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SM Higgs boson phenomenology

17. About the delta function influence

Status
colourYellow
titleto 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
colourYellow
titleTo 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
body2\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
body0.167587\ \text{pb}

LaTeX Math Inline
body1.52302\ \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
bodys
).

Here is the table for all masses:

LaTeX Math Inline
bodyM_A, \text{GeV}


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

LaTeX Math Inline
body2\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
body250

LaTeX Math Inline
body0.167587

LaTeX Math Inline
body1.52302

LaTeX Math Inline
body9.08797

LaTeX Math Inline
body400

LaTeX Math Inline
body0.138187

LaTeX Math Inline
body1.32066

LaTeX Math Inline
body9.55702

LaTeX Math Inline
body1000

LaTeX Math Inline
body0.00017508

LaTeX Math Inline
body0.00127272

LaTeX Math Inline
body7.26935

new notebook - int.nb

15.

Status
colourYellow
titleto be solved


View file
nameint.nb
height150

  • 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
    bodypp \rightarrow Zh
    is higher than
    LaTeX Math Inline
    bodyb\bar{b} \rightarrow Zh
    ? - not clear
  • There is a problem with units

10. Question on FormCalc

Status
colourYellow
titleTo 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
titlesolved

At small

LaTeX Math Inline
bodys \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
bodyx
, 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
nameint.nb
height150

13. What are we going to compare?

Status
colourGrey
titlesolved

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

LaTeX Math Inline
bodys
, 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
colourGrey
titlesolved

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
bodys = (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 (sad))

Status
titlesolved

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
bodys
is still the CM energy of 
LaTeX Math Inline
bodyb\bar{b}
squared
LaTeX Math Inline
bodys = (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
bodyf(x)
- PDF, this function gives the probability for quark of a certain type to carry the fraction
LaTeX Math Inline
bodyx
of initial proton momentum. To perform the integration we should rewrite
LaTeX Math Inline
bodys
in terms of fractions
LaTeX Math Inline
bodyx_1
and
LaTeX Math Inline
bodyx_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
bodyx_1 \bar{k}_1 = -x_2 \bar{k}_2
 
LaTeX Math Inline
body\Rightarrow
LaTeX Math Inline
bodyx_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
    bodys
    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
colourGrey
titlesolved

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

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

Cannot create directory

8. Questions about FeynArts:

Status
colourGrey
titlesolved

  • What is
    LaTeX Math Inline
    bodyG^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
namebb-Zh_SM.nb
height150

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

Status
titlesolved

View file
namebb-Zh_SM.nb
height150

6. BW for

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

5. How to understand the K-factor structure?

Status
titlesolved

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

LaTeX Math Inline
bodyM_A = 1000\ \text{GeV}
 
Status
titlesolved

  • LaTeX Math Inline
    bodyg_{AH^{\pm}W^{\mp}} \sim 1
    , is it true?
  • How to understand the onion-like structure of
    LaTeX Math Inline
    bodyA \rightarrow h^{\pm}W^{\mp}
    ?
  • Is it right that 
    LaTeX Math Inline
    bodyt\bar{t}
    and
    LaTeX Math Inline
    bodyb\bar{b}
    decay channels are cancel each other when we are looking at the picture of
    LaTeX Math Inline
    bodyZh
    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
bodyM_A = 1000\ \text{GeV}
Status
titlesolved - see the logbook section 4

Find attached sushi and 2HDMC output files for

LaTeX Math Inline
bodyM_A = 1000\ \text{GeV}
. According to those outputs total width has value in order of
LaTeX Math Inline
body7 \times 10^2
GeV, which is suspicious (?)

View file
name2HDMC_1000.out
height150
View file
namesushi_1000.out
height150

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

Status
colourGrey
titlesolved

LaTeX Math Block
anchorinit_L
alignmentcenter
\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
anchorfinal_L
alignmentcenter
\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
alignmentcenter
\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
colourGrey
titlesolved - for corrected files see logbook 3.2

View file
namemy_file.in
height150
View file
namerun
height150
View file
nameslharoutines.pm
height150