$2^{++}$ Tensor Di-Gluonium from Laplace Sum Rules at NLO

We evaluate the next-to-leading (NLO) corrections to the perturbative (PT) and $<\alpha_s G^2>$ condensate and the LO constant term of the $$ contributions to the $2^{++}$ tensor di-gluonium two-point correlator. Using these results into the inverse Laplace transform sum rules (LSR) moments and their ratio, we estimate the mass and coupling of the lowest ground state. We obtain\,: $M_T=3028(287)$ MeV and the renormalization group invariant (RGI) coupling $\hat f_T=224(33)$ MeV within a vacuum saturation estimate of the $D=8$ dimension gluon condensates ($k_G=1$). We study the effect of $k_G$ on the result and find: $M_T=3188(337)$ MeV and $\hat f_T$=245(32) MeV for $k_G=(3\pm 2)$. Our result does not favour the pure gluonia/glueball nature of the observed $f_2(2010,2300,2340)$ states.


Introduction
Since the pioneering work of Novikov et al. (NSVZ) [1], some efforts have been done for improving the determination of the 2 ++ tensor di-gluonium mass and coupling either using a least-square fit method [2] or stability criteria.[3,4].To Lowest Order (LO) of perturbative QCD (PT) and including the dimension d = 8 condensates estimated by (SVZ) [5] using vacuum saturation, the up-to-date results are [4] 1 : (20) MeV, M T | LO = 2.0(1) GeV, M T | LO ≤ 2.7(4) GeV. ( In this paper, we shall improve these LO results by including NLO corrections and checking the effect of the violation of vacuum saturation on the results.
built from the gluon component of the energy-momentum tensor 3 : with: η µν ≡ g µν − q µ q ν /q 2 : P µνρσ P µνρσ = 2(n 2 − n − 2) , (4) where: n = 4 + 2ϵ is the space-time dimension used for dimensional regularization and renormalization.To LO and up to dimension D = 8 gluon condensates, the QCD expression is [1]: where a s ≡ α s /π and : . Using the vacuum saturation hypothesis (k g = 1), it reads [1]: We shall test the effect of this assumption by taking : for an eventual violation of the factorization assumption like the one found for the D = 6 four-quark condensates (see e.g.[3,[7][8][9]) where this assumption is violated by about a factor 5-6.

N ¯ext-to-Leading (NLO) contribution
We use two approaches to perform the calculation :

D . iagrammatic renormalization
This approach has been initiated in [10] for QCD sum-rule correlation functions.It requires an isolation of the subdivergences arising from the one-loop subdiagram(s) of an individual bare NLO diagram (see e.g., Ref. [11]) .Counterterm diagrams generated from the subdivergences are then calculated and subtracted from the bare diagram to obtain the renormalized diagram.A self-consistency check of the method is the cancellation of non-local divergences in each diagram.
We shall be concerned with the bare diagrams a-g listed in Table 1 and their corresponding individual diagrammaticallyrenormalized contributions parametrized in Feynman gauge as: with : a s ≡ α s /π.The sum of the contributions of the bare diagrammatically-renormalized diagrams a-g in Table 1 leads to the renormalized NLO two-point function for n f flavours : with : L = log Q 2 ν 2 .Note that diagram h from Table 1 is not used in the diagrammatic renormalization method, but is crucial in the conventional renormalization approach for the cancellation of the non-local (1/ϵ) log(Q 2 /ν 2 ) as we shall see in the next section.

T . he conventional approach
Here, we calculate each QCD diagram using the standard Feynman approach (see e.g.[3,7,12]).We consider the renormalization of the gluonic current using the renormalization constant obtained in Ref. [13] for the current θ µν g /α s : for n = 4 + 2ϵ dimensions to which corresponds the anomalous dimension : Taking into account the renormalization of α s : one can deduce the anomalous dimension of the current θ µν g : The diagrams appearing in Table 1e) to g) are due to the nonabelian property of QCD where: The diagram in Table 1h) is induced by the off-diagonal term which arises due to the mixing of the qq and G 2 currents.Following [13,14], such terms are necessary to cancel the nonlocal (1/ϵ) log (Q 2 /ν 2 ) divergent terms appearing in the calculations given in Table 1.It is remarkable to notice that there is a systematic factor two difference for the coefficient A of diagrams a) to g) from the two approaches.The sum of the individual diagrams in Table 1 gives for the current normalized in Eq. 3 :

N . LO PT results
We have shown in the previous sections, that the diagrammatic and conventional approaches lead to the same result.The renormalized two-point function for n f flavours reads : where ψ pert T | LO can be deduced from Eq. ( 5).One can notice that for gluodynamics (n f = 0), we recover the earlier result of [15].

Dimension-four gluon ⟨α
One can notice from Eq. 5 that, unlike the case of scalar and pseudoscalar gluonia [1], the contributions of the gluon condensates are only due to the D = 8 dimension.

L ¯O contribution
From the diagram in Fig. 2, we have checked that to LO the leading log-term does not contribute to the two-point function.
The LO contribution comes from the constant term: where we have used two different approaches (plane wave and conventional one using the projection in Eq. 2).The nonzero value of this constant term raises the question of the validity of the null result obtained in Ref. [16] based on instantons for dual/antidual background fields.However, this term is harmless in the LSR analysis as it will disappear when one takes the different derivatives of the two-point functions.

N ¯LO contribution
The leading-log.contribution at NLO, can be derived from the renormalization group equation (RGE).Using the fact that the ⟨α s G 2 ⟩ obeys the RGE [17] (see different applications in section 4.4 of [12]) 4 : where t ≡ (1/2)log(Q 2 /ν 2 ) and γ θ ψ is the anomalous dimension defined in Eq. 15.Writing the α s expansion as : and considering that ⟨α s G 2 ⟩ is a constant, one deduces : We calculate the coefficients of the ⟨gG 3 ⟩ contribution using the conventional approach and the projection in Eq. 4. We show the different contributions in Fig. 3 where the total sum is zero (log .coefficient and constant term) in agreement with the result of [16].

C ¯heck of the result
We recompute the G 3 coefficient of the scalar gluonium twopoint function using the same method.We recover the result of Ref. [1] which is an indirect test of our result.6. Laplace Sum Rule (LSR) analysis

Q ¯CD expression
Collecting the previous results, we obtain for n f = 3 flavours to order α s and up to dimension-8 condensates: We shall be concerned with the following inverse Laplace transform moments and their ratio [5,[18][19][20][21]: t> dt e −tτ t Im ψT (t, ν) t> dt e −tτ Im ψT (t, ν) , To get the lowest moment L c 0 , we take the 3rd derivative of the two-point function which is superconvergent while for the L c 1 moment, we take the 4th derivative of Q 2 ψT (Q 2 ).The NLO QCD expressions of the moments for n f = 3 flavours are : and from which one can deduce the ratio R c 10 (τ).γ E = 0.5772... is the Euler constant and:

S ¯trategies P . arametrization of the spectral function
To a first approximation, we have parametrized the spectral function using the minimal duality ansatz (MDA): where we assume that the QCD expression of the spectral function above the continuum threshold t c smears all radial excitation contributions.f T is normalized as f π = 132 MeV.In the MDA parametrization:

O . ptimization procedure
One can notice that there are three free parameters in the analysis, namely the LSR variable τ, the continuum threshold t c and the perturbative subtraction constant ν.The later quantity is eliminated when one works with different derivatives of the two-point function for taking its inverse Laplace transform and working with the running QCD parameters.The optimal results will be extracted at the minimum or inflexion points in τ while we shall fix the range of t c in a conservative region from the beginning of τ-stability until the (approximate) t c -stability.

Q . CD input parameters
We shall work with the QCD input parameters [8,9]:

D ¯i-gluonium mass and coupling at Lowest Order (LO)
In this section, we redo the analysis in Ref. [4] using the expression in 5 that one shall explicitly compare with the one including the new NLO terms.
W . e show the determination of M T from R c 10 in Fig. 4, where the vacuum saturation estimate of the D = 8 gluon condensates is assumed.We show the t c -behaviour of the optimal values on τ in Fig. 5.The final optimal results are obtained for the set (τ, t c ) from (0.18,4.5) to (0.68,12) (GeV −2 , GeV 2 ) and are respectively 1857 and 2324 MeV.They lead to the mean : W . e show the analysis of the coupling f T from the moment L c 0 in Fig. 6.One obtains: These results agree within the errors with the ones in Eq. 1 obtained at slightly low value of t c ≃ 5.5 GeV 2 .The large error obtained here is due to the most conservative choice of the t crange.We obtain the upper bounds: The bound on the mass is comparable with the one in Eq.The behaviour of the mass is shown in Fig 7 where we have assumed the factorization of the dimension 8 gluon condensates.The stabilities in τ are reached for the set (τ, t c ) = (0.12, 9.5) to (0.36,20) (GeV −2 ,GeV 2 ) to which correspond the mass values 2746 and 3309 MeV.We deduce the mean value: The errors come mainly from t c .From Fig. 7, one can also deduce the optimal upper bound from the positivity of the ratio of moments.We obtain:

C . omparison with the LO results within factorization
We notice that the PT NLO corrections increase the central value of the mass by 561 MeV from its LO value while the ⟨α s G 2 ⟩ ones provide an additional increase of 376 MeV.

C . omparison with some other LSR results
I .n Ref. [22], the result: has been obtained to LO PT but including the NLO ⟨α s G 2 ⟩ term and the LO constant term of the ⟨gG 3 ⟩ condensates.Unfortunately, our results summarized in Eq. 23 do not agree with the coefficients of these condensates.The difference of these coefficients may come from the different current used by Ref. [22].R .esult within instanton liquid model is about 1525 MeV [23] which is lower than the above result.

E . ffect of the D = 8 condensates
Now, we study the effect of the estimate of the D = 8 gluon condensates on the mass determination assuming that the factorization can be violated like in the case of the four-quark condensate.The analysis is similar to the one in Fig 7 .We show the optimal results in Fig. 8 versus t c for different values of the violation factor k G .One can notice that the value of the mass is a smooth increasing function of k G .From the vacuum saturation estimate (k G = 1) to 5 (if one takes the same value as the violation of the four-quark condensate), the value of the mass moves from 3028(287) MeV to 3347(295) MeV thus an increase of about 319 MeV.For definiteness, we shall work with the conservative range : Then, we deduce the final estimate : Our result for the ground state mass is in line with the ones from some other approaches [24] and ADS/QCD [25] where its mass is expected to be above 2 GeV.It is slightly higher than recent lattice calculations in the range (2.27 ∼ 2.67) GeV [26][27][28].
T ¯he 2 ++ ground state di-gluonium coupling at NLO We introduce the renormalization group invariant (RGI) coupling fT which is related to the running coupling f T (ν) associated to the two-point correlator ψT as: We shall extract the coupling from the lowest moment L c 0 .

B . eyond the factorization of the D = 8 gluon condensates
We extract the value of the coupling corresponding to the k Gfactor in Eq. 37. The curves are similar to the ones in Fig. 9.One obtains for the set of (τ, t c ) : (0.We notice that,like the mass, the value of the coupling is weakly affected by the value of the D = 8 gluon condensates.

Summary and conclusions
W . e have computed the perturbative and ⟨α s G 2 ⟩ NLO corrections to the 2 ++ tensor di-gluonium two-point correlator and use the method of Laplace transform sum rules (LSR) to revisite the estimate of the mass and coupling of the lowest ground state.
W . e find that the LO ⟨α s G 2 ⟩ coefficient has no imaginary part like found by NSVZ [1] but the constant term is not zero in contrast to NSVZ who have used dual/anti-dual field arguments.Thus, the use of the RGE allows to fix its log(Q 2 /ν 2 ) NLO coefficient from this LO constant term.
W . e note that our LO coefficient of ⟨g 3 G 3 ⟩ disagrees with the one of [22] but agrees with the one of NSVZ.This disagreement may be related to the choice of current.As an indirect check of our result, we recalculate the ⟨g 3 G 3 ⟩ coefficient in the scalar gluonium channel and recover the one of NSVZ.
A .ssuming vacuum saturation for the estimate of the D = 8 gluon condensates, we found the lowest ground state mass M T = 3028(287) MeV and RGI coupling fT = 224(33) MeV.
W . e study the effect of the estimate of the D = 8 gluon condensates and find M T = 3188(337) MeV and fT = 245(32) MeV for the violation factor k G = (3 ± 2).
O .ur result is in line with the ones from some other approaches [24] and ADS/QCD [25] where its mass is expected to be above 2 GeV.However, the central value of our mass is slightly higher than lattice calculations in the range (2.27 ∼ 2.67) GeV [26][27][28].
O .ur result does not favour the interpretation of the observed f 2 (2010, 2300, 2340) states as pure gluonia/glueball candidates (see e.g.[29]).Moreover, we do not expect that an eventual meson-gluonium mixing will affect our result as this mixing is expected to be small (θ ≃ −10 0 ) [13].

3 .
PT expression of the two-point function up to NLO L ¯owest Order (LO) contribution It comes from the diagram in Fig. 1 and reads :

Figure 4 :
Figure 4: Behaviour of the 2 ++ tensor di-gluonium mass from the ratio of moments R c 10 versus τ for different values of t c at LO.

Figure 7 :
Figure 7: Behaviour of the 2 ++ di-gluonium mass from the ratio of moments R c 10 versus τ for different values of t c at NLO.

Figure 8 :
Figure 8: Behaviour of the 2 ++ tensor di-gluonium mass versus t c for different values of the factorization factor k G .

D = 8 Figure 9 :
Figure 9: Behaviour of the 2 ++ tensor RGI coupling from the moment L c 0 versus τ for different values of t c at NLO within factorization.