StandardModel: SM.fr

File SM.fr, 24.8 KB (added by Céline Degrande, 12 years ago)
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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 -> "1.4.6",
20 Date -> "15. 04. 2014",
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(* v1.4.6: NLO variable added. *)
41(* v1.4.5: Added widths for ghosts. *)
42(* v1.4.4: Changed widths of goldstone bosons to be the same as for the W and Z bosons *)
43(* v1.4.3: Updated conventions for the symmetric structure constants of SU3. *)
44(* v1.4.2: Set FeynmanGauge=True as default again. *)
45(* v1.4: Added SU(2) representation. *)
46(* -> Modification in the field declarations (doublets are added) *)
47(* -> Modification in the Lagrangian (much simpler). *)
48(* v1.3: Added yukawa couplings for all fermions for gauge invariance. *)
49(* Added yukawa couplings for 1st generation fermions to Massless.rst. *)
50(* Updated parameters to PDG 2010. *)
51(* v1.2: Set FeynmanGauge=True as default. *)
52(* Set Gluonic ghosts to be included in both gauges. *)
53(* v1.1: Fixed yukawa couplings in Feynman gauge. *)
54(* Changed yd[n] CKM[n,m] to yd[m] CKM[n,m]. *)
55(* Changed yu[n] Conjugate[CKM[m,n]] to yu[m] Conjugate[CKM[m,n]]. *)
56
57(* ************************** *)
58(* ***** vevs ***** *)
59(* ************************** *)
60M$vevs = { {Phi[2],vev} };
61
62(* ************************** *)
63(* ***** Gauge groups ***** *)
64(* ************************** *)
65M$GaugeGroups = {
66 U1Y == {
67 Abelian -> True,
68 CouplingConstant -> g1,
69 GaugeBoson -> B,
70 Charge -> Y
71 },
72 SU2L == {
73 Abelian -> False,
74 CouplingConstant -> gw,
75 GaugeBoson -> Wi,
76 StructureConstant -> Eps,
77 Representations -> {Ta,SU2D},
78 Definitions -> {Ta[a_,b_,c_]->PauliSigma[a,b,c]/2, FSU2L[i_,j_,k_]:> I Eps[i,j,k]}
79 },
80 SU3C == {
81 Abelian -> False,
82 CouplingConstant -> gs,
83 GaugeBoson -> G,
84 StructureConstant -> f,
85 Representations -> {T,Colour},
86 SymmetricTensor -> dSUN
87 }
88};
89
90
91(* ************************** *)
92(* ***** Indices ***** *)
93(* ************************** *)
94
95IndexRange[Index[SU2W ]] = Unfold[Range[3]];
96IndexRange[Index[SU2D ]] = Unfold[Range[2]];
97IndexRange[Index[Gluon ]] = NoUnfold[Range[8]];
98IndexRange[Index[Colour ]] = NoUnfold[Range[3]];
99IndexRange[Index[Generation]] = Range[3];
100
101IndexStyle[SU2W, j];
102IndexStyle[SU2D, k];
103IndexStyle[Gluon, a];
104IndexStyle[Colour, m];
105IndexStyle[Generation, f];
106
107
108(* ************************** *)
109(* *** Interaction orders *** *)
110(* *** (as used by mg5) *** *)
111(* ************************** *)
112
113M$InteractionOrderHierarchy = {
114 {QCD, 1},
115 {QED, 2}
116};
117
118
119(* ************************** *)
120(* **** Particle classes **** *)
121(* ************************** *)
122M$ClassesDescription = {
123
124(* Gauge bosons: physical vector fields *)
125 V[1] == {
126 ClassName -> A,
127 SelfConjugate -> True,
128 Mass -> 0,
129 Width -> 0,
130 ParticleName -> "a",
131 PDG -> 22,
132 PropagatorLabel -> "a",
133 PropagatorType -> W,
134 PropagatorArrow -> None,
135 FullName -> "Photon"
136 },
137 V[2] == {
138 ClassName -> Z,
139 SelfConjugate -> True,
140 Mass -> {MZ, 91.1876},
141 Width -> {WZ, 2.4952},
142 ParticleName -> "Z",
143 PDG -> 23,
144 PropagatorLabel -> "Z",
145 PropagatorType -> Sine,
146 PropagatorArrow -> None,
147 FullName -> "Z"
148 },
149 V[3] == {
150 ClassName -> W,
151 SelfConjugate -> False,
152 Mass -> {MW, Internal},
153 Width -> {WW, 2.085},
154 ParticleName -> "W+",
155 AntiParticleName -> "W-",
156 QuantumNumbers -> {Q -> 1},
157 PDG -> 24,
158 PropagatorLabel -> "W",
159 PropagatorType -> Sine,
160 PropagatorArrow -> Forward,
161 FullName -> "W"
162 },
163 V[4] == {
164 ClassName -> G,
165 SelfConjugate -> True,
166 Indices -> {Index[Gluon]},
167 Mass -> 0,
168 Width -> 0,
169 ParticleName -> "g",
170 PDG -> 21,
171 PropagatorLabel -> "G",
172 PropagatorType -> C,
173 PropagatorArrow -> None,
174 FullName -> "G"
175 },
176
177(* Ghosts: related to physical gauge bosons *)
178 U[1] == {
179 ClassName -> ghA,
180 SelfConjugate -> False,
181 Ghost -> A,
182 QuantumNumbers -> {GhostNumber -> 1},
183 Mass -> 0,
184 Width -> 0,
185 PropagatorLabel -> "uA",
186 PropagatorType -> GhostDash,
187 PropagatorArrow -> Forward
188 },
189 U[2] == {
190 ClassName -> ghZ,
191 SelfConjugate -> False,
192 Ghost -> Z,
193 QuantumNumbers -> {GhostNumber -> 1},
194 Mass -> {MZ,91.1876},
195 Width -> {WZ, 2.4952},
196 PropagatorLabel -> "uZ",
197 PropagatorType -> GhostDash,
198 PropagatorArrow -> Forward
199 },
200 U[31] == {
201 ClassName -> ghWp,
202 SelfConjugate -> False,
203 Ghost -> W,
204 QuantumNumbers -> {GhostNumber -> 1, Q -> 1},
205 Mass -> {MW,Internal},
206 Width -> {WW, 2.085},
207 PropagatorLabel -> "uWp",
208 PropagatorType -> GhostDash,
209 PropagatorArrow -> Forward
210 },
211 U[32] == {
212 ClassName -> ghWm,
213 SelfConjugate -> False,
214 Ghost -> Wbar,
215 QuantumNumbers -> {GhostNumber -> 1, Q -> -1},
216 Mass -> {MW,Internal},
217 Width -> {WW, 2.085},
218 PropagatorLabel -> "uWm",
219 PropagatorType -> GhostDash,
220 PropagatorArrow -> Forward
221 },
222 U[4] == {
223 ClassName -> ghG,
224 SelfConjugate -> False,
225 Indices -> {Index[Gluon]},
226 Ghost -> G,
227 PDG -> 82,
228 QuantumNumbers ->{GhostNumber -> 1},
229 Mass -> 0,
230 Width -> 0,
231 PropagatorLabel -> "uG",
232 PropagatorType -> GhostDash,
233 PropagatorArrow -> Forward
234 },
235
236(* Gauge bosons: unphysical vector fields *)
237 V[11] == {
238 ClassName -> B,
239 Unphysical -> True,
240 SelfConjugate -> True,
241 Definitions -> { B[mu_] -> -sw Z[mu]+cw A[mu]}
242 },
243 V[12] == {
244 ClassName -> Wi,
245 Unphysical -> True,
246 SelfConjugate -> True,
247 Indices -> {Index[SU2W]},
248 FlavorIndex -> SU2W,
249 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]}
250 },
251
252(* Ghosts: related to unphysical gauge bosons *)
253 U[11] == {
254 ClassName -> ghB,
255 Unphysical -> True,
256 SelfConjugate -> False,
257 Ghost -> B,
258 Definitions -> { ghB -> -sw ghZ + cw ghA}
259 },
260 U[12] == {
261 ClassName -> ghWi,
262 Unphysical -> True,
263 SelfConjugate -> False,
264 Ghost -> Wi,
265 Indices -> {Index[SU2W]},
266 FlavorIndex -> SU2W,
267 Definitions -> { ghWi[1] -> (ghWp+ghWm)/Sqrt[2], ghWi[2] -> (ghWm-ghWp)/(I*Sqrt[2]), ghWi[3] -> cw ghZ+sw ghA}
268 } ,
269
270(* Fermions: physical fields *)
271 F[1] == {
272 ClassName -> vl,
273 ClassMembers -> {ve,vm,vt},
274 Indices -> {Index[Generation]},
275 FlavorIndex -> Generation,
276 SelfConjugate -> False,
277 Mass -> 0,
278 Width -> 0,
279 QuantumNumbers -> {LeptonNumber -> 1},
280 PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
281 PropagatorType -> S,
282 PropagatorArrow -> Forward,
283 PDG -> {12,14,16},
284 ParticleName -> {"ve","vm","vt"},
285 AntiParticleName -> {"ve~","vm~","vt~"},
286 FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"}
287 },
288 F[2] == {
289 ClassName -> l,
290 ClassMembers -> {e, mu, ta},
291 Indices -> {Index[Generation]},
292 FlavorIndex -> Generation,
293 SelfConjugate -> False,
294 Mass -> {Ml, {Me,5.11*^-4}, {MMU,0.10566}, {MTA,1.777}},
295 Width -> 0,
296 QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
297 PropagatorLabel -> {"l", "e", "mu", "ta"},
298 PropagatorType -> Straight,
299 PropagatorArrow -> Forward,
300 PDG -> {11, 13, 15},
301 ParticleName -> {"e-", "mu-", "ta-"},
302 AntiParticleName -> {"e+", "mu+", "ta+"},
303 FullName -> {"Electron", "Muon", "Tau"}
304 },
305 F[3] == {
306 ClassName -> uq,
307 ClassMembers -> {u, c, t},
308 Indices -> {Index[Generation], Index[Colour]},
309 FlavorIndex -> Generation,
310 SelfConjugate -> False,
311 Mass -> {Mu, {MU, 2.55*^-3}, {MC,1.27}, {MT,172}},
312 Width -> {0, 0, {WT,1.50833649}},
313 QuantumNumbers -> {Q -> 2/3},
314 PropagatorLabel -> {"uq", "u", "c", "t"},
315 PropagatorType -> Straight,
316 PropagatorArrow -> Forward,
317 PDG -> {2, 4, 6},
318 ParticleName -> {"u", "c", "t" },
319 AntiParticleName -> {"u~", "c~", "t~"},
320 FullName -> {"u-quark", "c-quark", "t-quark"}
321 },
322 F[4] == {
323 ClassName -> dq,
324 ClassMembers -> {d, s, b},
325 Indices -> {Index[Generation], Index[Colour]},
326 FlavorIndex -> Generation,
327 SelfConjugate -> False,
328 Mass -> {Md, {MD,5.04*^-3}, {MS,0.101}, {MB,4.7}},
329 Width -> 0,
330 QuantumNumbers -> {Q -> -1/3},
331 PropagatorLabel -> {"dq", "d", "s", "b"},
332 PropagatorType -> Straight,
333 PropagatorArrow -> Forward,
334 PDG -> {1,3,5},
335 ParticleName -> {"d", "s", "b" },
336 AntiParticleName -> {"d~", "s~", "b~"},
337 FullName -> {"d-quark", "s-quark", "b-quark"}
338 },
339
340(* Fermions: unphysical fields *)
341 F[11] == {
342 ClassName -> LL,
343 Unphysical -> True,
344 Indices -> {Index[SU2D], Index[Generation]},
345 FlavorIndex -> SU2D,
346 SelfConjugate -> False,
347 QuantumNumbers -> {Y -> -1/2},
348 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]] }
349 },
350 F[12] == {
351 ClassName -> lR,
352 Unphysical -> True,
353 Indices -> {Index[Generation]},
354 FlavorIndex -> Generation,
355 SelfConjugate -> False,
356 QuantumNumbers -> {Y -> -1},
357 Definitions -> { lR[sp1_,ff_] :> Module[{sp2}, ProjP[sp1,sp2] l[sp2,ff]] }
358 },
359 F[13] == {
360 ClassName -> QL,
361 Unphysical -> True,
362 Indices -> {Index[SU2D], Index[Generation], Index[Colour]},
363 FlavorIndex -> SU2D,
364 SelfConjugate -> False,
365 QuantumNumbers -> {Y -> 1/6},
366 Definitions -> {
367 QL[sp1_,1,ff_,cc_] :> Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],
368 QL[sp1_,2,ff_,cc_] :> Module[{sp2,ff2}, CKM[ff,ff2] ProjM[sp1,sp2] dq[sp2,ff2,cc]] }
369 },
370 F[14] == {
371 ClassName -> uR,
372 Unphysical -> True,
373 Indices -> {Index[Generation], Index[Colour]},
374 FlavorIndex -> Generation,
375 SelfConjugate -> False,
376 QuantumNumbers -> {Y -> 2/3},
377 Definitions -> { uR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] uq[sp2,ff,cc]] }
378 },
379 F[15] == {
380 ClassName -> dR,
381 Unphysical -> True,
382 Indices -> {Index[Generation], Index[Colour]},
383 FlavorIndex -> Generation,
384 SelfConjugate -> False,
385 QuantumNumbers -> {Y -> -1/3},
386 Definitions -> { dR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] dq[sp2,ff,cc]] }
387 },
388
389(* Higgs: physical scalars *)
390 S[1] == {
391 ClassName -> H,
392 SelfConjugate -> True,
393 Mass -> {MH,125},
394 Width -> {WH,0.00407},
395 PropagatorLabel -> "H",
396 PropagatorType -> D,
397 PropagatorArrow -> None,
398 PDG -> 25,
399 ParticleName -> "H",
400 FullName -> "H"
401 },
402
403(* Higgs: physical scalars *)
404 S[2] == {
405 ClassName -> G0,
406 SelfConjugate -> True,
407 Goldstone -> Z,
408 Mass -> {MZ, 91.1876},
409 Width -> {WZ, 2.4952},
410 PropagatorLabel -> "Go",
411 PropagatorType -> D,
412 PropagatorArrow -> None,
413 PDG -> 250,
414 ParticleName -> "G0",
415 FullName -> "G0"
416 },
417 S[3] == {
418 ClassName -> GP,
419 SelfConjugate -> False,
420 Goldstone -> W,
421 Mass -> {MW, Internal},
422 QuantumNumbers -> {Q -> 1},
423 Width -> {WW, 2.085},
424 PropagatorLabel -> "GP",
425 PropagatorType -> D,
426 PropagatorArrow -> None,
427 PDG -> 251,
428 ParticleName -> "G+",
429 AntiParticleName -> "G-",
430 FullName -> "GP"
431 },
432
433(* Higgs: unphysical scalars *)
434 S[11] == {
435 ClassName -> Phi,
436 Unphysical -> True,
437 Indices -> {Index[SU2D]},
438 FlavorIndex -> SU2D,
439 SelfConjugate -> False,
440 QuantumNumbers -> {Y -> 1/2},
441 Definitions -> { Phi[1] -> -I GP, Phi[2] -> (vev + H + I G0)/Sqrt[2] }
442 }
443};
444
445
446(* ************************** *)
447(* ***** Gauge ***** *)
448(* ***** Parameters ***** *)
449(* ***** (FeynArts) ***** *)
450(* ************************** *)
451
452GaugeXi[ V[1] ] = GaugeXi[A];
453GaugeXi[ V[2] ] = GaugeXi[Z];
454GaugeXi[ V[3] ] = GaugeXi[W];
455GaugeXi[ V[4] ] = GaugeXi[G];
456GaugeXi[ S[1] ] = 1;
457GaugeXi[ S[2] ] = GaugeXi[Z];
458GaugeXi[ S[3] ] = GaugeXi[W];
459GaugeXi[ U[1] ] = GaugeXi[A];
460GaugeXi[ U[2] ] = GaugeXi[Z];
461GaugeXi[ U[31] ] = GaugeXi[W];
462GaugeXi[ U[32] ] = GaugeXi[W];
463GaugeXi[ U[4] ] = GaugeXi[G];
464
465
466(* ************************** *)
467(* ***** Parameters ***** *)
468(* ************************** *)
469M$Parameters = {
470
471 (* External parameters *)
472 aEWM1 == {
473 ParameterType -> External,
474 BlockName -> SMINPUTS,
475 OrderBlock -> 1,
476 Value -> 127.9,
477 InteractionOrder -> {QED,-2},
478 Description -> "Inverse of the EW coupling constant at the Z pole"
479 },
480 Gf == {
481 ParameterType -> External,
482 BlockName -> SMINPUTS,
483 OrderBlock -> 2,
484 Value -> 1.16637*^-5,
485 InteractionOrder -> {QED,2},
486 TeX -> Subscript[G,f],
487 Description -> "Fermi constant"
488 },
489 aS == {
490 ParameterType -> External,
491 BlockName -> SMINPUTS,
492 OrderBlock -> 3,
493 Value -> 0.1184,
494 InteractionOrder -> {QCD,2},
495 TeX -> Subscript[\[Alpha],s],
496 Description -> "Strong coupling constant at the Z pole"
497 },
498 ymdo == {
499 ParameterType -> External,
500 BlockName -> YUKAWA,
501 OrderBlock -> 1,
502 Value -> 5.04*^-3,
503 Description -> "Down Yukawa mass"
504 },
505 ymup == {
506 ParameterType -> External,
507 BlockName -> YUKAWA,
508 OrderBlock -> 2,
509 Value -> 2.55*^-3,
510 Description -> "Up Yukawa mass"
511 },
512 yms == {
513 ParameterType -> External,
514 BlockName -> YUKAWA,
515 OrderBlock -> 3,
516 Value -> 0.101,
517 Description -> "Strange Yukawa mass"
518 },
519 ymc == {
520 ParameterType -> External,
521 BlockName -> YUKAWA,
522 OrderBlock -> 4,
523 Value -> 1.27,
524 Description -> "Charm Yukawa mass"
525 },
526 ymb == {
527 ParameterType -> External,
528 BlockName -> YUKAWA,
529 OrderBlock -> 5,
530 Value -> 4.7,
531 Description -> "Bottom Yukawa mass"
532 },
533 ymt == {
534 ParameterType -> External,
535 BlockName -> YUKAWA,
536 OrderBlock -> 6,
537 Value -> 172,
538 Description -> "Top Yukawa mass"
539 },
540 yme == {
541 ParameterType -> External,
542 BlockName -> YUKAWA,
543 OrderBlock -> 11,
544 Value -> 5.11*^-4,
545 Description -> "Electron Yukawa mass"
546 },
547 ymm == {
548 ParameterType -> External,
549 BlockName -> YUKAWA,
550 OrderBlock -> 13,
551 Value -> 0.10566,
552 Description -> "Muon Yukawa mass"
553 },
554 ymtau == {
555 ParameterType -> External,
556 BlockName -> YUKAWA,
557 OrderBlock -> 15,
558 Value -> 1.777,
559 Description -> "Tau Yukawa mass"
560 },
561 cabi == {
562 ParameterType -> External,
563 BlockName -> CKMBLOCK,
564 OrderBlock -> 1,
565 Value -> 0.227736,
566 TeX -> Subscript[\[Theta], c],
567 Description -> "Cabibbo angle"
568 },
569
570 (* Internal Parameters *)
571 aEW == {
572 ParameterType -> Internal,
573 Value -> 1/aEWM1,
574 InteractionOrder -> {QED,2},
575 TeX -> Subscript[\[Alpha], EW],
576 Description -> "Electroweak coupling contant"
577 },
578 MW == {
579 ParameterType -> Internal,
580 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
581 TeX -> Subscript[M,W],
582 Description -> "W mass"
583 },
584 sw2 == {
585 ParameterType -> Internal,
586 Value -> 1-(MW/MZ)^2,
587 Description -> "Squared Sin of the Weinberg angle"
588 },
589 ee == {
590 ParameterType -> Internal,
591 Value -> Sqrt[4 Pi aEW],
592 InteractionOrder -> {QED,1},
593 TeX -> e,
594 Description -> "Electric coupling constant"
595 },
596 cw == {
597 ParameterType -> Internal,
598 Value -> Sqrt[1-sw2],
599 TeX -> Subscript[c,w],
600 Description -> "Cosine of the Weinberg angle"
601 },
602 sw == {
603 ParameterType -> Internal,
604 Value -> Sqrt[sw2],
605 TeX -> Subscript[s,w],
606 Description -> "Sine of the Weinberg angle"
607 },
608 gw == {
609 ParameterType -> Internal,
610 Definitions -> {gw->ee/sw},
611 InteractionOrder -> {QED,1},
612 TeX -> Subscript[g,w],
613 Description -> "Weak coupling constant at the Z pole"
614 },
615 g1 == {
616 ParameterType -> Internal,
617 Definitions -> {g1->ee/cw},
618 InteractionOrder -> {QED,1},
619 TeX -> Subscript[g,1],
620 Description -> "U(1)Y coupling constant at the Z pole"
621 },
622 gs == {
623 ParameterType -> Internal,
624 Value -> Sqrt[4 Pi aS],
625 InteractionOrder -> {QCD,1},
626 TeX -> Subscript[g,s],
627 ParameterName -> G,
628 Description -> "Strong coupling constant at the Z pole"
629 },
630 vev == {
631 ParameterType -> Internal,
632 Value -> 2*MW*sw/ee,
633 InteractionOrder -> {QED,-1},
634 Description -> "Higgs vacuum expectation value"
635 },
636 lam == {
637 ParameterType -> Internal,
638 Value -> MH^2/(2*vev^2),
639 InteractionOrder -> {QED, 2},
640 Description -> "Higgs quartic coupling"
641 },
642 muH == {
643 ParameterType -> Internal,
644 Value -> Sqrt[vev^2 lam],
645 TeX -> \[Mu],
646 Description -> "Coefficient of the quadratic piece of the Higgs potential"
647 },
648 yl == {
649 ParameterType -> Internal,
650 Indices -> {Index[Generation], Index[Generation]},
651 Definitions -> {yl[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
652 Value -> {yl[1,1] -> Sqrt[2] yme / vev, yl[2,2] -> Sqrt[2] ymm / vev, yl[3,3] -> Sqrt[2] ymtau / vev},
653 InteractionOrder -> {QED, 1},
654 ParameterName -> {yl[1,1] -> ye, yl[2,2] -> ym, yl[3,3] -> ytau},
655 TeX -> Superscript[y, l],
656 Description -> "Lepton Yukawa couplings"
657 },
658 yu == {
659 ParameterType -> Internal,
660 Indices -> {Index[Generation], Index[Generation]},
661 Definitions -> {yu[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
662 Value -> {yu[1,1] -> Sqrt[2] ymup/vev, yu[2,2] -> Sqrt[2] ymc/vev, yu[3,3] -> Sqrt[2] ymt/vev},
663 InteractionOrder -> {QED, 1},
664 ParameterName -> {yu[1,1] -> yup, yu[2,2] -> yc, yu[3,3] -> yt},
665 TeX -> Superscript[y, u],
666 Description -> "Up-type Yukawa couplings"
667 },
668 yd == {
669 ParameterType -> Internal,
670 Indices -> {Index[Generation], Index[Generation]},
671 Definitions -> {yd[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
672 Value -> {yd[1,1] -> Sqrt[2] ymdo/vev, yd[2,2] -> Sqrt[2] yms/vev, yd[3,3] -> Sqrt[2] ymb/vev},
673 InteractionOrder -> {QED, 1},
674 ParameterName -> {yd[1,1] -> ydo, yd[2,2] -> ys, yd[3,3] -> yb},
675 TeX -> Superscript[y, d],
676 Description -> "Down-type Yukawa couplings"
677 },
678(* N. B. : only Cabibbo mixing! *)
679 CKM == {
680 ParameterType -> Internal,
681 Indices -> {Index[Generation], Index[Generation]},
682 Unitary -> True,
683 Value -> {CKM[1,1] -> Cos[cabi], CKM[1,2] -> Sin[cabi], CKM[1,3] -> 0,
684 CKM[2,1] -> -Sin[cabi], CKM[2,2] -> Cos[cabi], CKM[2,3] -> 0,
685 CKM[3,1] -> 0, CKM[3,2] -> 0, CKM[3,3] -> 1},
686 TeX -> Superscript[V,CKM],
687 Description -> "CKM-Matrix"}
688};
689
690(* ************************** *)
691(* ***** Lagrangian ***** *)
692(* ************************** *)
693
694LGauge := Block[{mu,nu,ii,aa},
695 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]];
696
697LFermions := Block[{mu},
698 ExpandIndices[I*(
699 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]),
700 FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]}];
701
702LHiggs := Block[{ii,mu, feynmangaugerules},
703 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
704
705 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
706 ];
707
708LYukawa := Block[{sp,ii,jj,cc,ff1,ff2,ff3,yuk,feynmangaugerules},
709 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
710
711 yuk = ExpandIndices[
712 -yd[ff2, ff3] CKM[ff1, ff2] QLbar[sp, ii, ff1, cc].dR [sp, ff3, cc] Phi[ii] -
713 yl[ff1, ff3] LLbar[sp, ii, ff1].lR [sp, ff3] Phi[ii] -
714 yu[ff1, ff2] QLbar[sp, ii, ff1, cc].uR [sp, ff2, cc] Phibar[jj] Eps[ii, jj], FlavorExpand -> SU2D];
715 yuk = yuk /. { CKM[a_, b_] Conjugate[CKM[a_, c_]] -> IndexDelta[b, c], CKM[b_, a_] Conjugate[CKM[c_, a_]] -> IndexDelta[b, c]};
716 yuk+HC[yuk]/.feynmangaugerules
717 ];
718
719LGhost := Block[{LGh1,LGhw,LGhs,LGhphi,mu, generators,gh,ghbar,Vectorize,phi1,phi2,togoldstones,doublet,doublet0},
720 (* Pure gauge piece *)
721 LGh1 = -ghBbar.del[DC[ghB,mu],mu];
722 LGhw = -ghWibar.del[DC[ghWi,mu],mu];
723 LGhs = -ghGbar.del[DC[ghG,mu],mu];
724
725 (* Scalar pieces: see Peskin pages 739-742 *)
726 (* phi1 and phi2 are the real degrees of freedom of GP *)
727 (* Vectorize transforms a doublet in a vector in the phi-basis, i.e. the basis of real degrees of freedom *)
728 gh = {ghB, ghWi[1], ghWi[2], ghWi[3]};
729 ghbar = {ghBbar, ghWibar[1], ghWibar[2], ghWibar[3]};
730 generators = {-I/2 g1 IdentityMatrix[2], -I/2 gw PauliSigma[1], -I/2 gw PauliSigma[2], -I/2 gw PauliSigma[3]};
731 doublet = Expand[{(-I phi1 - phi2)/Sqrt[2], Phi[2]} /. MR$Definitions /. vev -> 0];
732 doublet0 = {0, vev/Sqrt[2]};
733 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}];
734 togoldstones := {phi1 -> (GP + GPbar)/Sqrt[2], phi2 -> (-GP + GPbar)/(I Sqrt[2])};
735 LGhphi=Plus@@Flatten[Table[-ghbar[[kkk]].gh[[lll]] Vectorize[generators[[kkk]].doublet0].Vectorize[generators[[lll]].(doublet+doublet0)],{kkk,4},{lll,4}]] /.togoldstones;
736
737ExpandIndices[ LGhs + If[FeynmanGauge, LGh1 + LGhw + LGhphi,0], FlavorExpand->SU2W]];
738
739LSM:= LGauge + LFermions + LHiggs + LYukawa + LGhost;