SM Higgs boson phenomenology
17. About the delta function influence
Status 

colour  Yellow 

title  to be solved 


We are interested in
LaTeX Math Inline 

body  \sigma (pp \rightarrow Zh)(s) = \int_0^1 dx_1 dx_2\ f_1(x_1) f_2 (x_2)\ \delta(\hat{s}x_1 x_2  s)\ \sigma (b\bar{b} \rightarrow Zh)(\hat{s}x_1 x_2) 


, which can be rewritten:
LaTeX Math Block 

\sigma (pp \rightarrow Zh)(s) = \sigma (b\bar{b} \rightarrow Zh)(s) \int_0^1 \frac{dx_1}{\hat{s}x_1} f_1(x_1) f_2 (\frac{s}{\hat{s}}x_1)\ \theta(\hat{s}x_1  s) 
This is probably also wrong, since this integration is giving result
LaTeX Math Inline 

body  \sim 10^{3}\ \text{pb} 


16.
Status 

colour  Yellow 

title  To be solved 


Mathematica notebook  int.nb
Notebook by Stefan:int_SL.nb
2HDMC & SusHi out files  2HDMC_250.out & sushi_250.out & sushi_400.out & sushi_1000.out
The employed bquark masses are
for mA=250GeV: 2.8831183569691126 GeV
for mA=400GeV: 2.7980373758125761 GeV
for mA=1000GeV: 2.6512880401491685 GeV
Results for cross section differ
times:
LaTeX Math Inline 

body  \sigma_{\text{SusHi}}(250\ \text{GeV}) \times \text{BR} 


 LaTeX Math Inline 

body  2\int dx_1 dx_2 f_1(x_1) f_2(x_2) \sigma_{\text{anal}} (x_1 x_2 \hat{s}) 




 
But I am not sure whatever should I trust it or not since NIntegrate is giving the following warnings:
Also I am not sure that I understand why those two answers for cross section should be the same. The first one is just hadronic cross section for one particular invariant mass 250 Gev, while the second one is something like a total cross section for given hadron CM energy
(it includes all kinetically allowed
).
Here is the table for all masses:
 LaTeX Math Inline 

body  \sigma_{\text{SusHi}}(M_A) \times \text{BR} 


 LaTeX Math Inline 

body  2\int dx_1 dx_2 f_1(x_1) f_2(x_2) \sigma_{\text{anal}} (x_1 x_2 \hat{s}) 


 ratio 

   
   
   
new notebook  int.nb
15.
Status 

colour  Yellow 

title  to be solved 


 Does
LaTeX Math Inline 

body  \frac{d\sigma}{d\sqrt{s}} 


in the BW approximation formula really mean the cross section derivative? I am wondering since I have got the following plots, where the pattern of the analytical cross section fits the LaTeX Math Inline 

body  \frac{d\sigma}{d\sqrt{s}} 


pattern from BW.
The result of BW formula and the integrated over result
 Bare analytical result and after the PDF's integration


LaTeX Math Inline 

body  \frac{d\sigma}{d\sqrt{s}} 


units are LaTeX Math Inline 

body  \text{pb} \times \text{GeV}^{1} 


, units are  units are LaTeX Math Inline 

body  \text{GeV}^{2} \approx 0.389379 \times 10^{9}\ \text{pb} 



 Is it OK that cross section for is higher than
LaTeX Math Inline 

body  b\bar{b} \rightarrow Zh 


?  not clear  There is a problem with units
10. Question on FormCalc
Status 

colour  Yellow 

title  To be solved 


 When the sum over polarizations is done is the numerical factor also taken into account by FormCalc?  seems like not, it calculates just bare polarization sum
 Since Z boson has 3 polarization states, should I multiply the polarization sum by ?
14. Bad behavior of the cross section. Integrals do not coverage.
At small
the analytical expression for the cross section behaves like
LaTeX Math Inline 

body  \sigma \sim \frac{1}{s} = \frac{1}{x_1 x_2 \hat{s}} 


, the PDFs are also rise at small
, that is why integrals do not coverage. May be we got this problem because of the our assumption of zero mass for quarks?
13. What are we going to compare?
The SusHi+2HDMC result for the BW is function of
, while analytical answer will be the function of
. How to compare this two results? or should they be the same?
12. Cross section formula
According to Peskin's book the cross section is given with
LaTeX Math Block 

d \sigma = \frac{1}{2s}\times \frac{\bar{k}_1}{16 \pi^2 \sqrt{s}}\times M(s)^2d \Omega, 
where
is the CM Energy squared and
is the magnitude of the momentum of one of the outgoing particles. In our case we do not have an angle dependence, so after angular integration
LaTeX Math Block 

\sigma = \frac{1}{s}\times \frac{\bar{k}_1}{8 \pi \sqrt{s}}\times M(s)^2. 
Questions:
 What to do with outgoing momentum in the formula?
LaTeX Math Block 

s = \left[
\begin{pmatrix}
E_Z\\
\bar{k}
\end{pmatrix} +
\begin{pmatrix}
E_h\\
\bar{k}
\end{pmatrix}
\right]^2 = (E_Z + E_h)^2 = m_Z^2 + m_h^2 + 2\bar{k}^2 + 2\sqrt{m_Z^2 + \bar{k}^2}\sqrt{m_h^2 + \bar{k}^2} 
for
we can obtain
LaTeX Math Block 

\bar{k} \to \frac{\sqrt{2 s m_h^22 m_h^2 m_Z^2+m_h^42 s m_Z^2+m_Z^4+s^2}}{2 \sqrt{s}} 
 Should we now substitute this expression in
LaTeX Math Inline 

body  \sigma(b\bar{b} \rightarrow Zh)(s) 


and differentiate with respect to , since we are interested in LaTeX Math Inline 

body  \frac{d \sigma (b\bar{b} \rightarrow Zh)}{d \sqrt{s}} 


?  Why
LaTeX Math Inline 

body  \frac{d \sigma (b\bar{b} \rightarrow Zh)}{d \sqrt{s}} 


not just LaTeX Math Inline 

body  \sigma(b\bar{b} \rightarrow Zh)(s) 


?
11. Difficulties with the BW formula (yes, again )
BW formula is given by
LaTeX Math Block 

\frac{d\sigma(b\bar b\to ZH_{125})}{d\sqrt{s}} = \sigma(b\bar b\to A)(s)\frac{2\sqrt{s}}{(sm_A^2)^2+m_A^2\Gamma_A^2}\frac{\sqrt{s}\Gamma(A\to ZH_{125})(s)}{\pi} 
When I applied it previously I used SusHi results and, since SusHi integrates cross section with parton distribution functions, I actually used
LaTeX Math Inline 

body  \sigma \left(pp \rightarrow A \right) 


. That is why probably it is better to say that I have calculated
LaTeX Math Inline 

body  \frac{d \sigma (pp \rightarrow Zh)}{d \sqrt{s}} 


, where
is still the CM energy of
squared
.
Now I am trying to understand what is
LaTeX Math Inline 

body  \sigma \left(pp \rightarrow A \right) 


:
LaTeX Math Block 

\sigma(pp \rightarrow A) = \int_0^1 dx_1 dx_2 f_b(x_1) f_{\bar{b}}(x_2) \sigma(b\bar{b} \rightarrow A)(s), 
where
 PDF, this function gives the probability for quark of a certain type to carry the fraction
of initial proton momentum. To perform the integration we should rewrite
in terms of fractions
and
:
LaTeX Math Block 

s = (p_1 + p_2)^2 = \left(
\begin{pmatrix}
\frac{1}{2}E_{CM} \\
x_1\bar{k}_1
\end{pmatrix} +
\begin{pmatrix}
\frac{1}{2}E_{CM} \\
x_2\bar{k}_2
\end{pmatrix}
\right)^2 = E_{CM}^2  (x_1 \bar{k}_1 + x_2 \bar{k}_2)^2 
here
and
are the proton momenta in quark CM frame
LaTeX Math Inline 

body  x_1 \bar{k}_1 = x_2 \bar{k}_2 


LaTeX Math Inline 

body  x_1 \bar{k}_1 = x_2 \bar{k}_2 


and we are obtaining the following result:
LaTeX Math Block 

\sigma (pp \rightarrow A) = \sigma(b\bar{b} \rightarrow A)(s) \times \int_0^1 d x f_b(x) f_\bar{b} \left(x\frac{\bar{k}_1}{\bar{k}_2}\right) 
Questions:
 Is it correct?
 I've got the dependence on
LaTeX Math Inline 

body  \frac{\bar{k}_1}{\bar{k}_2} 


, how to get rid of it?  I was assuming that quark from the first proton and antiquark from the second proton participate in the reaction, but the opposite situation is also possible how to take this into account correctly? (multiply by factor 2?)
 Here I was assuming that PDFs do not depend on but only on factorization scale which is constant , is it right?
9. Failed with FormCalc compilation
./compile: 1: ./compile: : Permission denied
mkdir: cannot create directory ‘’: No such file or directory
Cannot create directory
8. Questions about FeynArts:
 What is (scalar line)?
 What is the logic in particles numeration? Is there any table of particles and codes?
 How to extract amplitude from the FeynAmpList?
 How to make FeynArts output amplitudes readable? {Tried to use FeynCalc  FCFAConver[] }
 What are EL, MW, CW, SW?
7. Vdiagram in FeynArts output  how to get rid of it?
6. BW for
LaTeX Math Inline 

body  M_A = 1000\ \text{GeV} 


: the nlocurve below while the nnlocurve is above locurve (?) 5. How to understand the Kfactor structure?
4. Discussion of the graphs for heavy Higgses and graphs for
LaTeX Math Inline 

body  M_A = 1000\ \text{GeV} 


LaTeX Math Inline 

body  g_{AH^{\pm}W^{\mp}} \sim 1 


, is it true? How to understand the onionlike structure of
LaTeX Math Inline 

body  A \rightarrow h^{\pm}W^{\mp} 


?  Is it right that and decay channels are cancel each other when we are looking at the picture of channel? (they have different magnitude)
 How to understand the heavy Higgs graphs for BR? Why Higgses masses changed them in this way?
 What other channels might have a significant contribution?
3. Total decay width for
LaTeX Math Inline 

body  M_A = 1000\ \text{GeV} 


Status 

title  solved  see the logbook section 4 


Find attached sushi and 2HDMC output files for
LaTeX Math Inline 

body  M_A = 1000\ \text{GeV} 


. According to those outputs total width has value in order of GeV, which is suspicious (?)
View file 

name  2HDMC_1000.out 

height  150 


View file 

name  sushi_1000.out 

height  150 


1. I've a problem when making a transition from the Lagrangian
LaTeX Math Block 

anchor  init_L 

alignment  center 


\begin{equation}
\mathcal{L}^{hadr}_Y =  \left(\bar{Q}^\prime \phi {h_D}^\prime D^\prime + \bar{D}^\prime \phi_c^\dagger {{h_D}^\prime}^\dagger Q^\prime\right)  \left(\bar{Q}^\prime \phi {h_U}^\prime U^\prime + \bar{U}^\prime \phi_c^\dagger {{h_U}^\prime}^\dagger Q^\prime\right)
\end{equation} 
to the Lagrangian (see lecture notes by G. Ridolfi pp. 1819)
LaTeX Math Block 

anchor  final_L 

alignment  center 


\begin{equation}
\mathcal{L}^{hadr}_Y =  \frac{1}{\sqrt{2}} \left(v + H \right) \sum_{f = 1}^n \left( h_D^f \bar{d}^f d^f + h_U^f \bar{u}^f u^f \right).
\end{equation} 
To do this transition I use eq. 2.2.42, 4448 and definition 2.1.39 from G. Ridolfi lecture notes.
For example, in the initial Lagrangian there is a term
LaTeX Math Block 


\bar{Q}^\prime \phi {h_D}^\prime D^\prime = \left( \bar{u}_L^\prime \bar{d}_L^\prime\right) \frac{1}{\sqrt{2}} \begin{pmatrix}0 \\ v + H(x)\end{pmatrix} V_L^D h_D {V_R^D}^\dagger d_R^\prime = \frac{1}{\sqrt{2}} (v + H) \bar{d}_L^\prime V_L^D h_D {V_R^D}^\dagger d_R^\prime = \frac{1}{\sqrt{2}} (v + H) \color{Orange}{\bar{d}_L^\prime V_L^D} h_D d_R\\
d_L^\prime = V_L^D d_L \rightarrow \bar{d}_L^\prime = d_L^\dagger {V_L^D}^\dagger \gamma_0 \rightarrow \color{Orange}{\bar{d}_L^\prime V_L^D} = d_L^\dagger {V_L^D}^\dagger \gamma_0 V_L^D. 
2. Problem with the script
Status 

colour  Grey 

title  solved  for corrected files see logbook 3.2 


View file 

name  slharoutines.pm 

height  150 

