Changes between Version 1 and Version 2 of topBSM


Ignore:
Timestamp:
09/03/13 20:39:40 (7 years ago)
Author:
stefankrastanov
Comment:

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  • topBSM

    v1 v2  
    1 == TopBSM ==
     1== Top Quark Decay to a Higgs and a Light Quark Operator ==
     2
     3### Motivation
     4
     5Neutral Flavor Changing couplings are absent in the Standard Model at tree
     6level. Moreover, at next-to-leading order they are supressed by the GIM
     7mechanism. Therefore a detection of such processes would be a strong hint at
     8new physics. Here we focus Neutral Flavor Change mediated by the Higgs boson
     9following [@zhang2013top].
     10
     11The lowest dimensional operators compatible with the symmetries of the Standard
     12Model are the following six-dimensional operators (for a comprehensive list of
     13all 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}
     22O^{1,3}_{uG} = y_t g_s (\bar{q} \sigma^{\mu\nu} T^a t) \bar{\phi} G^a_{\mu\nu}; \\
     23 \\
     24O^{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}
     35O^{1,3}_{u\phi} = - y_t^3 (\phi^\dagger \phi) (\bar{q} t) \bar{\phi}; \\
     36 \\
     37O^{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
     51Where 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
     60The 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
     69where $\Lambda$ is the new physics energy scale, $O_i$ is for the various
     70six-dimensional operators in consideration and  $c_i$ are relative couplings.
     71
     72The normalizations for the six-dimensional operators were chosen such that for
     73any new SM-like vertices the ratio of the new couplings to the SM couplings is
     74of the form $c_i\frac{m_t^2}{\Lambda^2}$.
     75
     76### Implementation and Validation
     77
     78The implementation is a straightforward transcription of the Lagrangian into
     79`FeynRules` format as no new fields need to be defined.
     80
     81The model was validated using the build-in checks in `FeynRules` and
     82`MadGraph5`. Moreover the decay widths were confirmed through `MadGraph5` and
     83compared to the analytical results.
     84
     85== Beyond-SM Operators with the Top Quark ==
     86
     87### Motivation
     88
     89This model is a reimplementation of the model behind the following paper:
     90[@frederix2009top]. The paper looks at top pair invariant mass distribution as
     91a window for new physics by studying the effects that various s-channel
     92resonance would exert. The original model was implemented in `MadGraph4`. Here
     93we provide a reimplementation in the `FeynRules`-`MadGraph5` toolset.
     94
     95The model is not restricted to use only for studying the top pair invariant
     96mass distribution as will be seen below. For each newly implemented particle we
     97will discuss how it couples to the Standard Model particles, how the model was
     98validated against [@frederix2009top] and other previous studies, and whether
     99there are any constraints on the versions of `FeynRules` and `MadGraph5` to be
     100used.
     101
     102In addition, the new model files provide the width of the particles (there is
     103no need for them to be computed separately). Also, the constraint on the
     104particle masses were lifted (the previous version provided certain couplings
     105only for certain mass ranges, and the couplings themselves were expressed only
     106as series expansions). The new model provides the exact expressions for all
     107masses.
     108
     109### Spin Zero, Color Singlet Particle
     110
     111The 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
     113for the left and for the right top. The effective vertex of gluon fusion through
     114a top loop is explicitly given in the Lagrangian as well.
     115
     116The coupling to the top operator is
     117
     118{{{
     119#!latex
     120\begin{equation}
     121\mathcal{L}_{S_0 t}\; =\;
     122c_{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
     127The gluon fusion effective operator must be added explicitly because it is a
     128beyond-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
     138or
     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
     148where $FS(G)$ is the field strength for the gluon field and ~ denotes a dual field.
     149
     150By comparing the vertices produces by these operators to the result of the
     151integrated top loop we get
     152
     153{{{
     154#!latex
     155\begin{equation}
     156c_{s0fuison\,scalar} = -c_{s0scalar} \frac{g_s^2}{12 \pi^2 v} \;
     157f_S\left(\left(\frac{2 m_t}{m_{S_0}}\right)^2\right)
     158\end{equation}
     159\begin{equation}
     160c_{s0fuison\,axial} = -c_{s0axial} \frac{g_s^2}{8 \pi^2 v} \;
     161f_A\left(\left(\frac{2 m_t}{m_{S_0}}\right)^2\right)
     162\end{equation}
     163}}}
     164
     165with
     166
     167{{{
     168#!latex
     169\begin{equation}
     170f_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
     178and
     179
     180{{{
     181#!latex
     182\begin{equation}
     183f_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
     191With appropriate branch cuts in the complex plane these expressions are
     192actually the same when $\arcsin$ is expressed in terms of $\log$. The
     193integration of the top loop was verified with the `FeynCalc` package and the
     194notebook is provided together with the models.
     195
     196Finally, given this Lagrangian the width of the new particles is:
     197
     198{{{
     199#!latex
     200\begin{equation}
     201W_{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}}
     203c_{s0scalar}^2 + \left(c_{s0axial}^2 +
     204c_{s0scalar}^2\right)\right)
     205\end{equation}
     206}}}
     207
     208#### Validation of the Model
     209
     210The 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
     212parameters in the old model are to be evaluated at each point in phase space,
     213which the `standalone` mode does not permit. A short patch is provided in the
     214annex with an explanation of the necessary changes.
     215
     216After the application of the patch, the model was validated against the old
     217implementation in `standalone` mode. The decay width and the cross-section in
     218various processes was validated as well, after taking into account the
     219differences at runtime between `MadGraph4` and `MadGraph5`.
     220
     221However the old model is only for heavy $S_0$ particles ($m_{S_0}>2m_t$). The
     222changes 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
     227were 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
     229interest.
     230
     231### Spin Zero, Color Octet Particle
     232
     233The 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
     235for the left and for the right top and there is an effective vertex of gluon fusion through
     236a top loop is explicitly given in the Lagrangian as well.
     237
     238The operators are:
     239
     240{{{
     241#!latex
     242\begin{equation}
     243\mathcal{L}_{O_0 t}\; =\;
     244c_{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
     259where $S_{SU3}^{abc}$ is the completely symmetric tensor and where
     260$c_{o0fusion\,scalar}$ and $c_{o0fusion\,axial}$ are the same as for `S0` with
     261coupling and masses appropriately substituted.
     262
     263Again, given this Lagrangian the width of the new particles is:
     264
     265{{{
     266#!latex
     267\begin{equation}
     268W_{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}}
     270c_{o0scalar}^2 + \left(c_{o0axial}^2 +
     271c_{o0scalar}^2\right)\right)
     272\end{equation}
     273}}}
     274
     275which is $\frac{1}{6}$ times the expression for $W_{S_0}$ with appropriately
     276substituted couplings and masses.
     277
     278#### Validation of the Model
     279
     280As 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
     282that mode, as well as in `MadEvent` mode: both the decay width and the
     283cross-sections of various processes were checked.
     284
     285The 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
     287build-in tools in `FeynRules` and `MadGraph5`.
     288
     289### Spin One, Color Singlet Particle
     290
     291The name for this resonance is `S1`. It has both vector and axial couplings to
     292all quarks and leptons. It is used mostly for a "model-independent" vector
     293boson ($Z^\prime$). For convenience the Lagrangian has exactly the same form as
     294the part of the Standard Model Lagrangian that governs the coupling of the SM Z
     295to the fermions. In addition to that each coupling is parametrized by coupling
     296constant 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
     305For example the coupling to neutrinos is
     306
     307{{{
     308#!latex
     309\begin{equation}
     310\mathcal{L}_{S_{1}\nu}\;=\;
     311c_{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
     317where $\theta_W$ is
     318the Weinberg angle, $e$ is the electric coupling constant, $L$ is the leptonic
     319doublet and $L_2$ is its second component.
     320
     321The width of the particle is calculated and provided in the model as well.
     322
     323#### Validation of the Model
     324
     325Besides the basic correctness tests provided by `FeynRules` and `MadGraph5` the
     326`S1` model was verified against the original `MadGraph4` model.
     327
     328In `standalone` mode both models produce the same differential cross-section
     329withing machine precision. In `MadEvent` mode the decay width is the same in
     330both 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
     332well.
     333
     334### Spin One, Color Octet Particle
     335
     336The name for this resonance is `O1`. The need for a `FeynRules` version of it
     337is what originally caused the request for reimplementation of the whole model.
     338This field lives in the same representation of the gauge group as the gluons.
     339It is used to represent color vector particle (coloron) or an color axial
     340particle (axigluon).
     341
     342The 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}}}
     351where $i$ goes over right and left handedness of the up and down quarks of each
     352generation. $T$ is the representation of the SU3 group generators and $g_s$ is
     353the strong coupling constant.
     354
     355The width of the particles is calculated and provided in the model as well.
     356
     357#### Validation of the Model
     358
     359Similar models are discussed in [@choudhury2007top] and [@antunano2008top].
     360Their results confirm both the width and the differential cross-section
     361calculated in the `FeynRules` model.
     362
     363Another `FeynRules` model is available that implements axigluons in
     364[@falkowski2012axigluon]. It produces the same vertices, however it differs in
     365that it provides for a mixing between the axigluons and the gluons.
     366
     367The original `MadGraph4` model gives the same results in the `standalone`
     368configuration. Both the decay width and the cross section of top pair
     369production were checked as well. Well accounting for the differences in the
     370`MadGraph4` and `MadGraph5` runtime they produce the same results. Details are
     371provided in the annex.
     372
     373
     374#### Technical Constraints
     375
     376During the implementation of this model a bug in the canonicalization routines
     377of `FeynRules` was encountered. Whenever a tensor contraction expression is
     378passes through `FeynRules` it needs to get into a canonical form (in order to
     379permit equality checks, pattern matching and simplifications) before the
     380canonical quantization is executed. The symmetric tensor for the SU3 group was
     381not taken into account in this canonicalization. Benjamin Fuks graciously and
     382quickly 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
     389As was mentioned above, the minimal version of `FeynRules` in which the models
     390are guaranteed to work is `1.7.178`.
     391
     392Moreover, there is a disaccord between the formats for saving models in the
     393current versions of `MadGraph5` and `FeynRules`. It should be fixed in the next
     394versions, however if a runtime error message concerning undefined Goldstone
     395bosons is raised by `MadGraph` it can be quickly fixed by manually modifying
     396the offending lines in `particles.py`. It can be done automatically with the
     397following command:
     398
     399`perl -pi -e 's/goldstone/GoldstoneBoson/g' ./models/topBSM_UFO/particles.py`.
     400
     401### Setting Mass Ranges
     402
     403The calculation of the widths of different particles (especially `S0` and `O0`)
     404as well as the effective couplings for gluon fusion vertices changes
     405qualitatively if the mass of the particle passes over or under two times the mass
     406of the top. This is implemented in `FeynRules` with a delayed rewrite rule,
     407however `MadGraph5` does not permit such branching. Hence if the need arises to
     408change 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
     415In `standalone` mode couplings are evaluated only once, before generating a
     416random phase space point at which to evaluate the matrix element. This does not
     417permit testing some of the more complicated models like the original
     418implementation of the `S0` and `O0` particles.
     419
     420As a workaround for this issue, one can modify the code so that `setparam` is
     421called after each generation of random phase space points. A patch that does
     422this automatically is provided with the models.