FeynCalc manual (development version)

FCDiracIsolate

FCDiracIsolate[exp] wraps chains of Dirac matrices into heads specified by the user.

See also

Overview, DiracGamma, Spinor.

Examples

FCDiracIsolate provides an easy way to extract the Dirac structures present in the expression (e.g. an amplitude)

amp = (Spinor[Momentum[p2], SMP["m_u"], 1] . (-I GA[\[Mu]] SMP["g_s"] SUNTF[{Glu2}, 
         Col3, Col5]) . (GS[-k1 + p2] + SMP["m_u"]) . (-I GA[\[Nu]] SMP["g_s"] SUNTF[{Glu4}, 
         Col5, Col1]) . Spinor[Momentum[p1], SMP["m_u"], 1] FAD[{k1 - p2, 
       SMP["m_u"]}, Dimension -> 4] FV[Polarization[k1, I], \[Mu]] FV[Polarization[k2, -I], 
      \[Nu]] + Spinor[Momentum[p2], SMP["m_u"], 1] . (-I GA[\[Nu]] SMP["g_s"] SUNTF[{Glu4}, 
         Col3, Col5]) . (GS[k2 + p2] + SMP["m_u"]) . (-I GA[\[Mu]] SMP["g_s"] SUNTF[{Glu2}, 
         Col5, Col1]) . Spinor[Momentum[p1], SMP["m_u"], 1] FAD[{-k2 - p2, SMP["m_u"]}, 
      Dimension -> 4] FV[Polarization[k1, I], \[Mu]] FV[Polarization[k2, 
       -I], \[Nu]] - Spinor[Momentum[p2], SMP["m_u"], 1] . (-I GA[Lor3] SMP["g_s"] SUNTF[{Glu5}, 
         Col3, Col1]) . Spinor[Momentum[p1], SMP["m_u"], 1] FAD[-k1 + k2, 
      Dimension -> 4] FV[Polarization[k1, I], \[Mu]] FV[Polarization[k2, -I], 
      \[Nu]] MT[Lor3, Lor4] (FV[2 k1 - k2, \[Nu]] MT[Lor4, \[Mu]] + FV[-k1 + 2 k2, \[Mu]] MT[Lor4, 
         \[Nu]] + FV[-k1 - k2, Lor4] MT[\[Mu], \[Nu]]) SMP["g_s"] SUNF[Glu2, Glu4, Glu5])

εˉμ(k1)εˉν(k2)(φ(p2,mu)).(igsγˉμTCol3  Col5Glu2).(γˉ(p2k1)+mu).(igsγˉνTCol5  Col1Glu4).(φ(p1,mu))(k1p2)2mu2+εˉμ(k1)εˉν(k2)(φ(p2,mu)).(igsγˉνTCol3  Col5Glu4).(γˉ(k2+p2)+mu).(igsγˉμTCol5  Col1Glu2).(φ(p1,mu))(k2p2)2mu21(k2k1)2gsgˉLor3  Lor4εˉμ(k1)εˉν(k2)fGlu2  Glu4  Glu5(gˉLor4μ(2k1k2)ν+gˉLor4ν(2k2k1)μ+gˉμν(k1k2)Lor4)(φ(p2,mu)).(igsγˉLor3TCol3  Col1Glu5).(φ(p1,mu))\frac{\bar{\varepsilon }^{\mu }(\text{k1}) \bar{\varepsilon }^{*\nu }(\text{k2}) \left(\varphi (\overline{\text{p2}},m_u)\right).\left(-i g_s \bar{\gamma }^{\mu } T_{\text{Col3}\;\text{Col5}}^{\text{Glu2}}\right).\left(\bar{\gamma }\cdot \left(\overline{\text{p2}}-\overline{\text{k1}}\right)+m_u\right).\left(-i g_s \bar{\gamma }^{\nu } T_{\text{Col5}\;\text{Col1}}^{\text{Glu4}}\right).\left(\varphi (\overline{\text{p1}},m_u)\right)}{(\overline{\text{k1}}-\overline{\text{p2}})^2-m_u^2}+\frac{\bar{\varepsilon }^{\mu }(\text{k1}) \bar{\varepsilon }^{*\nu }(\text{k2}) \left(\varphi (\overline{\text{p2}},m_u)\right).\left(-i g_s \bar{\gamma }^{\nu } T_{\text{Col3}\;\text{Col5}}^{\text{Glu4}}\right).\left(\bar{\gamma }\cdot \left(\overline{\text{k2}}+\overline{\text{p2}}\right)+m_u\right).\left(-i g_s \bar{\gamma }^{\mu } T_{\text{Col5}\;\text{Col1}}^{\text{Glu2}}\right).\left(\varphi (\overline{\text{p1}},m_u)\right)}{(-\overline{\text{k2}}-\overline{\text{p2}})^2-m_u^2}-\frac{1}{(\overline{\text{k2}}-\overline{\text{k1}})^2}g_s \bar{g}^{\text{Lor3}\;\text{Lor4}} \bar{\varepsilon }^{\mu }(\text{k1}) \bar{\varepsilon }^{*\nu }(\text{k2}) f^{\text{Glu2}\;\text{Glu4}\;\text{Glu5}} \left(\bar{g}^{\text{Lor4}\mu } \left(2 \overline{\text{k1}}-\overline{\text{k2}}\right)^{\nu }+\bar{g}^{\text{Lor4}\nu } \left(2 \overline{\text{k2}}-\overline{\text{k1}}\right)^{\mu }+\bar{g}^{\mu \nu } \left(-\overline{\text{k1}}-\overline{\text{k2}}\right)^{\text{Lor4}}\right) \left(\varphi (\overline{\text{p2}},m_u)\right).\left(-i g_s \bar{\gamma }^{\text{Lor3}} T_{\text{Col3}\;\text{Col1}}^{\text{Glu5}}\right).\left(\varphi (\overline{\text{p1}},m_u)\right)

ampIso = FCDiracIsolate[amp, Head -> diracS]

gs2εˉμ(k1)εˉν(k2)TCol5  Col1Glu2TCol3  Col5Glu4  diracS((φ(p2,mu)).γˉν.(γˉ(k2+p2)+mu).γˉμ.(φ(p1,mu)))(k2p2)2mu2gs2εˉμ(k1)εˉν(k2)TCol5  Col1Glu4TCol3  Col5Glu2  diracS((φ(p2,mu)).γˉμ.(γˉ(p2k1)+mu).γˉν.(φ(p1,mu)))(k1p2)2mu2+1(k2k1)2igs2gˉLor3  Lor4εˉμ(k1)εˉν(k2)TCol3  Col1Glu5fGlu2  Glu4  Glu5(gˉμν((k1+k2)Lor4)gˉLor4ν(k12k2)μ+gˉLor4μ(2k1k2)ν)  diracS((φ(p2,mu)).γˉLor3.(φ(p1,mu)))-\frac{g_s^2 \bar{\varepsilon }^{\mu }(\text{k1}) \bar{\varepsilon }^{*\nu }(\text{k2}) T_{\text{Col5}\;\text{Col1}}^{\text{Glu2}} T_{\text{Col3}\;\text{Col5}}^{\text{Glu4}} \;\text{diracS}\left(\left(\varphi (\overline{\text{p2}},m_u)\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \left(\overline{\text{k2}}+\overline{\text{p2}}\right)+m_u\right).\bar{\gamma }^{\mu }.\left(\varphi (\overline{\text{p1}},m_u)\right)\right)}{(-\overline{\text{k2}}-\overline{\text{p2}})^2-m_u^2}-\frac{g_s^2 \bar{\varepsilon }^{\mu }(\text{k1}) \bar{\varepsilon }^{*\nu }(\text{k2}) T_{\text{Col5}\;\text{Col1}}^{\text{Glu4}} T_{\text{Col3}\;\text{Col5}}^{\text{Glu2}} \;\text{diracS}\left(\left(\varphi (\overline{\text{p2}},m_u)\right).\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \left(\overline{\text{p2}}-\overline{\text{k1}}\right)+m_u\right).\bar{\gamma }^{\nu }.\left(\varphi (\overline{\text{p1}},m_u)\right)\right)}{(\overline{\text{k1}}-\overline{\text{p2}})^2-m_u^2}+\frac{1}{(\overline{\text{k2}}-\overline{\text{k1}})^2}i g_s^2 \bar{g}^{\text{Lor3}\;\text{Lor4}} \bar{\varepsilon }^{\mu }(\text{k1}) \bar{\varepsilon }^{*\nu }(\text{k2}) T_{\text{Col3}\;\text{Col1}}^{\text{Glu5}} f^{\text{Glu2}\;\text{Glu4}\;\text{Glu5}} \left(\bar{g}^{\mu \nu } \left(-\left(\overline{\text{k1}}+\overline{\text{k2}}\right)^{\text{Lor4}}\right)-\bar{g}^{\text{Lor4}\nu } \left(\overline{\text{k1}}-2 \overline{\text{k2}}\right)^{\mu }+\bar{g}^{\text{Lor4}\mu } \left(2 \overline{\text{k1}}-\overline{\text{k2}}\right)^{\nu }\right) \;\text{diracS}\left(\left(\varphi (\overline{\text{p2}},m_u)\right).\bar{\gamma }^{\text{Lor3}}.\left(\varphi (\overline{\text{p1}},m_u)\right)\right)

Now that all Dirac structures are wrapped into the head diracS it is easy to extract them to a separate list

Cases2[ampIso, diracS]

{diracS((φ(p2,mu)).γˉLor3.(φ(p1,mu))),diracS((φ(p2,mu)).γˉμ.(γˉ(p2k1)+mu).γˉν.(φ(p1,mu))),diracS((φ(p2,mu)).γˉν.(γˉ(k2+p2)+mu).γˉμ.(φ(p1,mu)))}\left\{\text{diracS}\left(\left(\varphi (\overline{\text{p2}},m_u)\right).\bar{\gamma }^{\text{Lor3}}.\left(\varphi (\overline{\text{p1}},m_u)\right)\right),\text{diracS}\left(\left(\varphi (\overline{\text{p2}},m_u)\right).\bar{\gamma }^{\mu }.\left(\bar{\gamma }\cdot \left(\overline{\text{p2}}-\overline{\text{k1}}\right)+m_u\right).\bar{\gamma }^{\nu }.\left(\varphi (\overline{\text{p1}},m_u)\right)\right),\text{diracS}\left(\left(\varphi (\overline{\text{p2}},m_u)\right).\bar{\gamma }^{\nu }.\left(\bar{\gamma }\cdot \left(\overline{\text{k2}}+\overline{\text{p2}}\right)+m_u\right).\bar{\gamma }^{\mu }.\left(\varphi (\overline{\text{p1}},m_u)\right)\right)\right\}

This way we obtain a sorted list of all unique Dirac structures in amp.

ClearAll[amp, ampIso, diracS]