# The two-loop energy–momentum tensor within the gradient-flow formalism

## Abstract

The gradient-flow formulation of the energy–momentum tensor of QCD is extended to NNLO perturbation theory. This means that the Wilson coefficients which multiply the flowed operators in the corresponding expression for the regular energy–momentum tensor are calculated to this order. The result has been obtained by applying modern tools of regular perturbation theory, reducing the occurring two-loop integrals, which also include flow-time integrations, to a small set of master integrals which can be calculated analytically.

## 1 Introduction

The gradient-flow formalism as introduced by Lüscher [1] and further formalized by Lüscher and Weisz [2] has proven useful in lattice QCD in many respects. One of its main virtues is that composite operators at finite flow time *t* do not require ultra-violet (UV) renormalization beyond the one of the involved parameters and fields. This means that the operators do not mix under renormalization-group running, which makes it particularly simple to combine results from different regularization schemes. This feature opens promising prospects for a cross-fertilization of lattice and perturbative calculations, such as a possible lattice determination of \(\alpha _s(M_Z)\), for example [3].

A particularly powerful way to exhibit this possible interplay is obtained by considering the expansion of composite operators in the limit of small flow time, which expresses flowed operators in terms of QCD operators at \(t=0\), with *t*-dependent Wilson coefficients [2]. This method has been used by Makino and Suzuki [4, 5] to derive a regularization-independent formula for the energy–momentum tensor (EMT) \(T_{\mu \nu }\) which has already led to promising results (see, e.g., Refs. [6, 7, 8, 9, 10]).

The universal Wilson coefficients that occur in the formula of Ref. [5] for the EMT have been calculated through next-to-leading order (NLO) in perturbation theory [4, 5]. This corresponds to a one-loop calculation in the sense that it involves integrals over a single *D*-dimensional momentum. In this paper, we will carry this calculation to the next perturbative order.^{1} It is important to note at this point that the integrals which occur in the gradient-flow formalism are of a more general type than in regular QCD. They involve additional exponential factors which depend on loop and external momenta, as well as on flow-time variables, some of which are also integrated over. Nevertheless, the first two-loop result was already obtained in Ref. [1], even in analytic form. The extension to the three-loop level required significant aid from computer algebra and numerical tools [3]. From the quantum-field theoretical point of view, it closely followed the steps of Ref. [1] by directly expressing the Green’s functions in terms of integrals with the help of Wick’s theorem. The integrals themselves were evaluated using sector-decomposition [12, 13] in order to isolate the poles in \(D-4\), whose coefficients were determined using high-precision numerical methods [14, 15].

In the current calculation, we apply a completely independent setup. On the one hand, it applies the gradient-flow formalism described in terms of a five-dimensional quantum field theory [2], which leads to well-defined, albeit non-standard Feynman rules. On the other hand, rather than evaluating the resulting integrals numerically, we express them in terms of master integrals using the integration-by-parts method of Chetyrkin and Tkachov [16]. This reduces the NLO calculation of the Wilson coefficients of the EMT to a single one-loop integral without flow-time integration. The next-to-next-to-leading order (NNLO) calculation leads to four two-loop master integrals without flow-time integration, and two two-loop master integrals with a single flow-time integration. All master integrals can be calculated analytically by standard means for general values of *D*, the number of space-time dimensions.

By suitable renormalization, the Wilson coefficients of the EMT can be defined in such a way that they are formally renormalization-scale independent. For a fixed-order perturbative result, this means that the renormalization-scale dependence is formally of higher order. This allows one to estimate the perturbative uncertainty on the Wilson coefficients through their residual dependence on the renormalization scale \(\mu \) around a particular “central” value. Based on the form of the analytical result, we argue for a specific choice of this central value. Our numerical study shows that the higher-order terms indeed lead to an appreciable reduction of the \(\mu \)-variation. However, by comparison of the successive higher-order terms, it appears that the uncertainty estimate from a variation within \(\mu \in [\mu _0/2,2\mu _0]\), as it is common practice in regular perturbative QCD calculation, might be too optimistic.

While we consider the NNLO expressions for the Wilson coefficients of the EMT as our main result, our calculation allows us to obtain a number of additional results that might be useful in a broader context. Among these are the flowed quark-field renormalization constant \(Z_\chi \), and the matrix of anomalous dimensions for the set of operators which form the energy–momentum tensor in regular (non-flowed) QCD through NNLO.

The remainder of the paper is structured as follows. After briefly introducing the perturbative gradient-flow formalism in order to define our notation in Sect. 2, we outline the approach of Refs. [4, 5] for using this formalism to define the EMT in Sect. 3. Technical details of our calculation are described in Sect. 4. Section 5 contains our main result, the Wilson coefficients through NNLO QCD in the \(\overline{\text{ MS }}\) scheme. As pointed out in Ref. [5], the trace anomaly of the EMT allows for a welcome check of the calculation; we briefly describe the derivation of the resulting relations among the coefficient functions in Sect. 6. Finally, in Sect. 7, we use the finiteness condition of the flowed operators in order to derive the anomalous-dimension matrix for the set of operators which occur in the EMT in regular QCD. Section 8 presents our conclusions.

## 2 QCD gradient flow in perturbation theory

*D*-dimensional Euclidean space-time with \(D=4-2\varepsilon \). The gradient-flow formalism continues the gluon and quark fields \(A^a_\mu (x)\) and \(\psi (x)\) of regular

^{2}QCD to \((D+1)\)-dimensional fields \(B^a_\mu (t,x)\) and \(\chi (t,x\)) through the boundary conditions

*t*is a parameter of mass dimension minus two, and \(\kappa \) is an additional gauge parameter which drops out of physical observables.

*D*-dimensional Lorentz indices. Color indices of the fundamental representation are suppressed throughout this paper, unless required by clarity. The symmetry generators \(T^a\) obey the commutation relation

The flow-field equation leads to a smearing of gauge-field configurations at finite flow time \(t>0\). As a consequence, composite operators at \(t>0\) do not require renormalization beyond the renormalization of the parameters and fields of the Lagrangian. For the strong coupling and the quark mass, the renormalization constants are identical to those at \(t=0\); the flowed gluon fields do not require renormalization at finite flow time as was pointed out in Ref. [2]. The renormalization constant for the flowed quark field through NNLO is a by-product of this paper and will be given below.

## 3 Energy–momentum tensor

*D*-dimensional space-time, the gauge invariant part of the EMT reads

*f*labels the \(n_F\) different quark flavors, \(m_{f,0}\) is the bare quark mass, andThe notation \(i_f\in \{i_1,\ldots ,i_{n_F}\}\) for the indices which label different flavors will be useful later on in this paper. In general, \(T_{\mu \nu }\) may contain gauge-dependent operators which vanish when evaluating physical matrix elements [17]. Here and in what follows, we implicitly assume that the vacuum expectations values of all composite operators have been subtracted

^{3}so that \(\langle {\mathscr {O}}_{i,\mu \nu }(x)\rangle \equiv 0\ \forall i\).

## 4 Calculation of the Wilson coefficients

### 4.1 Method of projectors

*t*and the renormalization scale \(\mu \), we can choose arbitrary values for all other dimensional parameters in this equation. Setting them to zero turns all higher-order corrections on the r.h.s. into massless tadpoles, so that Eq. (17) is only required to hold at tree-level. One thus obtains

*m*and

*p*collectively denote all masses and external momenta. The right-hand side thus results in vacuum diagrams whose only dimensional scale is

*t*.

^{4}where the momenta are defined to be outgoing. This suggests to use

*i*and

*j*are the corresponding indices. The trace appearing in the projectors \(P_{3_f}\) and \(P_{4_f}\) is taken w.r.t. the spinor indices of the Green’s function, and

*f*denotes the associated quark flavor. Note that \(P_{4_f}\) is constructed such that

### 4.2 Computational methods

The gradient-flow formalism in perturbation theory can be formulated in terms of a Lagrangian field theory, where the flow equations (2) are implemented with the help of Lagrange-multiplier fields [2]. The crucial difference between the regular QCD Feynman rules and those in the gradient-flow formalism is the occurrence of exponential factors \(\exp (-sp^2)\), where *s* is a “flow-time variable”, and *p* the linear combination of *D*-dimensional external and/or loop momenta. Vertices involving flowed fields induce an integration over all positive values of the corresponding flow-time variable, which is, however, bounded from above by “propagators” of the Lagrange-multiplier fields, since they introduce step functions of the flow-time variables.

*f*and

*l*is the number of flow-time and loop integrations, respectively, the \(b_j\) are polynomials in (rescaled) flow-time parameters \(u_i\) and the \(q_i\) are linear combinations of the loop momenta \(k_j\). For the problem and the perturbative order under consideration, it is \(0\le f\le 4\), \(1 \le l\le 2\), and \(0 \le n \le 3\), respectively. Note that the projectors defined in Eqs. (21) and (23) eliminate all dependence on external momenta and masses, so that, after making a suitable ansatz for the index structure of the integrals, we only have to evaluate scalar vacuum integrals. Using the identities [16]

^{5}Their analytical evaluation is possible along the lines of Ref. [1]:

A more detailed description of parts of our setup will be described in a forthcoming publication [32]. As a check, we evaluated the correlators \(\langle G_{\mu \nu }^aG_{\mu \nu }^a\rangle \), \(\langle {{\bar{\chi }}}\chi \rangle \) and Open image in new window through NLO. They lead to the same set of master integrals as given in Eq. (34). Comparing our results to Ref. [1] and Ref. [5]^{6}, we find full agreement.

## 5 Coefficient functions through NNLO QCD

*T*the trace normalization in the fundamental representation. For SU(\(N_c\)), it is \(C_F=(N_c^2-1)/(2N_c)\), \(C_A=N_c\), and \(T=1/2\).

^{7}

^{8}

^{9}

While the energy–momentum tensor \(T_{\mu \nu }(x)\) is renormalization-scheme independent, this is not necessarily the case for the operators \({\tilde{{\mathscr {O}}}}_{i,\mu \nu }(t,x)\) and the coefficient functions \(c_i(t)\). Since \({\tilde{{\mathscr {O}}}}_{1,\mu \nu }\) and \({\tilde{{\mathscr {O}}}}_{2,\mu \nu }\) do not require operator renormalization, their matrix elements as well as the coefficient function are indeed renormalization-scheme independent. On the other hand, using the quark-field renormalization \(Z_\chi \) of Eq. (43) in the \(\overline{\text{ MS }}\) scheme, matrix elements of \({\tilde{{\mathscr {O}}}}_{i,\mu \nu }\) and coefficient functions \(c_i(t)\) become explicitly dependent on the renormalization scale \(\mu \) for \(i\in \{3,4\}\).

^{10}

*g*. The decrease of the residual \(\mu \)-dependence is thus commonly used as a qualitative check of the perturbation expansion for the specific observable under consideration. We thus study the \(\mu \)-dependence of the four coefficients after dividing \(c_3\) and \(c_4\) by the ratio \(\zeta _\chi \) defined in Eq. (51). We fix a characteristic value for the flow time

*t*and vary the renormalization scale \(\mu \) around the central value \(\mu _0\), which we define such that \(L(\mu _0,t)=0\), cf. Eq. (49), i.e.

*t*, corresponding to \(\mu _0=3\) GeV and \(\mu _0=130\) GeV, respectively. In the former case, we set \(n_F=3\), in the latter \(n_F=5\). We use \(\alpha ^{(n_F=5)}_s(M_Z)=0.118\) in order to evaluate the input values for the couplings, \(g^{(n_F=3)}(3\,\text {GeV})=1.77\) and \(g^{(n_F=5)}(130\,\text {GeV})=1.19\). The \(\mu \)-variation of the strong coupling constant \(g(\mu )\) is determined by numerically solving the corresponding renormalization-group equation with the help of RunDec [34, 35] at one-, two-, and three-loop level for the LO, NLO, and the NNLO curve, respectively. In Fig. 1, the value of

*t*is chosen such that the central scale of Eq. (54) is \(\mu _0=3\) GeV. At this central scale, the NNLO corrections increase the modulus of the coefficients \(c_1\) and \(c_2\) by 10% and 13% relative to NLO, respectively. This is within twice the NLO uncertainty due to missing higher-order effects as estimated by varying \(\mu /\mu _0\) between 1/2 and 2, where one finds 7.3% for \(c_1\), and \(8.0\%\) for \(c_2\). We are therefore confident that the NNLO uncertainty estimated in the same way is rather reliable: it is given by 5.7% for \(c_1\) and 7.2% for \(c_2\). Note that the dominant contribution to these numbers comes from the downward variation of \(\mu \), where \(g(\mu )\) starts to become sensitive to the non-perturbative region. The behavior of \(c_1\) and \(c_2\) towards larger values of \(\mu \) seems to suggest that this uncertainty estimate may actually be too conservative.

As opposed to the gluonic coefficients \(c_1\) and \(c_2\), the coefficients of the fermionic operators \(c_3\) and \(c_4\) exhibit a residual scale dependence only from the NLO term onwards. One therefore expects a stronger \(\mu \)-dependence at NNLO for these terms. Nevertheless, for \(\mathring{c}_3\), the estimate of the theory uncertainty due to scale variation still decreases from 9.8% to 8.1%. The increase of the result due to the NNLO effects is 8.3% relative to the NLO result at \(\mu =\mu _0\).

The behavior of \(\mathring{c}_4\), on the other hand, is less satisfactory at \(\mu _0=3\) GeV. The NNLO effects more than double the NLO result in this case, and the uncertainty estimate due to scale variation actually *increases* from 47% to 71% when going from NLO to NNLO. Note, however, that \(c_4=0\) at LO, which means that this coefficient is numerically sub-dominant.

As one would expect, for \(\mu _0=130\) GeV, the perturbative behavior of all coefficients is significantly improved, cf. Fig. 2. For \(c_1\), \(c_2\), and \(\mathring{c}_3\), the scale uncertainty is at the sub-percent level already at NLO; at NNLO, it amounts to less than 0.2% in all three cases. The effect of the NNLO corrections relative to the NLO result is about 2% for \(c_1\) and \(c_2\), and 0.8% for \(\mathring{c}_3\). Also for \(\mathring{c}_4\), the situation improves significantly: the NNLO terms add 38% to the NLO result, and the uncertainty goes down from 10% at NLO to 5.8% at NNLO.

It is also worth pointing out that the choice of the central scale \(\mu _0\) as defined in Eq. (54) seems justified by the behavior of the successively higher orders. In almost all cases, the NLO and the NNLO corrections are *both* relatively small at \(\mu =\mu _0\). At the same time, the NNLO corrections relative to the NLO result are always smaller than the NLO corrections compared to the LO result. The only exception to this is \(\mathring{c}_4\) at \(\mu _0=3\) GeV, where, however, no choice of \(\mu \) seems to stand out over any other.

In summary, we conclude that the NNLO terms lead to a significant improvement of the perturbative accuracy of the Wilson coefficients.

## 6 Trace anomaly

*i*individually. Instead, only the weaker conditions

## 7 Operator renormalization

*D*. The renormalization matrix is then defined as

## 8 Conclusions

We have presented the universal Wilson coefficients for the gradient-flow definition of the energy–momentum tensor through NNLO QCD. The NNLO corrections modify the three numerically dominant coefficients \(c_1\), \(c_2\), \(\mathring{c}_3\) at the level of 10% (1-2%) for a central scale of \(\mu _0=3\) GeV (\(\mu _0=130\) GeV), where \(\mu _0\) is related to the flow time *t* according to Eq. (54). We observe a reduction of the theoretical uncertainty relative to the NLO result as derived from varying the renormalization scale by a factor of two around its central value. The behavior of the fourth coefficient \(\mathring{c}_4\) is less satisfactory, but its impact is expected to be numerically suppressed.

Aside from this main outcome, new results presented in this paper include the flowed quark-field renormalization constant to NNLO in the \(\overline{\text{ MS }}\) scheme, and the anomalous dimension matrix for the regular QCD operators which make up the EMT.

In conclusion, we hope that our results will help to improve the studies of the EMT on the lattice. They are the first outcome of a systematic setup for higher-order perturbative calculations within the gradient-flow formalism [32], which should prove useful also in other applications of this theoretical framework.

## Footnotes

- 1.
Note that we work in the limit of infinite volume. The inclusion of finite-volume effects requires different techniques, such as Numerical Stochastic Perturbation Theory, see Ref. [11].

- 2.
We will use the terms “flowed” and “regular” QCD to distinguish quantities defined at \(t>0\) from those defined at \(t=0\).

- 3.
In other words, the precise definition of \({\mathscr {O}}_{1,\mu \nu }\), for example, would be given by \(F_{\mu \rho }^aF_{\rho \nu }^a-\langle F_{\mu \rho }^aF_{\rho \nu }^a\rangle \).

- 4.
Feynman diagrams in this paper were drawn using Ti

*k*Z-Feynman [20]. - 5.
The reduction with Kira 1.0 takes about 20 minutes on 8 CPU threads and requires less than 13 GB of RAM.

- 6.
We compare to arXiv versions 2 and 5 of that paper.

- 7.
Note that since \(c_4=0\) at leading order, this coefficient only requires NLO renormalization.

- 8.
For convenience of the reader, we provide the expressions for \(c_1,\ldots ,c_4\) also in electronic form in an ancillary file with this paper.

- 9.
This parameter is motivated by the product of the typical factor \((8\pi t)^{\varepsilon }\) occurring in flow-time integrals [1], and the usual definition of the renormalization scale in the \(\overline{\text{ MS }}\) scheme, see Eq. (39): \((8\pi t)^\varepsilon (\mu ^2e^{\gamma _\text {E}}/(4\pi ))^\varepsilon = 1 + \varepsilon \,L(\mu ,t) + \mathcal{O}(\varepsilon ^2)\).

- 10.
Note that the calculation of \(\mathring{Z}_\chi \) also provided an independent check for \(Z_\chi \).

## Notes

### Acknowledgements

We are indebted to Johannes Artz and Mario Prausa for helping to establish the setup within which this calculation was performed. We would also like to thank Tobias Neumann for fruitful communication, and for providing his tools which enabled us to obtain \(\mathring{Z}_\chi \) before the publication of Ref. [32]. This work was supported by *Deutsche Forschungsgemeinschaft (DFG)*, project 386986591.

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