New predictions on the mass of the 1−+ light hybrid meson from QCD sum rules

We calculate the coefficients of the dimension-8 quark and gluon condensates in the current-current correlator of 1−+ light hybrid current gq¯xγνiGμνxqx\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ g\overline{q}(x){\gamma}_{\nu }i{G}_{\mu \nu }(x)q(x) $$\end{document}. With inclusion of these higher-power corrections and updating the input parameters, we re-analyze the mass of the 1−+ light hybrid meson from Monte-Carlo based QCD sum rules. Considering the possible violation of factorization of higher dimensional condensates and variation of 〈g3G3〉, we obtain a conservative mass range 1.72-2.60 GeV, which favors π1(2015) as a better hybrid candidate compared with π1(1600) and π1(1400).


I. INTRODUCTION
Mesons with exotic quantum numbers have long been attractive in hadron physics, among which are the J P C = 1 −+ isovector states π 1 (1400) , π 1 (1600) and π 1 (2015) identified in the experiments [1]. The construction of these states are not quite clear, four-quark states and hybrid states are most possible explanations. Theoretical studies via different methods have shown that some of these states can be considered as good light hybrid candidates. In the bag model, the predicted mass of 1 −+ light hybrid meson is around 1.5 GeV [2]; the mass from the flux tube model is found to be in the range 1.7-1.9 GeV [3]; the lattice QCD prediction of 1 −+ mass is 1.9-2.2 GeV [4]. Calculations based on QCD sum rules [5] have been conducted by different groups [6][7][8][9] to NLO of d ≦ 6 contributions, and the latest versions of the predicted mass are 1.80 ± 0.06 GeV in [10] and 1.71 ± 0.22 GeV in [11]. Although the hybrid explanation for π 1 (1600) is supported by the previous sum rule analysis, the hybrid assignment of π 1 (2015) is also proposed [10]. Thus the calculation of higher power corrections (HPC) of the OPE is interesting and of value. How and how much the HPC affect the mass prediction would lead to totally different conclusions.
In this paper, we focus on the mass prediction of the 1 −+ light hybrid meson using QCD sum rule method.Dimension-8 contributions will be taken into account, and we will use the least-square method to fit the sum rules following Leiweber's procedure [12]. For the explicit consideration of higher power corrections is not seen very often in previous sum rule calculations, we will give a slightly more detailed presentation of our calculation and analysis.

II. OPE FOR THE CURRENT-CURRENT CORRELATOR
We start from the two-point correlator Π µν (q 2 ) = i d 4 xe iqx 0 T j µ (x)j + ν (0) 0 (1) where j µ (x) = gq(x)γ ν iG µν (x)q(x), and the invariants Π v (q 2 ) and Π s (q 2 ) correspond respectively to 1 −+ and 0 ++ contributions. The correlator obeys the standard dispersion relation In this paper, we focus on the dimension-8 corrections to the 1 −+ mass. Before showing the higher power results we need to mention that coefficients of dimension-8 quark-related operators of the 1 −+ light hybrid two-point correlator have been calculated in [6] and [7]. In [6] there is only a factorized form of the total result and a complete result is given in [7]. We obtain a new complete result which is consistent with the former factorized form but different from the latter one.
As for dimension-8 gluon operators, there arise IR divergences in the calculation of the quark loops as the result of setting m q = 0 before calculating the integrals. These IR divergences can be canceled after taking operator mixing into account. This process can partly check the calculation about dimension-8 quark and gluon operators and modify the finite part of the coefficients of gluon condensates. Some good examples for the case ofqq scalar and vector currents are given in [13,14].
According to the numbers of quark operators in the condensates, dimension-8 quark condensates can be classified into two groups: two-quark d = 8 condensates and four-quark d = 8 condensates. Only the formers can be mixed to d = 8 gluon condensates in LO. We use the dimensional regularization in n = 4 − ǫ space-time dimensions, thus the O(ǫ) terms of the two-quark d = 8 condensates can be obtained, which are needed to be multiplied by the 1 ǫ subtractions to modify the finite part of the quark loop calculations (see Eq. (5)).
The dimension-8 quark contributions (corresponding to Feynman diagrams in Figure 1) are listed in Appendix. Dimension-8 contributions of gluon condensates come from the calculations of quark loops. Here we give the quark propagator up to term O(q −5 ) needed in the calculation of the quark loops: where D µ = ∂ µ − igA µ and S 0 (q) = 1 / q . For a massless quark Eq.(3) can be rewritten as Eq.(4) can also be seen in [15] and [14], but the last term of (4) is missed in [15] and not consistent with [14]. We use (3) rather than (4) in practical calculations for (3) is more convenient in program calculations. Gluon contributions from calculations of quark loops (the corresponding Feynman diagrams are depicted in Figure 2) are listed in Appendix.
The two-quark d = 8 condensates in (15) and (19) can be expanded in the basis {Q j } listed in Table I, using the equations of motion and charge conjugation transformation and setting m q = 0. And we also list the expanding coefficients in Table I and the mixing coefficients of quark condensates with gluon condensates in Table II [13]. After Notice that quark condensates have been factorized in (6) so as to conduct the sum rule analysis. As for the value of G 4 condensates(O 1 − O 4 ), one may probably think of using factorization. However, for reasons in [16], factorization hypothesis may not be reliable in G 4 case. Therefore we choose to use a modified factorization proposed in [17] and supported in [19], which suggests an overestimation of factorization and are based on two technologies: factorization of quartic quark condensates and heavy quark expansion. In the framework of this modified factorization, O 1 − O 4 can be expressed in terms of the condensate φ = TrG νµ G µρ TrG ντ G τ ρ , which has been clarified in [17] to reasonably satisfy the factorization approximation. Thus after fitting φ using factorization, O 1 − O 4 can be estimated as follows With regard to other d = 8 gluon condensates, a scale M 2 ≈ 0.3 GeV 2 is estimated in [18,19], which characterizes the average off-shellness of the vacuum gluons and quarks:

III. QCD SUM RULES FOR THE 1 −+ LIGHT HYBRID MESON
The d ≦ 6 contributions to Π v (q 2 ) including the NLO corrections to the perturbative and the α s G 2 and α s qq 2 terms can be found in [6][7][8][9]. Π d≦6 v (q 2 ) can be written as where α s (µ) = 4π/(9 ln(µ 2 /Λ 2 QCD )) is the running coupling constant for three flavors. In addition, Π d=8 v (q 2 ) can be obtained from (6),(7) and (8): with The Borel transformation of Π OPE v (q 2 ) can be written as By using the single narrow resonance spectral density ansatz ImΠ phen Then the master equation for QCDSR can be written as physical properties of the relevant hadron, i.e., m H , f H and s 0 , should satisfy Eq. (13). In order to present the influence of the d = 8 contributions, we will conduct the sum rule analysis both in d ≦ 6 and d ≦ 8 cases. Before those, we should clarify our criteria for establishing the sum rule window in which the mass prediction is reliable. On the OPE side, we wish the Borel parameter τ is as small as possible so that power series converge as quickly as possible. On the hadron spectrum side, our wish is the opposite, because a larger τ can better suppress contributions of the excited states and continuum. The common procedure without considering the higher power contributions is usually as follows: 1.keep the highest dimensional contributions (normally dimension-6 contributions) no more than 10% (or 15%) of the total OPE contributions to ensure the convergence of OPE, which gives the upper bound of τ ; 2.make sure that the contributions from the continuum are under 50% of the total contributions, which ensures the validity of the narrow resonance ansatz and gives the lower bound of τ . For our case, if we require dimension-8 contributions are less than 15 percent, it means we choose a window with a larger upper bound compared with d ≦6 case. This choice enhances suppression of excited states and continuum, but the convergence of OPE gets worse, which increases the uncertainties of the OPE side. On the other hand, if we still require the dimension-6 contributions are less than 15 percent, uncertainties from the truncation of OPE are indeed decreased(because the dimension-8 contributions are now taken into account), but the validity of the narrow resonance ansatz is not improved. Apparently, to keep a balance should be a good resolution. Our choice is that make sure both 1% < d = 8 contributions < 5% and 20% < d = 6 contributions < 30% (correspondingly the perturbative and d = 4 contributions would be totally 120% -135% because the signs of the d = 6 and d = 8 contributions are minus), which ensures the OPE series converge in a proper trend and also a larger upper bound of τ is obtained compared with d ≦6 case,thus uncertainties from both sides of the master equation are reduced.
In traditional QCDSR analysis, the continuum threshold s 0 cannot be completely constrained. To overcome this shortcoming and make our conclusion more reliable, we use a weighted-least-square method following Leinweber [12] to match the two sides of Eq.(13) in the sum rule window.
By randomly generating 200 sets of Gaussian distributed phenomenological input parameters with given uncertainties (10% uncertainties, which are typical uncertainties in QCDSR) at τ j = τ min + (τ max − τ min ) × (j − 1)/(n B − 1), where n B = 21, we can estimate the standard deviation σ OPE (τ j ) for Π OPE v (τ j ). Then,the phenomenological output parameters s 0 , f H and m H can be obtained by minimizing We use two sets of the parameters as the central values of inputs (see Table III) to conduct the matching procedures respectively. Values in set I are from a recent review of QCD sum rules [20], we choose this set of values to exclude the errors in artificial selections of each input. Our main conclusions are based on this set of inputs. We also notice that the main difference between the condensates in [20] and those in Narison's work on 1 −+ light mesons [10] is the value of g 3 G 3 . This value changes from 1.2 GeV 2 α s G 2 (from charmonium systems [19]) to 8.2 GeV 2 α s G 2 (from dilute gas instantons [21] and lattice calculations [22]), which largely affects the mass prediction. To make our conclusion more reliable and to provide a comparison of the results from our matching procedure and those from Narison's LSR analysis, we maintain the value of g 3 G 3 the small one in set II.
As in our previous paper [11], we generate 2000 sets of Gaussian distributed input parameters with 10% uncertainties, and for each set we minimize χ 2 to obtain a set of phenomenological output parameters, after this procedure is finished, we can estimate the uncertainties of s 0 , f H and m H . Finally, before proceeding with numerical calculations, renormalization-group (RG) improvement of the sum rules, i.e., substitutions µ 2 → 1/τ in Eq.(13), is needed [23]. In addition, the anomalous dimensions for condensate g 3 G 3 and qq gqGq also should be implemented by multiplying g 3 G 3 and qq gqGq by a factor L(µ 0 ) −23/27 and L(µ 0 ) 10/27 respectively, where L(µ 0 ) = [ln(1/(τ Λ 2 QCD ))/ ln(µ 2 0 /Λ 2 QCD )], µ 0 is the renormalization scale for condensates [5,24]. The coupling constant f H also should be multiplied by a factor L(m) −32/81 , f H then receives its value at hybrid mass shell. In this paper, we neglect the anomalous dimensions for operators O 1 − O 8 , which are not calculated yet and very likely to have small effects on the mass prediction. In Table IV, we list our matching results with input parameters in Set I. The upper bounds of sum rules window are obtained by demanding |HDO|/OPE<10%, <5%, <3% and <2% respectively. The matching results, including the medians and the asymmetric standard deviations from the medians for s 0 , m H and f 2 H , are reported. We find by inputting Gaussian distributed input parameters with 10% uncertainties, we obtain some Gaussian-like distribution results for s 0 , m H and f 2 H with uncertainties <10%, this implies the matching results are very stable with different input parameters. Following our criteria above for establishing the window, we think the phenomenological outputs obtained in the condition |HDO|/OPE<2% (d = 6 contributions are less than 29%) is the most reliable. In fact, we can see that the predictions are not very sensitive to the variation of the range of the window. All output parameters slightly decrease for stronger constraints on contributions from HDO. Therefore,we obtained a prediction of 1 −+ mass at 2.26 +0.05 −0.05 GeV with inclusion of dimension-8 contributions. And also, we list the predictions obtained from d ≦ 6 contributions in Table IV, among which the mass is 2.15 +0.05 −0.05 GeV. These results strongly suggest that π 1 (2015) is a good hybrid candidate. In addition, we also list the results deduced from d ≦ 6 contributions in the optimal window of d ≦ 8 case (|HDO|/OPE<2%) to show the variation of the sum rules in this region after considering the dimension-8 contributions.
As a supplement of our analysis, we also list the results obtained from the small value of g 3 G 3 in Table V. Again the outputs are not sensitive to the set of the window. Under the consideration of the convergence of the OPE series, we think the the best condition is setting HDO less than 5% (the dimension-6 contributions are less than 22% in this case). Thus the mass prediction is 2.08 +0.04 −0.05 GeV, which also favors π 1 (2015) as a hybrid candidate.Besides, all the outputs (s 0 ,m H and f H ) in d ≦ 6 case are consistent with Narison's [10] from LSR, which increases the reliability of our matching results.
More details of the weighted-least-square matching method can be seen in our previous work on the 1 −+ light hybrid meson [11]. In that work, we concentrate ourselves on the sum rule analysis based on the matching procedure, especially the uncertainty analysis.However,the dimension-8 coefficients used there are not a complete form(only the factorized quark condensates in [6]). And we follow our earlier works [9] in choosing the inputs there. Moreover, the sum rule window there is just established by keeping HDO<10% as the common procedure, lacking in an explicit consideration of the convergence of OPE. All these lead to the discrepancy of the predictions.

IV. SUMMARY
We have calculated the dimension-8 coefficients of the two-point correlator of the current gq(x)γ ν iG µν (x)q(x). We re-extract the mass of the 1 −+ light hybrid meson and find that the results in both large and small g 3 G 3 cases support π 1 (2015) as a hybrid candidate, which is different from previous sum rule analyses. What's more, in large g 3 G 3 case, even the result without considering the dimension-8 contributions favors this conclusion. The most reliable predicted mass in our anlysis is 2.26 +0.05 −0.05 GeV with 10% uncertainties for input parameters. Besides, our results reflect that 5%-9% underestimation of the mass would be led to by neglecting the contributions from d = 8 condensates in the case of 1 −+ light hybrid state.