OPE of the energy-momentum tensor correlator and the gluon condensate operator in massless QCD to three-loop order

The correlator of two gluonic operators plays an important role for example in transport properties of a Quark Gluon Plasma (QGP) or in sum rules for glueballs. In [1] an operator product expansion (OPE) at zero temperature was performed for the correlators of two scalar operators $O_1=-\frac{1}{4} G^{\mu \nu}G_{\mu \nu}$ and two QCD energy-momentum tensors $T^{\mu\nu}$. There we presented analytical two-loop results for the Wilson coefficient $C_1$ in front of the gluon condensate operator $O_1$. In this paper these results are extended to three-loop order. The three-loop Wilson coefficient $C_0$ in front of the unity operator $O_0=1$ was already presented in [1] for the $T^{\mu\nu}$-correlator. For the $O_1$-correlator the coefficient $C_0$ is known to four loop order from [2]. For the correlator of two pseudoscalar operators $\tilde{O}_1=\varepsilon_{\mu\nu\rho\sigma} G^{\mu \nu} G^{\rho \sigma}$ both coefficients $C_0$ and $C_1$ were computed in [3] to three-loop order. At zero temperature $C_0$ and $C_1$ are the leading Wilson coefficients in massless QCD.


Introduction and definitions
with large Q 2 := −q 2 > 0, i.e. in the Euclidean region of momentum space. The function Π(Q 2 ) is connected to the spectral density ImΠ(s) in the region of physical momenta through a dispersion relation (see e.g. [4]).
The leading contribution to Π(Q 2 ) can be computed perturbatively and is exactly the first Wilson coefficient in front of the unity operator O 0 = 1. In order to include nonperturbative effects as well the correlator (1.1) is expanded in a series of local operators with Wilson coefficients containing the dependence on q in momentum space or x in x-space   (1.4) where the index B marks bare quantities. The factor (Q 2 ) is constructed from the mass dimensions of the operators in order to make C i (q) dimensionless.
The perturbative contribution is separated from the non-perturbative condensates in an operator product expansion (OPE) and hence resides in the Wilson coefficients in front of local operators. These Wilson coefficients are calculated perturbatively using the method of projectors [6,7] and contain the perturbative contribution to the correlator in question. If we insert expansion (1.4) into (1.2) we are left with the task of determining the VEVs of the local operators [O i ], the so-called condensates [8], which contain the non-perturbative part. These need to be derived from low energy theorems or be calculated on the lattice.
Three gluonic operators with the quantum numbers J P C = 0 ++ , 0 −+ and 2 ++ are usually considered: 2 with the bare gluon field strength tensor where f abc are the structure constants and T a the generators of the SU(N c ) gauge group. As described in [1] for T µν we use the gauge invariant and symmetric energy-momentum tensor of (massless) QCD: (1.9) In [13] it was argued that if we are only interested in matrix elements of only gauge invariant operators it is not necessary to consider the ghost terms appearing in the full energy-momentum tensor of QCD. It was also proven that the energy-momentum tensor of QCD is a finite operator without further renormalization.
2 For details on the sum rule approach to glueballs with the same quantum numbers see e.g. [4]. An OPE at one-loop level has been performed for the scalar [9] and pseudoscalar [10] correlator. Recent discussions on glueballs using an OPE of these correlators can be found in [11,12].
The operator O 1 and the Wilson coefficients C 1 , however, have to be renormalized in the following way: (1.10) The renormalization constant was derived in [14,15] from the renormalization constant Z αs for α s . At first order in α s we find Z G = Z αs , which is not true in higher orders however. We take the definition for the β-function of QCD, which is available at four-loop level [16,17]. For the renormalization ofÕ B 1 , which mixes with a pseudoscalar fermionic operator under renormalization, and its OPE we refer to [3,18].
The correlators of O 1 and O µν T have been discussed in [1], where C 1 has been presented at two-loop level. The results of this work are derived within the same theoretical and methodical framework, which is why we can refer to this work for most technical details. C 0 is also known to three-loop level for the T µν -correlator [1] and at two-, three-and fourloop level for the O 1 -correlator from [19], [20] and [2] respectively. Three-loop results for C 0 and C 1 for the correlator of two operatorsÕ 1 have been derived in [3].
At zero temperature (1.15) has the asymptotic behaviour where the tensor structure of the correlator resides in the Wilson coefficients if we are ultimately only interested in the VEV of the correlator.
Local tensor operators can always be decomposed in a trace part and a traceless part, i.e. for two Lorentz indices where D is the dimension of the space time. The VEV of the traceless part vanishes due to the Lorentz invariance of the vacuum and only a local scalar operator O ρ ρ survives. The OPE of the correlator (1.16) reads (1.21)

Calculation and results
As discussed in [1] there are five independent tensor structures for (1.19) allowed by the symmetries µ ←→ ν, ρ ←→ σ and (µν) ←→ (ρσ) of (1.19). These are (2.1) Due to the fact that the energy-momentum tensor is conserved except for contact terms, i.e. q µ T µν;ρσ (q) = local contact terms, (2.2) and due to the irrelevance of these contact terms for physical applications we can reduce (2.1) to only two independent tensor structures, which have already been suggested in [30], after contact term subtraction: : The structure t µν;ρσ T (q) is traceless and orthogonal to t µν;ρσ S (q). Hence the latter corresponds to the part coming from the traces of the energy-momentum tensors. The Wilson coefficient in front of the local operator [O 1 ] has the form where the contact terms have to be ∝ t µν;ρσ . This was checked explicitly in our three-loop result.
Just like in [1] (see this paper for more details) the method of projectors [6,7] was used in order to compute the coefficient C µν;ρσ 1 (q). We apply the same projector to both sides of (1.3): The projector P is constructed in such a way that it maps every operator on the rhs of (1.3) to zero except for O B 1 , which is mapped to 1 and hence gives us the bare Wilson coefficient C B 1 on the lhs. For the T µν -correlator (1.15) this is done after contracting the free Lorentz indices with a tensort (r) µν;ρσ (q) composed of the momentum q and the metric g µν in order to get the scalar pieces in (2.4): 3 We use the following projector: 4 where the blue circle represents the the sum of all bare Feynman diagrams which become 1PI after formal gluing (depicted as a dotted line in (2.7)) of the two external lines representing the operators on the lhs of the OPE. These external legs carry the large Euclidean momentum q.
In order to produce all possible Feynman diagrams we have used the program QGRAF [32]. These propagator-like diagrams were computed with the FORM [33,34] package MINCER [35] after projecting them to scalar pieces. For the colour factors of the diagrams the FORM package COLOR [36] was used.
We now give the three-loop results for the Wilson coefficient C 1 of the correlators (1. 15) and (1.16) in the MS-scheme. In the following the abbreviations a s = αs π = g 2 s 4π 2 and l µq = ln µ 2 Q 2 are used, where µ is the MS renormalization scale. The number of active quark flavours is denoted by n f . Furthermore, C F and C A are the quadratic Casimir operators of the quark and the adjoint representation of the gauge group, d R is the dimension of the quark representation, n g is the number of gluons (dimension of the adjoint representation), T F is defined through the relation Tr T a T b = T F δ ab for the trace of two group generators. 5 In [1] it was shown that up to two-loop level the coefficient C (S) 1 , which corresponds to the trace of the two energy-momentum tensors in the correlator (1.15), can be written in the where the first factor β(a s ) is due to the trace anomaly (1.17). It is interesting to check weather we can find a similar structure in terms of the β-function at three-loop level. However, we do not find such an elegant representation at the next loop order. The closest we get is  (2.14) We find In [1] the three-loop logarithmic terms of (2.15) und (2.16) were constructed from the twoloop result and the requirement that µ 2 d dµ 2 C (S,T ) 1,RGI vanishes identically. and indeed we find the same result in this explicit calculation. This requirement also explains the absence of Logarithms in the lower-order terms [1].
As already observed at two-loop level [1] there are divergent contact terms in C GG 1 starting from O(α 2 s ). It is intersting to observe that these divergent terms can be expressed through the β-function coefficients from (1.13): This feature points to the possibility that the contact terms and hence the additive part of the renormalization of the Wilson coefficient C GG 1 could be expressed completely through the β-function. An explanation for this curious behaviour and its meaning for the O 1correlator remains to to be found in the future.
An unambiguous result can be obtained for the Adler function of C GG 1 , in which all contact terms, finite and divergent, vanish: In analogy to the construction above we can also find an RGI Wilson coefficient For the derivative of the Wilson coefficient wrt Q 2 we find

Numerics
Finally, we consider two cases which are interesting for applications numerically, that is gluodynamics (n f = 0) and QCD with only three light quarks (n f = 3). For this we choose the scale µ 2 = Q 2 , i.e. we set l µq = 0. For the correlator (1.15) we find The numerical impact of the higher order corrections can be seen by evaluating the RGI coefficients at µ = M Z , µ = 3.5 GeV and µ = 2 GeV, where for the cases n f = 5 and n f = 3 respectively. We find for the correlator (1.15). This shows that for the energy-momentum tensor the Wilson coefficient C (T )  1 is well convergent, even at µ = 2 GeV. The three-loop approximation for C (S) 1 at low scales is less good, but still acceptable. At µ = 3.5 GeV the three-loop correction is 50% of the two-loop correction but both together are only a 12% correction to the one-loop result.
For the correlator (1.16) we find with [39] (3.20) in addition to (3.13): Here the convergence at low scales is not so good as the two-loop correction becomes larger than the one-loop correction at µ = 5 GeV and the three-loop correction shifts the result by another 50% of the one-loop results. This suggests that higher order corrections should always be taken into account when this coefficient is used e.g. in sum rules and special care has to be taken with regard to the convergence of the perturbation series at the scale where perturbative and non-perturbative physics are separated in the OPE. With this in mind, extending C GG 1 to even higher orders in the future could therefore be an interesting task.

Conclusions
We have presented the missing three-loop corrections to the OPE of the correlator of two scalar gluonic operators [O 1 ] = − Z G 4 G Ba µν G Ba µν and of the correlator of two energymomentum tensors T µν in massless QCD at zero temperature.
These are the three-loop contributions to the coefficient C 1 in front of the local operator [O 1 ]. We have also constructed renormalization group invariant versions of these coefficients and confirmed the predictions made in [1] for the logarithmic part of these coefficients.
In the coefficient C GG 1 for the O 1 -correlator we observe the curious feature that divergent contact terms appear which are expressible through the QCD β-function. These constact terms as well as contact terms in the T µν -correlator proportional to the tensor structures t µν;ρσ 4 (q) and t µν;ρσ 5 (q) from (2.1) have to be subtracted. If we consider only derivatives wrt Q 2 of ambiguous Wilson coefficients these terms vanish automatically.