For illustration purposes set three coordinate systems , and set to represent a quantum operator
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(1.1) |
Let be the interaction Lagrangian
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(1.2) |
The expansion of in coordinates representation, computed by default up to order = 3 (you can change that using the option order = n), by definition containing all possible configurations of external legs, displaying the related Feynman Diagrams, is given by
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![%eval(S, order = 3) = 1+%FeynmanIntegral(lambda*_GF(_NP(phi(X), phi(X), phi(X), phi(X))), [[X]])+%FeynmanIntegral(16*lambda^2*_GF(_NP(phi(X), phi(X), phi(X), phi(Y), phi(Y), phi(Y)), [[phi(X), phi(Y)]])+96*lambda^2*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Y)], [phi(X), phi(Y)]])+72*lambda^2*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Y)]]), [[X], [Y]])+%FeynmanIntegral(1728*lambda^3*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y), phi(Z), phi(Z)), [[phi(X), phi(Z)], [phi(X), phi(Y)], [phi(Z), phi(Y)]])+2592*lambda^3*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y)), [[phi(X), phi(Z)], [phi(X), phi(Z)], [phi(Z), phi(Y)], [phi(Z), phi(Y)]])+10368*lambda^3*_GF(_NP(phi(X), phi(Y), phi(Z), phi(Z)), [[phi(X), phi(Y)], [phi(X), phi(Y)], [phi(X), phi(Z)], [phi(Y), phi(Z)]])+10368*lambda^3*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]])+3456*lambda^3*_GF(_NP(phi(X), phi(X)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]])+576*lambda^3*_GF(_NP(phi(X), phi(X), phi(X), phi(Y), phi(Y), phi(Z), phi(Z), phi(Z)), [[phi(X), phi(Y)], [phi(Y), phi(Z)]]), [[X], [Y], [Z]])](/view.aspx?sf=211829_post/46d7b6b42fe947db3259f0109df0f2f7.gif)
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(1.3) |
The expansion of in coordinates representation to a specific order shows in a compact way the topology of the underlying Feynman diagrams. Each integral is represented with a new command, FeynmanIntegral , that works both in coordinates and momentum representation. To each term of the integrands corresponds a diagram, and the correspondence is always clear from the symmetry factors.
In a typical situation, one wants to compute a specific term, or scattering process, instead of the S matrix up to some order with all possible configurations of external legs. For example, to compute only the terms of this result that correspond to diagrams with 1 loop use numberofloops = 1 (for tree-level, use numerofloops = 0)
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![%eval(S, [`=`(order, 3), `=`(loops, 1)]) = FeynmanDiagrams(L, numberofloops = 1, diagrams)](/view.aspx?sf=211829_post/d401f3d93827cbe3fa5b94870e9d2e8f.gif)
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![%eval(S, [order = 3, loops = 1]) = %FeynmanIntegral(72*lambda^2*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Y)]]), [[X], [Y]])+%FeynmanIntegral(1728*lambda^3*_GF(_NP(phi(X), phi(X), phi(Y), phi(Y), phi(Z), phi(Z)), [[phi(X), phi(Z)], [phi(X), phi(Y)], [phi(Z), phi(Y)]]), [[X], [Y], [Z]])](/view.aspx?sf=211829_post/b3a66b00214ed28af5c58d44b74c3cea.gif)
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(1.4) |
In the result above there are two terms, with 4 and 6 external legs respectively.
A scattering process with matrix element in momentum representation, corresponding to the term with 4 external legs (symmetry factor = 72), could be any process where the total number of incoming + outgoing parties is equal to 4. For example, one with 2 incoming and 2 outgoing particles. The transition probability for that process is given by
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![`#mfenced(mrow(mo("⁢"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo(",",mathcolor = "olive"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo(",",mathcolor = "olive"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo("⁢",mathcolor = "olive")),open = "⟨",close = "⟩")` = FeynmanDiagrams(L, incomingparticles = [phi, phi], outgoingparticles = [phi, phi], numberofloops = 1, diagrams)](/view.aspx?sf=211829_post/47a4809aa18767b326ab8c2322b0412d.gif)
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![`#mfenced(mrow(mo("⁢"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo(",",mathcolor = "olive"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo(",",mathcolor = "olive"),mi("φ",fontstyle = "normal",mathcolor = "olive"),mo("⁢",mathcolor = "olive")),open = "⟨",close = "⟩")` = %FeynmanIntegral((9/8)*lambda^2*Dirac(-P__3-P__4+P__1+P__2)/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1-P__2-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])+%FeynmanIntegral((9/8)*lambda^2*Dirac(-P__3-P__4+P__1+P__2)/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__3-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])+%FeynmanIntegral((9/8)*lambda^2*Dirac(-P__3-P__4+P__1+P__2)/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__4-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])](/view.aspx?sf=211829_post/6bef3fa7580ffed31b1094970d8910e4.gif)
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(1.5) |
When computing in momentum representation, only the topology of the corresponding Feynman diagrams is shown (i.e. the diagrams associated to the corresponding Feynman integral in coordinates representation).
The transition matrix element is related to the transition probability density (formula (13) of sec. 21.1 of ref.[1]) by
where represent the particle densities of each of the s particles in the initial state , the (Dirac) is the expected singular factor due to the conservation of the energy-momentum and the amplitude is related to via
To directly get the probability density instead of use the option output = probabilitydensity
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![FeynmanDiagrams(L, incomingparticles = [phi, phi], outgoingparticles = [phi, phi], numberofloops = 1, output = probabilitydensity)](/view.aspx?sf=211829_post/1ad51fcfc5068ea2217eb2f890ce4967.gif)
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![Physics:-FeynmanDiagrams:-ProbabilityDensity(4*Pi^2*%mul(n[i], i = 1 .. 2)*abs(F)^2*Dirac(-P__3-P__4+P__1+P__2)*%mul(dP_[f]^3, f = 1 .. 2), F = %FeynmanIntegral((9/8)*lambda^2/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1-P__2-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])+%FeynmanIntegral((9/8)*lambda^2/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__3-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])+%FeynmanIntegral((9/8)*lambda^2/(Pi^6*(E__1*E__2*E__3*E__4)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__4-p__2)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]]))](/view.aspx?sf=211829_post/e16f0e4008388246f8db0d0ecdca3214.gif)
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(1.6) |
In practice, the most common computations involve processes with 2 or 4 external legs. To restrict the expansion of the scattering matrix in coordinates representation to that kind of processes use the numberofexternallegs option. For example, the following computes the expansion of up to order = 3, restricting the outcome to the terms corresponding to diagrams with only 2 external legs
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![%eval(S, [`=`(order, 3), `=`(legs, 2)]) = FeynmanDiagrams(L, numberofexternallegs = 2, diagrams)](/view.aspx?sf=211829_post/aee5564d039720bc720da27faba8bdf6.gif)
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![%eval(S, [order = 3, legs = 2]) = %FeynmanIntegral(96*lambda^2*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Y)], [phi(X), phi(Y)]]), [[X], [Y]])+%FeynmanIntegral(3456*lambda^3*_GF(_NP(phi(X), phi(X)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]])+10368*lambda^3*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]]), [[X], [Y], [Z]])](/view.aspx?sf=211829_post/5efd257f07a2cd7622ed1f2ed121cea9.gif)
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(1.7) |
This result shows two Feynman integrals, with respectively 2 and 3 loops, the second integral with two terms. The transition probability density in momentum representation for a process related to the first integral (1 term with symmetry factor = 96) is then
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![FeynmanDiagrams(L, incomingparticles = [phi], outgoingparticles = [phi], numberofloops = 2, diagrams, output = probabilitydensity)](/view.aspx?sf=211829_post/2be471543d971595998cfd5eb2240040.gif)
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![Physics:-FeynmanDiagrams:-ProbabilityDensity((1/2)*%mul(n[i], i = 1 .. 1)*abs(F)^2*Dirac(-P__2+P__1)*%mul(dP_[f]^3, f = 1 .. 1)/Pi, F = %FeynmanIntegral(%FeynmanIntegral(((3/8)*I)*lambda^2/(Pi^7*(E__1*E__2)^(1/2)*(p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__3^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1-p__2-p__3)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]]), [[p__3]]))](/view.aspx?sf=211829_post/5a99842179310a116b5c80596e91cb80.gif)
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(1.8) |
In the above, for readability, the contracted spacetime indices in the square of momenta entering the amplitude F (as denominators of propagators) are implicit. To make those indices explicit, use the option putindicesinsquareofmomentum
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![F = %FeynmanIntegral(%FeynmanIntegral(((3/8)*I)*lambda^2*Dirac(-P__2[`~kappa`]+P__1[`~kappa`])/(Pi^7*(E__1*E__2)^(1/2)*(p__2[mu]*p__2[`~mu`]-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__3[nu]*p__3[`~nu`]-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1[beta]-p__2[beta]-p__3[beta])*(-P__1[`~beta`]-p__2[`~beta`]-p__3[`~beta`])-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]]), [[p__3]])](/view.aspx?sf=211829_post/0351f556dfd88c4895eb6116f1169042.gif)
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(1.9) |
This computation can also be performed to higher orders. For example, with 3 loops, in coordinates and momentum representations, corresponding to the other two terms and diagrams in (1.7)
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![%eval(S[3], [legs = 2, loops = 3]) = %FeynmanIntegral(3456*lambda^3*_GF(_NP(phi(X), phi(X)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]])+10368*lambda^3*_GF(_NP(phi(X), phi(Y)), [[phi(X), phi(Y)], [phi(X), phi(Z)], [phi(X), phi(Z)], [phi(Y), phi(Z)], [phi(Y), phi(Z)]]), [[X], [Y], [Z]])](/view.aspx?sf=211829_post/a8ba7e0b254346a29fe03b261ebb761a.gif)
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(1.10) |
A corresponding S-matrix element in momentum representation:
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![%eval(%Bracket(phi, S[3], phi), loops = 3) = %FeynmanIntegral(%FeynmanIntegral(%FeynmanIntegral((9/32)*lambda^3*Dirac(-P__2+P__1)/(Pi^11*(E__1*E__2)^(1/2)*(p__3^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__4^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__5^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-p__3-p__4-p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+P__2+p__3+p__4+p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__3]]), [[p__4]]), [[p__5]])+2*%FeynmanIntegral(%FeynmanIntegral(%FeynmanIntegral((9/32)*lambda^3*Dirac(-P__2+P__1)/(Pi^11*(E__1*E__2)^(1/2)*(p__3^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__4^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__5^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-p__3-p__4-p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+p__4+p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__3]]), [[p__4]]), [[p__5]])+%FeynmanIntegral(%FeynmanIntegral((1/2048)*lambda*Dirac(-P__2+P__1)*%FeynmanIntegral(576*lambda^2/((p__2^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-p__2-p__4-p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])/(Pi^11*(E__1*E__2)^(1/2)*(p__4^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*(p__5^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((-P__1+p__4+p__5)^2-m__phi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__4]]), [[p__5]])](/view.aspx?sf=211829_post/b5663ca9de7ea6f50c013a3ffb70a9e7.gif)
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(1.11) |
Consider the interaction Lagrangian of Quantum Electrodynamics (QED). To formulate this problem on the worksheet, start defining the vector field .
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(1.12) |
Set lowercase Latin letters from i to s to represent spinor indices (you can change this setting according to your preference, see Setup ), also the (anticommutative) spinor field will be represented by , so set as an anticommutativeprefix, and set and as quantum operators
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(1.13) |
The matrix indices of the Dirac matrices are written explicitly and use conjugate to represent the Dirac conjugate
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(1.14) |
Compute , only the terms with 4 external legs, and display the diagrams: all the corresponding graphs have no loops
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![%eval(S[2], `=`(legs, 4)) = FeynmanDiagrams(L__QED, numberofvertices = 2, numberoflegs = 4, diagrams)](/view.aspx?sf=211829_post/7e9ffcb31cb533ae7be48307ad18af9b.gif)
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![%eval(S[2], legs = 4) = %FeynmanIntegral(-2*alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(psi[k](X), A[mu](X), conjugate(psi[i](Y)), A[alpha](Y)), [[psi[l](Y), conjugate(psi[j](X))]])+alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(conjugate(psi[j](X)), psi[k](X), conjugate(psi[i](Y)), psi[l](Y)), [[A[mu](X), A[alpha](Y)]]), [[X], [Y]])](/view.aspx?sf=211829_post/c9ed2068a72c9db9075cc582117ea80d.gif)
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(1.15) |
The same computation but with only 2 external legs results in the diagrams with 1 loop that correspond to the self-energy of the electron and the photon (page 218 of ref.[1])
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![%eval(S[2], legs = 2) = %FeynmanIntegral(-2*alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(psi[k](X), conjugate(psi[i](Y))), [[A[mu](X), A[alpha](Y)], [psi[l](Y), conjugate(psi[j](X))]])-alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(A[mu](X), A[alpha](Y)), [[psi[l](Y), conjugate(psi[j](X))], [psi[k](X), conjugate(psi[i](Y))]]), [[X], [Y]])](/view.aspx?sf=211829_post/7a4a5f1d808b1f55aefc474d18e09afd.gif)
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(1.16) |
where the diagram with two spinor legs is the electron self-energy. To restrict the output furthermore, for example getting only the self-energy of the photon, you can specify the normal products you want:
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![%eval(S[2], [`=`(legs, 2), `=`(products, _NP(A, A))]) = FeynmanDiagrams(L__QED, numberofvertices = 2, numberoflegs = 2, normalproduct = _NP(A, A))](/view.aspx?sf=211829_post/eda3e5b7a5d6d7ddcce7e755c1d3bc9b.gif)
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![%eval(S[2], [legs = 2, products = _NP(A, A)]) = %FeynmanIntegral(alpha^2*Physics:-Dgamma[`~mu`][j, k]*Physics:-Dgamma[`~alpha`][i, l]*_GF(_NP(A[mu](X), A[alpha](Y)), [[conjugate(psi[j](X)), psi[l](Y)], [psi[k](X), conjugate(psi[i](Y))]]), [[X], [Y]])](/view.aspx?sf=211829_post/31b05a4b41ca8ed274952cf85da99ed9.gif)
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(1.17) |
The corresponding S-matrix elements in momentum representation
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![`#mfenced(mrow(mo("⁢"),mi("ψ",fontstyle = "normal"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("ψ",fontstyle = "normal"),mo("⁢")),open = "⟨",close = "⟩")` = FeynmanDiagrams(L__QED, incomingparticles = [psi], outgoing = [psi], numberofloops = 1, diagrams)](/view.aspx?sf=211829_post/a366e1dfcad5f8a184c9031ad9657bd4.gif)
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![`#mfenced(mrow(mo("⁢"),mi("ψ",fontstyle = "normal"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("ψ",fontstyle = "normal"),mo("⁢")),open = "⟨",close = "⟩")` = -%FeynmanIntegral((1/8)*Physics:-FeynmanDiagrams:-Uspinor[psi][i](P__1_)*conjugate(Physics:-FeynmanDiagrams:-Uspinor[psi][l](P__2_))*(-Physics:-g_[alpha, nu]+p__2[nu]*p__2[alpha]/m__A^2)*alpha^2*Physics:-Dgamma[`~alpha`][l, m]*Physics:-Dgamma[`~nu`][n, i]*((P__1[beta]+p__2[beta])*Physics:-Dgamma[`~beta`][m, n]+m__psi*Physics:-KroneckerDelta[m, n])*Dirac(-P__2+P__1)/(Pi^3*(p__2^2-m__A^2+I*Physics:-FeynmanDiagrams:-epsilon)*((P__1+p__2)^2-m__psi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])](/view.aspx?sf=211829_post/df0b06610e6fdabcff180db6f4bd3cda.gif)
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(1.18) |
In this result we see spinor (see ref.[2]), and the propagator of the field with a mass . To indicate that this field is massless use
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(1.19) |
Now the propagator for is the one of a massless vector field:
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(1.20) |
The self-energy of the photon:
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![`#mfenced(mrow(mo("⁢"),mi("A"),mo("⁢"),mo("|"),mo("⁢"),mi("S"),mo("⁢"),mo("|"),mo("⁢"),mi("A"),mo("⁢")),open = "⟨",close = "⟩")` = -%FeynmanIntegral((1/16)*Physics:-FeynmanDiagrams:-PolarizationVector[A][nu](P__1_)*conjugate(Physics:-FeynmanDiagrams:-PolarizationVector[A][alpha](P__2_))*(m__psi*Physics:-KroneckerDelta[l, n]+p__2[beta]*Physics:-Dgamma[`~beta`][l, n])*alpha^2*Physics:-Dgamma[`~alpha`][n, i]*Physics:-Dgamma[`~nu`][m, l]*((P__1[tau]+p__2[tau])*Physics:-Dgamma[`~tau`][i, m]+m__psi*Physics:-KroneckerDelta[i, m])*Dirac(-P__2+P__1)/(Pi^3*(E__1*E__2)^(1/2)*(p__2^2-m__psi^2+I*Physics:-FeynmanDiagrams:-epsilon)*((P__1+p__2)^2-m__psi^2+I*Physics:-FeynmanDiagrams:-epsilon)), [[p__2]])](/view.aspx?sf=211829_post/69bab5d47a40e33a627f1b68cd29ca3e.gif)
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(1.21) |
where is the corresponding polarization vector.
When working with non-Abelian gauge fields, the interaction Lagrangian involves derivatives. FeynmanDiagrams can handle that kind of interaction in momentum representation. Consider for instance a Yang-Mills theory with a massless field where a is a SU2 index (see eq.(12) of sec. 19.4 of ref.[1]). The interaction Lagrangian can be entered as follows
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(1.22) |
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(1.23) |
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![L := (1/2)*g*LeviCivita[a, b, c]*F__B[mu, nu, a]*B[mu, b](X)*B[nu, c](X)+(1/4)*g^2*LeviCivita[a, b, c]*LeviCivita[a, e, f]*B[mu, b](X)*B[nu, c](X)*B[mu, e](X)*B[nu, f](X)](/view.aspx?sf=211829_post/257fcce495259f66ec6f57f962ab2f61.gif)
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![(1/2)*g*Physics:-LeviCivita[a, b, c]*Physics:-`*`(Physics:-d_[mu](B[nu, a](X), [X])-Physics:-d_[nu](B[mu, a](X), [X]), B[`~mu`, b](X), B[`~nu`, c](X))+(1/4)*g^2*Physics:-LeviCivita[a, b, c]*Physics:-LeviCivita[a, e, f]*Physics:-`*`(B[mu, b](X), B[nu, c](X), B[`~mu`, e](X), B[`~nu`, f](X))](/view.aspx?sf=211829_post/893fb9d1153d8a9e1e98df402b802df7.gif)
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(1.24) |
The transition probability density at tree-level for a process with two incoming and two outgoing B particles is given by
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![FeynmanDiagrams(L, incomingparticles = [B, B], outgoingparticles = [B, B], numberofloops = 0, output = probabilitydensity, factor, diagrams)](/view.aspx?sf=211829_post/69e04f44467ba00951ce0c4a5763550d.gif)
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(1.25) |
![Physics:-FeynmanDiagrams:-ProbabilityDensity(4*Pi^2*%mul(n[i], i = 1 .. 2)*abs(F)^2*Dirac(-P__3[`~sigma`]-P__4[`~sigma`]+P__1[`~sigma`]+P__2[`~sigma`])*%mul(dP_[f]^3, f = 1 .. 2), F = (((1/8)*I)*Physics:-LeviCivita[a1, a3, h]*((-P__1[`~kappa`]-P__2[`~kappa`]-P__4[`~kappa`])*Physics:-g_[`~lambda`, `~tau`]+(P__1[`~lambda`]+P__2[`~lambda`]+P__3[`~lambda`])*Physics:-g_[`~kappa`, `~tau`]-Physics:-g_[`~kappa`, `~lambda`]*(P__3[`~tau`]-P__4[`~tau`]))*Physics:-LeviCivita[a2, d, g]*((P__1[`~beta`]+(1/2)*P__2[`~beta`])*Physics:-g_[`~alpha`, `~sigma`]+(-(1/2)*P__1[`~sigma`]+(1/2)*P__2[`~sigma`])*Physics:-g_[`~alpha`, `~beta`]-(1/2)*Physics:-g_[`~beta`, `~sigma`]*(P__1[`~alpha`]+2*P__2[`~alpha`]))*Physics:-g_[sigma, tau]*Physics:-KroneckerDelta[a2, a3]/((-P__1[chi]-P__2[chi])*(-P__1[`~chi`]-P__2[`~chi`])+I*Physics:-FeynmanDiagrams:-epsilon)-((1/16)*I)*((-P__1[`~beta`]+P__3[`~beta`]-P__4[`~beta`])*Physics:-g_[`~lambda`, `~tau`]+(P__1[`~lambda`]-P__2[`~lambda`]-P__3[`~lambda`])*Physics:-g_[`~beta`, `~tau`]+Physics:-g_[`~beta`, `~lambda`]*(P__2[`~tau`]+P__4[`~tau`]))*Physics:-LeviCivita[a1, a3, g]*((P__1[`~sigma`]+P__3[`~sigma`])*Physics:-g_[`~alpha`, `~kappa`]+(-2*P__1[`~kappa`]+P__3[`~kappa`])*Physics:-g_[`~alpha`, `~sigma`]+Physics:-g_[`~kappa`, `~sigma`]*(P__1[`~alpha`]-2*P__3[`~alpha`]))*Physics:-LeviCivita[a2, d, h]*Physics:-g_[sigma, tau]*Physics:-KroneckerDelta[a2, a3]/((-P__1[chi]+P__3[chi])*(-P__1[`~chi`]+P__3[`~chi`])+I*Physics:-FeynmanDiagrams:-epsilon)-((1/16)*I)*((-P__1[`~beta`]-P__3[`~beta`]+P__4[`~beta`])*Physics:-g_[`~kappa`, `~tau`]+(P__1[`~kappa`]-P__2[`~kappa`]-P__4[`~kappa`])*Physics:-g_[`~beta`, `~tau`]+Physics:-g_[`~beta`, `~kappa`]*(P__2[`~tau`]+P__3[`~tau`]))*Physics:-LeviCivita[a3, g, h]*((P__1[`~sigma`]+P__4[`~sigma`])*Physics:-g_[`~alpha`, `~lambda`]+(P__1[`~alpha`]-2*P__4[`~alpha`])*Physics:-g_[`~lambda`, `~sigma`]-2*Physics:-g_[`~alpha`, `~sigma`]*(P__1[`~lambda`]-(1/2)*P__4[`~lambda`]))*Physics:-LeviCivita[a1, a2, d]*Physics:-g_[sigma, tau]*Physics:-KroneckerDelta[a2, a3]/((-P__1[chi]+P__4[chi])*(-P__1[`~chi`]+P__4[`~chi`])+I*Physics:-FeynmanDiagrams:-epsilon)-((1/16)*I)*(Physics:-KroneckerDelta[g, h]*Physics:-KroneckerDelta[a1, d]*(Physics:-g_[`~alpha`, `~beta`]*Physics:-g_[`~kappa`, `~lambda`]+Physics:-g_[`~alpha`, `~kappa`]*Physics:-g_[`~beta`, `~lambda`]-2*Physics:-g_[`~alpha`, `~lambda`]*Physics:-g_[`~beta`, `~kappa`])+Physics:-KroneckerDelta[d, h]*(Physics:-g_[`~alpha`, `~beta`]*Physics:-g_[`~kappa`, `~lambda`]-2*Physics:-g_[`~alpha`, `~kappa`]*Physics:-g_[`~beta`, `~lambda`]+Physics:-g_[`~alpha`, `~lambda`]*Physics:-g_[`~beta`, `~kappa`])*Physics:-KroneckerDelta[a1, g]-2*(Physics:-g_[`~alpha`, `~beta`]*Physics:-g_[`~kappa`, `~lambda`]-(1/2)*Physics:-g_[`~beta`, `~kappa`]*Physics:-g_[`~alpha`, `~lambda`]-(1/2)*Physics:-g_[`~alpha`, `~kappa`]*Physics:-g_[`~beta`, `~lambda`])*Physics:-KroneckerDelta[d, g]*Physics:-KroneckerDelta[a1, h]))*g^2*conjugate(Physics:-FeynmanDiagrams:-PolarizationVector[B][kappa, h](P__3_))*conjugate(Physics:-FeynmanDiagrams:-PolarizationVector[B][lambda, a1](P__4_))*Physics:-FeynmanDiagrams:-PolarizationVector[B][alpha, d](P__1_)*Physics:-FeynmanDiagrams:-PolarizationVector[B][beta, g](P__2_)/(Pi^2*(E__1*E__2*E__3*E__4)^(1/2)))](/view.aspx?sf=211829_post/dd64cc8a5fc683b7a1a3ae1d1b460581.gif)
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(1.26) |
To simplify the repeated indices, us the option simplifytensorindices. To check the indices entering a result like this one use Check ; there are no free indices, and regarding the repeated indices:
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(1.27) |
This process can be computed with 1 or more loops, in which case the number of terms increases significantly. As another interesting non-Abelian model, consider the interaction Lagrangian of the electro-weak part of the Standard Model
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(1.28) |
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(1.29) |
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(1.30) |
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(1.31) |
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(1.32) |
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(1.33) |
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![L__WZ := I*g*cos(`θ__w`)*((Dagger(F__W[mu, nu])*W[mu](X)-Dagger(W[mu](X))*F__W[mu, nu])*Z[nu](X)+W[nu](X)*Dagger(W[mu](X))*F__Z[mu, nu])](/view.aspx?sf=211829_post/e16242b0e28a259e3ab36161f4b89873.gif)
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(1.34) |
This interaction Lagrangian contains six different terms. The S-matrix element for the tree-level process with two incoming and two outgoing W particles is shown in the help page for FeynmanDiagrams .
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