pNG: SM_with_pNG.fr

File SM_with_pNG.fr, 28.7 KB (added by Ankit Beniwal, 6 years ago)

FeynRules model file

Line 
1(****************************************************************************************************************************)
2(**** ****)
3(****** FeynRules (.fr) file for the SM + pseudo-Nambu-Goldstone (pNG) DM in a softly broken U(1) scalar singlet model ******)
4(****** Model based on: C. Gross, O. Lebedev and T. Toma, [arXiv:1708.02253] ******)
5(****** Model file based on: Models/SM/SM.fr ******)
6(****** Gauge choice: Unitary gauge ****)
7(**** ****)
8(****** Choose whether Feynman gauge is desired. ******)
9(****** If set to False, unitary gauge is assumed. ******)
10(****** Feynman gauge is especially useful for CalcHEP/CompHEP where the calculation is 10-100 times faster. ******)
11(****** Feynman gauge is not supported in MadGraph and Sherpa. ******)
12(**** ****)
13(****************************************************************************************************************************)
14
15(* ************************** *)
16(* ***** Information ***** *)
17(* ************************** *)
18M$ModelName = "SM_with_pNG";
19
20M$Information = {
21 Authors -> {"A. Beniwal"},
22 Version -> "1.0.0",
23 Date -> "April 2019",
24 Institutions -> {"Universite catholique de Louvain (CP3)"},
25 Emails -> {"ankit.beniwal@uclouvain.be"}
26};
27
28FeynmanGauge = False;
29
30(* ************************** *)
31(* ***** NLO Variables ****** *)
32(******************************)
33
34FR$LoopSwitches = {{Gf, MW}};
35FR$RmDblExt = { ymb -> MB, ymc -> MC, ymdo -> MD, yme -> Me,
36 ymm -> MMU, yms -> MS, ymt -> MT, ymtau -> MTA, ymup -> MU};
37
38(* ************************** *)
39(* ***** Change log ***** *)
40(* ************************** *)
41
42(* v1.4.7: Index issue with the ghost Lagrangian (special thanks to S. Iwamoto *)
43(* v1.4.6: NLO variable added. *)
44(* v1.4.5: Added widths for ghosts. *)
45(* v1.4.4: Changed widths of goldstone bosons to be the same as for the W and Z bosons *)
46(* v1.4.3: Updated conventions for the symmetric structure constants of SU3. *)
47(* v1.4.2: Set FeynmanGauge=True as default again. *)
48(* v1.4: Added SU(2) representation. *)
49(* -> Modification in the field declarations (doublets are added) *)
50(* -> Modification in the Lagrangian (much simpler). *)
51(* v1.3: Added yukawa couplings for all fermions for gauge invariance. *)
52(* Added yukawa couplings for 1st generation fermions to Massless.rst. *)
53(* Updated parameters to PDG 2010. *)
54(* v1.2: Set FeynmanGauge=True as default. *)
55(* Set Gluonic ghosts to be included in both gauges. *)
56(* v1.1: Fixed yukawa couplings in Feynman gauge. *)
57(* Changed yd[n] CKM[n,m] to yd[m] CKM[n,m]. *)
58(* Changed yu[n] Conjugate[CKM[m,n]] to yu[m] Conjugate[CKM[m,n]]. *)
59
60(* ************************** *)
61(* ***** vevs ***** *)
62(* ************************** *)
63M$vevs = { {Phi[2],vh}, {S,vs} };
64
65(* ************************** *)
66(* ***** Gauge groups ***** *)
67(* ************************** *)
68M$GaugeGroups = {
69 U1Y == {
70 Abelian -> True,
71 CouplingConstant -> g1,
72 GaugeBoson -> B,
73 Charge -> Y
74 },
75 SU2L == {
76 Abelian -> False,
77 CouplingConstant -> gw,
78 GaugeBoson -> Wi,
79 StructureConstant -> Eps,
80 Representations -> {Ta,SU2D},
81 Definitions -> {Ta[a_,b_,c_]->PauliSigma[a,b,c]/2, FSU2L[i_,j_,k_]:> I Eps[i,j,k]}
82 },
83 SU3C == {
84 Abelian -> False,
85 CouplingConstant -> gs,
86 GaugeBoson -> G,
87 StructureConstant -> f,
88 Representations -> {T,Colour},
89 SymmetricTensor -> dSUN
90 }
91};
92
93
94(* ************************** *)
95(* ***** Indices ***** *)
96(* ************************** *)
97
98IndexRange[Index[SU2W ]] = Unfold[Range[3]];
99IndexRange[Index[SU2D ]] = Unfold[Range[2]];
100IndexRange[Index[Gluon ]] = NoUnfold[Range[8]];
101IndexRange[Index[Colour ]] = NoUnfold[Range[3]];
102IndexRange[Index[Generation]] = Range[3];
103
104IndexStyle[SU2W, j];
105IndexStyle[SU2D, k];
106IndexStyle[Gluon, a];
107IndexStyle[Colour, m];
108IndexStyle[Generation, f];
109
110
111(* ************************** *)
112(* *** Interaction orders *** *)
113(* *** (as used by mg5) *** *)
114(* ************************** *)
115
116M$InteractionOrderHierarchy = {
117 {QCD, 1},
118 {QED, 2}
119};
120
121
122(* ************************** *)
123(* **** Particle classes **** *)
124(* ************************** *)
125M$ClassesDescription = {
126
127(* Gauge bosons: physical vector fields *)
128 V[1] == {
129 ClassName -> A,
130 SelfConjugate -> True,
131 Mass -> 0,
132 Width -> 0,
133 ParticleName -> "a",
134 PDG -> 22,
135 PropagatorLabel -> "a",
136 PropagatorType -> W,
137 PropagatorArrow -> None,
138 FullName -> "Photon"
139 },
140 V[2] == {
141 ClassName -> Z,
142 SelfConjugate -> True,
143 Mass -> {MZ, 91.1876},
144 Width -> {WZ, 2.4952},
145 ParticleName -> "Z",
146 PDG -> 23,
147 PropagatorLabel -> "Z",
148 PropagatorType -> Sine,
149 PropagatorArrow -> None,
150 FullName -> "Z"
151 },
152 V[3] == {
153 ClassName -> W,
154 SelfConjugate -> False,
155 Mass -> {MW, Internal},
156 Width -> {WW, 2.085},
157 ParticleName -> "W+",
158 AntiParticleName -> "W-",
159 QuantumNumbers -> {Q -> 1},
160 PDG -> 24,
161 PropagatorLabel -> "W",
162 PropagatorType -> Sine,
163 PropagatorArrow -> Forward,
164 FullName -> "W"
165 },
166 V[4] == {
167 ClassName -> G,
168 SelfConjugate -> True,
169 Indices -> {Index[Gluon]},
170 Mass -> 0,
171 Width -> 0,
172 ParticleName -> "g",
173 PDG -> 21,
174 PropagatorLabel -> "G",
175 PropagatorType -> C,
176 PropagatorArrow -> None,
177 FullName -> "G"
178 },
179
180(* Ghosts: related to physical gauge bosons *)
181 U[1] == {
182 ClassName -> ghA,
183 SelfConjugate -> False,
184 Ghost -> A,
185 QuantumNumbers -> {GhostNumber -> 1},
186 Mass -> 0,
187 Width -> 0,
188 PropagatorLabel -> "uA",
189 PropagatorType -> GhostDash,
190 PropagatorArrow -> Forward
191 },
192 U[2] == {
193 ClassName -> ghZ,
194 SelfConjugate -> False,
195 Ghost -> Z,
196 QuantumNumbers -> {GhostNumber -> 1},
197 Mass -> {MZ,91.1876},
198 Width -> {WZ, 2.4952},
199 PropagatorLabel -> "uZ",
200 PropagatorType -> GhostDash,
201 PropagatorArrow -> Forward
202 },
203 U[31] == {
204 ClassName -> ghWp,
205 SelfConjugate -> False,
206 Ghost -> W,
207 QuantumNumbers -> {GhostNumber -> 1, Q -> 1},
208 Mass -> {MW,Internal},
209 Width -> {WW, 2.085},
210 PropagatorLabel -> "uWp",
211 PropagatorType -> GhostDash,
212 PropagatorArrow -> Forward
213 },
214 U[32] == {
215 ClassName -> ghWm,
216 SelfConjugate -> False,
217 Ghost -> Wbar,
218 QuantumNumbers -> {GhostNumber -> 1, Q -> -1},
219 Mass -> {MW,Internal},
220 Width -> {WW, 2.085},
221 PropagatorLabel -> "uWm",
222 PropagatorType -> GhostDash,
223 PropagatorArrow -> Forward
224 },
225 U[4] == {
226 ClassName -> ghG,
227 SelfConjugate -> False,
228 Indices -> {Index[Gluon]},
229 Ghost -> G,
230 PDG -> 82,
231 QuantumNumbers ->{GhostNumber -> 1},
232 Mass -> 0,
233 Width -> 0,
234 PropagatorLabel -> "uG",
235 PropagatorType -> GhostDash,
236 PropagatorArrow -> Forward
237 },
238
239(* Gauge bosons: unphysical vector fields *)
240 V[11] == {
241 ClassName -> B,
242 Unphysical -> True,
243 SelfConjugate -> True,
244 Definitions -> { B[mu_] -> -sw Z[mu]+cw A[mu]}
245 },
246 V[12] == {
247 ClassName -> Wi,
248 Unphysical -> True,
249 SelfConjugate -> True,
250 Indices -> {Index[SU2W]},
251 FlavorIndex -> SU2W,
252 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]}
253 },
254
255(* Ghosts: related to unphysical gauge bosons *)
256 U[11] == {
257 ClassName -> ghB,
258 Unphysical -> True,
259 SelfConjugate -> False,
260 Ghost -> B,
261 Definitions -> { ghB -> -sw ghZ + cw ghA}
262 },
263 U[12] == {
264 ClassName -> ghWi,
265 Unphysical -> True,
266 SelfConjugate -> False,
267 Ghost -> Wi,
268 Indices -> {Index[SU2W]},
269 FlavorIndex -> SU2W,
270 Definitions -> { ghWi[1] -> (ghWp+ghWm)/Sqrt[2], ghWi[2] -> (ghWm-ghWp)/(I*Sqrt[2]), ghWi[3] -> cw ghZ+sw ghA}
271 } ,
272
273(* Fermions: physical fields *)
274 F[1] == {
275 ClassName -> vl,
276 ClassMembers -> {ve,vm,vt},
277 Indices -> {Index[Generation]},
278 FlavorIndex -> Generation,
279 SelfConjugate -> False,
280 Mass -> 0,
281 Width -> 0,
282 QuantumNumbers -> {LeptonNumber -> 1},
283 PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
284 PropagatorType -> S,
285 PropagatorArrow -> Forward,
286 PDG -> {12,14,16},
287 ParticleName -> {"ve","vm","vt"},
288 AntiParticleName -> {"ve~","vm~","vt~"},
289 FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"}
290 },
291 F[2] == {
292 ClassName -> l,
293 ClassMembers -> {e, mu, ta},
294 Indices -> {Index[Generation]},
295 FlavorIndex -> Generation,
296 SelfConjugate -> False,
297 Mass -> {Ml, {Me,5.11*^-4}, {MMU,0.10566}, {MTA,1.777}},
298 Width -> 0,
299 QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
300 PropagatorLabel -> {"l", "e", "mu", "ta"},
301 PropagatorType -> Straight,
302 PropagatorArrow -> Forward,
303 PDG -> {11, 13, 15},
304 ParticleName -> {"e-", "mu-", "ta-"},
305 AntiParticleName -> {"e+", "mu+", "ta+"},
306 FullName -> {"Electron", "Muon", "Tau"}
307 },
308 F[3] == {
309 ClassName -> uq,
310 ClassMembers -> {u, c, t},
311 Indices -> {Index[Generation], Index[Colour]},
312 FlavorIndex -> Generation,
313 SelfConjugate -> False,
314 Mass -> {Mu, {MU, 2.55*^-3}, {MC,1.27}, {MT,172}},
315 Width -> {0, 0, {WT,1.50833649}},
316 QuantumNumbers -> {Q -> 2/3},
317 PropagatorLabel -> {"uq", "u", "c", "t"},
318 PropagatorType -> Straight,
319 PropagatorArrow -> Forward,
320 PDG -> {2, 4, 6},
321 ParticleName -> {"u", "c", "t" },
322 AntiParticleName -> {"u~", "c~", "t~"},
323 FullName -> {"u-quark", "c-quark", "t-quark"}
324 },
325 F[4] == {
326 ClassName -> dq,
327 ClassMembers -> {d, s, b},
328 Indices -> {Index[Generation], Index[Colour]},
329 FlavorIndex -> Generation,
330 SelfConjugate -> False,
331 Mass -> {Md, {MD,5.04*^-3}, {MS,0.101}, {MB,4.7}},
332 Width -> 0,
333 QuantumNumbers -> {Q -> -1/3},
334 PropagatorLabel -> {"dq", "d", "s", "b"},
335 PropagatorType -> Straight,
336 PropagatorArrow -> Forward,
337 PDG -> {1,3,5},
338 ParticleName -> {"d", "s", "b" },
339 AntiParticleName -> {"d~", "s~", "b~"},
340 FullName -> {"d-quark", "s-quark", "b-quark"}
341 },
342
343(* Fermions: unphysical fields *)
344 F[11] == {
345 ClassName -> LL,
346 Unphysical -> True,
347 Indices -> {Index[SU2D], Index[Generation]},
348 FlavorIndex -> SU2D,
349 SelfConjugate -> False,
350 QuantumNumbers -> {Y -> -1/2},
351 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]] }
352 },
353 F[12] == {
354 ClassName -> lR,
355 Unphysical -> True,
356 Indices -> {Index[Generation]},
357 FlavorIndex -> Generation,
358 SelfConjugate -> False,
359 QuantumNumbers -> {Y -> -1},
360 Definitions -> { lR[sp1_,ff_] :> Module[{sp2}, ProjP[sp1,sp2] l[sp2,ff]] }
361 },
362 F[13] == {
363 ClassName -> QL,
364 Unphysical -> True,
365 Indices -> {Index[SU2D], Index[Generation], Index[Colour]},
366 FlavorIndex -> SU2D,
367 SelfConjugate -> False,
368 QuantumNumbers -> {Y -> 1/6},
369 Definitions -> {
370 QL[sp1_,1,ff_,cc_] :> Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],
371 QL[sp1_,2,ff_,cc_] :> Module[{sp2,ff2}, CKM[ff,ff2] ProjM[sp1,sp2] dq[sp2,ff2,cc]] }
372 },
373 F[14] == {
374 ClassName -> uR,
375 Unphysical -> True,
376 Indices -> {Index[Generation], Index[Colour]},
377 FlavorIndex -> Generation,
378 SelfConjugate -> False,
379 QuantumNumbers -> {Y -> 2/3},
380 Definitions -> { uR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] uq[sp2,ff,cc]] }
381 },
382 F[15] == {
383 ClassName -> dR,
384 Unphysical -> True,
385 Indices -> {Index[Generation], Index[Colour]},
386 FlavorIndex -> Generation,
387 SelfConjugate -> False,
388 QuantumNumbers -> {Y -> -1/3},
389 Definitions -> { dR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] dq[sp2,ff,cc]] }
390 },
391
392(**************** pNG DM model: Physical scalars ***************)
393(* Higgs: physical scalars *)
394 S[1] == {
395 ClassName -> h,
396 SelfConjugate -> True,
397 Mass -> {Mh, 125},
398 Width -> {wh, 0.00407},
399 PropagatorLabel -> "h",
400 PropagatorType -> D,
401 PropagatorArrow -> None,
402 PDG -> 25,
403 ParticleName -> "h",
404 FullName -> "SM-like Higgs"
405 },
406 S[2] == {
407 ClassName -> h2,
408 SelfConjugate -> True,
409 Mass -> {Mh2, 300},
410 Width -> {wh2, 1},
411 PropagatorLabel -> "h2",
412 PropagatorType -> D,
413 PropagatorArrow -> None,
414 PDG -> 29,
415 ParticleName -> "h2",
416 FullName -> "Second scalar"
417 },
418 S[3] == {
419 ClassName -> X,
420 SelfConjugate -> True,
421 Mass -> {mX, 100},
422 Width -> {wX, 0},
423 PropagatorLabel -> "X",
424 PropagatorType -> D,
425 PropagatorArrow -> None,
426 PDG -> 30,
427 ParticleName -> "~X",
428 FullName -> "pNG DM"
429 },
430
431(***************************************************************)
432(* Higgs: physical scalars *)
433 S[4] == {
434 ClassName -> G0,
435 SelfConjugate -> True,
436 Goldstone -> Z,
437 Mass -> {MZ, 91.1876},
438 Width -> {WZ, 2.4952},
439 PropagatorLabel -> "Go",
440 PropagatorType -> D,
441 PropagatorArrow -> None,
442 PDG -> 250,
443 ParticleName -> "G0",
444 FullName -> "G0"
445 },
446 S[5] == {
447 ClassName -> GP,
448 SelfConjugate -> False,
449 Goldstone -> W,
450 Mass -> {MW, Internal},
451 QuantumNumbers -> {Q -> 1},
452 Width -> {WW, 2.085},
453 PropagatorLabel -> "GP",
454 PropagatorType -> D,
455 PropagatorArrow -> None,
456 PDG -> 251,
457 ParticleName -> "G+",
458 AntiParticleName -> "G-",
459 FullName -> "GP"
460 },
461
462(* Higgs: unphysical scalars *)
463 S[11] == {
464 ClassName -> Phi,
465 Unphysical -> True,
466 Indices -> {Index[SU2D]},
467 FlavorIndex -> SU2D,
468 SelfConjugate -> False,
469 QuantumNumbers -> {Y -> 1/2},
470 Definitions -> { Phi[1] -> -I GP, Phi[2] -> (vh + (ct h + st h2) + I G0)/Sqrt[2] }
471 },
472 S[12] == {
473 ClassName -> S,
474 Unphysical -> True,
475 SelfConjugate -> False,
476 Definitions -> { S -> (vs + (-st h + ct h2) + I X)/Sqrt[2] }
477 }
478};
479
480
481(* ************************** *)
482(* ***** Gauge ***** *)
483(* ***** Parameters ***** *)
484(* ***** (FeynArts) ***** *)
485(* ************************** *)
486
487GaugeXi[ V[1] ] = GaugeXi[A];
488GaugeXi[ V[2] ] = GaugeXi[Z];
489GaugeXi[ V[3] ] = GaugeXi[W];
490GaugeXi[ V[4] ] = GaugeXi[G];
491GaugeXi[ S[1] ] = 1;
492GaugeXi[ S[2] ] = GaugeXi[Z];
493GaugeXi[ S[3] ] = GaugeXi[W];
494GaugeXi[ U[1] ] = GaugeXi[A];
495GaugeXi[ U[2] ] = GaugeXi[Z];
496GaugeXi[ U[31] ] = GaugeXi[W];
497GaugeXi[ U[32] ] = GaugeXi[W];
498GaugeXi[ U[4] ] = GaugeXi[G];
499
500
501(* ************************** *)
502(* ***** Parameters ***** *)
503(* ************************** *)
504M$Parameters = {
505
506 (* External parameters *)
507 aEWM1 == {
508 ParameterType -> External,
509 BlockName -> SMINPUTS,
510 OrderBlock -> 1,
511 Value -> 127.9,
512 InteractionOrder -> {QED,-2},
513 Description -> "Inverse of the EW coupling constant at the Z pole"
514 },
515 Gf == {
516 ParameterType -> External,
517 BlockName -> SMINPUTS,
518 OrderBlock -> 2,
519 Value -> 1.16637*^-5,
520 InteractionOrder -> {QED,2},
521 TeX -> Subscript[G,f],
522 Description -> "Fermi constant"
523 },
524 aS == {
525 ParameterType -> External,
526 BlockName -> SMINPUTS,
527 OrderBlock -> 3,
528 Value -> 0.1184,
529 InteractionOrder -> {QCD,2},
530 TeX -> Subscript[\[Alpha],s],
531 Description -> "Strong coupling constant at the Z pole"
532 },
533 ymdo == {
534 ParameterType -> External,
535 BlockName -> YUKAWA,
536 OrderBlock -> 1,
537 Value -> 5.04*^-3,
538 Description -> "Down Yukawa mass"
539 },
540 ymup == {
541 ParameterType -> External,
542 BlockName -> YUKAWA,
543 OrderBlock -> 2,
544 Value -> 2.55*^-3,
545 Description -> "Up Yukawa mass"
546 },
547 yms == {
548 ParameterType -> External,
549 BlockName -> YUKAWA,
550 OrderBlock -> 3,
551 Value -> 0.101,
552 Description -> "Strange Yukawa mass"
553 },
554 ymc == {
555 ParameterType -> External,
556 BlockName -> YUKAWA,
557 OrderBlock -> 4,
558 Value -> 1.27,
559 Description -> "Charm Yukawa mass"
560 },
561 ymb == {
562 ParameterType -> External,
563 BlockName -> YUKAWA,
564 OrderBlock -> 5,
565 Value -> 4.7,
566 Description -> "Bottom Yukawa mass"
567 },
568 ymt == {
569 ParameterType -> External,
570 BlockName -> YUKAWA,
571 OrderBlock -> 6,
572 Value -> 172,
573 Description -> "Top Yukawa mass"
574 },
575 yme == {
576 ParameterType -> External,
577 BlockName -> YUKAWA,
578 OrderBlock -> 11,
579 Value -> 5.11*^-4,
580 Description -> "Electron Yukawa mass"
581 },
582 ymm == {
583 ParameterType -> External,
584 BlockName -> YUKAWA,
585 OrderBlock -> 13,
586 Value -> 0.10566,
587 Description -> "Muon Yukawa mass"
588 },
589 ymtau == {
590 ParameterType -> External,
591 BlockName -> YUKAWA,
592 OrderBlock -> 15,
593 Value -> 1.777,
594 Description -> "Tau Yukawa mass"
595 },
596 cabi == {
597 ParameterType -> External,
598 BlockName -> CKMBLOCK,
599 OrderBlock -> 1,
600 Value -> 0.227736,
601 TeX -> Subscript[\[Theta], c],
602 Description -> "Cabibbo angle"
603 },
604
605 (* Internal Parameters *)
606 aEW == {
607 ParameterType -> Internal,
608 Value -> 1/aEWM1,
609 InteractionOrder -> {QED,2},
610 TeX -> Subscript[\[Alpha], EW],
611 Description -> "Electroweak coupling contant"
612 },
613 MW == {
614 ParameterType -> Internal,
615 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
616 TeX -> Subscript[M,W],
617 Description -> "W mass"
618 },
619 sw2 == {
620 ParameterType -> Internal,
621 Value -> 1-(MW/MZ)^2,
622 Description -> "Squared Sin of the Weinberg angle"
623 },
624 ee == {
625 ParameterType -> Internal,
626 Value -> Sqrt[4 Pi aEW],
627 InteractionOrder -> {QED,1},
628 TeX -> e,
629 Description -> "Electric coupling constant"
630 },
631 cw == {
632 ParameterType -> Internal,
633 Value -> Sqrt[1-sw2],
634 TeX -> Subscript[c,w],
635 Description -> "Cosine of the Weinberg angle"
636 },
637 sw == {
638 ParameterType -> Internal,
639 Value -> Sqrt[sw2],
640 TeX -> Subscript[s,w],
641 Description -> "Sine of the Weinberg angle"
642 },
643 gw == {
644 ParameterType -> Internal,
645 Definitions -> {gw->ee/sw},
646 InteractionOrder -> {QED,1},
647 TeX -> Subscript[g,w],
648 Description -> "Weak coupling constant at the Z pole"
649 },
650 g1 == {
651 ParameterType -> Internal,
652 Definitions -> {g1->ee/cw},
653 InteractionOrder -> {QED,1},
654 TeX -> Subscript[g,1],
655 Description -> "U(1)Y coupling constant at the Z pole"
656 },
657 gs == {
658 ParameterType -> Internal,
659 Value -> Sqrt[4 Pi aS],
660 InteractionOrder -> {QCD,1},
661 TeX -> Subscript[g,s],
662 ParameterName -> G,
663 Description -> "Strong coupling constant at the Z pole"
664 },
665 vh == {
666 ParameterType -> Internal,
667 Value -> 2*MW*sw/ee,
668 InteractionOrder -> {QED,-1},
669 TeX -> Subscript[v,h],
670 Description -> "SM-like Higgs VEV (GeV)"
671 },
672 yl == {
673 ParameterType -> Internal,
674 Indices -> {Index[Generation], Index[Generation]},
675 Definitions -> {yl[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
676 Value -> {yl[1,1] -> Sqrt[2] yme / vh, yl[2,2] -> Sqrt[2] ymm / vh, yl[3,3] -> Sqrt[2] ymtau / vh},
677 InteractionOrder -> {QED, 1},
678 ParameterName -> {yl[1,1] -> ye, yl[2,2] -> ym, yl[3,3] -> ytau},
679 TeX -> Superscript[y, l],
680 Description -> "Lepton Yukawa couplings"
681 },
682 yu == {
683 ParameterType -> Internal,
684 Indices -> {Index[Generation], Index[Generation]},
685 Definitions -> {yu[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
686 Value -> {yu[1,1] -> Sqrt[2] ymup/vh, yu[2,2] -> Sqrt[2] ymc/vh, yu[3,3] -> Sqrt[2] ymt/vh},
687 InteractionOrder -> {QED, 1},
688 ParameterName -> {yu[1,1] -> yup, yu[2,2] -> yc, yu[3,3] -> yt},
689 TeX -> Superscript[y, u],
690 Description -> "Up-type Yukawa couplings"
691 },
692 yd == {
693 ParameterType -> Internal,
694 Indices -> {Index[Generation], Index[Generation]},
695 Definitions -> {yd[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
696 Value -> {yd[1,1] -> Sqrt[2] ymdo/vh, yd[2,2] -> Sqrt[2] yms/vh, yd[3,3] -> Sqrt[2] ymb/vh},
697 InteractionOrder -> {QED, 1},
698 ParameterName -> {yd[1,1] -> ydo, yd[2,2] -> ys, yd[3,3] -> yb},
699 TeX -> Superscript[y, d],
700 Description -> "Down-type Yukawa couplings"
701 },
702(* N. B. : only Cabibbo mixing! *)
703 CKM == {
704 ParameterType -> Internal,
705 Indices -> {Index[Generation], Index[Generation]},
706 Unitary -> True,
707 Value -> {CKM[1,1] -> Cos[cabi], CKM[1,2] -> Sin[cabi], CKM[1,3] -> 0,
708 CKM[2,1] -> -Sin[cabi], CKM[2,2] -> Cos[cabi], CKM[2,3] -> 0,
709 CKM[3,1] -> 0, CKM[3,2] -> 0, CKM[3,3] -> 1},
710 TeX -> Superscript[V,CKM],
711 Description -> "CKM-Matrix"
712 },
713
714 (**************** pNG DM model: Free/dependent parameters ****************)
715 (* External parameters *)
716 vs == {
717 ParameterType -> External,
718 Value -> 300,
719 InteractionOrder -> {QED, -1},
720 TeX -> Subscript[v,s],
721 Description -> "Second scalar VEV (GeV)"
722 },
723 theta == {
724 ParameterType -> External,
725 Value -> 0.7854,
726 TeX -> \[Theta],
727 Description -> "Mixing angle (radians)"
728 },
729
730 (* Internal parameters *)
731 ct == {
732 ParameterType -> Internal,
733 Value -> Cos[theta],
734 TeX -> Cos \[Theta],
735 Description -> "Cosine of theta"
736 },
737 st == {
738 ParameterType -> Internal,
739 Value -> Sin[theta],
740 TeX -> Sin \[Theta],
741 Description -> "Sine of theta"
742 },
743 lambdaP == {
744 ParameterType -> Internal,
745 InteractionOrder -> {QED, 2},
746 Value -> ((Mh ct)^2 + (Mh2 st)^2)/(vh^2),
747 TeX -> Subscript[\[Lambda], \[CapitalPhi]],
748 Description -> "Higgs quartic coupling"
749 },
750 lambdaS == {
751 ParameterType -> Internal,
752 InteractionOrder -> {QED, 2},
753 Value -> ((Mh st)^2 + (Mh2 ct)^2)/(vs^2),
754 TeX -> Subscript[\[Lambda], S],
755 Description -> "Second scalar quartic coupling"
756 },
757 lambdaPS == {
758 ParameterType -> Internal,
759 InteractionOrder -> {QED, 2},
760 Value -> ((Mh2^2 - Mh^2) st ct)/(vh vs),
761 TeX -> Subscript[\[Lambda], \[CapitalPhi]S],
762 Description -> "Higgs portal coupling"
763 },
764 muSpsq == {
765 ParameterType -> Internal,
766 Value -> mX^2,
767 TeX -> Superscript[Subscript[\[Mu], S], \[Prime] 2],
768 Description -> "Soft breaking parameter"
769 },
770 muPsq == {
771 ParameterType -> Internal,
772 Value -> lambdaP vh^2 + lambdaPS vs^2,
773 TeX -> Superscript[Subscript[\[Mu], \[CapitalPhi]], 2],
774 Description -> "Higgs bare mass squared"
775 },
776 muSsq == {
777 ParameterType -> Internal,
778 Value -> lambdaS vs^2 + lambdaPS vh^2 - muSpsq,
779 TeX -> Superscript[Subscript[\[Mu], S], 2],
780 Description -> "Second scalar bare mass squared"
781 }
782 (*************************************************************************)
783};
784
785(* ************************** *)
786(* ***** Lagrangian ***** *)
787(* ************************** *)
788
789LGauge := Block[{mu,nu,ii,aa},
790 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]];
791
792LFermions := Block[{mu},
793 ExpandIndices[I*(
794 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]),
795 FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]}];
796
797LHiggs := Block[{ii,mu, feynmangaugerules},
798 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
799
800 ExpandIndices[DC[Phibar[ii],mu] DC[Phi[ii],mu] + muPsq/2 Phibar[ii] Phi[ii] - lambdaP/2 Phibar[ii] Phi[ii] Phibar[jj] Phi[jj], FlavorExpand->{SU2D,SU2W}]/.feynmangaugerules
801 ];
802
803LYukawa := Block[{sp,ii,jj,cc,ff1,ff2,ff3,yuk,feynmangaugerules},
804 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
805
806 yuk = ExpandIndices[
807 -yd[ff2, ff3] CKM[ff1, ff2] QLbar[sp, ii, ff1, cc].dR [sp, ff3, cc] Phi[ii] -
808 yl[ff1, ff3] LLbar[sp, ii, ff1].lR [sp, ff3] Phi[ii] -
809 yu[ff1, ff2] QLbar[sp, ii, ff1, cc].uR [sp, ff2, cc] Phibar[jj] Eps[ii, jj], FlavorExpand -> SU2D];
810 yuk = yuk /. { CKM[a_, b_] Conjugate[CKM[a_, c_]] -> IndexDelta[b, c], CKM[b_, a_] Conjugate[CKM[c_, a_]] -> IndexDelta[b, c]};
811 yuk+HC[yuk]/.feynmangaugerules
812 ];
813
814LGhost := Block[{LGh1,LGhw,LGhs,LGhphi,mu, generators,gh,ghbar,Vectorize,phi1,phi2,togoldstones,doublet,doublet0},
815 (* Pure gauge piece *)
816 LGh1 = -ghBbar.del[DC[ghB,mu],mu];
817 LGhw = -ghWibar[ii].del[DC[ghWi[ii],mu],mu];
818 LGhs = -ghGbar[ii].del[DC[ghG[ii],mu],mu];
819
820 (* Scalar pieces: see Peskin pages 739-742 *)
821 (* phi1 and phi2 are the real degrees of freedom of GP *)
822 (* Vectorize transforms a doublet in a vector in the phi-basis, i.e. the basis of real degrees of freedom *)
823 gh = {ghB, ghWi[1], ghWi[2], ghWi[3]};
824 ghbar = {ghBbar, ghWibar[1], ghWibar[2], ghWibar[3]};
825 generators = {-I/2 g1 IdentityMatrix[2], -I/2 gw PauliSigma[1], -I/2 gw PauliSigma[2], -I/2 gw PauliSigma[3]};
826 doublet = Expand[{(-I phi1 - phi2)/Sqrt[2], Phi[2]} /. MR$Definitions /. vh -> 0];
827 doublet0 = {0, vh/Sqrt[2]};
828 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}];
829 togoldstones := {phi1 -> (GP + GPbar)/Sqrt[2], phi2 -> (-GP + GPbar)/(I Sqrt[2])};
830 LGhphi=Plus@@Flatten[Table[-ghbar[[kkk]].gh[[lll]] Vectorize[generators[[kkk]].doublet0].Vectorize[generators[[lll]].(doublet+doublet0)],{kkk,4},{lll,4}]] /.togoldstones;
831
832ExpandIndices[ LGhs + If[FeynmanGauge, LGh1 + LGhw + LGhphi,0], FlavorExpand->SU2W]];
833
834LSM:= LGauge + LFermions + LHiggs + LYukawa + LGhost;
835
836(********************* pNG DM model: Model Lagrangian *********************)
837
838(* ****** S kinetic, quadratic and quartic terms * ******)
839LS := del[Sbar, mu] del[S, mu] + muSsq/2 Sbar S - lambdaS/2 (Sbar S)^2;
840
841(* ****** SM Higgs-scalar interaction via a Higgs portal term * ******)
842Lint := Block[{ii, feynmangaugerules},
843 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
844
845 ExpandIndices[ - lambdaPS Phibar[ii] Phi[ii] Sbar S, FlavorExpand->{SU2D}]/.feynmangaugerules
846];
847
848(****** * Soft-breaking term * ******)
849Lsoft := muSpsq/4 (S^2 + Sbar^2);
850
851LpNG := LS + Lint + Lsoft;
852(**************************************************************************)