# Changes between Version 1 and Version 2 of SingleTopNLO

Ignore:
Timestamp:
04/06/12 16:33:02 (7 years ago)
Comment:

--

Unmodified
Added
Removed
Modified
• ## SingleTopNLO

 v1 === 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. 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. === Results === The parameters used are * %$m_t=172$% GeV * %$m_b=4.7$% GeV * %$m_W=80.419$% GeV * %$m_Z=91.118$% GeV * %$\mu_R=\mu_F=m_t$% * $m_t=172$ 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. ==== 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'''  || * 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'''  || ||  $\sigma(2\to 3)$  ||  '''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  || ==== 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. * 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'''  |||| ||  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  || ||  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'''  |||| ||  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  || ||  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'''  |||| ||  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  || ||  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'''  |||| ||  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  || ||  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'''  |||| ||  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  || scale dependence at the LHC for independent scale variations for light and heavy fermion lines:
scale dependence at the LHC for independent scale variations for light and heavy fermion lines: [[br]] 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. 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. 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 %$m_b=m_c$%  ===== * 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 $m_b=m_c$  =====
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 %$m_b=4.7\textrm{ GeV}$%. 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. === To do === 1. '''Analytic Check of Born amplitude'''
1. '''Analytic Check of Born amplitude''' [[br]] We need to understand what is the analytic dependence of the short distance cross section as a function of the bmass. 1. '''Check of the calculation'''
1. '''Check of the calculation''' [[br]] 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. 1.  '''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. 1. '''Allow for different event-by-event scales in the process'''
1.  '''Check the mass effects in the 2->2 calculation''' [[br]] 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. 1. '''Allow for different event-by-event scales in the process''' [[br]] 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.