VPolarization: sm_loop_wpolar.fr

File sm_loop_wpolar.fr, 29.3 KB (added by Richard Ruiz, 7 months ago)

FeynRules model file for W 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 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 Unphysical -> True,
154 SelfConjugate -> False,
155 Mass -> {MWX, Internal},
156 Width -> {WWX, Internal},
157 ParticleName -> "W+",
158 AntiParticleName -> "W-",
159 QuantumNumbers -> {Q -> 1},
160 PDG -> 249,
161 PropagatorLabel -> "W",
162 PropagatorType -> Sine,
163 PropagatorArrow -> Forward,
164 FullName -> "W",
165 Definitions -> {W[mu_] -> W0[mu] + WT[mu] + WA[mu] + WX[mu]}
166 },
167 V[240] == {
168 ClassName -> W0,
169 SelfConjugate -> False,
170 Mass -> {MW0, Internal},
171 Width -> {WW0, Internal},
172 ParticleName -> "W0+",
173 AntiParticleName -> "W0-",
174 QuantumNumbers -> {Q -> 1},
175 PDG -> 240,
176 PropagatorLabel -> "W0",
177 PropagatorType -> Sine,
178 PropagatorArrow -> Forward,
179 FullName -> "W0"
180 },
181 V[241] == {
182 ClassName -> WT,
183 SelfConjugate -> False,
184 Mass -> {MWT, Internal},
185 Width -> {WWT, Internal},
186 ParticleName -> "WT+",
187 AntiParticleName -> "WT-",
188 QuantumNumbers -> {Q -> 1},
189 PDG -> 241,
190 PropagatorLabel -> "WT",
191 PropagatorType -> Sine,
192 PropagatorArrow -> Forward,
193 FullName -> "WT"
194 },
195 V[242] == {
196 ClassName -> WA,
197 SelfConjugate -> False,
198 Mass -> {MWA, Internal},
199 Width -> {WWA, Internal},
200 ParticleName -> "WA+",
201 AntiParticleName -> "WA-",
202 QuantumNumbers -> {Q -> 1},
203 PDG -> 242,
204 PropagatorLabel -> "WA",
205 PropagatorType -> Sine,
206 PropagatorArrow -> Forward,
207 FullName -> "WA"
208 },
209 V[243] == {
210 ClassName -> WX,
211 SelfConjugate -> False,
212 Mass -> {MW, Internal},
213 Width -> {WW, 2.085},
214 ParticleName -> "WX+",
215 AntiParticleName -> "WX-",
216 QuantumNumbers -> {Q -> 1},
217 PDG -> 24,
218 PropagatorLabel -> "WX",
219 PropagatorType -> Sine,
220 PropagatorArrow -> Forward,
221 FullName -> "WX"
222 },
223 V[4] == {
224 ClassName -> G,
225 SelfConjugate -> True,
226 Indices -> {Index[Gluon]},
227 Mass -> 0,
228 Width -> 0,
229 ParticleName -> "g",
230 PDG -> 21,
231 PropagatorLabel -> "G",
232 PropagatorType -> C,
233 PropagatorArrow -> None,
234 FullName -> "G"
235 },
236
237(* Ghosts: related to physical gauge bosons *)
238 U[1] == {
239 ClassName -> ghA,
240 SelfConjugate -> False,
241 Ghost -> A,
242 QuantumNumbers -> {GhostNumber -> 1},
243 Mass -> 0,
244 Width -> 0,
245 PropagatorLabel -> "uA",
246 PropagatorType -> GhostDash,
247 PropagatorArrow -> Forward
248 },
249 U[2] == {
250 ClassName -> ghZ,
251 SelfConjugate -> False,
252 Ghost -> Z,
253 QuantumNumbers -> {GhostNumber -> 1},
254 Mass -> {MZ,91.1876},
255 Width -> {WZ, 2.4952},
256 PropagatorLabel -> "uZ",
257 PropagatorType -> GhostDash,
258 PropagatorArrow -> Forward
259 },
260 U[31] == {
261 ClassName -> ghWp,
262 SelfConjugate -> False,
263 Ghost -> W,
264 QuantumNumbers -> {GhostNumber -> 1, Q -> 1},
265 Mass -> {MW,Internal},
266 Width -> {WW, 2.085},
267 PropagatorLabel -> "uWp",
268 PropagatorType -> GhostDash,
269 PropagatorArrow -> Forward
270 },
271 U[32] == {
272 ClassName -> ghWm,
273 SelfConjugate -> False,
274 Ghost -> Wbar,
275 QuantumNumbers -> {GhostNumber -> 1, Q -> -1},
276 Mass -> {MW,Internal},
277 Width -> {WW, 2.085},
278 PropagatorLabel -> "uWm",
279 PropagatorType -> GhostDash,
280 PropagatorArrow -> Forward
281 },
282 U[4] == {
283 ClassName -> ghG,
284 SelfConjugate -> False,
285 Indices -> {Index[Gluon]},
286 Ghost -> G,
287 PDG -> 82,
288 QuantumNumbers ->{GhostNumber -> 1},
289 Mass -> 0,
290 Width -> 0,
291 PropagatorLabel -> "uG",
292 PropagatorType -> GhostDash,
293 PropagatorArrow -> Forward
294 },
295
296(* Gauge bosons: unphysical vector fields *)
297 V[11] == {
298 ClassName -> B,
299 Unphysical -> True,
300 SelfConjugate -> True,
301 Definitions -> { B[mu_] -> -sw Z[mu]+cw A[mu]}
302 },
303 V[12] == {
304 ClassName -> Wi,
305 Unphysical -> True,
306 SelfConjugate -> True,
307 Indices -> {Index[SU2W]},
308 FlavorIndex -> SU2W,
309 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]}
310 },
311
312(* Ghosts: related to unphysical gauge bosons *)
313 U[11] == {
314 ClassName -> ghB,
315 Unphysical -> True,
316 SelfConjugate -> False,
317 Ghost -> B,
318 Definitions -> { ghB -> -sw ghZ + cw ghA}
319 },
320 U[12] == {
321 ClassName -> ghWi,
322 Unphysical -> True,
323 SelfConjugate -> False,
324 Ghost -> Wi,
325 Indices -> {Index[SU2W]},
326 FlavorIndex -> SU2W,
327 Definitions -> { ghWi[1] -> (ghWp+ghWm)/Sqrt[2], ghWi[2] -> (ghWm-ghWp)/(I*Sqrt[2]), ghWi[3] -> cw ghZ+sw ghA}
328 } ,
329
330(* Fermions: physical fields *)
331 F[1] == {
332 ClassName -> vl,
333 ClassMembers -> {ve,vm,vt},
334 Indices -> {Index[Generation]},
335 FlavorIndex -> Generation,
336 SelfConjugate -> False,
337 Mass -> 0,
338 Width -> 0,
339 QuantumNumbers -> {LeptonNumber -> 1},
340 PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
341 PropagatorType -> S,
342 PropagatorArrow -> Forward,
343 PDG -> {12,14,16},
344 ParticleName -> {"ve","vm","vt"},
345 AntiParticleName -> {"ve~","vm~","vt~"},
346 FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"}
347 },
348 F[2] == {
349 ClassName -> l,
350 ClassMembers -> {e, mu, ta},
351 Indices -> {Index[Generation]},
352 FlavorIndex -> Generation,
353 SelfConjugate -> False,
354 Mass -> {Ml, {Me,5.11*^-4}, {MMU,0.10566}, {MTA,1.777}},
355 Width -> 0,
356 QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
357 PropagatorLabel -> {"l", "e", "mu", "ta"},
358 PropagatorType -> Straight,
359 PropagatorArrow -> Forward,
360 PDG -> {11, 13, 15},
361 ParticleName -> {"e-", "mu-", "ta-"},
362 AntiParticleName -> {"e+", "mu+", "ta+"},
363 FullName -> {"Electron", "Muon", "Tau"}
364 },
365 F[3] == {
366 ClassName -> uq,
367 ClassMembers -> {u, c, t},
368 Indices -> {Index[Generation], Index[Colour]},
369 FlavorIndex -> Generation,
370 SelfConjugate -> False,
371 Mass -> {Mu, {MU, 2.55*^-3}, {MC,1.27}, {MT,173.3}},
372 Width -> {0, 0, {WT,1.350}},
373 QuantumNumbers -> {Q -> 2/3},
374 PropagatorLabel -> {"uq", "u", "c", "t"},
375 PropagatorType -> Straight,
376 PropagatorArrow -> Forward,
377 PDG -> {2, 4, 6},
378 ParticleName -> {"u", "c", "t" },
379 AntiParticleName -> {"u~", "c~", "t~"},
380 FullName -> {"u-quark", "c-quark", "t-quark"}
381 },
382 F[4] == {
383 ClassName -> dq,
384 ClassMembers -> {d, s, b},
385 Indices -> {Index[Generation], Index[Colour]},
386 FlavorIndex -> Generation,
387 SelfConjugate -> False,
388 Mass -> {Md, {MD,5.04*^-3}, {MS,0.101}, {MB,4.7}},
389 Width -> 0,
390 QuantumNumbers -> {Q -> -1/3},
391 PropagatorLabel -> {"dq", "d", "s", "b"},
392 PropagatorType -> Straight,
393 PropagatorArrow -> Forward,
394 PDG -> {1,3,5},
395 ParticleName -> {"d", "s", "b" },
396 AntiParticleName -> {"d~", "s~", "b~"},
397 FullName -> {"d-quark", "s-quark", "b-quark"}
398 },
399
400(* Fermions: unphysical fields *)
401 F[11] == {
402 ClassName -> LL,
403 Unphysical -> True,
404 Indices -> {Index[SU2D], Index[Generation]},
405 FlavorIndex -> SU2D,
406 SelfConjugate -> False,
407 QuantumNumbers -> {Y -> -1/2},
408 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]] }
409 },
410 F[12] == {
411 ClassName -> lR,
412 Unphysical -> True,
413 Indices -> {Index[Generation]},
414 FlavorIndex -> Generation,
415 SelfConjugate -> False,
416 QuantumNumbers -> {Y -> -1},
417 Definitions -> { lR[sp1_,ff_] :> Module[{sp2}, ProjP[sp1,sp2] l[sp2,ff]] }
418 },
419 F[13] == {
420 ClassName -> QL,
421 Unphysical -> True,
422 Indices -> {Index[SU2D], Index[Generation], Index[Colour]},
423 FlavorIndex -> SU2D,
424 SelfConjugate -> False,
425 QuantumNumbers -> {Y -> 1/6},
426 Definitions -> {
427 QL[sp1_,1,ff_,cc_] :> Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],
428 QL[sp1_,2,ff_,cc_] :> Module[{sp2,ff2}, CKM[ff,ff2] ProjM[sp1,sp2] dq[sp2,ff2,cc]] }
429 },
430 F[14] == {
431 ClassName -> uR,
432 Unphysical -> True,
433 Indices -> {Index[Generation], Index[Colour]},
434 FlavorIndex -> Generation,
435 SelfConjugate -> False,
436 QuantumNumbers -> {Y -> 2/3},
437 Definitions -> { uR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] uq[sp2,ff,cc]] }
438 },
439 F[15] == {
440 ClassName -> dR,
441 Unphysical -> True,
442 Indices -> {Index[Generation], Index[Colour]},
443 FlavorIndex -> Generation,
444 SelfConjugate -> False,
445 QuantumNumbers -> {Y -> -1/3},
446 Definitions -> { dR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] dq[sp2,ff,cc]] }
447 },
448
449(* Higgs: physical scalars *)
450 S[1] == {
451 ClassName -> H,
452 SelfConjugate -> True,
453 Mass -> {MH,125.7},
454 Width -> {WH,0.004170},
455 PropagatorLabel -> "H",
456 PropagatorType -> D,
457 PropagatorArrow -> None,
458 PDG -> 25,
459 ParticleName -> "H",
460 FullName -> "H"
461 },
462
463(* Higgs: physical scalars *)
464 S[2] == {
465 ClassName -> G0,
466 SelfConjugate -> True,
467 Goldstone -> Z,
468 Mass -> {MZ, 91.1876},
469 Width -> {WZ, 2.4952},
470 PropagatorLabel -> "Go",
471 PropagatorType -> D,
472 PropagatorArrow -> None,
473 PDG -> 250,
474 ParticleName -> "G0",
475 FullName -> "G0"
476 },
477 S[3] == {
478 ClassName -> GP,
479 SelfConjugate -> False,
480 Goldstone -> W,
481 Mass -> {MW, Internal},
482 QuantumNumbers -> {Q -> 1},
483 Width -> {WW, 2.085},
484 PropagatorLabel -> "GP",
485 PropagatorType -> D,
486 PropagatorArrow -> None,
487 PDG -> 251,
488 ParticleName -> "G+",
489 AntiParticleName -> "G-",
490 FullName -> "GP"
491 },
492
493(* Higgs: unphysical scalars *)
494 S[11] == {
495 ClassName -> Phi,
496 Unphysical -> True,
497 Indices -> {Index[SU2D]},
498 FlavorIndex -> SU2D,
499 SelfConjugate -> False,
500 QuantumNumbers -> {Y -> 1/2},
501 Definitions -> { Phi[1] -> -I GP, Phi[2] -> (vev + H + I G0)/Sqrt[2] }
502 }
503};
504
505
506(* ************************** *)
507(* ***** Gauge ***** *)
508(* ***** Parameters ***** *)
509(* ***** (FeynArts) ***** *)
510(* ************************** *)
511
512GaugeXi[ V[1] ] = GaugeXi[A];
513GaugeXi[ V[2] ] = GaugeXi[Z];
514GaugeXi[ V[3] ] = GaugeXi[W];
515GaugeXi[ V[4] ] = GaugeXi[G];
516GaugeXi[ S[1] ] = 1;
517GaugeXi[ S[2] ] = GaugeXi[Z];
518GaugeXi[ S[3] ] = GaugeXi[W];
519GaugeXi[ U[1] ] = GaugeXi[A];
520GaugeXi[ U[2] ] = GaugeXi[Z];
521GaugeXi[ U[31] ] = GaugeXi[W];
522GaugeXi[ U[32] ] = GaugeXi[W];
523GaugeXi[ U[4] ] = GaugeXi[G];
524
525
526(* ************************** *)
527(* ***** Parameters ***** *)
528(* ************************** *)
529M$Parameters = {
530
531 (* External parameters *)
532 aEWM1 == {
533 ParameterType -> External,
534 BlockName -> SMINPUTS,
535 OrderBlock -> 1,
536 Value -> 127.94,
537 InteractionOrder -> {QED,-2},
538 Description -> "Inverse of the EW coupling constant at the Z pole"
539 },
540 Gf == {
541 ParameterType -> External,
542 BlockName -> SMINPUTS,
543 OrderBlock -> 2,
544 Value -> 1.17456*^-5,
545 InteractionOrder -> {QED,2},
546 TeX -> Subscript[G,f],
547 Description -> "Fermi constant"
548 },
549 aS == {
550 ParameterType -> External,
551 BlockName -> SMINPUTS,
552 OrderBlock -> 3,
553 Value -> 0.1184,
554 InteractionOrder -> {QCD,2},
555 TeX -> Subscript[\[Alpha],s],
556 Description -> "Strong coupling constant at the Z pole"
557 },
558 ymdo == {
559 ParameterType -> External,
560 BlockName -> YUKAWA,
561 OrderBlock -> 1,
562 Value -> 5.04*^-3,
563 Description -> "Down Yukawa mass"
564 },
565 ymup == {
566 ParameterType -> External,
567 BlockName -> YUKAWA,
568 OrderBlock -> 2,
569 Value -> 2.55*^-3,
570 Description -> "Up Yukawa mass"
571 },
572 yms == {
573 ParameterType -> External,
574 BlockName -> YUKAWA,
575 OrderBlock -> 3,
576 Value -> 0.101,
577 Description -> "Strange Yukawa mass"
578 },
579 ymc == {
580 ParameterType -> External,
581 BlockName -> YUKAWA,
582 OrderBlock -> 4,
583 Value -> 1.27,
584 Description -> "Charm Yukawa mass"
585 },
586 ymb == {
587 ParameterType -> External,
588 BlockName -> YUKAWA,
589 OrderBlock -> 5,
590 Value -> 4.7,
591 Description -> "Bottom Yukawa mass"
592 },
593 ymt == {
594 ParameterType -> External,
595 BlockName -> YUKAWA,
596 OrderBlock -> 6,
597 Value -> 173.3,
598 Description -> "Top Yukawa mass"
599 },
600 yme == {
601 ParameterType -> External,
602 BlockName -> YUKAWA,
603 OrderBlock -> 11,
604 Value -> 5.11*^-4,
605 Description -> "Electron Yukawa mass"
606 },
607 ymm == {
608 ParameterType -> External,
609 BlockName -> YUKAWA,
610 OrderBlock -> 13,
611 Value -> 0.10566,
612 Description -> "Muon Yukawa mass"
613 },
614 ymtau == {
615 ParameterType -> External,
616 BlockName -> YUKAWA,
617 OrderBlock -> 15,
618 Value -> 1.777,
619 Description -> "Tau Yukawa mass"
620 },
621 cabi == {
622 ParameterType -> External,
623 BlockName -> CKMBLOCK,
624 OrderBlock -> 1,
625 Value -> 0.227591,
626 TeX -> Subscript[\[Theta], c],
627 Description -> "Cabibbo angle"
628 },
629 th13 == {
630 ParameterType -> External,
631 BlockName -> CKMBLOCK,
632 OrderBlock -> 2,
633 Value -> 0.003508,
634 TeX -> Subscript[\[Theta], 13],
635 Description -> "CKM Theta 13"
636 },
637 th23 == {
638 ParameterType -> External,
639 BlockName -> CKMBLOCK,
640 OrderBlock -> 3,
641 Value -> 0.041539,
642 TeX -> Subscript[\[Theta], 23],
643 Description -> "CKM theta 23"
644 },
645 del13 == {
646 ParameterType -> External,
647 BlockName -> CKMBLOCK,
648 OrderBlock -> 4,
649 Value -> 1.20,
650 TeX -> Subscript[\[Delta], 13],
651 Description -> "CKM delta 13"
652 },
653
654 (* Internal Parameters *)
655 aEW == {
656 ParameterType -> Internal,
657 Value -> 1/aEWM1,
658 InteractionOrder -> {QED,2},
659 TeX -> Subscript[\[Alpha], EW],
660 Description -> "Electroweak coupling contant"
661 },
662 MW == {
663 ParameterType -> Internal,
664 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
665 TeX -> Subscript[M,W],
666 Description -> "W mass"
667 },
668 MWX == {
669 ParameterType -> Internal,
670 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
671 TeX -> Subscript[M,W],
672 Description -> "W mass"
673 },
674 MW0 == {
675 ParameterType -> Internal,
676 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
677 TeX -> Subscript[M,W],
678 Description -> "W mass"
679 },
680 MWT == {
681 ParameterType -> Internal,
682 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
683 TeX -> Subscript[M,W],
684 Description -> "W mass"
685 },
686 MWA == {
687 ParameterType -> Internal,
688 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
689 TeX -> Subscript[M,W],
690 Description -> "W mass"
691 },
692 WWX == {
693 ParameterType -> Internal,
694 Value -> WW,
695 TeX -> Subscript[W,W],
696 Description -> "W width"
697 },
698 WW0 == {
699 ParameterType -> Internal,
700 Value -> WW,
701 TeX -> Subscript[W,W],
702 Description -> "W width"
703 },
704 WWT == {
705 ParameterType -> Internal,
706 Value -> WW,
707 TeX -> Subscript[W,W],
708 Description -> "W width"
709 },
710 WWA == {
711 ParameterType -> Internal,
712 Value -> WW,
713 TeX -> Subscript[W,W],
714 Description -> "W width"
715 },
716 sw2 == {
717 ParameterType -> Internal,
718 Value -> 1-(MW/MZ)^2,
719 Description -> "Squared Sin of the Weinberg angle"
720 },
721 ee == {
722 ParameterType -> Internal,
723 Value -> Sqrt[4 Pi aEW],
724 InteractionOrder -> {QED,1},
725 TeX -> e,
726 Description -> "Electric coupling constant"
727 },
728 cw == {
729 ParameterType -> Internal,
730 Value -> Sqrt[1-sw2],
731 TeX -> Subscript[c,w],
732 Description -> "Cosine of the Weinberg angle"
733 },
734 sw == {
735 ParameterType -> Internal,
736 Value -> Sqrt[sw2],
737 TeX -> Subscript[s,w],
738 Description -> "Sine of the Weinberg angle"
739 },
740 gw == {
741 ParameterType -> Internal,
742 Definitions -> {gw->ee/sw},
743 InteractionOrder -> {QED,1},
744 TeX -> Subscript[g,w],
745 Description -> "Weak coupling constant at the Z pole"
746 },
747 g1 == {
748 ParameterType -> Internal,
749 Definitions -> {g1->ee/cw},
750 InteractionOrder -> {QED,1},
751 TeX -> Subscript[g,1],
752 Description -> "U(1)Y coupling constant at the Z pole"
753 },
754 gs == {
755 ParameterType -> Internal,
756 Value -> Sqrt[4 Pi aS],
757 InteractionOrder -> {QCD,1},
758 TeX -> Subscript[g,s],
759 ParameterName -> G,
760 Description -> "Strong coupling constant at the Z pole"
761 },
762 vev == {
763 ParameterType -> Internal,
764 Value -> 2*MW*sw/ee,
765 InteractionOrder -> {QED,-1},
766 Description -> "Higgs vacuum expectation value"
767 },
768 lam == {
769 ParameterType -> Internal,
770 Value -> MH^2/(2*vev^2),
771 InteractionOrder -> {QED, 2},
772 Description -> "Higgs quartic coupling"
773 },
774 muH == {
775 ParameterType -> Internal,
776 Value -> Sqrt[vev^2 lam],
777 TeX -> \[Mu],
778 Description -> "Coefficient of the quadratic piece of the Higgs potential"
779 },
780 yl == {
781 ParameterType -> Internal,
782 Indices -> {Index[Generation], Index[Generation]},
783 Definitions -> {yl[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
784 Value -> {yl[1,1] -> Sqrt[2] yme / vev, yl[2,2] -> Sqrt[2] ymm / vev, yl[3,3] -> Sqrt[2] ymtau / vev},
785 InteractionOrder -> {QED, 1},
786 ParameterName -> {yl[1,1] -> ye, yl[2,2] -> ym, yl[3,3] -> ytau},
787 TeX -> Superscript[y, l],
788 Description -> "Lepton Yukawa couplings"
789 },
790 yu == {
791 ParameterType -> Internal,
792 Indices -> {Index[Generation], Index[Generation]},
793 Definitions -> {yu[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
794 Value -> {yu[1,1] -> Sqrt[2] ymup/vev, yu[2,2] -> Sqrt[2] ymc/vev, yu[3,3] -> Sqrt[2] ymt/vev},
795 InteractionOrder -> {QED, 1},
796 ParameterName -> {yu[1,1] -> yup, yu[2,2] -> yc, yu[3,3] -> yt},
797 TeX -> Superscript[y, u],
798 Description -> "Up-type Yukawa couplings"
799 },
800 yd == {
801 ParameterType -> Internal,
802 Indices -> {Index[Generation], Index[Generation]},
803 Definitions -> {yd[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
804 Value -> {yd[1,1] -> Sqrt[2] ymdo/vev, yd[2,2] -> Sqrt[2] yms/vev, yd[3,3] -> Sqrt[2] ymb/vev},
805 InteractionOrder -> {QED, 1},
806 ParameterName -> {yd[1,1] -> ydo, yd[2,2] -> ys, yd[3,3] -> yb},
807 TeX -> Superscript[y, d],
808 Description -> "Down-type Yukawa couplings"
809 },
810(* N. B. : only Cabibbo mixing! *)
811 CKM == {
812 ParameterType -> Internal,
813 Indices -> {Index[Generation], Index[Generation]},
814 Unitary -> True,
815 Value -> {CKM[1,1] -> Cos[cabi] Cos[th13], CKM[1,2] -> Sin[cabi] Cos[th13], CKM[1,3] -> Sin[th13] Exp[-I del13],
816 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],
817 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]},
818 TeX -> Superscript[V,CKM],
819 Description -> "CKM-Matrix"}
820};
821
822(* ************************** *)
823(* ***** Lagrangian ***** *)
824(* ************************** *)
825
826LWPolarMassBase := MW*MW * W0bar[mu] * (WT[mu] + WA[mu] + WX[mu]) \
827 + MW*MW * WTbar[mu] * (W0[mu] + WA[mu] + WX[mu]) \
828 + MW*MW * WAbar[mu] * (W0[mu] + WT[mu] + WX[mu]) \
829 + MW*MW * WXbar[mu] * (W0[mu] + WT[mu] + WA[mu]) ;
830LPolarMass := LWPolarMassBase;
831
832
833LGauge := Block[{mu,nu,ii,aa},
834 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]];
835
836LFermions := Block[{mu},
837 ExpandIndices[I*(
838 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]),
839 FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]}];
840
841LHiggsBase := Block[{ii,mu, feynmangaugerules},
842 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
843
844 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
845 ];
846LHiggs := Simplify[LHiggsBase - LPolarMass];
847
848
849LYukawa := Block[{sp,ii,jj,cc,ff1,ff2,ff3,yuk,feynmangaugerules},
850 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
851
852 yuk = ExpandIndices[
853 -yd[ff2, ff3] CKM[ff1, ff2] QLbar[sp, ii, ff1, cc].dR [sp, ff3, cc] Phi[ii] -
854 yl[ff1, ff3] LLbar[sp, ii, ff1].lR [sp, ff3] Phi[ii] -
855 yu[ff1, ff2] QLbar[sp, ii, ff1, cc].uR [sp, ff2, cc] Phibar[jj] Eps[ii, jj], FlavorExpand -> SU2D];
856 yuk = yuk /. { CKM[a_, b_] Conjugate[CKM[a_, c_]] -> IndexDelta[b, c], CKM[b_, a_] Conjugate[CKM[c_, a_]] -> IndexDelta[b, c]};
857 yuk+HC[yuk]/.feynmangaugerules
858 ];
859
860LGhost := Block[{LGh1,LGhw,LGhs,LGhphi,mu, generators,gh,ghbar,Vectorize,phi1,phi2,togoldstones,doublet,doublet0},
861 (* Pure gauge piece *)
862 LGh1 = -ghBbar.del[DC[ghB,mu],mu];
863 LGhw = -ghWibar[ii].del[DC[ghWi[ii],mu],mu];
864 LGhs = -ghGbar[ii].del[DC[ghG[ii],mu],mu];
865
866 (* Scalar pieces: see Peskin pages 739-742 *)
867 (* phi1 and phi2 are the real degrees of freedom of GP *)
868 (* Vectorize transforms a doublet in a vector in the phi-basis, i.e. the basis of real degrees of freedom *)
869 gh = {ghB, ghWi[1], ghWi[2], ghWi[3]};
870 ghbar = {ghBbar, ghWibar[1], ghWibar[2], ghWibar[3]};
871 generators = {-I/2 g1 IdentityMatrix[2], -I/2 gw PauliSigma[1], -I/2 gw PauliSigma[2], -I/2 gw PauliSigma[3]};
872 doublet = Expand[{(-I phi1 - phi2)/Sqrt[2], Phi[2]} /. MR$Definitions /. vev -> 0];
873 doublet0 = {0, vev/Sqrt[2]};
874 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}];
875 togoldstones := {phi1 -> (GP + GPbar)/Sqrt[2], phi2 -> (-GP + GPbar)/(I Sqrt[2])};
876 LGhphi=Plus@@Flatten[Table[-ghbar[[kkk]].gh[[lll]] Vectorize[generators[[kkk]].doublet0].Vectorize[generators[[lll]].(doublet+doublet0)],{kkk,4},{lll,4}]] /.togoldstones;
877
878ExpandIndices[ LGhs + If[FeynmanGauge, LGh1 + LGhw + LGhphi,0], FlavorExpand->SU2W]];
879
880LSM:= LGauge + LFermions + LHiggs + LYukawa + LGhost;