from tensor integral to ibp - uni-bielefeld.de · 2012. 9. 11. · [chetyrkin, tkachov ’81]...
TRANSCRIPT
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From Tensor Integral to IBP
Mohammad Assadsolimani,
in collaboration with
P. Kant, B. Tausk and P. Uwer
11. Sep. 2012
Mohammad Assadsolimani From Tensor Integral to IBP 1
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Contents
Motivation
NNLO
Tensor Integral
Tarasov’s method
Projection method
Applications
Heavy Quark Form Factors
Single Top Quark Production
Conclusions
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Multi loop calculation
Multi loop calculation
More precision in calculated results
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More precision
Ex.: Total cross section for Higgs production in gluon fusion
10-3
10-2
10-1
1
100 120 140 160 180 200 220 240 260 280 300
σ(pp →H+X) [pb]
MH [GeV]
LONLONNLO
√s = 2 TeV
1
10
100 120 140 160 180 200 220 240 260 280 300
σ(pp→H+X) [pb]
MH [GeV]
LONLONNLO
√s = 14 TeV
[R. Harlander, W. Kilgore Nov. ’02]
Perturbative convergence LO → NLO(≈ 70%) and
NLO → NNLO(≈ 30%)
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Multi loop calculation
Multi loop calculation
More precision in calculated results
New effects
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New effects
Ex.: forward–backward charge asymmetry of the top quark
LO (CTEQ6L1)NLO (CTEQ6M)
pT,jet > 20GeV
√s = 1.96TeV
pp → tt+jet+X
µ/mt
σ[pb]
1010.1
6
5
4
3
2
1
0
LO (CTEQ6L1)NLO (CTEQ6M)
pT,jet > 20GeV
√s = 1.96TeV
pp → tt+jet+X
µ/mt
AtFB
1010.1
0.04
0.02
0
−0.02
−0.04
−0.06
−0.08
−0.1
−0.12
[S. Dittmaier, P. Uwer , S. Weinzierl Apr. ’08]
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Multi loop calculation
Multi loop calculation
More precision in calculated results
New effects
Exact NLO or NNLO calculations of σhard needed because of:
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Multi loop calculation
Multi loop calculation
More precision in calculated results
New effects
Exact NLO or NNLO calculations of σhard needed because of:
Accurate and reliable predictions of parton–level
observables.
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Multi loop calculation
Multi loop calculation
More precision in calculated results
New effects
Exact NLO or NNLO calculations of σhard needed because of:
Accurate and reliable predictions of parton–level
observables.
Backgrounds for New Physics Searches
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NNLO
When do we need NNLO?
When NLO corrections are large
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NNLO
When do we need NNLO?
When NLO corrections are large
When truly high precision is needed
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NNLO
When do we need NNLO?
When NLO corrections are large
When truly high precision is needed
When the precision of NLO–calculation has to be verified
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NNLO
When do we need NNLO?
When NLO corrections are large
When truly high precision is needed
When the precision of NLO–calculation has to be verified
For instance single top quark production :
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NNLO Single Top Quark Production
Single top quark production
Process√S σLO(pb) σNLO (pb)
t–channel 2.0 TeV pp 1.068 1.062
14.0 TeV pp 152.7 155.9
[B.Harris, E. Laenen, L.Phaf, Z. Sullivan, S. Weinzierl ’02]
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NNLO Single Top Quark Production
Single top quark production
Process√S σLO(pb) σNLO (pb)
t–channel 2.0 TeV pp 1.068 1.062
14.0 TeV pp 152.7 155.9
[B.Harris, E. Laenen, L.Phaf, Z. Sullivan, S. Weinzierl ’02]
No colour exchange at NLO:
W W
Only vertex corrections contribute:
W
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NNLO Single Top Quark Production
Single top quark production
Process√S σLO(pb) σNLO (pb)
t–channel 2.0 TeV pp 1.068 1.062
14.0 TeV pp 152.7 155.9
[B.Harris, E. Laenen, L.Phaf, Z. Sullivan, S. Weinzierl ’02]
No colour exchange at NLO:
W W
Only vertex corrections contribute:
W
Colour exchange at NNLO:
W W 0
W0W
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NNLO Single Top Quark Production
The true uncertainty due to missing higher order correction may
be greater because new colour exchange diagrams first
contribute at NNLO.
A significant effect on kinematical distributions
Important for studies of the V–A structure
Source of polarised top quarks
Access to the b quark pdfs
· · ·
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Tensor integral
In the NNLO–corrections occur tensor integrals:
I(d , a1, · · · , an)[1,kµ1 ,kν2 ,··· ] =
∫
ddk1
∫
ddk2
∏ij k
µi
1 kνj
2
Pa11 · · ·Pan
n
Possibilities to reduce tensor integrals to scalar integrals:
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Tensor integral
In the NNLO–corrections occur tensor integrals:
I(d , a1, · · · , an)[1,kµ1 ,kν2 ,··· ] =
∫
ddk1
∫
ddk2
∏ij k
µi
1 kνj
2
Pa11 · · ·Pan
n
Possibilities to reduce tensor integrals to scalar integrals:
By Schwinger parametrization
[O. V. Tarasov, Phys. Rev.’96, Nucl. Phys. ’81]
By projection method
[T. Binoth, E.W.N. Glover, P. Marquard and J.J. van der Bij ’02; E.W.N. Glover ’04]
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Tensor integral
In the NNLO–corrections occur tensor integrals:
I(d , a1, · · · , an)[1,kµ1 ,kν2 ,··· ] =
∫
ddk1
∫
ddk2
∏ij k
µi
1 kνj
2
Pa11 · · ·Pan
n
Possibilities to reduce tensor integrals to scalar integrals:
By Schwinger parametrization
[O. V. Tarasov, Phys. Rev.’96, Nucl. Phys. ’81]
By projection method
[T. Binoth, E.W.N. Glover, P. Marquard and J.J. van der Bij ’02; E.W.N. Glover ’04]
Tensor reduction ⇒ various scalar integrals with the same structure
of the integrand with different powers of propagators
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Integration by Parts
Possible approaches:
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Integration by Parts
Possible approaches:
solve each integral individually,
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Integration by Parts
Possible approaches:
solve each integral individually,
express all scalar integrals as a linear combination of some
basic master integrals, Integration by parts (IBP).
[Chetyrkin, Tkachov ’81]
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Integration by Parts
Possible approaches:
solve each integral individually,
express all scalar integrals as a linear combination of some
basic master integrals, Integration by parts (IBP).
[Chetyrkin, Tkachov ’81]
Reduction techniques:
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Integration by Parts
Possible approaches:
solve each integral individually,
express all scalar integrals as a linear combination of some
basic master integrals, Integration by parts (IBP).
[Chetyrkin, Tkachov ’81]
Reduction techniques:
Laporta: efficient algorithm to solve linear system of
IBP–Identities
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Integration by Parts
Possible approaches:
solve each integral individually,
express all scalar integrals as a linear combination of some
basic master integrals, Integration by parts (IBP).
[Chetyrkin, Tkachov ’81]
Reduction techniques:
Laporta: efficient algorithm to solve linear system of
IBP–Identities
AIR [Anastasiou, Lazopoulos ’04]
FIRE [Smirnov ’08]
Crusher [Marquard, Seidel (to be published)]
REDUZE 1&2 [Studerus ’09; Manteuffel, Studerus ’12]
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Tarasov’s method
Tensor reduction leads to a very large number of scalar integrals
which are shifted in dimension and have other powers of propagators
I(d , a1, · · · , an)[kµ1 kν2 ,··· ] → gµν∑
i
I(d + xi , ai1, · · · , a
in)[1]
Example for two loop corrections to Axial Vector Form Factors
I(d , 1, 1, 1, 1, 1, 1)[1,kµ11 k
µ21 k
ν12 k
ν22 ] → I(2+ d , 2, 1, 1, 1, 1, 2)+
· · · + I(4+ d , 1, 1, 1, 2, 3, 1) + · · · + I(8+ d , 3, 3, 3, 2, 1, 2)
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Tarasov’s method
Shift in the dimension
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Tarasov’s method
Shift in the dimension
An arbitrary scalar Feynman integral:
I(d)({si },{m2s })∝
∏Nj=1 cj
∫∞0· · ·
∫∞0
dαjαaj−1
j
[D(α)]d2
ei
[
Q({si },α)
D(α)−∑N
l=1αl (m
2l−iǫ)
]
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Tarasov’s method
Shift in the dimension
An arbitrary scalar Feynman integral:
I(d)({si },{m2s })∝
∏Nj=1 cj
∫∞0· · ·
∫∞0
dαjαaj−1
j
[D(α)]d2
ei
[
Q({si },α)
D(α)−∑N
l=1αl (m
2l−iǫ)
]
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Tarasov’s method
Shift in the dimension
An arbitrary scalar Feynman integral:
I(d)({si },{m2s })∝
∏Nj=1 cj
∫∞0· · ·
∫∞0
dαjαaj−1
j
[D(α)]d2
ei
[
Q({si },α)
D(α)−∑N
l=1αl (m
2l−iǫ)
]
D(
∂∂m2
j
)
(polynomial differential operator) obtained from D(α) by
substituting αi → ∂j ≡ ∂/∂m2j . The application of D(∂i ) to the
scalar integral:
I(d−2)({si }, {m2s }) ∝ D(∂j) I(d)({si }, {m
2s }),
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Tarasov’s method
Shift in the dimension
An arbitrary scalar Feynman integral:
I(d)({si },{m2s })∝
∏Nj=1 cj
∫∞0· · ·
∫∞0
dαjαaj−1
j
[D(α)]d2
ei
[
Q({si },α)
D(α)−∑N
l=1αl (m
2l−iǫ)
]
D(
∂∂m2
j
)
(polynomial differential operator) obtained from D(α) by
substituting αi → ∂j ≡ ∂/∂m2j . The application of D(∂i ) to the
scalar integral:
I(d−2)({si }, {m2s }) ∝ D(∂j) I(d)({si }, {m
2s }),
apply this to master integrals
Imaster (d − 2, a1, · · · , an) =∑
i
ciI(d , ai1, · · · , a
in),
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Tarasov’s method
all scalar integrals in rhs. of that equation have to be replaced by
master integrals.
i.e.
Imaster (d − 2, a1, · · · , an) =∑
j
DjImaster (d , aj1, · · · , a
jn),
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Tarasov’s method
all scalar integrals in rhs. of that equation have to be replaced by
master integrals.
i.e.
Imaster (d − 2, a1, · · · , an) =∑
j
DjImaster (d , aj1, · · · , a
jn),
we have for all master integrals:
Id−21...
Id−2l
master
= Dll ·
Id1...
Idl
master
where l is the number of master integrals.
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Projection method
By this method we get scalar products between loop momenta and
external momenta and no shift in the dimension of integrals
I(d , a1, · · · , an)[1,kµ1 ,kµ2 ,··· ] → gµν
∑
ij
I(d , a1, · · · , an)[1]kipj
Example for two loop corrections to Axial Vector Form Factors
I(d , 1, 1, 1, 1, 1, 1)[1,kµ11 k
µ21 k
ν12 k
ν22 ] → I(d ,−2, 1, 1, 1, 1, 1) + · · ·
+I(d ,−1, 0, 1, 1, 1, 1) + · · · + I(d , 0, 1, 1, 1,−2,−2)
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Projection method
The general tensor structure for the amplitude A:
A =
n∑
i=1
Bi(t, u, s)Si ,
where t, u and s are the Mandelstam variables and Si are the Dirac
structures.
Projectors for the tensor coefficients:
S†j A =
n∑
i=1
Bi(t, u, s)(
S†j Si
︸︷︷︸Mji
)
⇒ Bi(t, u, s) =∑
i
M−1ij
(
S†j A)
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Projection method
The general tensor structure for the amplitude A:
A =
n∑
i=1
Bi(t, u, s)Si ,
where t, u and s are the Mandelstam variables and Si are the Dirac
structures.
Projectors for the tensor coefficients:
S†j A =
n∑
i=1
Bi(t, u, s)(
S†j Si
︸︷︷︸Mji
)
⇒ Bi(t, u, s) =∑
i
M−1ij
(
S†j A)
essential for this method : to be able to calculate the inverse matrix
M−1ij
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Tarasov’s method vs. Projection method
Tarasov’s method
Positive powers for propagators (the sum of the powers of all
propagators is large )
Calculate the inverse matrix in order to shift back the dimension
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Tarasov’s method vs. Projection method
Tarasov’s method
Positive powers for propagators (the sum of the powers of all
propagators is large )
Calculate the inverse matrix in order to shift back the dimension
Projection method
Negative powers for propagators
Calculate the inverse matrix for the projector coefficients
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Tarasov’s method vs. Projection method
Tarasov’s method
Positive powers for propagators (the sum of the powers of all
propagators is large )
Calculate the inverse matrix in order to shift back the dimension
Projection method
Negative powers for propagators
Calculate the inverse matrix for the projector coefficients
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Two loop corrections to Heavy Quark Form Factors
We implemented both methods to calculate the two loop corrections to
Heavy Quark Vector and axial Vector Form Factors:
[W. Bernreuther,R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, P. Mastrolia, E. Remiddi ’04]
[J. Gluza, A. Mitov, S. Moch, T. Riemann ’09 ]
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Two loop corrections to Heavy Quark Form Factors
q1
p2
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Two loop corrections to Heavy Quark Form Factors
q1
p2
There are 6 Dirac structures
(Heavy Quark Vector and axial Vector
Form Factors):S1 = u(q1)(1+ γ5)u(p2)p
µ2
S2 = u(q1)(1− γ5)u(p2)pµ2
S3 = u(q1)(1+ γ5)u(p2)qµ1
S4 = u(q1)(1− γ5)u(p2)qµ1
S5 = u(q1)(1+ γ5)γµu(p2)
S6 = u(q1)(1− γ5)γµu(p2)
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Two loop corrections to Heavy Quark Form Factors
q1
p2
There are 6 Dirac structures
(Heavy Quark Vector and axial Vector
Form Factors):S1 = u(q1)(1+ γ5)u(p2)p
µ2
S2 = u(q1)(1− γ5)u(p2)pµ2
S3 = u(q1)(1+ γ5)u(p2)qµ1
S4 = u(q1)(1− γ5)u(p2)qµ1
S5 = u(q1)(1+ γ5)γµu(p2)
S6 = u(q1)(1− γ5)γµu(p2)
Projection Tarasov
number of integrals 564 671
max sum of powers
of propagators 6 14
max sum of negative
powers of propagators 4 0
reduction time 7500 s 433260 s
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Two loop corrections to Heavy Quark Form Factors
q1
p2
There are 6 Dirac structures
(Heavy Quark Vector and axial Vector
Form Factors):S1 = u(q1)(1+ γ5)u(p2)p
µ2
S2 = u(q1)(1− γ5)u(p2)pµ2
S3 = u(q1)(1+ γ5)u(p2)qµ1
S4 = u(q1)(1− γ5)u(p2)qµ1
S5 = u(q1)(1+ γ5)γµu(p2)
S6 = u(q1)(1− γ5)γµu(p2)
Projection Tarasov
number of integrals 564 671
max sum of powers
of propagators 6 14
max sum of negative
powers of propagators 4 0
reduction time 7500 s 433260 s
Now one may come to the conclusion:
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Two loop corrections to Heavy Quark Form Factors
q1
p2
There are 6 Dirac structures
(Heavy Quark Vector and axial Vector
Form Factors):S1 = u(q1)(1+ γ5)u(p2)p
µ2
S2 = u(q1)(1− γ5)u(p2)pµ2
S3 = u(q1)(1+ γ5)u(p2)qµ1
S4 = u(q1)(1− γ5)u(p2)qµ1
S5 = u(q1)(1+ γ5)γµu(p2)
S6 = u(q1)(1− γ5)γµu(p2)
Projection Tarasov
number of integrals 564 671
max sum of powers
of propagators 6 14
max sum of negative
powers of propagators 4 0
reduction time 7500 s 433260 s
Now one may come to the conclusion:
⇒ projection method is an alternative
method for the multi loop calculation!
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Two loop corrections to Heavy Quark Form Factors
q1
p2
There are 6 Dirac structures
(Heavy Quark Vector and axial Vector
Form Factors):S1 = u(q1)(1+ γ5)u(p2)p
µ2
S2 = u(q1)(1− γ5)u(p2)pµ2
S3 = u(q1)(1+ γ5)u(p2)qµ1
S4 = u(q1)(1− γ5)u(p2)qµ1
S5 = u(q1)(1+ γ5)γµu(p2)
S6 = u(q1)(1− γ5)γµu(p2)
Projection Tarasov
number of integrals 564 671
max sum of powers
of propagators 6 14
max sum of negative
powers of propagators 4 0
reduction time 7500 s 433260 s
Now one may come to the conclusion:
⇒ projection method is an alternative
method for the multi loop calculation!
but · · ·
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Two loop corrections to single Top Quark Production
There are three topological families:
t
W
t t
W W
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Two loop corrections to single Top Quark Production
There are three topological families:
t
W
t t
W W
and 11 Dirac structures:S1 = u(q1) γ7 u(p2)u(q2) γ6γq1 u(p1)
S2 = u(q1) γ6γp1 u(p2)u(q2) γ6γq1 u(p1)
S3 = u(q1) γ6γµ1u(p2)u(q2) γ6γµ1
u(p1)
S4 = u(q1) γ7γµ1γp1 u(p2)u(q2) γ6γµ1
u(p1)
S5 = u(q1) γ7γµ1γµ2
u(p2)u(q2) γ6γµ1γµ2
γq1 u(p1)
S6 = u(q1) γ6γµ1γµ2
γp1 u(p2)u(q2) γ6γµ1γµ2
γq1 u(p1)
S7 = u(q1) γ6γµ1γµ2
γµ3u(p2)u(q2) γ6γµ1
γµ2γµ3
u(p1)
S8 = u(q1) γ7γµ1γµ2
γµ3γp1 u(p2)u(q2) γ6γµ1
γµ2γµ3
u(p1)
S9 = u(q1) γ7γµ1γµ2
γµ3γµ4
u(p2)u(q2) γ6γµ1γµ2
γµ3γµ4
γq1 u(p1)
S10 = u(q1) γ6γµ1γµ2
γµ3γµ4
γp1 u(p2)u(q2) γ6γµ1γµ2
γµ3γµ4
γq1 u(p1)
S11 = u(q1) γ6γµ1γµ2
γµ3γµ4
γµ5u(p2)u(q2) γ6γµ1
γµ2γµ3
γµ4γµ5
u(p1)
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Two loop corrections to single Top Quark Production
Vertex corrections: both methods
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Vertex corrections
+ +
=
[
N2CTC
Fg4s
{
S1
[
1
m5t
−8−11t +t2
2ǫ(t −1)3−
−71−275t +44t2
12(t −1)3+O(ǫ)
−
44t
3m3t (t −1)3
+O(ǫ)
−
2nf3m3
t (t −1)+O(ǫ)
+
1
m3t
−−8−11t +t2
3ǫ(t −1)3+
−103−226t +41t2
18(t −1)3+O(ǫ)
+
2nf3mt (t −1)
+O(ǫ)
+
12t
mt (t −1)3+O(ǫ)
−
(
2mt (t +1)
3(t−1)+O(ǫ)
) ]
+S3
[
1
m4t
−11+12t +13t2
24ǫ(t −1)2+
−72+169t −324t2 +303t3 +76t4 −24t5
72(t −1)3t+O(ǫ)
+1
m2t
(
23ǫ
−46+28t
9t+O(ǫ)
)
+1
m2t
2t3ǫ(t −1)
+2t(1+5t −14t2)
9(t −1)3+O(ǫ)
+1
m2t
nf(5−19t)
36(t −1)−
nf(t +1)
6ǫ(t −1)+O(ǫ)
+1
m2t
5t2 +12t +19
36ǫ(t −1)2+
29t2 −86t −87
72(t −1)2+O(ǫ)
+(
−nf /9+nf /(3ǫ)+O(ǫ))
+
nft
3ǫ(t −1)−
nf(3+t)
9(t −1)+O(ǫ)
+(
2/t −t/3+O(ǫ))
+
(1+3t +3t2 −t3)
3(t −1)2+O(ǫ)
−
m2t (3t2 +19t +28)
9+O(ǫ)
−
m2t t(3t
2 +4t +1)
9(t −1)+O(ǫ)
]}
+N2CC2F
{
· · ·
}
+N2CCF CA
{
· · ·
}]1
m2t
(
t −m2W
)
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Two loop corrections to single Top Quark Production
Vertex corrections: both methods X
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Two loop corrections to single Top Quark Production
Vertex corrections: both methods X
Planar double boxes : projection method
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Two loop corrections to single Top Quark Production
Vertex corrections: both methods X
Planar double boxes : projection method
problem: build the inverse matrix Mji = S†j Si
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Two loop corrections to single Top Quark Production
Vertex corrections: both methods X
Planar double boxes : projection method
problem: build the inverse matrix Mji = S†j Si
try with common computer algebra system, e.g.
Mathematica or Maple
runtime ≈ 1 month with 64GB RAM
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Two loop corrections to single Top Quark Production
Vertex corrections: both methods X
Planar double boxes : projection method
problem: build the inverse matrix Mji = S†j Si
try with common computer algebra system, e.g.
Mathematica or Maple
runtime ≈ 1 month with 64GB RAM
We calculated M−1ji and all Planar double boxes diagrams X
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Two loop corrections to single Top Quark Production
Vertex corrections: both methods X
Planar double boxes : projection method
problem: build the inverse matrix Mji = S†j Si
try with common computer algebra system, e.g.
Mathematica or Maple
runtime ≈ 1 month with 64GB RAM
We calculated M−1ji and all Planar double boxes diagrams X
Non Planar double boxes: a challenge !
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Status of the calculation for single Top Quark Production
Topology # Diagrams reduction performed checks
Vertex corrections 29 X X
Planar double boxes 6 X work in progress
Non Planar double boxes 12 work in progress –
There are two most complicated topologies, which could not be
reduced completely until now :
t
t
t
W
t t
W
t
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Conclusions
We have seen two possibilities to reduce tensor integrals to
scalar integrals
The choice of reductions method determines how difficult the
next step (IBP) is
As a test of our setup, we have calculated the O(α2s )
contributions to the Heavy Quark Vector and Axial Vector Form
Factors, confirming the results of Bernreuther et al. and
Gluza et al.
We have also calculated the two loop vertex corrections to single
Top Quark Production
Mohammad Assadsolimani From Tensor Integral to IBP 26