ttbar (top pair production) cross-section in different decay channels

Hi,

They are multiple reasons.

1) the narrow width approximation is … an approximation. And therefore you expect that the two number are not identical
but close to each other. (the difference should be of the order of Gamma/M)

2) You need to be careful about the value of the width that you use for the computation. Did you use the LO width value or the NLO
one. In order to have a value close to one for such computation, you should use LO value.

3) you need to set bwcutoff to a very large value, since this cuts the tale of the off-shell effects and therefore reduce significantly the results:
for example:
with (the default) bwcutoff =15: I have 504.6
while with bwcutoff =1000: I have 532.8
(with my setup the p p > t t~ cross-section is at 555 +- 0.8424 pb)

Cheers,

Olivier

On Sep 30, 2013, at 3:57 PM, Pratishruti Saha <email address hidden> wrote:

> New question #236617 on MadGraph5:
> https://answers.launchpad.net/madgraph5/+question/236617
>
> Dear MadGraph Team,
>
> I am using MadGraph5 version 1.5.12. and am facing a rather queer problem.
> I have generated the ttbar cross-section using two sets of commands :
>
> ===========================================================
> SET-1:
> import model sm
> generate p p > t t~
> ===========================================================
>
> ===========================================================
> SET-2:
> import model sm
> define l+ = l+ ta+
> define l- = l- ta-
> generate p p > t t~, (t > b w+, w+ > j j), (t~ > b~ w-, w- > l- vl~) @1
> add process p p > t t~, (t > b w+, w+ > l+ vl), (t~ > b~ w-, w- > j j) @2
> add process p p > t t~, (t > b w+, w+ > l+ vl), (t~ > b~ w-, w- > l- vl~) @3
> add process p p > t t~, (t > b w+, w+ > j j), (t~ > b~ w-, w- > j j) @4
> ===========================================================
>
> I expect to get the same value for total cross-section by both these methods
> since SET-2 includes all possible decay modes for the top pair. But I do not get that.
> To be specific, I get the total cross-section to be 603.9 pb for SET-1 and 551.1 pb for SET-2
> (using CTEQ6L1 pdfs and with renormalization and factorization scales set to 173 GeV).
>
> Is there a reason to expect the cross-sections to have different values in the two cases?
>
> I use the same run_card.dat for both SET-1 & SET-2 where all cuts have been switched off
> by setting
>
> F = cut_decays ! Cut decay products
>
> and by setting variables related to etmiss/ptll/ptheavy/ht/sorted cuts to zero/-1
> to switch them off as well as discussed in
> https://answers.launchpad.net/madgraph5/+question/236296 .
>
> In this way, I find that the ttbar cross-section in the different channels correspond
> to the correct branching fraction when compared to the total cross-section obtained
> for SET-2 but not when compared to the total cross-section obtained for SET-1.
>
> Could you shed some light on what might be the problem here?
>
>
> with regards,
> Pratishruti Saha
>
> --
> You received this question notification because you are a member of
> MadTeam, which is an answer contact for MadGraph5.

Hello Olivier,

Thank you for your response. I followed up on both of your suggestions.

1) I was, in fact, using the LO widths for the top and the W. The values were the ones in the param_card.dat to start with
     and these are in excellent agreement with the 2-body decay widths calculated using MadGraph5 itself.

2) I also tried setting larger values for bwcutoff. This does improve the agreement to a certain extent - the discrepancy is
     reduced to ~ 5% from the earlier ~9% ( I went up to bwcutoff = 100000). But it does not remove the mismatch altogether.

     Moreover, in trying this out, I noticed something very surprising. When I increase the value of bwcutoff, I would expect
     the cross-section to steadily increase and then plateau after a certain value of bwcutoff. Instead, I find that the cross-section
     first increases, then decreases, then increases again and plateaus. I am pasting my data file below. The run parameters are
     the same as decribed earlier. Only bwcutoff is varied and everything else is held constant, including the seed for the random
     number generator.

     iseed bwcutoff SET-1 SET-2
     21 15 603.88 \pm 0.796 551.1 \pm 1.4
     21 20 603.88 \pm 0.796 562.98 \pm 1.63
     21 50 603.88 \pm 0.796 578.72 \pm 1.75
     21 100 603.88 \pm 0.796 538.63 \pm 1.42
     21 150 603.88 \pm 0.796 567.75 \pm 1.72
     21 200 603.88 \pm 0.796 561.73 \pm 1.35
     21 500 603.88 \pm 0.796 576.18 \pm 1.72
     21 1000 603.88 \pm 0.796 575.14 \pm 1.81
     21 5000 603.88 \pm 0.796 575.06 \pm 1.82
     21 10000 603.88 \pm 0.796 575.06 \pm 1.82
     21 100000 603.88 \pm 0.796 575.06 \pm 1.82

     I am puzzled as to what might explain this.

3) In trying to understand all this, I calculated the 2-body and 3-body decay widths for the top using MadGraph5.
     Surprisingly enough the 2-body decay width comes out to be larger than the 3-body decay width. Further,
     if the 3-body decay is specified in two steps, the width turns out to be even lower. This, of course, would be
     an effect of the narrow-width approximation, as you said in your last response. But it is not undone by taking
     a very large value of bwcutoff. Once again my numbers are presented below. I hope you can help me to
     understand them.

    process bwcutoff width (GeV)
    t > b w+ 15 1.4915 \pm 0.00315 (GeV)
    t > b w+ 10000 1.4915 \pm 0.00315 (GeV)

    t > b j j + t > b l+ vl 15 1.4758 \pm 0.00452 (GeV)
    t > b j j + t > b l+ vl 10000 1.4758 \pm 0.00452 (GeV)

    t > b w+, w+ > j j + t > b w+, w+ > l+ vl 15 1.4501 \pm 0.00479 (GeV)
    t > b w+, w+ > j j + t > b w+, w+ > l+ vl 10000 1.4638 \pm 0.00498 (GeV)

with thanks and regards,
Pratishruti

I noticed that somehow the formatting of the data tables has got screwed up.
Is it possible to upload the files directly instead of copy-pasting them?

Here are the links to the data files.

For cross-sections :
https://docs.google.com/file/d/0Bxy_j0AOtO0jQWloNkNLMjRnTEU/edit?usp=sharing

For widths :
https://docs.google.com/file/d/0Bxy_j0AOtO0jWU9aeEFLWUMxYk0/edit?usp=sharing

Hi,

Concerning the width,

> In trying to understand all this, I calculated the 2-body and 3-body decay widths for the top using MadGraph5.

> Further, if the 3-body decay is specified in two steps, the width turns out to be even lower.

As you can see with bwcutoff =10k the difference between the two number are not not statistically significant (you have a 1 sigma deviation).
Of course with the value at 15, you observe a reduction of the width as expected since you cut on the W invariant mass.

> This, of course, would be
> an effect of the narrow-width approximation, as you said in your last response. But it is not undone by taking
> a very large value of bwcutoff.

The narrow width approximation and the value of bwcutoff are two different things.
The narrow width approximation stands the following:
Width(top, 3body) = Width(top, 2 body) * Width(W, 2 body) + O(Width_W/Mass_W)

Bwcutoff is a simple cut on the invariant mass of the W which will reduces the width/cross-section.
When this value is very large, this has no impact.

> Surprisingly enough the 2-body decay width comes out to be larger than the 3-body decay width.

This is not surprising to me since I expect that both computation should differ by an order of Width_W/Mass_W.
The difference expected is therefore of the order of 2%.
By looking at your table the difference is 1.8% so this sounds the expected type of error made by the narrow width approximation.

> 2) I also tried setting larger values for bwcutoff. This does improve the agreement to a certain extent - the discrepancy is
> reduced to ~ 5% from the earlier ~9% ( I went up to bwcutoff = 100000). But it does not remove the mismatch altogether.

Since you have two W and two top, the error from each NWA should add up together. a linear combination of the error gives a scale of expected error
of 6%. So this sounds reasonable.

> Moreover, in trying this out, I noticed something very surprising. When I increase the value of bwcutoff, I would expect
> the cross-section to steadily increase and then plateau after a certain value of bwcutoff. Instead, I find that the cross-section
> first increases, then decreases, then increases again and plateaus. I am pasting my data file below. The run parameters are
> the same as decribed earlier. Only bwcutoff is varied and everything else is held constant, including the seed for the random
> number generator.

I would guess a Phase-Space inefficiency for some the intermediate value. The value at 100 is clearly wrong, I guess that one of the numerical integration
fails for one channel of integration. This can happen sometimes.

Cheers,

Olivier

On Oct 4, 2013, at 11:26 PM, Pratishruti Saha <email address hidden> wrote:

> Question #236617 on MadGraph5 changed:
> https://answers.launchpad.net/madgraph5/+question/236617
>
> Pratishruti Saha gave more information on the question:
> Here are the links to the data files.
>
> For cross-sections :
> https://docs.google.com/file/d/0Bxy_j0AOtO0jQWloNkNLMjRnTEU/edit?usp=sharing
>
> For widths :
> https://docs.google.com/file/d/0Bxy_j0AOtO0jWU9aeEFLWUMxYk0/edit?usp=sharing
>
> --
> You received this question notification because you are a member of
> MadTeam, which is an answer contact for MadGraph5.

Hello Olivier,

Could you please clarify some of the things from your previous reply.

01. " As you can see with bwcutoff =10k the difference between the two
          number are not not statistically significant (you have a 1 sigma deviation)."

Are you referring to the following entries in the table ?

process bwcutoff width (GeV)
t > b j j + t > b l+ vl 10000 1.4758 \pm 0.00452 (GeV)

t > b w+, w+ > j j + t > b w+, w+ > l+ vl 10000 1.4638 \pm 0.00498 (GeV)

The difference here is ~ 2.4 sigma.

02. "The narrow width approximation and the value of bwcutoff are two different things."

I understood from your previous response that increasing the value for bwcutoff
implies taking more and more of the Breit-Wigner tail into account and hence
reducing the impact of the narrow width approximation. Is that not correct?

03. "The narrow width approximation stands the following:
       Width(top, 3body) = Width(top, 2 body) * Width(W, 2 body) + O(Width_W/Mass_W)"

Since the above expression is dimensionally inconsistent, I presume what you mean is
Width(top,3-body) = Width(top,2-body) * ( 1 + O(Width_W/Mass_W) )

With the narrow-width approximation, I would expect to miss the O(Width_W/Mass_W)
contribution. But that still does not explain why the 3-body width is LESS than the
2-body width. I would expect it to be AT LEAST AS MUCH as the 2-body width even with
the narrow-width approximation for the W.

One more question :
In the .lhe event file generated for (t > b j j + t > b l+ vl) with bwcutoff=15,
I find some events in which information is listed for only 4 particles, i.e.
momentum information for the W+ is missing. How is this to be interpreted?

with thanks and regards,
Pratishruti

Hi,

> The difference here is ~ 2.4 sigma.

I personally compare like this:
a-b = (1.4758 - 1.4638) +- (0.00452+0.00498)
a-b = 0.012 +- 0.0095
sigma = 0.012/ 0.0095 = 1.3

> 02. "The narrow width approximation and the value of bwcutoff are two
> different things."
>
> I understood from your previous response that increasing the value for bwcutoff
> implies taking more and more of the Breit-Wigner tail into account.

Yes this correct.

> and hence
> reducing the impact of the narrow width approximation. Is that not correct?

No this is not. First because we don't apply the narrow width approximation.
What is True is that if you want to compare the NWA with MG, you have to set this value to a very large value.
Because this is the only case where the two computations are comparable since then both methods integrates
over the full tail.
But the integration over the tail in the NWA is anyway approximate since it use the following approximation:
 \int dm |M(m)|^2 = |M(m_pole|^2*\int dm Breit-Wigner(m) + O(Gamma/m_pole)

If you want to minimize the Narrow-width approximation intrinsic error, you need to change the coupling such that the width are going to be smaller.
Then it that case, you should expect to have a better agreement between the NWA value and the exact computation done by MadGraph.

> 03. "The narrow width approximation stands the following:
> Width(top, 3body) = Width(top, 2 body) * Width(W, 2 body) + O(Width_W/Mass_W)"
>
> Since the above expression is dimensionally inconsistent, I presume what you mean is
> Width(top,3-body) = Width(top,2-body) * ( 1 + O(Width_W/Mass_W) )
>

Yes you right, I would have write it like this:
> Width(top, 3body) = Width(top, 2 body) * Width(W, 2 body) / Total_width(W) + O(Width_W/Mass_W)"

But this should be the same.

> With the narrow-width approximation, I would expect to miss the O(Width_W/Mass_W)
> contribution. But that still does not explain why the 3-body width is LESS than the
> 2-body width. I would expect it to be AT LEAST AS MUCH as the 2-body width even with
> the narrow-width approximation for the W.

Why do you expect that (AT LEAST AS MUCH)?
I have personally no idea, a priori, of what is the sign of the correction.

 One more question :
> In the .lhe event file generated for (t > b j j + t > b l+ vl) with bwcutoff=15,
> I find some events in which information is listed for only 4 particles, i.e.
> momentum information for the W+ is missing. How is this to be interpreted?

When a particle is consider to be off-shell by MG, the particle is not written in the lhe file.
This help the shower programs to have a better description of the events simulation.
Since they are then allowed to shower the events away from the w mass of the associated events and
therefore increase the simulation procedure. (Of course this is a very small effect)

Cheers,

Olivier

On Oct 15, 2013, at 10:11 PM, Pratishruti Saha <email address hidden> wrote:

> Question #236617 on MadGraph5 changed:
> https://answers.launchpad.net/madgraph5/+question/236617
>
> Status: Answered => Open
>
> Pratishruti Saha is still having a problem:
> Hello Olivier,
>
> Could you please clarify some of the things from your previous reply.
>
>
> 01. " As you can see with bwcutoff =10k the difference between the two
> number are not not statistically significant (you have a 1 sigma deviation)."
>
> Are you referring to the following entries in the table ?
>
> process bwcutoff width (GeV)
> t > b j j + t > b l+ vl 10000 1.4758 \pm 0.00452 (GeV)
>
> t > b w+, w+ > j j + t > b w+, w+ > l+ vl 10000
> 1.4638 \pm 0.00498 (GeV)
>
> The difference here is ~ 2.4 sigma.
>
>
> 02. "The narrow width approximation and the value of bwcutoff are two
> different things."
>
> I understood from your previous response that increasing the value for bwcutoff
> implies taking more and more of the Breit-Wigner tail into account and hence
> reducing the impact of the narrow width approximation. Is that not correct?
>
>
> 03. "The narrow width approximation stands the following:
> Width(top, 3body) = Width(top, 2 body) * Width(W, 2 body) + O(Width_W/Mass_W)"
>
> Since the above expression is dimensionally inconsistent, I presume what you mean is
> Width(top,3-body) = Width(top,2-body) * ( 1 + O(Width_W/Mass_W) )
>
> With the narrow-width approximation, I would expect to miss the O(Width_W/Mass_W)
> contribution. But that still does not explain why the 3-body width is LESS than the
> 2-body width. I would expect it to be AT LEAST AS MUCH as the 2-body width even with
> the narrow-width approximation for the W.
>
>
> One more question :
> In the .lhe event file generated for (t > b j j + t > b l+ vl) with bwcutoff=15,
> I find some events in which information is listed for only 4 particles, i.e.
> momentum information for the W+ is missing. How is this to be interpreted?
>
>
> with thanks and regards,
> Pratishruti
>
> --
> You received this question notification because you are a member of
> MadTeam, which is an answer contact for MadGraph5.

Hello,

Once again, thank you for the prompt response.

With regard to your question :
> Why do you expect that (AT LEAST AS MUCH)?
> I have personally no idea, a priori, of what is the sign of the correction.

Consider the width Gamma for the process t -> b 1 2.
where 1 & 2 have presumably come from the W+.
Let m12 be the invariant mass of the particles 1 & 2.
I can now compute the distribution d(Gamma)/d(m12),
summing over all possibilities for 1 & 2.
My contention is that integrating this distribution
over its entire range should give the 3-body decay-width
of the top and integrating it in the range
m12 = Mass_W \pm Width_W/2 should give the 2-body
decay-width (t -> b W+).

Would you agree?

If this picture is correct then obviously one expects
the 3-body decay-width to be larger than the 2-body
decay-width (though only by a small amount).
That's why I am confused.

regards,
Pratishruti

Hi,

> My contention is that integrating this distribution
> over its entire range should give the 3-body decay-width
> of the top

That's obviously correct.

> and integrating it in the range
> m12 = Mass_W \pm Width_W/2 should give the 2-body
> decay-width (t -> b W+).

That is just not True.
Let's write the equation
1) the top 2-body decay:
\Gamma_{top, 2body} = \int d\Phi_{t} |M_{t, 2body}|^2

so d\Phi_{t} is the phase space measure
and M_{t, 2body} is the matrix element for the top 2 body decay.

2) the w decay:
\Gamma_{W, 2body} = \int d\Phi_{W} |M_{W, 2body}|^2

so d\Phi_{W} is the phase space measure
and M_{W, 2body} is the matrix element for the W 2 body decay.

3) The three body decay of the top is:

\Gamma_{top, 3body} = \int d\Phi_{t} d\Phi_{W} dM_W/J |M_{t, 3body}|^2

M_{t, 3body} is the matrix element for the top 3 body decay.
M_W is the W mass
J is the appropriate jacobian factor

d(Gamma)/d(M_W)= \int d\Phi_{t} d\Phi_{W} |M_{t, 3body}|^2 / J

Don't see how if you integrate that function over any range you will find back the equation from 1.

Cheers,

Olivier

On Oct 16, 2013, at 7:21 PM, Pratishruti Saha <email address hidden> wrote:

> Question #236617 on MadGraph5 changed:
> https://answers.launchpad.net/madgraph5/+question/236617
>
> Status: Answered => Open
>
> Pratishruti Saha is still having a problem:
> Hello,
>
> Once again, thank you for the prompt response.
>
> With regard to your question :
>> Why do you expect that (AT LEAST AS MUCH)?
>> I have personally no idea, a priori, of what is the sign of the correction.
>
> Consider the width Gamma for the process t -> b 1 2.
> where 1 & 2 have presumably come from the W+.
> Let m12 be the invariant mass of the particles 1 & 2.
> I can now compute the distribution d(Gamma)/d(m12),
> summing over all possibilities for 1 & 2.
> My contention is that integrating this distribution
> over its entire range should give the 3-body decay-width
> of the top and integrating it in the range
> m12 = Mass_W \pm Width_W/2 should give the 2-body
> decay-width (t -> b W+).
>
> Would you agree?
>
> If this picture is correct then obviously one expects
> the 3-body decay-width to be larger than the 2-body
> decay-width (though only by a small amount).
> That's why I am confused.
>
>
> regards,
> Pratishruti
>
> --
> You received this question notification because you are a member of
> MadTeam, which is an answer contact for MadGraph5.

Hello,

> Don't see how if you integrate that function over any range you will find back the equation from 1.

Well, it is certainly not demostrated by writing the widths schematically as above.
However, this does not tell us that once the expressions for |M|^2 are substituted
and the integrals are done the answers will not come out to be the same.
That, of course, is a somewhat tedious calculation.

However, just intuitively, when m12 approaches Mass_W, the decay is effectively
a 2-body decay. And hence, in that limit I would expect the 3-body width to approach
the 2-body width. Where is the flaw in this?

regards,
Pratishruti

Hi,

> Well, it is certainly not demostrated by writing the widths schematically as above.
> However, this does not tell us that once the expressions for |M|^2 are substituted
> and the integrals are done the answers will not come out to be the same.
> That, of course, is a somewhat tedious calculation.

Sure we can do physics like that, they are no proof that those two numbers are not different so let assume that they are the same!
I agree that sometimes it might be the way to go but then you should at least present some examples where it works otherwise this is
not science. Your idea to integrate from Mass_W \pm Width_W/2 is something that I have never heard about it and without any strong
justification, I will never (and nobody) buy it.

> However, just intuitively, when m12 approaches Mass_W, the decay is effectively
> a 2-body decay. And hence, in that limit I would expect the 3-body width to approach
> the 2-body width. Where is the flaw in this?

I agree with the idea of the argument but
1) When you performed the 3-body integration, you need to integrate over the full mass-spectrum and this means that your
argument didn't apply for the full phase-space.
2) This argument gives the intuition that the number should be of the same order but not that they should be identical (especially because of my first point)
3) This arguments didn't justify at all why to integrate between Mass_W \pm Width_W/2 why not Mass_W \pm Width_W/4?

Cheers,

Olivier

On Oct 22, 2013, at 10:06 PM, Pratishruti Saha <email address hidden> wrote:

> Question #236617 on MadGraph5 changed:
> https://answers.launchpad.net/madgraph5/+question/236617
>
> Status: Answered => Open
>
> Pratishruti Saha is still having a problem:
> Hello,
>
>> Don't see how if you integrate that function over any range you will
> find back the equation from 1.
>
> Well, it is certainly not demostrated by writing the widths schematically as above.
> However, this does not tell us that once the expressions for |M|^2 are substituted
> and the integrals are done the answers will not come out to be the same.
> That, of course, is a somewhat tedious calculation.
>
> However, just intuitively, when m12 approaches Mass_W, the decay is effectively
> a 2-body decay. And hence, in that limit I would expect the 3-body width to approach
> the 2-body width. Where is the flaw in this?
>
>
> regards,
> Pratishruti
>
> --
> You received this question notification because you are a member of
> MadTeam, which is an answer contact for MadGraph5.

Hello,

Thank you once again for the detailed reply.

regards,
Pratishruti