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Some Problems About the Source of Mass in the Minimal Standard Model Volume 5, Issue 10, October – 2020 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 Some Problems about the Source of Mass in the Minimal Standard Model Ting-Hang Pei Institute of Astronomy and Astrophysics, Academia Sinica, Taiwan Abstract –The source of mass is a long-time problem. The infinite lifetime. Einstein’s special relativity reveals that mass and energy Keywords:- The Electroweak Theory, Higgs Mechanism, can transfer to each other and this truth has been proved Gauge Boson, Fermion, Mass, Yukawa Coupling by many things such as the nuclear power. In the PACS: 11.15.Ex, 11.30.Qc, 12.10.Dm, 12.15.Ff, 14.70.-e electroweak theory, the Higgs mechanism makes the gauge bosons obtain their mass. If the vacuum states of the Higgs I. INTRODUCTION fields are the sources of mass for the massive gauge bosons W and Z even electron e-, then this lowest energy(√ℏ풄)풗of Quantum field theory (QFD) combines quantum the Higgs field must be smaller than the Higgs boson of mechanics, special relativity, and classical field theoryto form 125 GeV and even the electron’s rest mass 0.511 MeV. concepts and tools for describing the characteristics of the Surprisingly, (√ℏ풄)풗 is equal to 246 GeV revealed in high-energy particles [1-5]. In the early 1950s, based on the some textbooks. Furthermore, the scalar Higgs boson is success of quantum electrodynamics (QED), QFT was charge-zero (q=0) and spin-zero (S=0) so the vacuum believed by many theorists that it could eventually describe states of the Higgs fields have the same characteristics. and explain all microscopically physical phenomena, not just However, the W and Z bosons are all spin-1 (S=1) and the interactions between electrons, positrons, and photons. moreover, W bosons are charged. Therefore, how to However, the renormalization process cannot be used constitute those massive gauges bosons from the Higgs universally. All infinite values from the perturbation vacuum states becomes a questionable concept. On the calculationsin QED can be removed by redefining a few other hand, due to the local gauge invariance, all mass physical quantities but this method doesn’t fit to many theories. terms are removed and the Yukawa coupling can provide In 1954, Chen-Ning Yang and Robert Mills generalized the electron’s mass through the Higgs mechanism. How to local symmetry of QED to build the non-Abelian gauge theory, constitute the electron’s mass and other fermions from the or the so-called Yang-Mills theory [1,2]. In 1961, Sheldon vacuum states is another serious problem. Especially, the Glashowtried to combine the weakinteraction with the electron-positron pair production from two photons electromagnetic interaction and established a theory [1,2], but directly reveals electron and positron from the photon it had not yet considered the mechanism of fieldsabove 1.022 MeV, not from the Higgs field, and no spontaneoussymmetry breaking. The Higgs mechanism was direct coupling exists between the photon and Higgs fields proposed by Peter Higgs in 1964 [1,2]. Its in the standard model. The mass-energy equivalence spontaneous-symmetry-breaking mechanism shows that the shows the mass of particle coming from energy such as gauge symmetry in Yang-Mills theory can be broken. In 1967, high-energy photons, and the vacuum states of Higgs fields Steven Weinberger and Abdus Salam built the unified theory don’t possess stable energy to support electron with almost of the weak and electromagnetic interactions based on the IJISRT20OCT211 www.ijisrt.com 405 Volume 5, Issue 10, October – 2020 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 Yang-Mills field theory, and introduced the Higgs mechanism electron-positron pair production from two photons seems to into the Glashow's electroweak theory [1,2]. Thus, the directly tell us that the mass of electron and positron is from electroweak theory was obtained and finished, the form as we the photon fields through the coupling between the electron see nowadays. Weinberger further proposed that the mass of field and the photon field [1,2]. The mass-energy equivalence quarks and leptons can also be obtained from the vacuum also reveals the relation between energy and mass [1,2,6]. If states of the Higgs scalar fields. Therefore, the Higgs the mass of electron comes from the vacuum states of the mechanism is widely believed to explain the mass sources of Higgs fields, then the electron-positron pair production and its particles, including W and Z bosons, and fermions [1-5]. inverse process cause our confusion. Therefore, we try to discuss the source of mass in the electroweak theory from However, according to the success of QED, the different viewpoints in this paper. II. REVIEW OF THE GAUGE THEORY AND STANDARD MODEL In this section, we briefly review the gauge theory and the minimal standard modelin which ћ=c=1 are used in the most places. ћ is the reduced Planck’s constant and c is the speed of light in free space. Theoretically speaking,the gauge fields can proceed the gauge transformation by gauge groups.The Lagrangian is invariant under the gauge transformation. SU(3) isthe gauge group for the strong interaction, and the electroweak interaction is described by the SU(2)×U(1) group. Whatis so-called the standard model is described by the SU(3)×SU(2)×U(1) group [1-5]. In the U(1) Higgs mechanism, first consideringthe Klein-Gordon Lagrangian[3,5] ∗ ∗ 퐿 = (휕훼훷) (휕훼훷) + 푉(훷 훷), (1) Where Φ is the complex Higgs field and the mass term is removed due to the gauge invariance. U(1) gauge transformation involves phase transformation in which the transformation group is Abelian. Φsatisfies the gauge phase transformation [3-5] 훷 → 훷′ = 푒−푖훬훷, (2) where Λ is the phase. This gauge theory requires Lagrangian havingglobal gauge invariance after gauge transformation when Λ is a constant. The prove is as follows ′ −푖훬 ∗ 휇 −푖훬 푖훬 −푖훬 퐿 → 퐿 = [휕휇(푒 훷)] [휕 (푒 훷)] + 푉(푒 훷푒 훷) = 퐿. (3) When Λ is a variable, in order to satisfy this requirement, the partial derivative 휕훼has to change to the covariant derivative Dμ, so we have [1-5] 퐷휇 ≡ 휕휇 + 푖푞퐴휇, (4) whereAμ is the gauge vector field and its local gauge transformation is [1-5] ′ 퐴휇 → 퐴휇 + 휕휇훬. (5) Hence, the Lagrangianpossesses invariance shown as follows IJISRT20OCT211 www.ijisrt.com 406 Volume 5, Issue 10, October – 2020 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 ′ ′ 푖푞훬 ∗ 휇 휇′ 푖푞훬 −푖푞훬 ∗ 푖푞훬 퐿 → 퐿 = [(휕휇 + 푖푞퐴휇)(푒 훷)] [(휕 + 푖푞퐴 )(푒 훷)] + 푉(푒 훷 푒 훷) 푖푞훬 ∗ 휇 휇 휇 푖푞훬 ∗ = [(휕휇 + 푖푞퐴휇 − 푖푞휕휇훬)(푒 훷)] [(휕 + 푖푞퐴 − 푖푞휕 훬)(푒 훷)] + 푉(훷 훷) ∗ 휇 ∗ = (퐷휇훷) (퐷 훷) + 푉(훷 훷) = 퐿. (6) Furthermore, the local gauge invariance also requires additional Lagrangian describing this free-propagation gauge vector field,which is [3-5] 1 1 퐿 = − 퐹 퐹훼훽 + 푚2퐴 퐴훼 , (7) 푝 4 훼훽 2 훼 where [1-6] 퐹훼훽 = 휕훼퐴훽 − 휕훽퐴훼. (8) In order to satisfy the local gauge invariance, the m-term has to be removed as mentioned previously. The new Lagrangian satisfying the local gauge invariance is [1-5] 퐿 ∗ 1 = (퐷 훷) (퐷휇훷) − 퐹 퐹훼훽 − 푉(훷∗훷). (9) 휇 4 훼훽 It means that all gauge bosons describing here are zero mass which are correct for the photon in the electromagnetic interaction and the gluon in the strong interaction but not for W+, W-,and Z bosons. Therefore, the Higgs mechanism is used on a larger symmetrical group to solve this unsatisfied problem. Next, the SU(2)×U(1) gauge transformation is reviewed in the electroweak theory. In the lepton section, we mainly follow the contents in Ref. 3.In the first generation of the lepton section, it consists of a left-handed doublet of fermion [1-5] 훹 1 휈퐿 5 훹푒휈 훹퐿 = ( ) = (1 − 훾 ) ( ), (10) 훹푒퐿 2 훹푒 and a right-handed Fermi singlet [1-5] 1 훹 = (1 + 훾5)훹 . (11) 푅 2 푒 푖 The Lagrangian also consists of a doublet of complex scalars, three 퐴휇 gauge bosons related to the SU(2) symmetry, and one Bμ gauge boson related to the U(1) symmetry. The lepton and scalar doublets follow the transformation [3-5] ′ 훹 → 훹 1 1 푖푔 휎 훼 푖푔′ 훽퐘 = 푒⏟ 2 푖 푖 푒⏟ 2 훹, (12) 푈2 푈1 ’ Where i runs from 1 to 3,g and g are respectively two coupling constants for the SU(2) and U(1) gauge groups, αi and β are independent rotation angles, σi are the Pauli matrices, and Y is the hypercharge operator [3-5].Un-similar to the doublet, the gauge IJISRT20OCT211 www.ijisrt.com 407 Volume 5, Issue 10, October – 2020 International Journal of Innovative Science and Research Technology ISSN No:-2456-2165 transformation for this singlet is 1 ′ ′ −푖 푔 훽퐘 훹푅 = 푒 2 훹푅 = 푈1훹푅. (13) The gauge transformations for the gauge fields are [3-5] ′푖 푖 푗 퐴휇 = 퐴휇 + 휕휇훼푖 − 푔휖푖푗푘퐴휇훼푘, (14) and ′ 퐵휇 = 퐵휇 + 퐘휕휇훽. (15) The covariant derivative now is [3-5] 1 퐷⃑⃑ = 휕 + 푖푔퐀 + 푖 푔′퐘퐵 , (16) 휇 휇 휇 2 휇 where 1 퐀 = 휎 퐴푖 . (17) 휇 2 푖 휇 The arrow “⟶” means the derivative acting on the right and “⟵” means the derivative acting on the left. The double arrow “⟷” means the derivative actingon both sidesand equal to right minus left.If we define a gauge field tensor [3-5] 퐅휇휈 = 퐷휇퐀휈 − 퐷휈퐀휇, (18) then it satisfies gauge invariance under thegauge transformation.
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