Mass and residue of $ \Lambda(1405)$ as hybrid and excited ordinary baryon

TL;DR

This paper investigates the nature of the $ ext{Lambda}(1405)$ particle, calculating its mass and residue using QCD sum rules to determine if it is a hybrid baryon or an excited ordinary baryon.

Contribution

The study provides the first QCD sum rule calculations of the $ ext{Lambda}(1405)$ as both a hybrid and an excited baryon, offering insights into its internal structure.

Findings

Mass results support the hybrid baryon interpretation.

Residue calculations aid in understanding decay properties.

Comparison with experimental data favors the hybrid model.

Abstract

The nature of the $ \Lambda(1405) $ has been a puzzle for decades, whether it is a standard three-quark baryon, a hybrid baryon or a baryon-meson molecule. More information on the decay channels of this particle and its strong, weak and electromagnetic interactions with other hadrons is needed to clarify its internal organization. The residue of this particle is one of the main inputs in investigation of its decay properties in many approaches. We calculate the mass and residue of $\Lambda(1405)$ state in the context of QCD sum rules considering it as a hybrid baryon with three-quark---one gluon content as well as an excited ordinary baryon with quantum numbers $I(J^{P})=0(1/2^{-})$. The comparison of the obtained results on the mass with the average experimental value presented in PDG allows us to interpret this state as a hybrid baryon.

Tables

Table 1. Table 1: Input parameters.

Parameters	Values
$m_s$	${96}_{-4}^{+8}\mathrm{MeV}$
$⟨\overline{q}q⟩$	${\left(-0.24\pm 0.01\right)}^3$ ${\mathrm{GeV}}^3$
$⟨\overline{s}s⟩$	$0.8⟨\overline{q}q⟩$
$m_0^2$	$\left(0.8\pm 0.1\right)$ ${\mathrm{GeV}}^2$
$⟨\overline{s}{g}_s\sigma Gs⟩$	$m_0^2⟨\overline{s}s⟩$
$⟨\frac{{\alpha }_s{G}^2}{\pi }⟩$	$\left(0.012\pm 0.004\right)$ ${\mathrm{GeV}}^4$

Table 2. Table 2: Values for the mass and residue of the Λ ​ ( 1405 ) Λ 1405 \Lambda(1405) state.

	Present Work	Kisslinger:2009dr	Experiment Patrignani
$m_{\Lambda }$	${1403}_{-32}^{+33}\mathrm{MeV}$	$1407\mathrm{MeV}$	$\left({1405.1}_{-1.0}^{+1.3}\right)\mathrm{MeV}$
${\lambda }_{\Lambda }$	${0.52}_{-0.04}^{+0.05}\times {10}^{-3}{\mathrm{GeV}}^5$	$-$	$-$

Table 3. Table 3: The sum rule results for the mass and residue of the positive and negative parity Λ Λ \Lambda baryon with the spin- 1 / 2 1 2 1/2 .

	$\Lambda$	$\tilde{\Lambda }$
$M^2({\mathrm{GeV}}^2$)	$1.8-3.6$	$1.8-3.6$
$s_0({\mathrm{GeV}}^2$)	$1.8-2.0$	$2.1-2.3$
$m\left(\mathrm{MeV}\right)$	${1116}_{+28}^{-29}$	${1435}_{-31}^{+32}$
$\lambda \cdot {10}^2\left({\mathrm{GeV}}^3\right)$	${1.01}_{-0.07}^{+0.08}$	${0.81}_{-0.05}^{+0.06}$

Equations

$${\Pi }_{{\Lambda }_H}(q)=i\int d^{4}x{e}^{iq\cdot x}⟨0|\mathcal{T}\{{\eta }_{{\Lambda }_H}(x){\bar{\eta }}_{{\Lambda }_H}(0)\}|0⟩,$$

$${\eta }_{{\Lambda }_H}(x)$$

$$G^{\mu\nu}=\sum_{A=1}^{8}\frac{\lambda^{A}}{2}G_{A}^{\mu\nu},\$$

$${\Pi }_{{\Lambda }_H}^{\mathrm{Had}}(q)=\frac{⟨0|{\eta }_{{\Lambda }_H}|\Lambda (q)⟩⟨\Lambda (q)|{\bar{\eta }}_{{\Lambda }_H}|0⟩}{m_{\Lambda }^2-q^2}+\cdots ,$$

$$⟨0|{\eta }_{{\Lambda }_H}|\Lambda (q)⟩={\lambda }_{\Lambda }{u}_{{\Lambda }_H}(q,s).$$

$$\sum _s{u}_{{\Lambda }_H}(q,s){\bar{u}}_{{\Lambda }_H}(q,s)$$

$${\Pi }_{{\Lambda }_H}^{\mathrm{Had}}(q)=\frac{{\lambda }_{\Lambda }^2}{m_{\Lambda }^2-q^2}(/q+m_{\Lambda })+\cdots .$$

$${\hat{\mathcal{B}}}_{q^2}{\Pi }_{{\Lambda }_H}^{\mathrm{Had}}(q)={\lambda }_{\Lambda }^2{e}^{-\frac{m_{\Lambda }^2}{M^2}}\left(/q+m_{\Lambda }\right)+\cdots .$$

$${\Pi }_{{\Lambda }_H}^{\mathrm{OPE}}(q)$$

$${\tilde{S}}_{s,u,d}(x)=C{S}_{s,u,d}^T(x)C.$$

$$S_q^{ab}(x)=i{\delta }_{ab}\frac{/x}{2{\pi }^2{x}^4}-{\delta }_{ab}\frac{m_q}{4{\pi }^2{x}^2}-{\delta }_{ab}\frac{⟨\bar{q}q⟩}{12}$$

$$+i{\delta }_{ab}\frac{/x{m}_q⟨\bar{q}q⟩}{48}-{\delta }_{ab}\frac{x^2}{192}⟨\bar{q}{g}_s\sigma Gq⟩+i{\delta }_{ab}\frac{x^2/x{m}_q}{1152}$$

$$\times ⟨\bar{q}{g}_s\sigma Gq⟩-i\frac{g_s{G}_{ab}^{\alpha \beta }}{32{\pi }^2{x}^2}\left[/x{\sigma }_{\alpha \beta }+{\sigma }_{\alpha \beta }/x\right]$$

$$-i{\delta }_{ab}\frac{x^2/x{g}_s^2⟨\bar{q}q{⟩}^2}{7776}-{\delta }_{ab}\frac{x^4⟨\bar{q}q⟩⟨g_s^2{G}^2⟩}{27648}+\cdots .$$

$$⟨0|G_{\mu \alpha }(x)G_{\nu {\alpha }^{\prime }}(0)|0⟩=\frac{1}{2{\pi }^2{x}^4}[g_{\alpha {\alpha }^{\prime }}(g_{\mu \nu }-\frac{4x_{\mu }{x}_{\nu }}{x^2})$$

$$+(\alpha ,{\alpha }^{\prime })\leftrightarrow (\mu ,\nu )-\alpha \leftrightarrow \mu -{\alpha }^{\prime }\leftrightarrow \nu ],$$

$$⟨0|G_{\mu \alpha }(x)G_{\nu {\alpha }^{\prime }}(0)|0⟩=\frac{⟨g_s^2{G}^2⟩}{96}[g_{\mu \nu }{g}_{\alpha {\alpha }^{\prime }}-g_{\mu {\alpha }^{\prime }}{g}_{\nu \alpha }],$$

$${\Pi }^{\mathrm{OPE}}(q^2)={\int }_{m_s^2}^{\infty }\frac{{\rho }^{\mathrm{OPE}}(s)}{s-q^2}ds+...,$$

$${\rho }^{\mathrm{OPE}}(s)={\rho }^{pert.}(s)+\sum _{k=3}^{10}{\rho }_k(s),$$

$${\rho }^{pert.}(s)$$

$${\rho }_3(s)$$

$${\rho }_4(s)$$

$${\rho }_5(s)$$

$${\rho }_6(s)$$

$${\rho }_7(s)$$

$${\rho }_8(s)$$

$${\rho }_9(s)$$

$${\rho }_{10}(s)$$

$$m_{\Lambda }^2=\frac{{\int }_{m_s^2}^{s_0}dss{\rho }^{\mathrm{OPE}}(s)e^{-s/M^2}}{{\int }_{m_s^2}^{s_0}ds{\rho }^{\mathrm{OPE}}(s)e^{-s/M^2}}.$$

$${\lambda }_{\Lambda }^2{e}^{-m_{\Lambda }^2/M^2}={\int }_{m_s^2}^{s_0}ds{\rho }^{\mathrm{OPE}}(s)e^{-s/M^2}.$$

$$1.8{\mathrm{GeV}}^2\le M^2\le 3.6{\mathrm{GeV}}^2,$$

$$2.1{\mathrm{GeV}}^2\le s_0\le 2.3{\mathrm{GeV}}^2,$$

$$\Pi(q)=i\int d^{4}xe^{iq\cdot x}\langle 0|\mathcal{T}\Big{\{}J_{\Lambda}(x)\bar{J}_{\Lambda}(0)\Big{\}}|0\rangle,$$

$$J_{\Lambda }$$

$${\Pi }^{\mathrm{Phys}}(q)$$

$$⟨0|J|\Lambda (q,s)⟩$$

Full text

I Introduction

It was already suggested that, in addition to the standard particles, there might exist hadrons with different quark-gluon structures, which cannot be included into the ordinary $q\bar{q}$ and $qqq$ schemes. Due to their unconventional nature these states were included into a group of hadrons known as exotic particles Olsen:2014mea . The discoveries of the exotic hadrons by various collaborations, and collected experimental information on their mass, width and decay channels have made investigation of these states one of the central topics in high energy physics. Starting from the first observation of the $X(3872)$ resonance in 2003 by the Belle Collaboration  Choi:2003ue , numerous experimental groups planned to search for and detect resonances with unusual properties, and in fact, measured their mass and width; and determined their quantum numbers. The hadrons with unusual internal structures, i.e. exotic resonances were classified as $XYZ$ particles Shen:2016nth , glueballs Pimikov:2016pag , hybrids Ho:2016owu ; Chanowitz:1982qj , meson molecules Tuan:1977bg , tetraquarks Goerke:2016hxf , pentaquarks Chen:2016otp and dibaryons Gal:2014zia . Like other groups of exotic particles, an identification and classification of hybrid hadrons and calculation of their spectroscopic parameters are important for both revealing their inner organization and gaining new information on quantum chromodynamics (QCD).

The existence of hybrid mesons was first suggested by Jaffe and Johnson in 1976 Jaffe:1975fd . The main ingredients of hybrid mesons ($\bar{q}gq$) are a color-octet quark-antiquark pair and an excited gluonic field. A system with these constituents may have all $J^{PC}$ quantum numbers, implying that one of the fruitful ways to search for hybrids is to study these states with exotic quantum numbers, which are forbidden for the $q\bar{q}$ states. The light hybrid mesons were studied in the framework of different theoretical methos, such as the Bag model, flux tube model, lattice QCD and QCD sum rules (see for instance Barnes:1982tx ; Barnes:1995hc ; Close:1994hc ; Hedditch:2005zf ; Jin:2002rw and references therein). Unfortunately, the predictions for the masses of the hybrids obtained within these approaches differ considerably from each others. The properties of the heavy quarkonium hybrids were also calculated using various methods. Thus, relevant explorations were carried out in the constituent gluon model, the flux tube model, QCD sum rules, nonrelativistic QCD and lattice  (for instance see Refs. Horn:1977rq ; Govaerts:1986pp ; Kleiv:2014fda ; Chen:2013eha ; Chen:2014fza ; Petrov:1998ha ; Liu:2012ze and references therein).

Hybrid baryons can be defined in two different ways: as particles containing three valence quarks and a gluon; and three quarks moving is an excited adiabatic potential. Despite clear theoretical definitions, the experimental identification of the hybrid baryons is a more difficult task compared to the hybrid mesons. Since for the baryons, unfortunately there are not $J^{PC}$ exotics, so one must use other features of these particles to determine whether or not they are hybrids (for more information see Ref.Barnes:2000vn ).

One of the candidates to hybrid baryons is the $\Lambda(1405)$ baryon, which for many decades has attracted interest of physicists. More than forty years ago experiments showed that there was a state with the spin $1/2$ Engler:1965zz , which was predicted to be a $\bar{K}-N$ resonance with quantum numbers $I(J^{P})=0(1/2^{-})$ Dalitz:1967fp using a SU(3) meson-baryon potential. The $\Lambda(1405)$ was experimentally observed in the low energy exclusive reactions, where, as usual, the kaon and pion beams were used Thomas:1973uh ; Hemingway:1984pz . Recently, many high-statistics data are available by the LEPS, CLAS and HADES Collaborations Niiyama:2008rt ; Moriya:2013hwg ; Agakishiev:2012xk . The spin-parity $I(J^{P})=0(1/2^{-})$ was then experimentally confirmed for this particle by the CLAS collaboration Moriya:2014kpv (for detailed information, see for instance Ref.Cho:2017dcy ).

The mass and different decay properties of $\Lambda(1405)$ were studied using different theoretical methods including lattice QCD Jido:2009jf ; Hyodo:2008ek ; Akaishi:2010wt ; An:2010wb ; Jido:2010ag ; MartinezTorres:2010zv ; An:2010tv ; Takahashi:2010nj ; Sekihara:2010uz ; Hyodo:2011ur ; Sekihara:2011hw ; Menadue:2011pd ; MartinezTorres:2012yi ; Revai:2012fx ; Oller:2013zda ; Nakamura:2013boa ; Sekihara:2013sma ; Menadue:2013xqa ; Nakamura:2013qda ; Dote:2014ema ; Ohnishi:2014iba ; Hall:2014uca ; Sekihara:2014ica ; Hall:2014gqa ; Nam:2015yoa ; Miyahara:2015bya ; Miyahara:2015cja ; Miyahara:2015uya ; He:2015cca ; Ohnishi:2015iaq ; Miyahara:2015eyq ; Fernandez-Ramirez:2015fbq ; Hyodo:2015rnm ; Molina:2015uqp ; Kamiya:2016jqc ; Liu:2016wxq ; Dong:2016auh ; Hall:2016kou ; Kim:2017nxg ; Meissner1 ; Meissner2 ; Meissner3 ; Meissner4 ; Meissner5 . Despite a lot of experimental and theoretical studies on the properties of $\Lambda(1405)$ state, unfortunately, there remain important questions about its nature and internal quark organization whether it is a standard three-quark baryon, a hybrid baryon or a baryon-meson molecule with one or two-pole structure. Hence, more experimental and theoretical studies are needed to clarify the physical properties of $\Lambda(1405)$ .

In the present study we are going to calculate the mass and residue of the $\Lambda(1405)$ considering it as a hybrid baryon with three-quark—one gluon content as well as an excited ordinary baryon with quantum numbers $I(J^{P})=0(1/2^{-})$ in the framework of QCD sum rule. The mass of this state had already been calculated in Ref. Kisslinger:2009dr using the same method. In Ref. Kisslinger:2009dr the $\Lambda(1405)$ was also considered in two different pictures: as a hybrid and as a mixed hybrid/three-quark strange baryon. The calculations on the mass of this state and comparison of the obtained result with the experimental data allowed the authors to conclude that this state is consistent with being a strange hybrid baryon rather than a mixed state. The main aim here is to calculate the residue of this state in the considered pictures besides its mass. The residue is one of the main inputs in calculations of many parameters related to the strong, weak and electromagnetic decays of $\Lambda(1405)$ in many theoretical approaches. Investigations of such decay channels may help us better understand the internal organization of $\Lambda(1405)$ resonance and hopefully solve the puzzle on its nature.

This work is structured in the following way. In sec. II we derive two-point QCD sum rules for the mass and residue of the $\Lambda(1405)$ baryon by considering it as a hybrid baryon. In sec. III we perform similar computations by treating the $\Lambda(1405)$ as an excited ordinary three-quark strange baryon in the $\Lambda$ channel. Section IV is reserved for our concluding remarks.

II $\Lambda(1405)$ as a Hybrid Baryon

To calculate the mass and residue of the $\Lambda(1405)$ state in the framework of two-point QCD sum rule, we start with the correlation function

[TABLE]

where ${\eta}_{\Lambda_{H}}$ is the interpolating current for the hybrid $\Lambda(1405)$ baryon and $\mathcal{T}$ indicates the time ordering operator. In the three-quark—one gluon picture, one of the acceptable interpolating currents having the quantum numbers $I(J^{P})=0(1/2^{-})$ is

[TABLE]

where $a,b,c$ are color indices, $C$ is the charge conjugation operator, $u,d,s$ are light quark fields and

[TABLE]

with $\lambda_{A}$ being the generators of the color SU(3) group.

The correlation function $\Pi_{\Lambda_{H}}(q)$ can be calculated in two different ways. From phenomenological or physical side it is obtained in terms of hadronic parameters. From theoretical or QCD side, it is evaluated in terms of quark’s and gluon’s degrees of freedom by the help of the operator product expansion (OPE) in deep Euclidean region. The QCD sum rules for the physical observables such as the mass and residue are obtained equating the coefficients of the same structure in both representations of the correlation function. Finally, the continuum subtraction and Borel transformation are performed in order to suppress the contribution of the higher states and continuum.

First we calculate the correlation function in terms of the hadronic degrees of freedom. By inserting a complete set of hadronic state into Eq. (1) and performing integral over $x$, we get

[TABLE]

with $m_{\Lambda}$ being the mass of the $\Lambda(1405)$ state. Here the dots indicate contributions to the correlation function arising from the higher resonances and continuum states. We define the residue $\lambda_{\Lambda}$ using the matrix element

[TABLE]

Then performing the summation over spins in accordance with

[TABLE]

for the physical side of the correlation function we get

[TABLE]

The Borel transformation with respect to $q^{2}$ applied to $\Pi_{\Lambda_{H}}^{\mathrm{Had}}(q)$ leads to the final form of the hadronic representation:

[TABLE]

The OPE side of the correlation function is calculated at large space-like region, where $q^{2}\ll 0$ in terms of quark-gluon degrees of freedom. For this end, we substitute the interpolating current given by Eq. (2) into Eq. (1), and contract the relevant quark fields. As a result, we get

[TABLE]

where $Tr[\lambda^{A}\lambda^{B}]=2\delta^{AB}$ has been used. In Eq. (9) $S_{s,u,d}^{ab}(x)$ are the light quarks’ propagators and we have used the notation

[TABLE]

We work with the light quark propagator $S_{q}^{ab}(x)$ defined in the form

[TABLE]

Let us emphasize that in calculations we set the light quark masses $m_{u}$ and $m_{d}$ equal to zero, preserving at the same time dependence of the propagator $S_{s}^{ab}(x)$ on the $m_{s}$.

We will treat with $\langle 0|G_{\mu\alpha}(x)G_{\nu\alpha^{\prime}}(0)|0\rangle$ in Eq. (9) in two different ways: first we will replace it by the gluon full-propagator in space representation, i.e,

[TABLE]

and do all calculations. Such calculations are equivalent to the diagrams with valence-gluon as a full propagator. Secondly, we will write it in terms of gluon condensate using

[TABLE]

which represents the diagrams containing the gluon interacting with the QCD vacuum.

The correlation function $\Pi_{\Lambda_{H}}^{\mathrm{OPE}}(q)$ can be decomposed over the Lorentz structures $\sim$ $\!\not\!{q}$ and $\sim I$. In calculations, we choose the terms $\sim$ $\!\not\!{q}$.

The chosen invariant amplitude $\Pi^{\mathrm{OPE}}(q^{2})$ can be written down as the dispersion integral

[TABLE]

where $\rho^{\mathrm{OPE}}(s)$ is the two-point spectral density obtained after lengthy calculations on OPE side and taking the imaginary part of the obtained result. The spectral density corresponding to the structure $\!\not\!{q}$ is obtained as

[TABLE]

where $\rho^{\mathrm{pert.}}(s)$ is the perturbative part of the obtained result and by $\rho_{k}(s)$ we denote the nonperturbative contributions to $\rho^{\mathrm{OPE}}(s)$. The perturbative and nonperturbative parts of the spectral density are obtained as:

[TABLE]

Now applying the Borel transformation to $\Pi^{\mathrm{OPE}}(q^{2})$, equating the obtained expression with the relevant part of the function $\mathcal{B}_{q^{2}}\Pi_{\Lambda_{H}}^{\mathrm{Had}}(q)$, and subtracting the continuum contribution we get the required sum rules. Thus the mass of the $\Lambda$ state can be evaluated from the sum rule

[TABLE]

To extract the residue $\lambda_{\Lambda}$ we can employ the sum rule

[TABLE]

The QCD sum rules for the mass and residue of the $\Lambda(1405)$ contain various parameters that should be fixed in accordance with the standard procedures. Thus, for numerical computation of the $m_{\Lambda}$ and $\lambda_{\Lambda}$ we need values of the quark, gluon and mixed condensates as well as the $s$ quark mass. The values of these parameters can be found in Table 1. The QCD sum rules for the physical quantities under consideration additionally depend on the continuum threshold $s_{0}$ and Borel parameter $M^{2}$. One needs to fix some regions, where physical quantities are practically independent of or demonstrate weak dependence on these auxiliary parameters according to the standard prescription. To find the working window for the Borel parameter, we require the convergence of the operator product expansion as well as adequate suppression of the contributions arising from the higher resonances and continuum. As a result we find the interval

[TABLE]

for the Borel mass parameter. Our analyses show that in the interval

[TABLE]

the results relatively weakly depend on the continuum threshold $s_{0}$. By varying the parameters $M^{2}$ and $s_{0}$ within the allowed ranges, as well as taking into account the errors coming from other input parameters we estimate uncertainties of the whole calculations. The mass $m_{\Lambda}$ and residue $\lambda_{\Lambda}$ are depicted as functions of the Borel and threshold parameters in Figs. 1 and 2. From these figures we see that the results demonstrate good stability with respect to the helping parameters $M^{2}$ and $s_{0}$ in their working windows.

Obtained from our analyses, the average values of the mass and residue for $\Lambda(1405)$ are depicted in table 2.

For comparison, we also depict the QCD sum rules prediction on the mass of this state from Ref. Kisslinger:2009dr and the average experimental value from PDG Patrignani in the same table. From this table we see that our result on the mass is nicely consistent with the prediction of Kisslinger:2009dr and the average experimental value. Our prediction on the residue of $\Lambda(1405)$ in the considered picture may be checked via different approaches.

III $\Lambda(1405)$ as an excited ordinary three-quark strange baryon

In this section, we consider the $\Lambda(1405)$ as excited P-wave ordinary three-quark strange baryon with the same quantum numbers as the previous section. As both the ground (with positive parity) and orbitally excited P-wave (with negative parity) $\Lambda$ baryons couple to the same current, we will calculate the parameters of both these states and compare with the existing experimental data. In order to calculate the mass and residue of the positive and the negative parity spin$-1/2$ $\Lambda$ baryons, we start again with the following two point correlation function:

[TABLE]

where $J_{\Lambda}(x)$ is the interpolating current for $\Lambda$ state with spin $J=1/2$. This interpolating current for ordinary three-quark baryon has the following form

[TABLE]

where the superscript T denotes the transpose operator, and $\beta$ is an arbitrary parameter with $\beta=-1$ corresponding to the Ioffe current.

To derive the mass sum rules for the $\Lambda$ baryon we calculate this correlation function again using both the hadronic and OPE languages. By equating these two representations, one can get the QCD sum rules for the physical quantities of the baryon under consideration. The hadronic side of the correlation function is again obtained by inserting complete sets of intermediate states with both parities. After performing the four-dimensional integration over $x$ we get

[TABLE]

where $m$, $\widetilde{m}$ and $s$, $\widetilde{s}$ are the masses and spins of the positive and negative parity $\Lambda$ baryons, respectively. The dots denote contributions of higher resonances and continuum states. In Eq. (23) the summations over the spins $s$, $\widetilde{s}$ are implied.

We proceed by introducing the matrix elements

[TABLE]

Here $\lambda$ and $\widetilde{\lambda}$ are the residues of the positive and negative parity $\Lambda$ baryons, respectively. Using Eqs. (23) and (24) and carrying out summation over the spins of the baryons by means of the equality

[TABLE]

we obtain

[TABLE]

The Borel transformation of this expression is:

[TABLE]

The OPE side of the aforementioned correlation function is again calculated in terms of the QCD degrees of freedom in deep Euclidean region. After inserting the explicit form of the interpolating current given by Eq. (22) into the correlation function in Eq. (21) and performing contractions via Wick’s theorem, we get the OPE side in terms of the light quark propagators. By using light quark propagator in the coordinate space and performing the Fourier and Borel transformations, as well as applying the continuum subtraction, after lengthy calculations we obtain

[TABLE]

where, as an example, the $\mathcal{B}\Pi_{1}^{\mathrm{OPE}}(q)$ is given as:

[TABLE]

Here $\gamma_{E}\simeq 0.577$ is the Euler constant and $\Lambda$ is a scale parameter. $s_{0}$ is the continuum threshold, and for simplicity, we ignored to present the terms containing the $u$ and $d$ quarks masses and those proportional to $m_{0}^{4}$ . Note that we only ignored to present such terms in the above formula and we will take into account their contributions in the numerical calculations.

Having calculated both the hadronic and OPE sides of the correlation function, we match the coefficients of the structures $\not{q}$ and $I$ from these two sides and obtain the following sum rules that will be used to extract the masses and residues of the ground and first excited states:

[TABLE]

As is seen from Eq. (27) in order to obtain the numerical values of the mass and residue of the orbitally excited $\Lambda$ baryon, we need the values of the mass and residue of the ground state. Therefore, we first calculate the mass and residue of the ground state $\Lambda$ baryon by choosing an appropriate threshold parameter $s_{0}$ in accordance with the standard prescriptions.

QCD sum rules contain three auxiliary parameters namely the continuum threshold $s_{0}$, Borel parameter $M^{2}$ and $\beta$ arbitrary parameter. We find their working windows such that the physical quantities under consideration be roughly independent of these parameters.

Predictions obtained for the mass and pole residue of the positive and negative parity ordinary three-quark $\Lambda$ baryons, as well as the working ranges of the parameters $M^{2}$ and $s_{0}$ are collected in Table 3. These results are obtained by varying the parameter $\beta=\mathrm{tan}\theta$ within the limits

[TABLE]

By comparison of the obtained results on the masses with the experimental data presented in PDG we see that the ground state’s mass is in good consistency with the experimental value, however, our prediction on the mass of the $\Lambda(1405)$ as an excited ordinary baryon is considerably high compared to the average experimental value presented in PDG.

IV Concluding Remarks

In this letter we reported the QCD sum rules predictions on the mass and residue of the $\Lambda(1405)$ considering it as a hybrid baryon with three-quark—one gluon as well as an excited ordinary three-quark strange baryon with quantum numbers $I(J^{P})=0(1/2^{-})$. We found that the mass of the $\Lambda(1405)$ baryon obtained by considering it as a hybrid baryon is in a good agreement with the average experimental value. However, we found a mass considerably high compared to the experimental data when we considered it as an excited ordinary three-quark P-wave strange baryon. Hence, our predictions allow us to interpret this state as a hybrid baryon rather than the excited ordinary baryons. Our predictions for the residue of $\Lambda(1405)$ can be tested via different theoretical approaches and may be used as input information to study the strong, weak and electromagnetic interactions of $\Lambda(1405)$ state with other particles, and determine widths of its various decays. Such calculations, especially investigation of internal charge distribution of the $\Lambda(1405)$ baryon and its multipole moments together with the comparison of the obtained predictions on its mass and width with experimental data will allow us with high confidence level to determine whether the $\Lambda(1405)$ is a hybrid baryon, an excited ordinary three-quark P-wave strange baryon or whether it has other quark-gluon organizations.

ACKNOWLEDGEMENTS

K. A. and B. B. thank TÜBİTAK for financial support provided under the grant no: 115F183.