Revisiting the proton mass decomposition

Different decompositions (sum rules) for the proton mass have been proposed in the literature. All of them are related to the energy-momentum tensor in quantum chromodynamics. We review and revisit these decompositions by paying special attention to recent developments with regard to the renormalization of the energy-momentum tensor. The connection between the sum rules is discussed as well. We present numerical results for the various terms of the mass decompositions up to 3 loops in the strong coupling, and consider their scheme dependence. We also elaborate on the role played by the trace anomaly and the sigma terms.


I. INTRODUCTION
The most important properties of hadrons are global quantities such as their mass and spin. It is therefore natural to ask how these properties can be understood in quantum chromodynamics (QCD), the microscopic theory of the strong interaction. In this context, the QCD energy-momentum tensor (EMT) plays a key role. The matrix elements of the EMT give direct access to the mass, spin [1], angular momentum [2][3][4], and even pressure and shear distributions inside hadrons [5][6][7][8][9]. Extracting information on the EMT from experiment is difficult, though first proof-of-principle studies exist [10,11]. The EMT form factors, which parameterize the matrix elements of the EMT, have been computed in different models (see Ref. [12] for a review) and from first principles in lattice QCD (LQCD) [13][14][15][16][17][18].
In the present study, we will focus on the mass of the proton and how it could be decomposed into contributions from the mass and the energies of the partons. Different decompositions (sum rules) of the proton mass can be found in the literature [19][20][21][22][23][24]. All of them are related to the EMT and focus on either its 00-component (energy component) or its trace, including the EMT trace anomaly [25][26][27]. From the experimental point of view, input on the relevant matrix elements comes from the parton momentum fractions and, potentially, from photo-and electro-production of quarkonia close to the kinematical threshold [28][29][30][31][32][33][34][35][36].
Specifically, we explore the following mass sum rules: (i) a decomposition by Ji into four terms [19,20]; (ii) two decompositions by Lorcé, one with two terms and one with four terms [22]; (iii) a decomposition by Hatta, Rajan, Tanaka into two terms [23,24]. We review and, to some extent, revisit these sum rules. An important element of the discussion is the decomposition of the trace of the EMT into quark and gluon contributions for which a renormalization scheme must be chosen [23,24]. The EMT renormalization leads to the fact that perturbative QCD enters the decomposition of the proton mass. We have obtained numerical results for the sum rules up to 3 loops in the strong coupling, where perturbation theory enters because of the renormalization of the EMT. The mentioned scheme dependence influences the numerics of all the mass decompositions. For the analytical part of our work we follow closely our recent paper in which we have explored these mass sum rules for an electron target in QED [37]. We also refer to [38] for a related early study of the electron mass decomposition.
We discuss the differences and the similarities of the various sum rules. In fact, using suitable and properly renormalized operators fully reveals the overlap between the mass decompositions. We also emphasize that different (partial) operators showing up in the EMT provide the same forward matrix elements, which makes it difficult to quantify the contribution of a specific operator to the mass decomposition. This feature also applies to the operator associated with the EMT trace anomaly. Related to that point, we argue that any potential decomposition of the proton mass can at most have two independent terms. This is due to the fact that, in the forward limit, the EMT has two independent form factors only, as has already been emphasized in Ref. [22]. While the numerical values of the sum rules are quite stable upon including higher-order terms in perturbation theory, we observe a considerable scheme dependence of the results. We also discuss the role played by the quark mass terms (sigma terms) for the proton mass. Nonzero sigma terms are a consequence of the Higgs mechanism. According to current phenomenology, the proton mass receives a sizable contribution from the sigma term for charm quarks.
The paper is organized as follows: In Sec. II, we review the EMT in QCD by paying particular attention to its renormalization. In Sec. III, we discuss the different mass sum rules for the proton and present numerical results for the various terms up to 3 loops in the strong coupling. We summarize our work in Sec. IV. In Appendix A, we give a brief account of the decomposition of the EMT trace into quark and gluon contributions in the MS and MS renormalization schemes.

II. THE ENERGY-MOMENTUM TENSOR
For any field theory, the (canonical) EMT is the Noether current associated with the space-time translational invariance of the Lagrangian. It therefore satisfies the continuity equation for an arbitrary space-time point x = (t, x). Generally, the canonical EMT is not symmetric under the exchange µ ↔ ν. However, the Belinfante-Rosenfeld procedure allows one to symmetrize the EMT by adding a superpotential [39][40][41]. Here we focus on the symmetric EMT in QCD since the antisymmetric part of the EMT does not contribute to forward matrix elements which matter for the mass sum rules.
The (symmetric) EMT of QCD is decomposed into a quark and gluon part according to where in Eq. (3) a summation over quark flavors is understood, and in Eq. (4) a summation over gluon colors is understood. In Eq.
4π and the SU(3) color matrix T a . Because of the covariant derivative, the quark part of the EMT also contains a gluonic component.
Note that in Eqs. (3) and (4) renormalization of the parameters of the QCD Lagrangian is implied. The (conserved) total EMT is not renormalized, i.e., it requires no renormalization beyond the one of the Lagrange density. On the other hand, the individual quark and gluon parts of the EMT must be renormalized. For the discussion of this point we follow closely Refs. [23,24] and introduce the following operators/notation: 1 which allows us to write The four operators in Eqs. (5), (6) mix under renormalization. Specifically, one has The renormalization of the full EMT involves 10 renormalization constants. Note that the operator O 4 is not renormalized. The constants Z F and Z C are associated with the renormalization of the EMT trace, which is given by the well-known result [25][26][27]42] 1 To simplify the notation, we omit the tensor indices in the operators O i .
with the anomalous dimension for the quark mass γ m (g) and the QCD beta function β(g).
Taking the trace of the EMT in Eq. (2) and applying the classical equation of motion (iγ µ D µ − m)ψ = 0 would lead to mψψ. Hence the additional terms on the r.h.s. of Eq. (12) are pure quantum effects and denoted as trace anomaly. The renormalization constants Z L,T,Q,ψ are given by the evolution equations for the second moment of the flavor-singlet unpolarized parton distributions. This leaves us with Z M,S,B,K which, in dimensional regularization, can be computed through the other 6 renormalization constants via [24] which follow from the fact that the total EMT is invariant under renormalization.
Note that Eqs. (17)- (20) do not impose any constraint on x and y. These two parameters show up in the trace of the renormalized quark part T q,R and gluon part T g,R of the EMT, In other words, choosing x and y corresponds to a choice for the traces of the quark and gluon contributions of the EMT. Generally, fixing the finite parts of the renormalization constants is equivalent to selecting a renormalization scheme. The minimal subtraction (MS) scheme was picked in Refs. [23,24]. Here we will consider the following four schemes: • MS scheme: see Appendix A for details about fixing x and y.
• Modified minimal subtraction (MS) scheme: see Appendix A for more details.
• D2 scheme: x = y = 0. In this scheme, the entire trace anomaly is attributed to the trace of the renormalized gluon part T g,R of the EMT.
In the two D-type schemes, one finds the most natural decompositions of the total trace in Eq. (12) into the quark and gluon contributions in Eqs. (21), (22).

III. DECOMPOSITIONS OF THE PROTON MASS
We now proceed to discuss the sum rules for the proton mass. To this end, we consider the (forward) matrix element of the total EMT, and of its quark and gluon contributions.
For the full EMT we have where the proton state is characterized by the 4-momentum P µ = (P 0 , P) (with P 2 = M 2 ).
The forward matrix element of the EMT neither depends on the proton spin nor on the spacetime point x at which T µν is evaluated. Note that Eq. (23) holds in this form provided that the single-particle state is normalized according to P ′ |P = 2P 0 (2π) 3 δ (3) (P ′ − P). For the individual contributions to the EMT one finds [1] with the EMT form factors A i (0) andC i (0) (i = q, g) evaluated at vanishing momentum transfer ∆ = (P − P ) 2 = 0. The general form of the decomposition in Eq. (24) is valid for both the matrix element of the renormalized operators, T µν i,R , and the one of the bare operators, T µν i . (For ease of notation, we write A i (C i ) instead of A i,R (C i,R ) throughout this work.) The conservation of the total EMT imposes the following constraints on these form factors: In other words, the relations in (25) must be satisfied so that the sum of the quark and gluon contribution provides the r.h.s. of Eq. (23).
According to Eq. (23), the matrix elements of both the trace of the EMT and the component T 00 are directly related to the proton mass. Specifically, we find 1 2M as well as where the Eqs.
Therefore, in the rest frame of the proton, Eq. (27) is equivalent to the intuitive relation that is, the mass is just the expectation value of the QCD Hamiltonian. 2 Note that in Eq. (29) the delta-function divergence of the norm P |P is cancelled by a corresponding divergence in the numerator which arises from the spacial integral in Eq. (28).
We repeat that we will review the following mass sum rules for the proton: a four-term decomposition proposed by Ji in Ref. [19,20], a two-term and a four-term decomposition by Lorcé [22], as well as a two-term decomposition by Hatta, Rajan, Tanaka [23,24]. All these sum rules take as starting point either Eq. (26) or Eq. (27) (or equivalently Eq. (29)). For some more details on these sum rules we also refer to our previous study in Ref. [37]. We note in passing that very recent work on the mass structure of the nucleon as function of the parton momenta can be found in Refs. [43,44] A. Two-term decompositions We start with the two-term decomposition in Refs. [23,24], which is based on Eq. (26) and the decomposition of the trace of the EMT into its quark and gluon parts according to Eqs. (21), (22), The quark and gluon contributions in this sum rule have a simple relation to the form factors In contrast, the two-term sum rule of Ref. [22] focusses on T 00 , and decomposes this component of the EMT into the individual parton contributions, In terms of the EMT form factors one finds for the quark and gluon energies, which shows thatM i = U i (unlessC i = 0). On the other hand, the constraints in Eq. (25) guarantee the validity of both mass decompositions when summing over all partons. The (renormalized) operators associated with the U i have been identified in Ref. [37]. They read where the first term on the r.h.s. of Eq. (34) is the quark mass (so-called sigma term) contribution to M. The second term in that equation is typically referred to as kinetic plus potential energy of the quarks [19,20], while the operator in Eq. (35) represents the (total) energy stored in the gluon field. 3 Comparing both two-term decompositions in Eqs. (30), (32) and the underlying operators we find the relation that is, the sum of the quark energy and the gluon field energy coincides with the anomaly contribution to the proton mass.

B. Four-term decompositions
We now turn our attention to the four-term sum rules of the nucleon mass, and begin with the one proposed in Refs. [19,20]. In that case the focus is on the Hamiltonian density T 00 , and the EMT is decomposed into a traceless part and trace part according to with the trace term given byT µν = 1 4 g µν T α α . This immediately implies By means of Eq. (26)  Following Refs. [19,20] we use the decomposition into traceless and trace parts of the EMT in (37) also for individual partons, Because of Eqs. (21), (22), for both quarks and gluons the trace and the traceless part of the EMT involve mixing and an additional scheme dependence [23,24,37]. We repeat that this is a new development which was not taken into account in Refs. [19,20], even though the final form of the sum rule does not depend on this point. Using the expressions in Eqs. (21), (22) the quark and gluon contributions to the traceless and trace parts of 00-component of the The four terms in Eqs. (40)- (43) look simplest in the D1 scheme or D2 scheme discussed above. Their sum gives the proton mass according to but, in the spirit of Refs. [19,20], we want to form suitable linear combinations in order to recover the intuitive expressions of the quark and gluon energies from Eqs. (34), (35). To this end we consider with the constants , c ga = 1 .
where the individual operators are given by [37] We argue that Eqs. (51)- (53) are the appropriate operators for the mass sum rule if one follows the general logic of Ji's original work [19,20], which is (ultimately) based on a decomposition of the full 00-component of the EMT. The (renormalized) operators associated with this mass decomposition formally coincide with what one would find without renormalizing T 00 . We have arrived at a decomposition with three nontrivial terms only. Since the focus is on T 00 , the operators related to the trace, and in particular the trace anomaly, drop out from the final result. Let us briefly discuss this point which may be surprising at first sight.
In dimensional regularization, the entire anomaly derives from the bare gluon operatorsee, for instance, Ref. [23]. The time dimension, however, is left untouched in dimensional regularization, which implies that T 00 is rather special compared to the spatial components of the EMT trace. In fact, a careful analysis reveals that the anomaly is entirely contained in the spatial part of the EMT.
The main difference with regard to the work in Refs. [19,20] is the following. As mentioned in the previous paragraph, a priori, the trace anomaly should not show up in T 00 .
In Refs. [19,20], the total trace in Eq. (12) was used for computing the trace termT µν in Eq. (37). On the other hand, Eq. (12) was not used when subtracting the trace in order to findT µν in Eq. (37). As a consequence, at the operator level the results in Refs. [19,20] and those in Eqs. (51)-(53) differ by 1 4 of the trace anomaly -see also Ref. [37]. If one is just concerned with expectation values of the operators, which in principle is sufficient, then one could replace H q + H g (or part of it) in the sum rule through the anomaly operator by making use of Eq. (36). Based on our discussion above in relation to Eqs. (26), (27), (37), (38), one could also just use the traceless part or the trace part of the EMT for writing down a mass sum rule. In the former case all the operators in (51)- (54) plus the operator of the trace anomaly emerge (with appropriate weight factors), while in the latter case one just finds the relation between the proton mass and the EMT trace based on Eqs. (12), (26).
The three-term decomposition defined through the operators in Eqs. (51)-(53) can be considered a more detailed version of the two-term sum rule by Lorcé that is given by Eqs. (34), (35). The gluon sector is actually identical in both cases, but T 00 q,R in (34) is split into the quark energy term given by H q and the quark mass term given by H m . In Ref. [22] it has been emphasized that only the T 00 i,R have a clean interpretation as energy densities, whereas the traceless partsT 00 i,R and the trace partsT 00 i,R , which underly Ji's mass sum rule, get admixtures from pressure contributions. To see this, recall that according to Eq. (33) the T 00 i,R are related to the form factor combinations A i (0) +C i (0). On the other hand, the spacial diagonal elements of the EMT, that are associated with pressure, have the where the index j is not summed over. The form factorsC i (0) are thus directly related to pressures. We also have  48), on the gluon sector we actually add the traceless part and the trace part to recover T 00 g,R , so that H g in (53) has a clean interpretation as operator for the gluon energy density. However, the same does not apply for the quark sector, so that both H q in (51) and H m in (52), strictly speaking, indeed describe mixtures of energy densities and pressures. Nevertheless, it appears meaningful to split T 00 q,R in (34) into H q and H m . In this context recall also that the quark mass term H m has been studied intensively in lattice QCD and other approaches.
Finally, we address the four-term decomposition that has been discussed in Ref. [22], This sum rule also starts from the separation of the T 00 i,R into traceless and trace parts, and uses linear combinations in the spirit of Eqs. (45)- (48). However, in contrast to the fourterm sum rule discussed above, one picks from the traceless and the trace parts just the (fractional) contribution to the total energy density which, according to Eqs.
with the U i given in Eq. (33). For the gluon sector this logic just leads to the same result we have seen before for the two-term sum rule of Ref. [22] and the four-term sum rule expressed through Eqs. (51)- (54). The situation is different though for the quark sector withŨ q anď U q . While by construction these two terms can be interpreted as energy contributions, they merely represent linear combinations of total parton energies that are given by the two operators T 00 i,R in Eqs. (34), (35). Also, the (total) operators associated withŨ q andǓ q contain the scheme-dependent coefficients c qm and c qa = −c ma given in (49). For example, there is a nonzero contribution of U g toŨ q andǓ q in both the MS and MS schemes, which cancels out when summing the two terms. In contrast, simple expressions emerge in the According to (64), in the D2 scheme this decomposition actually coincides with the two-term decomposition of Ref. [22].
We conclude this subsection with a brief discussion about how many terms are actually independent for the various sum rules. For the two-term sum rule in Eq. (30), that is based on the trace of the EMT, there is of course just one independent term -once the quark contribution to this sum rule is known the gluon contribution is fixed as well and vice versa.
The same applies to the two-term sum rule in Eq. terms. This is actually the maximum possible number of independent contributions for any mass decomposition one could think of because the EMT in the forward limit is fully fixed by the form factors A i (0) andC i (0), which are subject to the constraints in (25). This means, there is two independent form factors only. A closely related discussion can be found in Ref. [22].

C. Numerical results
The previous paragraph implies that two independent numerical inputs suffice to fix all the terms of the different sum rules. One input/constraint comes from the parton momentum fractions a i in the proton through [19,20] where a q is a shorthand notation for the sum of the momentum fractions of all active quark flavors. The a i therefore determine the expectation values of the traceless operatorsT 00 i,R in Eqs. (40), (42). They satisfy the sum rule a q + a g = 1, which is equivalent to the constraint for the form factors A i (0) in (25). We take the quark mass term as the second independent input. Specifically, we define a parameter b according to Using Eqs. (12), (26) we then find for the gluon operator of the trace anomaly We also refer to [45] for a recent attempt to directly compute the gluon contribution to the EMT trace in lattice QCD. With these ingredients the terms in Eq. (44) can be written as while the three nonzero terms of Eq. (50) read Now we have collected all the equations that are needed for obtaining numbers for the various sum rules discussed above.
Our numerical results are for the scale µ = 2 GeV. We take the parton momentum fractions from the CT18NNLO parameterization [46], which in the case of four active quark flavors gives a q = 0.586 ± 0.013 , a g = 1 − a q = 0.414 ± 0.013 .
Other phenomenological fits of parton distributions provide very similar numbers -see, for instance, Refs. [47][48][49][50][51]. In order to fix the parameter b in Eq. (66) we use input for the quark mass term (sigma term), up to and including charm quarks, withm = (m u + m d )/2. For b we actually consider two cases. In the first, we take the sigma terms from an analysis in chiral perturbation theory in Refs. [52,53] for the three lightest quark flavors, The independent phenomenological determination in Ref. [54] gives a very similar value for σ πN . In the second scenario, we use results from lattice QCD (LQCD) which also include a sigma term for charm quarks [55], Other LQCD calculations, performed at (nearly) physical quark masses, mostly provide similar results for the sigma terms of the light quarks [56][57][58][59]. To the best of our knowledge, Ref. [55] is the only LQCD study which reports a value for the charm sigma term. The numerical values for σ πN and σ s are quite different for ChPT and LQCD. However, the difference for the sum σ πN + σ s is small and irrelevant for our purpose. On the other hand, including or not σ c has a clear impact on our numerics for the mass sum rules. To summarize this discussion, we consider numbers for the mass decompositions for the following two scenarios: • Scenario A: a i from (75); b from ChPT sigma terms in (77). • Scenario B: a i from (75); b from LQCD sigma terms in (78) including charm.
We will show results at 1-loop, 2-loop and 3-loop accuracies. For this we need the QCD beta function and the anomalous dimension of the quark mass through O(α 3 s ), where the explicit expressions for the coefficients β i and γ mi are given in Refs. [60,61]. we have explored two commonly used MS subtractions. They lead to somewhat different numbers for the proton mass decomposition.) There is a conceptual difference between the MS scheme and the D-type schemes in the context of our study. In principal, a fully consistent calculation in the MS scheme could be done, since all the numerical input that we use could be obtained in the MS scheme.
(Comparing the numerics for the MS scheme and, in particular, the MS scheme(s) should therefore be done with care.) In contrast, the D-type schemes have no meaning beyond fixing x and y, which means that the numbers in these two schemes cannot be "improved." However, according to Tab. II, the numerical values obtained in the MS scheme and the D-type schemes are not very different. All the following results in Tabs. III-VI are in the MS scheme, which was used in the previous studies in Refs. [23,24]. We also point out that obtaining results in the MS scheme(s) is somewhat cumbersome for the present study -see Appendix A for more details.
In Tabs Ref. [46]. On the other hand, we do not expect this point to have a significant impact on the qualitative outcome of a mild sensitivity to the loop-order. of, for instance, the expectation value (F αβ F αβ ) R could therefore help to clarify the role played by the Higgs mechanism for the numerics of the proton mass decomposition.
The contribution of the gluon energy M g to the proton mass is at least 50%. However, we repeat that the precise number depends on the renormalization scheme. We also find some negative contributions to mass um rules, namely the quark termM q in Tab is not renormalized, this contribution has no renormalization scheme dependence at all.

IV. SUMMARY
This work deals with the phenomenology of the decomposition of the proton mass in QCD.
We have reviewed and, to some extent, revisited several sum rules for the proton mass from the literature. All of them are based on forward matrix elements of certain components of the EMT in QCD. A key ingredient is the recently discussed decomposition of the EMT trace into quark and gluon contributions, which exhibits an additional dependence on a renormalization scheme [23,24]. We have obtained numerical results for the sum rules up to 3 loops in the strong coupling. The mentioned scheme dependence influences the numerics of all the mass decompositions. The analytical part of the present work is closely related to our recent paper in which we studied the very same mass sum rules for an electron in QED [37].
The following are the most important findings of our work: First, there is a close connection between the various sum rules, provided that properly constructed and renormalized operators are used -see also Ref. [37]. Here we discuss the decomposition of the trace of the EMT into individual contributions from quarks and gluons, which requires to fix x and y that show up in Eqs. (21), (22). In the D-type schemes presented in Sec. II we make motivated choices for these two parameters.
The focus of this appendix is on finding x and y in the MS and the MS schemes, where for the former we just outline the essential steps that were already given in Refs. [23,24].
We repeat that, according to Eqs. In the MS scheme, the renormalization constants take the form Z X = δ X,T + δ X,ψ + δ X,F + a X ǫ where δ X,X ′ denotes the Kronecker symbol, and a X , b X , c X are constants depending on α s , the number of quark flavors and color factors. In order to fix the values of x and y by means of Eqs. (13), (14), we need the results of the four renormalization constants Z T,F,L,C which can be found in Ref. [24] through O(α 3 s ). By taking the Laurent expansion of both sides of Eqs. (13), (14) about ǫ = 0, and collecting the O(ǫ 0 ) terms we find the relations where different conventions for the quantity S ǫ can be found in the literature. The definition in Ref. [63], to which we refer as MS 1 scheme, is given by Another frequently used convention, which we call MS 2 scheme [64], is Comparing Eqs. (A6) and (A7) shows that both schemes differ at O(ǫ 2 ) (and higher), which causes numerical differences for the present study of the proton mass decomposition -see the results for the MS 1 and MS 2 schemes in Tab. II. In the following we present the main steps that are needed to get x and y in a MS scheme, by showing the relevant equations for just the MS 1 scheme. In general, using as starting point the results for the renormalization constants in the MS scheme from Ref. [24], it is easier to find x and y in the MS scheme than in a MS scheme.
We first note that the divergent terms on the r.h.s. of Eqs. (A4) and (A5) must be identical, which implies The parameters x and y appear in the constant term of renormalization constants, which in the MS 1 scheme take the general form C MS 1 = α sā1 δ UV + α 2 s 2b 1 δ UV + 2b 2 δ 2 UV + π 2 6b 2