... | ... | @@ -27,19 +27,20 @@ In order to derive the results for the Wilson coefficients, PreSARAH needs an in |
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- `2Fermion1Vector`
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- `NameProcess` : A string to uniquely define the process
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- `ExternalFields` : The external fields. Possible names are `ChargedLepton`, `Neutrino`, `DownQuark`, `UpQuark`, `ScalarHiggs`, `PseudoScalar`, `Zboson`, `Wboson` <ref>
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The `particles.m` file is used to define for each model which particle corresponds to SM states using the `Description` statement together with `Leptons`, `Neutrinos`, `Down-Quarks`, `Up-Quarks`, `Higgs`, `Pseudo-Scalar Higgs`, `Z-Boson`, `W-Boson`. If there is a mixture between the SM particles and other states (like in*R*-parity violating SUSY or in models with additional vector quarks/leptons) the combined state has to be labeled according to the description for the SM state. Notice that in the SM `Pseudo-Scalar Higgs` is just the neutral Goldstone boson. If an external state is not present in a given model or has not been defined as such in the `particles.m` file the corresponding Wilson coefficients are not calculated by SPheno.
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The `particles.m` file is used to define for each model which particle corresponds to SM states using the `Description` statement together with `Leptons`, `Neutrinos`, `Down-Quarks`, `Up-Quarks`, `Higgs`, `Pseudo-Scalar Higgs`, `Z-Boson`, `W-Boson`. If there is a mixture between the SM particles and other states (like in *R*-parity violating SUSY or in models with additional vector quarks/leptons) the combined state has to be labeled according to the description for the SM state. Notice that in the SM `Pseudo-Scalar Higgs` is just the neutral Goldstone boson. If an external state is not present in a given model or has not been defined as such in the `particles.m` file the corresponding Wilson coefficients are not calculated by SPheno.
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</ref>
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- `FermionOrderExternal` : the fermion order to apply the Fierz transformation (see the FormCalc manual for more details)
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- `NeglectMasses` : which external masses can be neglected (a list of integers counting the external fields)
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- `ColorFlow` : defines the color flow in the case of four quark operators. To contract the colors of external fields, `ColorDelta` is used, i.e `ColorFlow = ColorDelta[1,2]*ColorDelta[3,4]` assigns(*q̄*<sup>*α*</sup>*Γ**q*<sub>*α*</sub>)(*q̄*<sup>*β*</sup>*Γ*′*q*<sub>*β*</sub>).
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- `ColorFlow` : defines the color flow in the case of four quark operators. To contract the colors of external fields, `ColorDelta` is used, i.e `ColorFlow = ColorDelta[1,2]*ColorDelta[3,4]` assigns $`(\bar{q}^\alpha \Gamma
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q_\alpha)(\bar{q}^\beta \Gamma' q_\beta)`$.
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- `AllOperators` : a list with the definition of the operators. This is a two dimensional list, where the first entry defines the name of the operator and the second one the Lorentz structure. The operators are expressed in the chiral basis and the syntax for Dirac chains in FormCalc is used:
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- `6` for$P_L = \\frac{1}{2}(1-\\gamma_5)$, `7` for$P_R = \\frac{1}{2}(1-\\gamma_5)$
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- `Lor[1]`, `Lor[2]` for*γ*<sub>*μ*</sub>,*γ*<sub>*ν*</sub>
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- `6` for$`P_L = \frac{1}{2}(1-\gamma_5)`$, `7` for$`P_R = \frac{1}{2}(1-\gamma_5)`$
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- `Lor[1]`, `Lor[2]` for $`\gamma_\mu`$, $`\gamma_\nu`$.
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- `ec[3]` for the helicity of an external gauge boson.
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- `k[N]` for the momentum of the external particle `N` (`N` is an integer).
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- `Pair[A,B]` is used to contract Lorentz indices. For instance, `Pair[k[1],ec[3]]` stands for*k*<sub>*μ*</sub><sup>1</sup>*ϵ*<sup>*μ*, \*</sup>
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- `Pair[A,B]` is used to contract Lorentz indices. For instance, `Pair[k[1],ec[3]]` stands for *k*<sub>*μ*</sub><sup>1</sup>*ϵ*<sup>*μ*, \*</sup>
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- A Dirac chain starting with a negative first entry is taken to be anti-symmetrized.
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See the FormCalc manual for more details.
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... | ... | @@ -58,7 +59,7 @@ In order to derive the results for the Wilson coefficients, PreSARAH needs an in |
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- `Filters = {NoPenguins}` might be useful for processes which at the 1-loop level are only induced by box diagrams
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- `Filters = {NoCrossedDiagrams}` is used to drop diagrams which only differ by a permutation of the external fields.
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For instance, the PreSARAH input to calculate the coefficient of the$(\\bar{\\ell}\\Gamma \\ell)(\\bar{d} \\Gamma' d)$ operator reads
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For instance, the PreSARAH input to calculate the coefficient of the$`(\bar{\ell}\Gamma \ell)(\bar{d} \Gamma' d)`$ operator reads
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NameProcess="2L2d";
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ConsideredProcess = "4Fermion";
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... | ... | @@ -94,13 +95,12 @@ For instance, the PreSARAH input to calculate the coefficient of the$(\\bar{\\el |
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Filters = {};
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Here, we neglect all external masses in the operators (`NeglectMasses={1,2,3,4}` ), and the different coefficients of the scalar operators$(\\bar{\\ell}P_X \\ell)(\\bar{d} P_Y d)$ are called `OllddSXY`, the ones for the vector operators$(\\bar{\\ell} P_X
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\\gamma_\\mu \\ell)(\\bar{d} P_Y \\gamma^\\mu d)$ are called `OllddVYX` and the ones for the tensor operators$(\\bar{\\ell} P_X \\sigma_{\\mu
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\\nu} \\ell)(\\bar{d} \\sigma^{\\mu \\nu} P_Y d)$ `OllddTYX`, with X,Y=L,R. Notice that FormCalc returns the results in form of*P*<sub>*X*</sub>*γ*<sub>*μ*</sub> while in the literature the order*γ*<sub>*μ*</sub>*P*<sub>*X*</sub> is often used. Finally, SPheno will not calculate all possible combinations of external states, but only some specific cases
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*μ**e**d**d*
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,*τ**e**d**d*,*τ**μ**d**d*,*μ**e**s**s*,*τ**e**s**s*,*τ**μ**s**s* [1].
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Here, we neglect all external masses in the operators (`NeglectMasses={1,2,3,4}` ), and the different coefficients of the scalar operators $`(\bar{\ell}P_X \ell)(\bar{d} P_Y d)`$ are called `OllddSXY`, the ones for the vector operators $`(\bar{\ell} P_X
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\gamma_\mu \ell)(\bar{d} P_Y \gamma^\mu d)`$ are called `OllddVYX` and the ones for the tensor operators$`(\bar{\ell} P_X \sigma_{\mu
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\nu} \ell)(\bar{d} \sigma^{\mu \nu} P_Y d)`$ `OllddTYX`, with X,Y=L,R. Notice that FormCalc returns the results in form of*P*<sub>*X*</sub>*γ*<sub>*μ*</sub> while in the literature the order*γ*<sub>*μ*</sub>*P*<sub>*X*</sub> is often used. Finally, SPheno will not calculate all possible combinations of external states, but only some specific cases
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$`\mu e d d`$, $`\tau e d d`$, $`\tau \mu d d`$, $`\mu e s s`$, $`\tau e s s`$, $`\tau \mu s s`$ [1]. Here we used $`d`$ for the first generation of down-type quarks while in the rest of this manual it is used to summarize all three families. [2]
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The input file to calculate the coefficients of theℓ − ℓ − *Z* operators$(\\bar{\\ell} \\gamma_\\mu P_{L,R} \\ell) Z^\\mu$ and$(\\bar{\\ell} p_\\mu P_{L,R} \\gamma_\\mu \\ell) Z^\\mu$ is
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The input file to calculate the coefficients of the ℓ − ℓ − *Z* operators $`(\bar{\ell} \gamma_\mu P_{L,R} \ell) Z^\mu$ and $`(\bar{\ell} p_\mu P_{L,R} \gamma_\mu \ell) Z^\mu`$ is
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NameProcess="Z2l";
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... | ... | @@ -147,7 +147,7 @@ Furthermore, for the dipole operators, defined by |
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we are using the results obtained by FeynArts and FormCalc and have implemented all special cases for the involved loop integrals (</math>C_0, C_{00}, C_1, C_2, C_{11}, C_{12}, C_{22}</math>) with identical or vanishing internal masses in SPheno. This guarantees the numerical stability of the results [2].
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The monopole operators of the form*q*<sup>2</sup>(*f̄**γ*<sub>*μ*</sub>*f*)*V*<sup>*μ*</sup> are only non-zero for off-shell external gauge bosons, while PreSARAH always treats all fields as on-shell. Because of this, and to stabilize the numerical evaluation later on, these operators are treated differently to all other operators: the coefficients are not calculated by FeynArts and FormCalc but instead we have included the generic expressions in PreSARAH using a special set of loop functions in SPheno. In these loop functions the resulting Passarino-Veltman integrals are already combined, leading to well-known expressions in the literature, see . They have been cross-checked with the package `Peng4BSM@LO`. To get the coefficients for the monopole operators, these have to be given always in the form
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The monopole operators of the form *q*<sup>2</sup>(*f̄**γ*<sub>*μ*</sub>*f*)*V*<sup>*μ*</sup> are only non-zero for off-shell external gauge bosons, while PreSARAH always treats all fields as on-shell. Because of this, and to stabilize the numerical evaluation later on, these operators are treated differently to all other operators: the coefficients are not calculated by FeynArts and FormCalc but instead we have included the generic expressions in PreSARAH using a special set of loop functions in SPheno. In these loop functions the resulting Passarino-Veltman integrals are already combined, leading to well-known expressions in the literature, see . They have been cross-checked with the package `Peng4BSM@LO`. To get the coefficients for the monopole operators, these have to be given always in the form
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{MonopoleL,Op[6,ec[3]] Pair[k[3],k[3]]},
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{MonopoleR,Op[7,ec[3]] Pair[k[3],k[3]]}
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... | ... | @@ -197,4 +197,4 @@ Notes |
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[1] Here we used*d* for the first generation of down-type quarks while in the rest of this manual it is used to summarize all three families.
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[2] We note that the coefficients for the operators defined above (</math>\\bar{f} \\gamma_\\mu f \\, V^\\mu</math>) are by a factor of 2 (4) larger than the coefficients of the standard definition for the dipole operators*f̄**σ*<sub>*μ**ν*</sub>*P*<sub>*L*</sub>*f**q*<sup>*ν*</sup>*V*<sup>*μ*</sup> (</math>\\bar{f} \\sigma_{\\mu\\nu}P_L f F^{\\mu\\nu}</math>). |
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\ No newline at end of file |
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[2] We note that the coefficients for the operators defined above ($`\bar{f} \gamma_\mu f \, V^\mu`$) are by a factor of 2 (4) larger than the coefficients of the standard definition for the dipole operators *f̄**σ*<sub>*μ**ν*</sub>*P*<sub>*L*</sub>*f**q*<sup>*ν*</sup>*V*<sup>*μ*</sup> ($`\bar{f} \sigma_{\mu\nu}P_L f F^{\mu\nu}`$). |
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\ No newline at end of file |