Changes between Version 1 and Version 2 of topBSM

Timestamp:

Sep 3, 2013, 8:39:40 PM (12 years ago)

Author:

stefankrastanov

Comment:

--

topBSM

@@ -1 @@
-== TopBSM ==
+== Top Quark Decay to a Higgs and a Light Quark Operator ==
+
+### Motivation
+
+Neutral Flavor Changing couplings are absent in the Standard Model at tree
+level. Moreover, at next-to-leading order they are supressed by the GIM
+mechanism. Therefore a detection of such processes would be a strong hint at
+new physics. Here we focus Neutral Flavor Change mediated by the Higgs boson
+following [@zhang2013top].
+
+The lowest dimensional operators compatible with the symmetries of the Standard
+Model are the following six-dimensional operators (for a comprehensive list of
+all six-dimensional operators compatible with Standard Model symmetries consult
+[@grzadkowski2010dimension]):
+
+- chromomagnetic operator $O_{uG}$
+
+{{{
+#!latex
+\begin{equation}
+\begin{matrix}
+O^{1,3}_{uG} = y_t g_s (\bar{q} \sigma^{\mu\nu} T^a t) \bar{\phi} G^a_{\mu\nu}; \\
+ \\
+O^{3,1}_{uG} = y_t g_s (\bar{Q} \sigma^{\mu\nu} T^a u) \bar{\phi} G^a_{\mu\nu};
+\end{matrix}
+\end{equation}
+}}}
+
+- dimension-six Yukawa interaction $O_{u\phi}$
+
+{{{
+#!latex
+\begin{equation}
+\begin{matrix}
+O^{1,3}_{u\phi} = - y_t^3 (\phi^\dagger \phi) (\bar{q} t) \bar{\phi}; \\
+ \\
+O^{3,1}_{u\phi} = - y_t^3 (\phi^\dagger \phi) (\bar{Q} u) \bar{\phi};
+\end{matrix}
+\end{equation}
+}}}
+
+- To each (1,3) operator corresponds a (3,1) operator where the flavors are
+  reversed.
+
+- To each operator (e.g. (1,3)) corresponds another where the up quark is
+  exchanged for a charm quark (e.g. (2,3)).
+
+- The hermitian conjugates of the above-mentioned operators contributing with
+  the opposite chirality.
+
+Where we denoted:
+
+- $\phi$ is the Higgs doublet;
+- $Q$ and $q$ are respectively the 1st (or 2nd) and the 3th left-handed quark
+  doublet;
+- $u$ (or $c$) and $t$ are the right-handed quarks;
+- $\bar{\phi} = i \sigma^2 \phi$
+- $y_t = \sqrt{2}\frac{m_t}{v}$ the top quark Yukawa coupling.
+
+The complete Lagrangian takes the form:
+
+{{{
+#!latex
+\begin{equation}
+\mathcal{L}_{eff} = \mathcal{L}_{SM} + \sum_i \frac{c_i O_i}{\Lambda^2},
+\end{equation}
+}}}
+
+where $\Lambda$ is the new physics energy scale, $O_i$ is for the various
+six-dimensional operators in consideration and  $c_i$ are relative couplings.
+
+The normalizations for the six-dimensional operators were chosen such that for
+any new SM-like vertices the ratio of the new couplings to the SM couplings is
+of the form $c_i\frac{m_t^2}{\Lambda^2}$. 
+
+### Implementation and Validation
+
+The implementation is a straightforward transcription of the Lagrangian into
+`FeynRules` format as no new fields need to be defined. 
+
+The model was validated using the build-in checks in `FeynRules` and
+`MadGraph5`. Moreover the decay widths were confirmed through `MadGraph5` and
+compared to the analytical results.
+
+== Beyond-SM Operators with the Top Quark ==
+
+### Motivation
+
+This model is a reimplementation of the model behind the following paper:
+[@frederix2009top]. The paper looks at top pair invariant mass distribution as
+a window for new physics by studying the effects that various s-channel
+resonance would exert. The original model was implemented in `MadGraph4`. Here
+we provide a reimplementation in the `FeynRules`-`MadGraph5` toolset.
+
+The model is not restricted to use only for studying the top pair invariant
+mass distribution as will be seen below. For each newly implemented particle we
+will discuss how it couples to the Standard Model particles, how the model was
+validated against [@frederix2009top] and other previous studies, and whether
+there are any constraints on the versions of `FeynRules` and `MadGraph5` to be
+used.
+
+In addition, the new model files provide the width of the particles (there is
+no need for them to be computed separately). Also, the constraint on the
+particle masses were lifted (the previous version provided certain couplings
+only for certain mass ranges, and the couplings themselves were expressed only
+as series expansions). The new model provides the exact expressions for all
+masses.
+
+### Spin Zero, Color Singlet Particle 
+
+The name used in [@frederix2009top] for this resonance is `S0` for "color
+[S]inglet, spin [Zero]". It is coupled only to the top with different couplings
+for the left and for the right top. The effective vertex of gluon fusion through
+a top loop is explicitly given in the Lagrangian as well.
+
+The coupling to the top operator is
+
+{{{
+#!latex
+\begin{equation}
+\mathcal{L}_{S_0 t}\; =\;
+c_{s0scalar}\, \frac{m_t}{v} S_0\, \bar{t}.t \;
++ \; i\, c_{s0axial}\, \frac{m_t}{v} S_0\, \bar{t}.\gamma^5.t.
+\end{equation}
+}}}
+
+The gluon fusion effective operator must be added explicitly because it is a
+beyond-tree-level effect. In general, such an operator takes the form
+
+{{{
+#!latex
+\begin{equation}
+\mathcal{L}_{G\,fusion\,scalar\,S_0}\; =\;
+-\frac{1}{4} c_{s0fusion\,scalar} S_0 \; FS(G)_{\mu \nu}^a \; FS(G)^{\mu \nu a}
+\end{equation}
+}}}
+
+or
+
+{{{
+#!latex
+\begin{equation}
+\mathcal{L}_{G\,fusion\,axial\,S_0}\; =\;
+-\frac{1}{4} c_{s0fusion\,axial} S_0 \; FS(G)_{\mu \nu}^a \; \widetilde{FS}(G)^{\mu \nu a}
+\end{equation}
+}}}
+
+where $FS(G)$ is the field strength for the gluon field and ~ denotes a dual field.
+
+By comparing the vertices produces by these operators to the result of the
+integrated top loop we get
+
+{{{
+#!latex
+\begin{equation}
+c_{s0fuison\,scalar} = -c_{s0scalar} \frac{g_s^2}{12 \pi^2 v} \;
+f_S\left(\left(\frac{2 m_t}{m_{S_0}}\right)^2\right)
+\end{equation}
+\begin{equation}
+c_{s0fuison\,axial} = -c_{s0axial} \frac{g_s^2}{8 \pi^2 v} \;
+f_A\left(\left(\frac{2 m_t}{m_{S_0}}\right)^2\right)
+\end{equation}
+}}}
+
+with
+
+{{{
+#!latex
+\begin{equation}
+f_S(t) = 
+  \begin{cases} 
+    \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 \\
+    \frac{3}{2} t \left(1 + \left(1 - t\right) \arcsin{\left(\frac{1}{\sqrt{t}}\right)}^2\right) & 1 \leq t.
+  \end{cases}
+\end{equation}
+}}}
+
+and
+
+{{{
+#!latex
+\begin{equation}
+f_A(t) = 
+  \begin{cases} 
+    - \frac{t}{4} \left(\log{\left(\frac{\sqrt{1 - t} + 1}{1 - \sqrt{1 - t}}\right)} - i \pi\right)^2 & t \leq 1 \\
+    t \arcsin{\left(\frac{1}{\sqrt{t}}\right)}^2 & 1 \leq t.
+  \end{cases}
+\end{equation}
+}}}
+
+With appropriate branch cuts in the complex plane these expressions are
+actually the same when $\arcsin$ is expressed in terms of $\log$. The
+integration of the top loop was verified with the `FeynCalc` package and the
+notebook is provided together with the models. 
+
+Finally, given this Lagrangian the width of the new particles is:
+
+{{{
+#!latex
+\begin{equation} 
+W_{S_0}\;=\;
+\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}}
+c_{s0scalar}^2 + \left(c_{s0axial}^2 +
+c_{s0scalar}^2\right)\right)
+\end{equation}
+}}}
+
+#### Validation of the Model
+
+The first step is to compare the old and the new implementations through the
+`standalone` mode. However this is complicated by the fact that certain
+parameters in the old model are to be evaluated at each point in phase space,
+which the `standalone` mode does not permit. A short patch is provided in the
+annex with an explanation of the necessary changes.
+
+After the application of the patch, the model was validated against the old
+implementation in `standalone` mode. The decay width and the cross-section in
+various processes was validated as well, after taking into account the
+differences at runtime between `MadGraph4` and `MadGraph5`.
+
+However the old model is only for heavy $S_0$ particles ($m_{S_0}>2m_t$). The
+changes permitting work with light $S_0$ particles:
+
+- correct calculation of the width when decay to top pair is impossible
+- correct expression for the effective gluon fusion vertex
+
+were not major and were validated using the build-in tools in `FeynRules` and
+`MadGraph5`. Moreover studies for such light particles are probably of minor
+interest.
+
+### Spin Zero, Color Octet Particle 
+
+The name for this resonance is `O0` for "color
+[O]ctet, spin [Zero]". Like `S0` it is coupled only to the top with different couplings
+for the left and for the right top and there is an effective vertex of gluon fusion through
+a top loop is explicitly given in the Lagrangian as well.
+
+The operators are:
+
+{{{
+#!latex
+\begin{equation}
+\mathcal{L}_{O_0 t}\; =\;
+c_{o0scalar}\, \frac{m_t}{v} O_0^a\, \bar{t}.T^a.t \;
++ \; i\, c_{o0axial}\, \frac{m_t}{v} O_0^a\, \bar{t}.\gamma^5.T^a.t.
+\end{equation}
+
+\begin{equation}
+\mathcal{L}_{G\,fusion\,scalar\,O_0}\; =\;
+-\frac{1}{4} c_{o0fusion\,scalar} S_{SU3}^{abc} O_0^a \; FS(G)_{\mu \nu}^b \; FS(G)^{\mu \nu c}
+\end{equation}
+
+\begin{equation}
+\mathcal{L}_{G\,fusion\,axial\,O_0}\; =\;
+-\frac{1}{4} c_{o0fusion\,axial} S_{SU3}^{abc} O_0^a \; FS(G)_{\mu \nu}^b \; \widetilde{FS}(G)^{\mu \nu c}
+\end{equation}
+}}}
+
+where $S_{SU3}^{abc}$ is the completely symmetric tensor and where
+$c_{o0fusion\,scalar}$ and $c_{o0fusion\,axial}$ are the same as for `S0` with
+coupling and masses appropriately substituted.
+
+Again, given this Lagrangian the width of the new particles is:
+
+{{{
+#!latex
+\begin{equation} 
+W_{O_0}\;=\;
+\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}}
+c_{o0scalar}^2 + \left(c_{o0axial}^2 +
+c_{o0scalar}^2\right)\right)
+\end{equation}
+}}}
+
+which is $\frac{1}{6}$ times the expression for $W_{S_0}$ with appropriately
+substituted couplings and masses.
+
+#### Validation of the Model
+
+As with `S0` a patch is necessary before one can proceed with validation in the
+`standalone` mode. The model was validated against the old implementation in
+that mode, as well as in `MadEvent` mode: both the decay width and the
+cross-sections of various processes were checked.
+
+The new model permits the use of light `O0` unlike the old implementation for
+`MadGraph4`. As in the case of `S0` this part was validated only through the
+build-in tools in `FeynRules` and `MadGraph5`.
+
+### Spin One, Color Singlet Particle
+
+The name for this resonance is `S1`. It has both vector and axial couplings to
+all quarks and leptons. It is used mostly for a "model-independent" vector
+boson ($Z^\prime$). For convenience the Lagrangian has exactly the same form as
+the part of the Standard Model Lagrangian that governs the coupling of the SM Z
+to the fermions. In addition to that each coupling is parametrized by coupling
+constant with default value of 1.
+
+- `s1uleft` for the coupling to up, charm and top left quarks;
+- `s1dleft` for the coupling to down, strange and bottom left quarks;
+- `s1uright` and `s1dright` for the corresponding right quarks;
+- `s1eleft` for the left electron, muon and tau-lepton;
+- `s1eright` for the right charged leptons;
+- `s1nu` for the neutrinos.
+
+For example the coupling to neutrinos is
+
+{{{
+#!latex
+\begin{equation}
+\mathcal{L}_{S_{1}\nu}\;=\;
+c_{s1nu}\;
+\frac{e}{2\sin{\theta_W}\cos{\theta_W}}\; S_1^\mu\;
+\underset{f=e,mu,tau}{\sum}\bar{L}_2^f.\gamma_\mu.L_2^f
+\end{equation}
+}}}
+
+where $\theta_W$ is
+the Weinberg angle, $e$ is the electric coupling constant, $L$ is the leptonic
+doublet and $L_2$ is its second component.
+
+The width of the particle is calculated and provided in the model as well.
+
+#### Validation of the Model
+
+Besides the basic correctness tests provided by `FeynRules` and `MadGraph5` the
+`S1` model was verified against the original `MadGraph4` model.
+
+In `standalone` mode both models produce the same differential cross-section
+withing machine precision. In `MadEvent` mode the decay width is the same in
+both cases. When accounting for the differences at runtime in `MadGraph4` and
+`MadGraph5` the cross sections of the various tested processes are the same as
+well.
+
+### Spin One, Color Octet Particle
+
+The name for this resonance is `O1`. The need for a `FeynRules` version of it
+is what originally caused the request for reimplementation of the whole model.
+This field lives in the same representation of the gauge group as the gluons.
+It is used to represent color vector particle (coloron) or an color axial
+particle (axigluon).
+
+The Lagrangian is of the form
+
+{{{
+#!latex
+\begin{equation}
+\mathcal{L}_{O_1}\; =\;
+\sum_i c_i g_s O_1^{\mu a} \; \bar{q_i}.\gamma_\mu.T^a.q_i
+\end{equation}
+}}}
+where $i$ goes over right and left handedness of the up and down quarks of each
+generation. $T$ is the representation of the SU3 group generators and $g_s$ is
+the strong coupling constant.
+
+The width of the particles is calculated and provided in the model as well.
+
+#### Validation of the Model
+
+Similar models are discussed in [@choudhury2007top] and [@antunano2008top].
+Their results confirm both the width and the differential cross-section
+calculated in the `FeynRules` model.
+
+Another `FeynRules` model is available that implements axigluons in
+[@falkowski2012axigluon]. It produces the same vertices, however it differs in
+that it provides for a mixing between the axigluons and the gluons.
+
+The original `MadGraph4` model gives the same results in the `standalone`
+configuration. Both the decay width and the cross section of top pair
+production were checked as well. Well accounting for the differences in the
+`MadGraph4` and `MadGraph5` runtime they produce the same results. Details are
+provided in the annex.
+
+
+#### Technical Constraints
+
+During the implementation of this model a bug in the canonicalization routines
+of `FeynRules` was encountered. Whenever a tensor contraction expression is
+passes through `FeynRules` it needs to get into a canonical form (in order to
+permit equality checks, pattern matching and simplifications) before the
+canonical quantization is executed. The symmetric tensor for the SU3 group was
+not taken into account in this canonicalization. Benjamin Fuks graciously and
+quickly fixed the issue, however for the model to work correctly at least
+`FeynRules 1.7.178` or later is necessary.
+
+## General Technical Constraints
+
+### Required Versions
+
+As was mentioned above, the minimal version of `FeynRules` in which the models
+are guaranteed to work is `1.7.178`.
+
+Moreover, there is a disaccord between the formats for saving models in the
+current versions of `MadGraph5` and `FeynRules`. It should be fixed in the next
+versions, however if a runtime error message concerning undefined Goldstone
+bosons is raised by `MadGraph` it can be quickly fixed by manually modifying
+the offending lines in `particles.py`. It can be done automatically with the
+following command:
+
+`perl -pi -e 's/goldstone/GoldstoneBoson/g' ./models/topBSM_UFO/particles.py`.
+
+### Setting Mass Ranges
+
+The calculation of the widths of different particles (especially `S0` and `O0`)
+as well as the effective couplings for gluon fusion vertices changes
+qualitatively if the mass of the particle passes over or under two times the mass
+of the top. This is implemented in `FeynRules` with a delayed rewrite rule,
+however `MadGraph5` does not permit such branching. Hence if the need arises to
+change the mass of these particles it is important to change it from
+`FeynRules` and not from `MadGraph5`.
+
+# Annex
+
+## Patching the `standalone` Mode
+
+In `standalone` mode couplings are evaluated only once, before generating a
+random phase space point at which to evaluate the matrix element. This does not
+permit testing some of the more complicated models like the original
+implementation of the `S0` and `O0` particles. 
+
+As a workaround for this issue, one can modify the code so that `setparam` is
+called after each generation of random phase space points. A patch that does
+this automatically is provided with the models.