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 $\mu_R^l=\mu_Fl=\mu_R^h=\mu_Fh=m_t$.
• For $2\to 3$ we choose $\mu_R^l=\mu_Fl=m_t/2$ and $\mu_R^h=\mu_Fh=m_t/4$.

$\sigma(2\to 2)$	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:

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:

scale dependence at the LHC for independent scale variations for light and heavy fermion lines:

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

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 $\alpha_s^2\log2(m_b/m_t)$) 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 $m_b=4.7\textrm{ GeV}$.

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

• all--TeV.pdf​: pt, eta of the t and bbar quarks at the tevatron
• all--LHCtop.pdf​: pt, rapidity of the t and bbar quarks at the LHC
• all--LHCantitop.pdf​: pt, rapidity of the tbar and b quarks at the LHC

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.

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