Quark and gluon contributions to the QCD trace anomaly

We show that, in dimensional regularization in the minimal subtraction scheme, the QCD trace anomaly can be unambiguously decomposed into two parts coming from the renormalized quark and gluon energy momentum tensors. We carry out this decomposition at the two-loop level. The result can be used to constrain the renormalization group properties of the nucleon’s twist-four gravitational form factor C¯q,g\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\overline{C}}_{q,g} $$\end{document}.


Introduction
It is well known that the QCD Lagrangian is classically scale invariant if the small quark masses are neglected, but the invariance is broken at the quantum level. A common way to express this phenomenon is to compute the trace of the QCD energy momentum tensor T µν In addition to the expected quark mass contribution, terms proportional to the betafunction β and the mass anomalous dimension γ m appear. This is called the trace anomaly and is fundamentally important in QCD as it signals the generation of a nonperturbative mass scale. In particular, its nucleon matrix element P |T α α |P (relative to the vacuum expectation value) is proportional to the nucleon mass squared (see (3.8) below).
In this paper, we address the following question. The energy momentum tensor consists of the quark part and the gluon part T µν = T µν q + T µν g . Which part of the anomaly (1.1) comes from T µν q and the rest from T µν g ? Until recently, there has not been enough motivation to ask this question. First of all, it is not clear a priori whether such a decomposition is well-defined, and even if the answer is yes, it appears to be a purely conceptual problem without any phenomenological implications. More technically, while the total anomaly is renormalization group (RG) invariant, the individual terms T α qα and T α gα are not. Furthermore, the decomposition may depend on the regularization scheme one is using to handle the UV divergences.
However, in recent years the necessity to understand the QCD energy momentum tensor has intensified significantly. It has become a common practice to parametrize the 'gravitational form factor' of hadrons separately for quarks and gluons P ′ |T µν q,g |P [1][2][3][4]. Very importantly, understanding the origin of the nucleon mass has emerged as one of the main objectives of the future Electron-Ion Collider [5], and an independent experiment JHEP12(2018)008 dedicated to this issue has been proposed at the Jefferson Laboratory [6]. Specifically, JLab proposes to measure the near-threshold photoproduction of J/ψ in ep scattering. It has been shown in [7,8] that the cross section of this process is sensitive to the F 2 part of the trace anomaly (1.1). However, the extraction of the forward matrix element P |F 2 |P from the experimental data is complicated by the fact that near the threshold, the momentum transfer ∆ = P ′ −P is not negligible. In order to facilitate the extrapolation to ∆ → 0, it is convenient to express P ′ |F 2 |P in terms of the gravitational form factors P ′ |(T q,g ) α α |P . In ref. [8], this relation has been worked out at the level of the bare operators. In this paper, we show that the relation gets modified once one considers the renormalized operators, and compute the correction to two loops in dimensional regularization in the minimal subtraction scheme. This clarifies the relation between the bare and renormalized operators (T q,g ) α α . As an important application of our result, we elucidate the renormalization group property of the twist-four gravitational form factorC q,g .

The trace anomaly
Our starting point is the gauge-invariant, symmetric QCD energy momentum tensor which is given by We have neglected the ghost and gauge fixing terms as they do not affect our final results. In the last equality, we decomposed the total energy momentum tensor into the gluon and quark parts as T µν is conserved and therefore it is a finite, scale-independent operator. However, T µν g and T µν q are not conserved separately and are subject to regularization and renormalization.
We work in d = 4 − 2ǫ spacetime dimensions. Let us decompose T µν into the traceless T µν and traceT µν parts.
The second equality follows from the equation of motion. (m is the quark mass.) The corresponding decomposition for the bare operators T µν q,g is The operator F 2 is divergent and has to be regularized. We shall use the (modified) minimal subtraction scheme, and in this scheme the renormalization of F 2 has been well understood JHEP12(2018)008 in the literature [9,10]. Denoting renormalized operators with a sub-or super-script R, one finds mψψ = (mψψ) R and − 2ǫ ∂ ln µ is the mass anomalous dimension. We thus arrive at the standard result The above derivation makes it clear that, in dimensional regularization, the anomaly entirely comes from the bare gluon energy momentum tensor, while the bare quark energy momentum tensor only contributes to the mass term Noting that the right hand sides of (2.8) and (2.9) are both renormalization-group (RG) invariant, one can also write where all the quantities and fields are bare. Note that such a clean separation of the trace anomaly into the quark and gluon parts may not be unambiguously done in other regularization schemes. For example, in the Pauli-Villars regularization, the anomaly comes from the energy-momentum tensor of the massive regulator field which is neither T q nor T g . 1 The goal of this paper is to derive the corresponding formulas for the renormalized operators (T qR ) α α and (T gR ) α α .

Nucleon gravitational form factors
The scale-dependence of T R q,g essentially determines the scale dependence of the so-called gravitational form factors. The nonforward nucleon matrix element of T µν q,g and (T µν q,g ) R can be parametrized as

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where ∆ µ = P ′µ − P µ is the momentum transfer andP µ ≡ P µ +P ′µ 2 . M is the nucleon mass. The conservation of the energy momentum tensor implies that A for all values of ∆. The renormalized operators are given by and the form factors A R , B R , C R ,C R are renormalized at scale µ. Naively, since (F 2 ) R is now a finite operator, one might think that (T gR ) α α = 0. However, this is not the case, because renormalization and the trace operation do not commute in dimensional regularization (see, e.g., [11,12]). Taking the trace as well as the forward limit, we find and The mass of the nucleon is given by (3.8) Note that, in the chiral limit,C q = − 1 4 A q [13]. As suggested in [13], this relation does not hold for the renormalized quantities.
The µ-dependence of A R q,g (µ) is well known. Since A q,g are the matrix elements of the twist-two, spin-2 quark and gluon operators, their evolution is closed under evolution. To one-loop order, one finds [To simplify the notation, in the following we write α R s = g 2 R 4π ≡ α s for the renormalized coupling.] On the other hand, the µ-dependence ofC q,g can be obtained as follows. First one uses the identity [4,14] where the terms which vanish due to the equation of motion have been neglected. Similarly, Note that (3.10) and (3.11) are compatible with the condition ∂ ν (T µν q + T µν g ) = 0 thanks to the equation of motion D α F αν = gψγ ν ψ. Taking the matrix element of (3.10), one finds

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Therefore, the µ-dependence ofC q,g is governed by the anomalous dimension of the twist-4 operators gψF µν γ ν ψ and F µ ν D α F αν . The latter can be computed either directly, using the well-established techniques in the literature [15][16][17][18][19][20][21], or simply by noticing that it must coincide with the anomalous dimension of T µν q by virtue of the identity (3.10). One finds, in the chiral limit m = 0 [22], from which one would conclude thatC R q,g (µ) → 0 as µ → ∞ (in this chiral limit). We have computed the correction due to the quark mass with the result This implies that, in the forward limit ∆ → 0, We note that although (3.12) and (3.13) make sense only at nonzero momentum transfer ∆ = 0, after removing the common factor ∆ µ the limit ∆ → 0 can be safely taken to arrive at (3.16). However, eq. (3.16) is actually problematic. One would expect that the value ofC q,g should be at least partly constrained by the trace anomaly, but (3.16) appears to be insensitive to it. As we shall see in the next section, one has to include certain two-loop contributions in order to obtain the correct asymptotic limit ofC R .
4 One-loop renormalization of T R q,g Since the right hand sides of (3.4) and (3.5) are both renormalization group invariant, one may naively think that A q,g + 4C q,g is invariant under renormalization, i.e., A q,g + 4C q,g = A R q,g (µ) + 4C R q,g (µ). However, this is not the case. In this section we show that this quantity receives a finite renormalization.
For notational simplicity, let us write Then the energy momentum tensor is

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We introduce the renormalization constants as 2 To one-loop order [10] The mass anomalous dimension is given by γ m = 3C F αs 2π to this order. From the condition T µν = T µν R , we get the following relations 14) Let us write where we have taken into account the fact that the trace operation and renormalization do not commute and parameterized the possible anomalous terms by the unknown constants 3 ) to this order. Then the twist-two quark operator becomes 3

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The renormalization relation is (cf., (3.9)), From these two equations, we find (4.28) Combining these relations with (4.12)-(4.15), we obtain the unique solution to this set of equations and We thus arrive at

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In terms of the matrix element, Taking the ∂ µ -derivative of (4.8), we find Noting that Z B − Z Q 4 = 0 to one-loop, we see that the relation (3.15) is reproduced. On the other hand, (4.8) can be written as Taking the trace and the forward matrix element, we get In the last term we may replace F 2 → (F 2 ) R since the difference is O(α 2 s ). Then (4.40) becomes consistent with (4.37) after taking into account (3.5).
Eq. (4.40) shows that A R q +4C R q is RG-invariant to O(α s ), but it gets a finite renormalization with respect to the bare quantities. 4 From the RG equation ∂ ∂ ln µ (A q,g + dC q,g ) = 0, we can deduce that This can be rewritten as Using (4.42), one may write (4.41) But the combination Aq + dCq is more useful as it is directly related to the mass term as in (3.5).

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where A R q (∞) = n f 4C F +n f and in the second line we used A R q (µ) + 4C R q (µ) = A R q (∞) + 4C R q (∞) to this order. In the third line, we used (4.37). Formally, (4.44) is consistent with (3.16) to O(α s ) accuracy because P |mψψ|P 2M 2 − 1 = O(α s ) due to the relation (3.8). However, the perturbative result (3.16) misses the fact that the trace anomaly converts naively O(α s ) terms α s F 2 , α s mψψ into O(1) quantities. The asymptotic limit ofC R q (µ) in the chiral limit in this one-loop approximation can be directly read off from (4.37) . This is an order of magnitude larger than and has an opposite sign from the result of [22]. While the two results are not necessarily inconsistent, as they are obtained at different scales, a more detailed study is needed to clarify this issue (see (5.30) and (5.31) below).

Renormalization at two-loop
It is straightforward to generalize the result of the previous section to two-loop. The beta-function and the mass anomalous dimension to this order are 10 3 C A n f . The two-loop evolution of the twist-two matrix elements reads [23,24] This can be integrated as

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For the renormalization constants, we now have [10] and Using Mathematica, we have repeated the calculation in the previous section. The result for the renormalization constants is 14) JHEP12(2018)008 The anomaly coefficients in (4.17) and (4.18) are given by This leads to the main result of this paper, which is the two-loop version of (4.34)-(4.37) and, similarly, (4.40) now becomes

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On the other hand, in the forward limit ∆ = 0 we have an additional constraint, A R q (µ) + A R g (µ) = 1. Using this and (5.20), we can eliminate A R q,g from (5.23) and find 5 Finally, we use (3.8) to eliminate P | F 2 R |P , yielding In the chiral limit, the solution of this equation approaches (4.45) asymptotically.
The above derivation makes it clear that the µ-dependence ofC R q,g is completely fixed by the condition T µν = T µν R and the anomalous dimension of the twist-two matrix elements A R q,g . In view of this, the RG equation ∂C R q,g /∂ ln µ = · · · is somewhat redundant and can be even misleading as the naive counting in α s does not work. We actually know the explicit solution of this equation including the integration constants, see (4.36)-(4.37) and (5.19)-(5.20).
Even more explicit formulas can be derived by using the well-known expression with a certain starting scale µ 0 . Here, the ellipses denote the next-to-leading contributions associated with β 1 and Y of (5.2), namely, the order α 2 s contributions when expanded in 5 If we instead use A R q (µ) → 1 − A R g (µ) and (5.19) in (5.23), we obtain (5.26) up to extra terms that vanish with the use of the relation (3.8).

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the power series in α s . Substituting (3.8) into (5.20) to eliminate P | F 2 R |P , we obtain which reproduces (4.45), now taking into account the quark mass effect. Numerically, we haveC Thus, the important correction comes from the evolution of the twist-two form factor A R q , while the other corrections play a minor (∼a few percent) role.

Conclusions
In this paper we have studied the renormalization of the QCD trace anomaly separately for the quark and gluon parts of the energy momentum tensor. While the renormalization of the total anomaly T = T q + T g is well understood in the literature [10], our analysis at the quark and gluon level has revealed some interesting new features. The bare and renormalized (T q,g ) α α differ by finite operators, and this difference can be systematically computed order by order in α s . It is interesting to notice that, at one loop, the renormalized T q gives the n f part of the beta function. However, this property no longer holds at twoloop, see (5.19). Besides, the partition of the total anomaly can be different if one uses other regularization schemes (see, e.g., the 'gradient flow' regularization [25]), and it is interesting to study their mutual relations. We have also found thatC q,g (µ) does not go to zero as µ → ∞ even in the chiral limit, contrary to what one would naively expect from the one-loop calculation (3.16).

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Our result has interesting phenomenological implications. In [8], the relation between F 2 and (T q,g ) α α has been worked out for the bare quantities. If a more careful analysis reveals that one should use the renormalized relation, the numerical result in [8] may have to be revised. Another place whereC q,g plays a role is the nucleon's transverse spin sum rule. It has been shown in [26][27][28] (see also [29]) that Ji's sum rule [1] does not hold for a transversely polarized nucleon unless the nucleon is at rest. One has, for the quark/gluon total angular momentum J q,g , J q,g = 1 2 (A q,g + B q,g ) + f (P z )C q,g (6.1) where f (P z ) is a frame-dependent function (depends on the nucleon longitudinal momentum P z ) which vanishes at P z = 0 and approaches 1 2 as P z → ∞. Asymptotically, 1 2 (A q + B q ) = 0.18 whileC q = −C g ≈ −0.15 for n f = 3, so the effect of the last term can be actually quite significant.