# A Demonstration of Hadron Mass Origin from QCD Trace Anomaly

AA Demonstration of Hadron Mass Origin from QCD Trace Anomaly

Fangcheng He , Peng Sun , Yi-Bo Yang , , ( χ QCD Collaboration) CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China Nanjing Normal University, Nanjing, Jiangsu, 210023, China School of Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China International Centre for Theoretical Physics Asia-Paciﬁc, Beijing/Hangzhou, China

Quantum chromodynamics (QCD) claims that the major source of the nucleon invariant mass isnot the Higgs mechanism but QCD energy momentum tensor trace anomaly. Although experimentaland theoretical results support such conclusion, a direct demonstration is still absent. We presentthe ﬁrst Lattice QCD calculation of the quark and gluon trace anomaly contributions to the hadronmasses, using the overlap fermion on the 2+1 ﬂavor dynamical Domain wall quark ensemble at m π = 340 MeV and lattice spacing a =0.1105 fm. The result shows that the gluon trace anomalycontributes most of the nucleon mass, and the contribution in the pion state is smaller than thatin others nearly by a factor 10. The quark mass and gluon anomalous dimensions are comparablewith the continuum theoretical prediction under MS scheme at the scale µ = 1 /a =1.785 GeV. Introduction:

Searching the origin of the mass is anessential topic for physicists. Although the Higgs mecha-nism explains the origin of quark masses, we still need an-other theory to explain the mass of hadrons. The theorystarts from the QCD energy momentum tensor (EMT)with quark ﬁeld ψ and gluon ﬁeld strength tensor F µν ,which can have an anomaly in its trace [1–4], T µµ = (1 + γ m ) H m + β g F , (1)where the quark mass term H m = (cid:80) q m q ¯ ψψ is the clas-sical trace of EMT, γ m = π α s + O ( α s ) is the anomalousdimension of the quark mass m q , g is the strong couplingconstant and α s = g π , β is the anomalous dimension of g , β g = ( − π + N f π ) α s + O ( α s ) and F = F µν F µν . Boththe γ m and β/g terms are purely quantum correctionsand absent in the classical level of the EMT.Since the hadron mass can be understood as its matrixelement of the EMT trace, we have the relation [5, 6] M H = (cid:104) T µµ (cid:105) H = (1 + γ m ) (cid:104) H m (cid:105) H + β g (cid:104) F (cid:105) H , (2)with (cid:104) O (cid:105) H ≡ (cid:104) H | O | H (cid:105) . In a weak coupling theory, thecontribution from trace anomaly will be much smallerthan that from the classic term (cid:104) H m (cid:105) H . But Ref. [5]found that (cid:104) F (cid:105) H term in Eq. (2) should contribute allthe nucleon mass in the chiral limit m q →

0. Based onthe EQ (2) the trace anomaly contribution in the pionmass should also approaches zero in the chiral limit butshould be O (Λ QCD ) in the other hadrons. Similar picturecan also be used to understand the QED anomaly eﬀectin the hydrogen atom mass [7]. In the realistic nucleon, the contributions from the (cid:104) H m (cid:105) N of three light quarks are around 90 MeV [8, 9],and then that from the three heavy quarks will be ∼ m Q ¯ ψ Q ψ Q −−−−−→ m Q →∞ − α s π F + O ( α s ) , (3)where ψ Q is the heavy quark ﬁeld and m Q is the corre-sponding quark mass. Thus even we count the contribu-tion from all the six quarks, the trace anomaly should stillcontribute around 2/3 of the nucleon mass. It means thatthe nucleon mass or observable matter, mainly comesfrom the condensate of the massless gluon in the nucleonthrough the quantum correction of QCD, not the classi-cal Higgs coupling of fundamental quarks. On the otherhand, the heavy quark contribution exactly cancels theﬂavor dependence in β at the leading order, and then thematrix element (cid:104) F (cid:105) H is insensitive to the numbers ofthe heavy quark ﬂavors in the theory [5].Even though we can calculate M H and (cid:104) H m (cid:105) N via lat-tice QCD precisely, the gluon matrix element on latticeQCD is very noisy. In order to improve the signal, wehave to apply the smearing on the gluon ﬁeld. After suchan operation, it is too complicated to calculate the β an-alystically under the lattice regularization. At the sametime, the additive renormalization of the quark mass inmost of the discretized fermion actions can introduce ad-ditional terms in Eq. (2), it leads to additional diﬃcultiesto control systemic uncertainty. Thus a ﬁrst principle di-rect check on the trace anomaly sum rule requires an ac-curate gluon matrix element calculation using the chiralfermion. It is a very challenging since the chiral fermionis much more expensive than others on lattice QCD. On a r X i v : . [ h e p - l a t ] J a n the other hand, the experimental check is also highly non-trivial even though there is a related proposal [10, 11] forthe coming Electron-Ion Collider (EIC).In this work, we present a demonstration of the traceanomaly sum rule in Eq. (2), based on the direct latticeQCD calculations with the chiral fermion, and both thequark and gluon contributions to kinds of the hadronsare considered. The γ m and β are determined by solv-ing Eq. (2) in the pseudo scalar and vector meson stateswith relatively heavy quark mass. Since γ m and β areindependent on the quark mass and hadron states, wecan apply them into all hadron states with the arbitraryquark mass. Then we ﬁnd that Eq. (2) is satisﬁed in allcases we calculated in this work. In addition, we alsoﬁnd that the gluon trace anomaly contribution in thepion turns out to be much smaller than that in the otherhadron likes nucleon and vector meson, as predicted bythe trace anomaly sum rule. TABLE I. Information of the RBC ensemble [12] used in cal-culation. The pion and kaon masses are in unit of MeV.Symbol L × T a (fm) 6 /g m π m K N cfg

24I 24 ×

64 0.1105(3) 2.13 340 593 203

Numerical setup : We preform the calculation on a24 ×

64 2+1 ﬂavor Domain-wall fermion ensemble fromthe RBC collaboration [12], with the pion mass m π =340 MeV which is heavier than the physical one but stillmuch smaller than the corresponding nucleon mass. Theother informations of the ensemble we use summarized inTable I. For the valence quark, we use the chiral fermionthrough the overlap approach [13] to avoid the additionalterm in the trace of EMT under the lattice regularization.In order to determine the γ m and β precisely and makean accurate test of the sum rule, we consider the partiallyquenched gauge theory (PQGT) [14] which allows thevalence quark mass to be diﬀerent from that in the gaugeensemble. In such a case, Eq. (2) becomes M H = (1 + γ m ) (cid:0) (cid:104) H m (cid:105) vH + (cid:104) (cid:88) i m i ¯ ψ i ψ i (cid:105) H (cid:1) + β g (cid:104) F (cid:105) H , (4)where (cid:104) H m (cid:105) vH includes the connected quark diagram withthe operator m v ¯ ψ v ψ v only, and the index i just includesthe ﬂavors exist in the gauge conﬁgurations: degeneratedlight up/down and also strange quark. When m v equalsto one of the quark masses in the gauge ensemble, itcomes back to the standard trace anomaly sum rule.In this work, we considered the cases with 5 quarkmasses m v a =0.0160, 0.0576, 0.1, 0.2, and 0.3, and thelightest two quark masses correspond to the light andstrange quark masses in the gauge ensemble we used.During the calculation with the overlap fermion, deﬂatingthe long-distance subspace of the Dirac operator using itseigenvectors v ( λ ) with | λ | < Λ QCD are essential to obtainthe light quark propagator eﬃciently [15]. At the same time, we can build the light quark loop operator O q ( t )only via those eigenvectors v ( λ ) with little cost, and thesystematic uncertainty from the rest is negligible [8], O q ( t ) ≡ (cid:88) (cid:126)x S ( m q ; (cid:126)x, t ; (cid:126)x, t ) (cid:39) (cid:88) (cid:126)x S L ( m q ; (cid:126)x, t ; (cid:126)x, t ) ,S L ( m q ) ≡ (cid:90) i Λ QCD − i Λ QCD d λ v ( λ ) v † ( λ ) λ + m q . (5)As implemented in Ref. [16], we calculate thetwo point function C ( t f ) and three point func-tions C q,g ( t f ) using the coulomb gauge ﬁxed wallsource propagator S w ( (cid:126)y, t ; t ) = (cid:80) (cid:126)x S ( (cid:126)y, t ; (cid:126)x, t ),the summed current sequential propagator [17]˜ S c ( (cid:126)y, t ; t ) = m q (cid:80) (cid:126)x,t S ( (cid:126)y, t ; (cid:126)x, t ) S w ( (cid:126)x, t ; t ), lightquark loops O q , and position summed gluon operator O F = (cid:80) (cid:126)x,t ∈ ( t f , F ( (cid:126)x, t ), C ( t f ; M ) = C M ( S w , S w , t f , I ) ,C q ( t f ; M ) = C M ( S c , S w , t f , I ) + C M ( S w , S c , t f , I ) , + C M ( S w , S w , t f , (cid:88) q = u,d,s O q ) C g ( t f ; M ) = C M ( S w , S w , t f , O F ) ,C M ( S , S , t f , O )= (cid:88) (cid:126)y (cid:104) Tr (cid:16) γ S † ( (cid:126)y, t f ; 0) γ Γ S ( (cid:126)y, t f ; 0)Γ (cid:17) O(cid:105) , (6)for the meson M (Γ) with the interpolation ﬁeld ¯ ψ Γ ψ ,where I is the unitary matrix, and S ( (cid:126)y, t ; (cid:126)x, t ) is thequark propagator from ( (cid:126)x, t ) to ( (cid:126)y, t ). The gluon ﬁeldtensor F µν is deﬁned through the clover deﬁnition, F µν ( x ) = i a g (cid:2) P [ µ,ν ] + P [ ν, − µ ] + P [ − µ, − ν ] + P [ − ν,µ ] (cid:3) ( x ) , P µ,ν ( x ) = U µ ( x ) U ν ( x + a ˆ µ ) U † µ ( x + a ˆ ν ) U † ν ( x ) , (7)where U − ν ( x ) = U † ν ( x − a ˆ ν ) and P [ µ,ν ] ≡ P µ,ν − P ν,µ .The nucleon case with the interpolation ﬁled ( u T ˜ Cd ) u is more or less similar and can be obtained without muchmodiﬁcations, C ( t f ; N ) = C N ( S w , S w , S w , t f , I ) ,C q ( t f ; N ) = C N ( S c , S w , S w , t f , I ) + C N ( S w , S c , S w , t f , I ) , + C N ( S w , S w , S c , t f , I ) , + C N ( S w , S w , S w , t f , (cid:88) q = u,d,s O q ) C g ( t f ; N ) = C N ( S w , S w , S w , t f , O F ) ,C N ( S , S , S , t , t , O ) = (cid:104) (cid:15) abc (cid:15) a (cid:48) b (cid:48) c (cid:48) (cid:88) (cid:126)y Tr (cid:16) Γ m S aa (cid:48) w ( (cid:126)y, t f ; 0) (cid:17) Tr (cid:16) S bb (cid:48) w (( (cid:126)y, t f ; 0) S cc (cid:48) w ( (cid:126)y, t f ; 0) (cid:17) + Tr (cid:16) Γ m S aa (cid:48) w ( (cid:126)y, t f ; 0) S bb (cid:48) w ( (cid:126)y, t f ; 0) S cc (cid:48) w ( (cid:126)y, t f ; 0) (cid:17) (cid:105) , (8)where S is deﬁned by ( ˜ CS ˜ C − ) T , ˜ C = γ γ γ , and Γ m = (1 + γ ) is the unpolarized projector.For each hadron, we carry out a joint correlated ﬁtof C ( t f ; H ) and C q,g ( t f ; H ) to extract the M H , (cid:104) H m (cid:105) H and (cid:104) F (cid:105) H simultaneously [16]: C q,g ( t f ; H ) = e − M H t f (cid:0) C t f (cid:104) O q,g (cid:105) H + C q,g e − δmt f + C q,g t f e − δmt f + C q,g (cid:1) C ( t f ; H ) = C e − M H t f (1 + C e − δmt f ) , (9)where (cid:104) O q (cid:105) H = (cid:104) H m (cid:105) H , (cid:104) O g (cid:105) H = (cid:104) F (cid:105) H , the e − δmt f terms are introduced to account for the contaminationfrom higher states, δm , C , C and C q,g ,..., are free pa-rameters. The above form is equivalent to extract thedesired quantities in the large t f limits, M H = log C ( t f − H ) C ( t f ; H ) + O ( e − δmt f ) , (cid:104) H m (cid:105) H = ∆ R q ( t f ; H ) + O ( e − δmt f ) , (cid:104) F (cid:105) H = ∆ R g ( t f ; H ) + O ( e − δmt f ) (10)where ∆ R i = q,g ( t f ) ≡ C i ( t f ; H ) C ( t f ; H ) − C i ( t f − H ) C ( t f − H ) . Note thatone need to replace C , ( t f ) into C , ( t f ) + C , ( T a − t f )for the PS meson case, to include the loop around eﬀectand describe the data around t f ∼ T a/ Result:

In this work, we calculated the S w and ˜ S c for 5 quark masses m v a =0.0160, 0.0576, 0.1, 0.2, and0.3 respectively, and construct the C q,g , for the nucleon(N), pseudoscalar (PS), and vector (V) mesons. We alsolooped over all the T = 64 time slides to suppress thestatistical uncertainty, and applied 5 steps of the HYPsmearing on the gauge operator. 300 pairs of the lowlying eigenpairs of the Dirac operator with λ ≤ . R q,g ( t f ) de-ﬁned in Eq. (10) for the PS (blue) and V (red) mesonswith m v a =0.3. The disconnected light quark contri-butions are not shown here as their contributions aresmall. The M H , (cid:104) H m (cid:105) H and (cid:104) F (cid:105) H are obtained us-ing the joint ﬁt deﬁned in Eq. 9 with χ /d.o.f. ∼

1, andthe bands of ∆ R q,g ( t f ) predicted by the joint ﬁt are alsoshown in Fig. 1 and agree with the data perfectly. Wecan see that the gluon trace anomaly matrix element −(cid:104) F (cid:105) H = − ∆ R g ( t f ) | t f →∞ in the V meson is more thantwo times of that in the PS meson, even though theirquark mass terms (cid:104) H m (cid:105) H = ∆ R q ( t f ) | t f →∞ just diﬀerfrom each other by 10%.Since the anomalous dimension γ m and β should beindependent to the hadron states, we solve the equations M P S − (1 + γ m ) (cid:104) H m (cid:105) P S − β g (cid:104) F (cid:105) P S | m v a =0 . = 0 ,M V − (1 + γ m ) (cid:104) H m (cid:105) V − β g (cid:104) F (cid:105) V | m v a =0 . = 0 , (11) □ □ □ □ □ □ □ □ □ □ □ □ □ □ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ ○ □ -< F > V = ( ) GeV ○ -< F > PS = ( ) GeV < H M > V = ( ) GeV < H M > PS = ( ) GeV0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.001234567 t f ( fm ) Δ R { q , g } ( t f )( GeV ) FIG. 1. The diﬀerential ratio ∆ R q,gPS,V ( t f ) of the pseu-doscalar and vector mesons with m v a = 0 .

3, which shouldapproaches to the ground state matrix elements (cid:104) H m (cid:105) PS,V and −(cid:104) F (cid:105) PS,V respectively at the t f → ∞ limit. The bandsshows the joint ﬁt prediction using the form deﬁned in Eq. (9)and they agree with the data well. using the (cid:104) H m (cid:105) P S/V and (cid:104) F (cid:105) P S/V with m v a =0.3, andobtain that the bare γ m = 0 . β g = − . γ m is close to that at MS 1 /a =1.785 GeVwhich is 0.325[18]. However, the value of β g at same MSscale is -0.146, which is much lower than our prediction.Precisely speaking, the γ m is consistent with the predic-tion of MS at 1.5 GeV, but the corresponding MS scalefor β g is about 4 GeV where the strong coupling constant α s (cid:39) .

223 is similar to the bare one . ∗ π (cid:39) . F operator and complications in the F def-inition, the diﬀerences is acceptable while worths furtherstudy in the future. PSVN0.0 0.1 0.2 0.3 0.4 0.5 0.60.60.81.01.21.4 m v ( GeV ) R H FIG. 2. The ratio of the hadron mass from the left and righthand side of Eq. (2), using the γ m and β/g obtained from thePS and V mesons with m v a =0.3. It is consistent with onewithin the uncertainties in all the cases. With the above values of γ m and β g , we calculate thehadron mass M H , quark mass term (cid:104) H m (cid:105) H and (cid:104) F (cid:105) H of the pseudoscalar, vector meson and nucleon with dif-ferent m v , and plot the ratio, R H ( m v ) = (1 + γ m ) (cid:0) (cid:104) H m (cid:105) H + β g (cid:104) F (cid:105) H M H , (12)in Fig. 2. We can see that the R H in all the cases areconsistent with one within the uncertainties. Note thatthe PS and V meson cases with m v a = 0 . γ m and β are universal. PSVN0.0 0.1 0.2 0.3 0.4 0.5 0.60.00.20.40.60.81.01.21.4 m v ( GeV ) H ga ( G e V ) FIG. 3. The gluon trace anomaly contribution to the hadronmass. We can see that it is is always small in the PS meson,while approaches to ∼

800 MeV for the nucleon and vectormeson in the chiral limit m v → The resulting gluonic trace anomaly contribution (cid:104) β g F (cid:105) H in the 2+1 ﬂavor ensemble are plotted in Fig. 3.We can see that (cid:104) β g F (cid:105) H in the pion state is generallysmaller than that in the other states, especially at theunitary point with 340 MeV pion mass. In such a case,the gluon trace anomaly contributes about 100 MeV ofthe pion mass which is ∼ ∼

800 MeV in the ρ meson and also nucleon. It is exactly what we expectedfrom the trace anomaly sum rule: The trace anomalycontributes most of the hadron masses, except the pioncase.The diﬀerence will be much more signiﬁcant at thephysical quark mass. If we use the GMOR relation m π ∝ m q and the Feynman-Hellman theorem (cid:104) H m (cid:105) H = m q ∂m π ∂m q = m π + O ( m π ), we can estimate the gluon traceanomaly contribution in the physical pion state to be (1 − γ m ) m π = 43(2) MeV. On the other hand, thatin the nucleon at the physical light quark mass will be816(10) MeV if we just consider the quark mass contri-bution from three light quarks [8, 9]. Due to the subtlety on whether the heavy quark massterm should be considered as the quark contribution,the gluon trace anomaly contribution in the hadron willchange when all the six quark ﬂavors are taken into ac-count. But thanks to the relation between heavy quarkloop and F in Eq. 3, the matrix element (cid:104) F (cid:105) H or (cid:104) m Q ¯ ψ Q ψ (cid:105) H are insensitive to the amount of the heavyﬂavors, and then the ratio of the (cid:104) F (cid:105) H between diﬀer-ent hadrons is also insensitive to this subtlety. Thus ourresults show a direct and clear evidences that the gluontrace anomaly in the nucleon and ρ meson, will be O (10)times larger than that in the pion. Summary and outlook : In this work, we calculatedthe quark mass term (cid:104) m ¯ ψψ (cid:105) H and gluon action terms (cid:104) F (cid:105) H in the hadron. Based on the EMT trace anomalysum rule in the PS and V meson states with m v a =0.3, wedetermined the bare anomalous dimensions of the quarkmass and gluon coupling constant as γ m = 0 . β g = − . γ m and β g , we ﬁndthat the gluon trace anomaly contribution in the PS me-son mass is always much smaller than that in the otherhadrons, especially around the chiral limit. Thus we ver-iﬁed that the major part of the proton mass comes fromthe trace anomaly mechanism but not the Higgs couplingwith the quarks.For PS meson, its correlation function with interpola-tion ﬁeld ¯ ψγ ψ corresponds to the norm Tr( S † S ) of thepropagator S . Such a norm term is cancelled in the othermesons, it leads that the decay rate of other mesons’ cor-relation function is larger than that of the PS meson andalso the PS ground state mass is smaller than others [19].However, it is still an open question how it is related tothe trace anomaly, further investigation is needed but itis beyond the study in this work. ACKNOWLEDGEMENT

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Proceedings, 37th International Sympo-sium on Lattice Field Theory (Lattice 2019): Wuhan,China, June 16-22 2019 , PoS