4LQ: SM4LQ.fr

File SM4LQ.fr, 30.9 KB (added by Céline Degrande, 5 weeks ago)
Line 
1(* ************************** *)
2(* ***** Information ***** *)
3(* ************************** *)
4M$ModelName = "SM4LQ_MFV";
5
6M$Information = {
7 Authors -> {"M.Maltoni"},
8 Institutions -> {"CP3 (UCLouvain)"},
9 Emails -> {"matteo.maltoni@uclouvain.be"}
10};
11
12FeynmanGauge = True;
13
14(* ************************** *)
15(* ***** NLO Variables ****** *)
16(******************************)
17
18FR$LoopSwitches = {{Gf, MW}};
19FR$RmDblExt = { ymb -> MB, ymc -> MC, ymdo -> MD, yme -> Me,
20 ymm -> MMU, yms -> MS, ymt -> MT, ymtau -> MTA, ymup -> MU};
21
22(* ************************** *)
23(* ***** vevs ***** *)
24(* ************************** *)
25M$vevs = { {Phi[2],vev} };
26
27(* ************************** *)
28(* ***** Gauge groups ***** *)
29(* ************************** *)
30M$GaugeGroups = {
31 U1Y == {
32 Abelian -> True,
33 CouplingConstant -> g1,
34 GaugeBoson -> B,
35 Charge -> Y
36 },
37 SU2L == {
38 Abelian -> False,
39 CouplingConstant -> gw,
40 GaugeBoson -> Wi,
41 StructureConstant -> Eps,
42 Representations -> {Ta,SU2D},
43 Definitions -> {Ta[a_,b_,c_]->PauliSigma[a,b,c]/2, FSU2L[i_,j_,k_]:> I Eps[i,j,k]}
44 },
45 SU3C == {
46 Abelian -> False,
47 CouplingConstant -> gs,
48 GaugeBoson -> G,
49 StructureConstant -> f,
50 Representations -> {T,Colour},
51 SymmetricTensor -> dSUN
52 }
53};
54
55(* ************************** *)
56(* ***** Indices ***** *)
57(* ************************** *)
58
59IndexRange[Index[SU2W ]] = Unfold[Range[3]];
60IndexRange[Index[SU2D ]] = Unfold[Range[2]];
61IndexRange[Index[Gluon ]] = NoUnfold[Range[8]];
62IndexRange[Index[Colour ]] = NoUnfold[Range[3]];
63IndexRange[Index[Generation]] = Range[3];
64
65IndexStyle[SU2W, j];
66IndexStyle[SU2D, k];
67IndexStyle[Gluon, a];
68IndexStyle[Colour, m];
69IndexStyle[Generation, f];
70
71(* ************************** *)
72(* *** Interaction orders *** *)
73(* *** (as used by mg5) *** *)
74(* ************************** *)
75
76M$InteractionOrderHierarchy = {
77 {QCD,2},
78 {QED,4},
79 {NP,1}
80};
81
82M$InteractionOrderLimit = {
83 {NP,2}
84};
85
86(* ************************** *)
87(* **** Particle classes **** *)
88(* ************************** *)
89M$ClassesDescription = {
90
91(* Gauge bosons: physical vector fields *)
92 V[1] == {
93 ClassName -> A,
94 SelfConjugate -> True,
95 Mass -> 0,
96 Width -> 0,
97 ParticleName -> "a",
98 PDG -> 22,
99 PropagatorLabel -> "a",
100 PropagatorType -> W,
101 PropagatorArrow -> None,
102 FullName -> "Photon"
103 },
104 V[2] == {
105 ClassName -> Z,
106 SelfConjugate -> True,
107 Mass -> {MZ, 91.1876},
108 Width -> {WZ, 2.4952},
109 ParticleName -> "Z",
110 PDG -> 23,
111 PropagatorLabel -> "Z",
112 PropagatorType -> Sine,
113 PropagatorArrow -> None,
114 FullName -> "Z"
115 },
116 V[3] == {
117 ClassName -> W,
118 SelfConjugate -> False,
119 Mass -> {MW, Internal},
120 Width -> {WW, 2.085},
121 ParticleName -> "W+",
122 AntiParticleName -> "W-",
123 QuantumNumbers -> {Q -> 1},
124 PDG -> 24,
125 PropagatorLabel -> "W",
126 PropagatorType -> Sine,
127 PropagatorArrow -> Forward,
128 FullName -> "W"
129 },
130 V[4] == {
131 ClassName -> G,
132 SelfConjugate -> True,
133 Indices -> {Index[Gluon]},
134 Mass -> 0,
135 Width -> 0,
136 ParticleName -> "g",
137 PDG -> 21,
138 PropagatorLabel -> "G",
139 PropagatorType -> C,
140 PropagatorArrow -> None,
141 FullName -> "G"
142 },
143
144(* Ghosts: related to physical gauge bosons *)
145 U[1] == {
146 ClassName -> ghA,
147 SelfConjugate -> False,
148 Ghost -> A,
149 QuantumNumbers -> {GhostNumber -> 1},
150 Mass -> 0,
151 Width -> 0,
152 PropagatorLabel -> "uA",
153 PropagatorType -> GhostDash,
154 PropagatorArrow -> Forward
155 },
156 U[2] == {
157 ClassName -> ghZ,
158 SelfConjugate -> False,
159 Ghost -> Z,
160 QuantumNumbers -> {GhostNumber -> 1},
161 Mass -> {MZ,91.1876},
162 Width -> {WZ, 2.4952},
163 PropagatorLabel -> "uZ",
164 PropagatorType -> GhostDash,
165 PropagatorArrow -> Forward
166 },
167 U[31] == {
168 ClassName -> ghWp,
169 SelfConjugate -> False,
170 Ghost -> W,
171 QuantumNumbers -> {GhostNumber -> 1, Q -> 1},
172 Mass -> {MW,Internal},
173 Width -> {WW, 2.085},
174 PropagatorLabel -> "uWp",
175 PropagatorType -> GhostDash,
176 PropagatorArrow -> Forward
177 },
178 U[32] == {
179 ClassName -> ghWm,
180 SelfConjugate -> False,
181 Ghost -> Wbar,
182 QuantumNumbers -> {GhostNumber -> 1, Q -> -1},
183 Mass -> {MW,Internal},
184 Width -> {WW, 2.085},
185 PropagatorLabel -> "uWm",
186 PropagatorType -> GhostDash,
187 PropagatorArrow -> Forward
188 },
189 U[4] == {
190 ClassName -> ghG,
191 SelfConjugate -> False,
192 Indices -> {Index[Gluon]},
193 Ghost -> G,
194 PDG -> 82,
195 QuantumNumbers ->{GhostNumber -> 1},
196 Mass -> 0,
197 Width -> 0,
198 PropagatorLabel -> "uG",
199 PropagatorType -> GhostDash,
200 PropagatorArrow -> Forward
201 },
202
203(* Gauge bosons: unphysical vector fields *)
204 V[11] == {
205 ClassName -> B,
206 Unphysical -> True,
207 SelfConjugate -> True,
208 Definitions -> { B[mu_] -> -sw Z[mu]+cw A[mu]}
209 },
210 V[12] == {
211 ClassName -> Wi,
212 Unphysical -> True,
213 SelfConjugate -> True,
214 Indices -> {Index[SU2W]},
215 FlavorIndex -> SU2W,
216 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]}
217 },
218
219(* Ghosts: related to unphysical gauge bosons *)
220 U[11] == {
221 ClassName -> ghB,
222 Unphysical -> True,
223 SelfConjugate -> False,
224 Ghost -> B,
225 Definitions -> { ghB -> -sw ghZ + cw ghA}
226 },
227 U[12] == {
228 ClassName -> ghWi,
229 Unphysical -> True,
230 SelfConjugate -> False,
231 Ghost -> Wi,
232 Indices -> {Index[SU2W]},
233 FlavorIndex -> SU2W,
234 Definitions -> { ghWi[1] -> (ghWp+ghWm)/Sqrt[2], ghWi[2] -> (ghWm-ghWp)/(I*Sqrt[2]), ghWi[3] -> cw ghZ+sw ghA}
235 } ,
236
237(* Fermions: physical fields *)
238 F[1] == {
239 ClassName -> vl,
240 ClassMembers -> {ve,vm,vt},
241 Indices -> {Index[Generation]},
242 FlavorIndex -> Generation,
243 SelfConjugate -> False,
244 Mass -> 0,
245 Width -> 0,
246 QuantumNumbers -> {LeptonNumber -> 1},
247 PropagatorLabel -> {"v", "ve", "vm", "vt"} ,
248 PropagatorType -> S,
249 PropagatorArrow -> Forward,
250 PDG -> {12,14,16},
251 ParticleName -> {"ve","vm","vt"},
252 AntiParticleName -> {"ve~","vm~","vt~"},
253 FullName -> {"Electron-neutrino", "Mu-neutrino", "Tau-neutrino"}
254 },
255 F[2] == {
256 ClassName -> l,
257 ClassMembers -> {e, mu, ta},
258 Indices -> {Index[Generation]},
259 FlavorIndex -> Generation,
260 SelfConjugate -> False,
261 Mass -> {Ml, {Me,5.11*^-4}, {MMU,0.10566}, {MTA,1.777}},
262 Width -> 0,
263 QuantumNumbers -> {Q -> -1, LeptonNumber -> 1},
264 PropagatorLabel -> {"l", "e", "mu", "ta"},
265 PropagatorType -> Straight,
266 PropagatorArrow -> Forward,
267 PDG -> {11, 13, 15},
268 ParticleName -> {"e-", "mu-", "ta-"},
269 AntiParticleName -> {"e+", "mu+", "ta+"},
270 FullName -> {"Electron", "Muon", "Tau"}
271 },
272 F[3] == {
273 ClassName -> uq,
274 ClassMembers -> {u, c, t},
275 Indices -> {Index[Generation], Index[Colour]},
276 FlavorIndex -> Generation,
277 SelfConjugate -> False,
278 Mass -> {Mu, {MU, 2.55*^-3}, {MC,1.27}, {MT,172}},
279 Width -> {0, 0, {WT,1.50833649}},
280 QuantumNumbers -> {Q -> 2/3},
281 PropagatorLabel -> {"uq", "u", "c", "t"},
282 PropagatorType -> Straight,
283 PropagatorArrow -> Forward,
284 PDG -> {2, 4, 6},
285 ParticleName -> {"u", "c", "t" },
286 AntiParticleName -> {"u~", "c~", "t~"},
287 FullName -> {"u-quark", "c-quark", "t-quark"}
288 },
289 F[4] == {
290 ClassName -> dq,
291 ClassMembers -> {d, s, b},
292 Indices -> {Index[Generation], Index[Colour]},
293 FlavorIndex -> Generation,
294 SelfConjugate -> False,
295 Mass -> {Md, {MD,5.04*^-3}, {MS,0.101}, {MB,4.7}},
296 Width -> 0,
297 QuantumNumbers -> {Q -> -1/3},
298 PropagatorLabel -> {"dq", "d", "s", "b"},
299 PropagatorType -> Straight,
300 PropagatorArrow -> Forward,
301 PDG -> {1,3,5},
302 ParticleName -> {"d", "s", "b" },
303 AntiParticleName -> {"d~", "s~", "b~"},
304 FullName -> {"d-quark", "s-quark", "b-quark"}
305 },
306
307(* Fermions: unphysical fields *)
308 F[11] == {
309 ClassName -> LL,
310 Unphysical -> True,
311 Indices -> {Index[SU2D], Index[Generation]},
312 FlavorIndex -> SU2D,
313 SelfConjugate -> False,
314 QuantumNumbers -> {Y -> -1/2},
315 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]] }
316 },
317 F[12] == {
318 ClassName -> lR,
319 Unphysical -> True,
320 Indices -> {Index[Generation]},
321 FlavorIndex -> Generation,
322 SelfConjugate -> False,
323 QuantumNumbers -> {Y -> -1},
324 Definitions -> { lR[sp1_,ff_] :> Module[{sp2}, ProjP[sp1,sp2] l[sp2,ff]] }
325 },
326 F[13] == {
327 ClassName -> QL,
328 Unphysical -> True,
329 Indices -> {Index[SU2D], Index[Generation], Index[Colour]},
330 FlavorIndex -> SU2D,
331 SelfConjugate -> False,
332 QuantumNumbers -> {Y -> 1/6},
333 Definitions -> {
334 QL[sp1_,1,ff_,cc_] :> Module[{sp2}, ProjM[sp1,sp2] uq[sp2,ff,cc]],
335 QL[sp1_,2,ff_,cc_] :> Module[{sp2,ff2}, CKM[ff,ff2] ProjM[sp1,sp2] dq[sp2,ff2,cc]] }
336 },
337 F[14] == {
338 ClassName -> uR,
339 Unphysical -> True,
340 Indices -> {Index[Generation], Index[Colour]},
341 FlavorIndex -> Generation,
342 SelfConjugate -> False,
343 QuantumNumbers -> {Y -> 2/3},
344 Definitions -> { uR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] uq[sp2,ff,cc]] }
345 },
346 F[15] == {
347 ClassName -> dR,
348 Unphysical -> True,
349 Indices -> {Index[Generation], Index[Colour]},
350 FlavorIndex -> Generation,
351 SelfConjugate -> False,
352 QuantumNumbers -> {Y -> -1/3},
353 Definitions -> { dR[sp1_,ff_,cc_] :> Module[{sp2}, ProjP[sp1,sp2] dq[sp2,ff,cc]] }
354 },
355
356(* Higgs: physical scalars *)
357 S[1] == {
358 ClassName -> H,
359 SelfConjugate -> True,
360 Mass -> {MH,125},
361 Width -> {WH,0.00407},
362 PropagatorLabel -> "H",
363 PropagatorType -> D,
364 PropagatorArrow -> None,
365 PDG -> 25,
366 ParticleName -> "H",
367 FullName -> "H"
368 },
369
370(* Higgs: physical scalars *)
371 S[2] == {
372 ClassName -> G0,
373 SelfConjugate -> True,
374 Goldstone -> Z,
375 Mass -> {MZ, 91.1876},
376 Width -> {WZ, 2.4952},
377 PropagatorLabel -> "Go",
378 PropagatorType -> D,
379 PropagatorArrow -> None,
380 PDG -> 250,
381 ParticleName -> "G0",
382 FullName -> "G0"
383 },
384 S[3] == {
385 ClassName -> GP,
386 SelfConjugate -> False,
387 Goldstone -> W,
388 Mass -> {MW, Internal},
389 QuantumNumbers -> {Q -> 1},
390 Width -> {WW, 2.085},
391 PropagatorLabel -> "GP",
392 PropagatorType -> D,
393 PropagatorArrow -> None,
394 PDG -> 251,
395 ParticleName -> "G+",
396 AntiParticleName -> "G-",
397 FullName -> "GP"
398 },
399
400(* Higgs: unphysical scalars *)
401 S[11] == {
402 ClassName -> Phi,
403 Unphysical -> True,
404 Indices -> {Index[SU2D]},
405 FlavorIndex -> SU2D,
406 SelfConjugate -> False,
407 QuantumNumbers -> {Y -> 1/2},
408 Definitions -> { Phi[1] -> -I GP, Phi[2] -> (vev + H + I G0)/Sqrt[2] }
409 }
410};
411
412(* ************************** *)
413(* ***** Gauge ***** *)
414(* ***** Parameters ***** *)
415(* ***** (FeynArts) ***** *)
416(* ************************** *)
417
418GaugeXi[ V[1] ] = GaugeXi[A];
419GaugeXi[ V[2] ] = GaugeXi[Z];
420GaugeXi[ V[3] ] = GaugeXi[W];
421GaugeXi[ V[4] ] = GaugeXi[G];
422GaugeXi[ S[1] ] = 1;
423GaugeXi[ S[2] ] = GaugeXi[Z];
424GaugeXi[ S[3] ] = GaugeXi[W];
425GaugeXi[ U[1] ] = GaugeXi[A];
426GaugeXi[ U[2] ] = GaugeXi[Z];
427GaugeXi[ U[31] ] = GaugeXi[W];
428GaugeXi[ U[32] ] = GaugeXi[W];
429GaugeXi[ U[4] ] = GaugeXi[G];
430
431(* ************************** *)
432(* ***** Parameters ***** *)
433(* ************************** *)
434M$Parameters = {
435
436 (* External parameters *)
437 LambdaSMEFT == {
438 ParameterType -> External,
439 BlockName -> SMEFTcutoff,
440 Value -> 1000.0,
441 TeX -> \[CapitalLambda],
442 Description -> "Defining scale of the effective theory"
443 },
444
445 mueft == {
446 ParameterType -> External,
447 BlockName -> SMEFTLoop,
448 Value -> 200.0,
449 TeX -> Subscript[\[Mu],EFT],
450 Description -> "scale defining the renormalisation scheme"
451 },
452
453 cqq1 == {
454 ParameterType -> External,
455 BlockName -> SMEFT,
456 Value -> 0,
457 InteractionOrder -> {NP,1},
458 TeX -> Subsuperscript[c,qq,(1)]
459 },
460 cqq3 == {
461 ParameterType -> External,
462 BlockName -> SMEFT,
463 Value -> 0,
464 InteractionOrder -> {NP,1},
465 TeX -> Subsuperscript[c,qq,(3)]
466 },
467 cuu == {
468 ParameterType -> External,
469 BlockName -> SMEFT,
470 Value -> 0,
471 InteractionOrder -> {NP,1},
472 TeX -> Subscript[c,uu]
473 },
474 cdd == {
475 ParameterType -> External,
476 BlockName -> SMEFT,
477 Value -> 0,
478 InteractionOrder -> {NP,1},
479 TeX -> Subscript[c,dd]
480 },
481 cud8 == {
482 ParameterType -> External,
483 BlockName -> SMEFT,
484 Value -> 0,
485 InteractionOrder -> {NP,1},
486 TeX -> Subsuperscript[c,ud,(8)]
487 },
488
489 cud1 == {
490 ParameterType -> External,
491 BlockName -> SMEFT,
492 Value -> 0,
493 InteractionOrder -> {NP,1},
494 TeX -> Subsuperscript[c,ud,(1)]
495 },
496 cqu1 == {
497 ParameterType -> External,
498 BlockName -> SMEFT,
499 Value -> 0,
500 InteractionOrder -> {NP,1},
501 TeX -> Subsuperscript[c,qu,(1)]
502 },
503 cqd1 == {
504 ParameterType -> External,
505 BlockName -> SMEFT,
506 Value -> 0,
507 InteractionOrder -> {NP,1},
508 TeX -> Subsuperscript[c,qd,(1)]
509 },
510
511 cqu8 == {
512 ParameterType -> External,
513 BlockName -> SMEFT,
514 Value -> 0,
515 InteractionOrder -> {NP,1},
516 TeX -> Subsuperscript[c,qu,(8)]
517 },
518 cqd8 == {
519 ParameterType -> External,
520 BlockName -> SMEFT,
521 Value -> 0,
522 InteractionOrder -> {NP,1},
523 TeX -> Subsuperscript[c,qd,(8)]
524 },
525 aEWM1 == {
526 ParameterType -> External,
527 BlockName -> SMINPUTS,
528 OrderBlock -> 1,
529 Value -> 127.9,
530 InteractionOrder -> {QED,-2},
531 Description -> "Inverse of the EW coupling constant at the Z pole"
532 },
533 Gf == {
534 ParameterType -> External,
535 BlockName -> SMINPUTS,
536 OrderBlock -> 2,
537 Value -> 1.16637*^-5,
538 InteractionOrder -> {QED,2},
539 TeX -> Subscript[G,f],
540 Description -> "Fermi constant"
541 },
542 aS == {
543 ParameterType -> External,
544 BlockName -> SMINPUTS,
545 OrderBlock -> 3,
546 Value -> 0.1184,
547 InteractionOrder -> {QCD,2},
548 TeX -> Subscript[\[Alpha],s],
549 Description -> "Strong coupling constant at the Z pole"
550 },
551 ymdo == {
552 ParameterType -> External,
553 BlockName -> YUKAWA,
554 OrderBlock -> 1,
555 Value -> 5.04*^-3,
556 Description -> "Down Yukawa mass"
557 },
558 ymup == {
559 ParameterType -> External,
560 BlockName -> YUKAWA,
561 OrderBlock -> 2,
562 Value -> 2.55*^-3,
563 Description -> "Up Yukawa mass"
564 },
565 yms == {
566 ParameterType -> External,
567 BlockName -> YUKAWA,
568 OrderBlock -> 3,
569 Value -> 0.101,
570 Description -> "Strange Yukawa mass"
571 },
572 ymc == {
573 ParameterType -> External,
574 BlockName -> YUKAWA,
575 OrderBlock -> 4,
576 Value -> 1.27,
577 Description -> "Charm Yukawa mass"
578 },
579 ymb == {
580 ParameterType -> External,
581 BlockName -> YUKAWA,
582 OrderBlock -> 5,
583 Value -> 4.7,
584 Description -> "Bottom Yukawa mass"
585 },
586 ymt == {
587 ParameterType -> External,
588 BlockName -> YUKAWA,
589 OrderBlock -> 6,
590 Value -> 172,
591 Description -> "Top Yukawa mass"
592 },
593 yme == {
594 ParameterType -> External,
595 BlockName -> YUKAWA,
596 OrderBlock -> 11,
597 Value -> 5.11*^-4,
598 Description -> "Electron Yukawa mass"
599 },
600 ymm == {
601 ParameterType -> External,
602 BlockName -> YUKAWA,
603 OrderBlock -> 13,
604 Value -> 0.10566,
605 Description -> "Muon Yukawa mass"
606 },
607 ymtau == {
608 ParameterType -> External,
609 BlockName -> YUKAWA,
610 OrderBlock -> 15,
611 Value -> 1.777,
612 Description -> "Tau Yukawa mass"
613 },
614 cabi == {
615 ParameterType -> External,
616 BlockName -> CKMBLOCK,
617 OrderBlock -> 1,
618 Value -> 0.227736,
619 TeX -> Subscript[\[Theta], c],
620 Description -> "Cabibbo angle"
621 },
622
623 (* Internal Parameters *)
624 aEW == {
625 ParameterType -> Internal,
626 Value -> 1/aEWM1,
627 InteractionOrder -> {QED,2},
628 TeX -> Subscript[\[Alpha], EW],
629 Description -> "Electroweak coupling contant"
630 },
631 MW == {
632 ParameterType -> Internal,
633 Value -> Sqrt[MZ^2/2+Sqrt[MZ^4/4-Pi/Sqrt[2]*aEW/Gf*MZ^2]],
634 TeX -> Subscript[M,W],
635 Description -> "W mass"
636 },
637 sw2 == {
638 ParameterType -> Internal,
639 Value -> 1-(MW/MZ)^2,
640 Description -> "Squared Sin of the Weinberg angle"
641 },
642 ee == {
643 ParameterType -> Internal,
644 Value -> Sqrt[4 Pi aEW],
645 InteractionOrder -> {QED,1},
646 TeX -> e,
647 Description -> "Electric coupling constant"
648 },
649 cw == {
650 ParameterType -> Internal,
651 Value -> Sqrt[1-sw2],
652 TeX -> Subscript[c,w],
653 Description -> "Cosine of the Weinberg angle"
654 },
655 sw == {
656 ParameterType -> Internal,
657 Value -> Sqrt[sw2],
658 TeX -> Subscript[s,w],
659 Description -> "Sine of the Weinberg angle"
660 },
661 gw == {
662 ParameterType -> Internal,
663 Definitions -> {gw->ee/sw},
664 InteractionOrder -> {QED,1},
665 TeX -> Subscript[g,w],
666 Description -> "Weak coupling constant at the Z pole"
667 },
668 g1 == {
669 ParameterType -> Internal,
670 Definitions -> {g1->ee/cw},
671 InteractionOrder -> {QED,1},
672 TeX -> Subscript[g,1],
673 Description -> "U(1)Y coupling constant at the Z pole"
674 },
675 gs == {
676 ParameterType -> Internal,
677 Value -> Sqrt[4 Pi aS],
678 InteractionOrder -> {QCD,1},
679 TeX -> Subscript[g,s],
680 ParameterName -> G,
681 Description -> "Strong coupling constant at the Z pole"
682 },
683 vev == {
684 ParameterType -> Internal,
685 Value -> 2*MW*sw/ee,
686 InteractionOrder -> {QED,-1},
687 Description -> "Higgs vacuum expectation value"
688 },
689 lam == {
690 ParameterType -> Internal,
691 Value -> MH^2/(2*vev^2),
692 InteractionOrder -> {QED, 2},
693 Description -> "Higgs quartic coupling"
694 },
695 muH == {
696 ParameterType -> Internal,
697 Value -> Sqrt[vev^2 lam],
698 TeX -> \[Mu],
699 Description -> "Coefficient of the quadratic piece of the Higgs potential"
700 },
701 yl == {
702 ParameterType -> Internal,
703 Indices -> {Index[Generation], Index[Generation]},
704 Definitions -> {yl[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
705 Value -> {yl[1,1] -> Sqrt[2] yme / vev, yl[2,2] -> Sqrt[2] ymm / vev, yl[3,3] -> Sqrt[2] ymtau / vev},
706 InteractionOrder -> {QED, 1},
707 ParameterName -> {yl[1,1] -> ye, yl[2,2] -> ym, yl[3,3] -> ytau},
708 TeX -> Superscript[y, l],
709 Description -> "Lepton Yukawa couplings"
710 },
711 yu == {
712 ParameterType -> Internal,
713 Indices -> {Index[Generation], Index[Generation]},
714 Definitions -> {yu[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
715 Value -> {yu[1,1] -> Sqrt[2] ymup/vev, yu[2,2] -> Sqrt[2] ymc/vev, yu[3,3] -> Sqrt[2] ymt/vev},
716 InteractionOrder -> {QED, 1},
717 ParameterName -> {yu[1,1] -> yup, yu[2,2] -> yc, yu[3,3] -> yt},
718 TeX -> Superscript[y, u],
719 Description -> "Up-type Yukawa couplings"
720 },
721 yd == {
722 ParameterType -> Internal,
723 Indices -> {Index[Generation], Index[Generation]},
724 Definitions -> {yd[i_?NumericQ, j_?NumericQ] :> 0 /; (i =!= j)},
725 Value -> {yd[1,1] -> Sqrt[2] ymdo/vev, yd[2,2] -> Sqrt[2] yms/vev, yd[3,3] -> Sqrt[2] ymb/vev},
726 InteractionOrder -> {QED, 1},
727 ParameterName -> {yd[1,1] -> ydo, yd[2,2] -> ys, yd[3,3] -> yb},
728 TeX -> Superscript[y, d],
729 Description -> "Down-type Yukawa couplings"
730 },
731(* N. B. : only Cabibbo mixing! *)
732 CKM == {
733 ParameterType -> Internal,
734 Indices -> {Index[Generation], Index[Generation]},
735 Unitary -> True,
736 Value -> {CKM[1,1] -> Cos[cabi], CKM[1,2] -> Sin[cabi], CKM[1,3] -> 0,
737 CKM[2,1] -> -Sin[cabi], CKM[2,2] -> Cos[cabi], CKM[2,3] -> 0,
738 CKM[3,1] -> 0, CKM[3,2] -> 0, CKM[3,3] -> 1},
739 TeX -> Superscript[V,CKM],
740 Description -> "CKM-Matrix"}
741};
742
743(* ************************** *)
744(* ***** Lagrangian ***** *)
745(* ************************** *)
746
747(***** Dim-6 *****)
748
749Lqq1 := Block[{mu,sp1,sp2,sp3,sp4,ii,jj,ff1,ff2,cc1,cc2},
750 op = ExpandIndices[
751 QLbar[sp1,ii,ff1,cc1].QL[sp2,ii,ff1,cc1] Ga[mu,sp1,sp2]
752 QLbar[sp3,jj,ff2,cc2].QL[sp4,jj,ff2,cc2] Ga[mu,sp3,sp4]
753 +QLbar[sp1,ii,ff1,cc1].QL[sp2,ii,ff2,cc1] Ga[mu,sp1,sp2]
754 QLbar[sp3,jj,ff2,cc2].QL[sp4,jj,ff1,cc2] Ga[mu,sp3,sp4]
755 -Sum[QLbar[sp1,ii,ff1,cc1].QL[sp2,ii,ff1,cc1] Ga[mu,sp1,sp2]
756 QLbar[sp3,jj,ff1,cc2].QL[sp4,jj,ff1,cc2] Ga[mu,sp3,sp4],{ff1,1,3}],
757 FlavorExpand->{SU2W,SU2D}];
758 Return[op cqq1/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
759
760Lqq3 := Block[{mu,sp1,sp2,sp3,sp4,cc1,cc2,ii1,jj1,ii2,jj2,kk,ff1,ff2},
761 op = ExpandIndices[
762 QLbar[sp1,ii1,ff1,cc1].QL[sp2,jj1,ff1,cc1] Ga[mu,sp1,sp2]
763 QLbar[sp3,ii2,ff2,cc2].QL[sp4,jj2,ff2,cc2] Ga[mu,sp3,sp4]
764 PauliSigma[kk,ii1,jj1] PauliSigma[kk,ii2,jj2]
765 +QLbar[sp1,ii1,ff1,cc1].QL[sp2,jj1,ff2,cc1] Ga[mu,sp1,sp2]
766 QLbar[sp3,ii2,ff2,cc2].QL[sp4,jj2,ff1,cc2] Ga[mu,sp3,sp4]
767 PauliSigma[kk,ii1,jj1] PauliSigma[kk,ii2,jj2]
768 -Sum[QLbar[sp1,ii1,ff1,cc1].QL[sp2,jj1,ff1,cc1] Ga[mu,sp1,sp2]
769 QLbar[sp3,ii2,ff1,cc2].QL[sp4,jj2,ff1,cc2] Ga[mu,sp3,sp4]
770 PauliSigma[kk,ii1,jj1] PauliSigma[kk,ii2,jj2],{ff1,1,3}],
771 FlavorExpand->{SU2W,SU2D}];
772 Return[op cqq3/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
773
774Luu := Block[{mu,sp1,sp2,sp3,sp4,cc1,cc2,ff1,ff2},
775 op = ExpandIndices[
776 uRbar[sp1,ff1,cc1].uR[sp2,ff1,cc1] Ga[mu,sp1,sp2]
777 uRbar[sp3,ff2,cc2].uR[sp4,ff2,cc2] Ga[mu,sp3,sp4]
778 +uRbar[sp1,ff1,cc1].uR[sp2,ff2,cc1] Ga[mu,sp1,sp2]
779 uRbar[sp3,ff2,cc2].uR[sp4,ff1,cc2] Ga[mu,sp3,sp4]
780 -Sum[uRbar[sp1,ff1,cc1].uR[sp2,ff1,cc1] Ga[mu,sp1,sp2]
781 uRbar[sp3,ff1,cc2].uR[sp4,ff1,cc2] Ga[mu,sp3,sp4],{ff1,1,3}],
782 FlavorExpand->{SU2W,SU2D}];
783 Return[op cuu/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
784
785Ldd := Block[{mu,sp1,sp2,sp3,sp4,cc1,cc2,ff1,ff2},
786 op = ExpandIndices[
787 dRbar[sp1,ff1,cc1].dR[sp2,ff1,cc1] Ga[mu,sp1,sp2]
788 dRbar[sp3,ff2,cc2].dR[sp4,ff2,cc2] Ga[mu,sp3,sp4]
789 +dRbar[sp1,ff1,cc1].dR[sp2,ff2,cc1] Ga[mu,sp1,sp2]
790 dRbar[sp3,ff2,cc2].dR[sp4,ff1,cc2] Ga[mu,sp3,sp4]
791 -Sum[dRbar[sp1,ff1,cc1].dR[sp2,ff1,cc1] Ga[mu,sp1,sp2]
792 dRbar[sp3,ff1,cc2].dR[sp4,ff1,cc2] Ga[mu,sp3,sp4],{ff1,1,3}],
793 FlavorExpand->{SU2W,SU2D}];
794 Return[op cdd/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
795
796Lud8 := Block[{mu,sp1,sp2,sp3,sp4,cc1,cc2,cc3,cc4,aa,ff1,ff2},
797 op = ExpandIndices[
798 uRbar[sp1,ff1,cc1].uR[sp2,ff1,cc2] Ga[mu,sp1,sp2]
799 dRbar[sp3,ff2,cc3].dR[sp4,ff2,cc4] Ga[mu,sp3,sp4]
800 T[aa,cc1,cc2] T[aa,cc3,cc4],
801 FlavorExpand->{SU2D,SU2W}];
802 Return[op cud8/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
803
804Lud1 := Block[{mu,sp1,sp2,sp3,sp4,cc1,cc2,ff1,ff2},
805 op = ExpandIndices[
806 uRbar[sp1,ff1,cc1].uR[sp2,ff1,cc1] Ga[mu,sp1,sp2]
807 dRbar[sp3,ff2,cc2].dR[sp4,ff2,cc2] Ga[mu,sp3,sp4],
808 FlavorExpand->{SU2D,SU2W}];
809 Return[op cud1/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
810
811Lqu1 := Block[{mu,sp1,sp2,sp3,sp4,ii,cc1,cc2,ff1,ff2},
812 op = ExpandIndices[
813 QLbar[sp1,ii,ff1,cc1].QL[sp2,ii,ff1,cc1] Ga[mu,sp1,sp2]
814 uRbar[sp3,ff2,cc2].uR[sp4,ff2,cc2] Ga[mu,sp3,sp4],
815 FlavorExpand->{SU2W,SU2D}];
816 Return[op cqu1/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
817
818Lqd1 := Block[{mu,sp1,sp2,sp3,sp4,ii,cc1,cc2,ff1,ff2},
819 op = ExpandIndices[
820 QLbar[sp1,ii,ff1,cc1].QL[sp2,ii,ff1,cc1] Ga[mu,sp1,sp2]
821 dRbar[sp3,ff2,cc2].dR[sp4,ff2,cc2] Ga[mu,sp3,sp4],
822 FlavorExpand->{SU2W,SU2D}];
823 Return[op cqd1/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
824
825Lqu8 := Block[{mu,sp1,sp2,sp3,sp4,ii,cc1,cc2,cc3,cc4,aa,ff1,ff2},
826 op = ExpandIndices[
827 QLbar[sp1,ii,ff1,cc1].QL[sp2,ii,ff1,cc2] Ga[mu,sp1,sp2]
828 uRbar[sp3,ff2,cc3].uR[sp4,ff2,cc4] Ga[mu,sp3,sp4]
829 T[aa,cc1,cc2] T[aa,cc3,cc4],
830 FlavorExpand->{SU2W,SU2D}];
831 Return[op cqu8/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
832
833Lqd8 := Block[{mu,sp1,sp2,sp3,sp4,ii,cc1,cc2,cc3,cc4,aa,ff1,ff2},
834 op = ExpandIndices[
835 QLbar[sp1,ii,ff1,cc1].QL[sp2,ii,ff1,cc2] Ga[mu,sp1,sp2]
836 dRbar[sp3,ff2,cc3].dR[sp4,ff2,cc4] Ga[mu,sp3,sp4]
837 T[aa,cc1,cc2] T[aa,cc3,cc4],
838 FlavorExpand->{SU2W,SU2D}];
839 Return[op cqd8/LambdaSMEFT^2 /.CKM[a_,b_]->IndexDelta[a,b] //OptimizeIndex];];
840
841L4lq := Lqq1 + Lqq3 + Luu + Ldd + Lud8 + Lud1 + Lqu1 + Lqd1 + Lqu8 + Lqd8;
842
843(***** SM *****)
844
845LGauge := Block[{mu,nu,ii,aa},
846 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]];
847
848LFermions := Block[{mu},
849 ExpandIndices[I*(
850 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]),
851 FlavorExpand->{SU2W,SU2D}]/.{CKM[a_,b_] Conjugate[CKM[a_,c_]]->IndexDelta[b,c], CKM[b_,a_] Conjugate[CKM[c_,a_]]->IndexDelta[b,c]}];
852
853LHiggs := Block[{ii,mu, feynmangaugerules},
854 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
855
856 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
857 ];
858
859LYukawa := Block[{sp,ii,jj,cc,ff1,ff2,ff3,yuk,feynmangaugerules},
860 feynmangaugerules = If[Not[FeynmanGauge], {G0|GP|GPbar ->0}, {}];
861
862 yuk = ExpandIndices[
863 -yd[ff2, ff3] CKM[ff1, ff2] QLbar[sp, ii, ff1, cc].dR [sp, ff3, cc] Phi[ii] -
864 yl[ff1, ff3] LLbar[sp, ii, ff1].lR [sp, ff3] Phi[ii] -
865 yu[ff1, ff2] QLbar[sp, ii, ff1, cc].uR [sp, ff2, cc] Phibar[jj] Eps[ii, jj], FlavorExpand -> SU2D];
866 yuk = yuk /. { CKM[a_, b_] Conjugate[CKM[a_, c_]] -> IndexDelta[b, c], CKM[b_, a_] Conjugate[CKM[c_, a_]] -> IndexDelta[b, c]};
867 yuk+HC[yuk]/.feynmangaugerules
868 ];
869
870LGhost := Block[{LGh1,LGhw,LGhs,LGhphi,mu, generators,gh,ghbar,Vectorize,phi1,phi2,togoldstones,doublet,doublet0},
871 (* Pure gauge piece *)
872 LGh1 = -ghBbar.del[DC[ghB,mu],mu];
873 LGhw = -ghWibar[ii].del[DC[ghWi[ii],mu],mu];
874 LGhs = -ghGbar[ii].del[DC[ghG[ii],mu],mu];
875
876 (* Scalar pieces: see Peskin pages 739-742 *)
877 (* phi1 and phi2 are the real degrees of freedom of GP *)
878 (* Vectorize transforms a doublet in a vector in the phi-basis, i.e. the basis of real degrees of freedom *)
879 gh = {ghB, ghWi[1], ghWi[2], ghWi[3]};
880 ghbar = {ghBbar, ghWibar[1], ghWibar[2], ghWibar[3]};
881 generators = {-I/2 g1 IdentityMatrix[2], -I/2 gw PauliSigma[1], -I/2 gw PauliSigma[2], -I/2 gw PauliSigma[3]};
882 doublet = Expand[{(-I phi1 - phi2)/Sqrt[2], Phi[2]} /. MR$Definitions /. vev -> 0];
883 doublet0 = {0, vev/Sqrt[2]};
884 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}];
885 togoldstones := {phi1 -> (GP + GPbar)/Sqrt[2], phi2 -> (-GP + GPbar)/(I Sqrt[2])};
886 LGhphi=Plus@@Flatten[Table[-ghbar[[kkk]].gh[[lll]] Vectorize[generators[[kkk]].doublet0].Vectorize[generators[[lll]].(doublet+doublet0)],{kkk,4},{lll,4}]] /.togoldstones;
887
888ExpandIndices[ LGhs + If[FeynmanGauge, LGh1 + LGhw + LGhphi,0], FlavorExpand->SU2W]];
889
890LSM:= LGauge + LFermions + LHiggs + LYukawa + LGhost;
891
892L6:= LSM + L4lq;
893