VPolarization: sm_loop_vpolar.fr

File sm_loop_vpolar.fr, 32.1 KB (added by Richard Ruiz, 21 months ago)

FeynRules model file for W and 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 Unphysical -> True,
204 SelfConjugate -> False,
205 Mass -> {MWX, Internal},
206 Width -> {WWX, Internal},
207 ParticleName -> "W+",
208 AntiParticleName -> "W-",
209 QuantumNumbers -> {Q -> 1},
210 PDG -> 249,
211 PropagatorLabel -> "W",
212 PropagatorType -> Sine,
213 PropagatorArrow -> Forward,
214 FullName -> "W",
215 Definitions -> {W[mu_] -> W0[mu] + WT[mu] + WA[mu] + WX[mu]}
216 },
217 V[240] == {
218 ClassName -> W0,
219 SelfConjugate -> False,
220 Mass -> {MW0, Internal},
221 Width -> {WW0, Internal},
222 ParticleName -> "W0+",
223 AntiParticleName -> "W0-",
224 QuantumNumbers -> {Q -> 1},
225 PDG -> 240,
226 PropagatorLabel -> "W0",
227 PropagatorType -> Sine,
228 PropagatorArrow -> Forward,
229 FullName -> "W0"
230 },
231 V[241] == {
232 ClassName -> WT,
233 SelfConjugate -> False,
234 Mass -> {MWT, Internal},
235 Width -> {WWT, Internal},
236 ParticleName -> "WT+",
237 AntiParticleName -> "WT-",
238 QuantumNumbers -> {Q -> 1},
239 PDG -> 241,
240 PropagatorLabel -> "WT",
241 PropagatorType -> Sine,
242 PropagatorArrow -> Forward,
243 FullName -> "WT"
244 },
245 V[242] == {
246 ClassName -> WA,
247 SelfConjugate -> False,
248 Mass -> {MWA, Internal},
249 Width -> {WWA, Internal},
250 ParticleName -> "WA+",
251 AntiParticleName -> "WA-",
252 QuantumNumbers -> {Q -> 1},
253 PDG -> 242,
254 PropagatorLabel -> "WA",
255 PropagatorType -> Sine,
256 PropagatorArrow -> Forward,
257 FullName -> "WA"
258 },
259 V[243] == {
260 ClassName -> WX,
261 SelfConjugate -> False,
262 Mass -> {MW, Internal},
263 Width -> {WW, 2.085},
264 ParticleName -> "WX+",
265 AntiParticleName -> "WX-",
266 QuantumNumbers -> {Q -> 1},
267 PDG -> 24,
268 PropagatorLabel -> "WX",
269 PropagatorType -> Sine,
270 PropagatorArrow -> Forward,
271 FullName -> "WX"
272 },
273 V[4] == {
274 ClassName -> G,
275 SelfConjugate -> True,
276 Indices -> {Index[Gluon]},
277 Mass -> 0,
278 Width -> 0,
279 ParticleName -> "g",
280 PDG -> 21,
281 PropagatorLabel -> "G",
282 PropagatorType -> C,
283 PropagatorArrow -> None,
284 FullName -> "G"
285 },
286
287(* Ghosts: related to physical gauge bosons *)
288 U[1] == {
289 ClassName -> ghA,
290 SelfConjugate -> False,
291 Ghost -> A,
292 QuantumNumbers -> {GhostNumber -> 1},
293 Mass -> 0,
294 Width -> 0,
295 PropagatorLabel -> "uA",
296 PropagatorType -> GhostDash,
297 PropagatorArrow -> Forward
298 },
299 U[2] == {
300 ClassName -> ghZ,
301 SelfConjugate -> False,
302 Ghost -> Z,
303 QuantumNumbers -> {GhostNumber -> 1},
304 Mass -> {MZ,91.1876},
305 Width -> {WZ, 2.4952},
306 PropagatorLabel -> "uZ",
307 PropagatorType -> GhostDash,
308 PropagatorArrow -> Forward
309 },
310 U[31] == {
311 ClassName -> ghWp,
312 SelfConjugate -> False,
313 Ghost -> W,
314 QuantumNumbers -> {GhostNumber -> 1, Q -> 1},
315 Mass -> {MW,Internal},
316 Width -> {WW, 2.085},
317 PropagatorLabel -> "uWp",
318 PropagatorType -> GhostDash,
319 PropagatorArrow -> Forward
320 },
321 U[32] == {
322 ClassName -> ghWm,
323 SelfConjugate -> False,
324 Ghost -> Wbar,
325 QuantumNumbers -> {GhostNumber -> 1, Q -> -1},
326 Mass -> {MW,Internal},
327 Width -> {WW, 2.085},
328 PropagatorLabel -> "uWm",
329 PropagatorType -> GhostDash,
330 PropagatorArrow -> Forward
331 },
332 U[4] == {
333 ClassName -> ghG,
334 SelfConjugate -> False,
335 Indices -> {Index[Gluon]},
336 Ghost -> G,
337 PDG -> 82,
338 QuantumNumbers ->{GhostNumber -> 1},
339 Mass -> 0,
340 Width -> 0,
341 PropagatorLabel -> "uG",
342 PropagatorType -> GhostDash,
343 PropagatorArrow -> Forward
344 },
345
346(* Gauge bosons: unphysical vector fields *)
347 V[11] == {
348 ClassName -> B,
349 Unphysical -> True,
350 SelfConjugate -> True,
351 Definitions -> { B[mu_] -> -sw Z[mu]+cw A[mu]}
352 },
353 V[12] == {
354 ClassName -> Wi,
355 Unphysical -> True,
356 SelfConjugate -> True,
357 Indices -> {Index[SU2W]},
358 FlavorIndex -> SU2W,
359 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]}
360 },
361
362(* Ghosts: related to unphysical gauge bosons *)
363 U[11] == {
364 ClassName -> ghB,
365 Unphysical -> True,
366 SelfConjugate -> False,
367 Ghost -> B,
368 Definitions -> { ghB -> -sw ghZ + cw ghA}
369 },
370 U[12] == {
371 ClassName -> ghWi,
372 Unphysical -> True,
373 SelfConjugate -> False,
374 Ghost -> Wi,
375 Indices -> {Index[SU2W]},
376 FlavorIndex -> SU2W,
377 Definitions -> { ghWi[1] -> (ghWp+ghWm)/Sqrt[2], ghWi[2] -> (ghWm-ghWp)/(I*Sqrt[2]), ghWi[3] -> cw ghZ+sw ghA}
378 } ,
379
380(* Fermions: physical fields *)
381 F[1] == {
382 ClassName -> vl,
383 ClassMembers -> {ve,vm,vt},
384 Indices -> {Index[Generation]},
385 FlavorIndex -> Generation,
386 SelfConjugate -> False,
387 Mass -> 0,
388 Width -> 0,
389 QuantumNumbers -> {LeptonNumber -> 1},
390 PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
391 PropagatorType -> S,
392 PropagatorArrow -> Forward,
393 PDG -> {12,14,16},
394 ParticleName -> {"ve","vm","vt"},
395 AntiParticleName -> {"ve~","vm~","vt~"},
396 FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"}
397 },
398 F[2] == {
399 ClassName -> l,
400 ClassMembers -> {e, mu, ta},
401 Indices -> {Index[Generation]},
402 FlavorIndex -> Generation,
403 SelfConjugate -> False,
404 Mass -> {Ml, {Me,5.11*^-4}, {MMU,0.10566}, {MTA,1.777}},
405 Width -> 0,
406 QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
407 PropagatorLabel -> {"l", "e", "mu", "ta"},
408 PropagatorType -> Straight,
409 PropagatorArrow -> Forward,
410 PDG -> {11, 13, 15},
411 ParticleName -> {"e-", "mu-", "ta-"},
412 AntiParticleName -> {"e+", "mu+", "ta+"},
413 FullName -> {"Electron", "Muon", "Tau"}
414 },
415 F[3] == {
416 ClassName -> uq,
417 ClassMembers -> {u, c, t},
418 Indices -> {Index[Generation], Index[Colour]},
419 FlavorIndex -> Generation,
420 SelfConjugate -> False,
421 Mass -> {Mu, {MU, 2.55*^-3}, {MC,1.27}, {MT,173.3}},
422 Width -> {0, 0, {WT,1.350}},
423 QuantumNumbers -> {Q -> 2/3},
424 PropagatorLabel -> {"uq", "u", "c", "t"},
425 PropagatorType -> Straight,
426 PropagatorArrow -> Forward,
427 PDG -> {2, 4, 6},
428 ParticleName -> {"u", "c", "t" },
429 AntiParticleName -> {"u~", "c~", "t~"},
430 FullName -> {"u-quark", "c-quark", "t-quark"}
431 },
432 F[4] == {
433 ClassName -> dq,
434 ClassMembers -> {d, s, b},
435 Indices -> {Index[Generation], Index[Colour]},
436 FlavorIndex -> Generation,
437 SelfConjugate -> False,
438 Mass -> {Md, {MD,5.04*^-3}, {MS,0.101}, {MB,4.7}},
439 Width -> 0,
440 QuantumNumbers -> {Q -> -1/3},
441 PropagatorLabel -> {"dq", "d", "s", "b"},
442 PropagatorType -> Straight,
443 PropagatorArrow -> Forward,
444 PDG -> {1,3,5},
445 ParticleName -> {"d", "s", "b" },
446 AntiParticleName -> {"d~", "s~", "b~"},
447 FullName -> {"d-quark", "s-quark", "b-quark"}
448 },
449
450(* Fermions: unphysical fields *)
451 F[11] == {
452 ClassName -> LL,
453 Unphysical -> True,
454 Indices -> {Index[SU2D], Index[Generation]},
455 FlavorIndex -> SU2D,
456 SelfConjugate -> False,
457 QuantumNumbers -> {Y -> -1/2},
458 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]] }
459 },
460 F[12] == {
461 ClassName -> lR,
462 Unphysical -> True,
463 Indices -> {Index[Generation]},
464 FlavorIndex -> Generation,
465 SelfConjugate -> False,
466 QuantumNumbers -> {Y -> -1},
467 Definitions -> { lR[sp1_,ff_] :> Module[{sp2}, ProjP[sp1,sp2] l[sp2,ff]] }
468 },
469 F[13] == {
470 ClassName -> QL,
471 Unphysical -> True,
472 Indices -> {Index[SU2D], Index[Generation], Index[Colour]},
473 FlavorIndex -> SU2D,
474 SelfConjugate -> False,
475 QuantumNumbers -> {Y -> 1/6},
476 Definitions -> {
477 QL[sp1_,1,ff_,cc_] :> Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],
478 QL[sp1_,2,ff_,cc_] :> Module[{sp2,ff2}, CKM[ff,ff2] ProjM[sp1,sp2] dq[sp2,ff2,cc]] }
479 },
480 F[14] == {
481 ClassName -> uR,
482 Unphysical -> True,
483 Indices -> {Index[Generation], Index[Colour]},
484 FlavorIndex -> Generation,
485 SelfConjugate -> False,
486 QuantumNumbers -> {Y -> 2/3},
487 Definitions -> { uR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] uq[sp2,ff,cc]] }
488 },
489 F[15] == {
490 ClassName -> dR,
491 Unphysical -> True,
492 Indices -> {Index[Generation], Index[Colour]},
493 FlavorIndex -> Generation,
494 SelfConjugate -> False,
495 QuantumNumbers -> {Y -> -1/3},
496 Definitions -> { dR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] dq[sp2,ff,cc]] }
497 },
498
499(* Higgs: physical scalars *)
500 S[1] == {
501 ClassName -> H,
502 SelfConjugate -> True,
503 Mass -> {MH,125.7},
504 Width -> {WH,0.004170},
505 PropagatorLabel -> "H",
506 PropagatorType -> D,
507 PropagatorArrow -> None,
508 PDG -> 25,
509 ParticleName -> "H",
510 FullName -> "H"
511 },
512
513(* Higgs: physical scalars *)
514 S[2] == {
515 ClassName -> G0,
516 SelfConjugate -> True,
517 Goldstone -> Z,
518 Mass -> {MZ, 91.1876},
519 Width -> {WZ, 2.4952},
520 PropagatorLabel -> "Go",
521 PropagatorType -> D,
522 PropagatorArrow -> None,
523 PDG -> 250,
524 ParticleName -> "G0",
525 FullName -> "G0"
526 },
527 S[3] == {
528 ClassName -> GP,
529 SelfConjugate -> False,
530 Goldstone -> W,
531 Mass -> {MW, Internal},
532 QuantumNumbers -> {Q -> 1},
533 Width -> {WW, 2.085},
534 PropagatorLabel -> "GP",
535 PropagatorType -> D,
536 PropagatorArrow -> None,
537 PDG -> 251,
538 ParticleName -> "G+",
539 AntiParticleName -> "G-",
540 FullName -> "GP"
541 },
542
543(* Higgs: unphysical scalars *)
544 S[11] == {
545 ClassName -> Phi,
546 Unphysical -> True,
547 Indices -> {Index[SU2D]},
548 FlavorIndex -> SU2D,
549 SelfConjugate -> False,
550 QuantumNumbers -> {Y -> 1/2},
551 Definitions -> { Phi[1] -> -I GP, Phi[2] -> (vev + H + I G0)/Sqrt[2] }
552 }
553};
554
555
556(* ************************** *)
557(* ***** Gauge ***** *)
558(* ***** Parameters ***** *)
559(* ***** (FeynArts) ***** *)
560(* ************************** *)
561
562GaugeXi[ V[1] ] = GaugeXi[A];
563GaugeXi[ V[2] ] = GaugeXi[Z];
564GaugeXi[ V[3] ] = GaugeXi[W];
565GaugeXi[ V[4] ] = GaugeXi[G];
566GaugeXi[ S[1] ] = 1;
567GaugeXi[ S[2] ] = GaugeXi[Z];
568GaugeXi[ S[3] ] = GaugeXi[W];
569GaugeXi[ U[1] ] = GaugeXi[A];
570GaugeXi[ U[2] ] = GaugeXi[Z];
571GaugeXi[ U[31] ] = GaugeXi[W];
572GaugeXi[ U[32] ] = GaugeXi[W];
573GaugeXi[ U[4] ] = GaugeXi[G];
574
575
576(* ************************** *)
577(* ***** Parameters ***** *)
578(* ************************** *)
579M$Parameters = {
580
581 (* External parameters *)
582 aEWM1 == {
583 ParameterType -> External,
584 BlockName -> SMINPUTS,
585 OrderBlock -> 1,
586 Value -> 127.94,
587 InteractionOrder -> {QED,-2},
588 Description -> "Inverse of the EW coupling constant at the Z pole"
589 },
590 Gf == {
591 ParameterType -> External,
592 BlockName -> SMINPUTS,
593 OrderBlock -> 2,
594 Value -> 1.17456*^-5,
595 InteractionOrder -> {QED,2},
596 TeX -> Subscript[G,f],
597 Description -> "Fermi constant"
598 },
599 aS == {
600 ParameterType -> External,
601 BlockName -> SMINPUTS,
602 OrderBlock -> 3,
603 Value -> 0.1184,
604 InteractionOrder -> {QCD,2},
605 TeX -> Subscript[\[Alpha],s],
606 Description -> "Strong coupling constant at the Z pole"
607 },
608 ymdo == {
609 ParameterType -> External,
610 BlockName -> YUKAWA,
611 OrderBlock -> 1,
612 Value -> 5.04*^-3,
613 Description -> "Down Yukawa mass"
614 },
615 ymup == {
616 ParameterType -> External,
617 BlockName -> YUKAWA,
618 OrderBlock -> 2,
619 Value -> 2.55*^-3,
620 Description -> "Up Yukawa mass"
621 },
622 yms == {
623 ParameterType -> External,
624 BlockName -> YUKAWA,
625 OrderBlock -> 3,
626 Value -> 0.101,
627 Description -> "Strange Yukawa mass"
628 },
629 ymc == {
630 ParameterType -> External,
631 BlockName -> YUKAWA,
632 OrderBlock -> 4,
633 Value -> 1.27,
634 Description -> "Charm Yukawa mass"
635 },
636 ymb == {
637 ParameterType -> External,
638 BlockName -> YUKAWA,
639 OrderBlock -> 5,
640 Value -> 4.7,
641 Description -> "Bottom Yukawa mass"
642 },
643 ymt == {
644 ParameterType -> External,
645 BlockName -> YUKAWA,
646 OrderBlock -> 6,
647 Value -> 173.3,
648 Description -> "Top Yukawa mass"
649 },
650 yme == {
651 ParameterType -> External,
652 BlockName -> YUKAWA,
653 OrderBlock -> 11,
654 Value -> 5.11*^-4,
655 Description -> "Electron Yukawa mass"
656 },
657 ymm == {
658 ParameterType -> External,
659 BlockName -> YUKAWA,
660 OrderBlock -> 13,
661 Value -> 0.10566,
662 Description -> "Muon Yukawa mass"
663 },
664 ymtau == {
665 ParameterType -> External,
666 BlockName -> YUKAWA,
667 OrderBlock -> 15,
668 Value -> 1.777,
669 Description -> "Tau Yukawa mass"
670 },
671 cabi == {
672 ParameterType -> External,
673 BlockName -> CKMBLOCK,
674 OrderBlock -> 1,
675 Value -> 0.227591,
676 TeX -> Subscript[\[Theta], c],
677 Description -> "Cabibbo angle"
678 },
679 th13 == {
680 ParameterType -> External,
681 BlockName -> CKMBLOCK,
682 OrderBlock -> 2,
683 Value -> 0.003508,
684 TeX -> Subscript[\[Theta], 13],
685 Description -> "CKM Theta 13"
686 },
687 th23 == {
688 ParameterType -> External,
689 BlockName -> CKMBLOCK,
690 OrderBlock -> 3,
691 Value -> 0.041539,
692 TeX -> Subscript[\[Theta], 23],
693 Description -> "CKM theta 23"
694 },
695 del13 == {
696 ParameterType -> External,
697 BlockName -> CKMBLOCK,
698 OrderBlock -> 4,
699 Value -> 1.20,
700 TeX -> Subscript[\[Delta], 13],
701 Description -> "CKM delta 13"
702 },
703
704 (* Internal Parameters *)
705 aEW == {
706 ParameterType -> Internal,
707 Value -> 1/aEWM1,
708 InteractionOrder -> {QED,2},
709 TeX -> Subscript[\[Alpha], EW],
710 Description -> "Electroweak coupling contant"
711 },
712 MW == {
713 ParameterType -> Internal,
714 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
715 TeX -> Subscript[M,W],
716 Description -> "W mass"
717 },
718 MWX == {
719 ParameterType -> Internal,
720 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
721 TeX -> Subscript[M,W],
722 Description -> "W mass"
723 },
724 MW0 == {
725 ParameterType -> Internal,
726 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
727 TeX -> Subscript[M,W],
728 Description -> "W mass"
729 },
730 MWT == {
731 ParameterType -> Internal,
732 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
733 TeX -> Subscript[M,W],
734 Description -> "W mass"
735 },
736 MWA == {
737 ParameterType -> Internal,
738 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
739 TeX -> Subscript[M,W],
740 Description -> "W mass"
741 },
742 MZX == {
743 ParameterType -> Internal,
744 Value -> MZ,
745 TeX -> Subscript[M,Z],
746 Description -> "Z mass"
747 },
748 MZ0 == {
749 ParameterType -> Internal,
750 Value -> MZ,
751 TeX -> Subscript[M,Z],
752 Description -> "Z mass"
753 },
754 MZT == {
755 ParameterType -> Internal,
756 Value -> MZ,
757 TeX -> Subscript[M,Z],
758 Description -> "Z mass"
759 },
760 MZA == {
761 ParameterType -> Internal,
762 Value -> MZ,
763 TeX -> Subscript[M,Z],
764 Description -> "Z mass"
765 },
766 WWX == {
767 ParameterType -> Internal,
768 Value -> WW,
769 TeX -> Subscript[W,W],
770 Description -> "W width"
771 },
772 WW0 == {
773 ParameterType -> Internal,
774 Value -> WW,
775 TeX -> Subscript[W,W],
776 Description -> "W width"
777 },
778 WWT == {
779 ParameterType -> Internal,
780 Value -> WW,
781 TeX -> Subscript[W,W],
782 Description -> "W width"
783 },
784 WWA == {
785 ParameterType -> Internal,
786 Value -> WW,
787 TeX -> Subscript[W,W],
788 Description -> "W width"
789 },
790 WZX == {
791 ParameterType -> Internal,
792 Value -> WZ,
793 TeX -> Subscript[W,Z],
794 Description -> "Z width"
795 },
796 WZ0 == {
797 ParameterType -> Internal,
798 Value -> WZ,
799 TeX -> Subscript[W,Z],
800 Description -> "Z width"
801 },
802 WZT == {
803 ParameterType -> Internal,
804 Value -> WZ,
805 TeX -> Subscript[W,Z],
806 Description -> "Z width"
807 },
808 WZA == {
809 ParameterType -> Internal,
810 Value -> WZ,
811 TeX -> Subscript[W,Z],
812 Description -> "Z width"
813 },
814 sw2 == {
815 ParameterType -> Internal,
816 Value -> 1-(MW/MZ)^2,
817 Description -> "Squared Sin of the Weinberg angle"
818 },
819 ee == {
820 ParameterType -> Internal,
821 Value -> Sqrt[4 Pi aEW],
822 InteractionOrder -> {QED,1},
823 TeX -> e,
824 Description -> "Electric coupling constant"
825 },
826 cw == {
827 ParameterType -> Internal,
828 Value -> Sqrt[1-sw2],
829 TeX -> Subscript[c,w],
830 Description -> "Cosine of the Weinberg angle"
831 },
832 sw == {
833 ParameterType -> Internal,
834 Value -> Sqrt[sw2],
835 TeX -> Subscript[s,w],
836 Description -> "Sine of the Weinberg angle"
837 },
838 gw == {
839 ParameterType -> Internal,
840 Definitions -> {gw->ee/sw},
841 InteractionOrder -> {QED,1},
842 TeX -> Subscript[g,w],
843 Description -> "Weak coupling constant at the Z pole"
844 },
845 g1 == {
846 ParameterType -> Internal,
847 Definitions -> {g1->ee/cw},
848 InteractionOrder -> {QED,1},
849 TeX -> Subscript[g,1],
850 Description -> "U(1)Y coupling constant at the Z pole"
851 },
852 gs == {
853 ParameterType -> Internal,
854 Value -> Sqrt[4 Pi aS],
855 InteractionOrder -> {QCD,1},
856 TeX -> Subscript[g,s],
857 ParameterName -> G,
858 Description -> "Strong coupling constant at the Z pole"
859 },
860 vev == {
861 ParameterType -> Internal,
862 Value -> 2*MW*sw/ee,
863 InteractionOrder -> {QED,-1},
864 Description -> "Higgs vacuum expectation value"
865 },
866 lam == {
867 ParameterType -> Internal,
868 Value -> MH^2/(2*vev^2),
869 InteractionOrder -> {QED, 2},
870 Description -> "Higgs quartic coupling"
871 },
872 muH == {
873 ParameterType -> Internal,
874 Value -> Sqrt[vev^2 lam],
875 TeX -> \[Mu],
876 Description -> "Coefficient of the quadratic piece of the Higgs potential"
877 },
878 yl == {
879 ParameterType -> Internal,
880 Indices -> {Index[Generation], Index[Generation]},
881 Definitions -> {yl[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
882 Value -> {yl[1,1] -> Sqrt[2] yme / vev, yl[2,2] -> Sqrt[2] ymm / vev, yl[3,3] -> Sqrt[2] ymtau / vev},
883 InteractionOrder -> {QED, 1},
884 ParameterName -> {yl[1,1] -> ye, yl[2,2] -> ym, yl[3,3] -> ytau},
885 TeX -> Superscript[y, l],
886 Description -> "Lepton Yukawa couplings"
887 },
888 yu == {
889 ParameterType -> Internal,
890 Indices -> {Index[Generation], Index[Generation]},
891 Definitions -> {yu[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
892 Value -> {yu[1,1] -> Sqrt[2] ymup/vev, yu[2,2] -> Sqrt[2] ymc/vev, yu[3,3] -> Sqrt[2] ymt/vev},
893 InteractionOrder -> {QED, 1},
894 ParameterName -> {yu[1,1] -> yup, yu[2,2] -> yc, yu[3,3] -> yt},
895 TeX -> Superscript[y, u],
896 Description -> "Up-type Yukawa couplings"
897 },
898 yd == {
899 ParameterType -> Internal,
900 Indices -> {Index[Generation], Index[Generation]},
901 Definitions -> {yd[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
902 Value -> {yd[1,1] -> Sqrt[2] ymdo/vev, yd[2,2] -> Sqrt[2] yms/vev, yd[3,3] -> Sqrt[2] ymb/vev},
903 InteractionOrder -> {QED, 1},
904 ParameterName -> {yd[1,1] -> ydo, yd[2,2] -> ys, yd[3,3] -> yb},
905 TeX -> Superscript[y, d],
906 Description -> "Down-type Yukawa couplings"
907 },
908(* N. B. : only Cabibbo mixing! *)
909 CKM == {
910 ParameterType -> Internal,
911 Indices -> {Index[Generation], Index[Generation]},
912 Unitary -> True,
913 Value -> {CKM[1,1] -> Cos[cabi] Cos[th13], CKM[1,2] -> Sin[cabi] Cos[th13], CKM[1,3] -> Sin[th13] Exp[-I del13],
914 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],
915 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]},
916 TeX -> Superscript[V,CKM],
917 Description -> "CKM-Matrix"}
918};
919
920(* ************************** *)
921(* ***** Lagrangian ***** *)
922(* ************************** *)
923
924LWPolarMassBase := MW*MW * W0bar[mu] * (WT[mu] + WA[mu] + WX[mu]) \
925 + MW*MW * WTbar[mu] * (W0[mu] + WA[mu] + WX[mu]) \
926 + MW*MW * WAbar[mu] * (W0[mu] + WT[mu] + WX[mu]) \
927 + MW*MW * WXbar[mu] * (W0[mu] + WT[mu] + WA[mu]) ;
928LZPolarMassBase := (MZ*MZ/2) * Z0[mu] * (ZT[mu] + ZA[mu] + ZX[mu]) \
929 + (MZ*MZ/2) * ZT[mu] * (Z0[mu] + ZA[mu] + ZX[mu]) \
930 + (MZ*MZ/2) * ZA[mu] * (Z0[mu] + ZT[mu] + ZX[mu]) \
931 + (MZ*MZ/2) * ZX[mu] * (Z0[mu] + ZT[mu] + ZA[mu]) ;
932LPolarMass := LWPolarMassBase + LZPolarMassBase;
933
934
935LGauge := Block[{mu,nu,ii,aa},
936 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]];
937
938LFermions := Block[{mu},
939 ExpandIndices[I*(
940 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]),
941 FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]}];
942
943LHiggsBase := Block[{ii,mu, feynmangaugerules},
944 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
945
946 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
947 ];
948LHiggs := Simplify[LHiggsBase - LPolarMass];
949
950
951LYukawa := Block[{sp,ii,jj,cc,ff1,ff2,ff3,yuk,feynmangaugerules},
952 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
953
954 yuk = ExpandIndices[
955 -yd[ff2, ff3] CKM[ff1, ff2] QLbar[sp, ii, ff1, cc].dR [sp, ff3, cc] Phi[ii] -
956 yl[ff1, ff3] LLbar[sp, ii, ff1].lR [sp, ff3] Phi[ii] -
957 yu[ff1, ff2] QLbar[sp, ii, ff1, cc].uR [sp, ff2, cc] Phibar[jj] Eps[ii, jj], FlavorExpand -> SU2D];
958 yuk = yuk /. { CKM[a_, b_] Conjugate[CKM[a_, c_]] -> IndexDelta[b, c], CKM[b_, a_] Conjugate[CKM[c_, a_]] -> IndexDelta[b, c]};
959 yuk+HC[yuk]/.feynmangaugerules
960 ];
961
962LGhost := Block[{LGh1,LGhw,LGhs,LGhphi,mu, generators,gh,ghbar,Vectorize,phi1,phi2,togoldstones,doublet,doublet0},
963 (* Pure gauge piece *)
964 LGh1 = -ghBbar.del[DC[ghB,mu],mu];
965 LGhw = -ghWibar[ii].del[DC[ghWi[ii],mu],mu];
966 LGhs = -ghGbar[ii].del[DC[ghG[ii],mu],mu];
967
968 (* Scalar pieces: see Peskin pages 739-742 *)
969 (* phi1 and phi2 are the real degrees of freedom of GP *)
970 (* Vectorize transforms a doublet in a vector in the phi-basis, i.e. the basis of real degrees of freedom *)
971 gh = {ghB, ghWi[1], ghWi[2], ghWi[3]};
972 ghbar = {ghBbar, ghWibar[1], ghWibar[2], ghWibar[3]};
973 generators = {-I/2 g1 IdentityMatrix[2], -I/2 gw PauliSigma[1], -I/2 gw PauliSigma[2], -I/2 gw PauliSigma[3]};
974 doublet = Expand[{(-I phi1 - phi2)/Sqrt[2], Phi[2]} /. MR$Definitions /. vev -> 0];
975 doublet0 = {0, vev/Sqrt[2]};
976 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}];
977 togoldstones := {phi1 -> (GP + GPbar)/Sqrt[2], phi2 -> (-GP + GPbar)/(I Sqrt[2])};
978 LGhphi=Plus@@Flatten[Table[-ghbar[[kkk]].gh[[lll]] Vectorize[generators[[kkk]].doublet0].Vectorize[generators[[lll]].(doublet+doublet0)],{kkk,4},{lll,4}]] /.togoldstones;
979
980ExpandIndices[ LGhs + If[FeynmanGauge, LGh1 + LGhw + LGhphi,0], FlavorExpand->SU2W]];
981
982LSM:= LGauge + LFermions + LHiggs + LYukawa + LGhost;