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Leader E., Predazzi E. — An introduction to gauge theories and modern particle physics
Leader E., Predazzi E. — An introduction to gauge theories and modern particle physics

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Название: An introduction to gauge theories and modern particle physics

Авторы: Leader E., Predazzi E.


This book is a comprehensive and unified treatment of modern theoretical and experimental particle physics aimed at beginning graduate students. The emphasis throughout is on presenting underlying physical principles in a simple and intuitive way. In 1983, researchers detected the existence of W and Z bosons and many new results have followed. This is the first graduate-level textbook to deal with the "second generation" of particle physics after 1983. Features of this edition include: A detailed discussion of higher order electroweak effects; an expanded discussion of quark mixing; revised sections on charm and beauty and on jet physics; enlarged treatment of deep inelastic lepton-hadron scattering; detailed treatment of QCD corrections to the simple parton model; and an introduction to the nonperturbative or confinement region of QCD.

Язык: en

Рубрика: Физика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1996

Количество страниц: 543

Добавлена в каталог: 20.02.2014

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Предметный указатель
$A_{1}$, $A_{2}$ asymmetry parameters for $\gamma N\rightarrow \gamma N$      1.346—1.349 1.381 1.405
$a_{e}$, $a_{f}$, $a_{l}$ neutral weak axial vector coupling constants      1.68
$A_{\perp}$ transverse asymmetry in polarized DIS      1.350
$A_{\|}$ longitudinal asymmetry in polarized DIS      1.348
$B^{0}-\bar{B}^{0}$ CP-violation      2.48
$B^{0}-\bar{B}^{0}$ CP-violation, $B^{0}\rightarrow \psi K_{\mathcal{S}}$      2.51
$B^{0}-\bar{B}^{0}$ mixing      1.297 2.19
$B^{0}-\bar{B}^{0}$ mixing, $\Upsilon(4\mathcal{S})$ decay, in      2.20
$B^{0}-\bar{B}^{0}$ mixing, dynamics of      2.45
$B^{0}-\bar{B}^{0}$ mixing, incoherent production, in      2.20
$B_{K}$ in $K^{0}-\bar{K}^{0}$ CP-violation      2.39
$B_{K}$ in $K^{0}-\bar{K}^{0}$ CP-violation, Lattice QCD, in      2.305
$C_{2}(G)$ (QCD)      xxvi 2.119
$C_{2}(R)$ (QCD)      xxvi 2.130
$C_{A}$ (QCD)      xxvi 2.119
$c_{A}$, $c_{V}$ neutral weak couplings      1.71 1.75
$C_{F}$ (QCD)      xxvi 2.130
$D^{0}-\bar{D}^{0}$ mixing      2.19
$d_{abc}$ in SU(3)      1.460—1.463 2.370—2.373
$eN\rightarrow eX$      see "DIS"
$ep\rightarrow ep$      1.321—1.324
$ep\rightarrow ep$, electromagnetic form factors      1.322
$ep\rightarrow ep$, electromagnetic form factors, dipole form      1.323
$ep\rightarrow ep$, electromagnetic form factors, parton model, in      2.229
$e\nu_{e}$, $e\bar{\nu_{e}}$ elastic scattering      1.69—1.77
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow 2\gamma$      1.126
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, $\sigma_{L}$, $\sigma_{T}$      2.256
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, and the SM      1.132
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, angular distribution      1.134 2.255
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, asymmetry at $Z^{0}$      1.149
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, Breit — Wigner form      1.137 1.140
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, cross-section, total      1.129 1.136
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, helicity amplitudes      1.137
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, polarization of / in SM      1.153
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, polarized beams, with      1.151
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow f\bar{f}$, vector and axial couplings at Z°      1.150
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow hX$      2.255
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow hX$, $\sigma_{L}, \sigma_{T}$      2.256
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow hX$, angular distribution      2.255
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow hX$, polarized beams, with      2.258
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow Jets$      see "Jets in $e^{+}e^{-}$ physics"
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow q\bar{q}$      2.255
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow X$      1.134
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow X$, parton model for      1.426
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow \mu^{+}\mu^{-}$      1.126 1.129
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow \mu^{+}\mu^{-}$, angular distribution      1.129
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow \mu^{+}\mu^{-}$, forward-backward asymmetry      1.138
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow \pi^{+}\pi^{-}$      2.257
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow \tau^{+}\tau^{-}$      1.143—1.147
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow$ hadrons      see also "Monte Carlo for $e^{+}e^{-} \rightarrow$ hadrons" "R"
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow$ hadrons, colour, role of      1.174
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow$ hadrons, energy flow      2.280
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow$ hadrons, energy-energy correlations      2.280
$e^{+}e^{-}$ reactions, $e^{+}e^{-} \rightarrow$ hadrons, multiplicity      2.277
$e^{+}e^{-}$ reactions, Bhabha scattering      1.126
$e^{+}e^{-}$ reactions, energies $\ll$ $M_{Z}$, at      1.126
$e^{+}e^{-}$ reactions, LEP data      1.133
$e^{+}e^{-}$ reactions, Moller scattering      1.128
$e^{+}e^{-}$ reactions, QED, tests of      1.130
$e^{+}e^{-}$ reactions, R, $\sigma(e^{+}e^{-} \rightarrow X)/\sigma(e^{+}e^{-} \rightarrow \mu^{+}\mu^{-})_{QED}$      1.122 1.175 1.205 1.222 1.239 1.273 1.300 1.427 2.128 2.131—2.133 2.262
$e^{+}e^{-}$ reactions, SM, in      1.132
$e^{+}e^{-}$, colliders      1.118
$e^{+}e^{-}$, colliders, ADA      1.119
$e^{+}e^{-}$, colliders, CLIC      1.154
$e^{+}e^{-}$, colliders, LEP      1.122
$e^{+}e^{-}$, colliders, SLC      1.125
$e^{+}e^{-}$, colliders, TRISTAN      1.121
$e^{+}e^{-}$, luminosity      1.120
$e^{+}e^{-}$, polarization      1.120 1.123
$e^{+}e^{-}$, storage rings      1.118
$E_{S_{p}}$ (sphaleron energy)      2.336
$e_{\mu}\rightarrow eX$      1.329
$e_{\mu}\rightarrow e_{\mu}$      1.317
$e_{\mu}\rightarrow e_{\mu}$, Mott cross-section      1.320
$f_{abc}$ structure constants      xxvi 1.460—1.463 2.105 2.368—2.371
$F_{L}$ , longitudinal scaling function      2.198
$f_{\pi}$, $f_{K}$, $f_{D}$, pseudoscalar meson decay constants      1.286
$f_{\pi}$, $f_{K}$, $f_{D}$, pseudoscalar meson decay constants, CP-violation, role in      2.38
$f_{\pi}$, $f_{K}$, $f_{D}$, pseudoscalar meson decay constants, Lattice QCD, in      2.312
$g^{e}_{A}$, $g^{f}_{A}$, $g^{l}_{A}$ neutral weak axial, vector coupling constants      1.68 1.141 1.150
$g^{e}_{V}$, $g^{f}_{V}$, $g^{l}_{V}$ neutral weak vector, coupling constants      1.68 1.141 1.150
$G_{A}$, axial coupling constant      1.8—1.10
$G_{V}$, vector coupling constant      1.8—1.10
$J/\psi$      see "Narrow vector resonance"
$K^{0}$ reactions, $K^{0} \rightarrow 2\pi, 3\pi$      2.18 2.23
$K^{0}$ reactions, $K^{0} \rightarrow \pi l\nu$      1.288 2.17
$K^{0}-\bar{K}^{0}$ CP-violation      see "CP-violation"
$K^{0}-\bar{K}^{0}$ mixing, introductory      1.165
$K^{0}-\bar{K}^{0}$ mixing, introductory, detailed phenomenology      2.23
$K^{0}-\bar{K}^{0}$ mixing, introductory, general      2.19
$K^{0}-\bar{K}^{0}$ mixing, introductory, mass difference      2.14—2.18 2.30
$K^{\mu}$ current in QCD      2.326
$m_{H}$ (Higgs mass)      1.54
$m_{H}$ (Higgs mass), limits on      1.97
$m_{H}$ (Higgs mass), relation to Higgs width      1.91 1.92
$m_{t}$ (top mass), $m_{H}$, connection with      1.98
$m_{t}$ (top mass), CP-violation, role in      2.37 2.45
$m_{t}$ (top mass), muon lifetime, role in      1.109
$m_{t}$ (top mass), note added in proof      xxix
$m_{t}$ (top mass), value, from LEP data      1.146
$n^{*}_{f}$, number of active flavours      2.123
$n_{f}$, number of flavours      2.119
$pp \rightarrow pp$ at large $p_{T}$      2.230—2.233
$q\bar{q}$ potential      see "Quarkonium"
$q\bar{q}$ potential, Lattice QCD, in      2.311
$R_{\nu}$, $R_{\bar{\nu}}$      1.184 1.423
$Sp\bar{p}S$ collider      1.77 1.117
$SU(3)_{C}$      2.105
$SU(3)_{F}$ invariance      1.10
$SU(3)_{F}$ invariance, particle assignments      1.209
$SU(3)_{F}$ invariance, quark content of hadrons      1.209
$S^{2}$, $S^{3}$ (topological manifolds)      2.320
$t^{a}$-matrices, properties of      1.460 2.368
$v_{e}$, $v_{f}$, $v_{l}$ neutral weak vector, coupling constants      1.68
$x_{d}$, $x_{s}$ in $B^{0}-\bar{B}^{0}$ mixing      2.46—2.48
$Z^{0}$-boson, couplings      see "Coupling constants" "Lagrangian
$Z^{0}$-boson, discovery of      1.63
$Z^{0}$-boson, mass, $\bar{p}p$ collision data, from      1.82
$Z^{0}$-boson, mass, LEP data, for      1.134
$Z^{0}$-boson, mass, W mass, relation to      1.54 1.58
$Z^{0}$-boson, polarization vector, for      1.330
$Z^{0}$-boson, production, $\bar{p}p$ collision data on      1.82
$Z^{0}$-boson, production, B and L violating reactions, in      2.337
$Z^{0}$-boson, production, parton model for      1.440
$Z^{0}$-boson, production, QCD, in      2.216
$Z^{0}$-boson, production, theory, comparison with      2.217
$Z^{0}$-boson, production, transverse momentum distributions      2.217
$Z^{0}$-boson, production, widths      see also "Gauge symmetry"
$Z^{0}$-boson, production, widths, $Z^{0} \rightarrow f\bar{f}$ partial width in SM      1.144
$Z^{0}$-boson, production, widths, LEP data, on      1.134
$Z^{0}$-boson, production, widths, pp collision data, from      1.82
$Z^{0}$-boson, production, widths, QCD effects on      2.269
$Z^{0}$-boson, production, widths, SM, in      1.144
$Z_{g}$ coupling constant renormalization parameter      2.58
$Z_{g}$ coupling constant renormalization parameter, MPS scheme, in      2.67—2.71 2.76
$Z_{g}$ coupling constant renormalization parameter, MS, $\bar{MS}$ schemes, in      2.71—2.73 2.77
$Z_{m}$ mass renormalization parameter      2.58
$Z_{m}$ mass renormalization parameter, MPS scheme, in      2.67—2.71
$Z_{m}$ mass renormalization parameter, MS, $\bar{MS}$ schemes, in      2.71—2.73
$Z_{\phi}$ field renormalization parameter      2.58
$Z_{\phi}$ field renormalization parameter, MPS scheme, in      2.67—2.71
$Z_{\phi}$ field renormalization parameter, MS, $\bar{MS}$ schemes, in      2.71—2.73
$\alpha$, fine structure constant in QED      1.57
$\alpha$, fine structure constant in QED, effective or running      1.57
$\alpha_{1}$, $\alpha_{2}$, $\alpha_{3}$ couplings in GUT      2.343
$\alpha_{s}$, fine structure constant for QCD      1.207 1.260
$\alpha_{s}$, fine structure constant for QCD, effective or running      2.79
$\alpha_{s}$, fine structure constant for QCD, effective or running, formula to lowest order      2.84
$\alpha_{s}$, fine structure constant for QCD, effective or running, formula to next to leading order      2.124—2.126 2.269
$\alpha_{s}$, fine structure constant for QCD, experimental determination, $J/\Psi$ decay      1.260
$\alpha_{s}$, fine structure constant for QCD, experimental determination, $\Upsilon$ decay      1.260
$\alpha_{s}$, fine structure constant for QCD, experimental determination, jet production in $e^{+}e^{-}$ collisions      2.258 2.265 2.282
$\alpha_{s}$, fine structure constant for QCD, experimental determination, jet production in hadronic reactions      2.239
$\alpha_{s}$, fine structure constant for QCD, experimental determination, vector quarkonium states      1.260
$\beta$ parameter (Lattice QCD)      2.305
$\beta$-Decay      1.6—1.7 2.12
$\beta$-decay, hyperons, of      2.2
$\beta$-decay, kaons, of      2.2
$\beta$-decay, neutrinoless      2.345
$\beta$-decay, pions, of      1.14
$\beta$-decay, quark picture of      1.12
$\beta$-function      see also "Renormalization group"
$\beta$-function, b, b' in QCD      2.120
$\chi$ in $B^{0}-\bar{B}^{0}$ mixing      2.22 2.47
$\chi_{b}$ states      1.236—1.238 1.253
$\chi_{c}$ states      1.220—1.221 1.253
$\Delta I = \frac{1}{2}$ rule in non-leptonic K-decay      2.26
$\Delta r$ parameter in radiative corrections to SM      1.109
$\Delta S = \Delta Q$ rule for semileptonic decays      1.14
$\delta$, phase angle in KM matrix      1.163
$\delta$, phase angle in KM matrix, $B^{0}-\bar{B}^{0}$ mixing, in      2.46
$\delta$, phase angle in KM matrix, $K^{0}-\bar{K}^{0}$ CP-violation, from      2.27 2.39
$\Delta(x)$ singular function      1.464 2.352
$\delta_{0}$, $\delta_{2}$, $\pi\pi$ phase-shifts      2.23
$\epsilon$, $\epsilon^{'}$ in $K^{0}-\bar{K}^{0}$ CP-violation, CP-violating Hamiltonian, relation to      2.28
$\epsilon$, $\epsilon^{'}$ in $K^{0}-\bar{K}^{0}$ CP-violation, definitions      2.26
$\epsilon$, $\epsilon^{'}$ in $K^{0}-\bar{K}^{0}$ CP-violation, estimate in SM      2.35—2.40
$\epsilon$, $\epsilon^{'}$ in $K^{0}-\bar{K}^{0}$ CP-violation, numerical values      2.27—2.28
$\epsilon$, $\epsilon^{'}$ in $K^{0}-\bar{K}^{0}$ CP-violation, Wu — Yang convention      2.28—2.29
$\epsilon(x)$ step function      1.464 2.352
$\eta^{'}_{c}$ (3592)      1.219
$\eta_{+-}$, $\eta_{00}$ in $K^{0}-\bar{K}^{0}$ CP-violation      2.23—2.28
$\eta_{c}$ (2980)      1.219 1.253
$\Gamma$-function      see "Renormalization group"
$\gamma$-matrices      xxiv
$\gamma$-matrices, d-dimensions, in      2.65
$\gamma$-matrices, traces of      1.452 2.360
$\gamma$-reactions, $\gamma G \rightarrow q\bar{q}$      see "Parton-parton reactions"
$\gamma$-reactions, $^{'}\gamma^{'}N \rightarrow ^{'}\gamma^{'} N$, $A_{1}$, $A_{2}$ asymmetry parameters      1.346—1.349 1.381 1.405
$\gamma$-reactions, $^{'}\gamma^{'}N \rightarrow ^{'}\gamma^{'} N$, DIS, relation to      1.330
$\gamma$-reactions, $^{'}\gamma^{'}N \rightarrow ^{'}\gamma^{'} N$, longitudinal cross-section      1.330
$\gamma$-reactions, $^{'}\gamma^{'}N \rightarrow ^{'}\gamma^{'} N$, R, $\sigma_{L}/\sigma_{T}$      1.331
$\gamma$-reactions, $^{'}\gamma^{'}N \rightarrow ^{'}\gamma^{'} N$, transverse cross-section      1.330
$\mu N \rightarrow \mu X$      see "DIS"
$\mu$ lifetime, electroweak radiative corrections to      1.107
$\mu$ lifetime, lowest order, in      1.57
$\mu$ lifetime, QED radiative corrections to      1.57
$\mu$ lifetime, tests of corrections, experimental      1.112
$\mu$-decay      1.57
$\nu_{\tau}$ (tau neutrino)      1.302 1.312
$\nu_{\tau}$ (tau neutrino), helicity of      1.315
$\pi\pi$ phase shifts in $k^{0}-\bar{K}^{0}$, CP-violation      2.23—2.24
$\Psi^{'}$ (3685)      see "Narrow vector resonances"
$\rho$, ratio of real to imaginary parts of amplitude      2.289 2.293
$\rho_{0} = M^{2}_{W}/(M^{2}_{Z} \cos^{2}\theta_{W})$      1.76 1.423
$\Sigma_{C}$ charm baryon      1.277 1.279
$\Sigma_{C}$ charm baryon, mass relations      1.282
$\sigma_{L}$, $\sigma_{T}$ in $e^{+}e^{-} \rightarrow hX$      2.256
$\sigma_{L}$, $\sigma_{T}$ in $e^{+}e^{-} \rightarrow hX$ in $e^{+}e^{-} \rightarrow f\bar{f}$      2.257
$\sigma_{tot}$ in hadronic reactions      2.289
$\tau$-lepton      1.300
$\tau$-lepton, $\tau-e-\mu$, universality      1.309
$\tau$-lepton, decays, leptonic      1.310
$\tau$-lepton, decays, Michel parameter      1.312
$\tau$-lepton, decays, parity violation      1.314
$\tau$-lepton, decays, QCD, tests of      1.311
$\tau$-lepton, decays, semi-leptonic      1.310
$\tau$-lepton, decays, V-A structure      1.313
$\tau$-lepton, discovery of      1.301
$\tau$-lepton, lifetime      1.306
$\tau$-lepton, mass      1.304
$\tau$-lepton, point-like structure      1.308
$\tau$-lepton, polarization in $e^{+}e^{-} \rightarrow \tau^{+}\tau^{-}$      1.154
$\tau$-lepton, properties, table of      1.303
$\tau$-lepton, rare decays      1.314
$\tau$-lepton, spin      1.307
$\tau$-neutrino      1.302 1.312
$\tau$-neutrino, helicity of      1.312
$\tau$-neutrino, mass of      1.314
$\theta$ problem      2.329—2.332 2.340 2.349
$\theta$, TCP operator      2.384
$\theta$-vacuum, periodic potential in QM      2.314
$\theta$-vacuum, QCD, in      2.318 2.326
$\theta$-vacuum, QCD, in, axions      2.332
$\theta$-vacuum, QCD, in, chiral non-invariance      2.331
$\theta$-vacuum, QCD, in, mass generation      2.331
$\theta$-vacuum, QCD, in, Peccei — Quinn symmetry      2.332
$\theta$-vacuum, QCD, in, problems with      2.331
$\theta$-vacuum, QCD, in, strong CP-violation      2.329
$\theta$-vacuum, QCD, in, U(1) problem resolution      2.329
$\theta$-vacuum, QCD, in, value of $\theta$      2.331
$\theta(x)$ step function      1.444 2.352
$\theta(x)$ step function, integral representation for      2.382
$\theta_{C}$      see "Cabibbo angle"
$\theta_{W}$      see "Weinberg angle"
$\Upsilon$ (upsilon)      see "Narrow vector resonances"
$\Xi_{c}$ charm baryon      1.275
$^{'}\gamma^{'}$, virtual photon, $\gamma-Z^{0}$ interference in DIS      1.341 1.374
$^{'}\gamma^{'}$, virtual photon, flux of      1.330
$^{'}\gamma^{'}$, virtual photon, polarization vector for      1.330
5-matrix      1.449 2.357 2.384
5-matrix, Dyson expansion for      1.150
5-matrix, Feynman amplitude, relation to      1.451 2.359
5-matrix, T-matrix, relation to      2.384
Abelian gauge symmetries      see "Gauge symmetry"
Absorptive part of transition amplitude      2.33—2.34 2.383
Acoplanarity      2.254 2.266
action      1.29
Action, Euclidean      2.327 2.396
Action, Lattice QCD, for      2.305
Active flavours      2.123
ADA storage ring      1.119
Ademollo — Gatto theorem      2.3
Adler sum rule      1.401 2.146 2.154
AEEC      2.281
Altarelli — Parisi equations      2.148 2.185
Altarelli — Parisi equations, summary of      2.186
Angular momentum sum rule      1.408
Angular ordering      2.275 2.285
anomalies      see also "Triangle anomaly"
Anomalies, axial isosinglet current      2.330
Anomalies, baryon number axial current      2.341
Anomalies, baryon number vector current      2.334
Anomalies, evidence for      1.360
Anomalies, lepton number vector current      2.334
Anomalous dimensions      2.83 2.142
Anomalous magnetic moment of nucleon      1.321
Anti-linear (anti-unitary) operators      2.384
Anticommutators      1.445 2.353
Antipartons      1.359
Antishadowing      1.414
Applequist — Carrazone theorem      2.122
Asymmetries and polarization at the $Z^{0}$      1.149
asymptotic freedom      2.81
Asymptotic freedom, Lattice QCD, in      2.304
Asymptotic freedom, scaling, and      2.81
Away jet      2.221
axial anomaly      see "Anomalies"
Axial gauges      2.114
Axions      2.332 2.341
B mesons      see "Bottom mesons"
B, b' (QCD $\beta$-function)      2.120
B-L symmetry (baryon-lepton)      2.339
B-violation (baryon number), GUT, in      2.342
B-violation (baryon number), SM, in      2.332 2.336
B-violation (baryon number), SM, in, $qq \rightarrow 7\bar{q} + 3\bar{l} + n$ gauge bosons      2.337
B-violation (baryon number), SM, in, $qq \rightarrow 7\bar{q} + 3\bar{l} + n$ gauge bosons, perturbative corrections      2.337
B-violation (baryon number), SM, in, $qq \rightarrow 7\bar{q} + 3\bar{l} + n$ gauge bosons, unitarity bound      2.337
B-violation (baryon number), SM, in, $qq \rightarrow 7\bar{q} + 3\bar{l}$      2.336
B-violation (baryon number), SM, in, $\sigma_{tot}$ for      2.338
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