pion mass. In particular for a width of about 600 MeV is found.
Although has a mass identified with the first excitation of the string mode dual to state, it has a too large compared to the range of the experimental values. It is known that should mix strongly with other structures such as states (see e.g. ). Indeed we got a mixing angle of about  to be consistent with the experimental data for the partial decay width. (We do not determine the sign of the mixing angle.) Below we briefly discuss the basis of the present holographic model and substantiate quantitatively our claims.
The search for string dual models of Gauge theories was pursued since the pioneering work by t’ Hooft  and vigorously developed after the Maldacena’s Conjecture . Applications of AdS/CFT to describe QCD in a bottom-up approach started with the Hard Wall model  that provides a good description of form factors at high , glueball mass spectrum, but does not give the linear Regge trajectory presented by the mesonic data. This phenomenological result can be obtained by a spin dependence in the metric  or by introducing a dilaton field(see also). In particular, scalar mesons were analyzed in ( and ). However, both works do not include the sigma meson in their studies, while in  the sigma is associated to the ground state. Those models are not solutions of Einstein equations and also do not confine by the Wilson Loop criteria. We proposed a Dynamical AdS/QCD model  that is a solution of the Einstein equations, confines by the Wilson loop criteria and describes the scalar sector. We start from the Einstein-Hilbert action of five-dimensional gravity coupled to a dilaton field ,
where is the five-dimensional Newton constant, the dilaton field depends on the radial coordinate only and is the potential for the dilaton field. We restrict our metric to with diag. Our model belongs to the general class of “Improved AdS/QCD theories” proposed recently by Gürsoy, Kiritsis and Nitti .
The static solutions of the Einstein-dilaton coupled field equations in the fifth dimension found in  are given by and
The boundary condition on the physical brane restricts the geometry to asymptotically () space-times and thus ensures conformality in the ultraviolet (UV). The infrared (IR) behavior is chosen to obtain Regge-trajectories consistent with the Wilson-loop area-law for the gravity dual of a confining theory. The conformal invariance is broken in the IR by .
To calculate the spectrum of scalar mesons we start from the action
that describes a scalar mode propagating in the dilaton-gravity background. We factorize the holographic coordinate dependence as , .
The string modes of the massive scalar field can be rewritten in terms of the reduced amplitudes which satisfy the Sturm-Liouville equation
where the string-mode potential is
with . (Note that for the spin nonzero states .) The gauge/gravity dictionary identifies the eigenvalues with the squared meson mass spectrum of the boundary gauge theory.
The AdS/CFT correspondence states that the wave function should behave as , where (conformal dimension minus spin) is the twist dimension for the corresponding interpolating operator that creates the given state configuration . The five-dimensional mass chosen as  fixes the UV limit of the dual string amplitude with the twist dimension.
In reference , within the context of higher spin mesonic states, we show that the metric
as the leading infrared contribution to the effective potential. This leads to a satisfactory description of the meson mass spectrum with nearly universal Regge slopes, just using the natural scale of , without any further tuning of parameters. (The spin dependent factor in Eq. (5) is required by universality of effective IR potential (6)). A good analytical approximation to the spectrum calculated with GeV for and is , that provides an overall fit to the radial and high-spin excitations .
In this letter we propose a generalization of the Dynamical AdS/QCD model for scalar mesons based on the universality of the form of the effective potential in the IR limit. We implement a scale transformation in the holographic distance tailored to fit the slope of the scalar Regge trajectory. We assume the same universal form of the metric as given by
with from the fit (see figure 1). The slope of the Regge trajectory is decreased for the scalar excitations (see figure 1) in respect to as . In our model, the size of should be larger than the size of other light mesons. Therefore, comes as a broad resonance in pionic channels owing to a large overlap between the corresponding amplitudes. More on that will come on what follows (see figure 2 and Table I).
Solving the Sturm-Liouville equation (3) we obtain the scalar meson spectrum and wave functions. We used the lowest conformal dimension operator corresponding to a scalar with twist 2 . The twist dimension selects the minimal partonic content of a hadron that is attributed to its valence wave function. For operators which have the same twist dimension the model at this stage gives in principle a degenerate spectrum. In particular this difficulty appears in the scalar and pseudoscalar description as we are going to discuss further. We remark that in our model the scalar state corresponds to a meson and it is not identified with a dilaton fluctuation or to a glueball state that has a different twist dimension.
The trajectories found were in agreement of experimental data of family as shown in figure 1. In this picture the sigma meson is the fundamental state and the other ’s are radial excitation of the sigma. We have included in the plot all ’s present in the Particle Listing of PDG . That agreement may be fortuitous and to support our interpretation we test the physics brought by the structure of these mesons measured by the partial width, that reflects the overlap between the dual string amplitudes of the corresponding mesons. Therefore, the model should include a description of the string dual to the pion state, as well.
The first striking point is the slope of about 1 GeV for the pion Regge trajectory, with a value twice of the one found for the scalars. This indicates that the scaling factor of the holographic coordinate for the pseudoscalars should be changed in respect to the family. A scaling factor of makes the IR effective potential of the pion the same as the one found the higher spin mesons. By allowing a fine-tuning variation of about 15% to fit the actual data, we found .
The almost vanishing pion mass is implemented by rescaling the fifth dimensional mass according to (see ), where is uniquely determined as GeV. The parameter gives the strength of the pion coupling to a scalar background. It will be introduced later as the normalization scale of the scalar decay amplitude. The size of the pion wave function is considerably smaller than the size of sigma, as shown in figure 2. This reflects the larger slope seen for the pseudoscalar Regge trajectory in comparison to the scalar one.
The rescale of the string mass can be interpreted as an indication of the coupling between the given mode with those corresponding to higher twist operators. In a couple channel model this could decrease the mass of the ground state as the parameter does.
We emphasize that no new free parameter is really associated with the pion Regge trajectory, beyond the almost vanishing pion mass and the universality of the IR potential as dictated by the observed spectra of light-flavored mesons (apart from the family).
The ’s partial decay width into are calculated from the overlap integral of the normalized string amplitudes (Sturm-Liouville form) in the holographic coordinate dual to the scalars and pion states,
We have introduced the parameter in the transition amplitude considering that it gives the natural scale for the coupling between the pion and a scalar, as has been obtained through the pion mass shift. By dimensional analysis one has to consider, that the coupling has dimension of and therefore comes to bring it to the correct dimension. We find that GeV, for 0.3 GeV, giving the results shown in table I.
|0.4 - 1.2||0.86||600 - 1000||602|
|0.98 0.01||1.10||15 - 80||47|
|1.2 - 1.5||1.32||41 - 141||159|
|1.505 0.006||1.52||38 3||42|
|1.720 0.006||1.70||0 - 6||6|
|2.29 - 2.35||2.33||—||3.2|
The overlap integral is the dual representation of the transition amplitude and therefore the decay width is given by where is the pion momentum in the meson rest frame. The Sturm-Liouville amplitudes of the scalar (pseudoscalar) modes are normalized just as a bound state wave function in quantum mechanics [27, 28], which also corresponds to a normalization of the string amplitude
The overlap integral for the transition amplitude (8) is naturally damped by increasing the excitation as the destructive interference comes into scene within the holographic view. A qualitative understanding of that effect can be seen in figure 2, by observing that within the range of the pion wave function nodes of the higher scalar excitation takes place.
The two-pion partial decay widths for the ’s present in the particle listing of PDG, are calculated with Eq. (8) and shown in Table I. In particular for the model gives a width of about 600 MeV, while its mass is 860 MeV. The range of experimental values quoted in PDG for the sigma mass and width are quite large as depicted in Table I. A recent analysis of the sigma pole in the scattering amplitude from ref. gives 441 MeV and MeV, which in comparison to our results the width seems consistent while the model mass appears somewhat larger. The analysis of the E791 experiment gives 478 MeV and MeV , and the CLEO collaboration  quotes 513 MeV and MeV, both values of the width smaller than our result. Other analysis of the -pole in the scattering amplitude present in the decay of heavy mesons indicates a mass around 500 MeV . A rescaling of the string mass as seen necessary for the pion case can lower the sigma mass. We just observe that coupling between the pion with higher twist string duals can be a source for the effective decrease of the string mass as in the pion case. The width is not strongly affect as the shift in the string mass mainly dislocates the squared meson mass by a constant. We do not attempt to fine-tuning the model at this stage.
The is identified with the first excitation of the string model dual to state (see Table I). The model mass is shifted to a value above the experimental one, i.e., 1.1 GeV compared to 0.98 GeV. The shift of about -0.12 GeV can be attributed to a rescaling of the string mass as in the sigma case. By increasing the excitation of the scalar meson this shift tends to decrease (see in Table I). The experimental values of for is too small compared to our result. We introduced a mixing angle for of , that corresponds to a composite nature by mixing, e.g., with light non-strange quarks. The mixing angle absolute value between to fits within the experimental range.
The mass of the higher scalar excitations are consistent with the experimental data in Table I and figure 1. The two-pion partial decay widths of , and are in good agreement to experimental range, without any further assumption.
Before concluding, let us discuss the validity of (8) used to calculate the partial decay width. Normally, this formula is understood as a reasonable approximation when the width is small. For the zero excitation, the sigma or , the width is quite wide as shown in table I, and in this case the approximation could be questionable. A weighted average of the decay rate around the attributed sigma mass would be preferred, as the masses of two-pion states in the resonant decay channel spreads out around the resonance mass. That means that the string amplitude dual to the scalar meson should be coupled to the two-pion state in the continuum. However, even in this more detailed picture, the string amplitude dual to heals into a region dictated by the confinement scale in the fifth dimension, that should not be affected by the coupling to pions. Therefore, within our holographic model, we believe that the calculation of the width without resorting to a weighted average is a reasonable approximation even in the case of the wide sigma meson, because the relevant physics is dominated by the confinement scale.
In summary, we provide the basic framework to study the family given by excitations of the sigma meson as the string duals to states. The classification, spectroscopy and decay is guided by an Holographic view of the scalar string modes obtained from a Dynamical AdS/QCD model of coupled dilaton-gravity background solution of the Einstein equations. The deformation of the anti-de Sitter metric encodes confinement by the area law behavior of the Wilson loop. Assuming the universality of the metric, apart a scale deformation, for the scalar, pseudoscalar and higher spin string modes dual to states, the Regge-like spectrum of the radial excitations of was obtained and the slope about 0.5 GeV fitted to the data. The almost vanishing pion mass was obtained adding a rescaled string mass. The decrease of the string mass can also improve the description of the lower excitations of the family, which can be attributed to the coupling between states to more complex configurations.
A nice point about the interpretation of holographic QCD comes from a recent work of Brodsky and Téramond , that related the wave equation in the Sturm-Liouville form to the squared mass operator eigenvalue equation for the valence component of the meson light-front wave function. In that respect, the effective potential that we have derived should incorporate the coupling between the valence wave function to all other higher Fock-components of the wave function, indicating a possible new class of AdS/QCD models with the coupling between twist 2 and higher twist modes. The experimentally known partial decay width into give further support to the proposed classification scheme for , , , and , while in the particular case of it should mix with more exotic structures as with an estimated mixing angle around .
Our phenomenological study of a 5d dynamical model consistent with the available experimental data, may constitute an useful guidance for the construction of 10d supergravity theories, that allows an effective 5d perspective and incorporates QCD-like properties. A 10d supergravity theory dual to QCD is a wishful goal, but it is still undelivered (see  and ). Our bottom-up approach, looks promising from the point of view of the phenomenology, but still there is a long journey to a 10d theory.
We acknowledge partial support from the Brazilian agencies CAPES, CNPq and FAPESP.
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