wiki:SingleTopNLO

Version 3 (modified by md987, 8 years ago) (diff)

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t-channel single top production at NLO

Motivation

There are two ways to calculate t-channel single top production. The first is using a 2->2 process where the b-quark is taken in the initial state. By taking the b-quark from the PDF and setting the factorization scale equal to the top mass, the logarithms $\log(m_b/m_t)$ will be resummed to all orders. T-channel single top production can also be calculated without the b-quark PDF by including the gluon splitting into the diagram. The process becomes then a 2->3 process with also a massive b-quark in the final state. The goal is to calculate the the NLO corrections to the 2->3 process and provide in this way reliable predictions for the kinematics of this b-quark.

Talks

Single top talks

Results

The parameters used are

  • $m_t=172$ GeV
  • $m_b=4.7$ GeV
  • $m_W=80.419$ GeV
  • $m_Z=91.118$ GeV
  • $\mu_R=\mu_F=m_t$
  • PDF is CTEQ6L (LO) and CTEQ6M (NLO)
  • Jet definition: kT jet algorithm, with ptjetmin=15 GeV, etajetmax=8, Rcut=0.7, Inclusive.
  • no cuts whatsoever are applied

if not stated otherwise.

total cross section

Default scale choices:

  • For $2\to 2$ we choose $\mu_R^l=\mu_F^l=\mu_R^h=\mu_F^h=m_t$.
  • For $2\to 3$ we choose $\mu_R^l=\mu_F^l=m_t/2$ and $\mu_R^h=\mu_F^h=m_t/4$.
$\sigma(2\to 2)$ LO NLO k-factor
Tevatron (fb) 809.1 +- 0.6 891.9 +- 0.7 1.10
LHC top (pb) 142.1 +- 0.1 146.8 +- 0.1 1.03
LHC anti-top (pb) 81.93 +- 0.05 86.43 +- 0.06 1.05
$\sigma(2\to 3)$ LO NLO k-factor
Tevatron (fb) 595.8 +- 0.7 800 +- 1 1.34
LHC top (pb) 132.8 +- 0.3 132.0 +- 0.5 0.99
LHC anti-top (pb) 74.85 +- 0.15 76.3 +- 2 1.02

Note in particular the small k-factor for the 2->2 process and the large k-factor for the 2->3 at the tevatron. In the PDF's a bottom mass of 4.5 GeV is used. Lowering the bottom mass will slightly increase the cross section for the 2->3 process.

scale dependence

As the central choices for the renormalization and factorization scales we have chosen:

  • for the heavy line: $\mu_{F,R}^{h}=m_t/4$. This is chosen from the previous runs, where for $m_t/4$ the total cross section for the 2->3 is closest to the 2->2 and also the LO is very close to the NLO.
  • for the light line: $\mu_{F,R}^{l}=m_t/2$. This is close to the average value for the $Q^2$ of the W boson, which would be the natural choice for the massless line if we could use event-by-event scale choises. Also here the LO is very close to the NLO.

The results for the scale dependence can be found in the tables below:

light quark line $\mu_0=m_t/2$ heavy quark line $\mu_0=m_t/4$ 2->3 at NLO, LHC top 2->3 at NLO, LHC anti-top 2->3 at NLO, Tevatron top
ren. scale fac. scale ren. scale fac. scale tag cross section tag cross section tag cross section
1 1 1 1 LHCt_89 132.0 +- 0.5 LHCa_89 76.0 +- 0.3 tev_89 800 +- 1
1 1 1 1 LHCt_90 131.1 +- 0.6 LHCa_90 76.5 +- 0.4 tev_90 800 +- 1
1 1 1 1 LHCt_91 131.6 +- 0.6 LHCa_91 76.0 +- 0.3 tev_91 800 +- 1
1 1 1 1 LHCt_92 131.8 +- 0.5 LHCa_92 76.6 +- 0.3 tev_92 800 +- 1
1 1/2 1 1 LHCt_L1 131.3 +- 0.5 LHCa_L1 74.9 +- 0.4 tev_L1 779 +- 1
1/2 1/2 1 1 LHCt_L2 130.7 +- 0.5 LHCa_L2 74.9 +- 0.3 tev_L2 778 +- 1
2 1 1 1 LHCt_L3 131.8 +- 0.5 LHCa_L3 76.1 +- 0.3 tev_L3 799 +- 1
1/2 1 1 1 LHCt_L4 130.8 +- 0.6 LHCa_L4 76.1 +- 0.3 tev_L4 803 +- 1
2 2 1 1 LHCt_L5 131.3 +- 0.5 LHCa_L5 76.9 +- 0.3 tev_L5 813 +- 1
1 2 1 1 LHCt_L6 132.5 +- 0.6 LHCa_L6 76.7 +- 0.4 tev_L6 815 +- 1
1 1 1 1/2 LHCt_H1 133.1 +- 0.8 LHCa_H1 78.0 +- 0.4 tev_H1 860 +- 1
1 1 1/2 1/2 LHCt_H2 138.1 +- 0.8 LHCa_H2 79.1 +- 0.4 tev_H2 901 +- 2
1 1 2 1 LHCt_H3 126.3 +- 0.4 LHCa_H3 73.7 +- 0.3 tev_H3 752 +- 1
1 1 1/2 1 LHCt_H4 133.7 +- 0.7 LHCa_H4 76.8 +- 0.7 tev_H4 846 +- 1
1 1 2 2 LHCt_H5 125.7 +- 0.4 LHCa_H5 73.1 +- 0.3 tev_H5 704 +- 1
1 1 1 2 LHCt_H6 130.2 +- 0.6 LHCa_H6 76.0 +- 0.3 tev_H6 752 +- 1
1/4 1/4 1 1 LHCt_Ls1 130.2 +- 0.6 LHCa_Ls1 74.6 +- 0.3 tev_Ls1 729 +- 1
1/8 1/8 1 1 LHCt_Ls2 128.2 +- 0.6 LHCa_Ls2 74.8 +- 0.4 tev_Ls2 640 +- 2
4 4 1 1 LHCt_Ls3 132.1 +- 0.5 LHCa_Ls3 77.5 +- 0.4 tev_Ls3 812 +- 1
8 8 1 1 LHCt_Ls4 132.6 +- 0.5 LHCa_Ls4 78.8 +- 0.3 tev_Ls4 810 +- 1
1 1 1/4 1/4 LHCt_Hs1 143.3 +- 0.9 LHCa_Hs1 83.4 +- 0.8 tev_Hs1 972 +- 2
1 1 4 4 LHCt_Hs2 120.6 +- 0.4 LHCa_Hs2 69.2 +- 0.2 tev_Hs2 616 +- 1
1 1 8 8 LHCt_Hs3 114.9 +- 0.3 LHCa_Hs3 65.9 +- 0.2 tev_Hs3 542 +- 1
1 1 16 16 LHCt_Hs4 109.6 +- 0.3 LHCa_Hs4 62.5 +- 0.2 tev_Hs4 480 +- 1
1/8 1/8 1/4 1/4 LHCt_LH1 135.5 +- 0.9 LHCa_LH1 75.8 +- 0.5 tev_LH1 564 +- 2
1/4 1/4 1/2 1/2 LHCt_LH2 134.8 +- 0.7 LHCa_LH2 76.9 +- 0.5 tev_LH2 784 +- 2
2 2 4 4 LHCt_LH3 120.3 +- 0.4 LHCa_LH3 69.5 +- 0.2 tev_LH3 617 +- 1
4 4 8 8 LHCt_LH4 113.8 +- 0.3 LHCa_LH4 66.1 +- 0.2 tev_LH4 534 +- 1
8 8 16 16 LHCt_LH5 107.5 +- 0.3 LHCa_LH5 62.3 +- 0.2 tev_LH5 462 +- 1
light quark line $\mu_0=m_t/2$ heavy quark line $\mu_0=m_t/4$ 2->3 at LO, LHC top 2->3 at LO, LHC anti-top 2->3 at LO, Tevatron top
ren. scale fac. scale ren. scale fac. scale tag cross section tag cross section tag cross section
1/8 1/8 1/4 1/4 LHCt_1 194.8 +- 0.5 LHCa_1 108.4 +- 0.2 tev_1 1368 +- 2
1/4 1/4 1/2 1/2 LHCt_2 162.0 +- 0.4 LHCa_2 90.37 +- 0.19 tev_2 916.0 +- 1.1
1/2 1/2 1 1 LHCt_3 136.7 +- 0.3 LHCa_3 76.05 +- 0.14 tev_3 645.6 +- 0.8
1 1 2 2 LHCt_4 116.2 +- 0.3 LHCa_4 64.74 +- 0.12 tev_4 473.7 +- 0.6
2 2 4 4 LHCt_5 100.7 +- 0.2 LHCa_5 55.81 +- 0.10 tev_5 358.7 +- 0.4
4 4 8 8 LHCt_6 87.13 +- 0.18 LHCa_6 48.49 +- 0.09 tev_6 279.5 +- 0.3
8 8 16 16 LHCt_7 76.56 +- 0.15 LHCa_7 42.52 +- 0.08 tev_7 222.1 +- 0.3
1/8 1/8 1 1 LHCt_L1 146.5 +- 0.4 LHCa_L1 78.46 +- 0.17 tev_L1 782.1 +- 1.0
1/4 1/4 1 1 LHCt_L2 141.0 +- 0.3 LHCa_L2 77.21 +- 0.16 tev_L2 705.3 +- 0.9
1 1 1 1 LHCt_L3 132.8 +- 0.3 LHCa_L3 74.85 +- 0.15 tev_L3 595.8 +- 0.7
2 2 1 1 LHCt_L4 129.1 +- 0.3 LHCa_L4 73.46 +- 0.14 tev_L4 554.6 +- 0.7
4 4 1 1 LHCt_L5 125.9 +- 0.3 LHCa_L5 72.42 +- 0.14 tev_L5 518.6 +- 0.6
8 8 1 1 LHCt_L6 122.8 +- 0.3 LHCa_L6 71.36 +- 0.13 tev_L6 488.3 +- 0.6
1 1 1/4 1/4 LHCt_H1 180.0 +- 0.4 LHCa_H1 105.6 +- 0.2 tev_H1 1050 +- 1
1 1 1/2 1/2 LHCt_H2 153.0 +- 0.4 LHCa_H2 87.85 +- 0.18 tev_H2 773.4 +- 0.9
1 1 1 1 LHCt_H3 132.8 +- 0.3 LHCa_H3 74.85 +- 0.15 tev_H3 595.8 +- 0.7
1 1 4 4 LHCt_H4 103.5 +- 0.2 LHCa_H4 56.88 +- 0.11 tev_H4 386.9 +- 0.5
1 1 8 8 LHCt_H5 92.68 +- 0.22 LHCa_H5 50.59 +- 0.10 tev_H5 322.0 +- 0.4
1 1 16 16 LHCt_H6 83.93 +- 0.18 LHCa_H6 45.23 +- 0.09 tev_H6 272.3 +- 0.3
For both quark lines $\mu_0=m_t$, massive b-quark 2->2 at NLO, LHC top 2->2 at NLO, LHC anti-top 2->2 at NLO, Tevatron top
ren. scale fac. scale tag cross section tag cross section tag cross section
1/16 1/16 LHCt_1 195.7 +- 0.5 LHCa_1 110.3 +- 0.2 tev_1 1241 +- 2
1/8 1/8 LHCt_2 170.5 +- 0.4 LHCa_2 96.65 +- 0.19 tev_2 927.6 +- 1.1
1/4 1/4 LHCt_3 157.1 +- 0.3 LHCa_3 89.10 +- 0.14 tev_3 833.3 +- 0.8
1/2 1/2 LHCt_4 150.7 +- 0.3 LHCa_4 85.87 +- 0.12 tev_4 823.4 +- 0.6
1 1 LHCt_5 149.9 +- 0.2 LHCa_5 85.46 +- 0.10 tev_5 841.4 +- 0.4
2 2 LHCt_6 151.0 +- 0.2 LHCa_6 86.79 +- 0.09 tev_6 868.2 +- 0.3
4 4 LHCt_7 154.7 +- 0.2 LHCa_7 89.24 +- 0.08 tev_7 891.0 +- 0.3
1 1/2 LHCt_8 156.5 +- 0.2 LHCa_8 89.19 +- 0.12 tev_8 822.4 +- 0.5
2 1 LHCt_9 148.5 +- 0.2 LHCa_9 85.29 +- 0.10 tev_9 841.8 +- 0.4
1/2 1 LHCt_10 142.3 +- 0.2 LHCa_10 81.07 +- 0.11 tev_10 843.0 +- 0.5
1 2 LHCt_11 151.0 +- 0.2 LHCa_11 86.36 +- 0.10 tev_11 870.1 +- 0.4
For both quark lines $\mu_0=m_t$ 2->2 at NLO, LHC top 2->2 at NLO, LHC anti-top 2->2 at NLO, Tevatron top
ren. scale fac. scale tag cross section tag cross section tag cross section
1/16 1/16 LHCt_1 167.6 +- 0.1 LHCa_1 99.02 +- 0.07 tev_1 1257.3 +- 1.0
1/8 1/8 LHCt_2 153.3 +- 0.1 LHCa_2 90.58 +- 0.07 tev_2 977.7 +- 0.8
1/4 1/4 LHCt_3 146.6 +- 0.1 LHCa_3 86.51 +- 0.06 tev_3 890.7 +- 0.7
1/2 1/2 LHCt_4 145.1 +- 0.1 LHCa_4 85.47 +- 0.06 tev_4 877.9 +- 0.7
1 1 LHCt_5 146.8 +- 0.1 LHCa_5 86.43 +- 0.06 tev_5 891.9 +- 0.7
2 2 LHCt_6 150.6 +- 0.1 LHCa_6 88.65 +- 0.07 tev_6 911.0 +- 0.7
4 4 LHCt_7 155.5 +- 0.1 LHCa_7 91.64 +- 0.07 tev_7 929.2 +- 0.7
1 1/2 LHCt_8 144.2 +- 0.1 LHCa_8 84.93 +- 0.06 tev_8 877.7 +- 0.7
2 1 LHCt_9 147.3 +- 0.1 LHCa_9 86.68 +- 0.06 tev_9 891.0 +- 0.7
1/2 1 LHCt_10 146.3 +- 0.1 LHCa_10 86.17 +- 0.07 tev_10 892.1 +- 0.7
1 2 LHCt_11 149.2 +- 0.1 LHCa_11 87.91 +- 0.07 tev_11 914.3 +- 0.7
For both quark lines $\mu_0=m_t$ 2->2 at LO, LHC top 2->2 at LO, LHC anti-top 2->2 at LO, Tevatron top
ren. scale fac. scale tag cross section tag cross section tag cross section
1/16 1/16 LHCt_1 51.98 +- 0.03 LHCa_1 29.28 +- 0.02 tev_1 512.9 +- 0.4
1/8 1/8 LHCt_2 83.35 +- 0.06 LHCa_2 47.35 +- 0.03 tev_2 692.0 +- 0.5
1/4 1/4 LHCt_3 107.8 +- 0.1 LHCa_3 61.65 +- 0.04 tev_3 774.8 +- 0.6
1/2 1/2 LHCt_4 127.0 +- 0.1 LHCa_4 73.01 +- 0.05 tev_4 806.5 +- 0.6
1 1 LHCt_5 142.1 +- 0.1 LHCa_5 81.94 +- 0.05 tev_5 809.1 +- 0.6
2 2 LHCt_6 154.1 +- 0.1 LHCa_6 89.04 +- 0.06 tev_6 797.4 +- 0.6
4 4 LHCt_7 163.4 +- 0.1 LHCa_7 94.65 +- 0.06 tev_7 776.1 +- 0.6
1 1/2 LHCt_8 127.0 +- 0.1 LHCa_8 73.01 +- 0.05 tev_8 806.5 +- 0.6
2 1 LHCt_9 142.1 +- 0.1 LHCa_9 81.94 +- 0.05 tev_9 809.1 +- 0.6
1/2 1 LHCt_10 142.1 +- 0.1 LHCa_10 81.94 +- 0.05 tev_10 809.1 +- 0.6
1 2 LHCt_11 154.1 +- 0.1 LHCa_11 89.04 +- 0.06 tev_11 797.4 +- 0.6

The 2->2 at LO the cross section is independent of the ren. scale.








Old results without the interference terms:
https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/scale--TeV.png https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/scale--LHCtop.png https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/scale--LHCantitop.png

The above plots show the cross section as a function of the scales. The renormalization and factorization scales are set equal to eachother and varied simultaneously. For the 2->2 processes (in particular at NLO) the scale dependence is extremely small over a very large range of scale choices. The dependence for the 2->3 is slightly larger, but is still small and very reasonable. At LO the cross sections for the 2->2 and 2->3 process become equal for small scale choices. For the NLO this is no longer the case. The difference between the 2->2 and 2->3 at NLO is large over the whole range of scales.

  • A tarball with all the plots (including sources) can be found here: scale.tar.gz

Normalized:
https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/scale-norm2--TeV.png https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/scale-norm2--LHCtop.png

scale dependence at the LHC for independent scale variations for light and heavy fermion lines:
https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/scaleLH--LHCtop.png

The above plot shows the scale dependence for the 2->3 process at NLO. In this plot the renormalization scale is set equal to the factorization scale $\mu_R=\mu_F$, but the scale for the heavy fermion line is varied independently from the scale of the light fermion line. The black curves show the scale variation of the heavy fermion line, i.e. the scales for the light line are fixed, and vice versa for the red line. The value to which the scales are fixed for a particular curve can be read of from the point where this black (or red) line crosses a red (or black) line in a point.

  • It is obvious that the total scale dependence, i.e. the blue curve (which is the same as the blue curve in the plots above), is totally coming from the scale variations in the heavy fermion line.

b-mass dependence


https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/bmass--TeV.png https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/bmass--LHCtop.png https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/bmass--LHCantitop.png

In the above plots the cross section is plotted as a function of the mass of the (anti-)b quark (for the 2->3 process). From these plots it is clear the cross section is quite sensitive to the mass of the b-quark. In particular, using the running mass at the scale of the top quark or at the scale of the bottom quark could enhance the cross section by over 10-15%. We should figure out to which mass we should let the b-mass run. Note that the CTEQ6 PDF sets assume a bottom mass of 4.5 GeV.

  • The NLO plots lie perfectly on a straight line (within the statistical MC errors). This suggests that the 'large logarithms' at NLO (that would be the terms proportional to $\alpha_s^2\log^2(m_b/m_t)$) do not play an important role and that using the resummed calculation with the b-quark PDF is not the best estimation for this process.
scale dependence for $m_b=m_c$


https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/scale-mb=mc--TeV.png https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/scale-mb=mc--LHCtop.png https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/scale-mb=mc--LHCantitop.png

In the above plots the cross section as a function of the scales is plotted. The mass of the bottom quark is set equal to the mass of the charm. The scale dependence for the NLO calculation is still very small, but the k-factors are much larger as compared to the $m_b=4.7\textrm{ GeV}$.

Unfortunaly the 'wrong' charm mass has been used for the 2->3 process: for the 2->2 process the charm PDF has been used which has a charm mass of 1.3 GeV. For the 2->3 process the a charm mass of 1.5 GeV was used. Using a charm mass of 1.3 also for the 2->3 process would increase this cross section slightly. The wobbly lines for the 2->3 at NLO at the LHC are entirely due to lack of statistics. Due to the small bottom numerical instabilities start playing a role and numerical convergence becomes more difficult.

  • The NLO 2->3 processes are wrong, because the old version with the bug was used.
  • All the plots (including sources) for the b-mass dependence can be downloaded here: bmass.tar.gz. This includes also some plots for the 2->3 process with scale dependence for a 50 GeV b quark mass. The dependence on the scales is very similar as using the normal b-quark mass.

final state particle kinematics

To do

  1. Analytic Check of Born amplitude
    We need to understand what is the analytic dependence of the short distance cross section as a function of the bmass.
  2. Check of the calculation
    We check the calculation inderectly by evaluating the s-channel top qq>W*>tbg at NLO. The idea is then to set mb=mt and compare with Oleari's NLO calculation for e+e-> Z/gamma>bb~ g.
  3. Check the mass effects in the 2->2 calculation
    Just to be sure, it would be useful to also compare with the 2->2 calculation at NLO, where the 2->3 contribution is calculated with a finite mass for the b. This can be done by using the collinear subtraction in the massive MSbar scheme, as we have done for W+jets and in the W+c calculation.
  4. Allow for different event-by-event scales in the process
    We need to be able to check all possible factorization and renormalization scales. The main point is that we can treat the light quark line and the heavy quark line independently, since as in the 2->2 there is no talking between the lines. The aim is to have four scales: muf_light, mur_light, muf_heavy, mur_heavy. So first thing is to assess the real scale dependence of the results by varying these scale independently. Scott suggested that we used a dynamical scale for the heavy line. After some thought I think that min(mT(b),mT(t)) =~ mT(b) should be used. By doing so we will slightly overestimate the diagrams where the initial gluon splits into a ttbar pair. However this contribution is anyway very small and should have no impact on the final result. This is also similar to the choice made by PS MC.

https://server06.fynu.ucl.ac.be/projects/madgraph/raw-attachment/wiki/SingleTopNLO/63.png

  • MCFM-Stopb.tar.gz: Latest version of MCFM including the 2->3 t-channel single top (updated with interference terms)

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