DsixTools: the standard model effective field theory toolkit

The European Physical Journal C, Jun 2017

We present DsixTools, a Mathematica package for the handling of the dimension-six standard model effective field theory. Among other features, DsixTools allows the user to perform the full one-loop renormalization group evolution of the Wilson coefficients in the Warsaw basis. This is achieved thanks to the SMEFTrunner module, which implements the full one-loop anomalous dimension matrix previously derived in the literature. In addition, DsixTools also contains modules devoted to the matching to the \(\Delta B = \Delta S = 1,2\) and \(\Delta B = \Delta C = 1\) operators of the Weak Effective Theory at the electroweak scale, and their QCD and QED Renormalization group evolution below the electroweak scale.

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DsixTools: the standard model effective field theory toolkit

Eur. Phys. J. C DsixTools: the standard model effective field theory toolkit Alejandro Celis 2 Javier Fuentes-Martín 1 Avelino Vicente 1 Javier Virto 0 0 Albert Einstein Center for Fundamental Physics, Institute for Theoretical Physics, University of Bern , 3012 Bern , Switzerland 1 Instituto de Física Corpuscular, Universitat de València-CSIC , 46071 Valencia , Spain 2 Fakultät für Physik, Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universität München , 80333 Munich , Germany We present DsixTools, a Mathematica package for the handling of the dimension-six standard model effective field theory. Among other features, DsixTools allows the user to perform the full one-loop renormalization group evolution of the Wilson coefficients in the Warsaw basis. This is achieved thanks to the SMEFTrunner module, which implements the full one-loop anomalous dimension matrix previously derived in the literature. In addition, DsixTools also contains modules devoted to the matching to the B = S = 1, 2 and B = C = 1 operators of the Weak Effective Theory at the electroweak scale, and their QCD and QED Renormalization group evolution below the electroweak scale. - Contents 1 Introduction 2 DsixTools in a nutshell 4.1 SMEFTrunner module Options.dat SMInput.dat BlockWC1 QG = f ABC GμAν GνBρ GCμ ρ QG = f ABC GμAν GνBρ GCμ ρ QW = εI J K WμI ν WνJρ W K μ ρ QW = εI J K WμI ν WνJρ W K μ ρ BlockWC2 Qϕ = BlockWC3 ϕ†ϕ Qϕ D = · · · BlockWCUPHI Quϕ [1, 1] = Quϕ [1, 2] = = ϕ†ϕ ϕ†ϕ ϕ† Dμϕ ∗ ϕ† Dμϕ ϕ†ϕ (q¯1u1ϕ) ϕ†ϕ (q¯1u2ϕ) Quϕ [3, 3] = ϕ†ϕ (q¯3u3ϕ) βi Ci (μ) = Ci ( ) + 16π 2 log Fig. 4 EWmatchermoduleflowchart W r i t e W C s M a s s B a s i s O u t p u t F i l e [ O u t p u t _ f i l e ] A p p l y E W m a t c h i n g 4.3 WETrunner module s c a l e [ G e V ] WCsInput.dat B l o c k S C A L E S 2 8 0 . 3 8 5 # EW B l o c k B S 2 1 0 . 0 0 0 0 0 0 # C 1 s b 2 0 . 0 0 0 0 0 0 # C 2 s b 3 0 . 0 0 0 0 0 0 # C 3 s b 4 1 . 0 0 0 0 0 0 # C 4 s b 5 0 . 0 0 0 0 0 0 # C 5 s b 6 0 . 0 0 0 0 0 0 # C1sb ’ 7 0 . 0 0 0 0 0 0 # C2sb ’ 8 0 . 0 0 0 0 0 0 # C3sb ’ B l o c k B C 1 1 0 . 0 0 0 0 0 0 # CV ( e ) 2 0 . 0 0 0 0 0 0 # CV ( mu ) 3 0 . 0 0 0 0 0 0 # CV ( t a u ) 4 0 . 0 0 0 0 0 0 # CS ( e ) 5 0 . 0 0 0 0 0 0 # CS ( mu ) 6 0 . 0 0 0 0 0 0 # CS ( t a u ) 7 0 . 0 0 0 0 0 0 # CT ( e ) 8 0 . 0 0 0 0 0 0 # CT ( mu ) 9 0 . 0 0 0 0 0 0 # CT ( t a u ) 10 0 . 0 0 0 0 0 0 # CV ( e ) ’ 11 0 . 0 0 0 0 0 0 # CV ( mu ) ’ 12 0 . 0 0 0 0 0 0 # CV ( t a u ) ’ 13 0 . 0 0 0 0 0 0 # CS ( e ) ’ 14 0 . 0 0 0 0 0 0 # CS ( mu ) ’ 15 0 . 0 0 0 0 0 0 # CS ( t a u ) ’ B l o c k B S 1 H 1 0 . 0 0 0 0 0 0 # C1 ( s b u u ) . . . Fig. 5 WETrunner module flowchart Options.dat 9 2 # T y p e of i n p u t ( S M E F T ) or 2 ( W E T ) W C s : 1 5 Summary L = LS(M4)+ C (5) Q(5) k k + C (6) Q(6) k k +O 1 WμI ν W I μν − 4 λ + m2ϕ†ϕ − 2 ϕ†ϕ Table 1 Purely bosonic operators + i ¯D/ + e¯ D/ e + q¯ D/ q + u¯ D/ u + d¯ D/ d θ g 2 Lθ = 32π 2 Bμν Bμν θs gs2 GμAν GμAν , + 32π 2 calculated in [3]. The dual tensors are defined as X = 1 2 εμνρσ X ρσ (with ε0123 = +1). dCi 1 d log μ = 16π 2 1 γi j C j ≡ 16π 2 βi . † 3 Tr Cuϕ u † + 3 Tr Cdϕ d η2 = −6 Tr C ϕ(3q) u u † − 6 Tr C ϕ(3q) d d † † + 3 Tr Cϕud d u + c.c. , η3 = 3 Tr C ϕ(1q) d d † − 3 Tr C ϕ(1q) u u † + 9 Tr C ϕ(3q) d d † + 9 Tr C ϕ(3q) u u † † + 3 Tr Cϕu u u † − 3 Tr Cϕd d d † − 3 Tr Cϕud d u + c.c. + Tr C ϕ(1) e e† + 3 Tr C ϕ(3) e e† η4 = 12 Tr C ϕ(1q) d d † − 12 Tr C ϕ(1q) u u † † + 12 Tr Cϕu u u † − 12 Tr Cϕd d d † + 6 Tr Cϕud d u + c.c. + 4 Tr C ϕ(1) e e† † − 4 Tr Cϕe e e , 3 − c.c. − 2 i Tr − c.c. , Qduq Qqque Qqqq Qduue Table 5 Hypercharge assignments Baryon-number-violating d T Cu q T C q T Cq uT Ce εil ε jk qiT Cq j qkT C l d T Cu uT Ce −1 − c.c. as well as + Cϕ D † † u u + d d rs 4 C(8) Cq(1u)qd + 3 quqd 4 C(8) Cq(1u)qd + 3 quqd 4 [ξu ]pt = 2 Cq(1u) + 3 Cq(8u) βG = 15 gs2 CG , βG = 15 gs2 CG , g2 CW , (B.21) (B.22) (B.23) (B.24) (B.25) (B.10) (B.11) (B.12) (B.13) (B.14) [ u ]s∗r (B.15) [ d ]s∗r (B.16) (B.17) (B.18) (B.19) (B.20) βϕ = − 2 ϕ4 D2 + 2g2 Tr C(3) ϕ + 3 Tr Cϕ(3q) X2ϕ2 − 2 − 2 9 2 g − 14gs2 CϕG + 6 λCϕG 9 g2 CϕB + 3gg CϕW B + 6 λCϕB − 2 + g 5Tr CuW u† +Tr CdW d† † −ig −5Tr CuB u† +Tr CdB d (B.26) (B.27) (B.28) (B.29) (B.30) (B.31) − 136 Cq(8u) +6 Cq(1u)qd −2 d d† dCϕud +8 Cq(1u) + 43Cq(8u) † rs e e e +4 Cuϕ u† u † 4C(8) −2 Cq(1u)qd + 3 quqd −12 Cq(1u)qd † +5 u u†Cuϕ −2 dCdϕ u − Cdϕ d† u −2 d dCuϕ † 35g 2 + 4 27g2 +8gs2 Cuϕ − 12 10 2 g Cϕ + 23 g 2 − g2 CϕD βdϕ = 3 rs 23g 2 + 4 27g2 +8gs2 Cdϕ − 12 prvt prsv − 4 Cqque + 2 Cqqq + 2 Cqqq + 2 Cqqq + 2 Cqqq prwv − 4g2 Cqqq + Cqqq Cqqq u u† + d d† u u† + d d† u u† + d d† (B.77) (B.78) Cduq vwst + Cduue + 4 Cqqq pwrv wvst † † u u + d d Cqque + Cduue vswt vrwt pvwt − Cduq − Cduq + 2 Cduue + 4 Cqqq − Cqqq − Cqqq + 2 Cqque + 2 Cduq + 2 Cduq vswt wsvt vrst pvst wsrv prvt prsv − Cqque + 2 Cduq pwvt vspw − 2 Cqqq − 2 Cqqq + 2 Cqque wprv vrst prwv + Cqqq u u† + d d † pwrv + p ↔ r Baryon-number-violating 1 g 2 + 4gs2 = − 3g2 + 3 (B.76) = − 2g 2 + 4gs2 Cduue − 8 Cqque prwv d X 1 dt ≡ 16π 2 βX . these are given by 19 3 βg = − 6 g − 8 g βg = 461 g 3 − 8 g 2 CϕW , 2 Cϕ B , (B.80) (B.81) (B.82) βgs = −7gs3 − 8 gs 2 CϕG , + Tr e e† e e† − 16λ + 130 g2 Cϕ rs − 49 g2 − 152 g 2 − 8gs2 + 3 C ϕ(3q)† d Cϕ D − 2Cϕ (B.83) (B.84) (B.85) (B.86) Cq(1u) ptrs Cϕ D − 2Cϕ rs − 49 g2 − 1127 g 2 − 8gs2 + 3 C ϕ(3q)† u Cϕ D − 2Cϕ − 2 C (1eq)u Cq(1u)qd + 3 Cq(1u)qd u Cϕ†u Cq(8u) Cq(8u)qd r pts rspt + 3 Cq(1u)qd Cq(8d) Cq(8u)qd r pts ptrs r pts u Cϕud − 2 C ∗eqd Cq(1d) Cq(1u)qd r pts + Cϕ(1)† e + 3 Cϕ(3)† e eCϕ†e + 3 C edq rspt − 2 C e Cϕ D − 2Cϕ r pts − 3 C(1eq)u rspt (B.87) (B.89) (B.90) 128π 2 m2 g 2 2 Cϕ B , 128π 2 m2 g2 2 CϕW , Ci Oi +h.c. , B = S = 2 operators In the case of B = O1sbsb = (s¯γμ PL b) (s¯γ μ PL b), O5sbsb = (s¯α PL bβ ) (s¯β PR bα) , O2sbsb = (s¯ PL b) (s¯ PL b), O1sbsb = (s¯γμ PR b) (s¯γ μ PR b), O3sbsb = (s¯α PL bβ ) (s¯β PL bα) , O2sbsb = (s¯ PR b) (s¯ PR b), O4sbsb = (s¯ PL b) (s¯ PR b), O3sbsb = (s¯α PR bβ ) (s¯β PR bα), B = C = 1 operators The basis for the B = = c PR γ μ b ¯ γμ ν , Ocb 5 = c PL γ μ b ¯ γμ ν , Ocb 5 B = S = 1 operators = (c PR b) ¯ ν , = (c PL b) ¯ ν , • Magnetic penguins: Oisbqq = Oisbqq PL,R→PR,L O1sbss = (s¯ γμ PL b) (s¯γ μ s) , O1sbss = (s¯ γμ PR b) (s¯γ μ s) , O3sbss = (s¯ γμνρ PL b) (s¯γ μνρ s) , O3sbss = (s¯ γμνρ PR b) (s¯γ μνρ s) , O5sbss = (s¯ PL b)(s¯ s) , O5sbss = (s¯ PR b)(s¯ s) , O7sbss = (s¯ σ μν PL b)(s¯ σμν s) , O7sbss = (s¯ σ μν PR b)(s¯ σμν s) , O9sbss = (s¯ γμνρσ PL b) (s¯γ μνρσ s) , O9sbss = (s¯ γμνρσ PR b) (s¯γ μνρσ s) . • Semileptonic: LoadModule[moduleName] TurnOffMessages ReadInputFiles[options_file,WCsInput_file,{SMInput_ file}] Reads all input files. MyPrint[string] WriteInputFiles[options_file,WCsInput_file,{SMInput_ file},data] TurnOnMessages NewScale[scale,newvalue] H[mat] CC[x] D.2: SMEFTrunner routines InitializeSMEFTrunnerInput RunRGEsSMEFT Runs the SMEFT RGEs. GetBeta LoadBetaFunctions Arguments: parameter must be the name of a WET WC. RotateToMassBasis Biunitary[mat,dim] ApplyEWmatching Match[WC] Example: Match[CBS1[d][1]] prints the numerical value of the C sbdd WC. 1 MatchAnalytical[WC] Example: Match[CBS1[d][1]] prints the analytical expression of the C sbdd WC. 1 ExportEWmatcher Exports the EWmatcher results in data to Output_ file. D.4: WETrunner routines InitializeWETrunnerInput RunRGEsWET Runs the WET RGEs. ExportWETrunner Example: E.1: SMEFT parameters Table 6 SMEFT parameters. Position denotes the position of the parameter (or parameters for 2- and 4-fermion objects) in the Parameters global array. The column β function gives the name of the β function (obtained with β[parameter] after using the GetBeta routine). Type indicates the type of parameter (with nF standing for n-fermion) and Category denotes the index symmetry category of the coefficient, being relevant for 2- and 4-fermion WCs Parameter(s) DsixTools name Category Position 156–161 162–167 168–173 174–179 180–185 186–191 192–200 CW Cϕ(3q) WC[ϕl3] WC[ϕe] WC[ϕq1] WC[ϕq3] WC[ϕu] WC[ϕd] WC[ϕud] Elements ϕL3[i, j] ϕE[i, j] ϕQ1[i, j] ϕQ3[i, j] ϕU[i, j] ϕD[i, j] ϕUD[i, j] Position 201–227 228–254 255–281 282–326 327–371 372–392 393–419 420–446 447–491 492–536 537–581 582–626 627–671 672–716 717–761 762–806 807–851 852–896 897–941 942–986 987–1067 1068–1148 1149–1229 1230–1310 1311–1391 1392–1472 1473–1526 1527–1583 1584–1664 Cq(3q) Cu(8d) Cq(8u) Cq(1d) Cq(8d) Cq(8u)qd C(1eq)u C(3eq)u WC[ll] WC[qq1] WC[qq3] WC[lq1] WC[qd8] WC[ledq] WC[quqd1] WC[quqd8] WC[lequ1] WC[lequ3] WC[duql] WC[qque] WC[qqql] WC[duue] LL[i, j, k, l] QQ1[i, j, k, l] QQ3[i, j, k, l] LQ1[i, j, k, l] LQ3[i, j, k, l] EE[i, j, k, l] UU[i, j, k, l] DD[i, j, k, l] EU[i, j, k, l] ED[i, j, k, l] UD1[i, j, k, l] UD8[i, j, k, l] LE[i, j, k, l] LU[i, j, k, l] LD[i, j, k, l] QE[i, j, k, l] QU1[i, j, k, l] QU8[i, j, k, l] QD1[i, j, k, l] QD8[i, j, k, l] LEDQ[i, j, k, l] QUQD1[i, j, k, l] QUQD8[i, j, k, l] LEQU1[i, j, k, l] LEQU3[i, j, k, l] DUQL[i, j, k, l] QQUE[i, j, k, l] QQQL[i, j, k, l] DUUE[i, j, k, l] β[lequ3] β[duql] β[qque] β[qqql] β[duue] Table 6 continued Parameter(s) DsixTools name Elements Category Category Meaning 0F scalar object 2F general 3 × 3 matrix 2F Hermitian matrix 4F general 3 × 3 × 3 × 3 object 4F two identical ψ¯ ψ currents 4F two independent ψ¯ ψ currents 4F two identical ψ¯ ψ currents – special case Cee 4F Baryon-number-violating – special case Cqque 4F Baryon-number-violating – special case Cqqql {1, 1} {1, 2} {1, 3} {2, 1} {2, 2} {2, 3} {3, 1} {3, 2} {3, 3} {1, 1, 1, 1} {1, 1, 1, 2} {1, 1, 1, 3} {1, 1, 2, 1} {1, 1, 2, 2} {1, 1, 2, 3} {1, 1, 3, 1} {1, 1, 3, 2} {1, 1, 3, 3} {1, 2, 1, 1} {1, 2, 1, 2} {1, 2, 1, 3} {1, 2, 2, 1} {1, 2, 2, 2} {1, 2, 2, 3} {1, 2, 3, 1} {1, 2, 3, 2} {1, 2, 3, 3} {1, 3, 1, 1} {1, 3, 1, 2} {1, 3, 1, 3} {1, 3, 2, 1} {1, 3, 2, 2} {1, 3, 2, 3} {1, 3, 3, 1} {1, 3, 3, 2} {1, 3, 3, 3} {2, 1, 1, 1} {2, 1, 1, 2} {2, 1, 1, 3} {2, 1, 2, 1} {2, 1, 2, 2} {2, 1, 2, 3} {2, 1, 3, 1} {2, 1, 3, 2} {2, 1, 3, 3} {2, 2, 1, 1} {2, 2, 1, 2} {2, 2, 1, 3} {2, 2, 2, 1} {2, 2, 2, 2} {2, 2, 2, 3} {2, 2, 3, 1} {2, 2, 3, 2} {2, 2, 3, 3} {1,1,1,1} {1, 1, 1, 2} {1, 1, 1, 3} {1,1,2,2} {1, 1, 2, 3} {1,1,3,3} {1, 2, 1, 2} {1, 2, 1, 3} {1,2,2,1} {1, 2, 2, 2} {1, 2, 2, 3} {1, 2, 3, 1} {1, 2, 3, 2} {1, 2, 3, 3} {1, 3, 1, 3} {1, 3, 2, 2} {1, 3, 2, 3} {1,3,3,1} {1, 3, 3, 2} {1, 3, 3, 3} {2,2,2,2} {2, 2, 2, 3} {2,2,3,3} {2, 3, 2, 3} {2,3,3,2} {2, 3, 3, 3} {3,3,3,3} {1,1,1,1} {1, 1, 1, 2} {1, 1, 1, 3} {1,1,2,2} {1, 1, 2, 3} {1,1,3,3} {1, 2, 1, 1} {1, 2, 1, 2} {1, 2, 1, 3} {1, 2, 2, 1} {1, 2, 2, 2} {1, 2, 2, 3} {1, 2, 3, 1} {1, 2, 3, 2} {1, 2, 3, 3} {1, 3, 1, 1} {1, 3, 1, 2} {1, 3, 1, 3} {1, 3, 2, 1} {1, 3, 2, 2} {1, 3, 2, 3} {1, 3, 3, 1} {1, 3, 3, 2} {1, 3, 3, 3} {2,2,1,1} {2, 2, 1, 2} {2, 2, 1, 3} {2,2,2,2} {2, 2, 2, 3} {2,2,3,3} {2, 3, 1, 1} {2, 3, 1, 2} {2, 3, 1, 3} {2, 3, 2, 1} {2, 3, 2, 2} {2, 3, 2, 3} {2, 3, 3, 1} {2, 3, 3, 2} {2, 3, 3, 3} {3,3,1,1} {3, 3, 1, 2} {3, 3, 1, 3} {3,3,2,2} {3, 3, 2, 3} {3,3,3,3} {1,1,1,1} {1, 1, 1, 2} {1, 1, 1, 3} {1,1,2,2} {1, 1, 2, 3} {1,1,3,3} {1, 2, 1, 2} {1, 2, 1, 3} {1, 2, 2, 2} {1, 2, 2, 3} {1, 2, 3, 2} {1, 2, 3, 3} {1, 3, 1, 3} {1, 3, 2, 3} {1, 3, 3, 3} {2,2,2,2} {2, 2, 2, 3} {2,2,3,3} {2, 3, 2, 3} {2, 3, 3, 3} {3,3,3,3} {1, 1, 1, 1} {1, 1, 1, 2} {1, 1, 1, 3} {1, 1, 2, 1} {1, 1, 2, 2} {1, 1, 2, 3} {1, 1, 3, 1} {1, 1, 3, 2} {1, 1, 3, 3} {1, 2, 1, 1} {1, 2, 1, 2} {1, 2, 1, 3} {1, 2, 2, 1} {1, 2, 2, 2} {1, 2, 2, 3} {1, 2, 3, 1} {1, 2, 3, 2} {1, 2, 3, 3} {1, 3, 1, 1} {1, 3, 1, 2} {1, 3, 1, 3} {1, 3, 2, 1} {1, 3, 2, 2} {1, 3, 2, 3} {1, 3, 3, 1} {1, 3, 3, 2} {1, 3, 3, 3} {2, 2, 1, 1} {2, 2, 1, 2} {2, 2, 1, 3} {2, 2, 2, 1} {2, 2, 2, 2} {2, 2, 2, 3} {2, 2, 3, 1} {2, 2, 3, 2} {2, 2, 3, 3} {2, 3, 1, 1} {2, 3, 1, 2} {2, 3, 1, 3} {2, 3, 2, 1} {2, 3, 2, 2} {2, 3, 2, 3} {2, 3, 3, 1} {2, 3, 3, 2} {2, 3, 3, 3} {1, 1, 1, 1} {1, 1, 1, 2} {1, 1, 1, 3} {1, 1, 2, 1} {1, 1, 2, 2} {1, 1, 2, 3} {1, 1, 3, 1} {1, 1, 3, 2} {1, 1, 3, 3} {1, 2, 1, 1} {1, 2, 1, 2} {1, 2, 1, 3} {1, 2, 2, 1} {1, 2, 2, 2} {1, 2, 2, 3} {1, 2, 3, 1} {1, 2, 3, 2} {1, 2, 3, 3} {1, 3, 1, 1} {1, 3, 1, 2} {1, 3, 1, 3} {1, 3, 2, 1} {1, 3, 2, 2} {1, 3, 2, 3} {1, 3, 3, 1} {1, 3, 3, 2} {1, 3, 3, 3} {2, 1, 2, 1} {2, 1, 2, 2} {2, 1, 2, 3} {2, 1, 3, 1} {2, 1, 3, 2} {2, 1, 3, 3} {2, 2, 2, 1} {2, 2, 2, 2} {2, 2, 2, 3} {2, 2, 3, 1} {2, 2, 3, 2} {2, 2, 3, 3} {2, 3, 1, 1} {2, 3, 1, 2} {2, 3, 1, 3} {2, 3, 2, 1} {2, 3, 2, 2} {2, 3, 2, 3} Table 7 continued {2, 3, 1, 1} {2, 3, 1, 2} {2, 3, 1, 3} {2, 3, 2, 1} {2, 3, 2, 2} {2, 3, 2, 3} {2, 3, 3, 1} {2, 3, 3, 2} {2, 3, 3, 3} {3, 1, 1, 1} {3, 1, 1, 2} {3, 1, 1, 3} {3, 1, 2, 1} {3, 1, 2, 2} {3, 1, 2, 3} {3, 1, 3, 1} {3, 1, 3, 2} {3, 1, 3, 3} {3, 2, 1, 1} {3, 2, 1, 2} {3, 2, 1, 3} {3, 2, 2, 1} {3, 2, 2, 2} {3, 2, 2, 3} {3, 2, 3, 1} {3, 2, 3, 2} {3, 2, 3, 3} {3, 3, 1, 1} {3, 3, 1, 2} {3, 3, 1, 3} {3, 3, 2, 1} {3, 3, 2, 2} {3, 3, 2, 3} {3, 3, 3, 1} {3, 3, 3, 2} {3, 3, 3, 3} {3, 3, 1, 1} {3, 3, 1, 2} {3, 3, 1, 3} {3, 3, 2, 1} {3, 3, 2, 2} {3, 3, 2, 3} {3, 3, 3, 1} {3, 3, 3, 2} {3, 3, 3, 3} E.2: WET parameters {2, 3, 3, 1} {2, 3, 3, 2} {2, 3, 3, 3} {3, 1, 3, 1} {3, 1, 3, 2} {3, 1, 3, 3} {3, 2, 3, 1} {3, 2, 3, 2} {3, 2, 3, 3} {3, 3, 3, 1} {3, 3, 3, 2} {3, 3, 3, 3} Table 8 WET parameters. Position denotes the position of the parameter in the WETParameters global array Position Parameter DsixTools name Position Parameter DsixTools name Csbsb Csbsb Csbsb Csbsb Ccbee Ccbee Ccbee Csbuu CBS2[1] CBS2[2] CBS2[3] CBS2[4] CBS2[5] CBS2p[1] CBS2p[2] CBS2p[3] CBC1[e][1] CBC1[e][5] CBC1p[e][1] CBC1p[e][5] CBC1p[e][7] CBC1[μ][1] CBC1[μ][5] CBC1p[μ][1] CBC1p[μ][5] CBC1p[μ][7] CBC1[τ][1] CBC1[τ][5] CBC1p[τ][1] CBC1p[τ][5] CBC1p[τ][7] CBS1[u][1] CBS1[u][2] CBS1[u][3] CBS1[u][4] CBS1[u][5] CBS1[u][6] CBS1[u][7] CBS1[u][8] CBS1[u][9] CBS1[u][10] CBS1[d][1] CBS1[d][2] CBS1[d][3] CBS1[d][4] CBS1[d][5] CBS1[d][6] CBS1[d][7] CBS1[d][8] CBS1[d][9] CBS1[d][10] Csbee Csbuu Csbuu Csbuu Csbuu Csbdd Csbdd Csbcc Csbcc CBS1[e][9] CBS1[μ][1] CBS1[μ][3] CBS1[μ][5] CBS1[μ][7] CBS1[μ][9] CBS1[τ][1] CBS1[τ][3] CBS1[τ][5] CBS1[τ][7] CBS1[τ][9] CBS1p[u][1] CBS1p[u][2] CBS1p[u][3] CBS1p[u][4] CBS1p[u][5] CBS1p[u][6] CBS1p[u][7] CBS1p[u][8] CBS1p[u][9] CBS1p[u][10] CBS1p[d][1] CBS1p[d][2] CBS1p[d][3] CBS1p[d][4] CBS1p[d][5] CBS1p[d][6] CBS1p[d][7] CBS1p[d][8] CBS1p[d][9] CBS1p[d][10] CBS1p[c][1] CBS1p[c][2] CBS1p[c][3] CBS1p[c][4] CBS1p[c][5] CBS1p[c][6] CBS1p[c][7] CBS1p[c][8] CBS1p[c][9] CBS1p[c][10] CBS1p[s][1] CBS1p[s][3] Position Parameter DsixTools name Position Parameter DsixTools name CBS1[c][1] CBS1[c][2] CBS1[c][3] CBS1[c][4] CBS1[c][5] CBS1[c][6] CBS1[c][7] CBS1[c][8] CBS1[c][9] CBS1[c][10] CBS1[s][1] CBS1[s][3] CBS1[s][5] CBS1[s][7] CBS1[s][9] CBS1[b][1] CBS1[b][3] CBS1[b][5] CBS1[b][7] CBS1[b][9] CBS1[M][7] CBS1[M][8] CBS1[e][1] CBS1[e][3] CBS1[e][5] CBS1[e][7] Csbss Csbss Csbee Csbee Csbee CBS1p[s][5] CBS1p[s][7] CBS1p[s][9] CBS1p[b][1] CBS1p[b][3] CBS1p[b][5] CBS1p[b][7] CBS1p[b][9] CBS1p[M][7] CBS1p[M][8] CBS1p[e][1] CBS1p[e][3] CBS1p[e][5] CBS1p[e][7] CBS1p[e][9] CBS1p[μ][1] CBS1p[μ][3] CBS1p[μ][5] CBS1p[μ][7] CBS1p[μ][9] CBS1p[τ][1] CBS1p[τ][3] CBS1p[τ][5] CBS1p[τ][7] CBS1p[τ][9] Table 8 continued Table 9 Baryon-number-violating operators in the mass basis Operator Qduq Qqque Csbcc Csbcc Csbss Csbss Csbss Csbbb Csbee Csbee Csbee Appendix F: Baryon-number-violating operators in the mass basis Definition in the mass basis Cduue ijkl dRiCu Rj ukRCelR uL →VuL uL , dL → VdL dL , References P , PROMETEOII/ 2013 /007, SEV- 2014 -0398]. 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Alejandro Celis, Javier Fuentes-Martín, Avelino Vicente, Javier Virto. DsixTools: the standard model effective field theory toolkit, The European Physical Journal C, 2017, DOI: 10.1140/epjc/s10052-017-4967-6