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Review Article

STANDARD MODEL OF —A HEALTH PHYSICS PERSPECTIVE

J. J. Bevelacqua*

publications describing the operation of the Large Abstract—The of is reviewed Collider and popular books and articles describing Higgs with an emphasis on its relationship to the physics supporting the health physics profession. Concepts important to health , magnetic monopoles, , physics are emphasized and specific applications are pre- dark , dark , and new particle discoveries sented. The capability of the Standard Model to provide health (PDG 2008). Many of the students’ questions arise from physics relevant information is illustrated with application of conservation laws to and decay and in the misconceptions regarding the Standard Model and its rela- calculation of the neutron mean lifetime. tionship to the of health physics (Bevelacqua 2008a). Health Phys. 99(5):613–623; 2010 The increasing number and complexity of these questions Key words: accelerators; beta particles; computer calcula- and misconceptions of the Standard Model motivated the tions; author to write this review article. The intent of this paper is to present the Standard Model to health physicists in a manner that minimizes the INTRODUCTION mathematical complexity. This is a challenge because the THE THEORETICAL formulation describing the properties Standard Model of Particle Physics is a theory of interacting and of fundamental particles is embodied in fields. It contains the electroweak (Glashow the Standard Model of Particle Physics (Bettini 2008; 1961; Weinberg 1967; Salam 1969) and chromo- Cottingham and Greenwood 2007; Guidry 1999; Grif- dynamics (QCD) (Gross and Wilczek 1973; Politzer 1973). fiths 2008; Halzen and Martin 1984; Klapdor- QCD is a gauge field theory that describes the strong Kleingrothaus and Staudt 1995; PDG 2008; Peskin and interactions of colored and . Schroeder 1995; Quigg 1997). The Standard Model also The construction of the Standard Model has been provides a basis for the or consistent descrip- guided by principles that are supported by tion of the strong, electromagnetic, and weak interac- theory. Some of these models and underlying tions. However, it does not include . principles may be unfamiliar to health physicists. Ac- During presentation of the author’s health physics cordingly, this paper will address these models and their certification review courses, a steadily increasing interest applications. However, the presentation will omit much in the Standard Model has been noted. Student inquiries of the mathematical rigor required from a theoretical usually occur during presentations of , muon, and physics perspective and focus on those elements of the neutron decay characteristics, and discussion of acceler- Standard Model that are important to health physicists. ator types. Questions often involve the As part of the discussion, the properties of low-energy model, contemporary theories such as or particles are introduced and related to the fundamental supersymmetry, and the discovery of new particles. interactions. These properties are governed by the medi- These questions arise principally from recent graduates ators of the fundamental interactions. The low-energy and appear to be influenced by the content of introduc- tory physics and courses and various particles are also characterized in terms of their under- lying quark content, and the characteristics of the quarks are reviewed. * Bevelacqua Resources, 343 Adair Drive, Richland, WA 99352. For correspondence contact: Joseph Bevelacqua at the above An overview of the four fundamental interactions address, or email at [email protected]. (strong, electromagnetic, weak, and gravitational) and (Manuscript accepted 18 March 2010) 0017-9078/10/0 basic conservation laws governing particle interactions is Copyright © 2010 Health Physics Society also presented in the context of the Standard Model. This DOI: 10.1097/HP.0b013e3181de7127 overview provides a foundation for a discussion of the 613 614 Health Physics November 2010, Volume 99, Number 5 characteristics of particle decays and particle interac- bottom (b), and top (t). These designations are further tions, and the resultant radiation types. These character- defined in subsequent discussion, sometimes desig- istics and associated conservation laws are shown to nated by just the first letter of the quark name; determine which modes are allowed. ● generation—A grouping of quarks and . The The significance of symmetry is also reviewed. Standard Model classifies quarks and leptons into 3 gener- Symmetry principles are illustrated by focusing upon the ations. The first generation includes the u and d quarks and Ϫ electromagnetic interaction and the Maxwell equations. the e and ve and their . Second generation Ϫ Symmetry properties are also reviewed in terms of the particles include the s and c quarks and the ␮ and v␮ and field equations and the calculation of particle decay their antiparticles. The third generation includes the b and t Ϫ properties. A specific application of the Standard Model quarks and the ␶ and v␶ and their antiparticles; is illustrated by considering induced ● hadron—A particle that interacts primarily through the neutron decay and by calculating the mean lifetime of the . and are . neutron. Hadrons are complex particles having an internal quark structure; PARTICLE PROPERTIES AND ● —A fundamental particle that interacts primar- SUPPORTING TERMINOLOGY ily through the weak interaction. The and the electron are examples of leptons. Leptons can In order to ensure the reader is familiar with the be electrically charged or uncharged. Standard model background of the Standard Model, basic particle prop- are massless. Leptons have no internal struc- erties are reviewed. Terminology that supports particle ture and include 3 generations; characterization and facilitates the discussion is also ● —A middleweight particle normally composed presented. The properties of particles that are less famil- of a quark and an antiquark. The charged and neutral iar to some applied health physicists (e.g., and are examples of mesons. Standard Model me- ) are compared to particles of more familiarity sons are composed of quark-antiquark pairs; and (e.g., neutrons, , and ). ● quark—A particle having a fractional that interacts through the strong, electromagnetic, and Terminology. Specific terminology is introduced to weak interactions. Quarks were initially inferred from facilitate presentation of the basic physics associated high-energy electron- (e-p) scattering. The e-p with the Standard Model. These terms, as utilized in this scattering cross-section indicates the presence of review, are specific to the Standard Model and include: point-like structures inside the proton that have been ● —A heavy particle normally composed of three interpreted as quarks. The Standard Model incorpo- quarks. Protons and neutrons are baryons. Baryons can rates 6 quark flavors and 3 generations. be electrically charged or uncharged. Standard Model baryons are composed of three quarks; Basic particle properties. Table 1 provides a sum- ● —A particle having . The mediators mary of the properties of selected low-energy particles or carriers of each of the four fundamental interactions having a below 2,000 MeV-cϪ2 (PDG 2008). These are bosons. The , WϮ and Z0, and gluons are properties include the particle mass, mean lifetime, and mediators of the electromagnetic, weak, and strong dominant decay mode, and are provided for neutrinos interactions, respectively. Pions are also bosons. (electron, muon, and ), the electron (eϪ) and its Bosons can be electrically charged or uncharged; (eϩ), the muon (␮Ϫ) and its antiparticle ● charge—A general term used to assign a particular (␮ϩ), the tau (␶Ϫ) and its antiparticle (␶ϩ), three pions property to a particle or field quanta. Health physicists are (␲ϩ, ␲0, and ␲Ϫ), three kaons (Kϩ,K0, and KϪ), the most familiar with that influences pro- proton (p) and its antiparticle (p៮), and the neutron (n) cesses such as and governs the electromagnetic and its antiparticle (n៮). . Other types of charge exist including Neutrinos are neutral leptons, assumed to be mass- that governs the strong interaction, and that less in the Standard Model, but experimental data sup- manifests itself either as a charged or neutral weak ports a small, but certainly non-zero mass (PDG 2008). current, and these currents govern the weak interaction; There are three known generations of neutrinos (␯) and ● —A particle having half-integer spin. Neu- their corresponding antiparticles (antineutrinos, v៮). Spe-

trons, protons, and electrons are examples of . cifically included are the (ve) and its ៮ Fermions can be electrically charged or uncharged; antiparticle (ve), the (v␮) and its antipar- ● flavor—A designation for the type of quark. The ticle (v៮␮), and the (v␶) and its antiparticle (v៮␶). flavors are down (d), up (u), strange (s), (c), The electron and muon neutrinos are well studied, but Standard Model of Particle Physics ● J. J. BEVELACQUA 615

Table 1. Properties of selected low energy particles. Table 2. Properties of quarks within the Standard Model.a Mass Dominant decay Mass (MeV-cϪ2) Particle (MeV-cϪ2) Mean lifetime mode Generation Flavor Charge (e) Barea Effectiveb ␯ Ͻ Ͼ ba e 0.000002 300 s/eV ␯៮ Ͻ Ͼ ba First d 3to7 340 e 0.000002 300 s/eV 1 ba Ϫ ␯␮ Ͻ0.19 Ͼ15.4 s/eV 3 ba ␯៮ ␮ Ͻ0.19 Ͼ15.4 s/eV a,c First u 2 1.5−3.0 336 ␯␶ Ͻ18.2 Not yet ϩ determinedb 3 ␯៮ Ͻ a,c ␶ 18.2 Not yet Second s 1 95 Ϯ 25 486 determinedb Ϫ eϪ 0.511 Ͼ4.6 ϫ 1026 y Stable 3 ϩ e 0.511 Ͼ4.6 ϫ 1026 y Stable Second c 2 1,270 1,550 ␮Ϫ ϫ Ϫ6 ␮Ϫ 3 Ϫ ϩ ␯ ϩ ␯៮ ϩ 105.7 2.2 10 s e ␮ e 3 ␮ϩ ϫ Ϫ6 ␮ϩ 3 ϩ ϩ ␯៮ ϩ ␯ 105.7 2.2 10 s e ␮ e ␶Ϫ 1777 2.9 ϫ 10Ϫ13 s Multiple decay modes Third b 1 4,200 4,730 ϩ Ϫ Ϫ ␶ 1777 2.9 ϫ 10 13 s Multiple decay modes 3 Ϫ Ϫ8 Ϫ Ϫ ␲ 139.6 2.6 ϫ 10 s ␲ 3 ␮ ϩ ␯៮ ␮ 0 Ϫ17 0 Third t 2 171,200 177,000 ␲ 135.0 8.4 ϫ 10 s ␲ 3 ␥ ϩ ␥ ϩ ϩ Ϫ8 ϩ ϩ ␲ 139.6 2.6 ϫ 10 s ␲ 3 ␮ ϩ ␯␮ 3 Ϫ Ϫ8 Ϫ Ϫ a K 493.7 1.24 ϫ 10 s K 3 ␮ ϩ ␯៮ ␮ PDG (2008). K0d 497.6 d K0 3 ␲ϩ ϩ ␲Ϫ b Griffiths (2008). ϩ Ϫ8 ϩ ϩ K 493.7 1.24 ϫ 10 s K 3 ␮ ϩ ␯␮ p 938.3 Ͼ2.1 ϫ 1029 y Stable ៮ Ͼ ϫ 29 p 938.3 2.1 10 y Stable a 3 ϩ Ϫ ϩ ␯៮ Table 3. Properties of selected baryons within the Standard Model. n 939.6 885.7 s n p e e ៮ ៮ 3 ៮ ϩ ϩ ϩ ␯ n 939.6 885.7 s n p e e Quark Charge Mass Baryon structure (e) (MeV-cϪ2) Mean lifetime a Dependent on the degree of neutrino mixing. b The measured quantities depend upon the Standard Model’s mixing p uud ϩ1 938.3 Ͼ2.1 ϫ 1029 y parameters and to some extent on the experimental conditions (e.g., energy n udd 0 939.6 885.7 s resolution). ⌳ uds 0 1,115.6 2.63 ϫ 10Ϫ10 s c Decay mode not yet determined. ⌺ϩ uus ϩ1 1,189.4 8.02 ϫ 10Ϫ11 s 1 ⌺0 ϫ Ϫ20 d 0 0 0 0 0 0 uds 0 1,192.5 7.40 10 s K K K ϭ ͑K ϩ K Ϫ Ϫ10 The K particle is a superposition of two states S and L; ͱ S L) ⌺ dds Ϫ1 1,197.4 1.48 ϫ 10 s 2 Ϫ 0 ϭ ϫ Ϫ11 0 ϭ ϫ Ϫ8 ⌶0 uss 0 1,314.9 2.90 ϫ 10 10 s with lifetimes of KS 8.95 10 s and KL 5.12 10 s. ⌶Ϫ dss Ϫ1 1,321.7 1.64 ϫ 10Ϫ10 s a Derived from PDG (2008). much less is known about tau neutrinos. The leptons (electrons, muons, taus, and their associated neutrinos) Table 4. Properties of selected mesons within the appearing in Table 1 are fundamental and have no Standard Model.a discernable substructure. This is not true of the mesons Quark Charge Mass and baryons that have an underlying quark structure. The Meson structure (e) (MeV-cϪ2) Mean lifetime properties of these quarks and the composition of se- ␲ϩ ud៮ ϩ1 139.6 2.6 ϫ 10Ϫ8 s ␲Ϫ du៮ Ϫ1 139.6 2.6 ϫ 10Ϫ8 s lected baryons and mesons are summarized in Tables 2, Ϫ ␲0 ͑uu៮ Ϫ dd៮͒/ͱ2 0 135.0 8.4 ϫ 10 17 s 3, and 4, respectively. ϩ Ϫ K us៮ ϩ1 493.7 1.24 ϫ 10 8 s Table 2 summarizes the properties of quarks within K0 ds៮ 0 497.6 b the Standard Model. Both bare and effective are KϪ su៮ Ϫ1 493.7 1.24 ϫ 10Ϫ8 s provided. Bare quark masses are theoretical values based a Derived from PDG (2008). on an isolated or free quark flavor (PDG 2008). The b See Table 1. effective mass or the mass of a quark within a baryon or meson has a different value. These effective mass values are model dependent, include couplings, and may includes the weak interaction not be experimentally measurable. The results are based and the electromagnetic interaction includes the Max- on the currently accepted quark interaction spatial depen- well field equations that have U(1) symmetry. The U dence (PDG 2008; Griffiths 2008). and SU designations refer to unitary and special The effective masses and particle properties of unitary groups, respectively. The numbers in paren- Table 1 are derived from the interactions defining the thesis are the dimensionality of the groups. Standard Model. It is sufficient to state the elec- Tables 3 and 4 provide the properties of selected troweak interaction is characterized by SU(2) R U(1) Standard Model SU(3) baryons and mesons, respec- symmetry, and the strong interaction has SU(3) color tively. The quark structure, electric charge, mass, and symmetry (Bevelacqua 2008a and b; PDG 2008). The lifetime are provided. It should be noted that other 616 Health Physics November 2010, Volume 99, Number 5 symmetry group structures are possible for models that well-known radiation type, it is has a much deeper include physics beyond the Standard Model (PDG 2008). physical significance because its exchange defines the These alternative models are not a significant consider- electromagnetic interaction. In a similar fashion, the ation within the context of this review. exchange of gluons (of which there are 8) defines The particle’s charge can be derived from the the strong interaction. The weak interaction is also internal quark structures of Tables 3 and 4. In addition, complex because there are three particles (i.e., Wϩ,WϪ, the lifetime values are readily derived from the Standard and Z0) that are exchanged. Properties of the field bosons Model. Subsequent discussion will illustrate the neutron give each a distinctive character. mean lifetime calculation. For example, the weak interaction governs , As an illustration of the success of the Standard and manifests itself in the magnitude and dose profile of Model, the quark charge results of Table 2 and specific the neutrino effective dose (Bevelacqua 2004, 2008a). quark structures of Tables 3 and 4 can be used to predict The field boson mass also exhibits a distinctive nature. the charge of a particle. For example, the proton has a The photon, gluons, and are all massless. In uud structure (Table 3) with a resulting charge given by contrast, the weak interaction field bosons have masses the algebraic sum of the individual quark charges (i.e., in the 80–92 GeV-cϪ2 range. The field boson mass does ϩ Ϫ ϭ ϩ ៮ 2/3 e 2/3 e 1/3 e e) and the K structure is us not uniquely determine the nature of the fundamental (Table 4), which provides the expected charge (i.e., 2/3 interaction. It is the collective nature of the field boson’s ϩ ϭ e 1/3 e e). mass, charge, number of allowed states, lifetime, and constant that determines the unique character- FUNDAMENTAL INTERACTIONS istics of an interaction. Four fundamental interactions or describe the In Table 5, the source of the interaction refers to the phenomena observed in the . These are the strong, basic physical quantity that gives rise to the force. The electromagnetic, weak, and gravitational interactions, and four fundamental interactions arise from very different their properties are summarized in Table 5. The unique physical constructs. For example, the gravitational and aspects of the strong, weak, and electromagnetic interac- electromagnetic interactions are derived from mass tions govern particle decays and interactions, which influ- and electric charge, respectively. The concepts of mass ence the health physics consequences of the resulting and electric charge are well known to health physicists. radiation types. However, weak charge and color charge are not. The field boson is the mediator or the carrier of the It is well known from that a force. For example, the electromagnetic interaction is moving charge produces a current (Jackson 1999). mediated by . A photon is exchanged between the Therefore, weak charges in motion generate a weak two particles involved in an electromagnetic interaction. current. Weak currents produce weak forces that govern The field mediators have been directly observed or lepton interactions. Leptons have no color charge, and inferred from observed phenomena. All mediators are consequently do not participate in the strong interaction. based on significant experimental evidence with the Neutrinos have no electric charge so they experience no exception of the graviton that is inferred from gravita- electromagnetic force, but they do participate in the weak tional field theory. interaction. These field bosons give the various fundamental Color charge produces the strong interaction. How- interactions unique properties. Although the photon is a ever, color charge is considerably more complex than

Table 5. Fundamental interactions and their properties. Fundamental interaction

Property Gravitational Electromagnetic Weak Strong Field bosons Graviton Photon Wϩ,WϪ, and Z0 8 gluons Ϫ2 ϭ Mass of field boson (GeV-c )00MW 80.398 0 ϭ MZ 91.1876 Range of the interaction (m) ϱϱ10Ϫ18 Յ10Ϫ15 Source of the interaction Mass Electric charge Weak charge Color charge Strength (relative to the strong interaction) 10Ϫ39 10Ϫ2 10Ϫ5 1 Typical cross-section (m2) a 10Ϫ33 10Ϫ39b 10Ϫ30 Typical lifetime (s) a 10Ϫ20 10Ϫ10 10Ϫ23 a In view of the range and source of the gravitational interaction, the cross-section and lifetime are not well-defined quantities. b This cross-section is applicable to contemporary accelerator (Bevelacqua 2008a). Standard Model of Particle Physics ● J. J. BEVELACQUA 617 electric charge. Color charge is a property assigned to a the nucleus. It arises from the exchange of gluons quark or gluon, and it has three states (i.e., red, blue, and between quarks and governs a number of commonly green). There are eight gluons governing the strong observed processes including fission, fusion, and activa- interaction instead of one photon for the electromagnetic tion. The radiation hazards from these processes are well interaction. Since the gluons themselves carry a color known to health physicists. charge, they can directly interact with other gluons. This The electromagnetic force results from the exchange possibility is not available with the electromagnetic force of photons. It governs much of the physics encountered since photons do not have electric charge. Therefore, it is in our daily lives. For example, and not surprising that the strong and electromagnetic forces molecular chemistry are governed by the electromagnetic have different characteristics. interaction. This interaction also influences nuclear reac- In Table 5, the interaction strength is a measure of tions and competes with the strong force in nuclear the magnitude of the force as measured over its effective processes. The electromagnetic interaction depends on range. The term interaction strength is intrinsically am- the electric charge of the interacting particles. As a biguous because it depends on the measurement distance practical example, can be accelerated because they from the source. Accordingly, the strength values listed have an electric charge and the electromagnetic force in Table 5 may be quoted with different values by other governs their final energy. authors (Bettini 2008; Cottingham and Greenwood 2007; The weak force governs processes such as beta Griffiths 2008; Halzen and Martin 1984). Table 5 pro- decay and decay. Weak interactions also dictate vides the strength relative to the strong interaction. In the behavior of leptons. terms of decreasing strength, nature orders these interac- Although the fundamental interactions are distinct tions as follows: strong, electromagnetic, weak, and phenomena, they often appear collectively in nature. As gravitational. As noted previously, the Standard Model an example, consider the beta decay of 60Co: does not include the gravitational interaction. 60 3 60 ϩ Ϫ ϩ ៮ The cross-section describes the probability of a Co Ni e ve. (1) typical interaction that is solely governed by one of The nuclear energy levels in the 60Co and 60Ni nuclei are the fundamental interactions. The lifetime represents determined by the strong and electromagnetic interac- the over which an interaction occurs assuming the tions. The relative position of the energy levels in the interaction is governed solely by that fundamental 60Co and 60Ni nuclei, their specific properties (e.g., spin force. For example, strong interactions typically create and ), and conservation laws determine if the Ϫ27 2 particles with cross-sections in the mb (10 cm ) transition between a specific set of energy levels pro- Ϫ23 range that have lifetimes on the order of 10 s. The duces a beta particle. cross-section and lifetime of a created particle are During beta decay, a neutron single particle level in often clear indications of the type of force involved in 60Co (59Co ϩ n) transitions to a proton single particle an interaction. level in 60Ni (59Co ϩ p) with the emission of an electron Neutrino interaction cross-sections, governed by the (beta particle) and antielectron neutrino. weak interaction, are orders of magnitude smaller than From a nuclear transformation perspective, beta the typical strong or electromagnetic interaction cross- decay is described by: sections (PDG 2008). The weak interaction cross-section 3 ϩ Ϫ ϩ ៮ magnitude makes neutrino detection difficult for energies n p e ve. (2) encountered at contemporary facilities (Bevelacqua Within the Standard Model, beta decay is described as a 2004, 2008a). sequential process incorporating the WϪ boson: The gravitational interaction is an interaction affect- Ϫ ing massive objects such as , solar systems, and n 3 p ϩ W , (3) . The terms cross-section and lifetime are not Ϫ Ϫ W 3 e ϩ v៮ . (4) clearly defined within the context of the gravitational e interaction. Since the gravitational interaction is not Although eqns (1) through (4) represent the same included in the Standard Model, no further commentary physical process, they differ in terms of the type of is provided. model utilized in the description of the neutron decay process. Fundamental interactions and their health Eqns (1) through (4) may be accepted on face value, physics impacts but the reader should question why these are the physical The strong interaction binds quarks into mesons and beta decay modes. The Standard Model provides insight baryons, and is responsible for binding within into this question. 618 Health Physics November 2010, Volume 99, Number 5 CONSERVATION LAWS not affected by the strong interaction. There are six leptons, classified according their electric charge (Q),

Fundamental physics is governed by basic symme- electron number (Le), muon number (L␮), and tau number tries that are expressed in terms of a set of conservation (L␶). The leptons are naturally grouped into three families laws that permit certain reactions and forbid others. In or generations as summarized in Table 6. this section, the specific conservation laws that facilitate There are also six antileptons, with all the signs in an understanding of the processes that lead to radiation Table 6 reversed (i.e., ϩ to Ϫ and Ϫ to ϩ). The positron types of concern in health physics are examined. Four for example has an electric charge of ϩ1 and an electron conservation laws are useful in understanding the under- number of Ϫ1. Considering both particles and antiparti- lying physics of the Standard Model that is relevant to cles, there are 12 leptons in the Standard Model. health physics applications. Other conservation laws In a similar manner, there are six flavors of quarks (e.g., energy, linear , and ) (u, d, s, c, b, and t) (PDG 2008), with their quantum are also applicable, but this review focuses on the laws numbers classified according to their electric charge, specifically related to the Standard Model. These conser- upness (U), downness (D), (S), charm (C), vation laws include: (B), and (T). These labels are histor- 1. Conservation of Electric Charge: All three of the ical and have no underlying physical meaning. The fundamental interactions governing health physics quarks also fall into three generations as summarized in applications (strong, electromagnetic, and weak) con- Table 7. Again, all signs are reversed on a table of serve electric charge. Many particles participating in antiquarks. Since each quark and antiquark comes in the various fundamental interactions contain electric three colors, there are 36 distinct quarks in the Standard charge (e.g., protons, pions, muons, and electrons); Model. Table 5 and the subsequent discussion noted 8 2. Conservation of Color Charge. The electromagnetic mediators for the strong interaction (gluons), the photon and weak interactions do not affect color charge. for the electromagnetic interaction, and 3 mediators for Color charge is conserved in strong interactions. the weak interaction (Wϩ,WϪ, and Z0). This yields 12 Physical particles (e.g., baryons and mesons) are mediators for the Standard Model. colorless. This means mesons contain a quark of one A careful reader notes that one of the shortcomings color (red, blue, or green) and an antiquark of the of the Standard Model is the number of free parameters same anticolor (antired, antiblue, or antigreen). Bary- or elementary particles that it requires: 12 leptons, 36 ons consist of three quarks each of a different color; quarks, and 12 mediators. There is also at least one other 3. Conservation of . The total number of particle (the ). Therefore, there is a mini- quarks is a constant. Since baryons are composed of mum of 61 parameters to address. The Standard Model three quarks, the baryon number is just the quark number has been remarkably successful, but mounting evidence divided by 3. There is no corresponding conservation of [e.g., indication that neutrinos have mass and recent meson number since the mesons, composed of quark- publications regarding four quark mesons and five quark antiquark pairs, carry zero baryon number; and baryons (PDG 2008)] suggests that physics beyond the 4. Conservation of Electron Number, Muon Number, Standard Model is required to explain these and other and Tau Number. The strong interaction does not affect results. However, the Standard Model can provide results leptons. In a pure electromagnetic interaction, the same of relevance to health physics applications. particle comes out (accompanied by a photon) as went in. The weak interaction only mixes together leptons Consequences of the conservation laws and the from the same generation. Therefore, the lepton number, Standard Model. With knowledge of conservation laws muon number, and tau number are all conserved. and the Standard Model, we will illustrate how these These conservation laws provide a key input to understanding the decay schemes summarized in Table 1. An examination of the health physics consequences of Table 6. Lepton classification.a

particle decays and their associated radiation types is Generation Lepton Charge (e) Le L␮ L␶ possible when these conservation laws are combined First eϪ Ϫ1 100 ␯ with an understanding of the Standard Model of Particle First e 0 100 ␮Ϫ Ϫ Physics. Second 1 010 Second ␯␮ 0 010 For example, leptons interact primarily through the Third ␶Ϫ Ϫ1 001 ␯ weak interaction and electrically charged leptons experi- Third ␶ 0 001 ence the effects of the electromagnetic force. They are a Derived from PDG (2008). Standard Model of Particle Physics ● J. J. BEVELACQUA 619

a ␮Ϫ 3 Ϫ ϩ ␯ ϩ ␯៮ Table 7. Quark classification. Table 9. Muon decay ( e ␮ e). Generation Quark Charge (e) D U S C B T Final state Initial state First d 1 Ϫ10 00 00 ␮Ϫ Ϫ ␯ ␯៮ Ϫ e ␮ e 3 Baryon number 0 0 0 0 a Ϫ First u 2 01 00 00 Lepton number (Le) 0101 a 3 Lepton number (L␮) 1010 Electric charge Ϫe Ϫe0 0 Second s 1 00Ϫ10 00 Ϫ Color charge 0 0 0 0 3 a See Table 6. Second c 2 00 01 00 3 Third b 1 00 00Ϫ10 Ϫ 3 in terms of group properties. As an example, the Third t 2 00 00 01 3 electromagnetic, weak, and strong interaction field a Derived from PDG (2008). quanta are represented by the generators of the unitary group of 1 [U(1)], the of dimension 2 [SU(2)], and the special unitary group laws are satisfied for neutron decay and muon decay. of dimension 3 [SU(3)], respectively. Tables 8 and 9 summarize beta decay and muon decay, Within the Standard Model, the number of genera- respectively. The decays of various particles summarized in tors (N) of a group of dimension n (n Ͼ 1) is given by: Table 1 are not arbitrary, and are governed by the conser- ϭ 2 Ϫ vation laws that follow from the symmetries underlying the N n 1. (5) Standard Model. In particular, decay modes of the neutron For n ϭ 1, there is a single generator. These generators and muon are governed by the conservation of baryon are equivalent to the field bosons summarized in Table 5. number, lepton number (for each generation), electric ϭ Ϫ Therefore, it is expected that the electromagnetic (n 1), charge, and color charge. Other decay modes (e.g., ␮ 3 weak (n ϭ 2), and strong (n ϭ 3) interactions have 1, 3, ϩ Ϫ ␶Ϫ 3 ϩ ␮Ϫ 3 ϩ Ϫ n e , n , and n p e ) are excluded by and 8 field bosons, respectively. This is these conservation laws. observed experimentally with one field boson (photon) Tables 8 and 9 illustrate the application of conser- for the electromagnetic interaction, 3 field bosons (Wϩ, vation laws to predict a particle’s decay and its associated WϪ, and Z0) for the weak interaction, and 8 field bosons radiation types. These laws and the Standard Model are (8 gluons) for the strong interaction. The prediction of sufficient to predict the radiation types that occur in the number and characteristics of the field bosons for the particle decay and interaction processes of interest in electromagnetic, weak, and strong interactions is an health physics applications. An examination of the initial impressive success of the Standard Model of Particle and final states of Tables 8 and 9 illustrate the impact of Physics, and provides additional confidence in its ability conservation laws on allowed processes. to predict the radiation types and their intensity resulting Conservation laws are also implied by the funda- from the decay and interaction of fundamental particles. mental interactions and their underlying symmetry An additional illustration of the relationship between properties. Noether’s Theorem (Bevelacqua 2008a; symmetry groups and underlying physics is provided by Cottingham and Greenwood 2007; Griffiths 2008) considering the U(1) group. provides a mathematical proof of the relationship between a given symmetry and its conservation law. Within the Standard Model, symmetries are expressed Electrodynamics and U(1) symmetry The symmetry groups become more complex as

Ϫ their dimension increases. The simplest group involved Table 8. Beta decay (n 3 p ϩ e ϩ ␯៮ ). e in the Standard Model is the U(1) group describing Final state Initial state electrodynamics. U(1) is defined in terms of a single, Ϫ ␯៮ Conservation law n pe e real, and continuous parameter that is usually selected to Baryon number 1 1 0 0 be a rotation angle. The U(1) group can be applied to the a Ϫ Lepton number (Le) 0011 a electromagnetic interaction embodied in the Maxwell Lepton number (L␮) 0000 Electric charge 0 e Ϫe0 equations. Color charge 0 0 0 0 The symmetrized form of the Maxwell equations is a See Table 6. invariant under a duality transformation (Jackson 1999). 620 Health Physics November 2010, Volume 99, Number 5 For example, the electric (Eជ) and magnetic (Hជ ) fields are the various terms are noted to illustrate the complexity of related by (Bevelacqua 2008b; Jackson 1999): the Standard Model machinery. This approach is consistent ជ ϭ ជ Ј ␰ ϩ ជ Ј ␰ with the desire to minimize the mathematical complexity E E Cos H Sin while maximizing health physics relevance. The notation (6) and definition of terms of Cottingham and Greenwood ជ ϭϪជЈ ␰ ϩ ជ Ј ␰ H E Sin H Cos , (2007) are used in representing the Lagrangian density. where ␰ is a rotation angle and the primed variables are Eqn (9) illustrates how the components of the Lagrang- defined with respect to the coordinate system generated ian density relate to the neutron decay calculation: by this rotation. The form of eqn (6) is often an indication 1 1 1 ϭ Ѩ Ѩ␮ Ϫ 2 2 Ϫ ␮v ϩ ␸ 2͑ 2 ϩ 2͒ ␮ of the underlying U(1) symmetry group. L ␮h h m h Z␮v Z 0 g1 g2 Z␮Z In general, symmetries are related to specific phys- 2 4 4 ical properties when they follow from conservation laws. 1 1 ␮v ϩ ϩ ␮ ϩ␯ The conservation of energy, linear and angular momen- Ϫ A␮ A Ϫ ͓͑D␮W␯ ͒* Ϫ ͑D␯W␮ ͒*͔͓D W 4 v 2 tum, baryon number, and lepton number are examples. Similar relationships are derived by considering the higher ␯ ϩ␮ 1 2 2 Ϫ ϩ␮ Ϫ D W ͔ ϩ g ␸ W␮ W , (9) order symmetries involved in the Standard Model. For 2 2 0 example, baryons and mesons can be classified in terms of SU(3) group representations including the singlet, octet, and where h is the Higgs field, m is the mass of the considered 0 decuplet structures (Griffiths 2008; PDG 2008). particle, Z␮ is the vector field of the Z boson, Z␮v is a field 0 Ϫ The next section of this paper provides a subset of strength tensor associated with the Z boson, W␮ is the Ϫ ϩ the defining equations of the Standard Model. These vector field of the W boson, W␮ is the vector field of the ϩ ␸ equations further illustrate the relationship between sym- W boson, g1 and g2 are coupling constants, and 0 is metry and predicted physics. the of the Higgs field. The D␮ operator and A␮v tensor terms are defined in terms of vector STANDARD MODEL FORMALISM fields and their derivatives, and an angle defined by the masses of the weak interaction field bosons. As with many field theories, the Standard Model can As an illustration of symmetries inherent in eqn (9), be formulated in terms of a Lagrangian density (Bettini consider three specific parameters. These are a rotation 2008; Cottingham and Greenwood 2007; Griffiths 2008; angle determined by the ratio of weak mediator masses, and Halzen and Martin 1984). The Lagrangian density (L) is two specific vector quantities (A␮ and Z␮). Specifically, let written in terms of the fields ␸ and their derivatives Ѩ␮␸ , i i ␪ be a rotation angle () defined by: where w Ѩ␸ MW i cos␪w ϭ . (10) Ѩ␮␸ ϵ ␮ , (7) W i Ѩx Z ␮ ϭ An inherent symmetry of the Standard Model is illus- where i labels the field and 0, 1, 2, and 3 labels the 3 ␮ ϭ ␮ ϭ trated by introducing two vector fields (B␮ and W␮) coordinates (i.e., 0 is the time coordinate and 1, 3 (Cottingham and Greenwood 2007). The B␮ and W␮ 2, and 3 labels the spatial coordinates). The Lagrangian fields can be rewritten in terms of ␪ in a manner that is density is important because the specific field equation (e.g., w analogous to eqn (6) based on U(1) symmetry: Klein-Gordon for spin 0 particles, Dirac for spin 1/2, or

Proca for spin 1) is obtained from the Euler-Lagrange B␮ ϭ A␮cos␪w Ϫ Z␮sin␪w (11) equations (Bettini 2008; Cottingham and Greenwood 2007; 3 ϭ ␪ ϩ ␪ Griffiths 2008; Halzen and Martin 1984): W␮ A␮sin w Z␮cos w. (12) ␪ ѨL ѨL In eqns (11) and (12), w is defined by the Z and W boson ϪѨ ϭ Ѩ␸ ␮ͫѨ͑Ѩ ␸ ͒ͬ 0. (8) masses following eqn (10). This specific relationship can be i ␮ i contrasted with the U(1) electromagnetic example that noted The complete Standard Model Lagrangian density (Cotting- the rotation angle ␰ was dependent on the specific formula- ham and Greenwood 2007; PDG 2008) is quite detailed and tion of the Maxwell equations (Bevelacqua 2008a). How- includes electroweak and QCD components. For the pur- ever, the similarity of eqns 6, 11, and 12 is illustrative of the pose of this paper, only a portion of the Lagrangian density types of symmetries encountered in the Standard Model. relevant to a subsequent neutron decay example is provided Symmetry also influenced the of the without a detailed description of the mathematical relation- Standard Model. For example, the discovery of the Wϩ, ships comprising each term. The general characteristics of WϪ, and Z0 bosons and an accurate determination of their Standard Model of Particle Physics ● J. J. BEVELACQUA 621 masses verified the Standard Model’s prediction of the For a given particle (1) that decays into several other ␪ ⌫ w value. The verification of this key prediction further particles 2, 3, 4, … k, the decay rate ( ) is determined established the validity of the Standard Model (PDG from the amplitude and kinematic factors according to 2008) and its assumed symmetry characteristics. the relationship:

S NEUTRON DECAY AND MEAN ⌫ ϭ ͉ ͉2͑ ␲͒4 ប ͵ M 2 LIFETIME CALCULATION 2 m1

Neutron decay is a fundamental problem in both k and health physics. A recent theoret- ϫ ␦4 ͑ Ϫ Ϫ Ϫ ͒͹ ␲ p1 p2 p3 … pk 2 ical effort (Faber et al. 2009) provided a rigorous jϭ2 Standard Model calculation of the neutron mean lifetime. 4 d pj Details of this calculation are well beyond the scope of ϫ ␦ ͑p2 Ϫ m2c2͒␪͑p0͒ , (14) this review. However, the basic elements of a Standard j j j ͑2␲͒4 Model calculation of the neutron mean lifetime are th where mi is the mass of the i particle, is its four- outlined in subsequent discussion. ␪ 0 momentum, S is a statistical factor, and (pj )isthe The previous sections of this review provided the Heaviside step function. This step function [␪(z)] is 0 if basis for a calculation of the neutron lifetime based on z Ͻ 0 and is 1 if z Ͼ 0. The Dirac delta function (␦)isthe the Standard Model and the characteristics of the neutron derivative of the Heaviside step function. and associated decay particles. To illustrate the essential For neutron decay, the particles (label number, elements of the underlying physics, four neutron decay ␷៮ associated four-momentum) are: n (1, p1), e (2, p2), p (3, lifetime cases (I, II, III, and IV) are presented. p ), eϪ (4, p ), and WϪ (5, q). The mean lifetime (␶)isthe The Case I calculation uses eqns (3) and (4) to 3 4 Ϫ reciprocal of the decay rate. In subsequent mean lifetime include the W boson and its subsequent decay into an calculations, detailed derivations are not presented, but electron (eϪ) and antielectron neutrino (␷៮ ). In Case I, the e salient results of the Feynman calculus are quoted. neutron (n) and proton (p) are treated as point particles. With a assumption, a longer neutron mean lifetime is expected because no internal correlations are Neutron mean lifetime—Case I. Using Feynman included in the description of the neutron and proton. calculus, the Case I total decay rate expression is: 4 These correlations would enhance the neutron decay rate. 1 gw ⌫ ϭ ͩ ͪ ͑m c2͒5 A more physical calculation (Case II) is achieved by I 4␲3ប 2M c2 e including weak charge coupling as part of the WϪ interac- W tions of eqns (3) and (4). Case III improves the Case II 1 ϫ ͫ ͑2x4 Ϫ 9x2 Ϫ 8͒ͱx2 Ϫ 1 calculation with the inclusion of specific quark structures 15 into eqn (3). Following Table 3, the neutron has a ddu quark structure and the proton has a uud structure. In eqn (3), the ϩ x ͑x ϩ ͱx2 Ϫ ͒ͬ ud quark pair acts as spectators to the quark transition: ln 1 , (15) d 3 u ϩ WϪ. (13) where

Charge conservation is met because the d quark has a mn Ϫ mp charge of Ϫ1/3 e and u has a charge of ϩ2/3 e (see x ϭ (16) me Tables 2 and 3). Case IV includes a radiative correction based on a In eqn (16), mn, mp, and me are the neutron, proton, and detailed, theoretical analysis (Faber et al. 2009). electron masses, respectively. This result is stated with- out providing the intermediate calculation steps that do Calculational approach. The neutron mean life- not affect the health physics purpose of this paper. Using time is obtained using Feynman calculus (Halzen and the values in the PDG (2008) leads to the Case I neutron ␶ Martin 1984; Peskin and Schroeder 1995; Bettini 2008; mean lifetime ( ): Griffiths 2008) to calculate the amplitude (M). The 1 ␶ ϭ ϭ amplitude is an important quantity from a health physics I ⌫ 1315.7 s. (17) perspective. Once it is determined, relevant health phys- I ics quantities (e.g., mean lifetime, branching ratios, and As expected from previous discussion, the calculated cross-sections) are readily determined. mean lifetime exceeds the experimental value of 885.7 s 622 Health Physics November 2010, Volume 99, Number 5 (PDG 2008). However, improved results are expected for , vector, and tensor couplings. These factors have subsequent cases that refine the simple point particle recently been included in a neutron decay calculation by model of Case I. Faber et al. (2009). This calculation includes the afore- mentioned effects, including a 3.9% correction for radi- Neutron mean lifetime—Case II. Case I assumes ative effects, and calculates a neutron mean lifetime that the proton and neutron are point particles that value essentially in agreement with data. interact with the WϪ in the same way that leptons Case IV is defined by modifying the Case III results interact. However, this is a poor assumption because the with radiative corrections. Including the aforementioned proton and neutron are composite quark structures (see 3.9% radiative correction, the Case IV mean neutron Table 3). lifetime is obtained from the Case III lifetime: To account for the net coupling strength of the ␶ composite structure and the associated gluon III 950.4 s ␶IV ϭ ϭ ϭ 914.7 s, (20) interactions, weak charge is introduced into eqn (3) 1.039 1.039 through coupling coefficients (PDG 2008). The inclusion which is about 3% larger than the measured value of of the weak coupling coefficients enhances the decay by 885.7 s. This result is sufficient to illustrate the capability a factor of 1.4587 that leads to an improved mean of the Standard Model to solve a problem relevant to lifetime of: health physics. 1 1315.7 s The interested reader is referred to Faber et al. ␶ ϭ ϭ ϭ 902.0 s. (18) (2009) for additional details of their Standard Model II ⌫ 1.4587 II calculation. Details of the types of calculations involved in Cases I, II, and III are found in textbooks describing Neutron mean lifetime—Case III. Case III in- Feynman calculus (Halzen and Martin 1984; Peskin and cludes the enhancements of Case II, and accounts for the Ϫ Schroeder 1995; Bettini 2008; Griffiths 2008). d to u quark transition through the W boson noted in eqns (3), (4), and (13). The enhancement of Case III leads to an amplitude that is similar to Case II with the CONCLUSION ␪ addition of a factor dependent on the Cabibbo angle ( C) (Cabibbo 1963). Electroweak interactions account for the The Standard Model of Particle Physics provides a decay properties of quarks by including functions of the theoretical methodology for describing processes rele- Cabibbo angle that has a value of 13.04° (PDG 2008). vant to health physics applications including the interac- The net effect of including the d 3 u quark tion characteristics of fundamental particles. This review transition is to retard the neutron decay by a factor of provided specific applications explaining the decay cosϪ2(␪ ) and leads to a Case III mean lifetime of: modes of the neutron and muon in terms of conservation C laws and illustrated the relationship of symmetry to the 1 902.0 s ␶ ϭ ϭ ϭ formulation of the Standard Model. III 2 950.4 s. (19) ⌫III Cos ͑␪C͒ A simplified calculation using the Standard Model predicted a neutron mean lifetime about 3% larger than The careful reader will note that the more physically the measured value. Detailed Standard Model calcula- correct result of Case III yields a poorer result than Case tions yield results in agreement with experiments. These II. When such occurrences are encountered, they are an results suggest that the Standard Model yields credible indication that the calculation is not yet complete and is predictions relevant to health physics applications. omitting some essential physics.

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