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the light flavor scalar mesons present a remarkable exception for the naive quark model , and the structures of those mesons have not been unambiguously determined yet . the numerous candidates with @xmath1 below @xmath2 can not be accommodated in one @xmath3 nonet , some are supposed to be glueballs , molecular states and tetraquark states ( or their special superpositions ) @xcite . the @xmath4 and @xmath5 are good candidates for the @xmath6 molecular states @xcite , however , their cousins @xmath7 and @xmath8 lie considerably higher than the corresponding thresholds , it is difficult to identify them as the @xmath9 and @xmath10 molecular states , respectively . there may be different dynamics which dominate the @xmath11 mesons below and above @xmath12 respectively , and result in two scalar nonets below @xmath13 @xcite . the strong attractions between the diquark states @xmath14 and @xmath15 in relative @xmath16-wave may result in a nonet tetraquark states manifest below @xmath12 , while the conventional @xmath17 @xmath3 nonet have masses about @xmath18 , and the well established @xmath19 and @xmath20 @xmath3 nonets with @xmath21 and @xmath22 respectively lie in the same region . furthermore , there are enough candidates for the @xmath17 @xmath3 nonet mesons , @xmath23 , @xmath24 , @xmath25 , @xmath26 and @xmath27 @xcite .    in the tetraquark scenario , the structures of the nonet scalar mesons in the ideal mixing limit can be symbolically written as @xcite @xmath28 the four light isospin-@xmath29 @xmath30 resonances near @xmath31 , known as the @xmath8 mesons , have not been firmly established yet , there are still controversy about their existence due to the large width and nearby @xmath30 threshold @xcite .    in general , we may expect constructing the tetraquark currents and studying the nonet scalar mesons below @xmath12 as the tetraquark states with the qcd sum rules @xcite . for the conventional mesons and baryons , the `` single - pole + continuum states '' model works well in representing the phenomenological spectral densities , the continuum states are usually approximated by the contributions from the asymptotic quarks and gluons , the borel windows are rather large and reliable qcd sum rules can be obtained . however , for the light flavor multiquark states , we can not obtain a borel window to satisfy the two criteria ( pole dominance and convergence of the operator product expansion ) of the qcd sum rules @xcite . in ref.@xcite , t. v. brito et al take the quarks as the basic quantum fields , and study the scalar mesons @xmath7 , @xmath8 , @xmath5 and @xmath4 as the diquak - antidiquark states with the qcd sum rules , and can not obtain borel windows to satisfy the two criteria , and resort to a compromise between the two criteria . for the heavy tetraquark states and molecular states , the two criteria can be satisfied , but the borel windows are rather small @xcite . we can take the colored diquarks as point particles and describe them as the basic scalar , pseudoscalar , vector , axial - vector and tensor fields respectively to overcome the embarrassment @xcite . in this article , we construct the color singlet tetraquark currents with the scalar diquark fields , parameterize the nonperturbative effects with new vacuum condensates besides the gluon condensate , and perform the standard procedure of the qcd sum rules to study the nonet scalar mesons below @xmath12 . the qcd sum rules are `` new '' because the interpolating currents are constructed from the basic diquark fields instead of the quark and gluon fields . whether or not the colored diquarks can be taken as basic constituents is of great importance , because it provides a new spectroscopy for the mesons and baryons @xcite . the article is arranged as follows : we derive the new qcd sum rules for the nonet scalar mesons in sect.2 ; in sect.3 , we present the numerical results and discussions ; and sect.4 is reserved for our conclusions . in the following , we write down the interpolating currents for the nonet scalar mesons below @xmath12 , @xmath32 where @xmath33 the @xmath34 are color indices , the @xmath35 is the charge conjugation matrix , the @xmath36 , @xmath37 and @xmath38 are basic scalar diquark fields , while the @xmath39 , @xmath40 and @xmath41 are the corresponding scalar two - quark currents . in this article , we take the isospin limit for the @xmath42 and @xmath43 quarks , and denote the fields @xmath36 and @xmath37 as @xmath44 .    for the general color antitriplet bilinear quark - quark fields @xmath45 , @xmath40 and @xmath41 are recovered . ] and color singlet bilinear quark - antiquark fields @xmath46 , where the flavor , color and spin indexes are not shown explicitly for simplicity , we can project them into a local and a nonlocal part , after bosonization , the two parts are translated into a basic quantum field and a bound state amplitude , respectively , @xmath47 where the @xmath48 and @xmath49 denote the diquark and meson fields respectively , the @xmath50 and @xmath51 denote the corresponding bethe - salpeter amplitudes respectively @xcite . in ref.@xcite , we study the structures of the pseudoscalar mesons @xmath52 , @xmath53 and the scalar diquarks @xmath54 , @xmath55 , @xmath56 in the framework of the coupled rainbow schwinger - dyson equation and ladder bethe - salpeter equation using a confining effective potential , and observe that the dominant dirac spinor structure of the bethe - salpeter amplitudes of the scalar diquarks is @xmath57 . if we take the local limit for the nonlocal bethe - salpeter amplitudes , the dimension-1 scalar diquark fields @xmath54 , @xmath55 and @xmath56 are proportional to the dimension-3 scalar two - quark currents @xmath58 , @xmath59 and @xmath60 , respectively . a dimension-1 quantity @xmath61 can be introduced to represent the hadronization @xmath62 , @xmath63 and @xmath64 . the attractive interaction of one - gluon exchange favors formation of the diquarks in color antitriplet @xmath65 , flavor antitriplet @xmath66 and spin singlet @xmath67 @xcite . lattice qcd studies of the light flavors indicate that the strong attraction in the scalar diquark channels favors the formation of good diquarks , the weaker attraction ( the quark - quark correlation is rather weak ) in the axial - vector diquark channels maybe form bad diquarks , the energy gap between the axial - vector and scalar diquarks is about @xmath68 of the @xmath69-nucleon mass splitting , i.e. @xmath70 @xcite , which is also expected from the hypersplitting color - spin interaction @xmath71 @xcite . on the other hand , the studies based on the random instanton liquid model indicate that the instanton induced quark - quark interactions are weakly repulsive in the vector and axial - vector channels , strongly repulsive in the pseudoscalar channel , and strongly attractive in the scalar and tensor channels @xcite . so it is sensible to use the scalar diquark fields to construct the tetraquark currents . the two - point correlation functions @xmath72 can be written as @xmath73 denotes @xmath74 , @xmath75 , @xmath76 and @xmath77 . we can insert a complete set of intermediate hadronic states with the same quantum numbers as the current operators @xmath78 into the correlation functions @xmath79 to obtain the hadronic representation @xcite . isolating the ground state contributions from the pole terms of the nonet scalar mesons , we obtain the results , @xmath80 where the @xmath81 are the ground state masses , the @xmath82 are corresponding pole residues defined by @xmath83 , the thresholds @xmath84 in the channels @xmath85 , @xmath8 , @xmath7 respectively , the @xmath86 are the thresholds for the higher resonances and continuum states @xmath87 , and the @xmath88 are the corresponding hadronic spectral densities .    we introduce the following new lagrangian @xmath89 , @xmath90 where @xmath91 , and carry out the operator product expansion with the basic diquark fields @xmath92 , @xmath93 and @xmath16 instead of the quark fields @xmath42 , @xmath43 and @xmath94 , and we have neglected the terms concerning the heavy diquark fields in the lagrangian . in the qcd , the basic quantum fields are the quark and gluon fields , the attractive interactions in the color antitriplet @xmath65 , flavor antitriplet @xmath95 and spin singlet @xmath67 quark - quark channels favor formation of the scalar diquarks , we can absorb some effects of the quark - gluon interactions into the effective diquark masses , which are characterized by the correlation length @xmath96 . at the distance @xmath97 , the @xmath98 diquark state combines with one quark or one @xmath99 antidiquark to form a baryon state or a tetraquark state , while at the distance @xmath100 , the @xmath98 diquark states dissociate into asymptotic quarks and gluons gradually , the strength of the quark - quark correlations is very weak . just like the quarks , the diquarks have three colors , and can be gauged with the same @xmath0 color group to embody the residual quark - gluon interactions .        in calculations , we take into account all diagrams like the typical ones shown in fig.1 , introduce new vacuum diquark condensates @xmath101 and @xmath102 besides the gluon condensate to parameterize the nonperturbative qcd vacuum , and consider the vacuum condensates up to dimension four . in the qcd sum rules , the high dimensional vacuum condensates are usually suppressed by large denominators or additional powers of the inverse borel parameters @xmath103 , the net contributions are very small . for example , in the present case the contributions of the dimension-6 vacuum condensates can be estimated as @xmath104 , which is a tiny quantity . if additional suppressions originate from the denominators are taken into account , the contributions are even smaller , and can be safely neglected . once the analytical results are obtained , then we can take the dualities below the thresholds @xmath105 and perform the borel transform with respect to the variable @xmath106 , finally we obtain the following sum rules , @xmath107 where the @xmath108 denote the qcd spectral densities @xmath109 , @xmath110 and @xmath111 , @xmath112\delta(s-\widehat{m}_{q / s}^2)\ , , \end{aligned}\ ] ] @xmath113 , @xmath114 , @xmath115 in the channels @xmath85 , @xmath8 , @xmath7 respectively , and the @xmath116 is the borel parameter . the threshold parameters @xmath117 are different from the corresponding @xmath118 in eq.(6 ) , because we absorb some qcd interactions into the effective diquark masses . differentiate eq.(8 ) with respect to @xmath103 , then eliminate the pole residues @xmath119 , we can obtain the sum rules for the masses of the nonet scalar mesons , @xmath120 we estimate the vacuum diquark condensates @xmath101 and @xmath102 with the assumption of the vacuum saturation , which works well in the large @xmath121 limit , @xmath122 where the @xmath61 is a quantity has dimension of mass and can be taken as the confinement energy scale @xmath123 . at the energy scale @xmath124 , @xmath125 , @xmath126 , @xmath127 @xcite , and @xmath128 . the quark condensates play a special role being responsible for the spontaneous breaking of the chiral symmetry , and relate with the masses and decay constants of the light pseudoscalar mesons through the gell - mann - oakes - renner relation @xcite . the values of other vacuum condensates , such as the mixed condensates , the four quark condensates and the gluon condensates , can not be obtained from the first principles , we usually calculate them with the lattice qcd , the instanton models , or determine them empirically by fitting the qcd sum rules to the experimental data . in this article , we introduce the diquark condensates @xmath101 and @xmath102 to parameterize the nonperturbative qcd vacuum , and assume that they relate with the four quark condensates ( therefore they have implicit relations with the spontaneous breaking of the chiral symmetry ) , and take the dimension - one parameter @xmath61 to be the confinement energy scale , as the scalar mesons @xmath5 , @xmath4 , @xmath8 and @xmath7 are bound states which consist of the confined quarks and gluons , and some parameters are needed to embody the confinement . on the other hand , we can understand the parameter @xmath123 as a fitted value , which happens to be the confinement energy scale , the crude estimation works well . we take the updated values of the diquark masses from the qcd sum rules for consistency , where the interpolating currents @xmath39 , @xmath40 and @xmath41 are used @xcite , @xmath129 and @xmath130 ; the scalar diquarks were originally studied with the qcd sum rules about twenty years ago @xcite . there have been several theoretical approaches to estimate the diquark masses , for example , the simple constituent diquark mass plus hyperfine spin - spin interaction model @xcite . the @xmath5 and @xmath4 are well established , and the existence of the @xmath7 meson is confirmed , although there are controversy about its mass and width , the values listed in the review of particle physics are @xmath131 and @xmath132 respectively @xcite . as far as the @xmath8 are concerned , there are still controversy about their existence , we take them as the @xmath16-wave isospin-@xmath29 @xmath30 resonance with the breit - wigner mass about @xmath133 . the e791 collaboration observed a low - mass scalar @xmath30 resonance with the breit - wigner mass @xmath134 and width @xmath135 respectively in the decay @xmath136 @xcite , and the bes collaboration observed a clear low mass enhancement in the invariant @xmath30 mass distribution in the decay @xmath137 with the breit - wigner mass @xmath138 and width @xmath139 , respectively @xcite . recently , the bes collaboration reported the charged @xmath8 in the decay @xmath140 with the breit - wigner mass @xmath141 and width @xmath142 , respectively @xcite . it is sensible to estimate @xmath143 . on the other hand , the qcd sum rules for the tetraquark states indicate that @xmath144 and @xmath145 @xcite .    assuming the energy gap between the ground and first radial excited tetraquark states is about @xmath146 , we can tentatively determine the threshold parameters @xmath147 , @xmath148 , and @xmath149 . the convergence behavior of the operator product expansion is very good , the contributions from the different terms have the hierarchy : perturbative - term @xmath150 . in calculation , we take uniform minimum value for the borel parameters @xmath151 . the perturbative continuum @xmath152 is suppressed by the factor @xmath153 , the contributions from the pole terms are very large , see fig.2 . in this article , we take uniform maximum value for the borel parameters , @xmath154 , the contributions from the pole terms are about @xmath155 , @xmath156 and @xmath157 in the channels @xmath158 , @xmath8 and @xmath7 , respectively . the two criteria ( pole dominance and convergence of the operator product expansion ) of the qcd sum rules are well satisfied . if the conventional quark currents are chosen , the multiquark states i.e. tetraquark states , pentaquark states , hexaquark states , etc , have the spectral densities @xmath159 with the largest @xmath160 , the integral @xmath161 converges slowly @xcite . if one do nt want to release the criterion of pole dominance , we have to either postpone the threshold parameter @xmath105 to very large value or choose very small value for the borel parameter @xmath162 . with large value of the threshold parameter @xmath105 , for example , @xmath163 , here @xmath164 stands for the ground state , the contributions from the high resonance states and continuum states are included in , we can not use single - pole ( or ground state ) approximation for the spectral densities ; on the other hand , with very small value of the borel parameter @xmath162 , the borel window @xmath165 shrinks to zero or very small values .    the numerical values of the masses and pole residues are presented in table 1 and figs.3 - 4 . from table 1 , we can see that the present predictions are compatible with ( or not in conflict with ) the experimental data @xcite and theoretical estimations @xcite . the scalar tetraquark currents @xmath166 , @xmath76 and @xmath77 maybe have non - vanishing couplings with the scattering states @xmath9 , @xmath167 , @xmath30 , @xmath168 , @xmath169 , @xmath170 , etc , for example , @xmath171 if the couplings denoted by the @xmath172 and @xmath173 are strong enough , the contaminations from the continuum states are expected to be large . in the following , we study the contributions of the intermediate pseudoscalar meson loops to the correlation function @xmath174 in details as an example , @xmath175 where @xmath176\left [ ( p - q)^2-m_{\pi}^2\right ] } \ , , \nonumber\\ \sigma_{kk}(p)&=&\int~{d^4q\over(2\pi)^4}\frac{1}{\left [ q^2-m_{k}^2\right]\left [ ( p - q)^2-m_{k}^2\right ] } \ , , \end{aligned}\ ] ] the @xmath177 , @xmath178 are the strong coupling constants between the @xmath5 and the pseudoscalar meson pairs @xmath9 , @xmath167 respectively , and the @xmath179 , @xmath180 , @xmath181 , @xmath182 are the scattering amplitudes among the pseudoscalar meson pairs @xmath9 and @xmath167 . the couplings @xmath183 and @xmath173 are complicated functions of the @xmath184 , @xmath185 , @xmath186 , @xmath179 , @xmath187 , @xmath181 , @xmath188 , @xmath177 and @xmath178 , the explicit expressions are difficult to obtain . we should bear in mind that the intermediate meson loops contribute a self - energy to the scalar meson @xmath5 , and therefore the scalar meson @xmath5 develops a breit - wigner width . in fact , the scalar mesons @xmath85 , @xmath8 and @xmath7 below @xmath12 can be generated dynamically from the unitaried scattering amplitudes of the pseudoscalar mesons @xcite . we can take into account those meson loops effectively by taking the following replacement for the hadronic spectral density , @xmath189 here we neglect the complicated renormalization procedure , take the physical values , and ignore the energy scale dependence of the mass @xmath184 and pole residue @xmath186 for simplicity ; the approximation works well . the qcd sum rules for other scalar mesons are treated with the same routine . in ref.@xcite , dai et al perform the renormalization procedure in details to take into account the contributions from the continuum states . the widths listed in the review of particle physics are @xmath190 , @xmath191 , @xmath192 and @xmath193 , respectively @xcite . taking into account the finite widths , we can obtain the modified masses from the corresponding qcd sum rules , which are shown in table 1 and fig.5 . in calculation , we observe that the narrow width @xmath194 modifies the mass @xmath195 slightly and the effect can be neglected safely , while the broad widths @xmath196 and @xmath197 reduce the masses @xmath198 and @xmath199 about @xmath200 and @xmath201 , respectively . comparing with the experimental data @xcite , the modified masses are better . the scalar tetraquark currents @xmath166 , @xmath76 and @xmath77 maybe also have non - vanishing couplings with the higher resonances @xmath23 , @xmath24 , @xmath25 , @xmath26 and @xmath27 , which are supposed to be the @xmath202 @xmath3 states or glueballs , the couplings should be very small . furthermore , the threshold parameters are @xmath203 , @xmath204 and @xmath205 in the channels @xmath85 , @xmath8 and @xmath7 , respectively , the contaminations should be very small .    . the masses and the pole residues of the nonet @xmath11 tetraquark states . the star denotes the modified masses from the sum rules where the finite widths shown in the bracket are taken into account in the hadronic spectral densities . [ cols="^,^,^",options="header " , ]     , @xmath206 and @xmath35 correspond to the channels @xmath158 , @xmath8 and @xmath7 respectively . , width=302 ]    , @xmath206 and @xmath35 correspond to the channels @xmath158 , @xmath8 and @xmath7 respectively . , width=302 ]    , @xmath206 and @xmath35 correspond to the channels @xmath158 , @xmath8 and @xmath7 respectively . , width=302 ]    , @xmath206 and @xmath35 correspond to the channels @xmath158 , @xmath8 and @xmath7 respectively . the values in the bracket denote the finite widths in the hadronic spectral densities . , width=302 ] in this article , we take the scalar diquarks as the point particles and describe them as the basic quantum fields , then introduce the @xmath0 color gauge interactions , and construct the tetraquark currents which consist of the scalar fields to study the nonet scalar mesons as tetraquark states with the new qcd sum rules . the numerical values are compatible with ( or not in conflict with ) the experimental data and theoretical estimations . comparing with the conventional quark currents , the diquark currents have the outstanding advantage to satisfy the two criteria of the qcd sum rules more easily , the new sum rules can be extended to study other multiquark states . this work is supported by national natural science foundation of china , grant number 11075053 , and the fundamental research funds for the central universities . j. weinstein and n. isgur , phys . * 48 * ( 1982 ) 659 ; f. e. close , n. isgur and s. kumana , nucl . phys . * b389 * ( 1993 ) 513 ; n. n. achasov , v. v. gubin and v. i. shevchenko , phys . rev . * d56 * ( 1997 ) 203 . z. g. wang , eur . j. * c62 * ( 2009 ) 375 ; z. g. wang , eur . j. * c63 * ( 2009 ) 115 ; r. d. matheus , s. narison , m. nielsen and j. m. richard , phys . * d75 * ( 2007 ) 014005 ; z. g. wang , eur . phys . j. * c67 * ( 2010 ) 411 .
in this article , we take the scalar diquarks as point particles and describe them as basic quantum fields , then introduce the @xmath0 color gauge interaction and new vacuum condensates to study the nonet scalar mesons as tetraquark states with the qcd sum rules . comparing with the conventional quark currents , the diquark currents have the outstanding advantage to satisfy the two criteria of the qcd sum rules more easily .     + zhi - gang wang + department of physics , north china electric power university , baoding 071003 , p. r. china    pacs numbers : 12.38.lg ; 13.25.jx ; 14.40.cs    * key words : * nonet scalar mesons , qcd sum rules