1 | | == TopBSM == |
| 1 | == Top Quark Decay to a Higgs and a Light Quark Operator == |
| 2 | |
| 3 | ### Motivation |
| 4 | |
| 5 | Neutral Flavor Changing couplings are absent in the Standard Model at tree |
| 6 | level. Moreover, at next-to-leading order they are supressed by the GIM |
| 7 | mechanism. Therefore a detection of such processes would be a strong hint at |
| 8 | new physics. Here we focus Neutral Flavor Change mediated by the Higgs boson |
| 9 | following [@zhang2013top]. |
| 10 | |
| 11 | The lowest dimensional operators compatible with the symmetries of the Standard |
| 12 | Model are the following six-dimensional operators (for a comprehensive list of |
| 13 | all six-dimensional operators compatible with Standard Model symmetries consult |
| 14 | [@grzadkowski2010dimension]): |
| 15 | |
| 16 | - chromomagnetic operator $O_{uG}$ |
| 17 | |
| 18 | {{{ |
| 19 | #!latex |
| 20 | \begin{equation} |
| 21 | \begin{matrix} |
| 22 | O^{1,3}_{uG} = y_t g_s (\bar{q} \sigma^{\mu\nu} T^a t) \bar{\phi} G^a_{\mu\nu}; \\ |
| 23 | \\ |
| 24 | O^{3,1}_{uG} = y_t g_s (\bar{Q} \sigma^{\mu\nu} T^a u) \bar{\phi} G^a_{\mu\nu}; |
| 25 | \end{matrix} |
| 26 | \end{equation} |
| 27 | }}} |
| 28 | |
| 29 | - dimension-six Yukawa interaction $O_{u\phi}$ |
| 30 | |
| 31 | {{{ |
| 32 | #!latex |
| 33 | \begin{equation} |
| 34 | \begin{matrix} |
| 35 | O^{1,3}_{u\phi} = - y_t^3 (\phi^\dagger \phi) (\bar{q} t) \bar{\phi}; \\ |
| 36 | \\ |
| 37 | O^{3,1}_{u\phi} = - y_t^3 (\phi^\dagger \phi) (\bar{Q} u) \bar{\phi}; |
| 38 | \end{matrix} |
| 39 | \end{equation} |
| 40 | }}} |
| 41 | |
| 42 | - To each (1,3) operator corresponds a (3,1) operator where the flavors are |
| 43 | reversed. |
| 44 | |
| 45 | - To each operator (e.g. (1,3)) corresponds another where the up quark is |
| 46 | exchanged for a charm quark (e.g. (2,3)). |
| 47 | |
| 48 | - The hermitian conjugates of the above-mentioned operators contributing with |
| 49 | the opposite chirality. |
| 50 | |
| 51 | Where we denoted: |
| 52 | |
| 53 | - $\phi$ is the Higgs doublet; |
| 54 | - $Q$ and $q$ are respectively the 1st (or 2nd) and the 3th left-handed quark |
| 55 | doublet; |
| 56 | - $u$ (or $c$) and $t$ are the right-handed quarks; |
| 57 | - $\bar{\phi} = i \sigma^2 \phi$ |
| 58 | - $y_t = \sqrt{2}\frac{m_t}{v}$ the top quark Yukawa coupling. |
| 59 | |
| 60 | The complete Lagrangian takes the form: |
| 61 | |
| 62 | {{{ |
| 63 | #!latex |
| 64 | \begin{equation} |
| 65 | \mathcal{L}_{eff} = \mathcal{L}_{SM} + \sum_i \frac{c_i O_i}{\Lambda^2}, |
| 66 | \end{equation} |
| 67 | }}} |
| 68 | |
| 69 | where $\Lambda$ is the new physics energy scale, $O_i$ is for the various |
| 70 | six-dimensional operators in consideration and $c_i$ are relative couplings. |
| 71 | |
| 72 | The normalizations for the six-dimensional operators were chosen such that for |
| 73 | any new SM-like vertices the ratio of the new couplings to the SM couplings is |
| 74 | of the form $c_i\frac{m_t^2}{\Lambda^2}$. |
| 75 | |
| 76 | ### Implementation and Validation |
| 77 | |
| 78 | The implementation is a straightforward transcription of the Lagrangian into |
| 79 | `FeynRules` format as no new fields need to be defined. |
| 80 | |
| 81 | The model was validated using the build-in checks in `FeynRules` and |
| 82 | `MadGraph5`. Moreover the decay widths were confirmed through `MadGraph5` and |
| 83 | compared to the analytical results. |
| 84 | |
| 85 | == Beyond-SM Operators with the Top Quark == |
| 86 | |
| 87 | ### Motivation |
| 88 | |
| 89 | This model is a reimplementation of the model behind the following paper: |
| 90 | [@frederix2009top]. The paper looks at top pair invariant mass distribution as |
| 91 | a window for new physics by studying the effects that various s-channel |
| 92 | resonance would exert. The original model was implemented in `MadGraph4`. Here |
| 93 | we provide a reimplementation in the `FeynRules`-`MadGraph5` toolset. |
| 94 | |
| 95 | The model is not restricted to use only for studying the top pair invariant |
| 96 | mass distribution as will be seen below. For each newly implemented particle we |
| 97 | will discuss how it couples to the Standard Model particles, how the model was |
| 98 | validated against [@frederix2009top] and other previous studies, and whether |
| 99 | there are any constraints on the versions of `FeynRules` and `MadGraph5` to be |
| 100 | used. |
| 101 | |
| 102 | In addition, the new model files provide the width of the particles (there is |
| 103 | no need for them to be computed separately). Also, the constraint on the |
| 104 | particle masses were lifted (the previous version provided certain couplings |
| 105 | only for certain mass ranges, and the couplings themselves were expressed only |
| 106 | as series expansions). The new model provides the exact expressions for all |
| 107 | masses. |
| 108 | |
| 109 | ### Spin Zero, Color Singlet Particle |
| 110 | |
| 111 | The name used in [@frederix2009top] for this resonance is `S0` for "color |
| 112 | [S]inglet, spin [Zero]". It is coupled only to the top with different couplings |
| 113 | for the left and for the right top. The effective vertex of gluon fusion through |
| 114 | a top loop is explicitly given in the Lagrangian as well. |
| 115 | |
| 116 | The coupling to the top operator is |
| 117 | |
| 118 | {{{ |
| 119 | #!latex |
| 120 | \begin{equation} |
| 121 | \mathcal{L}_{S_0 t}\; =\; |
| 122 | c_{s0scalar}\, \frac{m_t}{v} S_0\, \bar{t}.t \; |
| 123 | + \; i\, c_{s0axial}\, \frac{m_t}{v} S_0\, \bar{t}.\gamma^5.t. |
| 124 | \end{equation} |
| 125 | }}} |
| 126 | |
| 127 | The gluon fusion effective operator must be added explicitly because it is a |
| 128 | beyond-tree-level effect. In general, such an operator takes the form |
| 129 | |
| 130 | {{{ |
| 131 | #!latex |
| 132 | \begin{equation} |
| 133 | \mathcal{L}_{G\,fusion\,scalar\,S_0}\; =\; |
| 134 | -\frac{1}{4} c_{s0fusion\,scalar} S_0 \; FS(G)_{\mu \nu}^a \; FS(G)^{\mu \nu a} |
| 135 | \end{equation} |
| 136 | }}} |
| 137 | |
| 138 | or |
| 139 | |
| 140 | {{{ |
| 141 | #!latex |
| 142 | \begin{equation} |
| 143 | \mathcal{L}_{G\,fusion\,axial\,S_0}\; =\; |
| 144 | -\frac{1}{4} c_{s0fusion\,axial} S_0 \; FS(G)_{\mu \nu}^a \; \widetilde{FS}(G)^{\mu \nu a} |
| 145 | \end{equation} |
| 146 | }}} |
| 147 | |
| 148 | where $FS(G)$ is the field strength for the gluon field and ~ denotes a dual field. |
| 149 | |
| 150 | By comparing the vertices produces by these operators to the result of the |
| 151 | integrated top loop we get |
| 152 | |
| 153 | {{{ |
| 154 | #!latex |
| 155 | \begin{equation} |
| 156 | c_{s0fuison\,scalar} = -c_{s0scalar} \frac{g_s^2}{12 \pi^2 v} \; |
| 157 | f_S\left(\left(\frac{2 m_t}{m_{S_0}}\right)^2\right) |
| 158 | \end{equation} |
| 159 | \begin{equation} |
| 160 | c_{s0fuison\,axial} = -c_{s0axial} \frac{g_s^2}{8 \pi^2 v} \; |
| 161 | f_A\left(\left(\frac{2 m_t}{m_{S_0}}\right)^2\right) |
| 162 | \end{equation} |
| 163 | }}} |
| 164 | |
| 165 | with |
| 166 | |
| 167 | {{{ |
| 168 | #!latex |
| 169 | \begin{equation} |
| 170 | f_S(t) = |
| 171 | \begin{cases} |
| 172 | \frac{3}{2} t \left(1 + \frac{1}{4} \left(t - 1\right) \left(\log{\left(\frac{\sqrt{1 - t} + 1}{1 - \sqrt{1 - t}}\right)} - i \pi\right)^2\right) & t \leq 1 \\ |
| 173 | \frac{3}{2} t \left(1 + \left(1 - t\right) \arcsin{\left(\frac{1}{\sqrt{t}}\right)}^2\right) & 1 \leq t. |
| 174 | \end{cases} |
| 175 | \end{equation} |
| 176 | }}} |
| 177 | |
| 178 | and |
| 179 | |
| 180 | {{{ |
| 181 | #!latex |
| 182 | \begin{equation} |
| 183 | f_A(t) = |
| 184 | \begin{cases} |
| 185 | - \frac{t}{4} \left(\log{\left(\frac{\sqrt{1 - t} + 1}{1 - \sqrt{1 - t}}\right)} - i \pi\right)^2 & t \leq 1 \\ |
| 186 | t \arcsin{\left(\frac{1}{\sqrt{t}}\right)}^2 & 1 \leq t. |
| 187 | \end{cases} |
| 188 | \end{equation} |
| 189 | }}} |
| 190 | |
| 191 | With appropriate branch cuts in the complex plane these expressions are |
| 192 | actually the same when $\arcsin$ is expressed in terms of $\log$. The |
| 193 | integration of the top loop was verified with the `FeynCalc` package and the |
| 194 | notebook is provided together with the models. |
| 195 | |
| 196 | Finally, given this Lagrangian the width of the new particles is: |
| 197 | |
| 198 | {{{ |
| 199 | #!latex |
| 200 | \begin{equation} |
| 201 | W_{S_0}\;=\; |
| 202 | \frac{3 m_t^2 m_{S_0}}{8 \pi v^2} \sqrt{1 - \frac{4 m_t^2}{m_{S_0}^2}} \left(-\frac{4 m_t^2}{m_{S_0}} |
| 203 | c_{s0scalar}^2 + \left(c_{s0axial}^2 + |
| 204 | c_{s0scalar}^2\right)\right) |
| 205 | \end{equation} |
| 206 | }}} |
| 207 | |
| 208 | #### Validation of the Model |
| 209 | |
| 210 | The first step is to compare the old and the new implementations through the |
| 211 | `standalone` mode. However this is complicated by the fact that certain |
| 212 | parameters in the old model are to be evaluated at each point in phase space, |
| 213 | which the `standalone` mode does not permit. A short patch is provided in the |
| 214 | annex with an explanation of the necessary changes. |
| 215 | |
| 216 | After the application of the patch, the model was validated against the old |
| 217 | implementation in `standalone` mode. The decay width and the cross-section in |
| 218 | various processes was validated as well, after taking into account the |
| 219 | differences at runtime between `MadGraph4` and `MadGraph5`. |
| 220 | |
| 221 | However the old model is only for heavy $S_0$ particles ($m_{S_0}>2m_t$). The |
| 222 | changes permitting work with light $S_0$ particles: |
| 223 | |
| 224 | - correct calculation of the width when decay to top pair is impossible |
| 225 | - correct expression for the effective gluon fusion vertex |
| 226 | |
| 227 | were not major and were validated using the build-in tools in `FeynRules` and |
| 228 | `MadGraph5`. Moreover studies for such light particles are probably of minor |
| 229 | interest. |
| 230 | |
| 231 | ### Spin Zero, Color Octet Particle |
| 232 | |
| 233 | The name for this resonance is `O0` for "color |
| 234 | [O]ctet, spin [Zero]". Like `S0` it is coupled only to the top with different couplings |
| 235 | for the left and for the right top and there is an effective vertex of gluon fusion through |
| 236 | a top loop is explicitly given in the Lagrangian as well. |
| 237 | |
| 238 | The operators are: |
| 239 | |
| 240 | {{{ |
| 241 | #!latex |
| 242 | \begin{equation} |
| 243 | \mathcal{L}_{O_0 t}\; =\; |
| 244 | c_{o0scalar}\, \frac{m_t}{v} O_0^a\, \bar{t}.T^a.t \; |
| 245 | + \; i\, c_{o0axial}\, \frac{m_t}{v} O_0^a\, \bar{t}.\gamma^5.T^a.t. |
| 246 | \end{equation} |
| 247 | |
| 248 | \begin{equation} |
| 249 | \mathcal{L}_{G\,fusion\,scalar\,O_0}\; =\; |
| 250 | -\frac{1}{4} c_{o0fusion\,scalar} S_{SU3}^{abc} O_0^a \; FS(G)_{\mu \nu}^b \; FS(G)^{\mu \nu c} |
| 251 | \end{equation} |
| 252 | |
| 253 | \begin{equation} |
| 254 | \mathcal{L}_{G\,fusion\,axial\,O_0}\; =\; |
| 255 | -\frac{1}{4} c_{o0fusion\,axial} S_{SU3}^{abc} O_0^a \; FS(G)_{\mu \nu}^b \; \widetilde{FS}(G)^{\mu \nu c} |
| 256 | \end{equation} |
| 257 | }}} |
| 258 | |
| 259 | where $S_{SU3}^{abc}$ is the completely symmetric tensor and where |
| 260 | $c_{o0fusion\,scalar}$ and $c_{o0fusion\,axial}$ are the same as for `S0` with |
| 261 | coupling and masses appropriately substituted. |
| 262 | |
| 263 | Again, given this Lagrangian the width of the new particles is: |
| 264 | |
| 265 | {{{ |
| 266 | #!latex |
| 267 | \begin{equation} |
| 268 | W_{O_0}\;=\; |
| 269 | \frac{m_t^2 m_{O_0}}{16 \pi v^2} \sqrt{1 - \frac{4 m_t^2}{m_{O_0}^2}} \left(-\frac{4 m_t^2}{m_{O_0}} |
| 270 | c_{o0scalar}^2 + \left(c_{o0axial}^2 + |
| 271 | c_{o0scalar}^2\right)\right) |
| 272 | \end{equation} |
| 273 | }}} |
| 274 | |
| 275 | which is $\frac{1}{6}$ times the expression for $W_{S_0}$ with appropriately |
| 276 | substituted couplings and masses. |
| 277 | |
| 278 | #### Validation of the Model |
| 279 | |
| 280 | As with `S0` a patch is necessary before one can proceed with validation in the |
| 281 | `standalone` mode. The model was validated against the old implementation in |
| 282 | that mode, as well as in `MadEvent` mode: both the decay width and the |
| 283 | cross-sections of various processes were checked. |
| 284 | |
| 285 | The new model permits the use of light `O0` unlike the old implementation for |
| 286 | `MadGraph4`. As in the case of `S0` this part was validated only through the |
| 287 | build-in tools in `FeynRules` and `MadGraph5`. |
| 288 | |
| 289 | ### Spin One, Color Singlet Particle |
| 290 | |
| 291 | The name for this resonance is `S1`. It has both vector and axial couplings to |
| 292 | all quarks and leptons. It is used mostly for a "model-independent" vector |
| 293 | boson ($Z^\prime$). For convenience the Lagrangian has exactly the same form as |
| 294 | the part of the Standard Model Lagrangian that governs the coupling of the SM Z |
| 295 | to the fermions. In addition to that each coupling is parametrized by coupling |
| 296 | constant with default value of 1. |
| 297 | |
| 298 | - `s1uleft` for the coupling to up, charm and top left quarks; |
| 299 | - `s1dleft` for the coupling to down, strange and bottom left quarks; |
| 300 | - `s1uright` and `s1dright` for the corresponding right quarks; |
| 301 | - `s1eleft` for the left electron, muon and tau-lepton; |
| 302 | - `s1eright` for the right charged leptons; |
| 303 | - `s1nu` for the neutrinos. |
| 304 | |
| 305 | For example the coupling to neutrinos is |
| 306 | |
| 307 | {{{ |
| 308 | #!latex |
| 309 | \begin{equation} |
| 310 | \mathcal{L}_{S_{1}\nu}\;=\; |
| 311 | c_{s1nu}\; |
| 312 | \frac{e}{2\sin{\theta_W}\cos{\theta_W}}\; S_1^\mu\; |
| 313 | \underset{f=e,mu,tau}{\sum}\bar{L}_2^f.\gamma_\mu.L_2^f |
| 314 | \end{equation} |
| 315 | }}} |
| 316 | |
| 317 | where $\theta_W$ is |
| 318 | the Weinberg angle, $e$ is the electric coupling constant, $L$ is the leptonic |
| 319 | doublet and $L_2$ is its second component. |
| 320 | |
| 321 | The width of the particle is calculated and provided in the model as well. |
| 322 | |
| 323 | #### Validation of the Model |
| 324 | |
| 325 | Besides the basic correctness tests provided by `FeynRules` and `MadGraph5` the |
| 326 | `S1` model was verified against the original `MadGraph4` model. |
| 327 | |
| 328 | In `standalone` mode both models produce the same differential cross-section |
| 329 | withing machine precision. In `MadEvent` mode the decay width is the same in |
| 330 | both cases. When accounting for the differences at runtime in `MadGraph4` and |
| 331 | `MadGraph5` the cross sections of the various tested processes are the same as |
| 332 | well. |
| 333 | |
| 334 | ### Spin One, Color Octet Particle |
| 335 | |
| 336 | The name for this resonance is `O1`. The need for a `FeynRules` version of it |
| 337 | is what originally caused the request for reimplementation of the whole model. |
| 338 | This field lives in the same representation of the gauge group as the gluons. |
| 339 | It is used to represent color vector particle (coloron) or an color axial |
| 340 | particle (axigluon). |
| 341 | |
| 342 | The Lagrangian is of the form |
| 343 | |
| 344 | {{{ |
| 345 | #!latex |
| 346 | \begin{equation} |
| 347 | \mathcal{L}_{O_1}\; =\; |
| 348 | \sum_i c_i g_s O_1^{\mu a} \; \bar{q_i}.\gamma_\mu.T^a.q_i |
| 349 | \end{equation} |
| 350 | }}} |
| 351 | where $i$ goes over right and left handedness of the up and down quarks of each |
| 352 | generation. $T$ is the representation of the SU3 group generators and $g_s$ is |
| 353 | the strong coupling constant. |
| 354 | |
| 355 | The width of the particles is calculated and provided in the model as well. |
| 356 | |
| 357 | #### Validation of the Model |
| 358 | |
| 359 | Similar models are discussed in [@choudhury2007top] and [@antunano2008top]. |
| 360 | Their results confirm both the width and the differential cross-section |
| 361 | calculated in the `FeynRules` model. |
| 362 | |
| 363 | Another `FeynRules` model is available that implements axigluons in |
| 364 | [@falkowski2012axigluon]. It produces the same vertices, however it differs in |
| 365 | that it provides for a mixing between the axigluons and the gluons. |
| 366 | |
| 367 | The original `MadGraph4` model gives the same results in the `standalone` |
| 368 | configuration. Both the decay width and the cross section of top pair |
| 369 | production were checked as well. Well accounting for the differences in the |
| 370 | `MadGraph4` and `MadGraph5` runtime they produce the same results. Details are |
| 371 | provided in the annex. |
| 372 | |
| 373 | |
| 374 | #### Technical Constraints |
| 375 | |
| 376 | During the implementation of this model a bug in the canonicalization routines |
| 377 | of `FeynRules` was encountered. Whenever a tensor contraction expression is |
| 378 | passes through `FeynRules` it needs to get into a canonical form (in order to |
| 379 | permit equality checks, pattern matching and simplifications) before the |
| 380 | canonical quantization is executed. The symmetric tensor for the SU3 group was |
| 381 | not taken into account in this canonicalization. Benjamin Fuks graciously and |
| 382 | quickly fixed the issue, however for the model to work correctly at least |
| 383 | `FeynRules 1.7.178` or later is necessary. |
| 384 | |
| 385 | ## General Technical Constraints |
| 386 | |
| 387 | ### Required Versions |
| 388 | |
| 389 | As was mentioned above, the minimal version of `FeynRules` in which the models |
| 390 | are guaranteed to work is `1.7.178`. |
| 391 | |
| 392 | Moreover, there is a disaccord between the formats for saving models in the |
| 393 | current versions of `MadGraph5` and `FeynRules`. It should be fixed in the next |
| 394 | versions, however if a runtime error message concerning undefined Goldstone |
| 395 | bosons is raised by `MadGraph` it can be quickly fixed by manually modifying |
| 396 | the offending lines in `particles.py`. It can be done automatically with the |
| 397 | following command: |
| 398 | |
| 399 | `perl -pi -e 's/goldstone/GoldstoneBoson/g' ./models/topBSM_UFO/particles.py`. |
| 400 | |
| 401 | ### Setting Mass Ranges |
| 402 | |
| 403 | The calculation of the widths of different particles (especially `S0` and `O0`) |
| 404 | as well as the effective couplings for gluon fusion vertices changes |
| 405 | qualitatively if the mass of the particle passes over or under two times the mass |
| 406 | of the top. This is implemented in `FeynRules` with a delayed rewrite rule, |
| 407 | however `MadGraph5` does not permit such branching. Hence if the need arises to |
| 408 | change the mass of these particles it is important to change it from |
| 409 | `FeynRules` and not from `MadGraph5`. |
| 410 | |
| 411 | # Annex |
| 412 | |
| 413 | ## Patching the `standalone` Mode |
| 414 | |
| 415 | In `standalone` mode couplings are evaluated only once, before generating a |
| 416 | random phase space point at which to evaluate the matrix element. This does not |
| 417 | permit testing some of the more complicated models like the original |
| 418 | implementation of the `S0` and `O0` particles. |
| 419 | |
| 420 | As a workaround for this issue, one can modify the code so that `setparam` is |
| 421 | called after each generation of random phase space points. A patch that does |
| 422 | this automatically is provided with the models. |