Finite-Width Gaussian Sum Rules for $0^{-+} $ Pseudoscalar Glueball

$Finite-Width Gaussian Sum Rules for $0^{-+} $ Pseudoscalar Glueball$

arXiv:1505.00211v2 [hep-ph] 31 Aug 2015 arn,gubls oevr h lealmyb mixed unusual be great may the of glueball of the be properties Moreover, to glueballs. the shown hadrons, de- generating and is in effect be [15,16] this significance should instantons order, leading by QCD the of scribed In vacua account. tunneling into degenerate the a taken particular, the be In between to hadrons. effect of recognized difficult attributes is greatly the is impact vacuum may on and hadrons, QCD structure, complicated of the with medium formation which and QCD, handle, the of to for dynamics responsible non-perturbative is that is balls sense. makes investi- Further still evi- by glueballs decisive [13,14]. confirmed on now no been gation to but has up glueballs time, research of long experimental existence a the for of [7,8,9,10,11,12] on dence analyses going rule been sum have and model [4,5,6] [1,2,3], in- researches Theoretical simulations unique QCD. lattice a of including give dynamics investigations may non-Abelian gluons Be- signal the only glueballs. into such of sight of world, composed existence quarkless sates the the in bound of are signal the glueballs for cause seek to is Introduction 1 -alades [email protected] address: E-mail iiewdhGusa u ue for rules sum Gaussian Finite-width eateto hsc,colo hsc cec n Techno and Science Physics of Physics,School of Liu Department Jueping and Chen, to Junlong Wang, Feng interference instanton-gluon function from correlation correction on based wl eisre yteeditor) the by inserted be (will No. manuscript EPJ a n fteosalsi hoeia eerhso glue- of researches theoretical in obstacles the of One (QCD) chromodynamics quantum in issue significant A uhrt hmalcrepnecssol eaddressed. be should correspondences all whom to Author r band n ge ihtepeoeooia analysis PACS. phenomenological the with agree and obtained, are as ea it n opigcntnsfrte0 the for r constants coupling sum and unsubtracted width fun and decay subtracted spectral mass, the the for between behavior consistency threshold A suitable approxima of a zero-width with positivity usual exc form the the is of restore Instead condensates counting. vacuum to double from role contribution great is negligible a contribution The interference has The and one. divergence, together perturbative function correlation the the and between in interference included the and the culated, from chromodynamic arising quantum contribution for the expansion semiclassical of work h 0 the Abstract. date version: Revised / date Received: − + ASky1.5H,1.5K,1.8L,12.39.Mk 12.38.Lg, 11.15.Kc, 11.55.Hx, PACS-key suoclrgubl sivsiae nafml ffinite- of family a in investigated is glueball pseudoscalar ae ncreto rmisatngunitreec oc to interference instanton-gluon from correction on Based a − oyWhnUniversity,430072,Wuhan,Chinalogy,Wuhan + eoac nwihtegubl rcini dominant is fraction glueball the which in resonance nssfrtevoaino both of re- violation (for mech- the interaction provide for instantons strong particular, the anisms In of [16,18]). physics impor- see an the views play to in believed role widely tant are QCD, in fields gluon [12,17]. complicated more making glueball numbers, the quantum of same identification the the of mesons usual with h C auma u fidpnetinstantons independent of sum ¯ a radius as with vacuum QCD the discovered [19]. recently RHIC plasma at called so quark-gluon the coupled of properties strongly unusual the underlie may and gluons quarks transition, between interactions deconfinement instanton-induced that the so through persist instantons iighdo assadi h eouino h famous the of deter- resolution in the important in U and be masses therefore hadron may mining and QCD, in metry h hrlsmer SC)i h xa rpe channel triplet axial the in of (SBCS) breaking symmetry the is chiral probably model the are The predictions the success. important phenomenological is far most its model So by liquid investigated. justified essentially instanton intensively gas the being diluted of the still correctness in density The instanton model. infinite an by caused (200MeV) (1) ihpr-lsia otiuinfo instantons from contribution pure-classical with ntnos stesrn oooia utain of fluctuations topological strong the as Instantons, nteisatnlqi oe,anro es describes sense narrow a model, liquid instanton the In ntnosadteqatmgunfilsi cal- is fields gluon quantum the and instantons infrtersnne,teuulBreit-Wigner usual the resonances, the for tion to ftefiiewdhrsnne sadopted. is resonances finite-width the of ction A . undt egueivrat reo infrared of free gauge-invariant, be to turned ue norcreainfnto oaodthe avoid to function correlation our in luded 0 QD nteisatnlqi background, liquid instanton the in (QCD) s h pcr ftefl orlto function. correlation full the of spectra the rbe.Frhroe twsrcnl hw that shown recently was it Furthermore, problem. lsi eywl utfid h auso the of values The justified. well very is ules − 4 + it asinsmrls nteframe- the In rules. sum Gaussian width 2] hsmdlaodteifae problem infrared the avoid model This [20]. ρ reainfnto,tepoete of properties the function, orrelation suoclrglueball pseudoscalar (600MeV) = − 1 n ffciedniy¯ density effective and U (1) A n hrlsym- chiral and n = 2 Feng Wang et al.: Title Suppressed Due to Excessive Length

[21,22] and the absence of Goldstone bosons in the axial tion is taken into account[40,7]. However, as comparing singlet channel. with the interference contribution, which we have recalcu- The non-perturbative effects of QCD is commonly casted lated in this paper, and the pure perturbative one in the into the description of non-perturbative vacuum, such as considered energy region, the topological screening effect quark and gluon condensates and the instanton configu- turns out to be negligible; and the pathology of positiv- rations. When contributions from both condensates and ity violation disappears when including the interference the instanton are included into the correlation function contribution (s. below). together, it leads to a double counting in a sense that Phenomenologically, the identification of the pseudoscalar they can be two alternative ways for parametrizing the glueball has been a matter of debate since the Mark II ex- nonperturbative vacuum [23,24]. Moreover, the instanton periment proposed glueball candidates [41]. Later, in the distribution is closely connected with the vacuum conden- mass region of the first radial excitation of the η and η′ sates since the mean size and density of instantons can be mesons, a supernumerous candidate, the η(1405) has been deduced from the quark and gluon condensates, and con- observed. It turns out to be clear that η(1405) is allowed versely, the values of condensates can be reproduced from as glueball dominated state mixed with isoscalar qq¯ states instanton distribution [25,26,27,28]. The contributions of due to its behavior in production and decays, namely, it instanton and those of the condensates should be equiv- has comparably large branching ratios in the J/ψ radiative alent from each other in some case, and thus including decay, but not been observed in γγ collisions[42,13,43]. A both contributions at the same time will cause the dou- review on the experimental status of the η(1405) is given in ble counting problem. To avoid it, many authors invoke Ref. [13]. However, this state lies considerably lower than some special techniques in dealing with the two contribu- the theoretical expectations: the lattice QCD predictions tions [29,30,31,32]. This issue is, however, really not set- suggest a glueball around 2.5GeV [44,45]; the mass scale tled. Fortunately, the contribution of condensates to the of the pseudoscalar glueball obtained in the QCD sum rule correlation function in the glueball channels is unusually approach is above 2GeV [37,38,40,7]. On the other hand, weak as demonstrated in this work and by many other there are attractive arguments for the approximately de- authors[33,34]. Therefore, it is assumed that the conden- generate in mass for the scalar and pseudoscalar glueballs sate contributions can be understood as a small fraction [46], and even the scenario that a pseudoscalar glueball of the corresponding instanton one in the local limit. To may be lower in mass than the scalar one is recently dis- prevent thoroughly from the problem of double counting, cussed in Ref. [47]. The possibly non-vanishing gluonium we choose to work in the instanton vacuum model of QCD, content of the ground state η and η′ mesons is discussed and carry out a semi-classical expansion in instanton back- in [48,49,50,12]. Up to now, only a topological model of ground fields as suggested in our previous works to ana- the glueball as closed flux tube [46] predicts degeneracy of ++ ++ + lyze the properties of the lowest 0 scalar glueball [35, the 0 and 0− glueball masses and admits the region + 36] and 0− pseudoscalar one [37,38], where the correla- 1.3-1.5GeV. tion function of the glueball currents are calculated by just including the contributions from the pure instantons, the To face a more realistic phenomenological situation, pure quantum gluons, and the interference between both, we now reexamine the correlation function in the instan- instead of working with both instantons and condensates ton liquid vacuum, and include all the resonances below at the same time. and near the η(1405) into the finite-width spectral function, and then achieve a series of results in traditional Zhang and Steele [39] have reported a disparity be- Gaussian sum rule analyses which are consistent with the tween their Laplace sum rule and Gaussian sum rule for phenomenology. On the other hand, as a crosscheck, these the pseudoscalar glueball. In this reference, the optimized results are almost same with the Laplace ones (paper is parameters of pseudoscalar glueball have been obtained submitted), because both Laplace and Gaussian sum rules from the Gaussian sum rule, while it fails to achieve a sat- are derived from the same underlying dynamical theory. isfying Laplace sum rule. This result seems strange, since The paper is organized as follows: In the second section, both Laplace and Gaussian sum rules are derived from the we present systematically the calculation of the contri- same underlying dynamical theory, and should, at least, bution to the correlation function due to the interference be approximately consistent. It reflects the inconsistency between instantons and quantum gluons. It is this inter- for including both condensate and instanton contributions ference between instantons and quantum gluons to serve in the correlation function and disregarding the important to be a mechanism to keep the positivity of the spectral contribution from the interaction between instantons and function in contrast with the so-called topological charge their quantum counterparts at the same time. screening effect stressed in Ref.[7]. The effect of the contri- + An other serious problem in the 0− glueball sum bution of topological charge screening is found to be negli- rule approach is that the fundamental spectral positivity gible as comparing with the interference one in the consid- bound is violated when including the strong repulsive pure ered energy region. The spectral function is constructed instanton contribution, and as a consequence, the signal in the similar way as in case of the 0++ glueball[36,51] for the pseudoscalar glueball disappears[39]. To cure this with a suitable threshold behavior in the third section. pathology of positivity violation, the topological charge In the fourth section, a family of Gaussian sum rules are screening effect in the QCD vacuum is added to the cor- constructed. The numerical simulations are carried out in relation function, and a suitable instanton size distribu- the fifth section, and the results are consistent with vari- Feng Wang et al.: Title Suppressed Due to Excessive Length 3 ous Gaussian sum rules and in accordance with the phe- Consequently, the pure-glue Euclidean action can be ex- nomenology. Finally, the main conclusions are given, and pressed as a discussion of some interesting issues is open. 4 1 a ab bc c S[B]= S0 d x L[A + a]+ a D D a − Z 2ξ µ µ ν ν

1 4 a ab bc abc b = S0 d x a D D δµν +2gf F 2 Correlation function − 2 Z µ λ λ µν 1 1 DabDbc ac 2gf a a D a − − ξ µ ν ν − abc µb νc µ,ad νd We are working in Euclidean QCD. The pseudoscalar 1 glueball current is defined as g2f a a f a a (6) − 2 abc µb νc ade µd νe a Op(x)= αsG (x)Gµν,a(x) (1) 2 2 µν where S0 = 8π /g is the one-instanton contribution to e the action, Fµνa is the instanton field strength tensor a where αs is the strong coupling constant, Gµν (x) is the F (A)= ∂ A ∂ A + g f A A , (7) gluon field strength tensor with the color index a and µν,a µ ν,a − ν µ,a s abc µ,b ν,c Lorentz indices µ and ν, and ab and Dµ (A) the covariant derivative associated with the a 1 classical instanton field Aµ G (x)= ǫ Ga (x) (2) µν,a 2 µνρσ ρσ Dab(A)= ∂ δ + gf Ac (8) e µ µ ab acb µ a is the dual of Gµν (x). The current Op(x) is a Lorentz- In addition, the background field gauge irreducible, gauge-invariant and local composite operator µ with the lowest dimension, and renormalization group in- D (A)aµ = 0 (9) variant at least to the leading order αs in the quarkless world. It is noticed that the current Op(x) is anti- is used with ξ being the corresponding gauge parameter, hermitian due to the involving of the imaginary time, and certainly, the corresponding Faddeev-Popov ghosts while its analytic continuation to the Minkowskian space- according to the standard rule should be added to restore time is hermitian. The QCD correlation function is defined the unitarity. We note here that the structure constants as fabc should be understood as ǫabc when any one of the color-indices a,b and c is associated with an instanton field 2 4 iq x Π(q )= d xe · Ω T Op(x)O† (0) Ω . (3) due to the property of the closure of any group. h | { p }| i Z According to the decomposition (5), the correlation function Π splits into three parts, namely the pure classi- where Op† is the hermitian conjugation of Op. The advan- cal part, the pure quantum part and the interference part tages to use the hermitian conjugation in definition are: in the leading order first, the spectral functions both in Euclidean space-time and in its analytic continuation into Minkowskian space- ΠQCD(Q2)= Π(cl)(Q2)+ Π(qu)(Q2)+ Π(int)(Q2). (10) time are, in principle, positively definite; and second, the relationship between the correlation functions both in Eu- where the superscript indicates that it is calculated in the clidean and Minkowskian formulations becomes very sim- underlying dynamical theory, QCD. It is important to note ple as that every part in rhs of (10) is gauge-invariant because the decomposition (5), in principle, has no impact on the 2 2 2 2 ΠE(Q = q ) ΠM (Q = q ), (4) gauge-invariance of the correlation function. The pure in- ↔ − staton contribution Π(cl)(Q2) and the perturbative contri- (qu) 2 because the overall minus sign arising from the analytic bution Π (Q ) up to three-loop level in the chiral limit µνρσ of QCD are shown in Eqs. (80) and (81) respectively in continuation due to (ǫµνρσ ǫµνρσ)E = (ǫµνρσǫ )M is just canceled with another minus sign− arising from the Appendix A. The contribution of the topological charge mentioned different hermiticity of the pseudoscalar glue- screening which is not included in rhs of (10) is given in 2 Appendix B. ball current. Therefore, the expressions for ΠE(Q ) and Π (Q2) are, in fact, the same function of Q2. One of our main tasks in this work is to calculate the M contribution Π(int)(Q2) in (10), which is arising from the In the framework of semiclassical expansion, the glue interference between the classical instantons and quantum potential field B(x) can be decomposed into a summation gluons in the framework of the semi-classical expansion of the classical instanton A and the corresponding quan- for QCD with the instanton background. After imposing tum gluon field a as the background covariant Feynman gauge (ξ = 1) for the quantum gluon fields, we are still free to choose a gauge Bµ(x)= Aµ(x)+ aµ(x), (5) for the background field A. In the following, the singular 4 Feng Wang et al.: Title Suppressed Due to Excessive Length gauge is chosen to the non-perturbative instanton field with P ab = iDab. Keeping only terms proportional to µ − µ configurations as F , one has[56] 2 Aµa(x)= ηaµν (x z)ν φ(x z), (11) 4 iq x ab iq (y z) 1 2 g − − d xe · µν (x, 0) =e · − δµν + gs Fµν (z) s Z D q2 q4 with (y z) F (z)q ρ2 ig − ρ ρσ σ δ (z)+ φ(x z)= , (12) − s q4 µν ··· − (x z)2[(x z)2 + ρ2] − − (17) and the corresponding field strength tensor is where the first term in rhs of the above equation is the 8 (x z) (x z) 1 F (x) = − µ − ρ δ pure-gluon propagator in the usual Feynman gauge, and µν,a −g (x z)2 − 4 µρ s − the second and third ones are the leading contribution ρ2 of the instanton field to the gluon propagator. For short η (µ ν), (13) × aνρ ((x z)2 + ρ2)2 − ↔ distance region, we assume that the contribution from a − single instanton is dominant over multi-instantons[57]. At with z and ρ denote respectively the center and size of the leading loop level, the gluon propagator (17) becomes the instanton, called collective coordinates together with the pure-gluon one. the color orientation, and ηaµν is the ’t Hooft symbol In calculation, we expand the current Op into terms which should be replaced with the anti-’t Hooft oneη ¯aµν which are the products of quantum gluon fields and their for an anti-instanton field. For the sake of simplicity, in derivatives with coefficients being composed of the instan- the practice the most used is the spike size distribution ton fields n(ρ)=nδ ¯ (ρ ρ¯), wheren ¯ is the overall instanton den- − 10 sity andρ ¯ is the average instanton size. The fact that 1 O (x)= ǫ α O (x) (18) the strong coupling constant gs emerging in the denom- P 2 µνρσ s i inator of the rhs of (11) reveals the nonperturbative na- Xi=1 ture of these classical configurations. In fact, instantons where the operators Oi in terms of instanton and quan- play quite important role in QCD sum rule. Early QCD tum gluon fields are listed in Appendix C. Eq.(15) can be sum rules neglecting instanton-induced continuum contri- rewritten as butions didn’t obtain reliable results in many cases, but they are then solved by including such instanton-induced (int) 2 1 2 Π (q )= α nǫ¯ ǫ ′ ′ ′ ′ effects [30,9]. − 2 s µνρσ µ ν ρ σ Before starting with the contraction between the quan- 4 4 iq x d z d xe · Ω T Oi(x)Oj (0) Ω tum fields, we note that the time-development of the in- × Z Z h | { }| i stanton vacuum produces the pre-exponential factor for Xi,j the distribution of the instantons[52,15,53], and Π(int) is 12 (int) 2 understood as taking ensemble average over the collective = Π (q )+ (19) i ··· coordinates besides taking the usual vacuum expectation Xi=1 value due to the separation (5) where the denotes the contributions from the products (int) ··· 3 Π (x)= of operators being proportional to gs , and the expressions of Π(int)(q2) in terms of O are shown in Appendix D. 4 (int) i i dρn(ρ) d z Ω T Op(x)Op† (0) Ω , (14) The corresponding twelve kinds of Feynman diagrams as Z Z h | { } | i XI,I¯ shown in the FIG. 1, where the contributions from the first three diagrams are of the order of α , and the contri- where the super index ’(int)’ indicates the corresponding s butions of the remainders are superficially of the order of quantity containing only the interference part between the α2, and those from the diagrams (4),(5) and (6), in fact, quantum and classical ones. Using the spike distribution, s are vanishing because of violating the conservation of the (14) becomes color-charge, namely (int) 4 (int) Π (x)=2¯n d z Ω T Op(x)Op† (0) Ω , (15) (int) 2 Z h | { } | i Πi (q )=0, for i =4, 5, 6 (20) where the factor 2 comes from the mutually equal contri- Now, we are in the position to evaluate the contribu- butions of both instanton and anti-instanton. Next impor- tions of the remainder diagrams in FIG. 1. Using the stan- tant step is to specify the form of the gluon propagator dard technique to regularizing the ultraviolet divergence which in the background field Feynman gauge can be read in the modified minimal subtraction scheme, the result for from the part of S[B] quadratic in a[54,55] the interference part of the correlation function is ab (x, y)= Ω T aa (x)ab (y) Ω int 2 2 2 Dµν h | { µ ν }| i Π (Q )= c0αsnπ¯ + αsn¯ c1 + c2(Qρ)− ab 2 1 2 2 Q = x y (16) + c (Qρ) + c + c (Qρ)− ln , (21) h | P 2δ 2F | i 3 4 5 µ2 µν − µν Feng Wang et al.: Title Suppressed Due to Excessive Length 5

the form

QCD 2 5 2 4 2 Π (Q )= 2 π ny¯ K2 (y) − 2 + αsn¯ c0π + c1 + c2y− 2 2 2 Q + c y + c + c y− ln 3 4 5 µ2 2 2 2 αs 4 Q Q + Q ln 2 a0 + a1 ln 2 π µ µ Q2 + a ln2 . (23) 2 µ2

with y = Qρ¯. Before going on, let us compare the roles of the various parts of the contributions in correlation function. The imaginary part of the correlation function (23) can be worked out to be

1 ImΠQCD(s) =16π3s2n¯ρ¯4J (¯ρ√s)Y (¯ρ√s) π 2 2 2 2 2 1 +α n¯ c ρ¯ s c + c (¯ρ s)− s 3 − 4 5 2 2 αs s s a0 +2a1 ln 2 − π µ s + 3 ln2 π2 a . (24) µ2 − 2

where the pure classical contribution (the first term on rhs of (24)) is most dominant, and the contribution of the interference terms (the second term on rhs) is of the second place, the pure perturbative contribution simply plays Fig. 1. Feynman diagrams for the interference contribution the role of the third place, as shown in FIG. 2 where the (int) 2 2 Π (Q ) up to order αs, where spiral lines, dotted lines and imaginary part of the correlation function is multiplied the lines with circles denote gluons, instantons and the instan- with a weight function exp ( (s sˆ)/4τ) as required by ton field strength tenser respectively, and cross stands for the the Gaussian sum rules, and in− accordance− with the spirit position of instantons. of the semiclassical expansion. All these three contributions are positively definite as expected. We note that the contribution from the topological charge screening, from where we have ignored terms being proportional to the 2 (84), is displayed in FIG. 2 as well, and its role is almost positive powers of q which vanish after Borel transforma- insignificant. Moreover, it is easy to see from FIG. 2 that tion, and the dimensionless coefficients ci are numerically the imaginary part of the correlation function is already determined to be positive from s =0.5GeV2 to s = 10GeV2 without including the contribution of the topological charge screening, c = 118.23, c = 3700.59α , c = 2394.47α , 0 − 1 − s 2 − s and the so-called positivity problem is no longer there. c3 = 11561.90αs, c4 = 1850.30αs, c5 = 1197.24αs. (22) Therefore, the interference contribution to the correlation function is significant important not only in its magnitude through a tedious calculation. It should be noted that but also in restoring the positivity to the spectral function. there is no infrared divergence as expected by the instan- We note here that the so-called condensate contribu- ton size being fixed in the liquid instanton vacuum model. tion to the correlation function of the pseudoscalar glue- Comparing Eq.(21) with our previous result [37,38], they ball current is proven to be very small in comparing with differ not only in some coefficients but also in the loga- the one of (23), as shown in Appendix E, where the com- rithms structures due to the fact that the newly improved parison between the real and imaginary parts of the cor- calculation is free of infrared divergence while our old one relation function and condensate contribution to it are were not, and it would need a corresponding cutoff to reg- made, and shown in FIG. 8 and 9. For the reasons given ularize the integral in the infrared limit. above, the contributions from the topological charge screen- Putting everything above together, our final correla- ing effect and the usual condensates are omitted in our tion function for the pseudoscalar glueball current is of sum rule analysis. 6 Feng Wang et al.: Title Suppressed Due to Excessive Length

20 gluon world, is a linear combination of three operators Im Πinstanton(s)/π ˆ8 ˆ0 ˆ Im Πperturbation(s)/π J5 , J5 and Op after renormalization. Therefore, one as- 15 − − 2 (s sˆ) /4τ× interference e Im Π (s)/π sumes that there may be some isosinglet quark-antiquark ] 4 Πtop π 10 Im (s)/ pseudoscalar states mixed with the pseudoscalar glueball Im ΠQCD(s)/π (without ImΠtop(s)/π ) )[GeV s 5 ground state G. ImΠ( 1 π Now we construct the spectral function for the corre- × 0 τ 4

/ lation function of the pseudoscalar glueball current. The 2 ) 4 ˆ s

− τ = 1GeV s ( −5 2 2 usual lowest one resonance plus a continuum model is used − s .

e ˆ = 1 405 GeV to saturate the phenomenological spectral function: −10

−15 1 PHEN HAD 1 QCD 0 1 2 3 4 5 6 7 8 9 10 ImΠ (s)= ρ (s) θ(s s0) ImΠ (s), (33) s [Gev2] π − − π Fig. 2. The contributions to the imaginary part of the corre- where s0 is the QCD-hadron duality threshold, θ(s s0) lation function from the pure instanton (cross line), interfer- the step function and ρHAD(s) the spectral function− for ence (dashed-dotted line), pure perturbative (dotted line) and the lowest pseudoscalar glueball state. In the usual zero- topological charge screening (dashed line) and the total con- width approximation, the spectral function for a single tribution without the topological charge screening one (solid line) versus s. resonance is assumed to be ρHAD(s)= F 2δ(s m2), (34) 3 Spectral function − where m is the mass of the lowest glueball, and F is the In the isosinglet channel there are five gauge-invariant coupling constant of the current to the glueball defined as + composite operators with the quantum numbers of 0− 0 O (0) G = F. (35) which are bilinear in the fundamental quark, antiquark h | p | i and gluon fields, namely the pseudoscalar quark densities, the divergences of the axial quark currents and the gluon The threshold behavior for ρHAD(s) is known to be anomaly: ˆ8,0 8,0 ρHAD(s) λ s, for s 0 (36) J5 = iqγ¯ 5(λ /2)q, (25) → 0 → ˆ8,0 8,0 ∂µJµ5 = ∂µ[¯qγµγ5(λ /2)q], (26) from the low-energy theorem in the world of no light quark flavors [58] or the one in the world with three light flavors ˆ a a Op = αsGµν Gµν , (27) and m m [59]. In fact, the threshold behavior (36) is u,d ≪ s where λ8 is the flavor Gell-Mann matrix,e and λ0 = 2/3I only proven to be valid near by the chiral limit; it may not be extrapolated far away. Therefore, instead of considering with I being the 3 3 flavor unit matrix. Only threep of these operators are independent× due to the two renormalization-the coupling F as a constant [7], we choose a model for F as invariant axial Ward identities 2 λ0s, for s

where mi and Γi being the mass and width of the i-th for the phenomenological contributions to the sum rules, resonance, respectively. For the sake of simplicity, all cou- and 2 pling constants Fi for s

4 Finite width Gaussian sum rules + Substituting the correlation function (23) of 0− pseudoscalar glueball into (47), one can derive the Gaussian In this section, we construct the appropriate sum rules + sum rules of k = 1, 0 and +1 of 0− pseudoscalar glueball current which has the form − due to dispersion relation

∞ 1 1 QCD 2 QCD ˆ 3 4 Π (Q )= ds 2 ImΠ(s). (40) 1 (ˆs, τ)= 16π n¯ρ¯ J2 ρ¯√s Y2 ρ¯√s s Z0 s + Q π G− I· 1 sˆ2 +nπα ¯ c exp where ImΠ(s) could be simulated by the phenomenolog- s 0 √ −4τ PHEN 4πτ ical one ImΠ (s) whithin an assumed model of the 2 spectral function in a spirit of sum rule approach. Using 2 1 sˆ +nα ¯ s (c1 + c3γ)exp the Borel transformation [61] √4πτ − −4τ 2 2 sˆ /8τ + c2ρ¯ √2τe− D 1( s/ˆ √2τ) N N − − ˆ ( 1) 4 N d 2 lim − (Q ) (41) 2 2 sˆ sˆ N 4 B≡ 4 (N 1)! dQ (c4/ρ¯ c5(γ 1)/ρ¯ ) exp Q →∞ 4 − − − − 2τ −4τ →∞ Q /N 4τ ≡ 1 + D 2( s/ˆ √2τ)[a0 (2γ 2)a1 to both sides of (40), a family of Gaussian sum rules can √4πτ − − − − be formed to be [61] 2 2 2 sˆ /8τ +0.5(6γ 12γ π )a ]2τe− , (48) − − 2 HAD(s , s,ˆ τ)= QCD(s , s,ˆ τ) Gk 0 Gk 0 1 sˆ2 + exp[ ]Π(0)δk, 1, (42) √4πτ −4τ − where s0 is the continuum threshold which hadronic physics QCD ˆ 3 4 2 0 (ˆs, τ)= 16π n¯ρ¯ J2 ρ¯√s Y2 ρ¯√s s is (locally) dual to QCD above it. G I· 2 2 1 2 sˆ /8τ +nα ¯ s c2ρ¯ 2τe− D 2( s/ˆ √2τ) 2 mumd √4πτ n − − Π(0) = (8π) qq¯ (43) 2 mu + md h i sˆ /8τ c3√2τe− D 1( s/ˆ √2τ) − − − comes from the low-energy theorem for QCD with three sˆ2 + (c /ρ¯2 c γ/ρ¯2)exp light ﬂavors[59], and 4 − 5 −4τ

s0 2 HAD 1 HAD 1 k (s sˆ) ρ (s) + D 3( s/ˆ √2τ)[2a0 + (6 4γ)a1 k (s0, s,ˆ τ)= dss exp[ − ] , √4πτ − − − G √4πτ Z0 − 4τ π 2 2 2 3/2 sˆ /8τ (44) + (6γ 18γ π + 6)a ](2τ) e− ,(49) − − 2 8 Feng Wang et al.: Title Suppressed Due to Excessive Length

and the values of the average instanton size and the overall instanton density are adopted from the instanton liquid QCD(ˆs, τ)= ˆ 16π3n¯ρ¯4J ρ¯√s Y ρ¯√s s3 G1 I· 2 2 model[25] 2 2 nα¯ s 2 3/2 sˆ /8τ + 2c2ρ¯ (2τ) e− D 3( s/ˆ √2τ) 4 4 n = 1fm− =0.0016GeV , √4πτ n − − 2 sˆ /8τ 1 1 c32τe− D 2( s/ˆ √2τ) ρ = fm 1.667GeV− . (57) − − 2 − 3 ≃ 2 sˆ /8τ + c5ρ¯− √2τe− D 1( s/ˆ √2τ) − − o The resonance parameters in Eq.(39) could be esti- 1 + D 4( s/ˆ √2τ)[6a0 + (22 12γ)a1 mated by matching both sides of sum rules Eq.(42) op- √4πτ − − − timally in the fiducial domain. In doing so, the parame- 2 2 2 2 sˆ /8τ ters ˆ and the threshold s0 should be determined priority. + (18γ 66γ 3π + 36)a2]4τ e− , (50) − − Firstly, it is obvious that s0 must be greater than the mass where square of the highest lying isolated resonance considered, 2 namely 1 ∞ (s sˆ) ˆ = ds exp − , (51) 2 I √4πτ Z0 − 4τ s m (58) 0 ≥ max and parabolic cylinder function D d 1(ˆs√τ) defined as − − in our multi-resonance assumption (or just the resonance 2 2 2 d z /2 y mass itself in the case of a single resonance assumption), d z /4 d e ∞ dye− √2( 1) e− z/√2 and should guarantee that there is a sum rule window D d 1(z)= − R , − − d! (dz)d for Gaussian sum rules. Secondly, the peak positions of QCD(s , s,ˆ τ) versuss ˆ curves should not change too much d 0. Gk 0 ≥(52) with moderate variation of s0. Here we do not mention about the values of τ in this condition, because the peak positions of these curves is not affected by appropriate val- 5 Numerical simulation ues of τ. It is found that the behavior of these curves can satisfy above requirements as shown in FIG. 3 if s0 lies The expressions for the three-loop running coupling in the interval of (4GeV2, 5GeV2). It is also remarkable 2 constant αs(Q ) with three massless flavors (Nf = 3) at that the peak positions has already indicated the approx- renormalization scale µ [62] imate mass of the hadron considered. Thus, we would ex- pect that the mass of the pseudoscalar glueball should be 2 (2) 2 αs(µ ) αs (µ ) 1 β1 2 β2 nearby the value √sˆ 1.449 GeV. In another way, if one = + 3 L1( ) + (53) QCD≃ π π (β0L) β0 β0 uses the curves of (s , s,ˆ τ) versus τ with fixeds ˆ and Gk 0 s0 to obtain the physical parameters through (43), then (2) 2 are used, where αs (µ )/π is the two-loop running cou- sˆ should be set approximately to bes ˆpeak of the curves pling constant with (Nf = 0) of QCD(s , s,ˆ τ) versus τ, so as to highlight the underly- Gk 0 (2) 2 ing hadron state in consideration and suppress the con- αs (µ ) 1 β1 ln L tributions from other states. Besides, it needs a sum rule = 2 (54) π β0L − β0 (β0L) and 4 s =4 GeV2 2 0 µ 2 τ = 1GeV4 3.5 s =4.5 GeV2 L = ln 2 ,L1 = ln L ln L 1, 0 Λ − − s =5 GeV2 3 0 ] 4 1 2 2.5 β0 = 11 Nf , 4 − 3 ) [GeV ˆ s, τ 2 , 0 s ( 1 38 1.5 QCD β1 = 102 Nf , 0 42 3 G − 1

1 2857 5033 325 0.5 β2 = 3 Nf + Nf , (55) 4 2 − 18 54 0 −3 −2 −1 0 1 2 3 4 5 6 sˆ [GeV2] with the color number Nc = 3 and the QCD renormal- 2 QCD ization invariant scale Λ = 120MeV. We take µ = √τ Fig. 3. The curves of 0 (s0, s,ˆ τ) versuss ˆ with diﬀerent s0 after calculating Borel transforms based on the renormal- at τ = 1 GeV4. G ization group improvement for Gaussian sum rules [63]. The subtraction constant Π(0) has been ﬁxed as [7] window which the hadron physical properties should be Π(0) 0.022GeV4, (56) stable in this region. For the upper limit τ of the sum ≃− max Feng Wang et al.: Title Suppressed Due to Excessive Length 9

+ rule window, the resonance contribution should be great The mass values of 0− pseudoscalar glueball in (64) are than the continuum one reasonably consist with the one obtained in Ref. [7] by adding the topological charge screening effect and per- QCD(s , s,ˆ τ ) CONT(s , s,ˆ τ ) (59) Gk 0 max ≥ Gk 0 max forming the so-called Gaussian-tail distribution for instanton size. However, after performing the Gaussian-tail dis- according to the standard requirement due to the fact that tribution for instanton size, the mass of the 0++ glueball is in the energy region above τ the perturbative contri- max lower to be around 1.25GeV [7] in contradiction with the bution is dominant. At τ , which lies in the low-energy min lattice simulation [64,45] and phenomenology [65]. The region, we require that the single instanton contribution mass scales in (64) locate at the strong repulsive potential should be relatively large so that region of energy (below 3.9GeV2) where the bound state inst(s , s,ˆ τ ) of glueball cannot form when working in the spike distri- Gk 0 min 50% (60) QCD ≥ bution which is motivated from the liquid instanton model k (s0, s,ˆ τmin) G of QCD vacuum in the large Nc limit. In fact, the funda- In the same time, to require that the multi-instanton cor- mental spectral positivity bound can be traced back to rections remain negligible, we simply adopt a rough esti- the definition of the correlation function (3). The spectral mate function should be positive even before taking the average 4 with any specific instanton-size distribution. It is difficult 4 2 τ 2¯ρ− . (61) min ≥ ≃ 0.6GeV for us to understand that there is no artificial in chang- ing the positivity behavior of the spectral function just by According to the above requirements, we find that in the performing the average with the Gaussian-tail distribution domain 4 for the instanton size. τ (0.5, 4.5)GeV , (62) QCD 2 ∈ From now on, the full correlation function Π (Q ) our sum rules work very well. Finally, in order to measure including the interference contribution Πint(Q2), (23), is the compatibility between both sides of the sum rules (42) used in our analysis of the Gaussian sum rules (42). In the in our numerical simulation, we divide the sum rule win- case of single-resonance plus continuum models, specified dow [τmin, τmax] into N = 100 segments of equal width, respectively by (34) and (38), for the spectral function, [τi, τi+1], with τ0 = τmin and τN = τmax, and introduce a the optimal parameters governing the sum rules with zero variation δ (called the matching measure) which is defined (the case B) and finite (the case C) widths are listed from as the fourth to ninth line of Tab. 1 and the corresponding N 2 curves for the lhs and rhs of (42) with k = 1, 0 and 1 [L(τi) R(τi)] δ = − , (63) +1 are displayed in FIG. 5 and 6 respectively. From− Tab. N L(τ )R(τ ) Xi=1 | i i | 1, the optical values of the pseudoscalar glueball mass, where L(τi) and R(τi) are lhs and rhs of (42) evaluated width, coupling and the duality threshold with the best at τi. matching are: Let us first consider the case of single-resonance plus continuum model (34) of the spectral function by exclud- m =1.644 0.194GeV, f =1.412 0.129GeV, ± 2 ± ing the interference contribution Πint(Q2) from ΠQCD(Q2) s0 =4.77 0.74GeV , (66) ± (the case A), in order to recover the results before. In this case, the imaginary part of ΠQCD(Q2), however, be- for one zero-width resonance model, and 2 comes negative for s below 3.9GeV , and the full interac- m =1.407 0.162 GeV, Γ =0.053 0.018 GeV tion plays a role of repulsive potential in the pseudoscalar ± ± f =1.687 0.145 GeV, s =4.63 0.62 GeV2, (67) channel. This is the reason why the authors in Ref. [39] ± 0 ± cannot find the signal for the pseudoscalar glueball. When for one finite-width resonance model. It is shown in FIG. we chose to work in the positively definite region, say 5 that the topological charge screening effect has little s (4, 10)GeV2, The mass of the 0 + glueball can be − impact on Gaussian sum rules indeed. worked∈ out by using the family of Gaussian sum rules In the numerical simulation for the case of the five (42). The fitting parameters are listed in the first three finite-width resonances plus continuum model (39) for the lines of Tab. 1 and the corresponding matching curves for spectral function (the case D), we just choose the data in k = 1, 0 and +1 are displayed in FIG. 4, respectively. PDG as the fitting parameters for masses and width of The− optical values of mass, coupling constant and s of 0 the resonances η(548), η(985), η(1295) and η(1475), and 0 + pseudoscalar glueball are − the result of the single resonance model (the case C) as m =2.010 0.299GeV, f =0.584 0.043GeV, the fitting parameters for η(1405); while the couplings of ± ± the five resonances to the current are chosen to be ap- s =5.21 0.43GeV2, (64) 0 ± proximately the same as that for η(1405) determined in where the errors are estimated from the uncertainties of case C because the pseudoscalar qurkonia can be directly the spread between the individual sum rules, and by vary- coupled to the gluon anomaly, and as a consequence, all ing value of Λ in the region of (the same for hereafter) five resonances should be coupled to the current with the strengths of almost the same magnitude of degree; finally, Λ = 120 200MeV. (65) the optimal parameters are determined by adjusting the ∼ 10 Feng Wang et al.: Title Suppressed Due to Excessive Length

chosen parameters so that the matching measure δ for 0.5 QCD k=−1 HAD both sides of the Gaussian sum rules (42) is minimal. The 0.45 optimal parameters governing the sum rules are listed in 0.4

the remaining lines of Tab. 1. The corresponding curves ] 4 for the lhs and rhs of (42) with k = 1, 0 and +1 are 0.35 − 0.3 sˆ = 2.0502GeV2 displayed in the FIG. 7. Taking the average, the optical 2 s0 = 5.26GeV + 0.25 m = 1.950GeV values of the widths of the ﬁve lowest 0− resonances in QCD/HAD [GeV f = 0.561GeV the world of QCD with three massless quarks, and the 0.2 corresponding optical ﬁt parameters are predicted to be 0.15

0.1 0 1 2 3 4 5 6 mη(548) =0.548 0.022 GeV,fη(548) =1.133 0.167 GeV, 4 ± ± τ [GeV ] 6 8 Γη(548) =1.3 10− 3.9 10− GeV (68) 2.2 × ± × QCD k=0 HAD 2

1.8 mη(985) =0.958 0.051 GeV,fη(985) =1.200 0.233 GeV, ] ± ± 4 3 5 1.6 Γη(985) =1.9 10− 5.7 10− GeV (69) × ± × 1.4 sˆ = 2.1002GeV2 2 s0 = 5.18GeV 1.2 m = 2.000GeV QCD/HAD [GeV m =1.295 0.075 GeV,f =1.202 0.112 GeV, f = 0.599GeV η(1295) ± η(1295) ± 1 Γ =0.055 0.018GeV (70) 0.8 η(1295) ±

0 1 2 3 4 5 6 τ [GeV4] mη(1405) =1.405 0.081 GeV,fη(1405) =1.313 0.105 GeV, ± ± 12 QCD Γη(1405) =0.051 0.017GeV (71) ± 11 k=1 HAD 10

] 9 mη(1475) =1.475 0.092 GeV,fη(1475) =1.023 0.097 GeV, 4 ± ± 8 Γη(1475) =0.085 0.028GeV (72) sˆ = 2.1102GeV2 7 2 ± s0 = 5.20GeV m = 2.080GeV with QCD/HAD [GeV 6 f = 0.592GeV 2 5 s0 =4.78 0.64 GeV . (73) ± 4

The FIG. 6 and 7 show the satisfactory compatibility be- 3 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 tween both sides of the sum rules over the whole ﬁducial τ [GeV4] region. These results are in good accordance with the experimental discovered resonance [14] Fig. 4. The lhs (dashed line) and rhs (solid line) of the sum rules (42) with k = 1, 0, 1 versus τ in the case where the correlation function Π−QCD(Q2) contains only pure perturbative mη(1405) = 1409.8 2.5MeV, Γη(1405) = 51.1 3.4MeV ± ± (74) contribution and pure instanton one, and a single zero-width resonance plus continuum model is adopted for the spectral function.

6 Discussion and Conclusion of semi-classical expansion. The imaginary part of the cor- The instanton-gluon interference and its role in finite- relation function including this interference contribution + width Gaussian sum rules for 0− pseudoscalar glueball turns to be positive without including of the topological are analyzed in this paper. Our main results can be sum- charge screening effect which is proven to be smaller than marized as follows: the perturbative contribution in the fiducial sum rule win- First, the contribution to the correlation function aris- dow, and negligible in comparison with the interference ing from the interference between the classical instanton effect. The so-called problem of positivity violations in fields and the quantum gluon ones is reexamined and de- the imaginary part of the correlation function, stressed rived in the framework of the semi-classical expansion of by H. Forkel[7], disappears. Moreover, it is excluded in the instanton liquid vacuum model of QCD. The resul- the correlation function the traditional condensate con- tant expression is gauge invariant, and free of the infrared tribution to avoid the double counting[7] because conden- divergence, and differs from our previous one not only in sates can be reproduced by the instanton distributions[25, some coefficients but also in the logarithms structures[37, 26,27,28]; Another cause to do so is that the usual con- 38]. Its magnitude is just between the larger contribu- densate contribution is proven to be unusually weak, and tion from pure classical instanton configurations and the cannot fully reflect the nonperturbative nature of the low- smaller one from the pure quantum fields, and plays a lying gluonia[66,7,37,38]; In our opinion, the condensate great role in sum rule analysis in accordance with the spirit contribution can be considered as a small fraction of the Feng Wang et al.: Title Suppressed Due to Excessive Length 11

Table 1. The optimal fitting values of the mass m, width Γ , coupling constant f, continuum threshold s0 ,s ˆ, and matching −+ measure δ for the possible 0 resonances in the sum rule window [τmin,τmax] for the best matching between lhs and rhs of the sum rules (42) with k = 1, 0, 1 are listed, where in case A, the correlation function ΠQCD(Q2) contains only pure perturbative contribution and pure− instanton one, and a single zero-width resonance plus continuum model is adopted for the spectral function, while all the contributions arising from pure instanton, pure perturbative and interference between both are included in the correlation function for cases B, C and D, in which a single zero-width resonance plus continuum model of the spectral function is adopted for case B, and a single finite-width resonances plus continuum model for case C, and the five finite-width resonances plus continuum model for case D, respectively.

2 4 −5 case k resonances √sˆ(GeV) m(GeV) Γ (GeV) f(GeV) s0(GeV ) [τmin,τmax](GeV ) δ/10 1 2.050 1.950 0.050 0 0.561 5.26 0.14 [0.8, 4.0] 22.60 + 1.46 A −0 2.100 2.000 ± 0.035 0 0.599 5.18 ± 0.12 [0.5, 6.0] 1.99 + 0.13 1 2.110 2.080 ± 0.040 0 0.592 5.20 ± 0.10 [0.8, 6.0] 7.56 + 0.45 ± ± 1 1.405 1.682 0.023 0 1.457 5.24 0.15 [1.0, 3.0] 49.72 + 2.98 B −0 1.490 1.620 ± 0.027 0 1.387 4.61 ± 0.09 [0.8, 2.5] 3.67 + 0.18 1 1.380 1.631 ± 0.031 0 1.392 4.45 ± 0.11 [1.4, 3.2] 8.97 + 0.43 ± ± 1 1.405 1.405 0.024 0.05 1.630 5.25 0.14 [0.8, 4.0] 2.44 + 0.12 C −0 1.350 1.400 ± 0.025 0.08 1.671 4.45 ± 0.12 [0.5, 4.3] 4.47 + 0.21 1 1.380 1.416 ± 0.024 0.03 1.760 4.19 ± 0.11 [0.5, 4.5] 2.19 + 0.11 ± ± η(548) 0.548 0.008 1.3 10−6 1.100 η(958) 0.958 ± 0.014 1.9 × 10−3 1.100 1 η(1295) 1.405 1.295 ± 0.020 0.055× 1.200 5.25 0.12 [0.5, 4.5] 4.88 + 0.23 − η(1405) 1.405 ± 0.021 0.051 1.330 ± η(1475) 1.475 ± 0.023 0.085 1.010 η(548) 0.548 ± 0.009 1.3 10−6 1.200 ± × −3 η(958) 0.958 0.016 1.9 10 1.300 0 η(1295) 1.500 1.295 ± 0.021 0.055× 1.195 4.79 0.11 [0.5, 4.3] 3.38 + 0.17 η(1405) 1.405 ± 0.022 0.051 1.300 ± D η(1475) 1.475 ± 0.025 0.085 1.011 ± −6 η(548) 0.548 0.010 1.3 10 1.100 η(958) 0.958 ± 0.017 1.9 × 10−3 1.200 1 η(1295) 1.405 1.295 ± 0.022 0.055× 1.210 4.30 0.10 [1.0, 4.0] 3.56 + 0.17 η(1405) 1.405 ± 0.023 0.051 1.310 ± η(1475) 1.475 ± 0.024 0.085 1.050 ±

corresponding instanton one, so it is naturally taken into each other. The main difference between this work and account already. our previous one[37,38] is that for the spectral function of + considered resonances, instead of the zero-width approx- Second, the properties of the lowest lying 0− pseu- imation of one gluonic resonance plus another low-lying doscalar glueball are systematically investigated in a fam- quark-antiquartk ones, the finite-width Breit-Wigner form ily of Gaussian sum rules in five different cases. In case QCD 2 with a correct threshold behavior for the lowest five res- A, the correlation function Π (Q ) contains only pure onances with the same quantum numbers is used in this perturbative contribution and pure instanton one, and work in order to compare with the phenomenology. The re- a single zero-width resonance plus continuum model is sultant Gaussian sum rules with k = 1, 0, +1 are carried adopted for the spectral function, and of course, the old re- out with a few of the QCD standard inputting− parameters, sults are recovered (even excluding the topological charge and really in accordance with the experimental data. screening contribution), and some pathology is explored. To go beyond the above constraint, all the contributions As a discussion, let us now identify where the low- + arising from pure instanton, pure perturbative and inter- est lying 0− pseudoscalar glueball is. The result of the ference between both are included in the correlation func- single-resonance plus continuum models B and C, namely tion for cases B, C and D, in which a single zero-width Eqs. (66) and (67), imply that the meson η(1405) may be + resonance plus continuum model of the spectral function the most fevered candidate for the lowest lying 0− pseu- is adopted for case B, and a single finite-width resonances doscalar glueball because the difference between the two plus continuum model for case C, and the five finite-width models is just the width of the resonances, and the latter resonances plus continuum model for case D, respectively. is of course believed to be more in accordance with the re- The optimal fitting values of the mass m, width Γ , cou- ality. This conclusion can further be justified by the result pling constant f, continuum threshold s0 for the possible of the five-resonances plus continuum model, namely Eqs. + 0− resonances are obtained, and quite consistent with (71), (72) and (73). Note that the first two resonances 12 Feng Wang et al.: Title Suppressed Due to Excessive Length

2 top 2.5 QCD with Π QCD k=−1 QCD k=−1 HAD 1.8 HAD

1.6 2 ] 4 2 ] s . 2

4 ˆ = 1 405 GeV 1.4 2 s0 = 5.25GeV 2 2 sˆ = 1.405 GeV m = 1.405GeV 1.2 2 1.5 s0 = 5.24GeV f = 1.630GeV m = 1.682GeV QCD/HAD [GeV Γ = 0.05GeV 1 f = 1.457GeV QCD/HAD [GeV

0.8 1

0.5 1 1.5 2 2.5 3 3.5 4 τ [GeV4] 0.5 0 1 2 3 4 5 6 7 8 5 τ [GeV4] QCD with Πtop k=0 QCD 4.5 HAD 4 QCD k=0 HAD 4 3.5 ] 4

3.5 s . 2 2 ] 3 ˆ = 1 350 GeV

2 2 4 2 sˆ = 1.490 GeV s0 = 4.45GeV 2 3 s0 = 4.61GeV m = 1.620GeV QCD/HAD [GeV f = 1.387GeV 2.5 m = 1.400GeV 2.5 f = 1.671GeV Γ = 0.08GeV

QCD/HAD [GeV 2 2

1.5 1.5 0.5 1 1.5 2 2.5 3 3.5 4 τ [GeV4] 1 0 1 2 3 4 5 6 11 τ [GeV4] QCD with Πtop k=1 QCD 10 HAD 11 QCD k=1 HAD 9 10 ] 4

8 9 sˆ = 1.3802GeV2 2 2 2 sˆ = 1.380 GeV s0 = 4.19GeV 2 8 ]

7 s0 = 4.45GeV 4 m . = 1 631GeV m .

QCD/HAD [GeV = 1 416GeV f = 1.392GeV 7 6 f = 1.760GeV Γ = 0.03GeV 6 5 QCD/HAD [GeV 5 4 0.5 1 1.5 2 2.5 3 3.5 4 4 τ [GeV4] 3 Fig. 5. The lhs without the topological charge screening contri- 2 bution (dashed line), rhs with the topological charge screening 0 1 2 3 4 5 6 4 contribution (dot line) and rhs (solid line) of the sum rules (42) τ [GeV ] with k = 1, 0, 1 versus τ in the case where the interference Fig. 6. The lhs (dashed line) and rhs (solid line) of the sum − QCD 2 contribution is included in the correlation function Π (Q ), rules (42) with k = 1, 0, 1 versus τ in the case where the and a single zero-width resonance plus continuum model is correlation function Π−QCD(Q2) contains the pure instanton, adopted for the spectral function. interference, and pure perturbative contributions, and a single ﬁnite-width resonance plus continuum model is adopted for the spectral function.

η(548) and η(985) are far away from the mass scale of state through a rotation η(1405), and usually considered as the superposition of the 0 Oˆ η 0 Oˆ η fundamental ﬂavor-singlet and octet pseudoscalar mesons h | p| 1295i h | p| 8i  0 Oˆ η  = U  0 Oˆ η  , (75) composed of quark-antiquark pair to have a dominant role h | p| 1475i h | p| 1i 0 Oˆ η 0 Oˆ G in responsible for the axial anomaly[67,68,69]. In order to  p 1405   p  explore the structures of the remainder three resonance h | | i h | | i η(1295), η(1405) and η(1475), we would like use the η-η′-G where the U is the mixing matrix [12,49] mixing formalism based on the anomalous Ward identity for transition matrix elements[49,12] to relate the physical cos ϕ sin ϕ 0 p − p states η, η′ and G to the fundamental ﬂavor-singlet and U =  cos φG sin ϕp cos ϕp sin φG sin φG  (76) octet quark-antiquark mesons and the lowest pure gluon cos φG sin ϕp cos ϕp sin φG cos φG  − −  Feng Wang et al.: Title Suppressed Due to Excessive Length 13

2.5 after normalization, respectively. As a relatively rough es- QCD k=−1 ˆ HAD timation, from Eq. (78) the values of the couplings of Op to the states η1, η8 and pseudoscalar glueball G are ob- 2 tained

] M1 = 0.548GeV f1 = 1.100GeV 4 −6 Γ1 = 1.3 × 10 GeV 3 M = 0.958GeV f = 1.100GeV ˆ 2 2 2 2 0 Op η1 = 0.117GeV , sˆ = 1.405 GeV Γ = 1.9 × 10−3GeV 1.5 2 2 h | | i − s0 = 5.25GeV M3 = 1.295GeV f3 = 1.200GeV ˆ 3 Γ3 = 0.055GeV 0 Op η8 =0.568GeV , M4 = 1.405GeV f4 = 1.330GeV h | | i QCD/HAD [GeV 3 Γ4 = 0.051GeV 0 Oˆ G =0.813GeV . (79) M5 = 1.475GeV f5 = 1.010GeV p 1 Γ5 = 0.085GeV h | | i by reversing Eq. (75). This shows that the coupling of + the 0− glueball current to the pure glueball state G is 0.5 0 1 2 3 4 5 6 7 8 ˆ 4 dominant, and the signs of the couplings 0 Op η1 and τ [GeV ] h | | i 0 Oˆp η8 are similar to those predicted by the scalar glueball- 6.5 h | | i QCD meson coupling theorems [70,58]. 6 HAD k=0 Furthermore, η(1405) and η(1475) could originate from 5.5 a single pole (see Amsler and Masoni’s review for eta(1405)

5 M1 = 0.548GeV f1 = 1.200GeV ] − 4 6 Γ1 = 1.3 × 10 GeV in PDG). To check whether the single pseudoscalar meson 4.5 M2 = 0.958GeV f2 = 1.300GeV −3 assumption may be consistent with our sum rule approach, Γ2 = 1.9 × 10 GeV 4 M3 = 1.295GeV f3 = 1.195GeV or the dependence of the results on the model selected for Γ3 = 0.055GeV 3.5 M4 = 1.405GeV f4 = 1.300GeV QCD/HAD [GeV Γ = 0.051GeV the spectral function, we add an analysis of four resonance 3 4 M5 = 1.475GeV f5 = 1.011GeV 2 2 model for the spectral function. The result is shown in Ap- sˆ = 1.500 GeV Γ5 = 0.085GeV 2.5 2 s0 = 4.79GeV pendix F. From FIG. 10 and Tab. 2 where it is easy to see 2 that the four resonance model for the spectral function 1.5 0 1 2 3 4 5 6 is inconsistent with our sum rule analysis because, ﬁrstly 4 τ [GeV ] the resultant mass scale of the fourth resonance is approx- 12 imately 1.41GeV in average which locates outsides of the QCD k=1 HAD 11 sum rule window; secondly the match degree δ of the four resonance model is obviously worse than the δ of the ﬁve 10 M1 = 0.548GeV f1 = 1.100GeV −6 Γ1 = 1.3 × 10 GeV resonance model. M2 = 0.958GeV f2 = 1.200GeV 9 − ] 3 4 Γ2 = 1.9 × 10 GeV In summary, our result suggests that η(1405) is a good M3 = 1.295GeV f3 = 1.210GeV + 8 Γ3 = 0.055GeV candidate for the lowest 0− pseudoscalar glueball with M4 = 1.405GeV f4 = 1.310GeV 7 Γ4 = 0.051GeV some mixture with the nearby excited isovector and isoscalar M5 = 1.475GeV f5 = 1.050GeV

QCD/HAD [GeV Γ5 = 0.085GeV qq¯ mesons. This is a ﬁrst theoretical support for the phe- 6 sˆ = 1.4052GeV2 2 nomenological estimation from the sum rule approach. s0 = 4.30GeV 5

3 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Acknowledgements τ [GeV4]

Fig. 7. The lhs (dashed line) and rhs (solid line) of the sum This work is supported by BEPC National Laboratory rules (42) with k = 1, 0, 1 versus τ in the case where the Project R&D and BES Collaboration Research Founda- correlation function Π−QCD(Q2) contains the pure instanton, in- tion. terference, and pure perturbative contributions, and ﬁve ﬁnite- width resonance plus continuum model is adopted for the spectral function. A Pure instanton and perturbative contribution where ϕp 40◦ and φG 22◦ is η η′ [49] mixing angle and the mixing≈ angle of the≈ pseudoscalar− glueball with η . ′ In instanton liquid model, the pure instanton contri- Then, one has + bution of 0− pseudoscalar glueball with spike instanton 0.766044 0.642788 0 distribution in Euclidean space-time is known as[7,71,72, − 73] U =  0.595982 0.286965 0.374607 (77) 0.595982 0.286965 0.927184 Π(cl)(Q2)= 25π2n¯ρ¯4Q4K2(Qρ¯). (80)  − −  E − 2 2 To be quantitative, the corresponding normalized cou- where K2 is the McDonald functions and n(ρ) is the size plings F to the three resonances η, η′ and G with masses distribution of instantons, and the appearance of the over- 1.295, 1.475, 1.405 GeV can be read from Tab. 1 to be all minus sign in rhs of (80) is due to the anti-hermitian property of the current, i.e. Op† = Op in Euclidean space- 0.51GeV3, 0.65GeV3, 0.56GeV3, (78) time. − 14 Feng Wang et al.: Title Suppressed Due to Excessive Length

The second part of (10) has already been calculated up C The operators Oi(x) to three-loop level in the MS dimensional regularization scheme[11,71,74] The operators Oi in terms of instanton and quantum gluon ﬁelds are

2 2 2 (qu) 2 αs 4 Q Q O1(x)= Fµν,a[A(x)]Fρσ,a[A(x)] Π (Q )= Q ln 2 a0 + a1 ln 2 π µ µ O2(x)=4Fµν,a[A(x)](∂ρaσa[A(x)]) 2 2 Q O3(x)=4gsfabcFµν,a[A(x)]Aρb(x)aσc(x) + a2 ln 2 , (81) µ O4(x) =4(∂µaνa(x))(∂ρaσa(x))

O5(x)=8gsfabcAρb(x)(∂µaνa(x))aσc(x) where µ is the renormalization scale, and the coefficients 2 O6(x)=4g fabcfadeAµb(x)Aρd(x)aνc(x)aσe(x) with the inclusion of the correct threshold effect are s O7(x)=2gsfabcFµν,a[A(x)]aρb(x)aσc(x) 2 αs αs O8(x)=4gsfabcaµb(x)aνc(x)(∂ρaσa(x)) a0 = 2 1+20.75 + 305.95 , 2 − π π O9(x)=4gs fabcfadeAρd(x)aµb(x)aνc(x)aσe(x) α 9 α O (x)= g2f f a (x)a (x)a (x)a (x) (85) a =2 s + 72.531 s , 10 s abc ade µb νc ρd σe 1 π 4 π where Fµν,a[A(x)] is the instanton field strength associ- 2 αs ated with the instanton field A. a2 = 10.1250 (82) − π (int) 2 for QCD with three massless quark flavors up to three- D The interference contributions Πi (q ) loop level. (int) 2 The expressions of the interference contributions Πi (q ) in terms of Oi are Π(cl+qu)(q2)= Tˆ Ω O (x)O (0) Ω B Topological charge screening 1 h | 2 2 | i Π(cl+qu)(q2)= Tˆ Ω O (x)O (0) Ω 2 h | 2 3 | i Π(cl+qu)(q2)= Tˆ Ω O (x)O (0) Ω The topological charge screening effect can be under- 3 h | 3 3 | i (cl+qu) 2 ˆ stood by tracing back to the anomaly of the axial-vector Π4 (q )= T Ω O4(x)O5(0) Ω 5 h | | i current J = Σ ψ¯ γ γ ψ with ψ being quark field of fla- (cl+qu) µ i i 5 µ i i Π (q2)= Tˆ Ω O (x)O (0) Ω vor i 5 h | 4 6 | i Π(cl+qu)(q2)= Tˆ Ω O (x)O (0) Ω 6 h | 4 7 | i 5 (cl+qu) 2 ∂µJµ =2Nf Q(x) (83) Π (q )= Tˆ Ω O (x)O (0) Ω 7 h | 5 5 | i (cl+qu) 2 Π (q )= Tˆ Ω O7(x)O7(0) Ω where the topology charge current Q(x) relates to the 8 h | | i (cl+qu) 2 ˆ pseudoscalar glueball current as Q(x) = O (x)/(8π). Eq. Π9 (q )= T Ω O5(x)O7(0) Ω p h | | i (83) indicates that instantons generate quark-antiquark (cl+qu) 2 Π (q )= Tˆ Ω O6(x)O7(0) Ω pairs, which then may form a light meson to mediate the 10 h | | i (cl+qu) 2 ˆ long-range multi-instanton interaction [16]. Kikuchi and Π (q )= T Ω O5(x)O6(0) Ω 11 h | | i Wudka [75] suggest that such a light meson, which can Π(cl+qu)(q2)= Tˆ Ω O (x)O (0) Ω (86) couple to the instanton in the way that the instanton gen- 12 h | 6 6 | i erates all light quark flavors with the identical probability, where be the meson η0, and constructs an effective Lagrangian 1 ˆ 2 ′ ′ ′ ′ 4 4 (see also [76,77]) which gives rise to the topological charge T α nǫ¯ µνρσ ǫµ ν ρ σ d z d x. (87) ≡− 2 s Z Z screening contribution Πtop to the correlation function of the pseudoscalar current as[7] E Comparison between the condensate 2 2 ′ top 2 Fη Fη contributions and the instanton-induced ones Π (Q )= 2 2 + 2 2 , (84) Q + m Q + m ′ η η The contributions arising from condensates up to the eighth dimensions are known to be as follows [7] where m and m ′ are the masses of the mesons η and η η 2 ′ 9 Q η′, and the two constants Fη and Fη are evaluated to be cond 2 2 2 2 Π (Q )=4αs αsG + αs αsG ln 2 16πnξ¯ sin φ and 16πnξ¯ cos φ, and ξ and φ 22◦ are the h i π h i µ ≈ coupling strength of η0 to an instanton and the η η′ 1 15π 1 − 8α2 gG3 + α2 α G2 2 , (88) mixing angle, respectively. − sh iQ2 2 sh s i Q4 Feng Wang et al.: Title Suppressed Due to Excessive Length 15 where F The results for the four finite-width resonance model from the Gaussian sum rules 2 4 αsG =0.05GeV h i The optimal parameters of the four finite-width reso- gG3 =0.27GeV2 α G2 , (89) h i h s i nance model for the spectral function are listed in Tab 2, and the corresponding plots are shown in FIG. 10. The imaginary part of condensates (88) has the form

1 Cond 9 2 2 2 3 3 ImΠ (s)= αs αsG 8αs gG δ(s) k=−1 QCD π −π h i− h i HAD 15π 2 2 2 2.5 + αs αsG δ′(s). (90)

2 h i ] M1 = 0.548GeV f1 = 1.100GeV 4 . × −6 2 Γ1 = 1 3 10 GeV M2 = 0.958GeV f2 = 1.100GeV × −3 The comparison between the imaginary part of the cor- Γ2 = 1.9 10 GeV M3 = 1.295GeV f3 = 1.200GeV relation function, (24), and condensate contribution to it 1.5 Γ3 = 0.055GeV M4 = 1.412GeV f4 = 1.340GeV QCD/HAD [GeV are shown in FIG. 8, while the comparison between the Γ4 = 0.051GeV

1 2 various real parts of (23) (which altogether are related sˆ = 1.4052GeV 2 with the imaginary part by the dispersion relation (40)) s0 = 5.30GeV 0.5 and condensate contribution to it are shown in FIG. 9. 0 0.5 1 1.5 2 2.5 3 3.5 4 τ [GeV4]

5.5 k=0 QCD 10 HAD 2 ΠCond π 5 −(s−sˆ) /4τ Im (s)/ 9 e × QCD top Im Π (s)/π (without Im Π (s)/π ) 4.5 8 M1 = 0.548GeV f1 = 1.200GeV −6 ] Γ = 1.3 × 10 GeV 4

] 1 4 7 4 4 M2 = 0.958GeV f2 = 1.300GeV τ −3 = 1GeV Γ2 = 1.9 × 10 GeV )[GeV

s 2 2 6 s . M3 = 1.295GeV f3 = 1.195GeV ˆ = 1 405 GeV 3.5 Γ3 = 0.055GeV ImΠ( 5 M = 1.420GeV f = 1.670GeV 1

π 4 4

× 3 Γ4 = 0.051GeV QCD/HAD [GeV

τ 4 4 / 2 ) ˆ s 2.5 2 2 − 3 sˆ = 1.500 GeV s

( 2

− s0 = 4.80GeV e 2 2

1 1.5 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 0 τ 4 0 1 2 3 4 5 6 7 8 9 10 [GeV ] s [Gev2] 8.5 k=1 QCD Fig. 8. The contributions to the imaginary part of the corre- 8 HAD lation function from the condensate (dashed-dotted line) and 7.5 M1 = 0.548GeV f1 = 1.100GeV the total contribution (solid line) versus s. − Γ = 1.3 × 10 6GeV ] 7 1 4 M2 = 0.958GeV f2 = 1.200GeV Γ = 1.9 × 10−3GeV 6.5 2 M3 = 1.295GeV f3 = 1.210GeV Γ3 = 0.055GeV 6 M4 = 1.410GeV f4 = 1.740GeV Γ4 = 0.051GeV QCD/HAD [GeV 5.5 sˆ = 1.4052GeV2 s = 4.29GeV2 5 0 10 Πinstanton 2 (Q ) 4.5 Πperturbation(Q2) 8 Π interference(Q2) 4 1 1.5 2 2.5 3 3.5 Πtop 2 (Q ) τ [GeV4] 6 ΠQCD(Q2) (without Πtop(Q2) ) ] 4 Fig. 10. The lhs (dashed line) and rhs (solid line) of the sum 4 ) [GeV

2 rules (43) with k = 1, 0, 1 versus τ in the case where the

(Q −QCD 2 Π correlation function Π (Q ) contains the pure instanton, 2 interference, and pure perturbative contributions, and the four

0 ﬁnite-width resonance plus continuum model is adopted for the spectral function.

−2 1 2 3 4 5 6 7 8 9 10 Q2 [Gev2] Fig. 9. The contributions to the correlation function arising from the pure instanton (cross line), interference (dashed- References dotted line), pure perturbative (dotted line) and topological charge screening (dashed line) and the total contribution with- 1. Apoorva Patel, Rajan Gupta, Gerald Guralnik, Gre- 2 out the topological charge screening one (solid line) versus Q . gory W. Kilcup, and Stephen R. Sharpe. An Improved Estimate of the Scalar Glueball Mass. Phys.Rev.Lett., 57:1288, 1986. 16 Feng Wang et al.: Title Suppressed Due to Excessive Length

Table 2. For the four ﬁnite-width resonances plus continuum model, the optimal ﬁtting values of the mass m, width Γ , −+ coupling constant f, continuum threshold s0 ,s ˆ, and matching measure δ for the possible 0 resonances in the sum rule −1 window [tmin,tmax] (t = τ ) for the best matching between lhs and rhs of the sum rules with k = 1, 0, 1 are listed. − 2 −4 −4 k resonances √sˆ(GeV) m(GeV) Γ (GeV) f(GeV) s0(GeV ) [tmin,tmax](GeV ) δ/10 η(548) 0.548 1.3 10−6 1.100 × −3 1 η(958) 1.405 0.958 1.9 10 1.100 5.30 [0.25, 1.00] 2.3 − η(1295) 1.295 0.055× 1.200 1.412 0.051 1.340 η(548) 0.548 1.3 10−6 1.200 × −3 0 η(958) 1.500 0.958 1.9 10 1.300 4.80 [0.29, 0.77] 2.2 η(1295) 1.295 0.055× 1.195 1.420 0.051 1.670 η(548) 0.548 1.3 10−6 1.100 × −3 1 η(958) 1.405 0.958 1.9 10 1.200 4.29 [0.42, 0.83] 9.3 η(1295) 1.295 0.055× 1.210 1.410 0.051 1.740

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