<p>Beyond Feynman Diagrams </p><p>Lecture 2 </p><p>Lance Dixon </p><p>Academic Training Lectures </p><p>CERN <br>April 24-26, 2013 </p><p>Modern methods for trees </p><p>1. Color organization (briefly) </p><p>2. Spinor variables </p><p>3. Simple examples 4. Factorization properties </p><p>5. BCFW (on-shell) recursion relations </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">2</li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>How to organize </p><p>gauge theory amplitudes </p><p>• Avoid tangled algebra of color and Lorentz indices </p><p>generated by Feynman rules </p><p>structure constants </p><p>• Take advantage of physical properties of amplitudes • Basic tools: </p><p>- dual (trace-based) color decompositions </p><p>- spinor helicity formalism </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">3</li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Color </p><p>Standard color factor for a QCD graph has lots of structure constants contracted in various orders; for example: </p><p>Write every <em>n</em>-gluon tree graph color factor as a sum of traces of&nbsp;matrices <em>T </em><sup style="top: -0.5em;"><em>a </em></sup>in the fundamental (defining) </p><p>representation of <em>SU(N</em><sub style="top: 0.41em;"><em>c</em></sub><em>): </em></p><p>+ all non-cyclic permutations </p><p>Use definition: </p><p></p><p>+ normalization: </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">4</li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Double-line picture (’t Hooft) </p><p>• In limit of large number of colors N<sub style="top: 0.41em;">c</sub>, a gluon is always a combination of a color and a different anti-color. <br>• Gluon tree amplitudes dressed by lines carrying color </p><p>indices, 1,2,3,…,N<sub style="top: 0.41em;">c</sub>. </p><p>• Leads to color ordering of the external gluons. • Cross section, summed over colors of all external gluons </p><p>= S |color-ordered amplitudes|<sup style="top: -0.5em;">2 </sup></p><p>• Can still use this picture at N<sub style="top: 0.41em;">c</sub>=3. </p><p>• Color-ordered amplitudes are still the building blocks. <br>• Corrections to the color-summed cross section, can be handled </p><p>exactly, but are suppressed by 1/ N<sub style="top: 0.41em;">c </sub></p><p>2</p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">5</li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Trace-based (dual) color decomposition </p><p>For <em>n</em>-gluon tree amplitudes, the color decomposition is </p><p>momenta helicities </p><p>color </p><p>color-ordered subamplitude only depends on momenta. </p><p>Compute separately for each cyclicly inequivalent helicity configuration </p><p>• Because </p><p>comes from planar diagrams with cyclic ordering of external legs fixed to 1,2,…,<em>n</em>, </p><p>it only has singularities in cyclicly-adjacent channels <em>s</em><sub style="top: 0.41em;"><em>i,i+</em>1 </sub>, … Similar decompositions for amplitudes with external quarks. </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">6</li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Far fewer factorization channels with color ordering </p><p></p><ul style="display: flex;"><li style="flex:1">3</li><li style="flex:1">2</li></ul><p></p><p><em>k+</em>1 </p><p>3<br>1</p><p>1</p><p><em>k</em></p><p>16 </p><p>(<sup style="top: -2.2275em;"><em>n</em></sup><sub style="top: 0.9467em;"><em>k </em></sub>) </p><p>only <em>n </em></p><p>partitions </p><p><em>n</em></p><p>4</p><p><em>k+</em>1 </p><p>2</p><p>…<br>…</p><p><em>n</em>-1 <em>k+</em>2 </p><ul style="display: flex;"><li style="flex:1"><em>n</em></li><li style="flex:1"><em>k+</em>2 </li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">7</li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Color sums </p><p>Parton model says to sum/average over final/initial colors (as well as helicities): </p><p>Insert: </p><p>and do color sums to get: </p><p><em>2</em></p><p> Up to 1<em>/N</em><sub style="top: 0.41em;"><em>c </em></sub>suppressed effects, squared subamplitudes have </p><p>definite color flow – important for development of parton shower </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">8</li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Spinor helicity formalism </p><p>Scattering amplitudes for massless plane waves of definite momentum: Lorentz 4-vectors <em>k</em><sub style="top: 0.41em;"><em>i</em></sub><sup style="top: -0.5em;">m </sup></p><p><em>k</em><sub style="top: 0.41em;"><em>i</em></sub><sup style="top: -0.5em;">2</sup>=0 </p><p>Natural to use Lorentz-invariant products (invariant masses): </p><p>But for elementary particles with <strong>spin </strong>(<em>e.g. </em>all except Higgs!) </p><p><strong>there is a better way: </strong></p><p><strong>Take “square root” </strong>of 4-vectors <em>k</em><sub style="top: 0.41em;"><em>i</em></sub><sup style="top: -0.5em;">m </sup>(spin 1) </p><p>use Dirac (Weyl) spinors <em>u</em><sub style="top: 0.41em;">a</sub>(<em>k</em><sub style="top: 0.41em;"><em>i</em></sub>) (spin ½) </p><p><em>q ,g</em>,g, all have 2 helicity states, </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">9</li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Massless Dirac spinors </p><p>• Positive and negative energy solutions to the massless <br>Dirac equation, </p><p>are identical up to normalization. </p><p>• Chirality/helicity eigenstates are • Explicitly, in the Dirac representation </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">10 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Spinor products </p><p>Instead of Lorentz products: </p><p>Use spinor products: </p><p>Identity These are <strong>complex square roots </strong>of Lorentz products (for real <em>k</em><sub style="top: 0.41em;"><em>i</em></sub>): </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">11 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>~ Simplest Feynman diagram of all </p><p>21<br>3add helicity information, </p><p>numeric labels </p><p><em>R</em></p><p><em>R</em></p><p>g</p><p>4</p><p><em>L</em></p><p><em>L</em></p><p>Fierz identity </p><p>helicity suppressed as 1&nbsp;|| 3&nbsp;or 2&nbsp;|| 4 </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">12 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Useful to rewrite answer </p><p>Crossing symmetry more manifest </p><p>if we switch to all-outgoing helicity labels </p><p>(flip signs of incoming helicities) </p><p>-</p><p>2</p><p>3<sup style="top: -0.5em;">+ </sup></p><p>useful identities: </p><p>-</p><p>1<sup style="top: -0.5em;">+ </sup></p><p>4</p><p>“holomorphic” </p><p>“antiholomorphic” </p><p>Schouten </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">13 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Symmetries for all other helicity config’s </p><p>-</p><p>3<sup style="top: -0.5em;">+ </sup></p><p>C</p><p>2</p><p></p><ul style="display: flex;"><li style="flex:1">2<sup style="top: -0.5em;">+ </sup></li><li style="flex:1">3<sup style="top: -0.5em;">+ </sup></li></ul><p></p><p>-</p><p>1<sup style="top: -0.5em;">+ </sup></p><p>4</p><p></p><ul style="display: flex;"><li style="flex:1">-</li><li style="flex:1">-</li></ul><p></p><p></p><ul style="display: flex;"><li style="flex:1">1</li><li style="flex:1">4</li></ul><p></p><p>P</p><p></p><ul style="display: flex;"><li style="flex:1">-</li><li style="flex:1">-</li></ul><p>-</p><p></p><ul style="display: flex;"><li style="flex:1">2</li><li style="flex:1">3</li></ul><p></p><p>2<sup style="top: -0.5em;">+ </sup></p><p>3</p><p></p><ul style="display: flex;"><li style="flex:1">1<sup style="top: -0.5em;">+ </sup></li><li style="flex:1">4<sup style="top: -0.5em;">+ </sup></li></ul><p></p><p>-</p><p>1</p><p>4<sup style="top: -0.5em;">+ </sup></p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">14 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Unpolarized, helicity-summed cross sections </p><p>(the norm in QCD) </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">15 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Helicity formalism for massless vectors </p><p>Berends, Kleiss, De Causmaecker, Gastmans, Wu (1981); De Causmaecker, Gastmans, Troost, Wu (1982); Xu, Zhang, Chang (1984); Kleiss, Stirling (1985); Gunion, Kunszt (1985) </p><p></p><ul style="display: flex;"><li style="flex:1">obeys </li><li style="flex:1">(required transversality) </li></ul><p>(bonus) under azimuthal rotation about <em>k</em><sub style="top: 0.41em;"><em>i </em></sub>axis, helicity +1/2 helicity -1/2 so as required for helicity +1 </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">16 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Next most famous pair </p><p>of Feynman diagrams </p><p>(to a higher-order QCD person) </p><p>gg</p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">17 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>(cont.) </p><p>Choose to remove 2<sup style="top: -0.5em;">nd </sup>graph </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">18 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>Properties of </p><p>1. Soft&nbsp;gluon behavior </p><p>Universal “eikonal” factors </p><p>for emission of soft gluon <em>s </em></p><p>between two hard partons <em>a </em>and <em>b </em></p><p>Soft emission is from the classical chromoelectric current: </p><p>independent of parton type (<em>q vs. g</em>) and helicity – only depends on momenta of <em>a,b</em>, and color charge: </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">19 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p><p>(cont.) </p><p>Properties of </p><p>2. Collinear&nbsp;behavior </p><p><em>z</em></p><p>Square root of Altarelli-Parisi splitting probablility </p><p><em>1-z </em></p><p>Universal collinear factors, or splitting amplitudes depend on parton type and helicity </p><p></p><ul style="display: flex;"><li style="flex:1">Beyond Feynman Diagrams </li><li style="flex:1">20 </li><li style="flex:1">Lecture 2 </li><li style="flex:1">April 25, 2013 </li></ul><p></p>