## Abstract

The QCD Sum Rules approach had proposed by Shifman, Vaishtein Zakharov Novikov, Okun and Voloshin (SVZNOV) in 1979 and has been used as a method for extracting useful properties of hadrons having the lowest mass, called as ground states. On the other hand, the most recent experimental results make it clear that the study of the excited states can help to solve many puzzles about the new XYZ mesons structure. In this paper, we propose a new method to study the first excited state of the vector mesons, in particular we focus our attention on the study of the ρ vector mesons, that have been studied previously by SVZNOV method. In principle, the method that we used is a simple modification to the shape of the spectral density of the SVZNOV method, which is written as “pole + continuum”, to a new functional form “pole + pole + continuum”. In this way, We may obtain the ρ and the ρ2S masses and also their decay constants.

### Keywords

- QCD Sum Rules
- double pole
- light quarks
- vector mesons

## 1. Introduction

The successful QCD sum rules was created in 1977 by Shifman, Vainshtein, Zakharov, Novikov, Okun and Voloshin [1, 2, 3, 4], and until today is widely used. Using this method, we may obtain many hadron parameters such as: hadrons masses, decay constant, coupling constant and form factors, all they giving in terms of the QCD parameters, it means, in terms of the quark masses, the strong coupling and nonperturbative parameters like quark condensate and gluon condensate.

The main point of this method is that the quantum numbers and content of quarks in hadron are presented by an interpolating current. So, to determine the mass and the decay constant of the ground state of the hadron, we use the two-point correlation function, where this correlation function is introduced in two different interpretations. The first one is the OPE’s interpretation, where the correlation function is presented in terms of the operator product expansion (OPE).

On the phenomenological side we can be written the correlation function in terms of the ground state and several excited states. The usual QCDSR method uses an ansatz that the phenomenological spectral density can be represented by a form “pole + continuum”, where it is assumed that the phenomenological and OPE spectral density coincides with each other above the continuum threshold. The continuum is represented by an extra parameter called * s*0, as being correlated with the onset of excited states [5].

In general, the resonances occurs with * ρ*meson spectrum, for example, the ansatz “pole + continuum” is a good approach, due to the large decay width of the

*or*ρ(2S)

*1450*ρ(

*, which allows to approximate the excited states as a continuum. For the*)

*meson [6], the value of*ρ

*1020*φ(

*meson the value is 1.41*)

*. We note that the values quoted above for*GeV

*1450*ρ(

*and*)

*1680*φ(

*. One interpretation of this result is due to the effect of the large decay width of these mesons.*)

Novikov et al. [1], in a pioneering paper, proposed, for the charmonium sum rule, that the phenomenological side with double pole (“pole + pole + continuum”) and* GeV,*while the experimental data had shown 2.83

*for this meson [1, 2, 4].*GeV

In general, by QCDSR, the excited states are studied in “pole + pole + continuum” ansatz with * ρ(*1

*2*S,

*mesons [8, 10, 11], nucleons [7, 12],*S)

*1*ηc(

*2*S,

*mesons [2],*S)

*1*ψ(

*2*S,

*mesons [1, 13] and*S)

*1*ϒ(

*2*S,

*mesons [14]. In this paper, we obtain the*S)

*1*ρ(

*2*S,

*mesons masses and their decay constants taking the “pole + pole + continuum” ansatz in QCD sum rules,*S)

## 2. The two point correlation function

As it is known, to determinate the hadron mass and the decay constant in QCD sum rules, we may use the two-point correlation function [3], that is given by

where, on the OPE side, this current density for

where the subscribe index * a*and

*represents the color index. Now, using Eq. (2) in Eq. (1) we have*b

Evaluating Eq. (3) in terms of the OPE [15], which can be written by a dispersion relation, where this relation depends on the QCD parameters, the correlation function takes the form:

with:

Note that

and * s*parameter to have an imaginary part of the

On the phenomenological side, the interpolating current density may be written considering just hadronic freedom degrees, it means, inserting a complete set of intermediate states among the operator, where they are the creation and annihilation describes by the interpolating current density. In this way we can use the following operator algebra

where * f*and

*are, respectively, the decay constant and the mass of the meson and*m

The Invariant part of Eq. (8),

where

Note that, we can introduce a minimum number of parameters in the calculus by the approach

so, * s*values greater than

Note that, by the Quark-Hadron duality we can develop the two-point correlation function in both different interpretation that are equivalent each other. I.e., we can match the correlation function by de OPE, Eq. (5), with the correlation function by the Phenomenological side, Eq. (10), through the Borel transformation.

## 3. Borel transformation

To macht the Eqs. (5) and (10) is not that simple, because in the OPE side the calculations of the all OPE terms is almost impossible, in this way, at someone moment we must truncate the series and, beyond this, guaranties its convergence. However, for the truncation of the series to be possible, the contributions of the terms of higher dimensions must be small enough to justify to be disregarded in the expansion.

Thereby, for both descriptions to be in fact equivalent, we must suppress both the contributions of the highest order terms of the OPE and the contributions of the excited states on the phenomenological side. It is can be done by the Borel transformation that is define as follow:

where * Borel Mass*.

Because of this, we can determine a region of the M2 space in which both the highest order contributions from OPE and those from excited states are suppressed, so that the phenomenological parameters associated with the hadron fundamental state can be determined. Therefore, we must determine an interval of M2 where this comparison is adequate, enabling the determination of reliable results. This interval is called Borel Window.

At the Phenomenological side, we introduce some approximations when we assume that the spectral density can be considering as a polo plus a continuum of excited states. So, we must suppress the continuum contributions for the result to be sufficiently dominated by de pole.

## 4. The double pole method

This method is consisted by the assumption that the spectral density at the phenomenological side can be given like [16]:

where

Note that, inserting Eq. (13) in Eq. (9) we get the following two-point phenomenological correlation function:

Applying the Borel transformation in Eqs. (14) and (5) we get:

By the Quark-Hadron duality we have

The contribution of the resonances is given by:

To develop Eq. (17) let us make a variable change taking

Now, taking de derivative of Eq. (19) with respect to * x*we get:

where, now, we are considering

We observe that the Eqs. (19) and (21) form a equations system in * x*variable. In this system we can make a new change of variables as follow:

this way we get the following system:

Solving the above system of equation for * A(x)*and

*we have:*B(x)

Note that, Eqs. (

This way we have the both solutions coupling each other. On the other hand, what we are looking for are mass solutions for the ground state and its first excited state independent each other. To do so, we take the second derivative of the Eq. (19) with respect to * x*and the result we divide by Eq. (23) for decoupling of the

Substituting Eq. (29) in Eq. (28) we obtain the following polynomial equation:

where * a, b*and

*are respectively:*c

Note that, for the

These results can be developed to obtain the masses of the ground state and its first excited state for any

## 5. Results for the ρ meson

For the

and

where

Following [19], for the

Using the mass of the * GeV*. That way, the excited state decay constant is a bit lower than the ground state decay constant. The minimum value for

*, because*GeV

MeV.

In this way, we can find the Borel window where the QCDSR is valid. In this case, the Borel window is shown in Figure 2 and it is calculated by the ratio between Eqs. (17) and (18) for a given

In Figure 3 we display the masses of the * GeV*(polygonal blue point), 1.61

*(red dot-dashed line), 1.56*GeV

*(diagonal green cross point) and the grey lines representing the masses of the ground state and the first excited state for the*GeV

*[16].*GeV

For the ground state, in Figure 3, we show that the mass average of the * MeV*for the experimental data.

To evaluate the decay constants, we use the experimental masses given by [16]. For the * GeV*and for the

*. In Figure 4, we display the decay constant for the*GeV

In Figure 4, on the left side, we have an average for the decay constant of the * MeV*, note that the maximum value for the decay constant is that one where

*. On the right side, we have an average for the decay constant of the*MeV

Furthermore, it is interesting note that in Ref. [20] we can see another way to extract the experimental decay constant of the

## 6. Conclusions

In this work, we made a little revision about the QCD Sum rules method and presented a new method for calculation of the hadronic parameters like mass and decay constant [19] as function of the Borel mass.

We show that the double pole method on QCDSR consists in a fit on the interpretation of the correlation function by the phenomenological side, where the relations dispersion is now presented with two poles plus a continuum of excited states, being these two poles representing the ground state and the first excited state.

For the ρ(1S, 2S) mesons we had a good approximation for the calculations of these masses comparing with the experimental data on the literature. Beyond that, for the decay constant of the * ρ(2S)*meson we had a good prediction like it is seen in [19] where

Our intention with this work consists on the studying of the vector mesons testing the accuracy of the double pole method and apply this method to analyze others kind of mesons such as scalar mesons.

## Acknowledgments

We would like to thank Colégio Militar de Fortaleza and the Universidade Federal de Campina Grande by technical and logistical support.