Beyond Feynman Diagrams Lecture 2

Beyond Feynman Diagrams Lecture 2

<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>

View Full Text

Details

  • File Type
    pdf
  • Upload Time
    -
  • Content Languages
    English
  • Upload User
    Anonymous/Not logged-in
  • File Pages
    36 Page
  • File Size
    -

Download

Channel Download Status
Express Download Enable

Copyright

We respect the copyrights and intellectual property rights of all users. All uploaded documents are either original works of the uploader or authorized works of the rightful owners.

  • Not to be reproduced or distributed without explicit permission.
  • Not used for commercial purposes outside of approved use cases.
  • Not used to infringe on the rights of the original creators.
  • If you believe any content infringes your copyright, please contact us immediately.

Support

For help with questions, suggestions, or problems, please contact us