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QCD Constituent quark masses as SM parameters S.I. Sukhoruchkin Petersburg Nuclear Physics Institute Gatchina Russia

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Introduction Nonstrange pion and ρ-meson are well-known participants of nuclear forces acting between nucleons. Nucleon masses themselves are result of the quark- gluon interaction estimated within QCD-lattice calculations [1]. They are parameters of NRCQM constituent quark model with the meson interaction between quarks ( for example, Goldston-boson interaction [2]). NRCQM is a part of QCD which itself is a part of Standard Model - theory of all interactions. Observed universal character of beta-transitions in many nuclei permitted a conclusion that "the nucleus is one specific case, the coldest and most symmetric one, of hadronic matter" [3]. The mean binding energy of the nucleon in nuclei is about 8 MeV, its mass is about 940-8=932 MeV. Recent understanding of a role of pion exchange (and corresponding tensor forces between nucleons) in observed systematic effects of nuclear structure was discussed by T.Otsuka [4] and used in this work during the correlation analysis of binding energies and excitations of a large scope of nuclei [5-8]. Clustering effects in the nuclear structure [9], unexpected stability of parameters of nucleon residual interaction [10], the interconnection between nuclear parameters and parameters of the Standard Model are considered in light of the suggestion by Y.Nambu [11] that empirical relations in particle masses could be used for the development of the Standard Model. Empirical relations in masses discussed [12-15] are considered together with recent result from ATLAS [16] on Higgs boson mass 126GeV (2.5 σ signif.)

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Figure 11. Observed intervals in nuclear binding energies of light nuclei (Z less 26) differing with 4 alpha.

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. Parameters of NRCQM Model Observed tuning effects in nuclear excitations and binding energies (parameter 2me etc.) were compared with relations in particle masses. Two aspects of this study are connected with NRQM parameters (1) and SM-parameters (2). 1a) Nambu pointed out relations 1:8 between masses of the pion and lambda hyperon (1952), checked in b) Samios (Part. Data Group) found coincidence of pion mass with Omega-Ksi hyperons (the periodic sequence with the period equal to the pion mass, nxmp expression). 1c) Sternheimer noticed that constant splitting in pseudoscalar mesons 409 MeV is p+2mpo 1d) Wick considered stable mass intervals Mq=m(omega)/2=(1/2)782 MeV=390 MeV of vector boson mass (close to three times the pion parameter 3fpi=3x130MeV=390 MeV). 1e) Sternheimer considered stable mass intervals Mq=441 MeV between masses of many particles, Md=436 MeV was used in NRCQM calculation of strange baryon masses by C. Itoh et al. [17]. 1f) Kropotkin noticed that Sternheimers parameter Mq is close to (1/3) of the mass of the Ksi-hyperon 1324 MeV/3=441 MeV. It is the result of a compensaton of the mass increase due to the doubled strangeness (2ms=2x150 MeV) and the mass decrease due to the residual interaction (2x147 MeV) between three initial baryon quarks (with values of 441 MeV each). The initial baryon quark mass M=1324 MeV=3x441 MeV is three times the parameter of the residual quark interaction DMD=147~MeV well known from nucleon Delta excitation (see the last line of Table 3,4 that follow). Relations 1a-1f are in agreement with NRQCM where estimates (1/2) and (1/3) are applied to real masses. According to NRCQM (Fig 13 that follows, bottom) three initial quarks (with values explained with QCD lattice calculations, Fig. 13 top) become Delta-baryon (parralell spins, S=3/2) and nucleons (S=1/2). The baryon quark mass Md=1232 MeV=3x410 is equal to the equidistancy in pseudoscalars (line 1c).

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Fig.12 (from [1]). QCD gluon-quark-dressing effect calculated with Dyson-Schwinger Equation, initial masses m; the constituent quark mass arises from a cloud of low-momentum gluons attaching themselves to the current-quark; this is dynamical chiral symmetry breacking: an nonperturbative effect that generates a quark mass from nothing even at chiral limit m=0 ( bottom).

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Fig.13 (from [2]). Calculation of nonstrange baryon masses (left) and lambda-baryon masses as a function of interaction strength within GBECQM – Goldstone Boson Exchange interaction Constituent Quark Model; initial baryon mass 1350 MeV=3x450 MeV=3Mq is near bottom + on left vertical axis.

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. Parameters of NRCQM Model (cont.) Observed tuning effect in nuclear data with parameters 161 keV (equall to 1/8 of nucleon mass splitting) and 170 keV (equal to 1/3 of the electron mass) was considered in the view of Y. Nambu suggestion to a search for relations in particle masses. The important role of the pion-exchange dynamics permitted to find out the relation in the nucleon masses which are parameters of the NRQM model. 1g) Above mentioned mass parameters (including pion mp and muon mu masses) can be expressed as integers of the doubled value of the pion beta-decay energy (2dmp – 2me) which is close to 16me=d (due to relation 9:1 between pion mass difference and the electron mass, 1968). The observed ratios are: (mu + me)/2(dmp – me) = fp=130.7 MeV (PDG) /2(dmp – me)=16.01 (mp(+-) – me)/ 2(dmp – me)=17.03 DMD/2(dmp-me)=18.02 equidistant interval in pseudiscalar mesons/2(dmp-me)=50.1 neutron mass +me/2(dmp-me)= h) The confirmation of NRCQM parameters comes from the stable character of difference of nuclear binding energies MeV and MeV in light and heavy nuclei, 441 MeV in all odd-odd nuclei. 1i) NRCQM parameters Mq=441 MeV and Mq=M(rho)/2=376 MeV form with vector boson masses (Mz=91,19 GeV and Mw=80.40 GeV) ratios and which are close to lepton ratio L=mu/me=207=13x18-1=23x9 1j) Using the exactly known ratio between masses of the proton and the electron the shift of neutron mass from 115 periods d=16 me was determined as exactly 161 keV equal to 1/8 of the nucleon mass splitting (see Table that follows). Such relation between particle masses and electron mass ( SM parameter) is supported by analysis by Frosch ( follows).

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Fig.14 Tests of the Empirical Mass Rormula m=Nx3me for Leptons and Hadrons, by R.Frosch (in ) Nuovo Cimento v.104A, 6, 913, 1991 Deapest maximum in the mean root deviations corresponds to the period 3me The experimental set of 47 masses was replaced by set of 47 random numbers… the probability for random masses to fit the 3me formula better than the experimental masses is only 2x10-4 See numbers close to the integers in the 2 nd column of the table that follows, they are marked by asterisk

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Table 3. Comparison of particle masses (PDG 2008) with periods 3me and 16me = δ = (2) (N - number of the period δ), me= (13) keV Part.m i, MeVm i /3m e N·16m e Nm i -N·16m e Comments µ (4)68.92* πoπo (6)88,05* ,0174 π±π± (4)91.04* ηoηo (24) (2) ω782.65(12) (12) φ (2)665.01* (2) Κ±Κ± (16)322.03* (16) p (1)612.05* (1) m e -9/8δm N n (1)612.89*115-m e (1) keV -m e -1/8δm N ΣoΣo (2)777.98* (2)-0.51·2=-1.02 ΞoΞo (20)857.71* (20)-0.51·3=-1.53

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Parameters of the Standard Model (I) It was noticed in 1972 that nonstatistical effects in nuclear excitation and spacing distributions can be presented as three structure (few-particle, fine and super- fine structures) with the corresponding periods MeV, 1.2 keV and 1.3 eV related as small QED radiative correction to the doubled value of three-fold Delta-excitions of the nucleon (or the empirical parameter 441 MeV introduced by Sternheimer and Kropotkin which coincides with the estimate of baryon constituent quark in NRCQM model developed many years later). It was also found later that such well-known factor alpha/2p=115.9x10-5 (QED radiative correction to the magnetic moment of the electron – Schwinger term) exactly coincides with the well-known ratio between two important parameters of the Standard Model mu/Z =115.9x10-5. It was suggested by Belokurov and Shirkov (see [18]) that the electron mass itself could contain a component proportional to this small factor. Discussed here empirical relations in particle masses and in nuclear data were represented as the common expression with different powers of such common dimentionless factor to fit a great number of empirically observed stable nuclear intervals (Tables 5-6). An example of such stable nuclear interval is given in Fig 15 and Table 7 for the above discussed (found by Schiffer et al. and explained by Otsuka) linear trend in valence proton excitations in Sb isotopes. Maximum at 160 keV in spacing distribution in two neighbour Sb-isotopes confirm a stable character of intervals related as QED correction to the pion mass mp=140 MeV (excitations were connected with pion-exchange dynamics).

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Parameters of the Standard Model (II) Lepton ratio as the distinguished parameter Earlier, as a realization of Nambus suggestion to search for empirical mass relations needed for SM-development, it was noticed that: 1) the well-known lepton ratio L=mμ/m e = becomes the integer 207=9x23=13x16-1 after a small QED radiative correction applied to m e (it becomes mμ/me(1-α/2π)=207.01) 2) the same ratio L=207 exists between masses of vector bosons M Z =91.188(2) GeV and M W =80.40(3) GeV and two discussed estimates of baryon/meson constituent quark masses Mq=441 MeV=mΞ¡/3=(3/2)(m Δ -m N ) and M˝ q =mρ/2=775.5(4) MeV/2 =387.8(2) MeV. These ratios are Mz/441 MeV=206.8 and Mw/(m ρ /2)= The origin of these effects should be considered in the complex analysis of tunung effects in particle masses and in nuclear data which would be performed after the determination of the mass of the scalar fields. Preliminary result Mh=126 GeV [16] is obtained by ATLAS collaboration (2011). We do not use earlier obtained results from LEP, namely, Mh=116 GeV (P.Igo-Kemenes, J.Phys G 388, 2006; also: ALEPH collabor. Phys.Lett. B526, 191, 2002; PDG Phys.Lett, B667, 1, 2008; see also: H.Schopper The Lord of the Collider Rings at CERN , Springer 2009).

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Table 5. Presentation of parameters of tuning effects in particle masses (three upper parts with x = -1,0,1) and in nuclear data (separately in binding energies x=0 and excitations x = 1,2) by expression (n·16me(α/2π)x)·m with QED parameter α = Higgs boson and pion masses, the discussed shift in neutron mass nδ-mn-me=161 keV, the parameter of nucleon Delta excitation and me/3 (which related as α/2π) are marked. xmn=1n=13n=16n=17n=18 3/2m t =171.2 GeV1M Z =91.2M H =115M H =126 1/2M L3 = m e =δm µ =105.7F π =130.7.m π -m e (m Δ -m N ) /2=147 MeV3M q ˝=m ρ /2M q ΄=420M q =441 3M q ˝΄=m ω /23ΔM Δ = Δ-ε o 106=ΔE B 130=ΔE B 140=ΔE B 147.2=ΔE B MeV3441.5=ΔE B 1 keV 1 8, 9.5 keV123 keVnδ-m n -m e =161.6(1) δm N = (1) 170=m e /3 m e = eV eV 22 eV 143 eV187 eV 375 eV

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Fig.15 Different estimates of constituent quark masses

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Figure 15 (continued). Different mass intervals and hadron/lepton masses are shown here by the two-dimensional mass-presentation with the horizontal axis in units δ=16me close to mω/6. Residuals Mi - n(16×16me=16δ) are plotted in the vertical direction (y-axis with units of δ =16me). Three values corresponding to different estimates of constituent quark masses are shown as lines with different slopes: 1) horizontal line corresponds to Wicks interval mω/2=M´´q=48δ; 2) crossed arrows correspond to MΔq=409 MeV=32+18=50δ and 3) parrallel lines with the large slope – to the interval considered by Sternheimer and Kropotkin Mq=441MeV=3x18=54δ. It is corresponding to the initial value of the baryon constituent quark which is equal to three-fold value of nucleon Delta-excitation parameter 147 MeV. Mass of the nucleon in nuclear medium which is less than free nucleon mass (N) by about 8 MeV is located on the stright line from omega-meson mass to the sigma- hyperon mass. This reduced value corresponds to 6x16+18=114δ.

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Fig.16. Spacing distribution in Sb-122,124 Table 7 : Nuclear excitations close to me/3 and nδmN/8 in Sn and Sb isotopes Appearance of stable intervals in nuclear excitations rational to me and δmN (named tuning effect) was found in data for many near-magic nuclei.

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Parameters of the Standard Model (III) It was discussed by many authors [19-21] that radiative correction of the type g/2pi could be useful for comparison of different parameters. A possible example of application of this method was given in [22,23]. Confirmation of preliminary ATLAS data on the SM-scalar mass permitted to check again empirical facts on presence of mass/energy intervals similar to the discussed parameters of NRCQM (147 MeV) and given above intervals multiple to it. Observed in nuclear data long-range correlations with the parameters me and εo=2me (see Table 2) should be considered as additional argument for a fundamental meaning of earlier found empirical relations in nuclear and neutron resonance data, for example, nonstatistical effects in low-lying and highly excited nuclear states found by V.Andreev, M.Ohkubo, K.Ideno, F.Belyaev and G.Rohr.

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Conclusions Presence of tuning effects in nuclear excitations and nuclear binding energies is confirmed with new analysis of data from PNPI compilations (Vols. I/19, I/22, I/25 LB Springer) based on the role of pion-exchange. Relation between observed stable nuclear intervals and particle masses can be connected with the properties of SM scalars if the preliminary ATLAS results (MH=126 GeV) will be confirmed. Ratios µ/Mz= α/2π and (1/3me)/MH=(α/2π)^2 could be a reflection of the fundamental relations between SM-parameters (possible super-duper model by R.Feynman [24]). Relation (N·16me – me –mn)/δmN =1/8.000 could be checked with new more accurate value of the mp/me ratio (this coincidence is important). Observed analogy between tuning effects in particle masses and in nuclear data should be theoretically based on QCD as a part of Standard Model and Nambu suggestion about the role of empirical relation for SM development. Scientific potential of nuclear physics can be connected with fundamental role of QED parameters including the electron mass and QED corrections.

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