2GravitonsSM: 2GravitonsSM.fr

File 2GravitonsSM.fr, 27.6 KB (added by Loc Tran Quang, 5 years ago)

Model file for 2GravitonsSM

Line 
1(***************************************************************************************************************)
2(****** This is the FeynRules mod-file for 2 Gravitons coupled to Standard Model (2GravitonsSM). ******)
3(****** ******)
4(****** Authors: Tran Quang Loc, developed from SM file of 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$ModelName = "2 Gravitons coupled to Standard Model";
16
17M$Information = {
18 Authors -> {"Tran Quang Loc"},
19 Version -> "1.0.0",
20 Date -> "05. 09. 2021",
21 Institutions -> {"DAMTP, Cambridge"},
22 Emails -> {"loctranq@gmail.com"},
23 URLs -> "https://feynrules.irmp.ucl.ac.be/wiki/2GravitonsSM"
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(* v1.0.0: Implement the first model *)
41
42(* ************************** *)
43(* ***** vevs ***** *)
44(* ************************** *)
45M$vevs = { {Phi[2],vev} };
46
47(* ************************** *)
48(* ***** Gauge groups ***** *)
49(* ************************** *)
50M$GaugeGroups = {
51 U1Y == {
52 Abelian -> True,
53 CouplingConstant -> g1,
54 GaugeBoson -> B,
55 Charge -> Y
56 },
57 SU2L == {
58 Abelian -> False,
59 CouplingConstant -> gw,
60 GaugeBoson -> Wi,
61 StructureConstant -> Eps,
62 Representations -> {Ta,SU2D},
63 Definitions -> {Ta[a_,b_,c_]->PauliSigma[a,b,c]/2, FSU2L[i_,j_,k_]:> I Eps[i,j,k]}
64 },
65 SU3C == {
66 Abelian -> False,
67 CouplingConstant -> gs,
68 GaugeBoson -> G,
69 StructureConstant -> f,
70 Representations -> {T,Colour},
71 SymmetricTensor -> dSUN
72 }
73};
74
75
76(* ************************** *)
77(* ***** Indices ***** *)
78(* ************************** *)
79
80IndexRange[Index[SU2W ]] = Unfold[Range[3]];
81IndexRange[Index[SU2D ]] = Unfold[Range[2]];
82IndexRange[Index[Gluon ]] = NoUnfold[Range[8]];
83IndexRange[Index[Colour ]] = NoUnfold[Range[3]];
84IndexRange[Index[Generation]] = Range[3];
85
86IndexStyle[SU2W, j];
87IndexStyle[SU2D, k];
88IndexStyle[Gluon, a];
89IndexStyle[Colour, m];
90IndexStyle[Generation, f];
91
92
93(* ************************** *)
94(* *** Interaction orders *** *)
95(* *** (as used by mg5) *** *)
96(* ************************** *)
97
98M$InteractionOrderHierarchy = {
99 {QCD, 1},
100 {QED, 2}
101};
102
103
104(* ************************** *)
105(* **** Particle classes **** *)
106(* ************************** *)
107M$ClassesDescription = {
108
109(* Gauge bosons: physical vector fields *)
110 V[1] == {
111 ClassName -> A,
112 SelfConjugate -> True,
113 Mass -> 0,
114 Width -> 0,
115 ParticleName -> "A",
116 PDG -> 22,
117 PropagatorLabel -> "A",
118 PropagatorType -> W,
119 PropagatorArrow -> None,
120 FullName -> "Photon"
121 },
122 V[2] == {
123 ClassName -> Z,
124 SelfConjugate -> True,
125 Mass -> {MZ, 91.1876},
126 Width -> {WZ, 2.4952},
127 ParticleName -> "Z",
128 PDG -> 23,
129 PropagatorLabel -> "Z",
130 PropagatorType -> Sine,
131 PropagatorArrow -> None,
132 FullName -> "Z"
133 },
134 V[3] == {
135 ClassName -> W,
136 SelfConjugate -> False,
137 Mass -> {MW, Internal},
138 Width -> {WW, 2.085},
139 ParticleName -> "W+",
140 AntiParticleName -> "W-",
141 QuantumNumbers -> {Q -> 1},
142 PDG -> 24,
143 PropagatorLabel -> "W",
144 PropagatorType -> Sine,
145 PropagatorArrow -> Forward,
146 FullName -> "W"
147 },
148 V[4] == {
149 ClassName -> G,
150 SelfConjugate -> True,
151 Indices -> {Index[Gluon]},
152 Mass -> 0,
153 Width -> 0,
154 ParticleName -> "g",
155 PDG -> 21,
156 PropagatorLabel -> "G",
157 PropagatorType -> C,
158 PropagatorArrow -> None,
159 FullName -> "G"
160 },
161
162(* Ghosts: related to physical gauge bosons *)
163 U[1] == {
164 ClassName -> ghA,
165 SelfConjugate -> False,
166 Ghost -> A,
167 QuantumNumbers -> {GhostNumber -> 1},
168 Mass -> 0,
169 Width -> 0,
170 PropagatorLabel -> "uA",
171 PropagatorType -> GhostDash,
172 PropagatorArrow -> Forward
173 },
174 U[2] == {
175 ClassName -> ghZ,
176 SelfConjugate -> False,
177 Ghost -> Z,
178 QuantumNumbers -> {GhostNumber -> 1},
179 Mass -> {MZ,91.1876},
180 Width -> {WZ, 2.4952},
181 PropagatorLabel -> "uZ",
182 PropagatorType -> GhostDash,
183 PropagatorArrow -> Forward
184 },
185 U[31] == {
186 ClassName -> ghWp,
187 SelfConjugate -> False,
188 Ghost -> W,
189 QuantumNumbers -> {GhostNumber -> 1, Q -> 1},
190 Mass -> {MW,Internal},
191 Width -> {WW, 2.085},
192 PropagatorLabel -> "uWp",
193 PropagatorType -> GhostDash,
194 PropagatorArrow -> Forward
195 },
196 U[32] == {
197 ClassName -> ghWm,
198 SelfConjugate -> False,
199 Ghost -> Wbar,
200 QuantumNumbers -> {GhostNumber -> 1, Q -> -1},
201 Mass -> {MW,Internal},
202 Width -> {WW, 2.085},
203 PropagatorLabel -> "uWm",
204 PropagatorType -> GhostDash,
205 PropagatorArrow -> Forward
206 },
207 U[4] == {
208 ClassName -> ghG,
209 SelfConjugate -> False,
210 Indices -> {Index[Gluon]},
211 Ghost -> G,
212 PDG -> 82,
213 QuantumNumbers ->{GhostNumber -> 1},
214 Mass -> 0,
215 Width -> 0,
216 PropagatorLabel -> "uG",
217 PropagatorType -> GhostDash,
218 PropagatorArrow -> Forward
219 },
220
221(* Gauge bosons: unphysical vector fields *)
222 V[11] == {
223 ClassName -> B,
224 Unphysical -> True,
225 SelfConjugate -> True,
226 Definitions -> { B[mu_] -> -sw Z[mu]+cw A[mu]}
227 },
228 V[12] == {
229 ClassName -> Wi,
230 Unphysical -> True,
231 SelfConjugate -> True,
232 Indices -> {Index[SU2W]},
233 FlavorIndex -> SU2W,
234 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]}
235 },
236
237(* Ghosts: related to unphysical gauge bosons *)
238 U[11] == {
239 ClassName -> ghB,
240 Unphysical -> True,
241 SelfConjugate -> False,
242 Ghost -> B,
243 Definitions -> { ghB -> -sw ghZ + cw ghA}
244 },
245 U[12] == {
246 ClassName -> ghWi,
247 Unphysical -> True,
248 SelfConjugate -> False,
249 Ghost -> Wi,
250 Indices -> {Index[SU2W]},
251 FlavorIndex -> SU2W,
252 Definitions -> { ghWi[1] -> (ghWp+ghWm)/Sqrt[2], ghWi[2] -> (ghWm-ghWp)/(I*Sqrt[2]), ghWi[3] -> cw ghZ+sw ghA}
253 } ,
254
255(* Fermions: physical fields *)
256 F[1] == {
257 ClassName -> vl,
258 ClassMembers -> {ve,vm,vt},
259 Indices -> {Index[Generation]},
260 FlavorIndex -> Generation,
261 SelfConjugate -> False,
262 Mass -> 0,
263 Width -> 0,
264 QuantumNumbers -> {LeptonNumber -> 1},
265 PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
266 PropagatorType -> S,
267 PropagatorArrow -> Forward,
268 PDG -> {12,14,16},
269 ParticleName -> {"ve","vm","vt"},
270 AntiParticleName -> {"ve~","vm~","vt~"},
271 FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"}
272 },
273 F[2] == {
274 ClassName -> l,
275 ClassMembers -> {e, mu, ta},
276 Indices -> {Index[Generation]},
277 FlavorIndex -> Generation,
278 SelfConjugate -> False,
279 Mass -> {Ml, {Me,5.11*^-4}, {MMU,0.10566}, {MTA,1.777}},
280 Width -> 0,
281 QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
282 PropagatorLabel -> {"l", "e", "mu", "ta"},
283 PropagatorType -> Straight,
284 PropagatorArrow -> Forward,
285 PDG -> {11, 13, 15},
286 ParticleName -> {"e-", "mu-", "ta-"},
287 AntiParticleName -> {"e+", "mu+", "ta+"},
288 FullName -> {"Electron", "Muon", "Tau"}
289 },
290 F[3] == {
291 ClassName -> uq,
292 ClassMembers -> {u, c, t},
293 Indices -> {Index[Generation], Index[Colour]},
294 FlavorIndex -> Generation,
295 SelfConjugate -> False,
296 Mass -> {Mu, {MU, 2.55*^-3}, {MC,1.27}, {MT,172}},
297 Width -> {0, 0, {WT,1.50833649}},
298 QuantumNumbers -> {Q -> 2/3},
299 PropagatorLabel -> {"uq", "u", "c", "t"},
300 PropagatorType -> Straight,
301 PropagatorArrow -> Forward,
302 PDG -> {2, 4, 6},
303 ParticleName -> {"u", "c", "t" },
304 AntiParticleName -> {"u~", "c~", "t~"},
305 FullName -> {"u-quark", "c-quark", "t-quark"}
306 },
307 F[4] == {
308 ClassName -> dq,
309 ClassMembers -> {d, s, b},
310 Indices -> {Index[Generation], Index[Colour]},
311 FlavorIndex -> Generation,
312 SelfConjugate -> False,
313 Mass -> {Md, {MD,5.04*^-3}, {MS,0.101}, {MB,4.7}},
314 Width -> 0,
315 QuantumNumbers -> {Q -> -1/3},
316 PropagatorLabel -> {"dq", "d", "s", "b"},
317 PropagatorType -> Straight,
318 PropagatorArrow -> Forward,
319 PDG -> {1,3,5},
320 ParticleName -> {"d", "s", "b" },
321 AntiParticleName -> {"d~", "s~", "b~"},
322 FullName -> {"d-quark", "s-quark", "b-quark"}
323 },
324
325(* Fermions: unphysical fields *)
326 F[11] == {
327 ClassName -> LL,
328 Unphysical -> True,
329 Indices -> {Index[SU2D], Index[Generation]},
330 FlavorIndex -> SU2D,
331 SelfConjugate -> False,
332 QuantumNumbers -> {Y -> -1/2},
333 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]] }
334 },
335 F[12] == {
336 ClassName -> lR,
337 Unphysical -> True,
338 Indices -> {Index[Generation]},
339 FlavorIndex -> Generation,
340 SelfConjugate -> False,
341 QuantumNumbers -> {Y -> -1},
342 Definitions -> { lR[sp1_,ff_] :> Module[{sp2}, ProjP[sp1,sp2] l[sp2,ff]] }
343 },
344 F[13] == {
345 ClassName -> QL,
346 Unphysical -> True,
347 Indices -> {Index[SU2D], Index[Generation], Index[Colour]},
348 FlavorIndex -> SU2D,
349 SelfConjugate -> False,
350 QuantumNumbers -> {Y -> 1/6},
351 Definitions -> {
352 QL[sp1_,1,ff_,cc_] :> Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],
353 QL[sp1_,2,ff_,cc_] :> Module[{sp2,ff2}, CKM[ff,ff2] ProjM[sp1,sp2] dq[sp2,ff2,cc]] }
354 },
355 F[14] == {
356 ClassName -> uR,
357 Unphysical -> True,
358 Indices -> {Index[Generation], Index[Colour]},
359 FlavorIndex -> Generation,
360 SelfConjugate -> False,
361 QuantumNumbers -> {Y -> 2/3},
362 Definitions -> { uR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] uq[sp2,ff,cc]] }
363 },
364 F[15] == {
365 ClassName -> dR,
366 Unphysical -> True,
367 Indices -> {Index[Generation], Index[Colour]},
368 FlavorIndex -> Generation,
369 SelfConjugate -> False,
370 QuantumNumbers -> {Y -> -1/3},
371 Definitions -> { dR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] dq[sp2,ff,cc]] }
372 },
373
374(* Higgs: physical scalars *)
375 S[1] == {
376 ClassName -> H,
377 SelfConjugate -> True,
378 Mass -> {MH,125},
379 Width -> {WH,0.00407},
380 PropagatorLabel -> "H",
381 PropagatorType -> D,
382 PropagatorArrow -> None,
383 PDG -> 25,
384 ParticleName -> "H",
385 FullName -> "H"
386 },
387
388(* Higgs: physical scalars *)
389 S[2] == {
390 ClassName -> G0,
391 SelfConjugate -> True,
392 Goldstone -> Z,
393 Mass -> {MZ, 91.1876},
394 Width -> {WZ, 2.4952},
395 PropagatorLabel -> "Go",
396 PropagatorType -> D,
397 PropagatorArrow -> None,
398 PDG -> 250,
399 ParticleName -> "G0",
400 FullName -> "G0"
401 },
402 S[3] == {
403 ClassName -> GP,
404 SelfConjugate -> False,
405 Goldstone -> W,
406 Mass -> {MW, Internal},
407 QuantumNumbers -> {Q -> 1},
408 Width -> {WW, 2.085},
409 PropagatorLabel -> "GP",
410 PropagatorType -> D,
411 PropagatorArrow -> None,
412 PDG -> 251,
413 ParticleName -> "G+",
414 AntiParticleName -> "G-",
415 FullName -> "GP"
416 },
417
418(* Higgs: unphysical scalars *)
419 S[11] == {
420 ClassName -> Phi,
421 Unphysical -> True,
422 Indices -> {Index[SU2D]},
423 FlavorIndex -> SU2D,
424 SelfConjugate -> False,
425 QuantumNumbers -> {Y -> 1/2},
426 Definitions -> { Phi[1] -> -I GP, Phi[2] -> (vev + H + I G0)/Sqrt[2] }
427 },
428(******************************* Spin 2 particles: graviton *****************************)
429
430 T[1] == {
431 ClassName -> h,
432 SelfConjugate -> True,
433 Symmetric -> True,
434 Mass -> {Mh, 500},
435 Width -> {Wh, 10.},
436 PropagatorLabel -> "h",
437 PropagatorType -> GhostDash,
438 PropagatorArrow -> None,
439 PDG -> 5000002,
440 TeX -> h,
441 FullName -> "Graviton" }
442};
443
444
445
446
447(* ************************** *)
448(* ***** Gauge ***** *)
449(* ***** Parameters ***** *)
450(* ***** (FeynArts) ***** *)
451(* ************************** *)
452
453GaugeXi[ V[1] ] = GaugeXi[A];
454GaugeXi[ V[2] ] = GaugeXi[Z];
455GaugeXi[ V[3] ] = GaugeXi[W];
456GaugeXi[ V[4] ] = GaugeXi[G];
457GaugeXi[ S[1] ] = 1;
458GaugeXi[ S[2] ] = GaugeXi[Z];
459GaugeXi[ S[3] ] = GaugeXi[W];
460GaugeXi[ U[1] ] = GaugeXi[A];
461GaugeXi[ U[2] ] = GaugeXi[Z];
462GaugeXi[ U[31] ] = GaugeXi[W];
463GaugeXi[ U[32] ] = GaugeXi[W];
464GaugeXi[ U[4] ] = GaugeXi[G];
465
466
467(* ************************** *)
468(* ***** Parameters ***** *)
469(* ************************** *)
470M$Parameters = {
471
472 (* External parameters *)
473 aEWM1 == {
474 ParameterType -> External,
475 BlockName -> SMINPUTS,
476 OrderBlock -> 1,
477 Value -> 127.9,
478 InteractionOrder -> {QED,-2},
479 Description -> "Inverse of the EW coupling constant at the Z pole"
480 },
481 Gf == {
482 ParameterType -> External,
483 BlockName -> SMINPUTS,
484 OrderBlock -> 2,
485 Value -> 1.16637*^-5,
486 InteractionOrder -> {QED,2},
487 TeX -> Subscript[G,f],
488 Description -> "Fermi constant"
489 },
490 aS == {
491 ParameterType -> External,
492 BlockName -> SMINPUTS,
493 OrderBlock -> 3,
494 Value -> 0.1184,
495 InteractionOrder -> {QCD,2},
496 TeX -> Subscript[\[Alpha],s],
497 Description -> "Strong coupling constant at the Z pole"
498 },
499 ymdo == {
500 ParameterType -> External,
501 BlockName -> YUKAWA,
502 OrderBlock -> 1,
503 Value -> 5.04*^-3,
504 Description -> "Down Yukawa mass"
505 },
506 ymup == {
507 ParameterType -> External,
508 BlockName -> YUKAWA,
509 OrderBlock -> 2,
510 Value -> 2.55*^-3,
511 Description -> "Up Yukawa mass"
512 },
513 yms == {
514 ParameterType -> External,
515 BlockName -> YUKAWA,
516 OrderBlock -> 3,
517 Value -> 0.101,
518 Description -> "Strange Yukawa mass"
519 },
520 ymc == {
521 ParameterType -> External,
522 BlockName -> YUKAWA,
523 OrderBlock -> 4,
524 Value -> 1.27,
525 Description -> "Charm Yukawa mass"
526 },
527 ymb == {
528 ParameterType -> External,
529 BlockName -> YUKAWA,
530 OrderBlock -> 5,
531 Value -> 4.7,
532 Description -> "Bottom Yukawa mass"
533 },
534 ymt == {
535 ParameterType -> External,
536 BlockName -> YUKAWA,
537 OrderBlock -> 6,
538 Value -> 172,
539 Description -> "Top Yukawa mass"
540 },
541 yme == {
542 ParameterType -> External,
543 BlockName -> YUKAWA,
544 OrderBlock -> 11,
545 Value -> 5.11*^-4,
546 Description -> "Electron Yukawa mass"
547 },
548 ymm == {
549 ParameterType -> External,
550 BlockName -> YUKAWA,
551 OrderBlock -> 13,
552 Value -> 0.10566,
553 Description -> "Muon Yukawa mass"
554 },
555 ymtau == {
556 ParameterType -> External,
557 BlockName -> YUKAWA,
558 OrderBlock -> 15,
559 Value -> 1.777,
560 Description -> "Tau Yukawa mass"
561 },
562 cabi == {
563 ParameterType -> External,
564 BlockName -> CKMBLOCK,
565 OrderBlock -> 1,
566 Value -> 0.227736,
567 TeX -> Subscript[\[Theta], c],
568 Description -> "Cabibbo angle"
569 },
570
571 GN == {
572 ParameterType -> External,
573 ParameterName -> GN,
574 InteractionOrder -> {QCD, 2},
575 Value -> 10^(-16),
576 Description -> "Newton Constant"},
577
578 (* Internal Parameters *)
579 aEW == {
580 ParameterType -> Internal,
581 Value -> 1/aEWM1,
582 InteractionOrder -> {QED,2},
583 TeX -> Subscript[\[Alpha], EW],
584 Description -> "Electroweak coupling contant"
585 },
586 MW == {
587 ParameterType -> Internal,
588 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
589 TeX -> Subscript[M,W],
590 Description -> "W mass"
591 },
592 sw2 == {
593 ParameterType -> Internal,
594 Value -> 1-(MW/MZ)^2,
595 Description -> "Squared Sin of the Weinberg angle"
596 },
597 ee == {
598 ParameterType -> Internal,
599 Value -> Sqrt[4 Pi aEW],
600 InteractionOrder -> {QED,1},
601 TeX -> e,
602 Description -> "Electric coupling constant"
603 },
604 cw == {
605 ParameterType -> Internal,
606 Value -> Sqrt[1-sw2],
607 TeX -> Subscript[c,w],
608 Description -> "Cosine of the Weinberg angle"
609 },
610 sw == {
611 ParameterType -> Internal,
612 Value -> Sqrt[sw2],
613 TeX -> Subscript[s,w],
614 Description -> "Sine of the Weinberg angle"
615 },
616 gw == {
617 ParameterType -> Internal,
618 Definitions -> {gw->ee/sw},
619 InteractionOrder -> {QED,1},
620 TeX -> Subscript[g,w],
621 Description -> "Weak coupling constant at the Z pole"
622 },
623 g1 == {
624 ParameterType -> Internal,
625 Definitions -> {g1->ee/cw},
626 InteractionOrder -> {QED,1},
627 TeX -> Subscript[g,1],
628 Description -> "U(1)Y coupling constant at the Z pole"
629 },
630 gs == {
631 ParameterType -> Internal,
632 Value -> Sqrt[4 Pi aS],
633 InteractionOrder -> {QCD,1},
634 TeX -> Subscript[g,s],
635 ParameterName -> G,
636 Description -> "Strong coupling constant at the Z pole"
637 },
638 vev == {
639 ParameterType -> Internal,
640 Value -> 2*MW*sw/ee,
641 InteractionOrder -> {QED,-1},
642 Description -> "Higgs vacuum expectation value"
643 },
644 lam == {
645 ParameterType -> Internal,
646 Value -> MH^2/(2*vev^2),
647 InteractionOrder -> {QED, 2},
648 Description -> "Higgs quartic coupling"
649 },
650 muH == {
651 ParameterType -> Internal,
652 Value -> Sqrt[vev^2 lam],
653 TeX -> \[Mu],
654 Description -> "Coefficient of the quadratic piece of the Higgs potential"
655 },
656 yl == {
657 ParameterType -> Internal,
658 Indices -> {Index[Generation], Index[Generation]},
659 Definitions -> {yl[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
660 Value -> {yl[1,1] -> Sqrt[2] yme / vev, yl[2,2] -> Sqrt[2] ymm / vev, yl[3,3] -> Sqrt[2] ymtau / vev},
661 InteractionOrder -> {QED, 1},
662 ParameterName -> {yl[1,1] -> ye, yl[2,2] -> ym, yl[3,3] -> ytau},
663 TeX -> Superscript[y, l],
664 Description -> "Lepton Yukawa couplings"
665 },
666 yu == {
667 ParameterType -> Internal,
668 Indices -> {Index[Generation], Index[Generation]},
669 Definitions -> {yu[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
670 Value -> {yu[1,1] -> Sqrt[2] ymup/vev, yu[2,2] -> Sqrt[2] ymc/vev, yu[3,3] -> Sqrt[2] ymt/vev},
671 InteractionOrder -> {QED, 1},
672 ParameterName -> {yu[1,1] -> yup, yu[2,2] -> yc, yu[3,3] -> yt},
673 TeX -> Superscript[y, u],
674 Description -> "Up-type Yukawa couplings"
675 },
676 yd == {
677 ParameterType -> Internal,
678 Indices -> {Index[Generation], Index[Generation]},
679 Definitions -> {yd[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
680 Value -> {yd[1,1] -> Sqrt[2] ymdo/vev, yd[2,2] -> Sqrt[2] yms/vev, yd[3,3] -> Sqrt[2] ymb/vev},
681 InteractionOrder -> {QED, 1},
682 ParameterName -> {yd[1,1] -> ydo, yd[2,2] -> ys, yd[3,3] -> yb},
683 TeX -> Superscript[y, d],
684 Description -> "Down-type Yukawa couplings"
685 },
686(* N. B. : only Cabibbo mixing! *)
687 CKM == {
688 ParameterType -> Internal,
689 Indices -> {Index[Generation], Index[Generation]},
690 Unitary -> True,
691 Value -> {CKM[1,1] -> Cos[cabi], CKM[1,2] -> Sin[cabi], CKM[1,3] -> 0,
692 CKM[2,1] -> -Sin[cabi], CKM[2,2] -> Cos[cabi], CKM[2,3] -> 0,
693 CKM[3,1] -> 0, CKM[3,2] -> 0, CKM[3,3] -> 1},
694 TeX -> Superscript[V,CKM],
695 Description -> "CKM-Matrix"},
696
697(******************************* Spin 2 particles: graviton *****************************)
698
699 kappa == {
700 TeX -> \[Kappa],
701 ParameterType -> Internal,
702 Value -> Sqrt[16 Pi GN]}
703};
704
705
706(************************************************************************************************)
707(******************* 2 Gravitons coupled to SM Lagrangian ************************)
708(************************************************************************************************)
709SqrtdetG := Block[{muS,nuS},
710 ExpandIndices[1 + kappa/2 h[muS,muS] - kappa^2/4 h[muS, nuS] h[muS, nuS] + kappa^2/8 h[muS, muS] h[nuS, nuS]]];
711
712G[mu_,nu_,rho_] := Block[{mu,nu,rho},
713 ExpandIndices[ME[mu,nu] - kappa h[mu,nu] + kappa^2 h[mu,rho]h[rho,nu]]];
714
715Sigma[mu_,nu_] := Block[{mu,nu}, ExpandIndices[ I/2 (Ga[mu].Ga[nu] - Ga[nu].Ga[mu])]];
716
717Vielbein[a_, mu_, lambda_] := Block[{a, mu,lambda},
718 ExpandIndices[ME[a, mu] - kappa/2 h[a, mu] + 3/8 kappa^2 h[a, lambda] h[lambda, mu]]];
719
720Lambda[mu_,a_,b_,lambda_,beta_] := Block[{a,b,mu,lambda,beta},
721 ExpandIndices[Normal[Series[I/8 kappa Sigma[a,b](del[h[b,mu],a]-del[h[a,mu],b])-I/16 kappa^2 Sigma[a,b](h[b,lambda](del[h[lambda,a],mu]+del[h[lambda,mu],a]-del[h[mu,a],lambda])+h[a,beta](del[h[b,mu],beta]-del[h[mu,beta],b])),{kappa,0,2}]]]] ;
722
723LGGauge := Block[{mu, nu, alpha, beta, ii, aa, itn1, itn2},
724 ExpandIndices[SqrtdetG * (-1/4 G[mu, nu, itn1] G[alpha, beta, itn2] FS[B, mu, alpha] FS[B, nu, beta] - 1/4 G[mu, nu, itn1] G[alpha, beta, itn2] FS[Wi, mu, alpha, ii] FS[Wi, nu, beta, ii] - 1/4 G[mu, nu, itn1] G[alpha, beta, itn2] FS[G, mu, alpha, aa] FS[G, nu, beta, aa]), FlavorExpand -> SU2W]];
725
726LGFermions := Block[{mu,a,b,c,alpha,beta,lambda,LGFermionsLL1,LGFermionsLL2,LGFermionslR1,LGFermionslR2},
727 LGFermionsLL1 = ExpandIndices[I * SqrtdetG * LLbar.Ga[a].Vielbein[a,mu,alpha].DC[LL, mu], FlavorExpand->{SU2W,SU2D}];
728 LGFermionsLL2 = ExpandIndices[I * SqrtdetG * LLbar.Ga[a].Vielbein[a,mu,alpha].Lambda[mu,b,c,lambda,beta].LL, FlavorExpand->{SU2W,SU2D}];
729 LGFermionslR1 = ExpandIndices[I * SqrtdetG * lRbar.Ga[a].Vielbein[a,mu,alpha].DC[lR, mu], FlavorExpand->{SU2W,SU2D}];
730 LGFermionslR2 = ExpandIndices[I * SqrtdetG * lRbar.Ga[a].Vielbein[a,mu,alpha].Lambda[mu,b,c,lambda,beta].lR, FlavorExpand->{SU2W,SU2D}];
731 LGFermionsQL1 = ExpandIndices[I * SqrtdetG * QLbar.Ga[a].Vielbein[a,mu,alpha].DC[QL, mu], FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]};
732 LGFermionsQL2 = ExpandIndices[I * SqrtdetG * QLbar.Ga[a].Vielbein[a,mu,alpha].Lambda[mu,b,c,lambda,beta].QL, FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]};
733 LGFermionsuR1 = ExpandIndices[I * SqrtdetG * uRbar.Ga[a].Vielbein[a,mu,alpha].DC[uR, mu], FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]};
734 LGFermionsuR2 = ExpandIndices[I * SqrtdetG * uRbar.Ga[a].Vielbein[a,mu,alpha].Lambda[mu,b,c,lambda,beta].uR, FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]};
735 LGFermionsdR1 = ExpandIndices[I * SqrtdetG * dRbar.Ga[a].Vielbein[a,mu,alpha].DC[dR, mu], FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]};
736 LGFermionsdR2 = ExpandIndices[I * SqrtdetG * dRbar.Ga[a].Vielbein[a,mu,alpha].Lambda[mu,b,c,lambda,beta].dR, FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]};
737 LGFermionsLL1 + LGFermionsLL2 + LGFermionslR1 + LGFermionslR2 + LGFermionsQL1 + LGFermionsQL2 + LGFermionsuR1 + LGFermionsuR2 + LGFermionsdR1 + LGFermionsdR2];
738
739LGHiggs := Block[{ii,mu,nu,rho,feynmangaugerules},
740 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
741
742 ExpandIndices[SqrtdetG * (G[mu,nu,rho] DC[Phibar[ii],mu] DC[Phi[ii],nu] + muH^2 Phibar[ii] Phi[ii] - lam Phibar[ii] Phi[ii] Phibar[jj] Phi[jj]), FlavorExpand->{SU2D,SU2W}]/.feynmangaugerules
743 ];
744
745LGYukawa := Block[{sp,ii,jj,cc,ff1,ff2,ff3,yuk,feynmangaugerules},
746 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
747
748 yuk = ExpandIndices[
749 SqrtdetG * (-yd[ff2, ff3] CKM[ff1, ff2] QLbar[sp, ii, ff1, cc].dR [sp, ff3, cc] Phi[ii] -
750 yl[ff1, ff3] LLbar[sp, ii, ff1].lR [sp, ff3] Phi[ii] -
751 yu[ff1, ff2] QLbar[sp, ii, ff1, cc].uR [sp, ff2, cc] Phibar[jj] Eps[ii, jj]), FlavorExpand -> SU2D];
752 yuk = yuk /. { CKM[a_, b_] Conjugate[CKM[a_, c_]] -> IndexDelta[b, c], CKM[b_, a_] Conjugate[CKM[c_, a_]] -> IndexDelta[b, c]};
753 SqrtdetG *(yuk+HC[yuk])/.feynmangaugerules
754 ];
755
756LGGhost := Block[{LGh1,LGhw,LGhs,LGhphi,mu,nu,rho,generators,gh,ghbar,Vectorize,phi1,phi2,togoldstones,doublet,doublet0},
757 (* Pure gauge piece *)
758 LGh1 = -ghBbar.G[mu,nu,rho].del[DC[ghB,mu],nu];
759 LGhw = -ghWibar[ii].G[mu,nu,rho].del[DC[ghWi[ii],mu],nu];
760 LGhs = -ghGbar[ii].G[mu,nu,rho].del[DC[ghG[ii],mu],nu];
761
762 (* Scalar pieces: see Peskin pages 739-742 *)
763 (* phi1 and phi2 are the real degrees of freedom of GP *)
764 (* Vectorize transforms a doublet in a vector in the phi-basis, i.e. the basis of real degrees of freedom *)
765 gh = {ghB, ghWi[1], ghWi[2], ghWi[3]};
766 ghbar = {ghBbar, ghWibar[1], ghWibar[2], ghWibar[3]};
767 generators = {-I/2 g1 IdentityMatrix[2], -I/2 gw PauliSigma[1], -I/2 gw PauliSigma[2], -I/2 gw PauliSigma[3]};
768 doublet = Expand[{(-I phi1 - phi2)/Sqrt[2], Phi[2]} /. MR$Definitions /. vev -> 0];
769 doublet0 = {0, vev/Sqrt[2]};
770 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}];
771 togoldstones := {phi1 -> (GP + GPbar)/Sqrt[2], phi2 -> (-GP + GPbar)/(I Sqrt[2])};
772 LGhphi=Plus@@Flatten[Table[-ghbar[[kkk]].gh[[lll]] Vectorize[generators[[kkk]].doublet0].Vectorize[generators[[lll]].(doublet+doublet0)],{kkk,4},{lll,4}]] /.togoldstones;
773
774ExpandIndices[SqrtdetG * LGhs + If[FeynmanGauge, SqrtdetG * (LGh1 + LGhw + LGhphi),0], FlavorExpand->SU2W]];
775
776LGGaugeO2:= Normal[Series[LGGauge,{kappa,0,2}]];
777LGFermionsO2:= Normal[Series[LGFermions,{kappa,0,2}]];
778LGHiggsO2:= Normal[Series[LGHiggs,{kappa,0,2}]];
779LGYukawaO2:= Normal[Series[LGYukawa,{kappa,0,2}]];
780LGGhostO2:= Normal[Series[LGGhost,{kappa,0,2}]];
781
782LGTotal:= LGGaugeO2 + LGFermionsO2 + LGHiggsO2 + LGYukawaO2 + LGGhostO2;