RSmodel: RS.fr

File RS.fr, 29.0 KB (added by Priscila de Aquino, 14 years ago)

RS model file

Line 
1(***************************************************************************************************************)
2(****** This is the FeynRules mod-file for the Large Extra Dimensions ******)
3(****** ******)
4(****** Author: Priscila de Aquino ******)
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
12M$ModelName = "RS";
13
14M$Information = {Authors -> {"Priscila de Aquino"},
15 Date -> "22.11.2011",
16 Institute -> {"Katholieke Universiteit Leuven & Universite Catholique Louvain - CP3"},
17 Emails -> {"priscila@itf.kuleuven.be"},
18 Version -> "2.1"};
19
20FeynmanGauge = False;
21
22
23(*****************************************************************************************)
24(****************************** Index definitions ****************************************)
25(*****************************************************************************************)
26
27IndexRange[ Index[Generation] ] = Range[3]
28
29IndexRange[ Index[Colour] ] = NoUnfold[Range[3]]
30
31IndexRange[ Index[Gluon] ] = NoUnfold[Range[8]]
32
33IndexRange[Index[SU2W]] = Unfold[Range[3]]
34
35IndexStyle[Colour, i]
36
37IndexStyle[Generation, f]
38
39IndexStyle[Gluon ,a]
40
41IndexStyle[SU2W ,k]
42
43(*****************************************************************************************)
44(************************************* Parameters ***************************************)
45(*****************************************************************************************)
46
47M$Parameters = {
48
49 (* External parameters *)
50
51 \[Alpha]EWM1== {
52 ParameterType -> External,
53 BlockName -> SMINPUTS,
54 ParameterName -> aEWM1,
55 InteractionOrder -> {QED, -2},
56 Value -> 127.9,
57 Description -> "Inverse of the electroweak coupling constant"},
58
59 Gf == {
60 ParameterType -> External,
61 BlockName -> SMINPUTS,
62 InteractionOrder -> {QED, 2},
63 Value -> 1.166 * 10^(-5),
64 Description -> "Fermi constant"},
65
66 \[Alpha]S == {
67 ParameterType -> External,
68 BlockName -> SMINPUTS,
69 ParameterName -> aS,
70 InteractionOrder -> {QCD, 2},
71 Value -> 0.118,
72 Description -> "Strong coupling constant at the Z pole."},
73
74 ymc == {
75 ParameterType -> External,
76 BlockName -> YUKAWA,
77 Value -> 1.42,
78 OrderBlock -> {4},
79 Description -> "Charm Yukawa mass"},
80
81 ymb == {
82 ParameterType -> External,
83 BlockName -> YUKAWA,
84 Value -> 4.2,
85 OrderBlock -> {5},
86 Description -> "Bottom Yukawa mass"},
87
88 ymt == {
89 ParameterType -> External,
90 BlockName -> YUKAWA,
91 Value -> 174.3,
92 OrderBlock -> {6},
93 Description -> "Top Yukawa mass"},
94
95 ymtau == {
96 ParameterType -> External,
97 BlockName -> YUKAWA,
98 Value -> 1.777,
99 OrderBlock -> {15},
100 Description -> "Tau Yukawa mass"},
101
102 LRS == {
103 ParameterType -> External,
104 Value -> 3000,
105 Description -> "Cutoff of the theory"},
106
107 (* Internal Parameters *)
108
109 \[Alpha]EW == {
110 ParameterType -> Internal,
111 Value -> 1/\[Alpha]EWM1,
112 ParameterName -> aEW,
113 InteractionOrder -> {QED, 2},
114 Description -> "Electroweak coupling contant"},
115
116 sw2 == {
117 ParameterType -> External,
118(* Value -> 1-(MW/MZ)^2, *)
119 Value -> 0.2312,
120 Description -> "Squared Sin of the Weinberg angle"},
121
122 ee == {
123 TeX -> e,
124 ParameterType -> Internal,
125 Value -> Sqrt[4 Pi \[Alpha]EW],
126 InteractionOrder -> {QED, 1},
127 Description -> "Electric coupling constant"},
128
129 cw == {
130 TeX -> Subscript[c, w],
131 ParameterType -> Internal,
132 Value -> Sqrt[1 - sw2],
133 Description -> "Cos of the Weinberg angle"},
134
135 sw == {
136 TeX -> Subscript[s, w],
137 ParameterType -> Internal,
138 Value -> Sqrt[sw2],
139 Description -> "Sin of the Weinberg angle"},
140
141 gw == {
142 TeX -> Subscript[g, w],
143 ParameterType -> Internal,
144 Value -> ee / sw,
145 InteractionOrder -> {QED, 1},
146 Description -> "Weak coupling constant"},
147
148 g1 == {
149 TeX -> Subscript[g, 1],
150 ParameterType -> Internal,
151 Value -> ee / cw,
152 InteractionOrder -> {QED, 1},
153 Description -> "U(1)Y coupling constant"},
154
155 gs == {
156 TeX -> Subscript[g, s],
157 ParameterType -> Internal,
158 Value -> Sqrt[4 Pi \[Alpha]S],
159 InteractionOrder -> {QCD, 1},
160 ParameterName -> G,
161 Description -> "Strong coupling constant"},
162
163 v == {
164 ParameterType -> Internal,
165 Value -> 2*MW*sw/ee,
166 InteractionOrder -> {QED, -1},
167 Description -> "Higgs VEV"},
168
169 \[Lambda] == {
170 ParameterType -> Internal,
171 Value -> MH^2/(2*v^2),
172 InteractionOrder -> {QED, 2},
173 ParameterName -> lam,
174 Description -> "Higgs quartic coupling"},
175
176 muH == {
177 ParameterType -> Internal,
178 Value -> Sqrt[v^2 \[Lambda]],
179 TeX -> \[Mu],
180 Description -> "Coefficient of the quadratic piece of the Higgs potential"},
181
182
183 yl == {
184 Indices -> {Index[Generation]},
185 AllowSummation -> True,
186 ParameterType -> Internal,
187 Value -> {yl[1] -> 0, yl[2] -> 0, yl[3] -> Sqrt[2] ymtau / v},
188 ParameterName -> {yl[1] -> ye, yl[2] -> ym, yl[3] -> ytau},
189 InteractionOrder -> {QED, 1},
190 ComplexParameter -> False,
191 Description -> "Lepton Yukawa coupling"},
192
193 yu == {
194 Indices -> {Index[Generation]},
195 AllowSummation -> True,
196 ParameterType -> Internal,
197 Value -> {yu[1] -> 0, yu[2] -> Sqrt[2] ymc / v, yu[3] -> Sqrt[2] ymt / v},
198 ParameterName -> {yu[1] -> yu, yu[2] -> yc, yu[3] -> yt},
199 InteractionOrder -> {QED, 1},
200 ComplexParameter -> False,
201 Description -> "U-quark Yukawa coupling"},
202
203 yd == {
204 Indices -> {Index[Generation]},
205 AllowSummation -> True,
206 ParameterType -> Internal,
207 Value -> {yd[1] -> 0, yd[2] -> 0, yd[3] -> Sqrt[2] ymb / v},
208 ParameterName -> {yd[1] -> yd, yd[2] -> ys, yd[3] -> yb},
209 InteractionOrder -> {QED, 1},
210 ComplexParameter -> False,
211 Description -> "D-quark Yukawa coupling"},
212
213 cabi == {
214 TeX -> Subscript[\[Theta], c],
215 ParameterType -> External,
216 BlockName -> CKMBLOCK,
217 OrderBlock -> {1},
218 Value -> 0.488,
219 Description -> "Cabibbo angle"},
220
221 CKM == {
222 Indices -> {Index[Generation], Index[Generation]},
223 TensorClass -> CKM,
224 Unitary -> True,
225 Value -> {CKM[1,1] -> 1,
226 CKM[1,2] -> 0,
227 CKM[2,1] -> 0,
228 CKM[2,2] -> 1,
229 CKM[1,3] -> 0,
230 CKM[3,1] -> 0,
231 CKM[2,3] -> 0,
232 CKM[3,2] -> 0,
233 CKM[3,3] -> 1},
234 Description -> "CKM-Matrix"},
235
236 kappa == {
237 TeX -> \[Kappa]_F,
238 ParameterType -> Internal,
239(* Value -> Sqrt[16 Pi GN] *)
240 InteractionOrder -> {QTD, 1},
241 Value -> 2/LRS}
242}
243
244TeXFormat[mphi, Subscript[m, phi]]
245TeXFormat[mpsi, Subscript[m, psi]]
246TeXFormat[mG, Subscript[m, G]]
247
248(*****************************************************************************************)
249(********************************* Gauge Groups ******************************************)
250(*****************************************************************************************)
251
252M$GaugeGroups = {
253
254 U1Y == {
255 Abelian -> True,
256 GaugeBoson -> B,
257 Charge -> Y,
258 CouplingConstant -> g1},
259
260 SU2L == {
261 Abelian -> False,
262 GaugeBoson -> Wi,
263 StructureConstant -> Eps,
264 CouplingConstant -> gw},
265
266 SU3C == {
267 Abelian -> False,
268 GaugeBoson -> G,
269 StructureConstant -> f,
270 SymmetricTensor -> dSUN,
271 Representations -> {T, Colour},
272 CouplingConstant -> gs}
273}
274(*****************************************************************************************)
275(******************************* Particle Classes ****************************************)
276(*****************************************************************************************)
277
278M$ClassesDescription = {
279
280(************************************ Fermions *******************************************)
281
282 (* Leptons (neutrino): I_3 = +1/2, Q = 0 *)
283 F[1] == {
284 ClassName -> vl,
285 ClassMembers -> {ve,vm,vt},
286 FlavorIndex -> Generation,
287 SelfConjugate -> False,
288 Indices -> {Index[Generation]},
289 Mass -> 0,
290 Width -> 0,
291 QuantumNumbers -> {LeptonNumber -> 1},
292 PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
293 PropagatorType -> S,
294 PropagatorArrow -> Forward,
295 PDG -> {12,14,16},
296 FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"} },
297
298 (* Leptons (electron): I_3 = -1/2, Q = -1 *)
299 F[2] == {
300 ClassName -> l,
301 ClassMembers -> {e, m, tt},
302 FlavorIndex -> Generation,
303 SelfConjugate -> False,
304 Indices -> {Index[Generation]},
305 Mass -> {Ml, {ME, 0}, {MM, 0}, {MTA, 1.777}},
306 Width -> 0,
307 QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
308 PropagatorLabel -> {"l", "e", "m", "tt"},
309 PropagatorType -> Straight,
310 ParticleName -> {"e-", "m-", "tt-"},
311 AntiParticleName -> {"e+", "m+", "tt+"},
312 PropagatorArrow -> Forward,
313 PDG -> {11, 13, 15},
314 FullName -> {"Electron", "Muon", "Tau"} },
315
316 (* Quarks (u): I_3 = +1/2, Q = +2/3 *)
317 F[3] == {
318 ClassMembers -> {u, c, t},
319 ClassName -> uq,
320 FlavorIndex -> Generation,
321 SelfConjugate -> False,
322 Indices -> {Index[Generation], Index[Colour]},
323 Mass -> {Mu, {MU, 0}, {MC, 1.42}, {MT, 174.3}},
324 Width -> {0, 0, {WT, 1.51013490}},
325 QuantumNumbers -> {Q -> 2/3},
326 PropagatorLabel -> {"uq", "u", "c", "t"},
327 PropagatorType -> Straight,
328 PropagatorArrow -> Forward,
329 PDG -> {2, 4, 6},
330 FullName -> {"u-quark", "c-quark", "t-quark"}},
331
332 (* Quarks (d): I_3 = -1/2, Q = -1/3 *)
333 F[4] == {
334 ClassMembers -> {d, s, b},
335 ClassName -> dq,
336 FlavorIndex -> Generation,
337 SelfConjugate -> False,
338 Indices -> {Index[Generation], Index[Colour]},
339 Mass -> {Md, {MD, 0}, {MS, 0}, {MB, 4.2}},
340 Width -> 0,
341 QuantumNumbers -> {Q -> -1/3},
342 PropagatorLabel -> {"dq", "d", "s", "b"},
343 PropagatorType -> Straight,
344 PropagatorArrow -> Forward,
345 PDG -> {1,3,5},
346 FullName -> {"d-quark", "s-quark", "b-quark"} },
347
348(************************************ Gauge Bosons ***************************************)
349
350 (* Gauge bosons: Q = 0 *)
351 V[1] == {
352 ClassName -> A,
353 SelfConjugate -> True,
354 Indices -> {},
355 Mass -> 0,
356 Width -> 0,
357 PropagatorLabel -> "a",
358 PropagatorType -> W,
359 PropagatorArrow -> None,
360 PDG -> 22,
361 FullName -> "Photon" },
362
363 V[2] == {
364 ClassName -> Z,
365 SelfConjugate -> True,
366 Indices -> {},
367 Mass -> {MZ, 91.5445000},
368 Width -> {WZ, 2.44639985},
369 PropagatorLabel -> "Z",
370 PropagatorType -> Sine,
371 PropagatorArrow -> None,
372 PDG -> 23,
373 FullName -> "Z" },
374
375 (* Gauge bosons: Q = -1 *)
376 V[3] == {
377 ClassName -> W,
378 SelfConjugate -> False,
379 Indices -> {},
380 Mass -> {MW, 80.2673592},
381 Width -> {WW, 2.03535570},
382 QuantumNumbers -> {Q -> 1},
383 PropagatorLabel -> "W",
384 PropagatorType -> Sine,
385 PropagatorArrow -> Forward,
386 ParticleName ->"W+",
387 AntiParticleName ->"W-",
388 PDG -> 24,
389 FullName -> "W" },
390
391V[4] == {
392 ClassName -> G,
393 SelfConjugate -> True,
394 Indices -> {Index[Gluon]},
395 Mass -> {mG,0},
396 Width -> 0,
397 PropagatorLabel -> G,
398 PropagatorType -> C,
399 PropagatorArrow -> None,
400 PDG -> 21,
401 FullName -> "G" },
402
403V[5] == {
404 ClassName -> Wi,
405 Unphysical -> True,
406 Definitions -> {Wi[mu_, 1] -> (W[mu] + Wbar[mu])/Sqrt[2],
407 Wi[mu_, 2] -> (Wbar[mu] - W[mu])/Sqrt[2]/I,
408 Wi[mu_, 3] -> cw Z[mu] + sw A[mu]},
409 SelfConjugate -> True,
410 Indices -> {Index[SU2W]},
411 FlavorIndex -> SU2W,
412 Mass -> 0,
413 PDG -> {1,2,3}},
414
415V[6] == {
416 ClassName -> B,
417 SelfConjugate -> True,
418 Definitions -> {B[mu_] -> -sw Z[mu] + cw A[mu]},
419 Indices -> {},
420 Mass -> 0,
421 Unphysical -> True},
422
423(********** Ghosts **********)
424 U[1] == {
425 ClassName -> ghA,
426 SelfConjugate -> False,
427 Indices -> {},
428 Ghost -> A,
429 Mass -> 0,
430 QuantumNumbers -> {GhostNumber -> 1},
431 PropagatorLabel -> uA,
432 PropagatorType -> GhostDash,
433 PropagatorArrow -> Forward},
434
435 U[2] == {
436 ClassName -> ghZ,
437 SelfConjugate -> False,
438 Indices -> {},
439 Mass -> {MZ, 91.188},
440 Ghost -> Z,
441 QuantumNumbers -> {GhostNumber -> 1},
442 PropagatorLabel -> uZ,
443 PropagatorType -> GhostDash,
444 PropagatorArrow -> Forward},
445
446 U[31] == {
447 ClassName -> ghWp,
448 SelfConjugate -> False,
449 Indices -> {},
450 Mass -> {MW, 80.2673592},
451 Ghost -> W,
452 QuantumNumbers -> {Q-> 1, GhostNumber -> 1},
453 PropagatorLabel -> uWp,
454 PropagatorType -> GhostDash,
455 PropagatorArrow -> Forward},
456
457 U[32] == {
458 ClassName -> ghWm,
459 SelfConjugate -> False,
460 Indices -> {},
461 Mass -> {MW, 80.2673592},
462 Ghost -> Wbar,
463 QuantumNumbers -> {Q-> -1, GhostNumber -> 1},
464 PropagatorLabel -> uWm,
465 PropagatorType -> GhostDash,
466 PropagatorArrow -> Forward},
467
468 U[4] == {
469 ClassName -> ghG,
470 SelfConjugate -> False,
471 Indices -> {Index[Gluon]},
472 Ghost -> G,
473 Mass -> 0,
474 QuantumNumbers -> {GhostNumber -> 1},
475 PropagatorLabel -> uG,
476 PropagatorType -> GhostDash,
477 PropagatorArrow -> Forward},
478
479 U[5] == {
480 ClassName -> ghWi,
481 Unphysical -> True,
482 Definitions -> {ghWi[1] -> (ghWp + ghWm)/Sqrt[2],
483 ghWi[2] -> (ghWm - ghWp)/Sqrt[2]/I,
484 ghWi[3] -> cw ghZ + sw ghA},
485 SelfConjugate -> False,
486 Ghost -> Wi,
487 Indices -> {Index[SU2W]},
488 FlavorIndex -> SU2W},
489
490 U[6] == {
491 ClassName -> ghB,
492 SelfConjugate -> False,
493 Definitions -> {ghB -> -sw ghZ + cw ghA},
494 Indices -> {},
495 Ghost -> B,
496 Unphysical -> True},
497
498(****************************** Scalar Fields *********************************************)
499
500(* physical Higgs: Q = 0 *)
501 S[1] == {
502 ClassName -> H,
503 SelfConjugate -> True,
504 Mass -> {MH, 100},
505 Width -> {WH, 0.004276087},
506 PropagatorLabel -> "H",
507 PropagatorType -> D,
508 PropagatorArrow -> None,
509 PDG -> 25,
510 TeXParticleName -> "\\phi",
511 TeXClassName -> "\\phi",
512 FullName -> "H" },
513
514S[2] == {
515 ClassName -> phi,
516 SelfConjugate -> True,
517 Mass -> {MZ, 91.188},
518 Width -> Wphi,
519 PropagatorLabel -> "Phi",
520 PropagatorType -> D,
521 PropagatorArrow -> None,
522 ParticleName ->"phi0",
523 PDG -> 250,
524 FullName -> "Phi",
525 Goldstone -> Z },
526
527S[3] == {
528 ClassName -> phi2,
529 SelfConjugate -> False,
530 Mass -> {MW, 80.2673592},
531 Width -> Wphi2,
532 PropagatorLabel -> "Phi2",
533 PropagatorType -> D,
534 PropagatorArrow -> None,
535 ParticleName ->"phi+",
536 AntiParticleName ->"phi-",
537 PDG -> 251,
538 FullName -> "Phi2",
539 TeXClassName -> "\\phi^+",
540 TeXParticleName -> "\\phi^+",
541 TeXAntiParticleName -> "\\phi^-",
542 Goldstone -> W,
543 QuantumNumbers -> {Q -> 1}},
544
545(******************************* Spin 2 particles: graviton *****************************)
546
547T[1] == {
548 ClassName -> Gr,
549 SelfConjugate -> True,
550 ParticleName ->"y",
551 PDG -> 39,
552 Symmetric -> True,
553 Mass -> {MGr, 1000},
554 Width -> {WGr,10.6689}}
555
556}
557
558(*****************************************************************************************)
559(* *)
560(* The Lagrangian *)
561(* *)
562(*****************************************************************************************)
563
564(* Some shorthands (for nicer printing) *)
565
566Format[mu, TraditionalForm] = \[Mu];
567Format[nu, TraditionalForm] = \[Nu];
568Format[lam, TraditionalForm] = \[Lambda];
569Format[rho, TraditionalForm] = \[Rho];
570
571psi = \[Psi];
572psibar = \[Psi]bar;
573phi = \[Phi];
574phibar = \[Phi]bar;
575phiK = \[Sigma];
576
577(******************** SM Lagrangian *************************************)
578
579(******************** Gauge F^2 Lagrangian terms*************************)
580(*Sign convention from Lagrangian in between Eq. (A.9) and Eq. (A.10) of Peskin & Schroeder.*)
581 LGauge = -1/4 (del[Wi[nu, i1], mu] - del[Wi[mu, i1], nu] + gw Eps[i1, i2, i3] Wi[mu, i2] Wi[nu, i3])*
582 (del[Wi[nu, i1], mu] - del[Wi[mu, i1], nu] + gw Eps[i1, i4, i5] Wi[mu, i4] Wi[nu, i5]) -
583
584 1/4 (del[B[nu], mu] - del[B[mu], nu])^2 -
585
586 1/4 (del[G[nu, a1], mu] - del[G[mu, a1], nu] + gs f[a1, a2, a3] G[mu, a2] G[nu, a3])*
587 (del[G[nu, a1], mu] - del[G[mu, a1], nu] + gs f[a1, a4, a5] G[mu, a4] G[nu, a5]);
588
589
590(********************* Fermion Lagrangian terms*************************)
591(*Sign convention from Lagrangian in between Eq. (A.9) and Eq. (A.10) of Peskin & Schroeder.*)
592 LFermions = Module[{Lkin, LQCD, LEWleft, LEWright},
593
594 Lkin = I uqbar.Ga[mu].del[uq, mu] +
595 I dqbar.Ga[mu].del[dq, mu] +
596 I lbar.Ga[mu].del[l, mu] +
597 I vlbar.Ga[mu].del[vl, mu];
598
599 LQCD = gs (uqbar.Ga[mu].T[a].uq +
600 dqbar.Ga[mu].T[a].dq)G[mu, a];
601
602 LBright =
603 -2ee/cw B[mu]/2 lbar.Ga[mu].ProjP.l + (*Y_lR=-2*)
604 4ee/3/cw B[mu]/2 uqbar.Ga[mu].ProjP.uq - (*Y_uR=4/3*)
605 2ee/3/cw B[mu]/2 dqbar.Ga[mu].ProjP.dq; (*Y_dR=-2/3*)
606
607 LBleft =
608 -ee/cw B[mu]/2 vlbar.Ga[mu].ProjM.vl - (*Y_LL=-1*)
609 ee/cw B[mu]/2 lbar.Ga[mu].ProjM.l + (*Y_LL=-1*)
610 ee/3/cw B[mu]/2 uqbar.Ga[mu].ProjM.uq + (*Y_QL=1/3*)
611 ee/3/cw B[mu]/2 dqbar.Ga[mu].ProjM.dq ; (*Y_QL=1/3*)
612
613 LWleft = ee/sw/2(
614 vlbar.Ga[mu].ProjM.vl Wi[mu, 3] - (*sigma3 = ( 1 0 )*)
615 lbar.Ga[mu].ProjM.l Wi[mu, 3] + (* ( 0 -1 )*)
616
617 Sqrt[2] vlbar.Ga[mu].ProjM.l W[mu] +
618 Sqrt[2] lbar.Ga[mu].ProjM.vl Wbar[mu]+
619
620 uqbar.Ga[mu].ProjM.uq Wi[mu, 3] - (*sigma3 = ( 1 0 )*)
621 dqbar.Ga[mu].ProjM.dq Wi[mu, 3] + (* ( 0 -1 )*)
622
623 Sqrt[2] uqbar.Ga[mu].ProjM.CKM.dq W[mu] +
624 Sqrt[2] dqbar.Ga[mu].ProjM.HC[CKM].uq Wbar[mu]
625 );
626
627 Lkin + LQCD + LBright + LBleft + LWleft];
628
629(******************** Higgs Lagrangian terms****************************)
630 Phi := If[FeynmanGauge, {-I phi2, (v + H + I phi)/Sqrt[2]}, {0, (v + H)/Sqrt[2]}];
631 Phibar := If[FeynmanGauge, {I phi2bar, (v + H - I phi)/Sqrt[2]} ,{0, (v + H)/Sqrt[2]}];
632
633
634
635 LHiggs := Block[{PMVec, WVec, Dc, Dcbar, Vphi},
636
637 PMVec = Table[PauliSigma[i], {i, 3}];
638 Wvec[mu_] := {Wi[mu, 1], Wi[mu, 2], Wi[mu, 3]};
639
640 (*Y_phi=1*)
641 Dc[f_, mu_] := I del[f, mu] + ee/cw B[mu]/2 f + ee/sw/2 (Wvec[mu].PMVec).f;
642 Dcbar[f_, mu_] := -I del[f, mu] + ee/cw B[mu]/2 f + ee/sw/2 f.(Wvec[mu].PMVec);
643
644 Vphi[Phi_, Phibar_] := -muH^2 Phibar.Phi + \[Lambda] (Phibar.Phi)^2;
645
646 (Dcbar[Phibar, mu]).Dc[Phi, mu] - Vphi[Phi, Phibar]];
647
648
649(*************** Yukawa Lagrangian***********************)
650LYuk := If[FeynmanGauge,
651
652 Module[{s,r,n,m,i}, -
653 yd[m] CKM[n,m] uqbar[s,n,i].ProjP[s,r].dq[r,m,i] (-I phi2) -
654 yd[n] dqbar[s,n,i].ProjP[s,r].dq[r,n,i] (v+H +I phi)/Sqrt[2] -
655
656 yu[n] uqbar[s,n,i].ProjP[s,r].uq[r,n,i] (v+H -I phi)/Sqrt[2] + (*This sign from eps matrix*)
657 yu[m] Conjugate[CKM[m,n]] dqbar[s,n,i].ProjP[s,r].uq[r,m,i] ( I phi2bar) -
658
659 yl[n] vlbar[s,n].ProjP[s,r].l[r,n] (-I phi2) -
660 yl[n] lbar[s,n].ProjP[s,r].l[r,n] (v+H +I phi)/Sqrt[2]
661 ],
662
663 Module[{s,r,n,m,i}, -
664 yd[n] dqbar[s,n,i].ProjP[s,r].dq[r,n,i] (v+H)/Sqrt[2] -
665 yu[n] uqbar[s,n,i].ProjP[s,r].uq[r,n,i] (v+H)/Sqrt[2] -
666 yl[n] lbar[s,n].ProjP[s,r].l[r,n] (v+H)/Sqrt[2]
667 ]
668 ];
669
670LYukawa := LYuk + HC[LYuk];
671
672
673
674(**************Ghost terms**************************)
675(* Now we need the ghost terms which are of the form: *)
676(* - g * antighost * d_BRST G *)
677(* where d_BRST G is BRST transform of the gauge fixing function. *)
678
679LGhost := If[FeynmanGauge,
680 Block[{dBRSTG,LGhostG,dBRSTWi,LGhostWi,dBRSTB,LGhostB},
681
682 (***********First the pure gauge piece.**********************)
683 dBRSTG[mu_,a_] := 1/gs Module[{a2, a3}, del[ghG[a], mu] + gs f[a,a2,a3] G[mu,a2] ghG[a3]];
684 LGhostG := - gs ghGbar[a].del[dBRSTG[mu,a],mu];
685
686 dBRSTWi[mu_,i_] := sw/ee Module[{i2, i3}, del[ghWi[i], mu] + ee/sw Eps[i,i2,i3] Wi[mu,i2] ghWi[i3] ];
687 LGhostWi := - ee/sw ghWibar[a].del[dBRSTWi[mu,a],mu];
688
689 dBRSTB[mu_] := cw/ee del[ghB, mu];
690 LGhostB := - ee/cw ghBbar.del[dBRSTB[mu],mu];
691
692 (***********Next the piece from the scalar field.************)
693 LGhostphi := - ee/(2*sw*cw) MW ( - I phi2 ( (cw^2-sw^2)ghWpbar.ghZ + 2sw*cw ghWpbar.ghA ) +
694 I phi2bar ( (cw^2-sw^2)ghWmbar.ghZ + 2sw*cw ghWmbar.ghA ) ) -
695 ee/(2*sw) MW ( ( (v+H) + I phi) ghWpbar.ghWp + ( (v+H) - I phi) ghWmbar.ghWm ) -
696 I ee/(2*sw) MZ ( - phi2bar ghZbar.ghWp + phi2 ghZbar.ghWm ) -
697 ee/(2*sw*cw) MZ (v+H) ghZbar.ghZ ;
698
699
700 (***********Now add the pieces together.********************)
701 LGhostG + LGhostWi + LGhostB + LGhostphi]
702
703, 0];
704
705(*********Total SM Lagrangian*******)
706LSM := LGauge + LHiggs + LFermions + LYukawa + LGhost;
707
708
709(********************** Gravitational Coupling ******************************************)
710
711(*****************************************************************************************)
712(********************** Defining the cov derivatives *************************************)
713(*****************************************************************************************)
714
715covdelU[field_, mu_] :=
716 Module[{j, a}, del[field, mu] - I gs G[mu, a] T[a].field
717 - I ee/cw 4/3 B[mu]/2 ProjP.field - I ee/cw/3 B[mu]/2 ProjM.field - I ee/sw/2 ProjM.field Wi[mu,3]];
718
719covdelD[field_, mu_] :=
720 Module[{j, a}, del[field, mu] - I gs G[mu, a] T[a].field
721 + I ee/cw 2/3 B[mu]/2 ProjP.field - I ee/cw/3 B[mu]/2 ProjM.field + I ee/sw/2 ProjM.field Wi[mu,3]];
722
723covdelE[field_, mu_] :=
724 Module[{j, a}, del[field, mu]
725 + I ee/cw 2 B[mu]/2 ProjP.field + I ee/cw B[mu]/2 ProjM.field + I ee/sw/2 ProjM.field Wi[mu,3]];
726
727(* Version 2.0 => fixed a sign problem and a factor 2 missing in the above derivative *)
728
729covdelN[field_, mu_] :=
730 Module[{j, a}, del[field, mu] + I ee/cw B[mu]/2 ProjM.field - I ee/sw/2 ProjM.field Wi[mu,3]];
731
732(*****************************************************************************************)
733(******************** Defining the field strenght tensors:********************************)
734(*****************************************************************************************)
735
736FG[mu_,nu_,a1_,a2_,a3_] := del[G[nu, a1], mu] - del[G[mu, a1], nu] + gs f[a1, a2, a3] G[mu, a2] G[nu, a3];
737
738FA[mu_,nu_] := del[B[nu], mu] - del[B[mu], nu];
739
740FW[mu_,nu_,i1_,i2_,i3_] := del[Wi[nu, i1], mu] - del[Wi[mu, i1], nu] + ee/sw Eps[i1, i2, i3] Wi[mu, i2] Wi[nu, i3];
741
742
743
744(*****************************************************************************************)
745(******************* Defining the energy-momentum tensor T[mu,nu] ************************)
746(*****************************************************************************************)
747
748(* Gauge bosons *)
749
750TG[mu_,nu_]:= ( -ME[mu,nu]. (-1/4 FA[rho, sig] FA[rho,sig] - 1/4 FW[rho,sig,i1,i2,i3] FW[rho,sig, i1,i4,i5] - 1/4 FG[rho,sig,a1,a2,a3] FG[rho,sig, a1,a4,a5])
751 -FA[mu,rho] FA[nu,rho] - FW[mu,rho,i1,i2,i3] FW[nu,rho, i1,i4,i5] - FG[mu,rho,a1,a2,a3] FG[nu,rho, a1,a2,a3]);
752
753(* Fermions *)
754
755TF[mu_,nu_] := (-ME[mu,nu] (I uqbar.(Ga[rho].covdelU[uq, rho]) -1/2 del[I uqbar.Ga[rho].uq, rho]
756 + I dqbar.(Ga[rho].covdelD[dq, rho]) -1/2 del[I dqbar.Ga[rho].dq, rho]
757 + I vlbar.(Ga[rho].covdelN[vl, rho]) -1/2 del[I vlbar.Ga[rho].vl, rho]
758 + I lbar.(Ga[rho].covdelE[l, rho] ) -1/2 del[I lbar.Ga[rho].l, rho]
759
760 + ee/sw/Sqrt[2] (uqbar.Ga[rho].ProjM.CKM.dq W[rho] + dqbar.Ga[rho].ProjM.HC[CKM].uq Wbar[rho]
761 + vlbar.Ga[rho].ProjM.l W[rho] + lbar.Ga[rho].ProjM.vl Wbar[rho]) )
762 + ( I/2 uqbar.Ga[mu].covdelU[uq, nu] - 1/4 I del[uqbar.Ga[nu].uq, mu]
763 + I/2 uqbar.Ga[nu].covdelU[uq, mu] - 1/4 I del[uqbar.Ga[mu].uq, nu]
764 + I/2 dqbar.Ga[mu].covdelD[dq, nu] - 1/4 I del[dqbar.Ga[nu].dq, mu]
765 + I/2 dqbar.Ga[nu].covdelD[dq, mu] - 1/4 I del[dqbar.Ga[mu].dq, nu]
766 + I/2 vlbar.Ga[mu].covdelN[vl, nu] - 1/4 I del[vlbar.Ga[nu].vl, mu]
767 + I/2 vlbar.Ga[nu].covdelN[vl, mu] - 1/4 I del[vlbar.Ga[mu].vl, nu]
768 + I/2 lbar.Ga[mu].covdelE[l, nu] - 1/4 I del[lbar.Ga[nu].l, mu]
769 + I/2 lbar.Ga[nu].covdelE[l, mu] - 1/4 I del[lbar.Ga[mu].l, nu] )
770
771 + ee/sw/2/Sqrt[2] (uqbar.Ga[mu].ProjM.CKM.dq W[nu] + dqbar.Ga[mu].ProjM.HC[CKM].uq Wbar[nu]
772 + uqbar.Ga[nu].ProjM.CKM.dq W[mu] + dqbar.Ga[nu].ProjM.HC[CKM].uq Wbar[mu]
773 + vlbar.Ga[mu].ProjM.l W[nu] + lbar.Ga[mu].ProjM.vl Wbar[nu]
774 + vlbar.Ga[nu].ProjM.l W[mu] + lbar.Ga[nu].ProjM.vl Wbar[mu]));
775
776(* Definitions for Higgs and Yukawa *)
777
778 Phi := If[FeynmanGauge, {-I phi2, (v + H + I phi)/Sqrt[2]}, {0, (v + H)/Sqrt[2]}];
779 Phibar := If[FeynmanGauge, {I phi2bar, (v + H - I phi)/Sqrt[2]} ,{0, (v + H)/Sqrt[2]}];
780
781 PMVec = Table[PauliSigma[i], {i, 3}];
782 Wvec[mu_] := {Wi[mu, 1], Wi[mu, 2], Wi[mu, 3]};
783
784 Dc[f_, mu_] := I del[f, mu] + ee/cw B[mu]/2 f + ee/sw/2 (Wvec[mu].PMVec).f;
785 Dcbar[f_, mu_] := -I del[f, mu] + ee/cw B[mu]/2 f + ee/sw/2 f.(Wvec[mu].PMVec);
786
787 Vphi[Phi_, Phibar_] := -muH^2 Phibar.Phi + \[Lambda] (Phibar.Phi)^2;
788
789
790(* Higgs *)
791
792TH[mu_, nu_] := (-ME[mu,nu].(Dcbar[Phibar, rho]).Dc[Phi, rho] + ME[mu,nu] Vphi[Phi, Phibar] +
793 (Dcbar[Phibar, mu]).Dc[Phi, nu] + (Dcbar[Phibar, nu]).Dc[Phi, mu] );
794
795(* Yukawa *)
796
797TYuk:= Module[{s,r,n,m,i}, -
798 yd[n] dqbar[s,n,i].ProjP[s,r].dq[r,n,i] (v+H)/Sqrt[2] -
799 yu[n] uqbar[s,n,i].ProjP[s,r].uq[r,n,i] (v+H)/Sqrt[2] -
800 yl[n] lbar[s,n].ProjP[s,r].l[r,n] (v+H)/Sqrt[2]];
801
802TY[mu_,nu_] := -ME[mu,nu](TYuk + HC[TYuk]);
803
804
805(* Gauge fixing term is here because Madgraph takes the Feynman gauge for massless gauge boson propagators and unitary gauge for massive gauge boson propagators. *)
806
807TGF[mu_, nu_]:= (-ME[mu,nu].(del[del[G[sig, a1], sig], rho].G[rho, a1] + del[del[A[sig], sig], rho].A[rho] +
808 1/2 del[G[rho, a1], rho].del[G[rho, a1], rho] + 1/2 del[A[rho], rho].del[A[rho], rho])
809 + (del[del[G[rho, a1], rho], mu].G[nu, a1] + del[del[A[rho], rho], mu].A[nu] +
810 del[del[G[rho, a1], rho], nu].G[mu, a1] + del[del[A[rho], rho], nu].A[mu] ));
811
812(*****************************************************************************************)
813(******************************* Writing the lagrangian *********************************)
814(*****************************************************************************************)
815
816LagH := -kappa/2 (Gr[mu,nu] TH[mu,nu]);
817
818LagG := -kappa/2 (Gr[mu,nu] (TG[mu,nu] + TGF[mu,nu]));
819
820LagF := -kappa/2 (Gr[mu,nu] TF[mu,nu]);
821
822LagY := -kappa/2 (Gr[mu,nu] TY[mu,nu]);
823
824LagRS := LagH + LagG + LagF + LagY;
825
826LagTot := LagRS + LSM;
827
828(*****************************************************************************************)
829