# On the hadron mass decomposition

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## Abstract

We argue that the standard decompositions of the hadron mass overlook pressure effects, and hence should be interpreted with great care. Based on the semiclassical picture, we propose a new decomposition that properly accounts for these pressure effects. Because of Lorentz covariance, we stress that the hadron mass decomposition automatically comes along with a stability constraint, which we discuss for the first time. We show also that if a hadron is seen as made of quarks and gluons, one cannot decompose its mass into more than two contributions without running into trouble with the consistency of the physical interpretation. In particular, the so-called quark mass and trace anomaly contributions appear to be purely conventional. Based on the current phenomenological values, we find that in average quarks exert a repulsive force inside nucleons, balanced exactly by the gluon attractive force.

## 1 Introduction

According to the standard model of particle physics, the masses of almost all known elementary particles are generated through the Brout–Englert–Higgs (BEH) mechanism. The current light quark masses obtained from such a mechanism correspond however to only about 1% of the nucleon mass. Therefore, the mass of the ordinary matter around us essentially finds its origin in the strong interactions which confine quarks and gluons inside hadrons, and not the BEH mechanism [1, 2].

Understanding the origin of the hadron mass in Quantum Chromodynamics (QCD) represents a formidable challenge owing to the relativistic, quantum and non-perturbative nature of the problem. The QCD lagrangian in the chiral limit appears to be scale invariant at the classical level. This implies in particular that all the hadron masses should vanish in that limit. Scale invariance is however an anomalous symmetry, in the sense that it appears to be broken at the quantum level by radiative corrections. This gives rise, via dimensional transmutation, to a dimensionful parameter \(\Lambda _\text {QCD}\approx 0.2\) GeV in the theory [3, 4], and therefore to a non-trivial spectrum.

Lattice QCD calculation of hadron masses obtained from the analysis of correlations functions in Euclidean time is in remarkable agreement with the experimental spectrum [5, 6]. Unfortunately, this method gives little insight on how these masses arise from quark and gluon contributions.

*m*is the quark mass matrix and \(\gamma _m\) is its anomalous dimension. The first term is known as the trace anomaly, which is a pure quantum effect as indicated by the \(\beta \) function factor. In the chiral limit \(m\rightarrow 0\), this term prevents the trace of the EMT to vanish, and hence prevents QCD to become a scale-invariant theory with trivial spectrum.

Mass decompositions can be obtained from the expectation value of the EMT. In the next Section, we briefly present the two standard decompositions used in the literature, and point out a couple of issues regarding their physical interpretation. To the best of our knowledge, some of these issues have not been addressed before, in particular regarding pressure effects. Based on the semiclassical picture, we propose in Sect. 3 a new decomposition free of these issues. Then we discuss in Sect. 4 the new picture in more detail and we estimate the various contributions using our current phenomenological knowledge. Finally, we summarize our results in Sect. 5.

## 2 A critique of the standard decompositions

### 2.1 Trace decomposition

*P*and spin \(j\le \tfrac{1}{2}\) reads [16]

*M*can then be defined via the trace of the EMT [16, 17, 18, 19, 20, 21, 22, 23]

In our opinion, frame dependence is not really a problem since physical quantities often find a simple interpretation only in some particular set of reference frames. For instance, using the light-front form of dynamics [28], Eq. (8) becomes \(\langle T^\mu _{\mu }\rangle =M^2/P^+\) which can be interpreted in the symmetric frame (defined by \(\varvec{P}_\perp =\varvec{0}_\perp \)) as twice the light-front energy \(\langle T^\mu _{\mu }\rangle |_{\varvec{P}_\perp =\varvec{0}_\perp }=2P^-\). Frame dependence^{1} seems to be a generic feature associated with many decompositions of a system. If one insists on manifest Lorentz invariance,^{2} it turns out that Lorentz non-invariant quantities defined in a given frame can be formally expressed as Lorentz scalars [7, 29]. By construction, the latter will give the same result in any frame, and will coincide with the original non-invariant quantities in the given frame. Take for example the energy \(p^0\) which is obviously not a Lorentz-invariant quantity. The rest-frame energy though can be written as a Lorentz scalar \(p^0|_\text {rest}=p^2/m\). More generally, the energy in a given reference frame determined by the time-like unit four-vector \(u^\mu \) is given by \(p\cdot u\). For the rest frame, we simply have \(u^\mu =p^\mu /m\). Other typical examples are the proper time and length which are Lorentz-invariant quantities interpreted respectively as the time and length measured in the rest frame of the system. In other words, it is in principle always possible to express quantities in a Lorentz-invariant form, but the price to pay is that the physical interpretation necessarily singles out a privileged frame. In the present case, the frame dependence of Eq. (8) originates from the fact that we are considering the operator \(\int \mathrm {d}^3r\,T^\mu _{\mu }(r)\), which is not Lorentz invariant unlike \(T^\mu _{\mu }(0)\). It is however in line with the standard expectation that mass effects in hadronic matrix elements should become negligible in the ultra-relativistic regime \(P^0\gg M\). If one insists on preserving manifest Lorentz invariance, one can alternatively consider the integral over the proper volume \(\frac{P^0}{M}\int \mathrm {d}^3r\,T^\mu _{\mu }(r)\). Frame dependence will disappear from the formal expressions but it will remain in their physical interpretation.

Beside the question of frame dependence, a more important point we would like to stress is that the physical interpretation should be directly based on the quantum operator. Providing an interpretation at the level of the matrix element may be misleading. For example, in Eq. (5) the hadron mass is connected to the trace of the EMT at the level of matrix elements, but the connection remains obscure at the operator level, especially when the operator is decomposed into several contributions. Indeed, the trace \(T^{\mu }_{\,\,\mu }\) involves beside energy \(T^{0}_{\,\,0}\) the spatial components \(T^{i}_{\,\,i}\) usually associated with normal stresses. It seems also counterintuitive that the contribution from the quark mass to the hadron mass is not proportional to *m*, but rather to \(\sqrt{m}\). In fact, there exist (infinitely) *many* operators whose forward matrix elements are proportional to some power of the hadron mass. Following the same logic as for the trace operator, one could then in principle argue that these operators can be used to provide alternative definitions (and hence alternative decompositions) of the hadron mass, raising the question of deciding which one is the “correct” one. This happens simply because the hadron mass gives the natural scale of the matrix elements. The only way out is to use a quantum operator with a clear connection to the concept of mass.

In view of all these caveats, we feel that claims about the physical origin of hadron mass based on Eq. (5) are somewhat misleading. Note also that the above remarks can similarly be applied to the Gell-Mann–Oakes–Renner formula [30].

### 2.2 Ji’s decomposition

^{3}but it is well suited for treating the trace anomaly contribution. Using the relativistic normalization (4) for the states, the corresponding forward matrix elements read

^{4}

Ji’s decomposition is sometimes criticized because it is performed in a specific frame and applies to massive states only [25, 26]. As we already argued in the previous section, the frame dependence is not really a problem but a general feature. Note also that Ji’s decomposition can formally be put in a covariant form by considering the Lorentz-invariant quantity \(\langle T^{0\mu }u_\mu \rangle \), where \(u^\mu \equiv P^\mu /M\) is interpreted as the hadron four-velocity. Although this form is frame independent, its physical interpretation becomes simple only in the rest frame, where \(u^\mu =(1,\varvec{0})\). For a massless state, since there is no rest frame, one can consider instead the energy decomposition in any frame. In this case, there is no contribution from the trace part since \(P^2=0\) and so only the coefficient \(a(\mu ^2)\) is needed.^{5}

The actual problem with Ji’s decomposition is that the separation of the EMT into traceless and trace parts is not inconsequential for the physical interpretation of the individual contributions.^{6} Namely, although \(T^{00}\), \(\bar{T}^{00}\) and \(\hat{T}^{00}\) all have the dimension of energy densities, they actually correspond to different thermodynamic potentials as we will show in the following. By focusing on the \(\mu =\nu =0\) component in the rest frame, Ji’s decomposition does not make any distinction between the Lorentz tensors \(P^\mu P^\nu \) and \(M^2\eta ^{\mu \nu }\), and therefore disregards pressure effects.

To sum up, although all terms in Ji’s decomposition (27) can formally be defined and evaluated, they cannot however be interpreted as pure mass contributions. In the next section, we will explain how to achieve a proper mass decomposition based on a more covariant treatment using the semiclassical picture.

## 3 A new decomposition

*M*and spin \(j\le \tfrac{1}{2}\) [7, 39, 40], we obtain in the forward limit

*V*denoting the hadron proper volume,

*M*/

*V*. In a generic frame, we can write

*V*by the subsystem

*i*pushing (\(p_i>0\)) or pulling (\(p_i<0\)) the rest of the system.

## 4 Discussion

In the previous section, we observed that the generic form of the EMT for a hadron with spin \(j\le \tfrac{1}{2}\) is characterized by two Lorentz scalar quantities similarly to that of an element of perfect fluid. We are of course not claiming that hadrons consist of a set of perfect fluids,^{7} let alone that a hydrodynamical description is quite difficult to justify in this context. Forward matrix elements of the EMT, like the ones we considered, allow us to determine only *static* mechanical properties of the state. In the semiclassical limit, these matrix elements can effectively be thought of as a continuum description averaged over time [34, 44, 45]. We simply adopted the terminology of Continuum Mechanics to identify the physical meaning of the various energy-momentum form factors.

*effective*(time-averaged) coupled multifluid picture of the hadron [46], where each species of the mixture is regarded as a separate continuum coexisting with the continuums made up of other species. Each continuum is then described by its own (partial) energy density \(\varepsilon _i\) and pressure \(p_i\). Since these quantities are averaged over the hadron proper volume, they trivially obey a barotropic equation of state for species

*i*

### 4.1 Back to the old decompositions

Considering the trace of Eq. (37) shows that the trace decomposition in Eq. (6) does not represent a decomposition of the total energy of the system, but rather a decomposition of the interaction measure \(M=\sum _i(U_i-3W_i)=\sum _i(\varepsilon _i-3p_i)V\). Each individual contribution involves, beside internal energy \(U_i\), the partial pressure–volume work \(W_i\). For a stable system, the total pressure–volume work \(\sum _iW_i=0\) has to vanish, explaining why one obtains at the end just the total mass *M* of the system.^{8}

*does not*mean however that gluons are at the origin of most of the light hadron masses. It actually indicates that

^{9}

*a*and

*b*. In the effective coupled two-fluid picture, they correspond therefore to different linear combinations of energy density and pressure owing to Eq. (40). This is of course not physically acceptable, in the sense that one is adding apples and oranges. It is however mathematically correct since the sum over species gives at the end just the hadron mass

*M*, once again thanks to the stability constraint \(\sum _iW_i=0\). The only way to make physical sense out of Ji’s decomposition within the effective coupled two-fluid picture is to define the quark mass and trace anomaly contributions \(\varepsilon _{m,a}\) from the onset, and to split the quark and gluon energies as follows

What Ji did actually in Eq. (13) is a covariant decomposition of the QCD EMT into *four* parts. He considered with Eqs. (19)–(22) that each quantity \(\bar{T}^{00}_q\), \(\bar{T}^{00}_g\), \(\hat{T}^{00}_m\), and \(\hat{T}^{00}_a\) should represent an energy density, and hence *implicitly* adopted an effective coupled four-fluid picture of the hadron. The problem with such a description is that e.g. gluons carrying kinetic and potential energies are considered as *different* from those involved in the trace anomaly, in the sense that they are effectively treated as two distinct continuums with their own equations of state. Our opinion is that this is not acceptable from a physical point of view. Since hadrons are composed of quarks and gluons, an effective description of hadrons in terms of only two continuums is more natural.

### 4.2 Virial decomposition

In the effective coupled two-fluid picture, the hadron mass is decomposed into quark and gluon internal energies \(M=U_q+U_g\), where \(U_{q,g}=\langle T^{00}_{q,g}\rangle |_{\varvec{P}=\varvec{0}}\) with \(T^{\mu \nu }_{q,g}\) given by Eqs. (49) and (50). The question now is whether one can further decompose into parton kinetic and potential energies, quark mass and trace anomaly contributions, similarly to Ji’s decomposition.

*c*is then chosen such that \(\tilde{T}^{00}_q\) takes the form \(\psi ^\dag (-i\varvec{D}\cdot \varvec{\alpha })\psi \) upon using the QCD equations of motion, leading us to

*i*naturally leads to Eq. (12). Keeping only the internal energy contributions leads to the following finer decomposition of the hadron mass in the effective coupled two-fluid picture

We refrain from interpreting \(\check{U}_q\) as quark mass contribution and \(\hat{U}_g\) as trace anomaly contribution. Indeed, in the classical limit \(\gamma _m\rightarrow 0\) we have \(\check{U}_q=U_q\) which should also include quark kinetic and potential energies. Moreover, the contribution \(\hat{U}_g\) does not vanish when the trace anomaly is set to zero \(\langle \hat{T}^{\mu \nu }_g\rangle =0\). By keeping only the internal energy contributions, we lost the direct connection with the matrix elements and hence a simple physical interpretation. At best, our finer hadron mass decomposition (57) can be seen as some sort of virial decomposition [31, 32, 33].

*i*, we obtain the analogue of Eq. (12) for pressure–volume work

The above finer decompositions are not based on the nature of the constituents but on Lorentz symmetry and this has a dramatic impact on the physical picture. Indeed, we were not able find a simple physical interpretation for the individual contributions to the decompositions (57) and (65), because the effective coupled two-fluid picture forced us to discard some of the terms in the matrix elements for consistency. In order to avoid this, one has to adopt a picture where the number of effective fluids is at least equal to the number of contributions. In the present case, we need at least an effective four-fluid picture, leading us back directly to Ji’s decomposition. We have however already argued that the four-fluid picture is not natural from the physical point of view, since gluons carrying kinetic and potential energies are the same as those involved in the trace anomaly. Similarly, quarks carrying kinetic and potential energies are the same as those characterized by mass *m*.

*fixed*equations of state. This can be seen as some sort of decomposition onto a basis. For example, through a decomposition of the EMT into traceless and trace parts \(T^{\mu \nu }_i=\bar{T}^{\mu \nu }_i+\hat{T}^{\mu \nu }_i\), the set \(\{\varepsilon _i,p_i\}\) can formally be replaced by \(\{\bar{\varepsilon }_i,\hat{\varepsilon }_i\}\) with \(\bar{\varepsilon }_i=\tfrac{3}{4}\left( \varepsilon _i+p_i\right) \) and \(\hat{\varepsilon }_i=\tfrac{1}{4}\left( \varepsilon _i-3p_i\right) \), since traceless EMT \(\bar{T}^{\mu \nu }_i\) are characterized by the equation of state \(\bar{p}_i=\tfrac{1}{3}\,\bar{\varepsilon }_i\) and pure trace EMT \(\hat{T}^{\mu \nu }_i\) are characterized by \(\hat{p}_i=-\hat{\varepsilon }_i\). In Ji’s decomposition, the gluon contribution is indeed divided into kinetic and potential energies treated as a pure radiation \(w_g=\tfrac{1}{3}\), and trace anomaly treated as a cosmological constant \(w_a=-1\). Since the choice of a basis is not unique, the decomposition based on Lorentz symmetry is purely conventional, and hence artificial. In fact, Lorentz symmetry has

*already*been used to provide a physical interpretation of the various components of the EMT, namely by distinguishing in our case energy density \(\varepsilon _i\) from pressure \(p_i\) in Eq. (37). One can then easily understand why using Lorentz symmetry

*again*to perform a decomposition amounts to choosing an arbitrary basis of EMT with fixed equations of state not determined by the physics of the problem.

### 4.3 Phenomenology

In practice one cannot extract directly from an experiment the matrix elements of the EMT owing to the weakness of the gravitational force. However, all that is needed to characterize the quark and gluon EMT (33) are the form factors \(A_{q,g}(0)\) and \(\bar{C}_{q,g}(0)\) in the forward limit, and these can be extracted from other more directly accessible physical amplitudes thanks to operator identities. Using the energy-momentum sum rules (35), we can reduce this set to e.g. the quark form factors \(A_q(0)\) and \(\bar{C}_q(0)\), or equivalently Ji’s coefficients *a* and *b* owing to Eq. (34). In the following, we will consider only the proton case and fix the renormalization scale to \(\mu =2\) GeV.

The parameter *a*, which is also interpreted as the average fraction of hadron momentum carried by quarks, can be extracted from deep-inelastic lepton-proton scatterings. A recent global analysis with leading-order parametrization obtained \(a=0.546\pm 0.005\) [54]. The parameter *b* is related to the scalar charge of the proton and has been estimated to \(b=0.113\pm 0.010\) by Gao et al. [2], based on a recent determination of the pion–nucleon \(\sigma \)-term [55], a recent lattice calculation of the strangeness content [56] and neglecting the heavy quark contributions. It has also been suggested to extract the parameter *b* from the trace anomaly using quarkonium-hadron scattering close to threshold [21, 57]. For completeness, the anomalous quark mass dimension is approximatively given by \(\gamma _m\approx -0.15\) for \(n_f=3\) active flavors [58]. The various decompositions obtained with these values are depicted in Figs. 1, 2, 3 and 4.

In Fig. 1 we represent the trace decomposition given by Eq. (6) and determined only by the parameter *b*. As already discussed, it is largely dominated by the gluon contribution and is at the origin of the claim that most of the nucleon mass comes from gluons [17, 24, 25, 26, 27]. The trace of the EMT being given by \(T^\mu _{\mu }=T^{00}-\sum _jT^{jj}\), what the quark and gluon contributions in Eq. (6) do actually represent are the combinations \(U_{q,g}-3W_{q,g}\). Since the total pressure–volume work vanishes, one artificially supresses one of the contributions in favor of the other with pressure effects. As we will see below, it turns out that \(W_q=-W_g>0\) which explains why the quark contribution appears to be much smaller than the gluon contribution. Because of these pressure effects, the sole trace of the QCD EMT does not provide enough information to determine the actual quark and gluon contributions to the hadron mass.

*a*and

*b*, is included in this decomposition. As argued in the previous sections, a decomposition of the hadron mass into four contributions is however artificial. In the effective coupled two-fluid picture, the four contributions appear to be combinations of internal energies and pressure–volume works

^{10}In the effective coupled two-fluid picture, they are explicitly given by

*a*and

*b*appear to characterize two different hadron properties of the hadron, namely mass and pressure–volume work. Accordingly, instead of a single decomposition of the hadron mass into four contributions like Ji’s, we propose to consider separately the decompositions of hadron mass and pressure–volume work into two contributions, given by Eq. (72) and represented in Fig. 4. We do not require any splitting into traceless and trace parts, and hence we do not fix a priori the relation between energy density and pressure. Note that all the standard energy conditions are satisfied [59, 60]

## 5 Conclusions

We used forward matrix elements of the energy-momentum tensor to characterize static mechanical properties of hadrons. The components of such an energy-momentum tensor can be interpreted semi-classically in terms of parton energy density and pressure averaged over time and the hadron proper volume.

We showed that, because of pressure effects, the physical interpretation of the standard decompositions of the hadron mass have to be considered with care. In the trace decomposition, a pressure–volume work contribution artificially emphasizes the role played by gluons. In Ji’s decomposition, the splitting of the quark and gluon energy-momentum tensors into traceless and trace parts mixes the information about the hadron mass budget with the pressure–volume work budget. This can be understood by the fact that Lorentz symmetry is already used to provide a physical interpretation of the various components of the energy-momentum tensor. It cannot be used a second time to define separate mass contributions without modifying the physical picture. In particular, from the point of view of physical interpretation, using Ji’s decomposition amounts to treating gluons involved in the trace anomaly and those carrying kinetic and potential energies as separate entities with different equations of state, which is not physically acceptable.

Since hadrons are made of quarks and gluons, it is natural to decompose their mass into two contributions only. Any further decomposition that is not based on the properties of the constituents will be somewhat arbitrary and will mix internal energy with pressure–volume work. This mixing is mathematically harmless because the total pressure–volume work vanishes for a stable system, but it is a problem for the physical interpretation of the individual contributions.

We proposed a new picture, where the hadron mass and pressure–volume work budgets are kept separate and expressed in terms of the sole quark and gluon contributions which are physically unambiguous. In particular, unlike Ji’s decomposition we do not fix a priori the equations of state for quarks and gluons.

Finally, we quantitatively compared the different decompositions based on recent phenomelogical estimates. It turned out that, as expected for highly relativistic systems, pressure–volume work contributions are of the same order of magnitude as internal energy contributions. In particular, quarks are responsible in average for the repulsive force and gluons for the attractive force inside nucleons.

Having clearly identified the pressure contributions opens many interesting applications related to compact stars. For example, determining the quark and gluon equations of state inside a nucleon may give important clues about the internal structure of compact stars.

## Footnotes

- 1.
Note that in the light-front form of dynamics, boost invariance is sometimes incorrectly called frame independence.

- 2.
Sometimes people consider that physical quantities have to be Lorentz invariant, and so they conclude that decompositions are often unphysical and hence uninteresting or irrelevant. Interestingly, this seems to be the dominant thought regarding diffeomorphism invariance in General Relativity, but alternative descriptions based on the concept of vielbein are more flexible on the matter. We feel that this is too strict a requirement, for that many measurable quantities like e.g. energy and spin would be considered as unphysical.

- 3.
The general trace part is defined as \(\hat{T}^{\mu \nu }_c=\hat{T}^{\mu \nu }+c\,\bar{T}^{\mu \nu }\) with

*c*an arbitrary constant. - 4.
Strictly speaking, the equality holds only at the level of matrix elements.

- 5.
Note that one could still formally use Ji’s decomposition based on the covariant quantities \(\lim _{P^2\rightarrow 0}\langle T^{0\mu }P_\mu \rangle /P^2\) or \(\partial \langle T^{0\mu }P_\mu \rangle /\partial P^2|_{P^2=0}\), which are in the spirit of Ji’s remark [32] that although the

*overall scale*is essentially determined by the anomaly in the light hadron sector, the*relative magnitudes*of the various contributions reflect essential aspects of the underlying quark-gluon dynamics in the non-perturbative regime. - 6.
Paying attention not to introduce spurious contributions that sum up to zero in a decomposition is a general problem. For example, the proper definition of angular momentum at the level of spatial distribution has recently been discussed in detail in [38], where the problem was solved by treating with care all contributions that vanish under integration.

- 7.
- 8.
In principle, one can construct infinitely many decompositions of the form \(M=\sum _i(U_i+c W_i)\) with

*c*an arbitrary constant. - 9.
- 10.
Note that the appearance of \(M_m\) in the expression for \(\mathcal W_q\) comes from the reshuffling the quark mass terms between the quark traceless and trace parts (54).

## Notes

### Acknowledgements

This work is a result of discussions held at the workshop “The Proton Mass: At the Heart of Most Visible Matter” at the ECT* Trento, on 3–7 April 2017. This work has been supported by the Agence Nationale de la Recherche Under the project ANR-16-CE31-0019.

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