Possible Light Neutral Boson and Particle Mass Quantization

  • Qualities of nucleons, such as the fundamental parameter mass, in extreme conditions might be modified relative to the isolate ones. We show the ratio of the EMC-effect tagged nucleon mass to that of the free one ($m^{\ast}/m$), which are derived from nuclear structure function ratio between heavy nuclei and deuterium measured in electron Deep Inelastic Scattering (DIS) reaction in 0.3$\leqslant x\leqslant $0.7. The increase of $m^{\ast}/m$ with $A^{-1/3}$ is phenomenological interpreted via the release of color-singlet cluster formed by sea quarks and gluons in bound nucleons holding high momentum in the nucleus, from which the mass and fraction of non-nucleonic components in nuclei are deduced. The mass of color-singlet cluster released from per short range correlated (SRC) proton in high momentum region ($k>$ 2 fm$^{-1}$) is extracted to be 16.890$\pm$0.016 MeV/c$^{2}$, which is an evidence of the possible indication of a light neutral boson and quantized mass of matter.
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Tao-Feng Wang, Zi-Ming Li and Xiao-Ting Yang. Possible Light Neutral Boson and Particle Mass Quantization[J]. Chinese Physics C.
Tao-Feng Wang, Zi-Ming Li and Xiao-Ting Yang. Possible Light Neutral Boson and Particle Mass Quantization[J]. Chinese Physics C. shu
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Possible Light Neutral Boson and Particle Mass Quantization

    Corresponding author: Tao-Feng Wang, tfwang@buaa.edu.cn
  • 1. School of Physics, Beihang University, Beijing 100191, China

Abstract: Qualities of nucleons, such as the fundamental parameter mass, in extreme conditions might be modified relative to the isolate ones. We show the ratio of the EMC-effect tagged nucleon mass to that of the free one ($m^{\ast}/m$), which are derived from nuclear structure function ratio between heavy nuclei and deuterium measured in electron Deep Inelastic Scattering (DIS) reaction in 0.3$\leqslant x\leqslant $0.7. The increase of $m^{\ast}/m$ with $A^{-1/3}$ is phenomenological interpreted via the release of color-singlet cluster formed by sea quarks and gluons in bound nucleons holding high momentum in the nucleus, from which the mass and fraction of non-nucleonic components in nuclei are deduced. The mass of color-singlet cluster released from per short range correlated (SRC) proton in high momentum region ($k>$ 2 fm$^{-1}$) is extracted to be 16.890$\pm$0.016 MeV/c$^{2}$, which is an evidence of the possible indication of a light neutral boson and quantized mass of matter.

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    I.   INTRODUCTION
    • The widely accepted physical framework, the Standard Model (SM), can explain the fundamental properties and force between leptons or baryons. It was confirmed substantially after the discovery of the Higgs boson that breaks the weak isospin symmetry of the electroweak interaction [1]. Theoretical complement of SM extensions [2] attribute to explain the astrophysical observations of evidences of dark matter [3-6], thought to account for a quarter of universe's total energy density and compose of subatomic particles. A new proposed gauge vector boson is the force mediator of Weakly Interacting Massive Particle (WIMP) annihilations [6]. This boson is expected to be a light neutral spin-1 boson with a small gauge coupling in the MeV/c$ ^{2} $ to hundred MeV/c$ ^{2} $ mass via allowed decays into leptons and hadrons [7].

      The differential cross section for electron Deep Inelastic Scattering (DIS) cross section is expressed in a form as,

      $ \frac{d\sigma}{d\Omega dE'} = \sigma_{Mott}[W_{2}(Q^{2},\nu)+2W_{1}(Q^{2},\nu)tan^{2}(\theta/2)] $

      (1)

      $ \sigma_{Mott} = \frac{\alpha^{2}cos^{2}(\theta/2)}{4E^{2}sin^{4}(\theta/2)} $

      (2)

      where $ \alpha $ is the fine-structure constant, E and $ \theta $ are the incident electron energy and the scattering angel, $ W_{1} $ and $ W_{2} $ are form factors. Under the limit of high energy and momentum transfer, the structure function have been found to scale with the form factors of $ F_{1}(x) = mW_{1}(x) $ and $ F_{2}(x) = \nu W_{2}(x) $, where m and $ \nu $ are the free nucleon mass and the transfer energy in the electron DIS reaction. The two structure functions are connected to each other by the CallanGross relation [8], $ F_{2}(x) = 2xF_{1}(x) $. Therefore, the ratio of differential cross sections depending on x is proportional to that of structure function $ F_{2}(x) $. The ratio of the per-nucleon lepton DIS cross section of heavier nuclei to that of deuterium less than one, known as the EMC effect [9-13], indicates a quenching of nuclear structure functions with respect to those of the isolate nucleon. Structure functions are interpreted as a linear combination of quarks distribution functions. The quarks distribution corresponding to EMC effect is essentially altered accordingly. Does the properties of the EMC-effect tagged nucleon, such as its mass, also change remarkably? which is an attractive topic for both experimental and theoretical sides of nuclear and particle physics. Rare experimental researches have been carried out due to the extreme difficulty for producing the properly nuclear in-medium state and using indirect method to deduce the dependent nucleon mass information. It is very striking to probe the mass of the nucleon in dependence on the EMC effect in the nucleus via a phenomenological framework based on the quark distribution functions.

      We mining the world-wide experimental EMC effect data to extract the nucleon mass in the light and heavy nuclei by using the Bjorken re-scaling method. A reducing trend of the mass of the EMC-effect tagged nucleon with A is exhibited clearly, which could be understood by the model of the releasing non-nucleonic components from the nucleon under the configuration with a strong nucleon-nucleon interaction. The detailed description of the analysis process and results are shown in the following.

    II.   X-RESCALING
    • The momentum fraction of the quark in the free nucleon is $ x = Q^{2}/2m\nu $, where $ Q^{2} = \overrightarrow{q}^{2}-\nu^{2} $, $ \overrightarrow{q} $ is 3-vector momentum transfer in the ($ e, e' $) reaction. The nucleon structure functions ($ F_{2}(x) $) in the nucleus detected by the charged lepton deep inelastic scattering (DIS) reaction differ significantly from those measured for the free nucleon, namely EMC effect, which probably indicate the essential modification of the quark distribution in bound nucleons. Therefore, the momentum fraction of quarks in the nucleon in the nucleus is formed as, $ x' = Q^{2}/2m^{\ast}\nu $, resulted from the altered quantity of the nucleon mass in the nucleus, especially the EMC-effect tagged nucleon. A x-rescaling was modeled as $ x' = xm/m^{\ast} = x\eta $ reflecting in the modification of the bound nucleon mass inside nuclei [14], even though the experiments detect only an momentum-averaged $ m^{\ast} $ for the EMC-effect. The ratio of differential cross sections of the DIS reaction for nuclei and deuterium is approximately equal to their nuclear structure function ratio. The x dependence of $ R(A) = 2F_{2}^{(A)}(x)/AF_{2}^{(D)}(x) $ for different targets were measured previously by electron DIS reaction [15]. Though $ m^{\ast}/m $ were extracted based on the electron DIS measurements by using x-rescaling with the parton model [14], $ F_{2}(x) $ of nuclei were roughly expressed by x parameterizations for valence quark distribution without gluon and anti-quark considerations [14]. We utilized the modern parton distribution function to analyse the new measured EMC-effect data, and obtained the different function of $ m^{\ast}/m $ depending on $ A^{-1/3} $. Remarkably, a quantized-mass relationship for the reducing mass of EMC-effect tagged nucleon and the mass of the well-known mesons are observed.

      To extract $ m^{\ast}/m $, the ratio of nuclear structure function is expressed as [17]

      $ \begin{aligned}[b] R(x,Q^{2}) =& \frac{F_{2}^{A}(x,Q^{2})}{F_{2}^{D}(x,Q^{2})} = \frac{5(u^{A}+\overline{u}^{A}+d^{A}+\overline{d}^{A})+4s^{A}}{5(u+\overline{u}+d+\overline{d})+4s}\\ &+\frac{3(\frac{2Z}{A}-1)(u^{A}+\overline{u}^{A}-d^{A}-\overline{d}^{A})}{5(u+\overline{u}+d+\overline{d})+4s} \end{aligned} $

      (3)

      where $ u,\overline{u},d,\overline{d} $ and s are parton distribution of the free proton [18], $ u^{A},\overline{u}^{A},d^{A},\overline{d}^{A} $ and $ s^{A} $ are nuclear parton distribution [16],

      $ \begin{aligned}[b] u_{v}^{A}(x) =& w_{u_{v}}(x,A,Z)\frac{Zu_{v}(x)+Nd_{v}(x)}{A}\\ d_{v}^{A}(x) =& w_{d_{v}}(x,A,Z)\frac{Zd_{v}(x)+Nu_{v}(x)}{A}\\ \overline{u}^{A}(x) =& w_{\overline{q}}(x,A,Z)\frac{Z\overline{u}(x)+N\overline{d}(x)}{A}\\ \overline{d}^{A}(x) =& w_{\overline{q}}(x,A,Z)\frac{Z\overline{d}(x)+N\overline{u}(x)}{A}\\ s^{A}(x) =& w_{\overline{q}}(x,A,Z)s(x) \end{aligned} $

      (4)

      where $ w_{i} $ is a weight function to indicate the nuclear modification for type-i parton distribution. Since $ w_{i} $ are x dependent function, we let $ w_{i} = 1 $ and utilized $ \eta x $ to replace $ x' $ to describe the nuclear modification.

      Recently $ R(A) $ of several nuclei are precisely re-measured in J-Lab [13], and the data are corrected for isoscalar, nuclear Coulomb field, inelastic scattering and Short Range Correlation (SRC) center-of-mass motion. We carried out the x-rescaling for the new measured $ R(A) $ as $ 2F_{2}^{(A)}(\eta x)/AF_{2}^{(D)}(\eta x) $, and adopted the new parameterization of nuclear parton distributions [16] including valence-, anti-, sea-quark descriptions for nuclear structure function [17, 18], then performed a fit (Fig. 1) to deduce the only parameter $ \eta $, namely $ m/m^{\ast} $. The new $ m^{\ast}/m $ depending on $ A^{-1/3} $ are plotted and made a fitting with a function of $ m^{\ast}/m = 0.994-0.069e^{-4.488A^{-1/3}} $, as shown in Fig. 2, which differs from the linear function given in Ref. [14]. The nucleon mass reduction is related to the density distributions of the nucleus which are usually described by a Woods-Saxon type distribution as

      Figure 1.  The fitting for EMC data [13, 15] in the range 0.35$ < x < $0.7, with x-rescaling of nuclear structure function via the parton model [14, 16, 17].

      Figure 2.  The ratio of $ m^{\ast}/m $ dependent on $ A^{-1/3} $: triangles denotes the ratio of average per nucleon mass calculated from mean field theory [19] to the free nucleon mass, solid circles indicates $ m^{\ast}/m $ extracted from $ 2\sigma_{A}/A\sigma_{2} $ experimental data in the range of $ 0.35\leq x \leq0.7 $ to the free nucleon mass [13, 15]. The square indicates the previous $ m^{\ast}/m $ value [14]. The solid curve is the fitting by the function of $ m^{\ast}/m = 0.994- 0.069e^{-4.488A^{-1/3}} $.

      $ \rho_{ws}(r) = \rho_{0}\left [1+exp\left (\frac{r-R}{a}\right )\right ]^{-1} $

      (5)

      with diffuseness parameter $ a\sim $ 0.53 fm and $ \rho_{0}\sim $ 0.17 fm$ ^{-1} $ is the density at the center of the nucleus. The radius parameter R is formed by mass number A as $ R\sim1.10A^{1/3} $ (fm). Therefore, an exponential function was utilized for the fitting of $ m^{\ast}/m $ dependent on $ A^{-1/3} $.

      The data of $ R(A) $ after the isospin correction are shown in Fig. 1. The directly measured EMC effect of $ R(A) $ of $ ^{3} $He is larger than one, those of the isospin symmetry and neutron rich nuclei are less than one, which indicate an enhancement of the structure function for the proton-excess nucleus and a shrinkage of the structure function for the isospin symmetry and neutron rich nuclei compared to that of the free nucleon. $ R(A) $ is the ratio of the structure function averaged by atomic number A, it reflects the characteristics of the major nucleons. On the other hand, the main mass of the nucleon come from the dynamic motion of quarks and gluon. The variation of the structure function correspond to the changed mass of the nucleon in EMC-effect region. Therefore, it is expected that the isospin splitting of the nucleon is $ m^{\ast}_{p}>m^{\ast}_{n} $ for the proton-excess nuclei and the isospin symmetry and neutron rich nuclei in the EMC-effect range of $ 0.35< x < 0.7 $.

    III.   COLOR SINGLET CLUSTER MODEL
    • The existence of light boson with mass $ m_{B}c^{2}<20 $ MeV, coupling to baryons of the Higgs boson or the proposed scalar partners of graviton, would give rise to a force with relatively long-range $ \hbar c/m_{B}c^{2}>10 $ fm [20]. This light boson is possibly produced from a color-singlet cluster formed by sea quarks and gluons in the bound nucleon. The color-singlet cluster might be released from the bound nucleons rather than the valence quarks that represent the quantum number of nucleons and are confined due to QCD potential. The released color-singlet clusters can exchange momentum within the nucleus. Those clusters are proposed to be the origin of the non-nucleonic components in the nucleus and give a significant influence to the nuclear effect on structure function [21]. If the non-nucleonic degree of freedom is combined with the x-rescaling model, a perfect description of the structure function ratio for the isospin scalar nucleus in the region $ x < 0.7 $ can be realized [21].

      The forming of the color-singlet clusters only connects with the color distribution and should not relate to the momentum distribution [21]. The parton momentum distribution in the color-singlet clusters is expressed as $ P_{i}'(x,Q^{2}) = c_{i}(Q^{2})P_{i}(x,Q^{2}) $, where $ i = s,g $ indicate sea quarks or gluons, $ P_{i}(x,Q^{2}) $ are their momentum distribution in the nucleon. $ c_{i} (Q^{2}) $ is the fraction of momentum distribution $ P_{i}(x,Q^{2}) $ in the released cluster. Assuming $ \beta = c_{s}(Q^{2})S+c_{g}(Q^{2})G $, where $ G = \int_{0}^{\infty}P_{G}(x)dx $ and $ S = \int_{0}^{\infty}P_{S}(x)dx $, $ \beta $ is $ Q^{2} $ independent. The nucleons at the center of nucleus are easier to produce color-singlet clusters since there are more surrounding nucleons than those on the nuclear surface to supply the nuclear medium environment. Hence, $ c_{i}(Q^{2}) $ are an averaged value, $ \beta $ becomes larger with A increasing. According to the uncertainty principle, the longitudinal size of a parton would extend over more than one nucleon and participate in the neighbor nucleon if x is much small, which results in the shadowing region of $ R(A) $. The momentum of the nucleus is therefore, $ P_{A} = A(1+\beta-\alpha)P_{N} $, where $ \alpha $ is the fraction of common parton in the nucleus, $ P_{N} $ is nucleon momentum distribution. The fraction of parton momentum in nucleon is

      $ x' = \frac{P_{parton}}{P_{N}} = \frac{P_{parton}}{P_{A}}\left(\frac{P_{N}}{P_{A}}\right)^{-1}= \frac{Q^{2}}{2M_{A}\omega}A(1+\beta-\alpha) = \eta x $

      (6)

      $ \eta = (1+\beta-\alpha)Am/M_{A} $, where $ M_{A} $ is the mass of a nucleus. Hence, the total fraction of the non-nucleonic color-singlet cluster $ (\beta-\alpha) $ in nuclei can be obtained, and its $ A^{-1/3} $ dependence is shown in Fig. 3. The corresponding fitting function is $ 0.0032+0.0901e^{-7.1428A^{-1/3}} $.

      Figure 3.  Fraction of non-nucleonic color-singlet cluster in nuclei, the corresponding fitting function is $ 0.0032+ 0.0901e^{-7.1428A^{-1/3}} $.

    IV.   RESULTS AND DISCUSSION
    • Two nucleons locating in a short-range correlation pair imply that their distance r is smaller than that of two normal nucleons inside a nucleus. The range of the relative momentum of the dominant neutron-proton pairs is from 1.5 fm$ ^{-1} $ to 5.0 fm$ ^{-1} $ [9]. Under this condition, the short-range correlated two nucleons are strongly overlapped, their interaction potential is positive and has the force mediated by $ \omega $ mesons. The effective nucleon mass has a dip at the relative momentum exactly over the Fermi surface [22]. This minimum mass possibly results from the releasing color-singlet cluster from the very close short-range correlated nucleons with high local density.

      The slope of EMC effect $ -dR_{EMC}/dx $ and the scaling factor $ a_{2} $ of SRC have a linear correlation for the light and medium nuclei, which imply that both stem from the high local density and large nucleon virtuality ($ \nu = P^{2}-m^{2} $, where P is the four-momentum). The difference of the free nucleon mass and the EMC-effect tagged nucleon mass ($ \delta_{m} = m-m^{\ast} $) has a linear function depending on the average proton number $ N_{p} $ in the high-momentum region above 2 fm$ ^{-1} $, as shown in Fig. 4. We calculated the average SRC-proton number for $ ^{3} $He, $ ^{4} $He, $ ^{9} $Be, $ ^{12} $C and $ ^{40} $Ca with their single nucleon momentum distributions [23], which are obtained from the many-body variational Monte Carlo calculations (VMC) using the phenomenological AV18 two-nucleon and Urbana-X three-nucleon potential (AV18+UX). We also calculate the average SRC-proton number for $ ^{56} $Fe with Ciofi and Simula's parameterization of momentum distribution [24]. The probability of short-range correlated $ np $ pairs is 20 times than that of $ nn $ pairs or $ pp $ pairs above Fermi momentum. For the isospin symmetry nuclei, the average SRC-proton number $ N_{p} $ is approximately equal to the average SRC-neutron number $ N_{n} $ over 2 fm$ ^{-1} $. In the case of isospin asymmetry nuclei, especially neutron rich nuclei, $ N_{n} $ is slightly larger than $ N_{p} $ due to the occupation of $ nn $ pairs. Here we adopt $ N_{p} $ dependence to show the mass of the color-singlet cluster released by the high-momentum short-range correlated proton. A linear function $ \delta_{m} = 6.141N_{p}+10.749 $ of fitting except $ ^{3} $He, as shown in Fig. 4, is performed and obtained the value of 16.890 $ \pm $ 0.016 MeV/c$ ^{2} $ when $ N_{p} $ equal to one. This mass quantity is surprisedly consistent with the measurement of possible indication of light neutral boson in $ ^{8} $Be [25]. The high consistence implies the existence of light neutral boson with a mass around 16.89 MeV/c$ ^{2} $, which is possibly produced by the two close touching correlated nucleon pair and is re-absorbed by the flying away two correlated nucleons.

      Figure 4.  The mass ($ \delta_{m} $) of the color-singlet cluster dependent on the short-range correlated proton number $ N_{p} $.

      The masses of non-nucleonic color-singlet clusters leaked out from bound nucleons might be mass-quantized by the multiples of unit of 17.35 MeV/c$ ^{2} $. A connection is found between Mac Gregor's Constituent Quark mass model and Nambu's empirical mass formula $ M = nm_{e}/4\alpha $, where n is a positive integer, $ m_{e} $ is the mass of electron and $ \alpha $ is the fine-structure constant [26]. All particles except stable leptons with mass below 1 GeV/c$ ^{2} $ obey this empirical mass criterion, as shown in Fig. 5. If $ n = 1 $, $ M = 17.35 $ MeV/c$ ^{2} $ which is very close to the mass (16.89 MeV/c$ ^{2} $) of the color singlet component released by the correlated nucleon pair. As long as $ n = 8 $, the value of M is similar to the mass of pion meson. The meson cloud around a nucleon probably consists of the pion meson and the new proposed light neutral boson. The mass quantity in the present study (close to 17.35 MeV/c$ ^{2} $) imply it might be one of the elementary building blocks in nature, and probably is a new light vector gauge boson generated in the short-distance correlated nucleon-nucleon interaction.

      Figure 5.  Particle masses M (MeV/c$ ^{2} $) dependent on quantum number n: star indicates the present deduced mass of a color-singlet cluster released from per proton in the momentum region over 2 fm$ ^{-1} $; solid circle denotes the mass of particle below 1 GeV/c$ ^{2} $ depending on quantum number n; line is the fitting with linear function for the M and n.

    V.   CONCLUSION
    • In summary, the mass ratio $ m^{\ast}/m $, deduced from the x-rescaling for the ratio of nuclear structure function in EMC effect region with the parton model, are interpreted by a configuration of non-nucleonic color-singlet cluster released from the short-distance correlated nucleons inside the nuclei. The mass of the color-singlet cluster is derived from the fitting of the mass difference between the mass $ m^{\ast} $ and free nucleon mass depending on SRC-proton number $ N_{p} $. This mass quantity is determined to be 16.890 $ \pm $ 0.016 MeV/c$ ^{2} $, which is an evidence of the possible indication of a light neutral boson and quantized mass of matter.

    • ACKNOWLEDGEMENTS

    • The authors gratefully appreciate financial support from China Scholarship Council.

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