A QCD ANALYSIS OF THE MASS STRUCTURE OF THE NUCLEON?
Center for Theoretical Physics Laboratory for Nuclear Science and Department of Physics Massachusetts Institute of Technology Cambridge, Massachusetts 02139
arXiv:hep-ph/9410274v1 12 Oct 1994
MIT-CTP: #2368 HEP-PH: #9410274 (Submitted to: Physical Review Letters October 1994)
From the deep-inelastic momentum sum rule and the trace anomaly of the energy-momentum tensor, I derive a separation of the nucleon mass into the contributions of the quark and gluon kinetic and potential energies, the quark masses, and the trace anomaly.
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work is supported in part by funds provided by the U.S. Department of Energy (D.O.E.) under cooperative agreement #DF-FC02-94ER40818.
The nucleon derives its mass (939 MeV) from the quark-gluon dynamics of its underlying structure. However, due to the complexity of the low-energy Quantum Chromodynamics (QCD), a more detailed understanding of the nucleon mass seems di?cult. The lattice QCD is successful in reproducing the measured mass from the fundamental lagrangian , but the approach provides little insight on how the number is partitioned between the nucleon’s quark and gluon content. Years after the advent of QCD, our knowledge of the nucleon’s mass structure mostly comes from models: non-relativistic quark models, Bag models, the Skyrme model, string models, the Nambu-Jona-Lasinio model, to just name a few. Though all the models are made to ?t the mass of the nucleon, they di?er considerably on the account of its origin. Depending on di?erent facets of QCD the models are created to emphasize, the interpretations of the nucleon mass often go opposite extremes. In this Letter I show that an insight on the mass structure of the nucleon can be produced within QCD with the help of the deep-inelastic momentum sum rule and the trace anomaly. The result is a separation of the nucleon mass into the contributions from the quark, antiquark, gluon kinetic and potential energies, the quark masses, the gluon trace anomaly. Numerically, the only large uncertainty is the size of P |ms ss|P , the strange ? scalar charge of the nucleon. Some implications of this break-up of the masses are discussed following the result. Let me begin with the energy-momentum tensor of QCD, 1 ? ?(? 1 T ?ν = ψi D γ ν) ψ + g ?ν F 2 ? F ?α F να , 2 4 (1)
where ψ is the quark ?eld with color, ?avor, and Dirac indices; F ?ν is the gluon ?eld strength with color indices and F 2 = F αβ Fαβ ; and all implicit indices are summed over. The covariant ? ? ?? →? ←? →? → ←? ← derivative D = D ? D , with D = ? + igA? and D = ? ? igA? , where A? = A? ta is a the gluon potential. The symmetrization of the indices ? and ν in the ?rst term is indicated by (?ν). Eq. (1) is quite formal, for it contains neither the gauge ?xing and ghost terms, nor the trace anomaly. The ?rst type of terms are BRST-exact  and have vanishing physical matrix elements according to the Joglekar-Lee theorems . I will add the trace anomaly explicitly when the renormalization issue is dealt with. A few results about the energy-momentum tensor are well-known. First of all, it is a symmetric and conserved tensor, T ?ν = T ν? , ?? T ?ν = 0. (2)
Because of the second property, the tensor is a ?nite operator and does not need an overall renormalization. All the ?elds and couplings in Eq. (1) are bare and their divergence are cancelled by the standard set of renormalization constants. The only complication is the gluon part of the tensor cannot be renormalized with a vanishing trace (see below). Second, the tensor de?nes the hamiltonian operator of QCD, HQCD = d3 x T 00 (0, x), (3)
which is also ?nite and scale-independent. Third, the matrix element of the tensor operator in the nucleon state is , P |T ?ν |P = P ? P ν /M, 2 (4)
where |P is the nucleon state with momentum P ? and is normalized according to P |P = (E/M)(2π)3 δ 3 (0), and E and M is the energy and mass of the nucleon, respectively. Lastly, the trace of the tensor is  1 β(g) 2 ? ? T ?ν = g ?ν (1 + γm )ψmψ + F , 4 2g (5)
where m is a quark mass matrix, γm is the anomalous dimension of the mass operator, and β(g) is the β-function of QCD. At the leading order β(g) = ?β0 g 3 /(4π)2 and β0 = (11 ?2nf /3), where nf is the number of ?avors. The second term is called the trace anomaly and is generated in the process of renormalization. According to the above, the mass of the nucleon is M= P | d3 x T 00 (0, x)|P ≡ T 00 , P |P (6)
in the nucleon’s rest frame. Although I formally work with the matrix elements of the nucleon, it actually is the di?erence of the nucleon matrix elements and the vacuum matrix elements that enters all the formula (the vacuum has zero measurable energy density). According to (6), a mass separation can be found through a decomposition of T ?ν into various parts, which are then evaluated with the deep-inelastic momentum sum rule and the scalar charge of the nucleon. [Note that the parts of the energy-momentum tensor are not separately conserved, so the breaking of the nucleon energy cannot be Lorentz covariant.] First of all, let me decompose the T ?ν into traceless and trace parts, ? ? T ?ν = T ?ν + T ?ν , ? where T ?ν is traceless. According to Eq. (4), I have, 1 ? P |T ?ν |P = (P ? P ? ? M 2 g ?ν )/M, 4 1 ?ν ? P |T ?ν |P = g M. 4 Combining Eq. (6) with the above three equations, I get, 3 ? T 00 = M, 4 1 ? T 00 = M. 4 (10) (11) (8) (9) (7)
Thus 3/4 of the nucleon mass comes from the traceless part of the energy-momentum tensor and 1/4 from the trace part. The magic number 4 is just the space-time dimension. This decomposition, a bit like the virial theorem, is valid for any bound states in ?eld theory! The traceless part of the energy-momentum tensor can be decomposed into the contribution from the quark and gluon parts, ? ? ? T ?ν = Tq?ν + Tg?ν , where 3 (12)
1 1 ? ?(? ? ? Tq?ν = ψi D γ ν) ψ ? g ?ν ψmψ, 2 4 1 ? Tg?ν = g ?ν F 2 ? F ?α F να . 4
? ? Although the sum of Tq?ν and Tg?ν with bare ?elds and bare couplings is ?nite (now neglecting the trace anomaly), the individual operators are divergent and must be renormalized. Under renormalization, they mix with each other and with other BRST-exact and the equations of motion operators, which have vanishing physical matrix elements [2,3]. For my purpose, I regard both operators renormalized and dependent on a renormalization scale ?2 . De?ne their matrix elements in the nucleon state, 1 ? P |Tq?ν |P = a(?2 )(P ? P ν ? g ?ν M 2 )/M, 4 1 ? P |Tg?ν |P = (1 ? a(?2 ))(P ? P ν ? g ?ν M 2 )/M, 4 (15) (16)
where I have used Eq. (8) to get the second equation. The constant a(?2 ) is related to the deep inelastic sum rule , a(?2 ) =
f 1 0
x[qf (x, ?2 ) + qf (x, ?2 )]dx, ?
where the sum is over all quark ?avors and qf (x, ?2 ) and qf (x, ?2 ) are quark momentum ? distributions inside the nucleon in the in?nite momentum frame or light-front coordinate system. Again, according the Eq. (6), I ?nd the contribution to the nucleon mass, 3 ? Tq00 = a(?2 )M, 4 3 00 ? Tg = (1 ? a(?2 ))M. 4 (18) (19)
? Finally, I turn to the trace part of the energy-momentum tensor T ?ν . According to ?ν ?ν ? ? Eq. (5), I decompose it into Tm and Ta , the mass term and trace anomaly term, respectively. Both operators are ?nite and scale independent. If I de?ne, ? 00 b = 4 Tm /M, then according to Eq. (11), the anomaly part contributes, 1 ? 00 Ta = (1 ? b)M. 4 (21) (20)
Thus, the energy-momentum tensor T ?ν can be separated into four gauge-invariant parts, ? ? ? ?ν ? ?ν Tq?ν , Tg?ν , Tm , and Ta . They contribute, respectively, 3a/4, 3(1 ? a)/4, b/4, and (1 ? b)/4 fractions of the nucleon mass. The corresponding breakdown for the hamiltonian is, HQCD = ′ ′ Hq + Hg + Hm + Ha , with
′ Hq =
′ Hm =
3? ? d3 x ψ(?iD · α)ψ + ψmψ , 4 1 2 2 3 d x (E + B ), 2 1? 3 d x ψmψ, 4 9αs 2 3 dx (E ? B2 ). 16π
(22) (23) (24) (25)
where I have consistently neglected γm and the higher-order terms in β(g). One can put them back if a higher precision analysis becomes necessary. I also have taken nf = 3. [Note that the heavy quarks do contribute to the mass term, the kinetic and potential energy term, and the trace anomaly term. However, the contributions cancel each other in the limit of mf → ∞, and thus for simplicity I neglect them.] If I rearrange the mass terms by de?ning, Hq = Hm = then the QCD hamiltonian becomes, HQCD = Hq + Hm + Hg + Ha . (28) ? d3 x ψ(?iD · α)ψ, ? d3 x ψmψ, (26) (27)
Here Hq (Eq. (26)) represents the quark and antiquark kinetic and potential energies and contributes 3(a ? b)/4 fraction of the nucleon mass. Hm (Eq. (27)) is the quark mass term and contributes b fraction of the mass. Hg (Eq. (23)) is the normal part of the gluon energy and contributes 3(1 ? a)/4 fraction of the mass. Finally, Ha (Eq. (25)) is the gluon energy from the trace anomaly. It contributes (1 ? b)/4 fraction of the mass. To determine the decomposition numerically, I need the matrix elements a and b. The deep-inelastic scattering experiments have determined a(?2 ) with an accuracy of a few percent. Using a recent ?t to the quark distributions , aMS (1GeV2 ) = 0.55 where MS refers to the modi?ed minimal subtraction scheme. Without the heavy quarks, the matrix element b is, ? bM = P |mu uu + md dd|P + P |ms ss|P ? ? (30) (29)
The ?rst term is the πN σ-term apart from a small isospin-violating contribution of order 2 MeV. A most recent analysis gave a magnitude of 45±5 MeV for this term . So the only unknown in our analysis is the strange scalar charge P |ms ss|P in the nucleon. There are ? model calculations for this quantity in the literature . Here I choose to estimate it using two standard approaches, though both of them are not completely satisfactory. In the ?rst approach, the strange quark mass is considered small in the QCD scale and so the chiral perturbation theory can be used to calculate the SU(3) symmetry breaking e?ects. A recent second-order analysis on the spectra of the baryon octet combined with the measured σ-term yields , 5
? P |?s|P ? 0.11 × P |?u + dd|P s u ? 0.77,
where in the second line, I have used (mu + md )/2 ? 7 MeV at the scale of 1 GeV2 . Taking the strange quark mass to be 150 MeV at the same scale, I get, bM ? 160 MeV, Using Eq. (11), I have, P| αs 2 F |P = ?693 MeV. π (34) (33)
In the second approach, the strange quark is considered heavy in the QCD scale. Using heavy-quark expansion, it was found , ? P |mQ QQ|P = ? Thus the strange quark contribution in β(g) 2 ? F |P = M, P |ψmψ + 2g (36) αs 1 P | F 2 |P . 12 π (35)
which is an explicit form of Eq. (11), cancels. From the above equation and the σ-term, I ?nd, P| αs 2 F |P = ?740 MeV. π (37)
This yields a strange matrix element P |ms ss|P = 62 MeV. Together with the σ-term, I ? determine, bM = 107 MeV. (38)
The complete result of the mass decomposition at the scale of ?2 = 1GeV2 , together with the two numerical estimates, is shown Table 1. I have not shown the errors due to omission of higher-order perturbative e?ects and errors on the σ-term and current quark masses. The total e?ect on individual numbers is about 5 to 10 MeV. Thus I have rounded up the numbers to nearest 10 MeV. The largest uncertainty is from the matrix element P |ms ss|P , ? which could be larger than the di?erence of the two estimates shown. Nevertheless, I will argue below that the total strange contribution to the nucleon mass is quite small and with a smaller uncertainty. The following comments can be made with regard to the numerical result. ? The quark kinetic and potential energies contribute about 1/3 of the nucleon mass. Because the quark kinetic energy must be very large when con?ned within a radius of 1 fm, there must exists a large cancellation between the kinetic and potential energies. This may not be entirely surprising in the presence of strong interactions between quarks and gluons. Such strong interactions are clearly at the origin of the chiral symmetry breaking, modeled, for instance, by the Nambu-Jona-Lasinio . 6
? The decomposition of the quark energy into di?erent ?avors is possible. Taking the number 270 MeV (the ms → 0 limit) as an example, I ?nd the up-quark energy in the proton is 250 MeV using the momentum fraction carried by up quark 0.375 , the down quark energy, 105 MeV, and strange quark energy, ?85 MeV. Further decomposition into valence and sea contribution cannot be made without knowledge of the separate valence and sea contributions to the scalar charge. ? The quark mass term accounts for about 1/8 of the nucleon mass. About half of which or more is carried by the strange quark. The contributions from the up and down quarks are well determined by the σ-term. ? The normal gluon energy is about 1/3 of the nucleon mass and the trace anomaly part contributes about 1/4. From these two, I deduce the color-electric and color-magnetic ?elds in the nucleon separately (take αs (1GeV)? 0.4), P |E2|P = 1700 MeV, P |B2|P = ?1050 MeV. (39) (40)
So the magnetic-?eld energy is negative in the nucleon! This of course is due to a cancellation between the quark’s magnetic ?eld and that of the vacuum. The electric ?eld in the vacuum is presumably small, however, it is large and positive in the nucleon. This behavior of the color ?elds in presence of quarks is very interesting, it may help to unravel the structure of the QCD vacuum. ? In the chiral limit, the gluon energy from the trace anomaly (M/4) corresponds exactly to the vacuum energy in the MIT bag model . The role of such energy in the model is to con?ne quarks. Thus we see here a clear way through which the scale symmetry breaking leads to quark con?nement. To keep the con?nement mechanism, a model must include Ha , the anomaly part of the hamiltonian. ? The strange quark contributes about ?60 MeV through trace anomaly. When adding together with the kinetic and potential energy contribution ?85 MeV and the mass term 115 MeV (the ms → 0 limit) the total strange contribution to the nucleon mass is a mere ?30 MeV. (The other limit gives a total of ?45 MeV.) The smallness of the contribution is, to a large extent, insensitive to the matrix element P |ms ss|P . ? To summarize, we have ?nd a separation of the nucleon mass into contributions from the quark kinetic and potential energy, gluon energy, and the trace anomaly. The largest uncertainty is from the strange matrix element P |ms ss|P . The result has interesting ? implications on the quark-gluon structure of the nucleon and on the response of the QCD vacuum to color charges. I wish to thank D. Freedman, F. Low, and K. Johnson for useful discussions and suggestions.
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TABLE I. A decomposition of the nucleon mass into di?erent contributions. The matrix elements a and b are de?ned in Eqs. (15) and (20). mass type quark energy quark mass gluon energy trace anomaly Hi ? ψ(?iD · α)ψ ? ψmψ
1 2 2 2 (E + B ) 9αs 2 2 16π (E ? B )
Mi 3(a ? b)/4 b 3(1 ? a)/4 (1 ? b)/4
ms → 0(MeV) 270 160 320 190
ms → ∞(MeV) 300 110 320 210