The structure, mixing angle, mass and couplings of the light scalar and mesons
The mixing angle, mass and couplings of the light scalar mesons and are calculated in the framework of QCD two-point sum rule approach by assuming that they are tetraquarks with diquark-antidiquark structures. The mesons are treated as mixtures of the heavy and light scalar diquark-antidiquark components. We extract from corresponding sum rules the mixing angles and of these states and evaluate the masses and couplings of the particles and .
1. Introduction. Light scalar mesons that reside in the region of the meson spectroscopy are sources of long-standing problems for the conventional quark model. The standard approach when treating mesons as bound states of a quark and an antiquark meets with evident troubles to include and , as well as some other light particles into this scheme: There are discrepancies between predictions of this model for a mass hierarchy of light scalars and measured masses of these particles. Therefore, for instance, the meson was already considered as a four-quark state with content Jaffe:1976ig .
During passed decades physicists made great efforts to understand features of the light scalar mesons: They were treated as meson-meson molecules Weinstein:1982gc ; Weinstein:1990gu ; Achasov:1996ei ; Branz:2007xp , or considered as diquark-antidiquark bound states Maiani:2004uc ; Hooft:2008we . These models stimulated not only qualitative analysis of the light scalar mesons, but also allowed one to calculate their parameters using different methods. Thus, in Ref. Ebert:2008id masses of the , , and mesons were evaluated in the context of the relativistic diquark-antidiquark approach and nice agreements with the data were found. There are growing understanding that the mesons from the light scalars’ nonet are exotic particles or at least contain substantial multiquark components: lattice simulations and experimental data seem support these suggestions. Further information on relevant theoretical ideas and models, as well as on experimental data can be found in original and review articles Alford:2000mm ; Amsler:2004ps ; Bugg:2004xu ; Jaffe:2004ph ; Klempt:2007cp .
Intensive studies of the light scalars as tetraquark states were carried out using QCD sum rules method Latorre:1985uy ; Narison:1986vw ; Brito:2004tv ; Wang:2005cn ; Chen:2007xr ; Lee:2005hs ; Sugiyama:2007sg ; Kojo:2008hk ; Wang:2015uha . Essential part of these investigations confirmed assignment of the light scalars as tetraquark states despite the fact that to explain experimental data in some of them authors had to introduce various modifications to a pure diquark-antidiquark picture and to treat the particles as a mixture of diquark-antidiquarks with different flavor structures Chen:2007xr , or as superpositions of diquark-antidiquark and components Sugiyama:2007sg ; Kojo:2008hk ; Wang:2015uha . There was also the article (see, Ref. Lee:2005hs ), results of which did not support an interpretation of the light scalars as diquark-antidiquark bound states.
As is seen, theoretical analyses performed even within the same method lead to different conclusions about the internal structures of the mesons from the light scalar nonet. One should add to this picture also large errors from which suffer experimental data on the masses and widths of these particles Patrignani:2016xqp to understand difficulty of existing problems.
2. Mixing schemes. An approach to the nonet of light scalars as mixtures of tetraquarks belonging to different representations of the color group was recently proposed in Ref. Kim:2017yvd . In accordance with this approach the nonet of the light spin-0 mesons can be considered as tetraquarks composed of the color () and flavor () antitriplet scalar diquarks. Then, these tetraquarks in the flavor space form a nonet of the scalar particles . In order to embrace the second nonet of the scalar mesons occupying the region above spin-1 diquarks belonging to the color-flavor representation () can be used. The tetraquarks built of the spin-1 diquarks have the same flavor structure as ones constructed from spin-0 diquarks, and therefore can mix with them.
In the present Letter we restrict ourselves by considering only the first nonet of the scalar particles. Therefore, in what follows we neglect their possible mixing with tetraquarks composed of the spin-1 diquarks. The flavor singlet and octet components of this nonet have the structures
that in the exact symmetry can be directly identified with the physical mesons. But the real scalar particles are mixtures of these states, and in the singlet-octet basis and one-angle mixing scheme have the decomposition
where, for the sake of simplicity, we denote and , and is the corresponding mixing angle. Alternatively, one can introduce the heavy-light basis
and for the physical mesons get the expansion
Here we use as the mixing angle in the heavy-light basis. An emerged situation is familiar to one from analysis of the mixing problems in the nonet of the pseudoscalar mesons, namely in the system Feldmann:1998vh ; Feldmann:1998sh ; Agaev:2014wna . The heavy-light basis in the case under consideration is similar to the quark-flavor basis employed there. The mixing angles in the two basis are connected by the simple relation
In general, one may introduce also two-angles mixing scheme if it leads to a better description of the experimental data
The couplings in the system can be defined in the form
We suggest that the couplings follow pattern of state mixing in both one- and two-angles scheme. In the general case of two-angles mixing scheme this implies fulfillment of the equality
where and may be formally interpreted as couplings of the ”particles” and
Currents and in Eq. (6) that correspond to and states are given by the expressions
where and are color indices and is the charge conjugation operator. Then the interpolating currents for physical states and take the forms
In the simple case of one-angle mixing scheme Eq. (10) transforms to the familiar superpositions
These currents or their more complicated forms in the two-angles mixing scheme may be used in QCD sum rule calculations to evaluate the masses and couplings of the mesons and .
3. Sum rules. At the first stage of our calculations we derive the sum rule for the mixing angle of the system. To this end, we use the heavy-light basis and one-angle mixing scheme and start from the correlation function Aliev:2010ra
The sum rule obtained using allow us to fix the mixing angle . In fact, because the currents and create only and mesons, respectively, a phenomenological expression for the correlator equals to zero. Then the second ingredient of the sum rule, namely expression of the correlation function calculated in terms of quark-gluon degrees of freedom should be equal to zero. Because depends on the mixing angle it is not difficult to find
In deriving of Eq. (13) we benefited from the fact that which can be proved by explicit calculations. After applying the Borel transformation and performing required continuum subtractions one can employ it to evaluate .
Having found the mixing angle we proceed and evaluate the spectroscopic parameters of the mesons and . The correlation functions
are appropriate for these purposes and can be utilized to derive the relevant sum rules. The expression of in terms of the physical parameters of the meson is given by the following simple formula
where the dots stand for contributions of the higher resonances and continuum states. Using the interpolating current and matrix elements of the meson from Eqs. (11) and (6) it is a easy task to show that
After calculating the correlation function and applying the Borel transformation in conjunction with continuum subtraction one gets the sum rule
where is the Borel transformed and subtracted expression of with and being the Borel and continuum threshold parameters, respectively. This sum rule and another one obtained from Eq. (16) by means of the standard operation can be used to evaluate the mass of the meson.
The similar analysis for yields
From Eqs. (16) and (17) it is also possible to extract and for evaluating of the couplings and , but they may suffer from large uncertainties: We instead evaluate and from sum rules for the scalar ”particles” and , using to this end correlation functions and given by Eq. (15), where and are replaced by and , respectively.
4. Numerical results. In calculations we utilize the light quark propagator
and take into account quark, gluon and mixed operators up to dimension twelve. The vacuum expectations values of the operators used in numerical computations are well known: , , , , , , where .
The working regions for the Borel and continuum threshold parameters are fixed in the following form
that satisfy standard requirements of sum rules computations. For example, at the lower limit of the Borel parameter the sum of the dimension-10, 11 and 12 terms in does not exceed of all contributions. At the upper bound of the working window for the pole contribution to the same quantity is larger than of the whole result, which is typical for multiquark systems. Variation of the auxiliary parameters and within the regions (19), as well as uncertainties of the other input parameters generate theoretical errors of sum rules computations. The extracted using Eq. (13), as is seen from Fig.1 (a), demonstrates a mild dependence on As a result, it is not difficult to estimate that
This value of in the heavy-light basis is equivalent to in the singlet-octet basis. Using Eq. (20) it is not difficult to evaluate the mesons’ masses in the one-angle mixing scheme that read
As is seen, the one-angle mixing scheme, if take into account the central values from Eq. (21), does not describe correctly the experimental data: it overshoots the mass of the meson and, at the same time, underestimates the mass of the meson. The agreement can be improved by introducing the two mixing angles and . It turns out that to achieve a nice agreement with the available experimental data it is enough to vary and within the limits (20):
For and the sum rules with two mixing angles and lead to predictions
which are compatible with experimental data. The theoretical errors in Eq. (23) accumulate uncertainties connected with and , as well as arising from other input parameters. The dependence of and on the auxiliary parameters and does not exceed limits allowed for such kind of calculations: In Figs. 1(b) and 1(c) we plot and as functions of the Borel parameter to confirm a stability of corresponding sum rules.
In the two-angle mixing scheme the system of the physical particles is characterized by four couplings (7). After determining the mixing angles and that fix the matrix , quantities which should be found from the relevant sum rules are only the couplings and . As we have mentioned above to this end we consider two additional sum rules by treating basic states and as real ”particles” and obtain
The mixing angles , the masses and the couplings complete the set of the spectroscopic parameters of the and mesons.
5. Concluding notes. The investigation performed in the present Letter has allowed us to calculate the mass and couplings of the and mesons by treating them as the mixtures of the diquark-antidiquarks and . We have demonstrated that by choosing the heavy-light basis and mixing angles and a reasonable agreement with experimental data can be achieved even information on the meson suffers from large uncertainties Patrignani:2016xqp . The assumption on structures of the light mesons made in the present work determines also their possible decay mechanisms. Indeed, it is known that the dominant decay channels of the and mesons are and processes. In experiments the decay was seen, as well. The mixing of the and diquark-antidiquark states to form the physical mesons implies that all of these decays can run through the superallowed Okubo-Zweig-Iizuka (OZI) mechanism: Without the mixing the decay can proceed due to one gluon exchange, whereas is still OZI superallowed process Brito:2004tv . The another problem that finds its natural explanation within the mixing framework is a large difference between the full width of the mesons and , which amount to and Patrignani:2016xqp , respectively. In fact, the strong couplings and that determine the width of the dominant partial decays and depend on the mixing angle in the form
In the model under consideration this dependence is a main source that generates the numerical difference between the partial width of aforementioned processes, and hence between the full width of the mesons and .
Analysis of the partial decays of the mesons and , as well as
calculation of the spectroscopic parameters of other light scalar mesons
deserves further detailed investigations results of which will be published
K. A. thanks TÜBITAK for the partial financial support provided under Grant No. 115F183.
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