PProton mass decomposition
Yi-Bo
Yang ,(cid:63) , Ying
Chen , Terrence
Draper , Jian
Liang , and Keh-Fei
Liu Department of Physics and Astronomy, Michigan State University, East Lansing, MI 48824, USA Institute of High Energy Physics and Theoretical Physics Center for Science Facilities,Chinese Academy of Sciences, Beijing 100049, China Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, USA
Abstract.
We report the results on the proton mass decomposition and also on the relatedquark and glue momentum fractions. The results are based on overlap valence fermionson four ensembles of N f = + u , d , and s quark masses contribute 9(2)% to the proton mass. The quark energy and glue fieldenergy contribute 31(5)% and 37(5)% respectively in the MS scheme at µ = u , d , s and glue momentumfractions in the MS scheme are consistent with the global analysis at µ = The Higgs boson provides the source of quark masses. But how it is related to the proton mass andthus the mass of nuclei and atoms is another question. The masses of the valence quarks in the protonare just ∼ / or with information from experiment. Withphenomenological input, the first decomposition was carried out by Ji over twenty years ago [1]. Asin Ref. [1, 2], the Hamiltonian of QCD can be decomposed as M = −(cid:104) T (cid:105) = (cid:104) H m (cid:105) + (cid:104) H E (cid:105) ( µ ) + (cid:104) H g (cid:105) ( µ ) + (cid:104) H a (cid:105) , (1)and the trace anomaly gives M = − (cid:104) ˆ T (cid:105) = (cid:104) H m (cid:105) + (cid:104) H a (cid:105) , (2)with H m , H E , and H g denoting the contributions from the quark mass, the quark energy, and the gluefield energy, respectively: H m = (cid:88) u , d , s ··· (cid:90) d x m ψψ, H E = (cid:88) u , d , s ... (cid:90) d x ψ ( (cid:126) D · (cid:126)γ ) ψ, H g = (cid:90) d x
12 ( B − E ) , (3) (cid:63) Speaker, e-mail: [email protected] a r X i v : . [ h e p - l a t ] O c t nd the QCD anomaly term H a is the joint contribution from the quantum anomaly of both glue andquark, H a = H a g + H γ m , H a g = (cid:90) d x − β ( g )4 g ( E + B ) , H γ m = (cid:88) u , d , s ··· (cid:90) d x γ m m ψψ. (4)All the (cid:104) H (cid:105) are defined by (cid:104) N | H | N (cid:105) / (cid:104) N | N (cid:105) where | N (cid:105) is the nucleon state in the rest frame. Note that (cid:104) H E + H g (cid:105) , (cid:104) H m (cid:105) and (cid:104) H a (cid:105) are scale and renormalization scheme independent, but (cid:104) H E (cid:105) ( µ ) and (cid:104) H g (cid:105) ( µ )separately have scale dependence.The nucleon mass M can be calculated from the nucleon two-point function. If one calculatesfurther (cid:104) H m (cid:105) and (cid:104) H E (cid:105) ( µ ), then (cid:104) H g (cid:105) ( µ ) and (cid:104) H a (cid:105) can be obtained through Eqs. (1) and (2). Theapproach has been adopted to decompose the S-wave meson masses to gain insight about contributionsof each term from light mesons to charmonuims [2]. In this work, the quark energy (cid:104) H E (cid:105) is obtainedfrom the quark momentum fraction from a local current and (cid:104) H m (cid:105) and with the help of the equationof motion, i.e. (cid:104) H E (cid:105) = (cid:104) x (cid:105) q M − (cid:104) H m (cid:105) . (5)Since there is an O ( a ) error in the equation of motion due to the fact that the local energy-momentumtensor operator adopted is not the conserved current, the concomitant systematic error can be up to20% for the light quark case in the meson mass study of the pseudoscalar meson [2]. In principle, itwould be better to use the conserved energy-momentum tensor (EMT) on the lattice to avoid the needfor normalization and attempts to construct such a conserved EMT on the lattice have been madeperturbatively and non-perturbatively [3] and recently by Suzuki [4, 5] with gradient flow at finitelattice spacing. However, they are complicated to construct. In the present work, we still use the localcurrent and will address the normalization in addition to renormalization and mixing.In addition to calculating the quark momentum fraction (cid:104) x (cid:105) q , we also calculate the glue momentumfraction (cid:104) x (cid:105) g . The latter is directly related to the glue field energy (cid:104) H g (cid:105) = (cid:104) x (cid:105) g M . (6)We will discuss the normalization, renormalization, and mixing of (cid:104) x (cid:105) q and (cid:104) x (cid:105) g later.In this proceeding, we will calculate the renormalized quark and glue momentum fractions in theproton on four lattice ensembles and interpolate the results to the physical pion mass with a globalfit including finite lattice spacing and volume corrections. Then we will combine the previous (cid:104) H m (cid:105) result [6] to obtain the full decomposition of the proton mass. We use overlap valence fermion on (2 +
1) flavor RBC / UKQCD DWF gauge configurations from fourensembles on 24 ×
64 (24I), 32 ×
64 (32I) [7], 32 ×
64 (32ID) and 48 ×
96 (48I) [8] lattices.These ensembles cover three values of the lattice spacing and volume, and four values of the quarkmass in the sea, and then allow us to implement a global fit on our results to control the systematicuncertainties as in Ref. [6, 9]. Other parameters of the ensembles used are listed in Table 1.The e ff ective quark propagator of the massive overlap fermion is the inverse of the operator ( D c + m ) [10, 11], where D c is chiral, i.e. { D c , γ } = m π ∈ (250, 400) MeV on the 24Iand 32I ensembles, and 6 quark masses from m π ∈ (140, 400) MeV on the other two ensembles whichhave larger volumes and thus allow a lighter pion mass with the constraint m π L > .
8. For all the able 1.
The parameters for the RBC / UKQCD configurations[8]: spatial / temporal size, lattice spacing, the seastrange quark mass under MS scheme at 2 GeV, the pion mass with the degenerate light sea quark (both in unitof MeV), and the number of configurations used in this proceeding. Symbol L × T a (fm) m ( s ) s m π N c f g
24I 24 ×
64 0.1105(3) 120 330 20332I 32 ×
64 0.0828(3) 110 300 30932ID 32 ×
64 0.1431(7) 89.4 171 20048I 48 ×
96 0.1141(2) 94.9 139 81quark propagators, 1 step of HYP smearing is applied on all the configurations to improve the signal.Numerical details regarding the calculation of the overlap operator, eigenmode deflation in inversionof the quark matrix, and the Z (3) grid smeared source with low-mode substitution (LMS) to increasestatistics are given in [13–15].The matrix elements we need are obtained from the ratio of the three-point function to the two-point function R ( t f , t ) = (cid:104) | (cid:82) d y Γ e χ ( (cid:126)y, t f ) O ( t ) (cid:80) (cid:126) x ∈ G ¯ χ S ( (cid:126) x , | (cid:105)(cid:104) | (cid:82) d y Γ e χ ( (cid:126)y, t f ) (cid:80) (cid:126) x ∈ G ¯ χ S ( (cid:126) x , | (cid:105) , (7)where χ is the standard proton interpolation field and ¯ χ S is the field with gaussian smearing appliedto all three quarks. All the correlation functions from the source points (cid:126) x in the grid G are combinedto improve the the signal-to-noise ratio (SNR). O ( t ) is the current operator located at time slice t and Γ e is the unpolarized projection operator of the nucleon. When t f is large enough, R ( t f , t ) approachesthe bare nucleon matrix element matrix element (cid:104) N |O| N (cid:105) .For each quark mass on each ensemble, we construct R ( t f , t ) for several sink-source separations t f from 0.7 fm to 1.5 fm and all the current insertion times t between the source and sink, combine allthe data to do the two-state fit, and then obtain the matrix elements we want with the excited-statescontamination under control. The more detailed discussion of the simulation setup and the two-statefit can be found in our previous work [6, 9, 16].To improve the signal in the disconnected insertion case needed for the gluon, strange and alsothe light sea quarks cases, all the time slices are looped over for the proton two-point functions. With5 steps of the HYP smearing, the signal of the glue operator is further improved as evidenced inRef. [16]. The quark and gluon momentum fractions in the nucleon can be defined by the traceless diagonal partof the EMT matrix element in the rest frame [17], (cid:104) x (cid:105) tr q ,g ≡ Tr[ Γ e (cid:104) N | ¯ T q ,g | N (cid:105) ] M Tr[ Γ e (cid:104) N | N (cid:105) ] , (8)¯ T q = (cid:90) d x ψ ( x ) 12 ( 34 γ ←→ D − (cid:88) i = , , γ i ←→ D i ) ψ ( x ) , ¯ T g = (cid:90) d x
12 ( B ( x ) − E ( x ) ) , here M is the proton mass, or alternatively by the o ff -diagonal part of the EMT matrix elements, (cid:104) x (cid:105) o ff q ,g ≡ Tr[ Γ e (cid:104) P | T q ,g i | P (cid:105) ] P i Tr[ Γ e (cid:104) P | P (cid:105) ] (9) T q i = (cid:90) d x ψ ( x ) 14 γ { ←→ D i } ψ ( x ) , T g i = (cid:90) d x (cid:15) i jk E j ( x ) B k ( x ) , where | P (cid:105) is the nucleon state with momentum P and P i is a non-zero component of P . These twodefinitions should give the same result in the continuum due to the rotational symmetry. But they canbe di ff erent under the lattice regularization which breaks this symmetry and should be renormalizedseparately to get consistent results.In Ref. [18], we provided the 1-loop renormalization and mixing calculation of ¯ T and ¯ T i . Therotational symmetry breaking e ff ects in the renormalization constant of the quark operator and themixing from quark to gluon are small, while that in the gluon to quark mixing case is large. The gluerenormalization constant turns out to be ∼ Figure 1.
The contributions of di ff erent quark flavors and glue to the proton momentum fraction. The left panelshows the lattice results renormalized in the MS scheme at 2 GeV with 1-loop perturbative calculation and propernormalization of the glue. The experimental values are illustrated in the right panel, as a function of the MS scale.Our results agree with the experimental values at 2 GeV. In view of the uncertainty in the glue renormalization, we calculate the renormalized quark mo-mentum fractions with the 1-loop perturbative calculation including the mixing of the bare glue mo-mentum fraction and apply the momentum sum rule to determine the renormalized glue momentumfraction. The resulting renormalized momentum fractions of the u , d , s quarks, and glue in the MSscheme at 2 GeV are illustrated in the left panel of Fig. 1, while the right panel shows the correspond-ing experimental values as a function of Q [20]. We note that they agree with each other well withinuncertainties. igure 2. The pie chart of the proton mass decomposition, in terms of the quark mass, quark energy, glue fieldenergy and trace anomaly.
With these momentum fractions, we can apply Eqs. (5) and (6) to obtain the quark and glueenergy contributions in the proton mass, and combine with the quark mass contribution [6] to obtainthe entire picture of the proton mass decomposition, as illustrated in Fig. 2.
In summary, we present a simulation strategy to calculate the proton mass decomposition. The renor-malization and mixing between the quark and glue energy can be calculated perturbatively or non-perturbatively, while the quark mass contribution and the trace anomaly are renormalization groupinvariant. Based on this strategy, the lattice simulation is processed on four ensembles with threelattice spacings and volumes, and several pion masses including the physical pion mass, to control thesystematic uncertainties. With 1-loop perturbative calculation and proper normalization on the glue,we obtained the proton mass decomposition, with the quark mass and trace anomaly contributing9(2)% and 23(1)% respectively, while the fractional contributions of the quark and glue field energiesare 31(5)% and 37(5)% in the MS scheme at 2 GeV. As a check of validity of the present calculation,e find that the individual u , d , s and glue momentum fractions compare favorably with those fromthe global fit in the MS scheme at 2 GeV. References [1] X.D. Ji, Phys. Rev. Lett. , 1071 (1995), hep-ph/9410274 [2] Y.B. Yang, Y. Chen, T. Draper, M. Gong, K.F. Liu, Z. Liu, J.P. Ma, Phys. Rev. D91 , 074516(2015), [3] S. Caracciolo, G. Curci, P. Menotti, A. Pelissetto, Annals Phys. , 119 (1990)[4] H. Suzuki, PTEP , 083B03 (2013), [Erratum: PTEP2015,079201(2015)], [5] H. Makino, H. Suzuki, PTEP , 063B02 (2014), [Erratum: PTEP2015,079202(2015)], [6] Y.B. Yang, A. Alexandru, T. Draper, J. Liang, K.F. Liu (xQCD), Phys. Rev.
D94 , 054503 (2016), [7] Y. Aoki et al. (RBC, UKQCD), Phys. Rev.
D83 , 074508 (2011), [8] T. Blum et al. (RBC, UKQCD), Phys. Rev.
D93 , 074505 (2016), [9] R.S. Sufian, Y.B. Yang, A. Alexandru, T. Draper, J. Liang, K.F. Liu, Phys. Rev. Lett. , 042001(2017), [10] T.W. Chiu, Phys. Rev.
D60 , 034503 (1999), hep-lat/9810052 [11] K.F. Liu, Int. J. Mod. Phys.
A20 , 7241 (2005), hep-lat/0206002 [12] T.W. Chiu, S.V. Zenkin, Phys. Rev.
D59 , 074501 (1999), hep-lat/9806019 [13] A. Li et al. ( χ QCD), Phys. Rev.
D82 , 114501 (2010), [14] M. Gong et al. ( χ QCD), Phys. Rev.
D88 , 014503 (2013), [15] Y.B. Yang, A. Alexandru, T. Draper, M. Gong, K.F. Liu, Phys. Rev.
D93 , 034503 (2016), [16] Y.B. Yang, R.S. Sufian, A. Alexandru, T. Draper, M.J. Glatzmaier, K.F. Liu, Y. Zhao, Phys. Rev.Lett. , 102001 (2017), [17] R. Horsley, R. Millo, Y. Nakamura, H. Perlt, D. Pleiter, P.E.L. Rakow, G. Schierholz, A. Schiller,F. Winter, J.M. Zanotti (UKQCD, QCDSF), Phys. Lett.
B714 , 312 (2012), [18] Y.B. Yang, M. Glatzmaier, K.F. Liu, Y. Zhao (2016), [19] Y.B. Yang, In preparation (2017)[20] S. Dulat, T.J. Hou, J. Gao, M. Guzzi, J. Huston, P. Nadolsky, J. Pumplin, C. Schmidt, D. Stump,C.P. Yuan, Phys. Rev.
D93 , 033006 (2016),, 033006 (2016),