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Two-Component Spinor Techniques and Feynman Rules for Quantum field Theory and Supersymmetry

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BN-TH-2008-12 SCIPP-08/08 arXiv:0812.1594v5 [hep-ph] Modifications and corrections added Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry DRAFT version 2.16 March 3, 2016 Herbi K. Dreiner1, Howard E. Haber2 and Stephen P. Martin3 1Bethe Center for Theoretical Physics and Physikalisches Institut der Universit¨at Bonn, Nußallee 12, 53115 Bonn, Germany 2Santa Cruz Institute for Particle Physics, University of California, Santa Cruz CA 95064 3Department of Physics, Northern Illinois University, DeKalb IL 60115 and Fermi National Accelerator Laboratory, P.O. Box 500, Batavia IL 60510 Abstract Two-component spinors are the basic ingredients for describing fermions in quantum field theory in 3 + 1 spacetime dimensions. We develop and review the techniques of the two- component spinor formalism and provide a complete set of Feynman rules for fermions using two-component spinor notation. These rules are suitable for practical calculations of cross- sections, decay rates, and radiative corrections in the Standard Model and its extensions, including supersymmetry, and many explicit examples are provided. The unified treatment presented in this review applies to massless Weyl fermions and massive Dirac and Majorana fermions. We exhibit the relation between the two-component spinor formalism and the more traditional four-component spinor formalism, and indicate their connections to the spinor helicity method and techniques for the computation of helicity amplitudes. Contents 1 Introduction 5 2 Essential conventions, notations and two-component spinor identities 8 3 Properties of fermion fields 24 3.1 The two-component fermion field and spinor wave functions ........... 24 3.2 Fermion mass diagonalization in a general theory . ............ 31 4 Feynman rules with two-component spinors 38 4.1 Externalfermionandbosonrules . ....... 38 4.2 Propagators..................................... 39 4.3 Fermion interactions with bosons . ........ 44 4.4 General structure and rules for Feynman graphs . .......... 50 4.5 Basic examples of writing down diagrams and amplitudes . ............ 52 4.5.1 Scalar boson decay to fermion pairs . ..... 53 4.5.2 Fermion pair annihilation into a scalar boson . ......... 54 4.5.3 Vector boson decay into fermion pairs . ...... 56 4.5.4 Two-body scattering of a boson and a neutral fermion . ......... 57 4.5.5 Two-body scattering of a boson and a charged fermion . ......... 59 4.5.6 Two-body fermion–fermion scattering . ....... 62 4.5.7 Non-relativistic potential due to scalar or pseudoscalar exchange . 63 4.6 Self-energy functions and pole masses for two-componentfermions . 65 5 Conventions for fermion and antifermion names and fields 73 6 Practical examples from the Standard Model and its supersymmetric extension 77 6.1 Top quark decay: t bW + .............................. 77 → 6.2 Z0 vector boson decay: Z0 ff¯ ........................... 79 → 6.3 Bhabha scattering: e e+ e e+ ........................... 81 − → − 6.4 Polarizedmuondecay .............................. 83 6.5 Neutral MSSM Higgs boson decays φ0 ff¯, for φ0 = h0,H0, A0 ......... 84 → 6.6 Sneutrino decay ν C+e .............................. 86 e → i − + + 6.7 Chargino decay Ci νee .............................. 87 e → e 6.8 Neutralino decays N φ0N , for φ0 = h0,H0, A0 ................. 88 i → j 0 e e 6.9 Ni Z Nj ....................................... 89 → e e 6.10 Selectron pair production in electron-electron collisions............... 91 e e 6.10.1 e e e−e− .................................. 91 − − → L R e e 2 6.10.2 e e e−e− .................................. 93 − − → R R 6.10.3 e−e− eL−eL− .................................. 94 → e e 6.11 e e+ νν ....................................... 95 − → ∗ + e e 6.12 e−e NiNj ...................................... 97 → ee 6.13 N N ff¯ ....................................... 100 1 1 → + e e + 6.14 e e C−C ..................................... 110 − → i j e e + 6.15 ud¯ C Nj ....................................... 112 → i e e 6.16 Ni NjNkNℓ ..................................... 114 → e e 6.17 Three-body slepton decays ℓ− ℓ−τ ± τ ∓ for ℓ = e, µ ............... 116 e e e e R → 1 6.18 Neutralino decay to photon and Goldstino: Ni γG ................ 119 e e → 6.19 Gluino pair production from gluon fusion: gg gg ................. 121 → + + e e 6.20 R-parity violating stau decay: τR e ν¯µ ...................... 125 → ee 6.21 R-parity-violating neutralino decay: Ni µ−ud¯ .................. 126 e → 6.22 Top-quark condensation from a Nambu-Jona-Lasinio model gap equation . 129 e 6.23 Electroweak vector boson self-energies from fermion loops ............. 130 6.24 Self-energy and pole mass of the top quark . ......... 134 6.25 Self-energy and pole mass of the gluino . .......... 141 6.26 Triangle anomaly from chiral fermion loops . ........... 143 Acknowledgments 155 Appendix A: Metric and sigma matrix conventions 156 Appendix B: Sigma matrix identities and Fierz identities 160 B.1 Two-component spinor Fierz identities . ........... 160 B.2 Sigma matrix identities in d 6= 4 dimensions ..................... 164 Appendix C: Explicit forms for the two-component spinor wave functions 166 C.1Fixed-axisspinorwavefunctions . ......... 167 C.2 Fixed-axis spinors in the non-relativistic limit . ............... 172 C.3 Helicity spinor wave functions . ........ 174 Appendix D: Matrix decompositions for fermion mass diagonalization 177 D.1 Singular value decomposition . ........ 178 D.2 Takagi diagonalization . ....... 180 D.3 Relation between Takagi diagonalization and singular value decomposition . 182 D.4 Reduction of a complex antisymmetric matrix to real normalform. 184 3 Appendix E: Lie group theoretical techniques for gauge theories 187 E.1 Basic facts about Lie groups, Lie algebras and their representations . 187 E.2 The quadratic and cubic index and Casimir operator . ........... 189 Appendix F: Path integral treatment of two-component fermion propagators 192 Appendix G: Correspondence to four-component spinor notation 195 G.1 Dirac gamma matrices and four-component spinors . ........... 195 G.2 Free-field four-component fermion Lagrangians . ............. 207 G.3 Gamma matrices and spinors in spacetimes of diverse dimensions and signatures . 210 G.4 Four-component spinor wave functions . ......... 215 G.5 Feynman rules for four-component fermions . ........... 219 G.6 Applications of four-component spinor Feynman rules . .............. 224 G.7 Self-energy functions and pole masses for four-componentfermions. 230 Appendix H: Covariant spin operators and the Bouchiat-Michel formulae 233 H.1 The covariant spin operators for a spin-1/2 fermion . ............. 233 H.2 Two-component spinor wave function relations . ............ 235 H.3 Two-component Bouchiat-Michel formulae . .......... 236 H.4 Four-component Bouchiat-Michel formulae . ........... 240 Appendix I: Helicity amplitudes and the spinor helicity method 242 I.1 The helicity amplitude technique of Hagiwara and Zeppenfeld............ 243 I.2Thespinorhelicitymethod . 247 Appendix J: The Standard Model and its seesaw extension 258 J.1: Standard Model fermion interaction vertices . ............. 258 J.2: Incorporating massive neutrinos into the Standard Model.............. 265 Appendix K: MSSM fermion interaction vertices 271 K.1 Higgs-fermion interaction vertices in the MSSM . ............ 271 K.2 Gauge interaction vertices for neutralinos and charginos............... 275 K.3 Higgs interactions with charginos and neutralinos . .............. 278 K.4 Chargino and neutralino interactions with fermions and sfermions . 280 K.5SUSY-QCDFeynmanrules . 286 Appendix L: Trilinear R-parity-violating Yukawa interactions 288 References 290 4 1 Introduction A crucial feature of the Standard Model of particle physics is the chiral nature of fermion quan- tum numbers and interactions. According to the modern understanding of the electroweak interactions, the fundamental degrees of freedom for quarks and leptons are two-component Weyl-van der Waerden fermions [1], i.e. two-component Lorentz spinors that transform as irre- ducible representations under the gauge group SU(2) U(1) . Furthermore, within the context L× Y of supersymmetric field theories, two-component spinors enter naturally, due to the spinorial na- ture of the symmetry generators themselves, and the holomorphic structure of the superpoten- tial. Despite this, most pedagogical treatments and practical calculations in high-energy physics continue to use the four-component Dirac spinor notation, which combines distinct irreducible representations of the Lorentz symmetry algebra. Parity-conserving theories such as QED and QCD are well-suited to the four-component fermion methods. There is also a certain perceived advantage to familiarity. However, as we progress to phenomena at and above the scale of elec- troweak symmetry breaking, it seems increasingly natural to employ two-component fermion notation, in harmony with the irreducible transformation properties dictated by the physics. One occasionally encounters the misconception that two-component fermion notations are somehow inherently ill-suited or unwieldy for practical use. Perhaps this is due in part to a lack of examples of calculations using two-component language in the pedagogical literature. In this review, we seek to dispel this idea by presenting Feynman rules for fermions using two- component spinor notation, intended for practical calculations of cross-sections, decays, and radiative corrections. This formalism employs a unified
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