ZeeBabu: sm.fr

File sm.fr, 26.0 KB (added by Richard Ruiz, 2 years ago)

FeynRules model file for Standard Model Lagrangian

Line 
1(***************************************************************************************************************)
2(****** This is the FeynRules mod-file for the Standard model ******)
3(****** ******)
4(****** Authors: N. Christensen, C. Duhr, B. Fuks ******)
5(****** ******)
6(****** Choose whether Feynman gauge is desired. ******)
7(****** If set to False, unitary gauge is assumed. ****)
8(****** Feynman gauge is especially useful for CalcHEP/CompHEP where the calculation is 10-100 times faster. ***)
9(****** Feynman gauge is not supported in MadGraph and Sherpa. ****)
10(***************************************************************************************************************)
11
12(* ************************** *)
13(* ***** Information ***** *)
14(* ************************** *)
15M$ModelName = "Standard Model";
16
17M$Information = {
18 Authors -> {"N. Christensen", "C. Duhr", "B. Fuks"},
19 Version -> "1.4.7",
20 Date -> "28. 09. 2016",
21 Institutions -> {"Michigan State University", "Universite catholique de Louvain (CP3)", "IPHC Strasbourg / University of Strasbourg"},
22 Emails -> {"neil@pa.msu.edu", "claude.duhr@uclouvain.be", "benjamin.fuks@cnrs.in2p3.fr"},
23 URLs -> "http://feynrules.phys.ucl.ac.be/view/Main/StandardModel"
24};
25
26FeynmanGauge = True;
27
28(* ************************** *)
29(* ***** NLO Variables ****** *)
30(******************************)
31
32FR$LoopSwitches = {{Gf, MW}};
33FR$RmDblExt = { ymb -> MB, ymc -> MC, ymdo -> MD, yme -> Me,
34 ymm -> MMU, yms -> MS, ymt -> MT, ymtau -> MTA, ymup -> MU};
35
36(* ************************** *)
37(* ***** Change log ***** *)
38(* ************************** *)
39
40(* 2018 0318: RR: Added CKM angles theta13, theta23, and delta13 *)
41(* v1.4.7: Index issue with the ghost Lagrangian (special thanks to S. Iwamoto *)
42(* v1.4.6: NLO variable added. *)
43(* v1.4.5: Added widths for ghosts. *)
44(* v1.4.4: Changed widths of goldstone bosons to be the same as for the W and Z bosons *)
45(* v1.4.3: Updated conventions for the symmetric structure constants of SU3. *)
46(* v1.4.2: Set FeynmanGauge=True as default again. *)
47(* v1.4: Added SU(2) representation. *)
48(* -> Modification in the field declarations (doublets are added) *)
49(* -> Modification in the Lagrangian (much simpler). *)
50(* v1.3: Added yukawa couplings for all fermions for gauge invariance. *)
51(* Added yukawa couplings for 1st generation fermions to Massless.rst. *)
52(* Updated parameters to PDG 2010. *)
53(* v1.2: Set FeynmanGauge=True as default. *)
54(* Set Gluonic ghosts to be included in both gauges. *)
55(* v1.1: Fixed yukawa couplings in Feynman gauge. *)
56(* Changed yd[n] CKM[n,m] to yd[m] CKM[n,m]. *)
57(* Changed yu[n] Conjugate[CKM[m,n]] to yu[m] Conjugate[CKM[m,n]]. *)
58
59(* ************************** *)
60(* ***** vevs ***** *)
61(* ************************** *)
62M$vevs = { {Phi[2],vev} };
63
64(* ************************** *)
65(* ***** Gauge groups ***** *)
66(* ************************** *)
67M$GaugeGroups = {
68 U1Y == {
69 Abelian -> True,
70 CouplingConstant -> g1,
71 GaugeBoson -> B,
72 Charge -> Y
73 },
74 SU2L == {
75 Abelian -> False,
76 CouplingConstant -> gw,
77 GaugeBoson -> Wi,
78 StructureConstant -> Eps,
79 Representations -> {Ta,SU2D},
80 Definitions -> {Ta[a_,b_,c_]->PauliSigma[a,b,c]/2, FSU2L[i_,j_,k_]:> I Eps[i,j,k]}
81 },
82 SU3C == {
83 Abelian -> False,
84 CouplingConstant -> gs,
85 GaugeBoson -> G,
86 StructureConstant -> f,
87 Representations -> {T,Colour},
88 SymmetricTensor -> dSUN
89 }
90};
91
92
93(* ************************** *)
94(* ***** Indices ***** *)
95(* ************************** *)
96
97IndexRange[Index[SU2W ]] = Unfold[Range[3]];
98IndexRange[Index[SU2D ]] = Unfold[Range[2]];
99IndexRange[Index[Gluon ]] = NoUnfold[Range[8]];
100IndexRange[Index[Colour ]] = NoUnfold[Range[3]];
101IndexRange[Index[Generation]] = Range[3];
102
103IndexStyle[SU2W, j];
104IndexStyle[SU2D, k];
105IndexStyle[Gluon, a];
106IndexStyle[Colour, m];
107IndexStyle[Generation, f];
108
109
110(* ************************** *)
111(* *** Interaction orders *** *)
112(* *** (as used by mg5) *** *)
113(* ************************** *)
114
115M$InteractionOrderHierarchy = {
116 {QCD, 1},
117 {QED, 2}
118};
119
120
121(* ************************** *)
122(* **** Particle classes **** *)
123(* ************************** *)
124M$ClassesDescription = {
125
126(* Gauge bosons: physical vector fields *)
127 V[1] == {
128 ClassName -> A,
129 SelfConjugate -> True,
130 Mass -> 0,
131 Width -> 0,
132 ParticleName -> "a",
133 PDG -> 22,
134 PropagatorLabel -> "a",
135 PropagatorType -> W,
136 PropagatorArrow -> None,
137 FullName -> "Photon"
138 },
139 V[2] == {
140 ClassName -> Z,
141 SelfConjugate -> True,
142 Mass -> {MZ, 91.1876},
143 Width -> {WZ, 2.4952},
144 ParticleName -> "Z",
145 PDG -> 23,
146 PropagatorLabel -> "Z",
147 PropagatorType -> Sine,
148 PropagatorArrow -> None,
149 FullName -> "Z"
150 },
151 V[3] == {
152 ClassName -> W,
153 SelfConjugate -> False,
154 Mass -> {MW, Internal},
155 Width -> {WW, 2.085},
156 ParticleName -> "W+",
157 AntiParticleName -> "W-",
158 QuantumNumbers -> {Q -> 1},
159 PDG -> 24,
160 PropagatorLabel -> "W",
161 PropagatorType -> Sine,
162 PropagatorArrow -> Forward,
163 FullName -> "W"
164 },
165 V[4] == {
166 ClassName -> G,
167 SelfConjugate -> True,
168 Indices -> {Index[Gluon]},
169 Mass -> 0,
170 Width -> 0,
171 ParticleName -> "g",
172 PDG -> 21,
173 PropagatorLabel -> "G",
174 PropagatorType -> C,
175 PropagatorArrow -> None,
176 FullName -> "G"
177 },
178
179(* Ghosts: related to physical gauge bosons *)
180 U[1] == {
181 ClassName -> ghA,
182 SelfConjugate -> False,
183 Ghost -> A,
184 QuantumNumbers -> {GhostNumber -> 1},
185 Mass -> 0,
186 Width -> 0,
187 PropagatorLabel -> "uA",
188 PropagatorType -> GhostDash,
189 PropagatorArrow -> Forward
190 },
191 U[2] == {
192 ClassName -> ghZ,
193 SelfConjugate -> False,
194 Ghost -> Z,
195 QuantumNumbers -> {GhostNumber -> 1},
196 Mass -> {MZ,91.1876},
197 Width -> {WZ, 2.4952},
198 PropagatorLabel -> "uZ",
199 PropagatorType -> GhostDash,
200 PropagatorArrow -> Forward
201 },
202 U[31] == {
203 ClassName -> ghWp,
204 SelfConjugate -> False,
205 Ghost -> W,
206 QuantumNumbers -> {GhostNumber -> 1, Q -> 1},
207 Mass -> {MW,Internal},
208 Width -> {WW, 2.085},
209 PropagatorLabel -> "uWp",
210 PropagatorType -> GhostDash,
211 PropagatorArrow -> Forward
212 },
213 U[32] == {
214 ClassName -> ghWm,
215 SelfConjugate -> False,
216 Ghost -> Wbar,
217 QuantumNumbers -> {GhostNumber -> 1, Q -> -1},
218 Mass -> {MW,Internal},
219 Width -> {WW, 2.085},
220 PropagatorLabel -> "uWm",
221 PropagatorType -> GhostDash,
222 PropagatorArrow -> Forward
223 },
224 U[4] == {
225 ClassName -> ghG,
226 SelfConjugate -> False,
227 Indices -> {Index[Gluon]},
228 Ghost -> G,
229 PDG -> 82,
230 QuantumNumbers ->{GhostNumber -> 1},
231 Mass -> 0,
232 Width -> 0,
233 PropagatorLabel -> "uG",
234 PropagatorType -> GhostDash,
235 PropagatorArrow -> Forward
236 },
237
238(* Gauge bosons: unphysical vector fields *)
239 V[11] == {
240 ClassName -> B,
241 Unphysical -> True,
242 SelfConjugate -> True,
243 Definitions -> { B[mu_] -> -sw Z[mu]+cw A[mu]}
244 },
245 V[12] == {
246 ClassName -> Wi,
247 Unphysical -> True,
248 SelfConjugate -> True,
249 Indices -> {Index[SU2W]},
250 FlavorIndex -> SU2W,
251 Definitions -> { Wi[mu_,1] -> (Wbar[mu]+W[mu])/Sqrt[2], Wi[mu_,2] -> (Wbar[mu]-W[mu])/(I*Sqrt[2]), Wi[mu_,3] -> cw Z[mu] + sw A[mu]}
252 },
253
254(* Ghosts: related to unphysical gauge bosons *)
255 U[11] == {
256 ClassName -> ghB,
257 Unphysical -> True,
258 SelfConjugate -> False,
259 Ghost -> B,
260 Definitions -> { ghB -> -sw ghZ + cw ghA}
261 },
262 U[12] == {
263 ClassName -> ghWi,
264 Unphysical -> True,
265 SelfConjugate -> False,
266 Ghost -> Wi,
267 Indices -> {Index[SU2W]},
268 FlavorIndex -> SU2W,
269 Definitions -> { ghWi[1] -> (ghWp+ghWm)/Sqrt[2], ghWi[2] -> (ghWm-ghWp)/(I*Sqrt[2]), ghWi[3] -> cw ghZ+sw ghA}
270 } ,
271
272(* Fermions: physical fields *)
273 F[1] == {
274 ClassName -> vl,
275 ClassMembers -> {ve,vm,vt},
276 Indices -> {Index[Generation]},
277 FlavorIndex -> Generation,
278 SelfConjugate -> False,
279 Mass -> 0,
280 Width -> 0,
281 QuantumNumbers -> {LeptonNumber -> 1},
282 PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
283 PropagatorType -> S,
284 PropagatorArrow -> Forward,
285 PDG -> {12,14,16},
286 ParticleName -> {"ve","vm","vt"},
287 AntiParticleName -> {"ve~","vm~","vt~"},
288 FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"}
289 },
290 F[2] == {
291 ClassName -> l,
292 ClassMembers -> {e, mu, ta},
293 Indices -> {Index[Generation]},
294 FlavorIndex -> Generation,
295 SelfConjugate -> False,
296 Mass -> {Ml, {Me,5.11*^-4}, {MMU,0.10566}, {MTA,1.777}},
297 Width -> 0,
298 QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
299 PropagatorLabel -> {"l", "e", "mu", "ta"},
300 PropagatorType -> Straight,
301 PropagatorArrow -> Forward,
302 PDG -> {11, 13, 15},
303 ParticleName -> {"e-", "mu-", "ta-"},
304 AntiParticleName -> {"e+", "mu+", "ta+"},
305 FullName -> {"Electron", "Muon", "Tau"}
306 },
307 F[3] == {
308 ClassName -> uq,
309 ClassMembers -> {u, c, t},
310 Indices -> {Index[Generation], Index[Colour]},
311 FlavorIndex -> Generation,
312 SelfConjugate -> False,
313 Mass -> {Mu, {MU, 2.55*^-3}, {MC,1.27}, {MT,173.3}},
314 Width -> {0, 0, {WT,1.350}},
315 QuantumNumbers -> {Q -> 2/3},
316 PropagatorLabel -> {"uq", "u", "c", "t"},
317 PropagatorType -> Straight,
318 PropagatorArrow -> Forward,
319 PDG -> {2, 4, 6},
320 ParticleName -> {"u", "c", "t" },
321 AntiParticleName -> {"u~", "c~", "t~"},
322 FullName -> {"u-quark", "c-quark", "t-quark"}
323 },
324 F[4] == {
325 ClassName -> dq,
326 ClassMembers -> {d, s, b},
327 Indices -> {Index[Generation], Index[Colour]},
328 FlavorIndex -> Generation,
329 SelfConjugate -> False,
330 Mass -> {Md, {MD,5.04*^-3}, {MS,0.101}, {MB,4.7}},
331 Width -> 0,
332 QuantumNumbers -> {Q -> -1/3},
333 PropagatorLabel -> {"dq", "d", "s", "b"},
334 PropagatorType -> Straight,
335 PropagatorArrow -> Forward,
336 PDG -> {1,3,5},
337 ParticleName -> {"d", "s", "b" },
338 AntiParticleName -> {"d~", "s~", "b~"},
339 FullName -> {"d-quark", "s-quark", "b-quark"}
340 },
341
342(* Fermions: unphysical fields *)
343 F[11] == {
344 ClassName -> LL,
345 Unphysical -> True,
346 Indices -> {Index[SU2D], Index[Generation]},
347 FlavorIndex -> SU2D,
348 SelfConjugate -> False,
349 QuantumNumbers -> {Y -> -1/2},
350 Definitions -> { LL[sp1_,1,ff_] :> Module[{sp2}, ProjM[sp1,sp2] vl[sp2,ff]], LL[sp1_,2,ff_] :> Module[{sp2}, ProjM[sp1,sp2] l[sp2,ff]] }
351 },
352 F[12] == {
353 ClassName -> lR,
354 Unphysical -> True,
355 Indices -> {Index[Generation]},
356 FlavorIndex -> Generation,
357 SelfConjugate -> False,
358 QuantumNumbers -> {Y -> -1},
359 Definitions -> { lR[sp1_,ff_] :> Module[{sp2}, ProjP[sp1,sp2] l[sp2,ff]] }
360 },
361 F[13] == {
362 ClassName -> QL,
363 Unphysical -> True,
364 Indices -> {Index[SU2D], Index[Generation], Index[Colour]},
365 FlavorIndex -> SU2D,
366 SelfConjugate -> False,
367 QuantumNumbers -> {Y -> 1/6},
368 Definitions -> {
369 QL[sp1_,1,ff_,cc_] :> Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],
370 QL[sp1_,2,ff_,cc_] :> Module[{sp2,ff2}, CKM[ff,ff2] ProjM[sp1,sp2] dq[sp2,ff2,cc]] }
371 },
372 F[14] == {
373 ClassName -> uR,
374 Unphysical -> True,
375 Indices -> {Index[Generation], Index[Colour]},
376 FlavorIndex -> Generation,
377 SelfConjugate -> False,
378 QuantumNumbers -> {Y -> 2/3},
379 Definitions -> { uR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] uq[sp2,ff,cc]] }
380 },
381 F[15] == {
382 ClassName -> dR,
383 Unphysical -> True,
384 Indices -> {Index[Generation], Index[Colour]},
385 FlavorIndex -> Generation,
386 SelfConjugate -> False,
387 QuantumNumbers -> {Y -> -1/3},
388 Definitions -> { dR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] dq[sp2,ff,cc]] }
389 },
390
391(* Higgs: physical scalars *)
392 S[1] == {
393 ClassName -> H,
394 SelfConjugate -> True,
395 Mass -> {MH,125.7},
396 Width -> {WH,0.004170},
397 PropagatorLabel -> "H",
398 PropagatorType -> D,
399 PropagatorArrow -> None,
400 PDG -> 25,
401 ParticleName -> "H",
402 FullName -> "H"
403 },
404
405(* Higgs: physical scalars *)
406 S[2] == {
407 ClassName -> G0,
408 SelfConjugate -> True,
409 Goldstone -> Z,
410 Mass -> {MZ, 91.1876},
411 Width -> {WZ, 2.4952},
412 PropagatorLabel -> "Go",
413 PropagatorType -> D,
414 PropagatorArrow -> None,
415 PDG -> 250,
416 ParticleName -> "G0",
417 FullName -> "G0"
418 },
419 S[3] == {
420 ClassName -> GP,
421 SelfConjugate -> False,
422 Goldstone -> W,
423 Mass -> {MW, Internal},
424 QuantumNumbers -> {Q -> 1},
425 Width -> {WW, 2.085},
426 PropagatorLabel -> "GP",
427 PropagatorType -> D,
428 PropagatorArrow -> None,
429 PDG -> 251,
430 ParticleName -> "G+",
431 AntiParticleName -> "G-",
432 FullName -> "GP"
433 },
434
435(* Higgs: unphysical scalars *)
436 S[11] == {
437 ClassName -> Phi,
438 Unphysical -> True,
439 Indices -> {Index[SU2D]},
440 FlavorIndex -> SU2D,
441 SelfConjugate -> False,
442 QuantumNumbers -> {Y -> 1/2},
443 Definitions -> { Phi[1] -> -I GP, Phi[2] -> (vev + H + I G0)/Sqrt[2] }
444 }
445};
446
447
448(* ************************** *)
449(* ***** Gauge ***** *)
450(* ***** Parameters ***** *)
451(* ***** (FeynArts) ***** *)
452(* ************************** *)
453
454GaugeXi[ V[1] ] = GaugeXi[A];
455GaugeXi[ V[2] ] = GaugeXi[Z];
456GaugeXi[ V[3] ] = GaugeXi[W];
457GaugeXi[ V[4] ] = GaugeXi[G];
458GaugeXi[ S[1] ] = 1;
459GaugeXi[ S[2] ] = GaugeXi[Z];
460GaugeXi[ S[3] ] = GaugeXi[W];
461GaugeXi[ U[1] ] = GaugeXi[A];
462GaugeXi[ U[2] ] = GaugeXi[Z];
463GaugeXi[ U[31] ] = GaugeXi[W];
464GaugeXi[ U[32] ] = GaugeXi[W];
465GaugeXi[ U[4] ] = GaugeXi[G];
466
467
468(* ************************** *)
469(* ***** Parameters ***** *)
470(* ************************** *)
471M$Parameters = {
472
473 (* External parameters *)
474 aEWM1 == {
475 ParameterType -> External,
476 BlockName -> SMINPUTS,
477 OrderBlock -> 1,
478 Value -> 127.94,
479 InteractionOrder -> {QED,-2},
480 Description -> "Inverse of the EW coupling constant at the Z pole"
481 },
482 Gf == {
483 ParameterType -> External,
484 BlockName -> SMINPUTS,
485 OrderBlock -> 2,
486 Value -> 1.17456*^-5,
487 InteractionOrder -> {QED,2},
488 TeX -> Subscript[G,f],
489 Description -> "Fermi constant"
490 },
491 aS == {
492 ParameterType -> External,
493 BlockName -> SMINPUTS,
494 OrderBlock -> 3,
495 Value -> 0.1184,
496 InteractionOrder -> {QCD,2},
497 TeX -> Subscript[\[Alpha],s],
498 Description -> "Strong coupling constant at the Z pole"
499 },
500 ymdo == {
501 ParameterType -> External,
502 BlockName -> YUKAWA,
503 OrderBlock -> 1,
504 Value -> 5.04*^-3,
505 Description -> "Down Yukawa mass"
506 },
507 ymup == {
508 ParameterType -> External,
509 BlockName -> YUKAWA,
510 OrderBlock -> 2,
511 Value -> 2.55*^-3,
512 Description -> "Up Yukawa mass"
513 },
514 yms == {
515 ParameterType -> External,
516 BlockName -> YUKAWA,
517 OrderBlock -> 3,
518 Value -> 0.101,
519 Description -> "Strange Yukawa mass"
520 },
521 ymc == {
522 ParameterType -> External,
523 BlockName -> YUKAWA,
524 OrderBlock -> 4,
525 Value -> 1.27,
526 Description -> "Charm Yukawa mass"
527 },
528 ymb == {
529 ParameterType -> External,
530 BlockName -> YUKAWA,
531 OrderBlock -> 5,
532 Value -> 4.7,
533 Description -> "Bottom Yukawa mass"
534 },
535 ymt == {
536 ParameterType -> External,
537 BlockName -> YUKAWA,
538 OrderBlock -> 6,
539 Value -> 173.3,
540 Description -> "Top Yukawa mass"
541 },
542 yme == {
543 ParameterType -> External,
544 BlockName -> YUKAWA,
545 OrderBlock -> 11,
546 Value -> 5.11*^-4,
547 Description -> "Electron Yukawa mass"
548 },
549 ymm == {
550 ParameterType -> External,
551 BlockName -> YUKAWA,
552 OrderBlock -> 13,
553 Value -> 0.10566,
554 Description -> "Muon Yukawa mass"
555 },
556 ymtau == {
557 ParameterType -> External,
558 BlockName -> YUKAWA,
559 OrderBlock -> 15,
560 Value -> 1.777,
561 Description -> "Tau Yukawa mass"
562 },
563 cabi == {
564 ParameterType -> External,
565 BlockName -> CKMBLOCK,
566 OrderBlock -> 1,
567 Value -> 0.227591,
568 TeX -> Subscript[\[Theta], c],
569 Description -> "Cabibbo angle"
570 },
571 th13 == {
572 ParameterType -> External,
573 BlockName -> CKMBLOCK,
574 OrderBlock -> 2,
575 Value -> 0.003508,
576 TeX -> Subscript[\[Theta], 13],
577 Description -> "CKM Theta 13"
578 },
579 th23 == {
580 ParameterType -> External,
581 BlockName -> CKMBLOCK,
582 OrderBlock -> 3,
583 Value -> 0.041539,
584 TeX -> Subscript[\[Theta], 23],
585 Description -> "CKM theta 23"
586 },
587 del13 == {
588 ParameterType -> External,
589 BlockName -> CKMBLOCK,
590 OrderBlock -> 4,
591 Value -> 1.20,
592 TeX -> Subscript[\[Delta], 13],
593 Description -> "CKM delta 13"
594 },
595
596 (* Internal Parameters *)
597 aEW == {
598 ParameterType -> Internal,
599 Value -> 1/aEWM1,
600 InteractionOrder -> {QED,2},
601 TeX -> Subscript[\[Alpha], EW],
602 Description -> "Electroweak coupling contant"
603 },
604 MW == {
605 ParameterType -> Internal,
606 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
607 TeX -> Subscript[M,W],
608 Description -> "W mass"
609 },
610 sw2 == {
611 ParameterType -> Internal,
612 Value -> 1-(MW/MZ)^2,
613 Description -> "Squared Sin of the Weinberg angle"
614 },
615 ee == {
616 ParameterType -> Internal,
617 Value -> Sqrt[4 Pi aEW],
618 InteractionOrder -> {QED,1},
619 TeX -> e,
620 Description -> "Electric coupling constant"
621 },
622 cw == {
623 ParameterType -> Internal,
624 Value -> Sqrt[1-sw2],
625 TeX -> Subscript[c,w],
626 Description -> "Cosine of the Weinberg angle"
627 },
628 sw == {
629 ParameterType -> Internal,
630 Value -> Sqrt[sw2],
631 TeX -> Subscript[s,w],
632 Description -> "Sine of the Weinberg angle"
633 },
634 gw == {
635 ParameterType -> Internal,
636 Definitions -> {gw->ee/sw},
637 InteractionOrder -> {QED,1},
638 TeX -> Subscript[g,w],
639 Description -> "Weak coupling constant at the Z pole"
640 },
641 g1 == {
642 ParameterType -> Internal,
643 Definitions -> {g1->ee/cw},
644 InteractionOrder -> {QED,1},
645 TeX -> Subscript[g,1],
646 Description -> "U(1)Y coupling constant at the Z pole"
647 },
648 gs == {
649 ParameterType -> Internal,
650 Value -> Sqrt[4 Pi aS],
651 InteractionOrder -> {QCD,1},
652 TeX -> Subscript[g,s],
653 ParameterName -> G,
654 Description -> "Strong coupling constant at the Z pole"
655 },
656 vev == {
657 ParameterType -> Internal,
658 Value -> 2*MW*sw/ee,
659 InteractionOrder -> {QED,-1},
660 Description -> "Higgs vacuum expectation value"
661 },
662 lam == {
663 ParameterType -> Internal,
664 Value -> MH^2/(2*vev^2),
665 InteractionOrder -> {QED, 2},
666 Description -> "Higgs quartic coupling"
667 },
668 muH == {
669 ParameterType -> Internal,
670 Value -> Sqrt[vev^2 lam],
671 TeX -> \[Mu],
672 Description -> "Coefficient of the quadratic piece of the Higgs potential"
673 },
674 yl == {
675 ParameterType -> Internal,
676 Indices -> {Index[Generation], Index[Generation]},
677 Definitions -> {yl[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
678 Value -> {yl[1,1] -> Sqrt[2] yme / vev, yl[2,2] -> Sqrt[2] ymm / vev, yl[3,3] -> Sqrt[2] ymtau / vev},
679 InteractionOrder -> {QED, 1},
680 ParameterName -> {yl[1,1] -> ye, yl[2,2] -> ym, yl[3,3] -> ytau},
681 TeX -> Superscript[y, l],
682 Description -> "Lepton Yukawa couplings"
683 },
684 yu == {
685 ParameterType -> Internal,
686 Indices -> {Index[Generation], Index[Generation]},
687 Definitions -> {yu[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
688 Value -> {yu[1,1] -> Sqrt[2] ymup/vev, yu[2,2] -> Sqrt[2] ymc/vev, yu[3,3] -> Sqrt[2] ymt/vev},
689 InteractionOrder -> {QED, 1},
690 ParameterName -> {yu[1,1] -> yup, yu[2,2] -> yc, yu[3,3] -> yt},
691 TeX -> Superscript[y, u],
692 Description -> "Up-type Yukawa couplings"
693 },
694 yd == {
695 ParameterType -> Internal,
696 Indices -> {Index[Generation], Index[Generation]},
697 Definitions -> {yd[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
698 Value -> {yd[1,1] -> Sqrt[2] ymdo/vev, yd[2,2] -> Sqrt[2] yms/vev, yd[3,3] -> Sqrt[2] ymb/vev},
699 InteractionOrder -> {QED, 1},
700 ParameterName -> {yd[1,1] -> ydo, yd[2,2] -> ys, yd[3,3] -> yb},
701 TeX -> Superscript[y, d],
702 Description -> "Down-type Yukawa couplings"
703 },
704(* N. B. : only Cabibbo mixing! *)
705 CKM == {
706 ParameterType -> Internal,
707 Indices -> {Index[Generation], Index[Generation]},
708 Unitary -> True,
709 Value -> {CKM[1,1] -> Cos[cabi] Cos[th13], CKM[1,2] -> Sin[cabi] Cos[th13], CKM[1,3] -> Sin[th13] Exp[-I del13],
710 CKM[2,1] -> -Sin[cabi] Cos[th23] - Cos[cabi] Sin[th23] Sin[th13] Exp[I del13], CKM[2,2] -> Cos[cabi] Cos[th23] - Sin[cabi] Sin[th23] Sin[th13] Exp[I del13], CKM[2,3] -> Sin[th23] Cos[th13],
711 CKM[3,1] -> Sin[cabi] Sin[th23] - Cos[cabi] Cos[th23] Sin[th13] Exp[I del13], CKM[3,2] -> -Cos[cabi] Sin[th23] - Sin[cabi] Cos[th23] Sin[th13] Exp[I del13], CKM[3,3] -> Cos[th23] Cos[th13]},
712 TeX -> Superscript[V,CKM],
713 Description -> "CKM-Matrix"}
714};
715
716(* ************************** *)
717(* ***** Lagrangian ***** *)
718(* ************************** *)
719
720LGauge := Block[{mu,nu,ii,aa},
721 ExpandIndices[-1/4 FS[B,mu,nu] FS[B,mu,nu] - 1/4 FS[Wi,mu,nu,ii] FS[Wi,mu,nu,ii] - 1/4 FS[G,mu,nu,aa] FS[G,mu,nu,aa], FlavorExpand->SU2W]];
722
723LFermions := Block[{mu},
724 ExpandIndices[I*(
725 QLbar.Ga[mu].DC[QL, mu] + LLbar.Ga[mu].DC[LL, mu] + uRbar.Ga[mu].DC[uR, mu] + dRbar.Ga[mu].DC[dR, mu] + lRbar.Ga[mu].DC[lR, mu]),
726 FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]}];
727
728LHiggs := Block[{ii,mu, feynmangaugerules},
729 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
730
731 ExpandIndices[DC[Phibar[ii],mu] DC[Phi[ii],mu] + muH^2 Phibar[ii] Phi[ii] - lam Phibar[ii] Phi[ii] Phibar[jj] Phi[jj], FlavorExpand->{SU2D,SU2W}]/.feynmangaugerules
732 ];
733
734LYukawa := Block[{sp,ii,jj,cc,ff1,ff2,ff3,yuk,feynmangaugerules},
735 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
736
737 yuk = ExpandIndices[
738 -yd[ff2, ff3] CKM[ff1, ff2] QLbar[sp, ii, ff1, cc].dR [sp, ff3, cc] Phi[ii] -
739 yl[ff1, ff3] LLbar[sp, ii, ff1].lR [sp, ff3] Phi[ii] -
740 yu[ff1, ff2] QLbar[sp, ii, ff1, cc].uR [sp, ff2, cc] Phibar[jj] Eps[ii, jj], FlavorExpand -> SU2D];
741 yuk = yuk /. { CKM[a_, b_] Conjugate[CKM[a_, c_]] -> IndexDelta[b, c], CKM[b_, a_] Conjugate[CKM[c_, a_]] -> IndexDelta[b, c]};
742 yuk+HC[yuk]/.feynmangaugerules
743 ];
744
745LGhost := Block[{LGh1,LGhw,LGhs,LGhphi,mu, generators,gh,ghbar,Vectorize,phi1,phi2,togoldstones,doublet,doublet0},
746 (* Pure gauge piece *)
747 LGh1 = -ghBbar.del[DC[ghB,mu],mu];
748 LGhw = -ghWibar[ii].del[DC[ghWi[ii],mu],mu];
749 LGhs = -ghGbar[ii].del[DC[ghG[ii],mu],mu];
750
751 (* Scalar pieces: see Peskin pages 739-742 *)
752 (* phi1 and phi2 are the real degrees of freedom of GP *)
753 (* Vectorize transforms a doublet in a vector in the phi-basis, i.e. the basis of real degrees of freedom *)
754 gh = {ghB, ghWi[1], ghWi[2], ghWi[3]};
755 ghbar = {ghBbar, ghWibar[1], ghWibar[2], ghWibar[3]};
756 generators = {-I/2 g1 IdentityMatrix[2], -I/2 gw PauliSigma[1], -I/2 gw PauliSigma[2], -I/2 gw PauliSigma[3]};
757 doublet = Expand[{(-I phi1 - phi2)/Sqrt[2], Phi[2]} /. MR$Definitions /. vev -> 0];
758 doublet0 = {0, vev/Sqrt[2]};
759 Vectorize[{a_, b_}]:= Simplify[{Sqrt[2] Re[Expand[a]], Sqrt[2] Im[Expand[a]], Sqrt[2] Re[Expand[b]], Sqrt[2] Im[Expand[b]]}/.{Im[_]->0, Re[num_]->num}];
760 togoldstones := {phi1 -> (GP + GPbar)/Sqrt[2], phi2 -> (-GP + GPbar)/(I Sqrt[2])};
761 LGhphi=Plus@@Flatten[Table[-ghbar[[kkk]].gh[[lll]] Vectorize[generators[[kkk]].doublet0].Vectorize[generators[[lll]].(doublet+doublet0)],{kkk,4},{lll,4}]] /.togoldstones;
762
763ExpandIndices[ LGhs + If[FeynmanGauge, LGh1 + LGhw + LGhphi,0], FlavorExpand->SU2W]];
764
765LSM:= LGauge + LFermions + LHiggs + LYukawa + LGhost;