Mass sum rule of hadrons in the QCD instanton vacuum
MMass sum rule of hadrons in the QCD instanton vacuum
Ismail Zahed ∗ Center for Nuclear Theory, Department of Physics and Astronomy,Stony Brook University, Stony Brook, New York 11794–3800, USA
We briefly review the key aspect of the QCD instanton vacuum in relation to the quan-tum breaking of conformal symmetry and the trace anomaly. We use Ji (cid:48) s invariant massdecomposition of the energy momentum tensor together with the trace anomaly, to discussthe mass budget of the nucleon and pion in the QCD instanton vacuum. A measure of thegluon condensate in the nucleon, is a measure of the compressibility of the QCD instantonvacuum as a dilute topological liquid. I. INTRODUCTION
A remarkable feature of QCD is that in the chiral limit it is a scale free theory. Yet,all hadrons are massive, composing most of the visible mass in the universe. The typicalhadronic scale is 1 fm, but where does it come from? The answer appears to be from asubtle quantum effect referred to as dimensional transmutation, and related to the quan-tum breaking of the conformal symmetry of QCD. This mechanism is non-perturbative.On the lattice, the lattice cutoff along with the running coupling combine to generate thisscale. In the continuum, to achieve this mechanism requires a non-perturbative descriptionof the vacuum state and its excitations.The QCD vacuum as a topological liquid of instantons and anti-instantons, offers byfar the most compelling non-perturbative description that is analytically tractable in thecontinuum, thanks to its QCD semi-classical origin and diluteness [1–3]. It is not the onlydescription. Other candidates based on center vortices and monopoles to cite a few [4],are also suggested and may as well be present in addition to the instantons. However, thelatters appear to trigger the dual breaking of conformal and chiral symmetry breaking, anddominate the vacuum state and its low-lying hadronic excitations. Center vortices maybeimportant for the disordering of the large Wilson loops and confinement, a mechanismlikely at work in the orbitally excited hadrons as they Reggeize.The spontaneous breaking of chiral symmetry rather than confinement drives the for-mation of the low-lying and stable hadrons such as the nucleon and pion. In the QCDinstanton vacuum conformal symmetry is broken by the density of instantons: their con-tinuous quantum rate of tunneling in the vacuum. The breaking of chiral symmetry followssimultaneously from the delocalization of the light quark zero modes, by leapfrogging theinstantons and anti-instantons much like electrons leapfrogging atoms in a metal. Detailed ∗ [email protected] a r X i v : . [ h e p - ph ] F e b numerical simulations of light hadronic correlators in the QCD instanton vacuum [5] showremarkable agreement with direct lattice measurements [6], and a wealth of correlatorsextracted from data [5]. The universal conductance fluctuations in the zero mode regionof the Dirac spectrum, predicted by random matrix theory [7] and confirmed by latticesimulations [8], show unequivocally the topological character of the origin of mass.In this note, we briefly review the salient aspects of the QCD instanton vacuum inrelation to the quantum breaking of conformal symmetry in section II. We then discussthe role of the trace anomaly in the nucleon and pion mass in section III. The quarkand gluon composition of the hadronic mass using Ji (cid:48) s decomposition [9] is discussed insection IV. In section V, we show that the gluon condensate in the nucleon is tied to theQCD vacuum compressibility, a measure of the diluteness of the QCD instanton vacuumas a topological liquid. Our conclusions are in section VI. II. QCD INSTANTON VACUUM
As we noted above, the chief aspect of the QCD vacuum (meaning quenched through-out) is its quantum breaking of conformal symmetry with the emergence of all lighthadronic scales. The nature of the gauge fields at the origin of this breaking were mys-terious and the subject of considerable debates and speculations for many decades, tillstunning pictures were developed by Leinweber and his collaborators using cooling and/orprojection techniques [10, 11]. Out of the fog of millions of gauge fluctuations, coolinghas revealed a stunning vacuum landscape composed of inhomogeneous and topologicallyactive gauge fields as shown in Fig. 1. Remarkably, the key features of this vacuum werepredicted long ago by Shuryak [12] n I +¯ I ≡ R ≈ ¯ ρR ≈
13 (1)for the instanton plus anti-instanton density and size, respectively. In other words, thehadronic scale R = 1 fm emerges as the mean quantum tunneling rate of the topologicalcharge in the QCD vacuum. The dimensionfull parameters (1) combine in the dimension-less packing parameter κ ≡ π ¯ ρ n I +¯ I ≈ .
1, a measure of the diluteness of the instanton-anti-instanton ensemble in the QCD vacuum. Fortunatly, the smallness of κ is what willallow us to do reliable analytical calculations. In the cooled landscape shown in Fig. 1,most hadronic correlations are left unchanged with those before the cooling takes place[2] (and references therein).The size distribution of the instantons and anti-instantons density (their tunneling rateper size) in the QCD vacuum is well captured semi-empirically by dn ( ρ ) dρ ∼ ρ (cid:0) ρ Λ QCD (cid:1) b e − ρ /R (2)with b = 11 N c / − N f / R ≈ → l s ≈ . R sets the scale for both the gluon and chiral condensates). FIG. 1. Instantons (yellow) and anti-instantons (blue) configurations in the cooled Yang-Millsvacuum [10]. They constitute the primordial gluon epoxy at the origin of the hadronic mass. Seetext.
A. Quantum conformal symmetry breaking and the trace anomaly
The quantum breaking of conformal symmetry is best captured through the trace ofthe energy momentum tensor. Indeed, consider its symmetric form T µν = 2 √− g δS δg µν = F aµλ F aνλ − g µν F + 14 ψγ [ µ i ←→ D ν ] + ψ (3)with ←→ D = −→ D − ←− D and [] + denotes symmetrization. It is conserved ∂ µ T µν = 0, with ananomalous trace T µµ = β ( g )4 g F aµν F aµν + mψψ (4)with the Gell-Mann-Low beta-function (2 loops) β ( g ) = − bg π − ¯ bg π ) + O ( g ) (5)Throughout, we use the rescaling g F → F for all operators in the instanton and anti-instanton gauge fields. In the QCD instanton vacuum, the gluon operator F / (32 π ) → ( N + + N − ) /V counts the number of instantons plus anti-instantons per 4-volume V . Inthe canonical ensemble with zero theta-angle, it is fixed by the instanton density with N ± /V = ¯ N / V . Therefore we have (cid:10) T µµ (cid:11) ≈ − b (cid:18) ¯ NV (cid:19) + m (cid:10) ψψ (cid:11) ≈ − b (cid:18) ¯ NV (cid:19)(cid:18) O ( mR ) (cid:19) ≈ −
10 fm − (6)setting the scale of all hadrons. The current mass m ≈ ρ ≈ . mR ≈ (8 MeV)(1 fm) ≈ / B = (cid:104) T µµ (cid:105) / ≈ − (250 MeV) , with no strict confinement at work. Note that B ≈ − b (cid:10) F (cid:11) , where the important overall negative sign inherited from the scale anomalyis ultimatly a quantum magnetic effect (sign of the beta function). The gluon condensate (cid:10) F (cid:11) is always positive in Euclidean signature. This is usually referred to as the gluonepoxy (a term coined by the late Gerry Brown).Some of the quantum scale fluctuations in QCD are captured in the QCD instantonvacuum using the grand-canonical description instead of the canical one. In the former,the instanton number N = N + + N − is allowed to fluctuate with the measure [3, 14, 15] P ( N ) = e bN (cid:18) NN (cid:19) bN (7)which is stronger than Poisson ( b/ → (cid:104) ( N − ¯ N ) (cid:105) P ¯ N = 4 b (8)expected from QCD low-energy theorems [16]. B. Spontaneous breaking of chiral symmetry and conductance fluctuations
The quantum breaking of conformal symmetry and the generation of the R = 1 fmand a gluon condensate, is a direct measure of the instanton tunneling rate or topologicaldensity in the vacuum. In the quenched approximation, it is solely a property of the gluonfields. This is a necessary but not a sufficient mechanism for hadronic mass generation.The sufficient mechanism, which relies on these topological fields, is the delocalizationof the quark zero modes and the ensuing spontaneous breaking of chiral symmetry. Theresult is a vacuum chiral condensate and the emergence of a quark mass, both of which arefixed by the same R = 1 fm scale at the origin of the gluon condensate. This topologicalmechanism for mass generation leaves behind a fingerprint: universal conductance-likefluctuations in the quark spectrum [7]. This is a lesser known fundamental phenomenon,that we now briefly discuss.Decades ago, Banks and Casher observed that the spontaneous breaking of chiral sym-metry with a finite chiral condensate (cid:104) ψψ (cid:105) , is associated to a huge accumulation of thequark zero modes near the zero point of the virtual quark spectrum as illustrated in Fig. 2(left), and captured by the relation [17] (cid:104) ψψ (cid:105) = − πν (0) ≡ − σ C (9)The quark density of states is ν ( λ ) = lim m → lim V →∞ V (cid:42)(cid:88) n δ ( λ − λ n [ A ]) (cid:43) A ≡ V λ (10)with the virtual quark eigenstates solution to( iD [ A ] + im ) q n [ A ] = ( λ n [ A ] + im ) q n [ A ]for a given gauge configuration. A finite ν (0) means that ∆ λ ≈ R /V , as opposed to∆ λ ≈ / √ V in the continuum. The quark spectrum is extremely dense near λ = 0, asthe disordering turn the quark zero modes to quasi-zero modes. FIG. 2. Sketch of the virtual Dirac spectrum for the disordered quark zero modes (left), and theuniversal spectral fluctuations measured in lattice generated gauge configurations (right) [8]. Seetext.
Remarkably, the Banks-Casher relation (9) resembles the Kubo formula for the DCconductivity in metals with σ C ↔ −(cid:104) ψψ (cid:105) and the zero-virtuality point F = λ = 0 playingthe role of the Fermi-surface. The QCD vacuum turns metallic when chiral symmetry isspontaneously broken. The same holds for the QCD instanton vacuum. In the unquenchedcase, this statement is extremely non-trivial. A too dilute or too dense topologically activevacuum would result into either a neutral gas of instanton-anti-instanton molecules, or acrystal arrangement of instantons-anti-instantons, instead of a liquid, with no spontaneousbreaking of chiral symmetry [1–3]. We are thankfull that nature has served us a diluteliquid!In fact there are infinitly many Banks-Casher-like formula which capture the fluc-tuations of the conductance σ C in the mesoscopic limit, with the connected moments σ nC = (cid:104) ( ψψ ) n (cid:105) C . These conductance fluctuations are universal, and follow from chiral ran-dom matrix theory, with the result for the mesoscopic spectral density at zero virtualitygiven by the master formula [7] ν s ( z = N λ ) = z (cid:18) J N f ( z ) − J N f +1 ( z ) J N f − ( z ) (cid:19) (11)with N, V → ∞ but fixed z = N λ and
N/V . These predicted conductance fluctuationswere later measured in numerically generated lattice gauge configurations as shown inFig. 2 [8] (right). This is one of the signature for the topological origin of the hadronicmass scale, as it centers on the quark zero mode zone. The zero modes only occur inthe presence of topological gauge fields such as the instantons and anti-instantons [18] (ortheir close cousins the instanton-dyons [19]), a consequence of the Atiyah-Singer indextheorem (or the Atiyah-Patodi-Singer for their cousins).
III. MASS IDENTITY
We now focus on the anomalous trace of the QCD energy momentum tensor, andits relation to the hadron mass. It is worth stressing that the ensuing relation to themass is just a bulk identity and not a mass decomposition. Having said that, the tracecouples to a scalar dilaton which sources a sigma (2-pion) meson and/or a 0 ++ scalarglueball field. QCD perturbative arguments suggest that this trace may be accessible inthe photo-production of charmonium at treshold [20, 21] (and references therein), althoughthe coupling to the 2 ++ tensor glueball may still be very active in the treshold region [22].Recall that the Reggeized form of the 2 ++ exchange transmutes to the Pomeron, and isdominant asymptotically. A. Nucleon
For a nucleon state | P (cid:105) with the standard normalization (cid:104) P | P (cid:48) (cid:105) = 2 E P (2 π ) δ ( P − P (cid:48) ),one has (cid:104) P | T µν | P (cid:105) = 2 P µ P ν . (12)with the trace in any frame (one-loop) (cid:104) P | T µµ | P (cid:105) = (cid:104) P | (cid:18) − b π F + m ¯ ψψ (cid:19) | P (cid:105) = 2 M N . (13)with g F → F for the strong instanton and anti-instanton gauge fields. It is renormal-ization group invariant. The identity (13) shows that the nucleon mass is the change ofthe conformal anomaly or gluon field in a nucleon state. However, the formation of thestate occurs only if chiral symmetry is spontaneously broken as we noted earlier. (13) isa QCD identity that is satisfied in the QCD instanton vacuum as we now show.In the rest frame, the gluon contribution in (13) follows from the normalized andconnected 3-point function asymptotically (cid:104) P | F | P (cid:105)(cid:104) P | P (cid:105) = lim T →∞ (cid:68) J † P ( T ) F J P ( − T ) (cid:69) C (cid:68) J † P ( T ) J P ( − T ) (cid:69) (14)with J P a pertrinent nucleon source. In the canonical description of the QCD instantonvacuum F / (32 π ) → ¯ N /V is a number. It factors out in the 3-point correlator in (14)(numerator), and the connected correlator vanishes.A non-vanishing contribution to the connected 3-point correlator follows from thegrand-canonical description, where N is allowed to fluctuate as we noted in (7). Withthis in mind, it is straightforward to see that (14) is dominated by the variance V π (cid:104) P | F | P (cid:105)(cid:104) P | P (cid:105) ≈ (cid:10) ( N − ¯ N ) (cid:11) P ∂∂ ¯ N Log (cid:18) lim T →∞ (cid:68) J † P ( T ) J P ( − T ) (cid:69)(cid:19) (15)with the higher moments suppressed by 1 /b ∼ /N c . The result (15) was noted in [14] (seeEq. 5.8) using a fermionization method, and in [15] (see Eqs. 91,93) using a bosonizationmethod, each of the QCD instanton vacuum in the 1 /N c approximation. The expectationvalue in the first bracket is carried using the distribution (7). (15) illustrates how thenucleon scoops the epoxy from the QCD instanton vacuum.All dimensions in the QCD instanton vacuum are fixed by the density ¯ N /V = 1 /R and the current quark masses. The nucleon mass is the sum of the chirally symmetric(invariant mass) plus the symmetry breaking contribution (pion-nucleon sigma term), M N = M inv + σ πN = C (cid:18) ¯ NV (cid:19) + ¯ Cm (cid:18) O ( mR ) (cid:19) (16)with [23–25] σ πN = (cid:10) P | mψψ | P (cid:11) (cid:104) P | P (cid:105) ≈
50 MeV (17)evaluated at the soft renormalization scale ¯ ρ = 0 . V T − b π (cid:104) P | F | P (cid:105)(cid:104) P | P (cid:105) = 4 ∂M N ∂ Log ¯ N = M inv (18)which is seen to satisfy the sum rule (cid:104) P | T µµ | P (cid:105) M N = M inv + (cid:10) P | mψψ | P (cid:11) M N = M N (19) B. Pion
The preceding arguments apply also to the pion, with one major difference, m π = C √ m (cid:18) ¯ NV (cid:19) (1 + O ( mR )) (20)since it is a Goldstone mode. The O ( mR ) corrections are small in the QCD instantonvacuum. It follows that V T − b π (cid:104) π | F | π (cid:105)(cid:104) π | π (cid:105) = 4 ∂m π ∂ Log ¯ N = 12 m π (21)which was first observed in [26], with the sum rule (cid:104) π | T µµ | π (cid:105) m π = 12 m π + (cid:10) π | mψψ | π (cid:11) m π = m π (22)satisfied, as expected. The pion sigma-term follows from chiral reduction or the Feynman-Hellmann theorem (cid:10) π | mψψ | π (cid:11) m π = ∂ E π ∂ log m = 12 m π (23)0 IV. JI MASS SUM RULE
The trace identity (12) reflects on the general fact that all hadron masses in QCDare tied to the quantum breaking of conformal symmetry as we noted earlier, and shouldbe enforced by any non-perturbative quantum description, wether numerical such as thelattice or analytical such as the QCD instanton vacuum. However, it does not specificallybudget this mass breaking in terms of the hadron constituents. In a strongly interactingtheory, this issue may be elusive, especially with a soft renormalization scale, as the gluonsare strongly untertwined with the light quarks. This is more so in the unquenched andscreened formulation.This notwithstanding, a specific and physically motivated proposal to budget the mass,was put forth by Ji in [9, 26], and since revisited by many [27, 28] (and references therein).The ensuing mass composition involves the sum of partonic contributions, some of whichmay be measurable using DIS experiments. The proposal relies on an invariant decompo-sition of the energy momentum tensor which we now detail.The energy-momentum tensor (3) can be decomposed as the sum of a traceless andtracefull part [26, 29] T µν ≡ ¯ T µν + ˆ T µν ≡ ¯ T µν + g µν T αα , (24)where the traceless part reads¯ T µν = (cid:18) − F aµτ F aντ + 14 g µν F (cid:19) + 14 ψγ [ µ i ←→ D ν ] + ψ − g µν m ¯ ψψ, (25)and the tracefull part isˆ T µν = g µν (cid:18) β ( g )4 g F + mψψ ≈ − b π F + mψψ (cid:19) (26)We note that this decomposition is commensurate with the analysis of the nucleon energymomentum tensor in holographic QCD, through dual gravitons in bulk [22]. (Holographyis a good example of a strong coupling description of a gauge theory via its gravity dual,where the partonic structure is elusive).The tracefull and traceless part of the energy momentum tensor (24-26) correspond tothe spin-2 and spin-0 representations of the Lorentz group, and do not mix under renor-malization by symmetry. Their renormalization at the instanton size scale ¯ ρ ≈ . /µ ≈ . (cid:10) P | ¯ T µν | P (cid:11) =2 (cid:18) P µ P ν − g µν M N (cid:19)(cid:68) P | ˆ T µν | P (cid:69) = 12 g µν M N (27)The corresponding Hamiltonian in Minkowski signature, follows from the 00-componentof (24-26) modulo BRST exact and gauge dependent contributions, H G = (cid:90) d x ¯ T G = (cid:90) d x
12 ( E + B ) H (cid:48) Q = (cid:90) d x ¯ T Q = (cid:90) d x (cid:18) ψγ · i ←→ D ψ + 34 m ¯ ψψ (cid:19) H (cid:48) A = (cid:90) d x ˆ T A = (cid:90) d x (cid:18) β ( g )4 g F + mψψ ≈ − b π F + mψψ (cid:19) (28)where the time t = 0 is subsumed. The mass term can be rearranged so that (28) reads H G = (cid:90) d x ¯ T G = (cid:90) d x
12 ( E + B ) H Q = (cid:90) d x ¯ T Q = (cid:90) d x (cid:18) ψγ · i ←→ D ψ (cid:19) H A = (cid:90) d x ˆ T A = (cid:90) d x (cid:18) β ( g )4 g F ≈ − b π F (cid:19) H m = (cid:90) d x ¯ T G = (cid:90) d x mψψ (29)The nucleon mass budget is then M N = (cid:104) P | H G + H Q + H A + H m | P (cid:105)(cid:104) P | P (cid:105) ≡ M NG + M NQ + M NA + M Nm (30)which shows that the combination M inv = M NG + M NQ + M NA (31)2is chirally symmetric and equal to the invariant mass in (16).In Euclidean signature, whether on the lattice or using the QCD instanton vacuum,(30) can be evaluated by trading T → T and t = 0 → i
0. In the dilute QCD instantonvacuum, the gluonic operator in (28-30) is the sum of multi-instanton contributions of theform¯ T G [ A ] = N ± (cid:88) I =1 ¯ T G [ A I ( ξ I )] + N ± (cid:88) I (cid:54) = J =1 ¯ T G [ A I ( ξ I ) , A J ( ξ J )] + ... = N ± (cid:88) I (cid:54) = J =1 ¯ T G [ A I ( ξ I ) , A J ( ξ J )] + ... (32)Since the first one-instanton contribution in (32) is composed of self-dual fields it vanishes.So we are left with only the two and higher multi-instanton contributions. When averagedover a measure of independent instantons, the remaining terms in (32) are suppressed bythe diluteness factor κ ≈ .
1. As a result, the contribution of M NG is parametrically smallin comparison to M NQ or 4 M NA , i.e. M NG /M NQ ≈ κ ≈ .
1. The contributions M NQ,m aresolely given in terms of the fermionic zero modes (modulo the instanton gauge fields inthe long derivative).With this in mind, the breakdown in the mass budget (30) for the nucleon yields theestimates M NQ M N ≈
34 11 + κ (cid:18) − σ πN M N (cid:19) ≈ M NG M N ≈ κ κ (cid:18) − σ πN M N (cid:19) ≈ M NA M N = 14 (cid:18) − σ πN M N (cid:19) ≈ M Nm M M = σ πN M N ≈
5% (33)with the empirical pion-nucleon sigma term (17). The anomalous contribution is scaleand scheme independent at one-loop order. The mass contribution is also renormalizationgroup invariant. (33) shows that in the QCD instanton vacuum, about 70% of the nucleonmass stems from the valence quarks (hopping zero modes), 25% from the gluon condensateor epoxy (displaced vacuum instanton field), and 7% from emerging valence gluons. Thenucleon is composed mostly of quark constituents hopping and dragging the gluon epoxy.The gluon epoxy in the nucleon is the quantum anomalous energy in the nucleon discussedrecently in [31].We note that the budgeting of the nucleon mass in (33) differs from the one reportedon the lattice in [30], with a noticeably larger valence gluon fraction in the lattice nucleon.3In our analysis, this can only be accomodated by a stronger instanton packing fractionof κ ≈ . κ that includeclose instanton-anti-instanton pairs, not responsible for the breaking of chiral symmetry,were reported when analyzing certain correlations at zero cooling time [32]). The harderrenormalization scale µ = 2 GeV used in the reported lattice results, is the likely source ofthe valence and perturbative gluon enhancement reported in the lattice nucleon. Quan-tum evolution will enhance M NG at the expense of M NQ , which in (33) would amount toeffectively dressing κ ≈ . → . µ = 2 GeV. Remarkably, the same shift is what isneeded to bridge the gap between the lattice data and the QCD instanton vacuum resultsfor the spin decomposition of the nucleon [33].Finally, a similar mass decomposition holds for the pion at the same soft renormaliza-tion scale of ¯ ρ = 0 . M πQ m π ≈
38 11 + κ ≈ M πG m π ≈ κ κ ≈ M πA m π = 18 ≈ M πm m π = 12 ≈
50% (34)About 85% of the pion mass stems from the valence quarks (hopping zero modes), 13%from the gluon condensate or epoxy (displaced vacuum instanton field), and 3% fromemerging valence gluons. Needless to say that all mass contributions in the pion vanishsmootly in the chiral limit. Again, quantum evolution will enhance M πG at the expense of M πQ , with effectively dressing κ ≈ . → . µ = 2 GeV. V. MEASURING THE QCD VACUUM COMPRESSIBILITY
While the present discussion has focused on some key aspects of the QCD vacuum andthe hadronic mass sum rule, it is worth noting that the results (15-21) can be recast inthe following form (cid:104) P (cid:12)(cid:12) F (0) (cid:12)(cid:12) P (cid:105) (4 π ( m N − σ πN / ≈ − σ F (35)with the QCD vacuum compressibility4 σ F = 132 π (cid:90) d x (cid:104) F ( x ) F (0) (cid:105) C (cid:104) F (0) (cid:105) (36)A measure of the gluon condensate or epoxy inside the proton (left hand-side) is a measureof the QCD vacuum compressibility σ F (right hand-side), modulo the pion-nucleon sigmaterm which is small. Since (35) is a nucleon connected matrix element, it is natural that itprobes the fluctuations of F . While in the vacuum state the gluon condensate is positive,(36) shows that it is negative in the nucleon state. The nucleon state carries less epoxy.The cooled Yang-Mills vacuum in Fig. 1 is composed of interacting topological charges.The vacuum compressibility σ F captures the squared variance of their interactions: σ F =1 for a non-interacting gas phase, σ F < σ F (cid:28) σ F ≈ /b ≈ /
11 (one-loop and quenched) [16], so the QCD instanton vacuum appears to be a dilutequantum topological liquid. A measure of σ F is a measure of a fundamental and universalparameter of the QCD vacuum. VI. CONCLUSIONS
The QCD instanton vacuum is populated with topological tunneling configurations,with each costing zero energy. The way a light quark can propagate coherently throughthis maze of tunneling configurations is through its zero mode, scattering and hopping froman instanton to an anti-instanton and so on. The scattering through the instanton flipschirality, an amazing effect caused by a non-perturbative vector interaction (a perturbativegluon interaction preserves chirality). The hopping generates a very dense band in thevirtual quark spectrum, reminiscent of the conduction band in conductors. As a result,chiral symmetry is spontaneously broken, a chiral condensate is formed and a runningconstituent quark mass emerges.The QCD instanton vacuum breaks simultaneously conformal symmetry, with a largeand negative vacuum energy density, or equivalently a large and positive gluon condensate(gluon epoxy). A hadronic excitation in this vacuum, whether a quark, a meson or abaryon costs energy or mass. A useful and physical way to budget this mass is Ji (cid:48) s massdecomposition of the energy momentum tensor [9, 26]. In the QCD instanton vacuum, wefind that the hadronic masses are largely due to the contribution of the valence quarks asthey hop and drag the gluon epoxy.Finally, a measure of the gluon condensate or epoxy in the nucleon, is a measure ofthe compressibility of the QCD instanton vacuum as a topological liquid. The dilutenessof this liquid is central in our non-perturbative understanding of the emergence of massin QCD using analytical methods. This gluonic content of the proton may be accessible5through treshold electromagnetic production of heavy quarkonia [21], and perhaps diffrac-tive cluster production in hadron-hadron collisions [34] at current and future facilities.
Acknowledgements
I thank Xiang-dong Ji, Zein-Eddine Meziani and Edward Shuryak for discussion. Thiswork is supported by the Office of Science, U.S. Department of Energy under ContractNo. DE-FG-88ER40388. [1] Dmitri Diakonov, “Chiral symmetry breaking by instantons,” Proc. Int. Sch. Phys. Fermi , 397–432 (1996), arXiv:hep-ph/9602375.[2] Thomas Sch¨afer and Edward V. Shuryak, “Instantons in QCD,” Rev. Mod. Phys. , 323–426(1998), arXiv:hep-ph/9610451.[3] Maciej A. Nowak, Mannque Rho, and I. Zahed, Chiral nuclear dynamics (1996).[4] Jeff Greensite, “Confinement from Center Vortices: A review of old and new results,” EPJWeb Conf. , 01009 (2017), arXiv:1610.06221 [hep-lat].[5] Edward V. Shuryak, “Probing the boundary of the nonperturbative QCD by small size in-stantons,” (1999), arXiv:hep-ph/9909458.[6] M. C. Chu, J. M. Grandy, S. Huang, and John W. Negele, “Correlation functions of hadroncurrents in the QCD vacuum calculated in lattice QCD,” Phys. Rev. D , 3340–3353 (1993),arXiv:hep-lat/9306002.[7] J. J. M. Verbaarschot and I. Zahed, “Spectral density of the QCD Dirac operator near zerovirtuality,” Phys. Rev. Lett. , 3852–3855 (1993), arXiv:hep-th/9303012.[8] Hartmut Wittig, “QCD on the Lattice,” in Particle Physics Reference Library: Volume 1:Theory and Experiments , edited by Herwig Schopper (2020) pp. 137–262.[9] Xiang-Dong Ji, “A QCD analysis of the mass structure of the nucleon,” Phys. Rev. Lett. ,1071–1074 (1995), arXiv:hep-ph/9410274.[10] Derek B. Leinweber, “Visualizations of the QCD vacuum,” in Workshop on Light-Cone QCDand Nonperturbative Hadron Physics (1999) pp. 138–143, arXiv:hep-lat/0004025.[11] James C. Biddle, Waseem Kamleh, and Derek B. Leinweber, “Visualization of center vortexstructure,” Phys. Rev. D , 034504 (2020), arXiv:1912.09531 [hep-lat].[12] Edward V. Shuryak, “The Role of Instantons in Quantum Chromodynamics. 1. PhysicalVacuum,” Nucl. Phys. B , 93 (1982).[13] Anna Hasenfratz, “Spatial correlation of the topological charge in pure SU(3) gauge theoryand in QCD,” Phys. Lett. B , 188–192 (2000), arXiv:hep-lat/9912053.[14] Dmitri Diakonov, Maxim V. Polyakov, and C. Weiss, “Hadronic matrix elements of gluon op- erators in the instanton vacuum,” Nucl. Phys. B , 539–580 (1996), arXiv:hep-ph/9510232.[15] M. Kacir, M. Prakash, and I. Zahed, “Hadrons and QCD instantons: A Bosonized view,”Acta Phys. Polon. B , 287–348 (1999), arXiv:hep-ph/9602314.[16] V.A. Novikov, Mikhail A. Shifman, A.I. Vainshtein, and Valentin I. Zakharov, “AreAll Hadrons Alike? DESY-check = Moscow Inst. Theor. Exp. Phys. Gkae - Itef-81-048(81,rec.jun.) 32 P and Nucl. Phys. B191 (1981) 301-369 and Moscow Inst. Theor. Exp. Phys.Gkae - Itef-81-042 (81,rec.apr.) 70 P . (104907),” Nucl. Phys. B , 301–369 (1981).[17] Tom Banks and A. Casher, “Chiral Symmetry Breaking in Confining Theories,” Nucl. Phys.B , 103–125 (1980).[18] Gerard ’t Hooft, “Computation of the Quantum Effects Due to a Four-Dimensional Pseu-doparticle,” Phys. Rev. D , 3432–3450 (1976), [Erratum: Phys.Rev.D 18, 2199 (1978)].[19] Yizhuang Liu, Edward Shuryak, and Ismail Zahed, “Light quarks in the screened dyon-antidyon Coulomb liquid model. II.” Phys. Rev. D , 085007 (2015), arXiv:1503.09148 [hep-ph].[20] Dmitri E. Kharzeev, “The mass radius of the proton,” (2021), arXiv:2102.00110 [hep-ph].[21] S. Joosten and Z. E. Meziani, “Heavy Quarkonium Production at Threshold: from JLab toEIC,” PoS QCDEV2017 , 017 (2018), arXiv:1802.02616 [hep-ex].[22] Kiminad A. Mamo and Ismail Zahed, “Diffractive photoproduction of
J/ψ and Υ using holo-graphic QCD: gravitational form factors and GPD of gluons in the proton,” Phys. Rev. D , 086003 (2020), arXiv:1910.04707 [hep-ph].[23] James V. Steele, Hidenaga Yamagishi, and Ismail Zahed, “The Pion - nucleon sigma termand the Goldberger-Treiman discrepancy,” (1995), arXiv:hep-ph/9512233.[24] C. Alexandrou et al. , “Nucleon axial and pseudoscalar form factors from lattice QCD at thephysical point,” (2020), arXiv:2011.13342 [hep-lat].[25] Martin Hoferichter, Jacobo Ruiz de Elvira, Bastian Kubis, and Ulf-G. Meißner, “Remarkson the pion–nucleon σ -term,” Phys. Lett. B , 74–78 (2016), arXiv:1602.07688 [hep-lat].[26] Xiang-Dong Ji, “Breakup of hadron masses and energy - momentum tensor of QCD,” Phys.Rev. D , 271–281 (1995), arXiv:hep-ph/9502213.[27] C´edric Lorc´e, “The origin of the nucleon mass,” Springer Proc. Phys. , 635–641 (2020),arXiv:1811.02803 [hep-ph].[28] Craig D. Roberts, “On Mass and Matter,” (2021), arXiv:2101.08340 [hep-ph].[29] Xiang-Dong Ji, “Proton mass decomposition: naturalness and interpretation,” (2021).[30] Yi-Bo Yang, Jian Liang, Yu-Jiang Bi, Ying Chen, Terrence Draper, Keh-Fei Liu, andZhaofeng Liu, “Proton Mass Decomposition from the QCD Energy Momentum Tensor,” Phys.Rev. Lett. , 212001 (2018), arXiv:1808.08677 [hep-lat].[31] Xiangdong Ji and Yizhuang Liu, “Quantum Anomalous Energy Effects on the Nucleon Mass,”(2021), arXiv:2101.04483 [hep-ph]. [32] A. Athenodorou, Ph. Boucaud, F. De Soto, J. Rodr´ıguez-Quintero, and S. Zafeiropoulos,“Instanton liquid properties from lattice QCD,” JHEP02