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## SM Higgs boson phenomenology

17. About the delta function influence

Status
colour Yellow 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 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 b-quark 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

LaTeX Math Inline
body \sim 9
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})

LaTeX Math Inline
body 0.167587\ \text{pb}

LaTeX Math Inline
body 1.52302\ \text{pb}

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

LaTeX Math Inline
body \hat{s}
(it includes all kinetically allowed
LaTeX Math Inline
body s
).

Here is the table for all masses:

LaTeX Math Inline
body M_A, \text{GeV}

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

LaTeX Math Inline
 body 250

LaTeX Math Inline
 body 0.167587

LaTeX Math Inline
 body 1.52302

LaTeX Math Inline
 body 9.08797

LaTeX Math Inline
 body 400

LaTeX Math Inline
 body 0.138187

LaTeX Math Inline
 body 1.32066

LaTeX Math Inline
 body 9.55702

LaTeX Math Inline
 body 1000

LaTeX Math Inline
 body 0.00017508

LaTeX Math Inline
 body 0.00127272

LaTeX Math Inline
 body 7.26935

new notebook - int.nb

15.

Status
colour Yellow to be solved

View file
name int.nb 150

• 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

LaTeX Math Inline
body \sqrt{s}
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}
,
LaTeX Math Inline
body \sigma
units are
LaTeX Math Inline
body \text{pb}

LaTeX Math Inline
body \sigma
units are
LaTeX Math Inline
body \text{GeV}^{-2} \approx 0.389379 \times 10^{9}\ \text{pb}

• Is it OK that cross section for
LaTeX Math Inline
body pp \rightarrow Zh
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 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
LaTeX Math Inline
body \frac{1}{9}
?

14. Bad behavior of the cross section. Integrals do not coverage.

Status
title solved

At small

LaTeX Math Inline
body s \rightarrow 0
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
LaTeX Math Inline
body x
, that is why integrals do not coverage. May be we got this problem because of the our assumption of zero mass for quarks?

View file
name int.nb 150

13. What are we going to compare?

Status
colour Grey solved

The SusHi+2HDMC result for the BW is function of

LaTeX Math Inline
body s
, while analytical answer will be the function of
LaTeX Math Inline
body \hat{s}
. How to compare this two results? or should they be the same?

12. Cross section formula

Status
colour Grey solved

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

LaTeX Math Inline
body s = (p_1 + p_2)^2
is the CM Energy squared and
LaTeX Math Inline
body |\bar{k}_1|
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

LaTeX Math Inline
body |\bar{k}|
we can obtain

LaTeX Math Block
|\bar{k}| \to \frac{\sqrt{-2 s m_h^2-2 m_h^2 m_Z^2+m_h^4-2 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
LaTeX Math Inline
body \sqrt{s}
, 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 )

Status
title solved

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}}{(s-m_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
LaTeX Math Inline
body s
is still the CM energy of
LaTeX Math Inline
body b\bar{b}
squared
LaTeX Math Inline
body s = (p_1 + p_2)^2
.

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

LaTeX Math Inline
body f(x)
- PDF, this function gives the probability for quark of a certain type to carry the fraction
LaTeX Math Inline
body x
of initial proton momentum. To perform the integration we should rewrite
LaTeX Math Inline
body s
in terms of fractions
LaTeX Math Inline
body x_1
and
LaTeX Math Inline
body x_2
:

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

LaTeX Math Inline
body \bar{k}_1
and
LaTeX Math Inline
body \bar{k}_2
are the proton momenta in quark CM frame
LaTeX Math Inline
body \Rightarrow

LaTeX Math Inline
body x_1 \bar{k}_1 = -x_2 \bar{k}_2

LaTeX Math Inline
body \Rightarrow
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
LaTeX Math Inline
body s
but only on factorization scale which is constant
LaTeX Math Inline
body \sim \frac{1}{4}M_A
, is it right?

9. Failed with FormCalc compilation

Status
colour Grey solved

./compile: 1: ./compile: : Permission denied

mkdir: cannot create directory ‘’: No such file or directory

Cannot create directory

Status
colour Grey solved

• What is
LaTeX Math Inline
body G^0
(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?

View file
name bb-Zh_SM.nb 150

7. V-diagram in FeynArts output - how to get rid of it?

Status
title solved

View file
name bb-Zh_SM.nb 150

6. BW for

LaTeX Math Inline
body M_A = 1000\ \text{GeV}
: the nlo-curve below while the nnlo-curve is above lo-curve (?)
Status
title solved

5. How to understand the K-factor structure?

Status
title solved

4. Discussion of the graphs for heavy Higgses and graphs for

LaTeX Math Inline
body M_A = 1000\ \text{GeV}

Status
title solved

• LaTeX Math Inline
body g_{AH^{\pm}W^{\mp}} \sim 1
, is it true?
• How to understand the onion-like structure of
LaTeX Math Inline
body A \rightarrow h^{\pm}W^{\mp}
?
• Is it right that
LaTeX Math Inline
body t\bar{t}
and
LaTeX Math Inline
body b\bar{b}
decay channels are cancel each other when we are looking at the picture of
LaTeX Math Inline
body Zh
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
LaTeX Math Inline
body 7 \times 10^2
GeV, which is suspicious (?)

View file
name 2HDMC_1000.out 150
View file
name sushi_1000.out 150

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

Status
colour Grey solved

LaTeX Math Block
anchor init_L 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. 18-19)

LaTeX Math Block
anchor final_L 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, 44-48 and definition 2.1.39 from G. Ridolfi lecture notes.

For example, in the initial Lagrangian there is a term

LaTeX Math Block
alignment center
\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 solved - for corrected files see logbook 3.2

View file
name my_file.in 150
View file
name run 150
View file
name slharoutines.pm 150