DsixTools: the standard model effective field theory toolkit
Eur. Phys. J. C
DsixTools: the standard model effective field theory toolkit
Alejandro Celis 2
Javier FuentesMartín 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 ValènciaCSIC , 46071 Valencia , Spain
2 Fakultät für Physik, Arnold Sommerfeld Center for Theoretical Physics, LudwigMaximiliansUniversität München , 80333 Munich , Germany
We present DsixTools, a Mathematica package for the handling of the dimensionsix standard model effective field theory. Among other features, DsixTools allows the user to perform the full oneloop renormalization group evolution of the Wilson coefficients in the Warsaw basis. This is achieved thanks to the SMEFTrunner module, which implements the full oneloop 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
Baryonnumberviolating
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
Baryonnumberviolating
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 4fermion 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 nfermion) and Category
denotes the index symmetry
category of the coefficient,
being relevant for 2 and
4fermion 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 Baryonnumberviolating – special case Cqque
4F Baryonnumberviolating – 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
Baryonnumberviolating
operators in the mass basis
Operator
Qduq
Qqque
Csbcc
Csbcc
Csbss
Csbss
Csbss
Csbbb
Csbee
Csbee
Csbee
Appendix F: Baryonnumberviolating 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]. J.F. also acknowledges VLCCAMPUS for an “Atracció de Talent” scholarship . A.V. acknowledges financial support from the “Juan de la Cierva” program (2713 463B731) funded by the Spanish MINECO as well as from the Spanish Grants FPA201458183P, Multidark CSD200900064, SEV2014 0398 and PROMETEOII/ 2014/084 (Generalitat Valenciana) . J.V. is funded by the Swiss National Science Foundation and acknowledges support from Explora project FPA201461478EXP .
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