VPolarization: sm_loop_zpolar.fr

File sm_loop_zpolar.fr, 28.8 KB (added by Richard Ruiz, 6 months ago)

FeynRules model file for Z with polarization

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 Unphysical -> True,
142 SelfConjugate -> True,
143 Mass -> {MZX, Internal},
144 Width -> {WZX, Internal},
145 ParticleName -> "Z",
146 PDG -> 239,
147 PropagatorLabel -> "Z",
148 PropagatorType -> Sine,
149 PropagatorArrow -> None,
150 FullName -> "Z",
151 Definitions -> {Z[mu_] -> Z0[mu] + ZT[mu] + ZA[mu] + ZX[mu]}
152 },
153 V[230] == {
154 ClassName -> Z0,
155 SelfConjugate -> True,
156 Mass -> {MZ0, Internal},
157 Width -> {WZ0, Internal},
158 ParticleName -> "Z0",
159 PDG -> 230,
160 PropagatorLabel -> "Z0",
161 PropagatorType -> Sine,
162 PropagatorArrow -> None,
163 FullName -> "Z0"
164 },
165 V[231] == {
166 ClassName -> ZT,
167 SelfConjugate -> True,
168 Mass -> {MZT, Internal},
169 Width -> {WZT, Internal},
170 ParticleName -> "ZT",
171 PDG -> 231,
172 PropagatorLabel -> "ZT",
173 PropagatorType -> Sine,
174 PropagatorArrow -> None,
175 FullName -> "ZT"
176 },
177 V[232] == {
178 ClassName -> ZA,
179 SelfConjugate -> True,
180 Mass -> {MZA, Internal},
181 Width -> {WZA, Internal},
182 ParticleName -> "ZA",
183 PDG -> 232,
184 PropagatorLabel -> "ZA",
185 PropagatorType -> Sine,
186 PropagatorArrow -> None,
187 FullName -> "ZA"
188 },
189 V[233] == {
190 ClassName -> ZX,
191 SelfConjugate -> True,
192 Mass -> {MZ, 91.1876},
193 Width -> {WZ, 2.4952},
194 ParticleName -> "ZX",
195 PDG -> 23,
196 PropagatorLabel -> "ZX",
197 PropagatorType -> Sine,
198 PropagatorArrow -> None,
199 FullName -> "ZX"
200 },
201 V[3] == {
202 ClassName -> W,
203 SelfConjugate -> False,
204 Mass -> {MW, Internal},
205 Width -> {WW, 2.085},
206 ParticleName -> "W+",
207 AntiParticleName -> "W-",
208 QuantumNumbers -> {Q -> 1},
209 PDG -> 24,
210 PropagatorLabel -> "W",
211 PropagatorType -> Sine,
212 PropagatorArrow -> Forward,
213 FullName -> "W"
214 },
215 V[4] == {
216 ClassName -> G,
217 SelfConjugate -> True,
218 Indices -> {Index[Gluon]},
219 Mass -> 0,
220 Width -> 0,
221 ParticleName -> "g",
222 PDG -> 21,
223 PropagatorLabel -> "G",
224 PropagatorType -> C,
225 PropagatorArrow -> None,
226 FullName -> "G"
227 },
228
229(* Ghosts: related to physical gauge bosons *)
230 U[1] == {
231 ClassName -> ghA,
232 SelfConjugate -> False,
233 Ghost -> A,
234 QuantumNumbers -> {GhostNumber -> 1},
235 Mass -> 0,
236 Width -> 0,
237 PropagatorLabel -> "uA",
238 PropagatorType -> GhostDash,
239 PropagatorArrow -> Forward
240 },
241 U[2] == {
242 ClassName -> ghZ,
243 SelfConjugate -> False,
244 Ghost -> Z,
245 QuantumNumbers -> {GhostNumber -> 1},
246 Mass -> {MZ,91.1876},
247 Width -> {WZ, 2.4952},
248 PropagatorLabel -> "uZ",
249 PropagatorType -> GhostDash,
250 PropagatorArrow -> Forward
251 },
252 U[31] == {
253 ClassName -> ghWp,
254 SelfConjugate -> False,
255 Ghost -> W,
256 QuantumNumbers -> {GhostNumber -> 1, Q -> 1},
257 Mass -> {MW,Internal},
258 Width -> {WW, 2.085},
259 PropagatorLabel -> "uWp",
260 PropagatorType -> GhostDash,
261 PropagatorArrow -> Forward
262 },
263 U[32] == {
264 ClassName -> ghWm,
265 SelfConjugate -> False,
266 Ghost -> Wbar,
267 QuantumNumbers -> {GhostNumber -> 1, Q -> -1},
268 Mass -> {MW,Internal},
269 Width -> {WW, 2.085},
270 PropagatorLabel -> "uWm",
271 PropagatorType -> GhostDash,
272 PropagatorArrow -> Forward
273 },
274 U[4] == {
275 ClassName -> ghG,
276 SelfConjugate -> False,
277 Indices -> {Index[Gluon]},
278 Ghost -> G,
279 PDG -> 82,
280 QuantumNumbers ->{GhostNumber -> 1},
281 Mass -> 0,
282 Width -> 0,
283 PropagatorLabel -> "uG",
284 PropagatorType -> GhostDash,
285 PropagatorArrow -> Forward
286 },
287
288(* Gauge bosons: unphysical vector fields *)
289 V[11] == {
290 ClassName -> B,
291 Unphysical -> True,
292 SelfConjugate -> True,
293 Definitions -> { B[mu_] -> -sw Z[mu]+cw A[mu]}
294 },
295 V[12] == {
296 ClassName -> Wi,
297 Unphysical -> True,
298 SelfConjugate -> True,
299 Indices -> {Index[SU2W]},
300 FlavorIndex -> SU2W,
301 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]}
302 },
303
304(* Ghosts: related to unphysical gauge bosons *)
305 U[11] == {
306 ClassName -> ghB,
307 Unphysical -> True,
308 SelfConjugate -> False,
309 Ghost -> B,
310 Definitions -> { ghB -> -sw ghZ + cw ghA}
311 },
312 U[12] == {
313 ClassName -> ghWi,
314 Unphysical -> True,
315 SelfConjugate -> False,
316 Ghost -> Wi,
317 Indices -> {Index[SU2W]},
318 FlavorIndex -> SU2W,
319 Definitions -> { ghWi[1] -> (ghWp+ghWm)/Sqrt[2], ghWi[2] -> (ghWm-ghWp)/(I*Sqrt[2]), ghWi[3] -> cw ghZ+sw ghA}
320 } ,
321
322(* Fermions: physical fields *)
323 F[1] == {
324 ClassName -> vl,
325 ClassMembers -> {ve,vm,vt},
326 Indices -> {Index[Generation]},
327 FlavorIndex -> Generation,
328 SelfConjugate -> False,
329 Mass -> 0,
330 Width -> 0,
331 QuantumNumbers -> {LeptonNumber -> 1},
332 PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
333 PropagatorType -> S,
334 PropagatorArrow -> Forward,
335 PDG -> {12,14,16},
336 ParticleName -> {"ve","vm","vt"},
337 AntiParticleName -> {"ve~","vm~","vt~"},
338 FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"}
339 },
340 F[2] == {
341 ClassName -> l,
342 ClassMembers -> {e, mu, ta},
343 Indices -> {Index[Generation]},
344 FlavorIndex -> Generation,
345 SelfConjugate -> False,
346 Mass -> {Ml, {Me,5.11*^-4}, {MMU,0.10566}, {MTA,1.777}},
347 Width -> 0,
348 QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
349 PropagatorLabel -> {"l", "e", "mu", "ta"},
350 PropagatorType -> Straight,
351 PropagatorArrow -> Forward,
352 PDG -> {11, 13, 15},
353 ParticleName -> {"e-", "mu-", "ta-"},
354 AntiParticleName -> {"e+", "mu+", "ta+"},
355 FullName -> {"Electron", "Muon", "Tau"}
356 },
357 F[3] == {
358 ClassName -> uq,
359 ClassMembers -> {u, c, t},
360 Indices -> {Index[Generation], Index[Colour]},
361 FlavorIndex -> Generation,
362 SelfConjugate -> False,
363 Mass -> {Mu, {MU, 2.55*^-3}, {MC,1.27}, {MT,173.3}},
364 Width -> {0, 0, {WT,1.350}},
365 QuantumNumbers -> {Q -> 2/3},
366 PropagatorLabel -> {"uq", "u", "c", "t"},
367 PropagatorType -> Straight,
368 PropagatorArrow -> Forward,
369 PDG -> {2, 4, 6},
370 ParticleName -> {"u", "c", "t" },
371 AntiParticleName -> {"u~", "c~", "t~"},
372 FullName -> {"u-quark", "c-quark", "t-quark"}
373 },
374 F[4] == {
375 ClassName -> dq,
376 ClassMembers -> {d, s, b},
377 Indices -> {Index[Generation], Index[Colour]},
378 FlavorIndex -> Generation,
379 SelfConjugate -> False,
380 Mass -> {Md, {MD,5.04*^-3}, {MS,0.101}, {MB,4.7}},
381 Width -> 0,
382 QuantumNumbers -> {Q -> -1/3},
383 PropagatorLabel -> {"dq", "d", "s", "b"},
384 PropagatorType -> Straight,
385 PropagatorArrow -> Forward,
386 PDG -> {1,3,5},
387 ParticleName -> {"d", "s", "b" },
388 AntiParticleName -> {"d~", "s~", "b~"},
389 FullName -> {"d-quark", "s-quark", "b-quark"}
390 },
391
392(* Fermions: unphysical fields *)
393 F[11] == {
394 ClassName -> LL,
395 Unphysical -> True,
396 Indices -> {Index[SU2D], Index[Generation]},
397 FlavorIndex -> SU2D,
398 SelfConjugate -> False,
399 QuantumNumbers -> {Y -> -1/2},
400 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]] }
401 },
402 F[12] == {
403 ClassName -> lR,
404 Unphysical -> True,
405 Indices -> {Index[Generation]},
406 FlavorIndex -> Generation,
407 SelfConjugate -> False,
408 QuantumNumbers -> {Y -> -1},
409 Definitions -> { lR[sp1_,ff_] :> Module[{sp2}, ProjP[sp1,sp2] l[sp2,ff]] }
410 },
411 F[13] == {
412 ClassName -> QL,
413 Unphysical -> True,
414 Indices -> {Index[SU2D], Index[Generation], Index[Colour]},
415 FlavorIndex -> SU2D,
416 SelfConjugate -> False,
417 QuantumNumbers -> {Y -> 1/6},
418 Definitions -> {
419 QL[sp1_,1,ff_,cc_] :> Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],
420 QL[sp1_,2,ff_,cc_] :> Module[{sp2,ff2}, CKM[ff,ff2] ProjM[sp1,sp2] dq[sp2,ff2,cc]] }
421 },
422 F[14] == {
423 ClassName -> uR,
424 Unphysical -> True,
425 Indices -> {Index[Generation], Index[Colour]},
426 FlavorIndex -> Generation,
427 SelfConjugate -> False,
428 QuantumNumbers -> {Y -> 2/3},
429 Definitions -> { uR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] uq[sp2,ff,cc]] }
430 },
431 F[15] == {
432 ClassName -> dR,
433 Unphysical -> True,
434 Indices -> {Index[Generation], Index[Colour]},
435 FlavorIndex -> Generation,
436 SelfConjugate -> False,
437 QuantumNumbers -> {Y -> -1/3},
438 Definitions -> { dR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] dq[sp2,ff,cc]] }
439 },
440
441(* Higgs: physical scalars *)
442 S[1] == {
443 ClassName -> H,
444 SelfConjugate -> True,
445 Mass -> {MH,125.7},
446 Width -> {WH,0.004170},
447 PropagatorLabel -> "H",
448 PropagatorType -> D,
449 PropagatorArrow -> None,
450 PDG -> 25,
451 ParticleName -> "H",
452 FullName -> "H"
453 },
454
455(* Higgs: physical scalars *)
456 S[2] == {
457 ClassName -> G0,
458 SelfConjugate -> True,
459 Goldstone -> Z,
460 Mass -> {MZ, 91.1876},
461 Width -> {WZ, 2.4952},
462 PropagatorLabel -> "Go",
463 PropagatorType -> D,
464 PropagatorArrow -> None,
465 PDG -> 250,
466 ParticleName -> "G0",
467 FullName -> "G0"
468 },
469 S[3] == {
470 ClassName -> GP,
471 SelfConjugate -> False,
472 Goldstone -> W,
473 Mass -> {MW, Internal},
474 QuantumNumbers -> {Q -> 1},
475 Width -> {WW, 2.085},
476 PropagatorLabel -> "GP",
477 PropagatorType -> D,
478 PropagatorArrow -> None,
479 PDG -> 251,
480 ParticleName -> "G+",
481 AntiParticleName -> "G-",
482 FullName -> "GP"
483 },
484
485(* Higgs: unphysical scalars *)
486 S[11] == {
487 ClassName -> Phi,
488 Unphysical -> True,
489 Indices -> {Index[SU2D]},
490 FlavorIndex -> SU2D,
491 SelfConjugate -> False,
492 QuantumNumbers -> {Y -> 1/2},
493 Definitions -> { Phi[1] -> -I GP, Phi[2] -> (vev + H + I G0)/Sqrt[2] }
494 }
495};
496
497
498(* ************************** *)
499(* ***** Gauge ***** *)
500(* ***** Parameters ***** *)
501(* ***** (FeynArts) ***** *)
502(* ************************** *)
503
504GaugeXi[ V[1] ] = GaugeXi[A];
505GaugeXi[ V[2] ] = GaugeXi[Z];
506GaugeXi[ V[3] ] = GaugeXi[W];
507GaugeXi[ V[4] ] = GaugeXi[G];
508GaugeXi[ S[1] ] = 1;
509GaugeXi[ S[2] ] = GaugeXi[Z];
510GaugeXi[ S[3] ] = GaugeXi[W];
511GaugeXi[ U[1] ] = GaugeXi[A];
512GaugeXi[ U[2] ] = GaugeXi[Z];
513GaugeXi[ U[31] ] = GaugeXi[W];
514GaugeXi[ U[32] ] = GaugeXi[W];
515GaugeXi[ U[4] ] = GaugeXi[G];
516
517
518(* ************************** *)
519(* ***** Parameters ***** *)
520(* ************************** *)
521M$Parameters = {
522
523 (* External parameters *)
524 aEWM1 == {
525 ParameterType -> External,
526 BlockName -> SMINPUTS,
527 OrderBlock -> 1,
528 Value -> 127.94,
529 InteractionOrder -> {QED,-2},
530 Description -> "Inverse of the EW coupling constant at the Z pole"
531 },
532 Gf == {
533 ParameterType -> External,
534 BlockName -> SMINPUTS,
535 OrderBlock -> 2,
536 Value -> 1.17456*^-5,
537 InteractionOrder -> {QED,2},
538 TeX -> Subscript[G,f],
539 Description -> "Fermi constant"
540 },
541 aS == {
542 ParameterType -> External,
543 BlockName -> SMINPUTS,
544 OrderBlock -> 3,
545 Value -> 0.1184,
546 InteractionOrder -> {QCD,2},
547 TeX -> Subscript[\[Alpha],s],
548 Description -> "Strong coupling constant at the Z pole"
549 },
550 ymdo == {
551 ParameterType -> External,
552 BlockName -> YUKAWA,
553 OrderBlock -> 1,
554 Value -> 5.04*^-3,
555 Description -> "Down Yukawa mass"
556 },
557 ymup == {
558 ParameterType -> External,
559 BlockName -> YUKAWA,
560 OrderBlock -> 2,
561 Value -> 2.55*^-3,
562 Description -> "Up Yukawa mass"
563 },
564 yms == {
565 ParameterType -> External,
566 BlockName -> YUKAWA,
567 OrderBlock -> 3,
568 Value -> 0.101,
569 Description -> "Strange Yukawa mass"
570 },
571 ymc == {
572 ParameterType -> External,
573 BlockName -> YUKAWA,
574 OrderBlock -> 4,
575 Value -> 1.27,
576 Description -> "Charm Yukawa mass"
577 },
578 ymb == {
579 ParameterType -> External,
580 BlockName -> YUKAWA,
581 OrderBlock -> 5,
582 Value -> 4.7,
583 Description -> "Bottom Yukawa mass"
584 },
585 ymt == {
586 ParameterType -> External,
587 BlockName -> YUKAWA,
588 OrderBlock -> 6,
589 Value -> 173.3,
590 Description -> "Top Yukawa mass"
591 },
592 yme == {
593 ParameterType -> External,
594 BlockName -> YUKAWA,
595 OrderBlock -> 11,
596 Value -> 5.11*^-4,
597 Description -> "Electron Yukawa mass"
598 },
599 ymm == {
600 ParameterType -> External,
601 BlockName -> YUKAWA,
602 OrderBlock -> 13,
603 Value -> 0.10566,
604 Description -> "Muon Yukawa mass"
605 },
606 ymtau == {
607 ParameterType -> External,
608 BlockName -> YUKAWA,
609 OrderBlock -> 15,
610 Value -> 1.777,
611 Description -> "Tau Yukawa mass"
612 },
613 cabi == {
614 ParameterType -> External,
615 BlockName -> CKMBLOCK,
616 OrderBlock -> 1,
617 Value -> 0.227591,
618 TeX -> Subscript[\[Theta], c],
619 Description -> "Cabibbo angle"
620 },
621 th13 == {
622 ParameterType -> External,
623 BlockName -> CKMBLOCK,
624 OrderBlock -> 2,
625 Value -> 0.003508,
626 TeX -> Subscript[\[Theta], 13],
627 Description -> "CKM Theta 13"
628 },
629 th23 == {
630 ParameterType -> External,
631 BlockName -> CKMBLOCK,
632 OrderBlock -> 3,
633 Value -> 0.041539,
634 TeX -> Subscript[\[Theta], 23],
635 Description -> "CKM theta 23"
636 },
637 del13 == {
638 ParameterType -> External,
639 BlockName -> CKMBLOCK,
640 OrderBlock -> 4,
641 Value -> 1.20,
642 TeX -> Subscript[\[Delta], 13],
643 Description -> "CKM delta 13"
644 },
645
646 (* Internal Parameters *)
647 aEW == {
648 ParameterType -> Internal,
649 Value -> 1/aEWM1,
650 InteractionOrder -> {QED,2},
651 TeX -> Subscript[\[Alpha], EW],
652 Description -> "Electroweak coupling contant"
653 },
654 MW == {
655 ParameterType -> Internal,
656 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
657 TeX -> Subscript[M,W],
658 Description -> "W mass"
659 },
660 MZX == {
661 ParameterType -> Internal,
662 Value -> MZ,
663 TeX -> Subscript[M,Z],
664 Description -> "Z mass"
665 },
666 MZ0 == {
667 ParameterType -> Internal,
668 Value -> MZ,
669 TeX -> Subscript[M,Z],
670 Description -> "Z mass"
671 },
672 MZT == {
673 ParameterType -> Internal,
674 Value -> MZ,
675 TeX -> Subscript[M,Z],
676 Description -> "Z mass"
677 },
678 MZA == {
679 ParameterType -> Internal,
680 Value -> MZ,
681 TeX -> Subscript[M,Z],
682 Description -> "Z mass"
683 },
684 WZX == {
685 ParameterType -> Internal,
686 Value -> WZ,
687 TeX -> Subscript[W,Z],
688 Description -> "Z width"
689 },
690 WZ0 == {
691 ParameterType -> Internal,
692 Value -> WZ,
693 TeX -> Subscript[W,Z],
694 Description -> "Z width"
695 },
696 WZT == {
697 ParameterType -> Internal,
698 Value -> WZ,
699 TeX -> Subscript[W,Z],
700 Description -> "Z width"
701 },
702 WZA == {
703 ParameterType -> Internal,
704 Value -> WZ,
705 TeX -> Subscript[W,Z],
706 Description -> "Z width"
707 },
708 sw2 == {
709 ParameterType -> Internal,
710 Value -> 1-(MW/MZ)^2,
711 Description -> "Squared Sin of the Weinberg angle"
712 },
713 ee == {
714 ParameterType -> Internal,
715 Value -> Sqrt[4 Pi aEW],
716 InteractionOrder -> {QED,1},
717 TeX -> e,
718 Description -> "Electric coupling constant"
719 },
720 cw == {
721 ParameterType -> Internal,
722 Value -> Sqrt[1-sw2],
723 TeX -> Subscript[c,w],
724 Description -> "Cosine of the Weinberg angle"
725 },
726 sw == {
727 ParameterType -> Internal,
728 Value -> Sqrt[sw2],
729 TeX -> Subscript[s,w],
730 Description -> "Sine of the Weinberg angle"
731 },
732 gw == {
733 ParameterType -> Internal,
734 Definitions -> {gw->ee/sw},
735 InteractionOrder -> {QED,1},
736 TeX -> Subscript[g,w],
737 Description -> "Weak coupling constant at the Z pole"
738 },
739 g1 == {
740 ParameterType -> Internal,
741 Definitions -> {g1->ee/cw},
742 InteractionOrder -> {QED,1},
743 TeX -> Subscript[g,1],
744 Description -> "U(1)Y coupling constant at the Z pole"
745 },
746 gs == {
747 ParameterType -> Internal,
748 Value -> Sqrt[4 Pi aS],
749 InteractionOrder -> {QCD,1},
750 TeX -> Subscript[g,s],
751 ParameterName -> G,
752 Description -> "Strong coupling constant at the Z pole"
753 },
754 vev == {
755 ParameterType -> Internal,
756 Value -> 2*MW*sw/ee,
757 InteractionOrder -> {QED,-1},
758 Description -> "Higgs vacuum expectation value"
759 },
760 lam == {
761 ParameterType -> Internal,
762 Value -> MH^2/(2*vev^2),
763 InteractionOrder -> {QED, 2},
764 Description -> "Higgs quartic coupling"
765 },
766 muH == {
767 ParameterType -> Internal,
768 Value -> Sqrt[vev^2 lam],
769 TeX -> \[Mu],
770 Description -> "Coefficient of the quadratic piece of the Higgs potential"
771 },
772 yl == {
773 ParameterType -> Internal,
774 Indices -> {Index[Generation], Index[Generation]},
775 Definitions -> {yl[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
776 Value -> {yl[1,1] -> Sqrt[2] yme / vev, yl[2,2] -> Sqrt[2] ymm / vev, yl[3,3] -> Sqrt[2] ymtau / vev},
777 InteractionOrder -> {QED, 1},
778 ParameterName -> {yl[1,1] -> ye, yl[2,2] -> ym, yl[3,3] -> ytau},
779 TeX -> Superscript[y, l],
780 Description -> "Lepton Yukawa couplings"
781 },
782 yu == {
783 ParameterType -> Internal,
784 Indices -> {Index[Generation], Index[Generation]},
785 Definitions -> {yu[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
786 Value -> {yu[1,1] -> Sqrt[2] ymup/vev, yu[2,2] -> Sqrt[2] ymc/vev, yu[3,3] -> Sqrt[2] ymt/vev},
787 InteractionOrder -> {QED, 1},
788 ParameterName -> {yu[1,1] -> yup, yu[2,2] -> yc, yu[3,3] -> yt},
789 TeX -> Superscript[y, u],
790 Description -> "Up-type Yukawa couplings"
791 },
792 yd == {
793 ParameterType -> Internal,
794 Indices -> {Index[Generation], Index[Generation]},
795 Definitions -> {yd[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
796 Value -> {yd[1,1] -> Sqrt[2] ymdo/vev, yd[2,2] -> Sqrt[2] yms/vev, yd[3,3] -> Sqrt[2] ymb/vev},
797 InteractionOrder -> {QED, 1},
798 ParameterName -> {yd[1,1] -> ydo, yd[2,2] -> ys, yd[3,3] -> yb},
799 TeX -> Superscript[y, d],
800 Description -> "Down-type Yukawa couplings"
801 },
802(* N. B. : only Cabibbo mixing! *)
803 CKM == {
804 ParameterType -> Internal,
805 Indices -> {Index[Generation], Index[Generation]},
806 Unitary -> True,
807 Value -> {CKM[1,1] -> Cos[cabi] Cos[th13], CKM[1,2] -> Sin[cabi] Cos[th13], CKM[1,3] -> Sin[th13] Exp[-I del13],
808 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],
809 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]},
810 TeX -> Superscript[V,CKM],
811 Description -> "CKM-Matrix"}
812};
813
814(* ************************** *)
815(* ***** Lagrangian ***** *)
816(* ************************** *)
817
818LZPolarMassBase := (MZ*MZ/2) * Z0[mu] * (ZT[mu] + ZA[mu] + ZX[mu]) \
819 + (MZ*MZ/2) * ZT[mu] * (Z0[mu] + ZA[mu] + ZX[mu]) \
820 + (MZ*MZ/2) * ZA[mu] * (Z0[mu] + ZT[mu] + ZX[mu]) \
821 + (MZ*MZ/2) * ZX[mu] * (Z0[mu] + ZT[mu] + ZA[mu]) ;
822LPolarMass := LZPolarMassBase;
823
824LGauge := Block[{mu,nu,ii,aa},
825 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]];
826
827LFermions := Block[{mu},
828 ExpandIndices[I*(
829 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]),
830 FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]}];
831
832LHiggsBase := Block[{ii,mu, feynmangaugerules},
833 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
834
835 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
836 ];
837LHiggs := Simplify[LHiggsBase - LPolarMass];
838
839LYukawa := Block[{sp,ii,jj,cc,ff1,ff2,ff3,yuk,feynmangaugerules},
840 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
841
842 yuk = ExpandIndices[
843 -yd[ff2, ff3] CKM[ff1, ff2] QLbar[sp, ii, ff1, cc].dR [sp, ff3, cc] Phi[ii] -
844 yl[ff1, ff3] LLbar[sp, ii, ff1].lR [sp, ff3] Phi[ii] -
845 yu[ff1, ff2] QLbar[sp, ii, ff1, cc].uR [sp, ff2, cc] Phibar[jj] Eps[ii, jj], FlavorExpand -> SU2D];
846 yuk = yuk /. { CKM[a_, b_] Conjugate[CKM[a_, c_]] -> IndexDelta[b, c], CKM[b_, a_] Conjugate[CKM[c_, a_]] -> IndexDelta[b, c]};
847 yuk+HC[yuk]/.feynmangaugerules
848 ];
849
850LGhost := Block[{LGh1,LGhw,LGhs,LGhphi,mu, generators,gh,ghbar,Vectorize,phi1,phi2,togoldstones,doublet,doublet0},
851 (* Pure gauge piece *)
852 LGh1 = -ghBbar.del[DC[ghB,mu],mu];
853 LGhw = -ghWibar[ii].del[DC[ghWi[ii],mu],mu];
854 LGhs = -ghGbar[ii].del[DC[ghG[ii],mu],mu];
855
856 (* Scalar pieces: see Peskin pages 739-742 *)
857 (* phi1 and phi2 are the real degrees of freedom of GP *)
858 (* Vectorize transforms a doublet in a vector in the phi-basis, i.e. the basis of real degrees of freedom *)
859 gh = {ghB, ghWi[1], ghWi[2], ghWi[3]};
860 ghbar = {ghBbar, ghWibar[1], ghWibar[2], ghWibar[3]};
861 generators = {-I/2 g1 IdentityMatrix[2], -I/2 gw PauliSigma[1], -I/2 gw PauliSigma[2], -I/2 gw PauliSigma[3]};
862 doublet = Expand[{(-I phi1 - phi2)/Sqrt[2], Phi[2]} /. MR$Definitions /. vev -> 0];
863 doublet0 = {0, vev/Sqrt[2]};
864 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}];
865 togoldstones := {phi1 -> (GP + GPbar)/Sqrt[2], phi2 -> (-GP + GPbar)/(I Sqrt[2])};
866 LGhphi=Plus@@Flatten[Table[-ghbar[[kkk]].gh[[lll]] Vectorize[generators[[kkk]].doublet0].Vectorize[generators[[lll]].(doublet+doublet0)],{kkk,4},{lll,4}]] /.togoldstones;
867
868ExpandIndices[ LGhs + If[FeynmanGauge, LGh1 + LGhw + LGhphi,0], FlavorExpand->SU2W]];
869
870LSM:= LGauge + LFermions + LHiggs + LYukawa + LGhost;