GluonsAndQCDQuarks 2009-01-30 05:31:10 2009-01-30 06:12:28 Topic fermions bosons nucleons hadrons nucleons baryons valence quarks sea quarks mesons Yukawa model flavors color charges charm quarks gluons and QCD color confinement color `charge' Gell--Mann--Zweig model Sheldon Lee Glashow and James Bjorken % this is the default PlanetPhysics preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amsmath, amssymb, amsfonts, amsthm, amscd, latexsym, enumerate} \usepackage{xypic, xspace} \usepackage[mathscr]{eucal} \usepackage[dvips]{graphicx} \usepackage[curve]{xy} % define commands here \theoremstyle{plain} \newtheorem{lemma}{Lemma}[section] \newtheorem{proposition}{Proposition}[section] \newtheorem{theorem}{Theorem}[section] \newtheorem{corollary}{Corollary}[section] \theoremstyle{definition} \newtheorem{definition}{Definition}[section] \newtheorem{example}{Example}[section] %\theoremstyle{remark} \newtheorem{remark}{Remark}[section] \newtheorem*{notation}{Notation} \newtheorem*{claim}{Claim} \renewcommand{\thefootnote}{\ensuremath{\fnsymbol{footnote}}} \numberwithin{equation}{section} \newcommand{\Ad}{{\rm Ad}} \newcommand{\Aut}{{\rm Aut}} \newcommand{\Cl}{{\rm Cl}} \newcommand{\Co}{{\rm Co}} \newcommand{\DES}{{\rm DES}} \newcommand{\Diff}{{\rm Diff}} \newcommand{\Dom}{{\rm Dom}} \newcommand{\Hol}{{\rm Hol}} \newcommand{\Mon}{{\rm Mon}} \newcommand{\Hom}{{\rm Hom}} \newcommand{\Ker}{{\rm Ker}} \newcommand{\Ind}{{\rm Ind}} \newcommand{\IM}{{\rm Im}} \newcommand{\Is}{{\rm Is}} \newcommand{\ID}{{\rm id}} \newcommand{\grpL}{{\rm GL}} \newcommand{\Iso}{{\rm Iso}} \newcommand{\rO}{{\rm O}} \newcommand{\Sem}{{\rm Sem}} \newcommand{\SL}{{\rm Sl}} \newcommand{\St}{{\rm St}} \newcommand{\Sym}{{\rm Sym}} \newcommand{\Symb}{{\rm Symb}} \newcommand{\SU}{{\rm SU}} \newcommand{\Tor}{{\rm Tor}} \newcommand{\U}{{\rm U}} \newcommand{\A}{\mathcal A} \newcommand{\Ce}{\mathcal C} \newcommand{\D}{\mathcal D} \newcommand{\E}{\mathcal E} \newcommand{\F}{\mathcal F} %\newcommand{\grp}{\mathcal G} \renewcommand{\H}{\mathcal H} \renewcommand{\cL}{\mathcal L} \newcommand{\Q}{\mathcal Q} \newcommand{\R}{\mathcal R} \newcommand{\cS}{\mathcal S} \newcommand{\cU}{\mathcal U} \newcommand{\W}{\mathcal W} \newcommand{\bA}{\mathbb{A}} \newcommand{\bB}{\mathbb{B}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bD}{\mathbb{D}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bG}{\mathbb{G}} \newcommand{\bK}{\mathbb{K}} \newcommand{\bM}{\mathbb{M}} \newcommand{\bN}{\mathbb{N}} \newcommand{\bO}{\mathbb{O}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bV}{\mathbb{V}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bfE}{\mathbf{E}} \newcommand{\bfX}{\mathbf{X}} \newcommand{\bfY}{\mathbf{Y}} \newcommand{\bfZ}{\mathbf{Z}} \renewcommand{\O}{\Omega} \renewcommand{\o}{\omega} \newcommand{\vp}{\varphi} \newcommand{\vep}{\varepsilon} \newcommand{\diag}{{\rm diag}} \newcommand{\grp}{{\mathsf{G}}} \newcommand{\dgrp}{{\mathsf{D}}} \newcommand{\desp}{{\mathsf{D}^{\rm{es}}}} \newcommand{\grpeod}{{\rm Geod}} %\newcommand{\grpeod}{{\rm geod}} \newcommand{\hgr}{{\mathsf{H}}} \newcommand{\mgr}{{\mathsf{M}}} \newcommand{\ob}{{\rm Ob}} \newcommand{\obg}{{\rm Ob(\mathsf{G)}}} \newcommand{\obgp}{{\rm Ob(\mathsf{G}')}} \newcommand{\obh}{{\rm Ob(\mathsf{H})}} \newcommand{\Osmooth}{{\Omega^{\infty}(X,*)}} \newcommand{\grphomotop}{{\rho_2^{\square}}} \newcommand{\grpcalp}{{\mathsf{G}(\mathcal P)}} \newcommand{\rf}{{R_{\mathcal F}}} \newcommand{\grplob}{{\rm glob}} \newcommand{\loc}{{\rm loc}} \newcommand{\TOP}{{\rm TOP}} \newcommand{\wti}{\widetilde} \newcommand{\what}{\widehat} \renewcommand{\a}{\alpha} \newcommand{\be}{\beta} \newcommand{\grpa}{\grpamma} %\newcommand{\grpa}{\grpamma} \newcommand{\de}{\delta} \newcommand{\del}{\partial} \newcommand{\ka}{\kappa} \newcommand{\si}{\sigma} \newcommand{\ta}{\tau} \newcommand{\lra}{{\longrightarrow}} \newcommand{\ra}{{\rightarrow}} \newcommand{\rat}{{\rightarrowtail}} \newcommand{\ovset}[1]{\overset {#1}{\ra}} \newcommand{\ovsetl}[1]{\overset {#1}{\lra}} \newcommand{\hr}{{\hookrightarrow}} \newcommand{\<}{{\langle}} \def\baselinestretch{1.1} \hyphenation{prod-ucts} %\grpeometry{textwidth= 16 cm, textheight=21 cm} \newcommand{\sqdiagram}[9]{$$ \diagram #1 \rto^{#2} \dto_{#4}& #3 \dto^{#5} \\ #6 \rto_{#7} & #8 \enddiagram \eqno{\mbox{#9}}$$ } \def\C{C^{\ast}} \newcommand{\labto}[1]{\stackrel{#1}{\longrightarrow}} %\newenvironment{proof}{\noindent {\bf Proof} }{ \hfill $\Box$ %{\mbox{}} \newcommand{\quadr}[4] {\begin{pmatrix} & #1& \\[-1.1ex] #2 & & #3\\[-1.1ex]& #4& \end{pmatrix}} \def\D{\mathsf{D}} \section{Quarks, Gluons and QCD} \subsection{Brief History} The quark model was independently proposed by theoretical/mathematical physicists Murray Gell--Mann and George Zweig in 1964. However, there was little experimental evidence for the physical reality of quarks until 1968, when electron–proton scattering experiments indicated that the electrons were scattering off three point-like constituents `inside' the proton. Gell--Mann `borrowed' the word quark from James Joyce's book ``Finnegans Wake'': ``{\em Three quarks for Muster Mark! Sure he has not got much of a bark And sure any he has it's all beside the mark.''} By 1995, when the top quark was detected in high--energy experiments at the Fermilab in Illinois, all six quark flavors have been finally observed. Gell-Mann and Zweig proposed in 1964--without any substantial experimental evidens-- that hadrons were not elementary particles, but they were instead composed of specific (triple) combinations of quarks and antiquarks. They also postulated independently that there are only three flavors of quarks--up, down and strange—to which they also ascribed properties such as spin and electric charge. Within a year, however, extensions of the Gell--Mann--Zweig model were proposed when two other physicists, Sheldon Lee Glashow and James Bjorken, predicted the existence of a fourth flavor of quark, which they referred to as \emph{charm}. This addition was needed because it expanded the power and self-consistency of the theory: it allowed a much improved and consistent description of the weak interaction when it was relaized that it provided the mechanism that causes the quarks to decay; interestingly, this new theoretical prediction also equalized the number of quarks with the number of known leptons, and led to a formula for predicting correctly the mass of known ($\pi$) mesons (that are hadrons with integer spin, or {\em bosons}, previously predicted theoretically by Yukawa in 1934 as the carriers of the nuclear interaction {\em via} their exchange). In 1968, deep inelastic electron scattering experiments at the Stanford Linear Accelerator Center (SLAC) showed that the proton was not an elementary particle, but instead contained much smaller, point-like objects, that were not so hastily identified with quarks. While this showed that hadrons indeed had a substructure, as predicted by the quark model, physicists remained reluctant to identify these smaller objects with quarks. Instead, they became known as `partons' (a term proposed by Richard Feynman, and supported by some experimental project reports). Such partons were later identified as the $up$ and $down$ quarks when other flavors were also detected. Their discovery is claimed to have `validated' the existence of the strange quark, because it was necessary in the predictions made by the Gell-Mann/Zweig model. In a 1970 paper, Glashow, John Iliopoulos and Luciano Maiani gave much more compelling theoretical arguments for the prediction of the as-yet undiscovered quark that had charm. The number of the predicted quark flavors thus grew from two to the current six in 1973, following the more complete predictions by Makoto Kobayashi and Toshihide Maskawa who noted that the experimental observation of CP violation could only be explained if there were another pair of quarks with different flavor from the ones already observed.[26] These two new quarks became known as `beauty' and `truth', but later re-named as `bottom' , $b$, and `top', $t$, respectively. Following a decade without experimental evidence supporting the actual existence of charm quarks, they were finally produced and observed almost simultaneously by two teams in November 1974 : one team working at the Stanford Linear Accelerator Center (SLAC) supervised by Burton Richter and the other at the Brookhaven National Laboratory supervised by Samuel Ting. The two teams had assigned the discovered particle two different names, the $J$ and the $\psi$. The particle hence became formally known as the $J/ \psi$ `meson' and it was considered a quark–antiquark pair with the charm flavor that Glashow and Bjorken had predicted, called the `charmonium' particle by the latter theoreticians. In 1977, the bottom quark was observed by Leon Lederman's team at Fermilab in Illinois. This indicated that a `top' quark should also exist, because the bottom quark would have been most strangely without a partner if it had not existed. However, it was not until 1995, that the top quark was finally detected after much effort and lengthy high-energy experimentation. The top quark's discovery was crucial; furthermore, it showed that the top quark was significantly more massive than predicted, `almost as heavy as a gold atom', and thus its presence was found at higher energies than those expected. The actual theoretical reasons for the top quark's larger mass remain to be determined. \subsection{Quarks, antiquarks, nucleons and hadrons} The building blocks of the atomic nucleus, called also `{\em nucleons}'--the proton and the neutron--are baryons. Stable quarks are then considered at present as the {\em `elementary particles'} found in nucleons, that is, `inside' protons and neutrons of atomic nuclei, as well as in mesons where they appear as quark-pairs. Unstable, high-energy quarks are present in the `physical' vacuum in virtual states, and also in other subatomic particles generated in particle accelerators. They are major constituents of matter, along with leptons (such as electrons and neutrinos). In theoretical physicsl terms, quarks are elementary fermions (of spin $1/2$) because they are subject to Fermi statistics and the Pauli exclusion principle. A critical limitation to the experimental and theoretical studies of quarks is the fact that quarks are never found as isolated, single particles; rather, they are bound either as quark-pairs or bound together in composite particles named hadrons, (with the most common hadrons being protons and neutrons, which are the basic building blocks of all atomic nuclei). For this reason, much of what is known about quarks has been inferred from observations on the hadrons themselves and observations of quark jets or pairs that are generated in particle accelerators at very high energies. Quarks (and antiquarks) are the only known particles whose electric charge comes as exactly one third of the elementary charge of the electron or proton. However this can never be directly observed as {\em hadrons} because they latter have always an integer charge. There are two known types of hadrons: {\em baryons}, formed of three quarks, and {\em mesons}, formed of a quark and an antiquark pair. The quarks (and antiquarks) which determine the quantum numbers of hadrons are called {\em valence quarks}. Apart from these, any hadron may contain an indefinite number of virtual quarks, antiquarks and gluons which do not influence their quantum numbers. Such virtual quarks are called {\em sea quarks} ({\em vide infra}). Remarkably, quarks are the only particles in the current Standard Model of physics (SUSY) to experience all four fundamental forces: strong, electromagnetic, electroweak and gravitational. There are currently six known different types of quarks, that are defined by their flavor: $up$ (symbols: $u$), $down$ ($d$), charm ($c$), strange ($s$), top ($t$) and bottom ($b$). Furthermore, the QCD theory holds the view that these are the only possible types of quarks found in nature or in the high-energy laboratory. The quarks with the lowest masses, the $up$ and the $down$ quark, are stable within nucleons of atomic nuclei where they are coupled with each other and interact strongly also {\em via} \emph{gluons}- the nuclear field carrier particles. The heavier charm, strange, top and bottom quarks are unstable and decay extremely rapidly; these can only be produced in high energy collisions, such as in particle accelerators and in cosmic rays. Quarks have defining property in addition to mass, electric charge, and spin which is unique to nuclear interactions--the {\em color `charge'}--which behaves somewhat like a very strong `magnetic' interaction, but with three `poles' instead of the `North and South' characteristic magnetic poles of the classical magnets derived from electron magnetic moment/spin interactions. For every quark flavor there is a corresponding antiparticle, called its antiquark flavor, which differs from the quark only in that its electrical charge has the opposite sign. Such antiparticles of quarks--called antiquarks-- are denoted by a bar over the designating letter for the quark, such as $u$ for a quark and $\overline{u}$ for an $up$ antiquark. As with all antimatter, general, antiquarks have the same mass, lifetime and spin as their respective quarks, but the electric charge and other charges have the opposite sign. Having electric charge, mass, spin, flavor and color charge, the quarks are the only known elementary particles that engage in all four fundamental interactions of contemporary physics: electromagnetism, weak interaction, strong interaction and gravitation. Gravitation, however, is not included in the theoretical Standard Model, because quantum gravity developments are yet to be completed, and also because gravitational interactions are extremely weak in comparison with all of the other three fundamental interactions. \subsection{Quark's electric charge} A quark has a precise fractional electric charge value from that of the electron or proton (considered as unity), that is, either −1⁄3 or +2⁄3 times the elementary charge of the latter. More specifically, the up, charm and top quarks --that are collectively referred to as $up$ or $u$-quarks-- have a charge of $+2⁄3 $each, whereas the down, strange and bottom quarks (down-type , or $d$-quarks) have a charge value of $−1⁄3$. The antiquarks, as explained above, have the opposite charge of their corresponding quark (the up-type antiquarks have charges of $−2⁄3$, and the down-type antiquarks have a charge value of $+1⁄3$). Since the electric charge of a hadron is the sum of the charges of the constituent quarks, the combinations of three quarks, or three anti-quarks, or a quark with an anti-quark, always result in an integer charge. The electric charge of quarks is important in the formation of atomic nuclei. The two stable hadron constituents of the atom, the neutron and the proton, have respectively charge values of $0$ and $+1$; thus, the quark model expects that the neutron is `composed of' two down quarks and one up quark, whereas the proton is `made of' two up quarks and a down quark. The total electric charge of a nucleus is given by the number of protons present inside the atomic nucleus, is known as the atomic number that spans the periodic table of elements. subsection{Quark Masses} There are presently two different terms in use when one describes the quark masses: the current quark mass refers to the mass of a quark `by itself', while `constituent quark mass' refers to the current quark mass plus the mass of the gluon particle field(s) surrounding the quark. The two values are typically quite different for several reasons the will be explained next. In a hadron, like a proton, most of the mass comes from the gluons that bind the constituent quarks together, rather than from the individual quarks. The mass of the quarks `in themselves', or `by themselves' is quite low (about one third) compared to the mass derived from the gluons' energy. While gluons are inherently massless, they possess energy, and it is this energy that contributes so greatly to the overall mass of the hadron through relativistic effects. This is readily demonstrated for the proton- the most common hadron. Composed of one $d$ and two $u$ quarks, the proton has an overall mass of approximately $938 MeV/c^2$, of which the mass of three valence quarks contributes around $11 MeV/c^2$, with the remainder coming from the quantum chromodynamics binding energy (QCBE) provided by sea quarks and gluons. This makes `direct' calculations of quark masses based on quantum chromodynamics quite difficult, and very often quite unreliable, as quantum perturbation methods that were very successful in quantum electrodynamics, fail most of the time in QCD. Mass value estimates can be however derived after obtaining from experimental data the difference in mass between two related hadrons that have opposing or complementary quark components. For example, by comparing the proton with the neutron, where the difference between the two particles is one down quark to one up quark, the relative masses and the mass differences can be measured by the difference in the overall mass of the two hadrons. The masses of most quarks were within such `predicted ranges' at the time of their discovery, with the notable exception of the top quark, which was found to have a mass approximately equal to that of a gold nucleus, significantly heavier than what was expected from the QCD theoretical estimates. Several hypotheses have been suggested to explain this very large mass miscalculation. The extended Standard Model postulates that elementary particles derive their masses through the Higgs mechanism, thus related to a so far unobserved `Higgs boson', hypothetical particle. \subsection{Flavor quantum numbers} In order to explain the phenomenology of strong and weak interactions, particle physicists assigned quantum numbers to the known baryons and mesons. The first such quantum number is known as the {\em isospin}, related to the symmetry properties determined by the Lie group $SU(2)$. This was introduced by Werner Heisenberg in 1932 to represent the remarkable similarity between the properties of the protons and neutrons other than their electric charge value, and the presence of three types of pions. The z-component, commonly denoted $I_z$, is related to the electric charge $Q$ and the baryon number ($+1$ for baryons, $0$ for mesons) of these particles. An additional quantum number, strangeness ($S$, which is not to be confused with the spin), was introduced in 1954 to explain the unexpectedly long lifetimes of particles such as $K$ mesons and $\xi$ baryons. This new, (strangeness) quantum number was unchanged by strong interactions, but not by the weak ones, which would explains the anomalously long life-times of the particles in question that can be pair-produced by the strong force, but can only decay {\em via} the electro-weak interactions. The formula chosen for the new hypercharge $Y$ was then : $$Y = B + S,$$ where $B$ is the baryon number and $S$ is the strangeness value. This equation is known as the (original) Gell--Mann--Nishijima formula. The connection with group theory become clear only in 1961 when Gell-Mann and Ne'emann showed that all the proposed quantum numbers could be explained by relating the fundamental $SU(3)$ triplet to the three lightest quarks: the up, down and strange quarks. Further advances through both theory and high energy physics experiments has led to the introduction of a three flavor quantum numbers, charmness (C), bottomness (B′) and topness (T), corresponding the charm, bottom and top quark respectively. An enlarged flavor symmetry group, $SU(6)$, and also unified $SU(5)$, or $SU(3) \times SU(2) \times U(1)$ groups are being considered to provide a unified `symmetry' for electromagnetic, electroweak and strong interactions. The modified Gell--Man--Nishijima formula generalizes the equation for all of the flavor quantum numbers and the electrical charge, with the modified hypercharge formula being $$Y = B + S + C + B′ + T,$$ that includes also the charm, the `truth and beauty' numbers, with the following notations: $J = \;spin $, $B =\; baryon$ number, $Q= $ electric charge, $I_z =\; isospin$, $\S = \;strangeness$, $C = \; charmness$, $B′ =\; bottomness$, and $T = \; topness$. \begin{thebibliography}{99} \bibitem{BR2k0} The ``Quark (subatomic particle)''. in {\em Encyclopedia Britannica}($http://www.britannica.com/EBchecked/topic/486323/quark$) \bibitem{RN2k9} R.Nave. ``Confinement of Quarks''. HyperPhysics. ($http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/quark.html$)., Georgia State University, Department of Physics and Astronomy. \bibitem{RN2k9} R.Nave. ``Quarks''. HyperPhysics. ($http://hyperphysics.phy-astr.gsu.edu/hbase/Particles/quark.html$). \bibitem{BC-PG2k9} B. Carithers, P. 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