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| <header id="title-block-header"> | ||
| <h1 class="title">FeynCalc manual (development version)</h1> | ||
| </header> | ||
| <h2 id="treatment-of-gamma5-in-d-dimensions">Treatment of gamma5 in D | ||
| dimensions</h2> | ||
| <h3 id="see-also">See also</h3> | ||
| <p><a href="FeynCalc.html">Overview</a>.</p> | ||
| <h3 id="nature-of-the-problem">Nature of the problem</h3> | ||
| <p>It is a well-known fact (cf. eg. <a | ||
| href="https://arxiv.org/pdf/hep-th/0005255">Jegerlehner:2000dz</a>) that | ||
| the definition of <span class="math inline">\gamma^5</span> in 4 | ||
| dimensions cannot be consistently extended to <span | ||
| class="math inline">D</span> dimensions without giving up either the | ||
| anticommutativity property</p> | ||
| <p><span class="math display">\begin{equation} | ||
| \{\gamma^5, \gamma^\mu\} = 0 | ||
| \end{equation}</span></p> | ||
| <p>or the cyclicity of the Dirac trace, e.g. that</p> | ||
| <p><span class="math display">\begin{equation} | ||
| \mathrm{Tr}( \gamma^{\mu_1} \ldots \gamma^{\mu_{2n}} \gamma^5 ) = | ||
| \mathrm{Tr}( \gamma^{\mu_2} \ldots \gamma^{\mu_{2n}} \gamma^5 | ||
| \gamma^{\mu_1} ) = \mathrm{Tr}( \gamma^{\mu_3} \ldots \gamma^{\mu_{2n}} | ||
| \gamma^5 \gamma^{\mu_1} \gamma^{\mu_2} ) = \ldots | ||
| \end{equation}</span></p> | ||
| <p>This explains the existence of multiple prescriptions (called <span | ||
| class="math inline">\gamma^5</span>-schemes) that aim at avoiding these | ||
| issues and obtaining physical results in the <em>given | ||
| calculation</em>.</p> | ||
| <p>Indeed, as of now there is no simple solution or cookbook recipe that | ||
| can be readily applied to any theory at any loop order in a fully | ||
| automatic fashion.</p> | ||
| <p>The reason for this is that calculations involving <span | ||
| class="math inline">\gamma^5</span> are not limited to the algebraic | ||
| manipulations of Dirac matrices. In general, once <span | ||
| class="math inline">\gamma^5</span> shows up in <span | ||
| class="math inline">D</span>-dimensional amplitudes, there is a high | ||
| chance that the final result will violate some of the essential | ||
| symmetries, such as generalized Ward identities or Bose symmetry.</p> | ||
| <p>Once this happens, symmetries violated due to the chosen <span | ||
| class="math inline">\gamma^5</span> scheme must be restored by hand, | ||
| e.g. by introducing special finite counterterms. Unfortunately, an | ||
| explicit determination of such counterterms for a given model is a | ||
| nontrivial task, especially beyond 1-loop. This explains why people | ||
| usually try to avoid this situation and would rather opt for figuring | ||
| out special tricks that work only for this particular calculation but | ||
| manage to preserve the symmetries.</p> | ||
| <p>Further discussions on this topic can be found e.g. in</p> | ||
| <ul> | ||
| <li>chapter D of <a | ||
| href="https://arxiv.org/pdf/1809.01830">Blondel:2018mad</a></li> | ||
| <li><a | ||
| href="https://arxiv.org/pdf/hep-ph/9504315.pdf">Trueman:1995ca</a></li> | ||
| <li><a | ||
| href="https://arxiv.org/pdf/1912.06823.pdf">Denner:2019vbn</a></li> | ||
| <li><a | ||
| href="https://arxiv.org/abs/1705.01827">Gnendiger:2017pys</a></li> | ||
| <li><a | ||
| href="https://arxiv.org/abs/2312.11291">Stockinger:2023ndm</a></li> | ||
| </ul> | ||
| <h3 id="feyncalc-implementation">FeynCalc implementation</h3> | ||
| <p>FeynCalc has built-in support for several <span | ||
| class="math inline">\gamma^5</span>-schemes in the sense that it can | ||
| manipulate <span class="math inline">D</span>-dimensional algebraic | ||
| expressions involving <span class="math inline">\gamma^5</span> in | ||
| accordance with the rules provided by the scheme authors.</p> | ||
| <p>The nonalgebraic part of a typical <span | ||
| class="math inline">\gamma^5</span>-calculation, e.g. checking for | ||
| violated symmetries and restoring them is <strong>not handled</strong> | ||
| by FeynCalc. This is also not something easy to automatize (due to the | ||
| reasons explained above) so that here we expect the user to employ their | ||
| understanding of physics and common sense.</p> | ||
| <p>The responsibility of FeynCalc is to ensure that algebraic | ||
| manipulations of Dirac matrices (including <span | ||
| class="math inline">\gamma^5</span>) are consistent within the chosen | ||
| scheme. For the purpose of dealing with <span | ||
| class="math inline">\gamma^5</span> in <span | ||
| class="math inline">D</span> dimensions FeynCalc implements three | ||
| different schemes.</p> | ||
| <h4 id="ndr">NDR</h4> | ||
| <p>The Naive or Conventional Dimensional Regularization (NDR or CDR | ||
| respectively) <a | ||
| href="https://doi.org/10.1016/0550-3213(79)90333-X">Chanowitz:1979zu</a> | ||
| simply <em>assumes</em> that one can define a <span | ||
| class="math inline">D</span>-dimensional <span | ||
| class="math inline">\gamma^5</span> that anticommutes with any other | ||
| Dirac matrix and does not break the cyclicity of the trace. For FeynCalc | ||
| this means that in every string of Dirac matrices all <span | ||
| class="math inline">\gamma^5</span> can be safely anticommuted to the | ||
| right end of the string. In the course of this operation FeynCalc can | ||
| always apply <span class="math inline">(\gamma^5)^2 = 1</span>.</p> | ||
| <p>Consequently, all Dirac traces with an even number of <span | ||
| class="math inline">\gamma^5</span> can be rewritten as traces that | ||
| involve only the first four <span | ||
| class="math inline">\gamma</span>-matrices and evaluated directly, | ||
| e.g.</p> | ||
| <p><span class="math display">\begin{equation} | ||
| \mathrm{Tr}( \gamma^{\mu_1} \gamma^{\mu_2} \gamma^5 \gamma^{\mu_3} | ||
| \ldots \gamma^{\mu_{2n}} \gamma^5 ) = | ||
| \mathrm{Tr}( \gamma^{\mu_1} \gamma^{\mu_2} \ldots \gamma^{\mu_{2n}} ) | ||
| \end{equation}</span></p> | ||
| <p>The problematic cases are <span | ||
| class="math inline">\gamma^5</span>-odd traces with an even number of | ||
| other Dirac matrices, where the <span | ||
| class="math inline">\mathcal{O}(D-4)</span> pieces of the result depend | ||
| on the initial position of <span class="math inline">\gamma^5</span> in | ||
| the string. Using the anticommutativity property they can be always | ||
| rewritten as traces of a string of other Dirac matrices and one <span | ||
| class="math inline">\gamma^5</span>. If the number of the other Dirac | ||
| matrices is odd, such a trace is put to zero i.e. <span | ||
| class="math display">\begin{equation} | ||
| \mathrm{Tr}(\gamma^{\mu_1} \ldots \gamma^{\mu_{2n-1}} \gamma^5) = 0, | ||
| \quad n \in \mathbb{N} | ||
| \end{equation}</span> If the number is even, the trace <span | ||
| class="math display">\begin{equation} | ||
| \mathrm{Tr}(\gamma^{\mu_1} \ldots \gamma^{\mu_{2n}} \gamma^5) | ||
| \end{equation}</span> is returned unevaluated, since FeynCalc does not | ||
| know how to calculate it in a consistent way. A user who knows how these | ||
| ambiguous objects should be treated in the particular calculation can | ||
| still take care of the remaining traces by hand. This ensures that the | ||
| output produced by FeynCalc is algebraically consistent to the maximal | ||
| extent possible in the NDR scheme without extra assumptions.</p> | ||
| <p>In FeynCalc, this scheme the default choice. It can also be | ||
| explicitly activated via</p> | ||
| <div class="sourceCode" id="cb1"><pre | ||
| class="sourceCode mathematica"><code class="sourceCode mathematica"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a>FCSetDiracGammaScheme<span class="op">[</span><span class="st">"NDR"</span><span class="op">]</span></span></code></pre></div> | ||
| <p>Sometimes <span class="math inline">\gamma^5</span> may show up in | ||
| the calculation as an artifact of using a particular set of operators or | ||
| projectors even though the results itself is not supposed to be affected | ||
| by the <span class="math inline">\gamma^5</span>-problem. For such cases | ||
| FeynCalc offers a variety of the NDR scheme, where all traces of the | ||
| form <span class="math display">\begin{equation} | ||
| \mathrm{Tr}(\gamma^{\mu_1} \ldots \gamma^{\mu_{2n}} \gamma^5) | ||
| \end{equation}</span> are simply put to zero. It can be used to | ||
| e.g. examine the effects of the chosen scheme on the final result and | ||
| can be activated via</p> | ||
| <div class="sourceCode" id="cb2"><pre | ||
| class="sourceCode mathematica"><code class="sourceCode mathematica"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a>FCSetDiracGammaScheme<span class="op">[</span><span class="st">"NDR-Discard"</span><span class="op">]</span></span></code></pre></div> | ||
| <h3 id="bmhv">BMHV</h3> | ||
| <p>FeynCalc also supports the Breitenlohner-Maison implementation <a | ||
| href="https://doi.org/10.1007/BF01609069">Breitenlohner:1977hr</a> of | ||
| the t’Hooft-Veltman <a | ||
| href="https://doi.org/10.1016/0550-3213(72)90279-9">tHooft:1972tcz</a> | ||
| prescription, often abbreviated as BMHV, HVBM, HV or BM scheme. In this | ||
| approach <span class="math inline">\gamma^5</span> is treated as a | ||
| purely 4-dimensional object, while <span | ||
| class="math inline">D</span>-dimensional Dirac matrices and 4-vectors | ||
| are decomposed into <span class="math inline">4</span>- and <span | ||
| class="math inline">D-4</span>-dimensional components. Following <a | ||
| href="https://doi.org/10.1016/0550-3213(90)90223-Z">Buras:1989xd</a> | ||
| FeynCalc typesets the former with a bar and the latter with a hat | ||
| e.g.</p> | ||
| <p><span class="math display">\begin{equation} | ||
| \gamma^\mu = \bar{\gamma}^\mu + \hat{\gamma}^\mu, \quad p^\mu = | ||
| \bar{p}^\mu + \hat{p}^\mu | ||
| \end{equation}</span></p> | ||
| <p>The main advantage of the BMHV scheme is that the Dirac algebra | ||
| (including traces) can be evaluated without any algebraic ambiguities. | ||
| However, calculations involving tensors from three different spaces | ||
| (<span class="math inline">D</span>, <span class="math inline">4</span> | ||
| and <span class="math inline">D-4</span>) often turn out to be rather | ||
| cumbersome, even when using computer codes. Moreover, this prescription | ||
| is known to artificially violate Ward identities in chiral theories, | ||
| which is something that can be often avoided when using NDR. Within BMHV | ||
| FeynCalc can simplify arbitrary strings of Dirac matrices and calculate | ||
| arbitrary traces out-of-the-box. The evaluation of <span | ||
| class="math inline">\gamma^5</span>-odd Dirac traces is performed using | ||
| the West-formula from <a | ||
| href="https://doi.org/10.1016/0010-4655(93)90011-Z">West:1991xv</a>. It | ||
| is worth noting that <span class="math inline">D-4</span>-dimensional | ||
| components of external momenta are not set to zero by default, as it is | ||
| conventionally done in the literature. If this is required, the user | ||
| should evaluate <code>Momentum[pi,D-4]=0</code> for each relevant | ||
| momentum <span class="math inline">p_i</span>. To remove such | ||
| assignments one should use <code>FCClearScalarProducts[]</code>.</p> | ||
| <p>This scheme is activated by evaluating</p> | ||
| <div class="sourceCode" id="cb3"><pre | ||
| class="sourceCode mathematica"><code class="sourceCode mathematica"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>FCSetDiracGammaScheme<span class="op">[</span><span class="st">"BMHV"</span><span class="op">]</span></span></code></pre></div> | ||
| <h3 id="larins-scheme">Larin’s scheme</h3> | ||
| <p>Larin’s scheme <a | ||
| href="https://arxiv.org/pdf/hep-ph/9302240.pdf">Larin:1993tq</a> is a | ||
| variety of the BMHV scheme that has been extensively used in QCD | ||
| calculations involving axial vector currents. The main idea is to | ||
| replace the products of <span class="math inline">\gamma^\mu</span> and | ||
| <span class="math inline">\gamma^5</span> in a chiral trace as in</p> | ||
| <p><span class="math display">\begin{equation} | ||
| \gamma^\mu \gamma^5 \to \frac{1}{6} i \varepsilon^{\mu \nu \rho \sigma} | ||
| \gamma_\nu \gamma_\rho \gamma_\sigma | ||
| \end{equation}</span></p> | ||
| <p>and then calculate the resulting trace. Then, all <span | ||
| class="math inline">\varepsilon^{\mu \nu \rho \sigma}</span>-tensors | ||
| occurring in the amplitude should be evaluated in <span | ||
| class="math inline">D</span> dimensions. Together with the correct | ||
| counterterm, this prescription is known to give the same result as when | ||
| using the full BMHV scheme.</p> | ||
| <p>FeynCalc implement the so-called Moch-Vermaseren-Vogt MVV formula | ||
| from <a href="https://arxiv.org/pdf/1506.04517.pdf">Moch:2015usa</a> for | ||
| calculating <span class="math inline">\gamma^5</span>-traces in this | ||
| scheme. The scheme itself is activated by setting</p> | ||
| <div class="sourceCode" id="cb4"><pre | ||
| class="sourceCode mathematica"><code class="sourceCode mathematica"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a>FCSetDiracGammaScheme<span class="op">[</span><span class="st">"Larin"</span><span class="op">]</span></span></code></pre></div> | ||
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