Post on 19-Jun-2020
transcript
The QCD static potential:
perturbative calculations
Xavier Garcia i Tormo
Argonne National Laboratory
(based on work done with Nora Brambilla, Joan Soto and Antonio Vairo)
Outline of the talk
QWG6. Nara - December 2 2008 – 2 / 19
■ The static potential
Outline of the talk
QWG6. Nara - December 2 2008 – 2 / 19
■ The static potential
■ Renormalon effects in the static potential
Outline of the talk
QWG6. Nara - December 2 2008 – 2 / 19
■ The static potential
■ Renormalon effects in the static potential
■ Comparison with lattice
Outline of the talk
QWG6. Nara - December 2 2008 – 2 / 19
■ The static potential
■ Renormalon effects in the static potential
■ Comparison with lattice
■ Conclusions
The static potential
QWG6. Nara - December 2 2008 – 3 / 19
We want to study the potential of a static quark and antiquarkseparated by a distance r (the QCD static potential).
The static potential
QWG6. Nara - December 2 2008 – 3 / 19
We want to study the potential of a static quark and antiquarkseparated by a distance r (the QCD static potential).
It is basic ingredient in a Schrodinger-like formulation of heavyquark bound states.
The static potential
QWG6. Nara - December 2 2008 – 3 / 19
We want to study the potential of a static quark and antiquarkseparated by a distance r (the QCD static potential).
It is basic ingredient in a Schrodinger-like formulation of heavyquark bound states.
A linear behavior at long distances is a signal for confinement.
The static potential
QWG6. Nara - December 2 2008 – 3 / 19
We want to study the potential of a static quark and antiquarkseparated by a distance r (the QCD static potential).
It is basic ingredient in a Schrodinger-like formulation of heavyquark bound states.
A linear behavior at long distances is a signal for confinement.
Here we are interested in the short distance region(r ≪ 1/ΛQCD), where perturbative (weak coupling) calculationsare reliable.
Vs = −CFαs(1/r)
r
(
1 + a1αs(1/r)
4π+ a2
(
αs(1/r)
4π
)2
+ · · ·
)
The static potential
QWG6. Nara - December 2 2008 – 3 / 19
We want to study the potential of a static quark and antiquarkseparated by a distance r (the QCD static potential).
It is basic ingredient in a Schrodinger-like formulation of heavyquark bound states.
A linear behavior at long distances is a signal for confinement.
Here we are interested in the short distance region(r ≪ 1/ΛQCD), where perturbative (weak coupling) calculationsare reliable.
(in this talk I will always refer to the weak coupling regime)
QWG6. Nara - December 2 2008 – 4 / 19
When calculated in perturbation theory infrared divergences arefound, starting at three loops Appelquist, Dine, Muzinich ’78
QWG6. Nara - December 2 2008 – 4 / 19
When calculated in perturbation theory infrared divergences arefound, starting at three loops Appelquist, Dine, Muzinich ’78
QWG6. Nara - December 2 2008 – 4 / 19
When calculated in perturbation theory infrared divergences arefound, starting at three loops Appelquist, Dine, Muzinich ’78
QWG6. Nara - December 2 2008 – 4 / 19
When calculated in perturbation theory infrared divergences arefound, starting at three loops Appelquist, Dine, Muzinich ’78
After selective resummation of certain type of diagrams,logarithmic contributions (starting at three loops) are generated
QWG6. Nara - December 2 2008 – 4 / 19
When calculated in perturbation theory infrared divergences arefound, starting at three loops Appelquist, Dine, Muzinich ’78
After selective resummation of certain type of diagrams,logarithmic contributions (starting at three loops) are generated
The use of Effective Field Theories allows us to calculate thosecontributions
QWG6. Nara - December 2 2008 – 5 / 19
Currents status of perturbative calculations
QWG6. Nara - December 2 2008 – 5 / 19
Currents status of perturbative calculations
Vs(r, µ) = −CF
rαs(1/r)
{
1 + (a1 + 2γEβ0)αs(1/r)
4π
+
[
a2 +
(
π2
3+ 4γ2
E
)
β20 + γE (4a1β0 + 2β1)
](
αs(1/r)
4π
)2
+
[
16π2
3C3
A ln rµ + a3
] (
αs(1/r)
4π
)3
+
[
aL24 ln2 rµ +
(
aL4 +
16
9π2 C3
Aβ0(−5 + 6 ln 2)
)
ln rµ
+a4
]
(
αs(1/r)
4π
)4}
QWG6. Nara - December 2 2008 – 5 / 19
Currents status of perturbative calculations
Vs(r, µ) = −CF
rαs(1/r)
{
1 + (a1 + 2γEβ0)αs(1/r)
4π
+
[
a2 +
(
π2
3+ 4γ2
E
)
β20 + γE (4a1β0 + 2β1)
](
αs(1/r)
4π
)2
+
[
16π2
3C3
A ln rµ + a[nf ]3 + a0
3 s
] (
αs(1/r)
4π
)3
+
[
aL24 ln2 rµ +
(
aL4 +
16
9π2 C3
Aβ0(−5 + 6 ln 2)
)
ln rµ
+a4
]
(
αs(1/r)
4π
)4}
Known Not known
QWG6. Nara - December 2 2008 – 5 / 19
Currents status of perturbative calculations
Vs(r, µ) = −CF
rαs(1/r)
{
1 + (a1 + 2γEβ0)αs(1/r)
4π
+
[
a2 +
(
π2
3+ 4γ2
E
)
β20 + γE (4a1β0 + 2β1)
](
αs(1/r)
4π
)2
+
[
16π2
3C3
A ln rµ + a3
] (
αs(1/r)
4π
)3
+
[
aL24 ln2 rµ +
(
aL4 +
16
9π2 C3
Aβ0(−5 + 6 ln 2)
)
ln rµ
+a4
]
(
αs(1/r)
4π
)4}
RG improved expressions also available for sub-leading ultrasoftlogs (given later in the talk)
a3
QWG6. Nara - December 2 2008 – 6 / 19
One and two loop coefficients have been known since ten years ago
a1 =31
9CA −
20
9TF nf Billoire’80
a2 =
(
4343
162+ 4π2
−π4
4+
22
3ζ(3)
)
C2A −
(
1798
81+
56
3ζ(3)
)
CATF nf −
−
(
55
3− 16ζ(3)
)
CF TF nf +
(
20
9TF nf
)2
Peter’97 Schroder’98
a3
QWG6. Nara - December 2 2008 – 6 / 19
One and two loop coefficients have been known since ten years ago
a1 =31
9CA −
20
9TF nf Billoire’80
a2 =
(
4343
162+ 4π2
−π4
4+
22
3ζ(3)
)
C2A −
(
1798
81+
56
3ζ(3)
)
CATF nf −
−
(
55
3− 16ζ(3)
)
CF TF nf +
(
20
9TF nf
)2
Peter’97 Schroder’98
Very recently the fermionic parts of a3 have been calculated
a3 = a(3)3 n3
f+a(2)3 n2
f+a(1)3 nf+a
(0)3 Smirnov, Smirnov, Steinhauser’08
a3
QWG6. Nara - December 2 2008 – 6 / 19
Very recently the fermionic parts of a3 have been calculated
a3 = a(3)3 n3
f+a(2)3 n2
f+a(1)3 nf+a
(0)3 Smirnov, Smirnov, Steinhauser’08
(a) (b) (c) (d)
(e) (f) (g) (h)
(Picture from A. V. Smirnov, V. A. Smirnov and M. Steinhauser, Phys. Lett. B 668, 293 (2008) [arXiv:0809.1927 [hep-ph]])
a3
QWG6. Nara - December 2 2008 – 6 / 19
Very recently the fermionic parts of a3 have been calculated
a3 = a(3)3 n3
f+a(2)3 n2
f+a(1)3 nf+a
(0)3 Smirnov, Smirnov, Steinhauser’08
(a) (b) (c) (d)
(e) (f) (g) (h)
(Picture from A. V. Smirnov, V. A. Smirnov and M. Steinhauser, Phys. Lett. B 668, 293 (2008) [arXiv:0809.1927 [hep-ph]])
The computation of a(0)3 is reported to be in progress
Logarithmic contributions
QWG6. Nara - December 2 2008 – 7 / 19
Consider the non-relativistic bound state scales
m (≫ ΛQCD) hard scale
p ∼ mv soft scale
E ∼ mv2 ultrasoft scale
v ≪ 1 (αs(mv) ∼ v)
Logarithmic contributions
QWG6. Nara - December 2 2008 – 7 / 19
Consider the non-relativistic bound state scales
m (≫ ΛQCD) hard scale
p ∼ mv soft scale
E ∼ mv2 ultrasoft scale
v ≪ 1 (αs(mv) ∼ v)
The expansion is organized around the Coulombic state
Logarithmic contributions
QWG6. Nara - December 2 2008 – 7 / 19
Consider the non-relativistic bound state scales
m (≫ ΛQCD) hard scale
p ∼ mv soft scale
E ∼ mv2 ultrasoft scale
v ≪ 1 (αs(mv) ∼ v)
The expansion is organized around the Coulombic state
Logarithmic contributions
QWG6. Nara - December 2 2008 – 7 / 19
Consider the non-relativistic bound state scales
m (≫ ΛQCD) hard scale
p ∼ mv soft scale
E ∼ mv2 ultrasoft scale
v ≪ 1 (αs(mv) ∼ v)
The expansion is organized around the Coulombic state
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
hard
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
hard soft
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
hard soft ultrasoft
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
We can construct Effective Field Theories to disentangle theeffects from those scales
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
We can construct Effective Field Theories to disentangle theeffects from those scales
QCD
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
We can construct Effective Field Theories to disentangle theeffects from those scales
QCDm≫mv,mv2
−→ NRQCD
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
We can construct Effective Field Theories to disentangle theeffects from those scales
QCDm≫mv, mv2
−→ NRQCDm≫mv≫mv2
−→ pNRQCD
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
We can construct Effective Field Theories to disentangle theeffects from those scales
pNRQCD can be organized as an expansion in r (multipoleexpansion) and 1/m
L =
Z
d3r Tr
8
>
>
>
<
>
>
>
:
S†
[i∂0 − Vs(r; µ)] S + O†
[iD0 − Vo(r; µ)]O
9
>
>
>
=
>
>
>
;
+
+VA(r; µ)Trn
O†r · gES + S
†r · gEO
o
+
+VB(r; µ)
2Tr
n
O†r · gEO + O
†Or · gE
o
−1
4F
aµνF
µν a
QWG6. Nara - December 2 2008 – 8 / 19
All those bound state scales will get entangled in a typicaldiagram
We can construct Effective Field Theories to disentangle theeffects from those scales
pNRQCD can be organized as an expansion in r (multipoleexpansion) and 1/m
Potentials appear as Wilson coefficients in the EFT
QWG6. Nara - December 2 2008 – 9 / 19
We obtain the potential by matching NRQCD to pNRQCD. Schematically (atorder r2)
QWG6. Nara - December 2 2008 – 9 / 19
We obtain the potential by matching NRQCD to pNRQCD. Schematically (atorder r2)
E0(r) = limT→∞
i
Tln
*
P exp
(
−ig
I
r×Tdz
µAµ(z)
)+
= Vs(r; µ) − ig2
Nc
Z
∞
0dt e
−it(Vo−Vs)〈r · E r · E〉 (µ)
QWG6. Nara - December 2 2008 – 9 / 19
We obtain the potential by matching NRQCD to pNRQCD. Schematically (atorder r2)
E0(r) = limT→∞
i
Tln
*
P exp
(
−ig
I
r×Tdz
µAµ(z)
)+
= Vs(r; µ) − ig2
Nc
V2A
Z
∞
0dt e
−it(Vo−Vs)〈r · E r · E〉 (µ)
Expectation value of Wilson loop operator
QWG6. Nara - December 2 2008 – 9 / 19
We obtain the potential by matching NRQCD to pNRQCD. Schematically (atorder r2)
E0(r) = limT→∞
i
Tln
*
P exp
(
−ig
I
r×Tdz
µAµ(z)
)+
= Vs(r; µ) − ig2
Nc
V2A
Z
∞
0dt e
−it(Vo−Vs)〈r · E r · E〉 (µ)
Expectation value of Wilson loop operator Matching coefficient
QWG6. Nara - December 2 2008 – 9 / 19
We obtain the potential by matching NRQCD to pNRQCD. Schematically (atorder r2)
E0(r) = limT→∞
i
Tln
*
P exp
(
−ig
I
r×Tdz
µAµ(z)
)+
= Vs(r; µ)−ig2
Nc
V2A
Z
∞
0dt e
−it(Vo−Vs)〈r · E r · E〉 (µ)
Expectation value of Wilson loop operator Matching coefficient Ultrasoft contribution (retardation effects)
QWG6. Nara - December 2 2008 – 9 / 19
We obtain the potential by matching NRQCD to pNRQCD. Schematically (atorder r2)
E0(r) = limT→∞
i
Tln
*
P exp
(
−ig
I
r×Tdz
µAµ(z)
)+
= Vs(r; µ)−ig2
Nc
V2A
Z
∞
0dt e
−it(Vo−Vs)〈r · E r · E〉 (µ)
Expectation value of Wilson loop operator Matching coefficient Ultrasoft contribution (retardation effects)
■ Left hand side must be µ independent (static energy)
QWG6. Nara - December 2 2008 – 9 / 19
We obtain the potential by matching NRQCD to pNRQCD. Schematically (atorder r2)
E0(r) = limT→∞
i
Tln
*
P exp
(
−ig
I
r×Tdz
µAµ(z)
)+
= Vs(r; µ)−ig2
Nc
V2A
Z
∞
0dt e
−it(Vo−Vs)〈r · E r · E〉 (µ)
Expectation value of Wilson loop operator Matching coefficient Ultrasoft contribution (retardation effects)
■ Left hand side must be µ independent (static energy)■ The logarithmic contribution at three loops can be deduced from the
leading ultrasoft contribution (Brambilla, Pineda, Soto, Vairo ’99), the logarithmic termsat four loops from the sub-leading contribution (Brambilla, X.G.T., Soto, Vairo ’06)
QWG6. Nara - December 2 2008 – 10 / 19
The ultrasoft logarithms can be resummed by solving the renormalizationgroup equations
QWG6. Nara - December 2 2008 – 10 / 19
The ultrasoft logarithms can be resummed by solving the renormalizationgroup equations
µd
dµVs = −
2
3
αsCF
π
(
1 + 6αs
πB)
V 2A (Vo − Vs)
3 r2
µd
dµVo =
1
N2c − 1
2
3
αsCF
π
(
1 + 6αs
πB)
V 2A (Vo − Vs)
3 r2
µd
dµαs = αsβ(αs)
µd
dµVA = 0
µd
dµVB = 0 (B =
−5nf + CA(6π2 + 47)
108)
QWG6. Nara - December 2 2008 – 11 / 19
The solution of the renormalization group equation for the singlet and octetpotential is
QWG6. Nara - December 2 2008 – 11 / 19
The solution of the renormalization group equation for the singlet and octetpotential is Pineda, Soto ’00 Brambilla, X.G.T., Soto, Vairo ’08 (in preparation)
Vs(µ) = Vs(1/r) + 2N2
c − 1
N2c
[(Vo − Vs)(1/r)]3 r2 γ(0)os
β0
{
lnαs(µ)
αs(1/r)
+
(
−β1
4β0+
γ(1)os
γ(0)os
)
[
αs(µ)
π−
αs(1/r)
π
]
}
Vo(µ) = Vo(1/r) −2
N2c
[(Vo − Vs)(1/r)]3 r2 γ(0)os
β0
{
lnαs(µ)
αs(1/r)
+
(
−β1
4β0+
γ(1)os
γ(0)os
)
[
αs(µ)
π−
αs(1/r)
π
]
}
(γ(0)os =
Nc
3γ(1)os = 2BNc)
Renormalon effects in the static potential
QWG6. Nara - December 2 2008 – 12 / 19
■ Vs does not present a good convergent behavior
0.1 0.2 0.3 0.4 0.5
r
r0
-7-6-5-4-3-2-1
0r0VsHrL
nf = 0, we use a Pade estimate (Chishtie, Elias ’01) for a(0)3
r0 ∼ .5fm, µ = 2.5r−10 , αs determined according to Capitani et al. ’99
Dotted blue:tree level; Dot-dashed magenta:1 loop; dashed brown:2 loops (plus leading us log
resummation); Solid green: 3 loops (plus next-to-leading us log resummation)
Renormalon effects in the static potential
QWG6. Nara - December 2 2008 – 12 / 19
■ Vs does not present a good convergent behavior
■ This bad convergence can be interpreted as coming from asingularity close to the origin in the Borel plane, andsignaling that non-perturbative contributions are important
Renormalon effects in the static potential
QWG6. Nara - December 2 2008 – 12 / 19
■ Vs does not present a good convergent behavior
■ This bad convergence can be interpreted as coming from asingularity close to the origin in the Borel plane, andsignaling that non-perturbative contributions are important
■ The strategy is to find operators that account for thenon-perturbative effects. Then we impose that theambiguities in the Borel transform are accounted for thoseoperators and reshuffle contributions from the perturbativeseries to the operators
Renormalon effects in the static potential
QWG6. Nara - December 2 2008 – 12 / 19
■ Vs does not present a good convergent behavior
■ This bad convergence can be interpreted as coming from asingularity close to the origin in the Borel plane, andsignaling that non-perturbative contributions are important
■ The strategy is to find operators that account for thenon-perturbative effects. Then we impose that theambiguities in the Borel transform are accounted for thoseoperators and reshuffle contributions from the perturbativeseries to the operators
■ We will implement the renormalon cancellation along thelines of the so-called RS scheme Pineda’01
QWG6. Nara - December 2 2008 – 13 / 19
The lower dimensional operators, that account for the ambiguities, are thoserelated to the residual mass term in HQET, which get inherited in pNRQCD
LHQET = hv (iD0 − δmQ)hv + O
(
1
mQ
)
QWG6. Nara - December 2 2008 – 13 / 19
The lower dimensional operators, that account for the ambiguities, are thoserelated to the residual mass term in HQET, which get inherited in pNRQCD
In the weak coupling regime at the static limit, we account for them with theshift
Vs,o → Vs,o + Λs,o Λs,o ∼ ΛQCD
QWG6. Nara - December 2 2008 – 13 / 19
The lower dimensional operators, that account for the ambiguities, are thoserelated to the residual mass term in HQET, which get inherited in pNRQCD
In the weak coupling regime at the static limit, we account for them with theshift
Vs,o → Vs,o + Λs,o Λs,o ∼ ΛQCD
The renormalization group properties of Λs,o fix the renormalon singularity upto a normalization constant Beneke ’94
QWG6. Nara - December 2 2008 – 14 / 19
The renormalization group equations for Λs,o are given by
QWG6. Nara - December 2 2008 – 14 / 19
The renormalization group equations for Λs,o are given byBrambilla, X.G.T., Soto, Vairo ’08 (in preparation)
µd
dµΛs = −2
αsCF
π
(
1 + 6αs
πB)
V 2Ar2 [(Vo − Vs) (1/r)]2 (Λo − Λs)
µd
dµΛo =
2
N2c − 1
αsCF
π
(
1 + 6αs
πB)
V 2Ar2 [(Vo − Vs) (1/r)]2 (Λo − Λs)
The ultrasoft effects introduce anomalous dimensions and mixing betweensinglet and octet
QWG6. Nara - December 2 2008 – 14 / 19
The renormalization group equations for Λs,o are given byBrambilla, X.G.T., Soto, Vairo ’08 (in preparation)
µd
dµΛs = −2
αsCF
π
(
1 + 6αs
πB)
V 2Ar2 [(Vo − Vs) (1/r)]2 (Λo − Λs)
µd
dµΛo =
2
N2c − 1
αsCF
π
(
1 + 6αs
πB)
V 2Ar2 [(Vo − Vs) (1/r)]2 (Λo − Λs)
And the solution is
QWG6. Nara - December 2 2008 – 14 / 19
The renormalization group equations for Λs,o are given byBrambilla, X.G.T., Soto, Vairo ’08 (in preparation)
µd
dµΛs = −2
αsCF
π
(
1 + 6αs
πB)
V 2Ar2 [(Vo − Vs) (1/r)]2 (Λo − Λs)
µd
dµΛo =
2
N2c − 1
αsCF
π
(
1 + 6αs
πB)
V 2Ar2 [(Vo − Vs) (1/r)]2 (Λo − Λs)
And the solution is
Λs(µ) = NsΛ + 2CF (No − Ns)Λ r2 [(Vo − Vs) (1/r)]2
×
(
2
β0lnαs(µ) + η0αs(µ)
)
Λo(µ) = NoΛ −1
Nc(No − Ns)Λ r2 [(Vo − Vs) (1/r)]2
×
(
2
β0lnαs(µ) + η0αs(µ)
)
QWG6. Nara - December 2 2008 – 15 / 19
We have to match those structures to the ambiguities in a proper definition ofthe Borel integral.
QWG6. Nara - December 2 2008 – 15 / 19
We have to match those structures to the ambiguities in a proper definition ofthe Borel integral.Without ultrasoft effects (no anomalous dimension) we have
Is,o = ρ4π
β0
∫
∞
0du e
−4π
β0
u
αs
×
{
Rs,o
(1 − 2u)1+b
[
1 + c1(1 − 2u) + c2(1 − 2u)2 + c3(1 − 2u)3 + . . .
}
And the coefficients ci are determined just by the coefficients in the betafunction
QWG6. Nara - December 2 2008 – 15 / 19
Without ultrasoft effects (no anomalous dimension) we have
Is,o = ρ4π
β0
Z
∞
0du e
−4π
β0
u
αs
8
>
>
>
<
>
>
>
:
Rs,o
(1 − 2u)1+b
h
1 + c1(1 − 2u) + c2(1 − 2u)2
+ c3(1 − 2u)3
+ . . .
9
>
>
>
=
>
>
>
;
QWG6. Nara - December 2 2008 – 15 / 19
Without ultrasoft effects (no anomalous dimension) we have
Is,o = ρ4π
β0
Z
∞
0du e
−4π
β0
u
αs
8
>
>
>
<
>
>
>
:
Rs,o
(1 − 2u)1+b
h
1 + c1(1 − 2u) + c2(1 − 2u)2
+ c3(1 − 2u)3
+ . . .
9
>
>
>
=
>
>
>
;
With ultrasoft effects we have
Is,o = ρ4π
β0
∫
∞
0du e
−4π
β0
u
αs
×
{ Rs,o
(1 − 2u)1+b
[
1 + c1(1 − 2u) + c2;s,o(1 − 2u)2
+c3;s,o(1 − 2u)3 + . . . + d1;s,o(1 − 2u)2 ln(1 − 2u)
+d2;s,o(1 − 2u)3 ln(1 − 2u) + . . .]
}
c2, c3, . . . are now different for singlet and octet and we have new non-analyticterms (di)
QWG6. Nara - December 2 2008 – 15 / 19
The previous expression tells us which terms we have to subtract from Vs,o, toget rid of the bad behavior of the perturbative series
QWG6. Nara - December 2 2008 – 15 / 19
The previous expression tells us which terms we have to subtract from Vs,o, toget rid of the bad behavior of the perturbative series
0.1 0.2 0.3 0.4 0.5
r
r0
-7-6-5-4-3-2-1
0r0VsHrL
Rs,o is determined (approximately) through Rs,o = V BTs,o (u)(1 − 2u)1+b|
u= 12
, ρ = µ = 2.5r−10
Dotted blue:tree level; Dot-dashed magenta:1 loop; dashed brown:2 loops (plus leading us log resummation); Solid green: 3 loops
(plus next-to-leading us log resummation). RS scheme implemented by just subtracting the most singular term
QWG6. Nara - December 2 2008 – 15 / 19
The previous expression tells us which terms we have to subtract from Vs,o, toget rid of the bad behavior of the perturbative series
0.1 0.2 0.3 0.4 0.5
r
r0
-7-6-5-4-3-2-1
0r0VsHrL
Rs,o is determined (approximately) through Rs,o = V BTs,o (u)(1 − 2u)1+b|
u= 12
, ρ = µ = 2.5r−10
Dotted blue:tree level; Dot-dashed magenta:1 loop; dashed brown:2 loops (plus leading us log resummation); Solid green: 3 loops
(plus next-to-leading us log resummation). RS scheme implemented by just subtracting the most singular term
The RS scheme provides us with a better perturbative behavior
Comparison with lattice
QWG6. Nara - December 2 2008 – 16 / 19
We will compare the singlet static potential to the lattice data(Necco, Sommer ’01)
Comparison with lattice
QWG6. Nara - December 2 2008 – 16 / 19
We will compare the singlet static potential to the lattice data(Necco, Sommer ’01)
We have to plot Vs + Λs as a function of r
Vs + Λs = Vs (r, µ, ρ) + K1 + K2f(r, µ)
(K1 = NsΛ and K2 = (No − Ns)Λ are the two integration constants coming from the diff. eqs.)
Comparison with lattice
QWG6. Nara - December 2 2008 – 16 / 19
We will compare the singlet static potential to the lattice data(Necco, Sommer ’01)
We have to plot Vs + Λs as a function of r
Vs + Λs = Vs (r, µ, ρ) + K1 + K2f(r, µ)
To have a definite way to organize the different terms we will usethe counting
1
r≫
αs
r≫ ΛQCD Λ ∼ NsΛ ∼ NoΛ ∼ ΛQCD ∼
α2s
r
Comparison with lattice
QWG6. Nara - December 2 2008 – 16 / 19
We will compare the singlet static potential to the lattice data(Necco, Sommer ’01)
We have to plot Vs + Λs as a function of r
Vs + Λs = Vs (r, µ, ρ) + K1 + K2f(r, µ)
To have a definite way to organize the different terms we will usethe counting
1
r≫
αs
r≫ ΛQCD Λ ∼ NsΛ ∼ NoΛ ∼ ΛQCD ∼
α2s
r
Starting at three loop level we have the presence of two arbitraryconstants.
Comparison with lattice
QWG6. Nara - December 2 2008 – 16 / 19
We will compare the singlet static potential to the lattice data(Necco, Sommer ’01)
We have to plot Vs + Λs as a function of r
Vs + Λs = Vs (r, µ, ρ) + K1 + K2f(r, µ)
To have a definite way to organize the different terms we will usethe counting
1
r≫
αs
r≫ ΛQCD Λ ∼ NsΛ ∼ NoΛ ∼ ΛQCD ∼
α2s
r
Starting at three loop level we have the presence of two arbitraryconstants. We fix the constants by forcing the curves to gothrough the first one or two lattice data points
QWG6. Nara - December 2 2008 – 17 / 19
0.1 0.2 0.3 0.4
r
r0
-1.5
-1.0
-0.5
r0HVsHrL+LsHrLL
Dotted blue:tree level; Dot-dashed magenta:1 loop; dashed brown:2 loops (plus leading us log resummation); Solid green: 3 loops
(plus next-to-leading us log resummation)
QWG6. Nara - December 2 2008 – 18 / 19
Uncertainties of the result
QWG6. Nara - December 2 2008 – 18 / 19
Uncertainties of the result
0.1 0.2 0.3 0.4
r
r0
-1.5
-1.0
-0.5
r0HVsHrL+LsHrLL
Impact of varying the Pade estimate for a(0)3 by 30%
QWG6. Nara - December 2 2008 – 18 / 19
Uncertainties of the result
0.1 0.2 0.3 0.4
r
r0
-1.5
-1.0
-0.5
r0HVsHrL+LsHrLL
Impact of varying the Pade estimate for a(0)3 by 30%
0.1 0.2 0.3 0.4
r
r0
-1.5
-1.0
-0.5
r0HVsHrL+LsHrLL
H LL
Impact of the variation of αs (ΛMS
= 0.602(48)r−10 )
QWG6. Nara - December 2 2008 – 18 / 19
Uncertainties of the result
0.1 0.2 0.3 0.4
r
r0
-1.5
-1.0
-0.5
r0HVsHrL+LsHrLL
Impact of varying the Pade estimate for a(0)3 by 30%
0.1 0.2 0.3 0.4
r
r0
-1.5
-1.0
-0.5
r0HVsHrL+LsHrLL
0.1 0.2 0.3 0.4
r
r0
-1.5
-1.0
-0.5
r0HVsHrL+LsHrLL
Impact of the variation of αs (ΛMS
= 0.602(48)r−10 ) Effect of higher order (α5
s) terms
Conclusions
QWG6. Nara - December 2 2008 – 19 / 19
■ Calculation of the static potential at short distances
Conclusions
QWG6. Nara - December 2 2008 – 19 / 19
■ Calculation of the static potential at short distances
■ New ingredients since last QWG meeting
Conclusions
QWG6. Nara - December 2 2008 – 19 / 19
■ Calculation of the static potential at short distances
■ New ingredients since last QWG meeting
◆ Fermionic part of the three loop coefficient a3 is known
Conclusions
QWG6. Nara - December 2 2008 – 19 / 19
■ Calculation of the static potential at short distances
■ New ingredients since last QWG meeting
◆ Fermionic part of the three loop coefficient a3 is known◆ Inclusion of ultrasoft effects in the renormalon analysis
Conclusions
QWG6. Nara - December 2 2008 – 19 / 19
■ Calculation of the static potential at short distances
■ New ingredients since last QWG meeting
◆ Fermionic part of the three loop coefficient a3 is known◆ Inclusion of ultrasoft effects in the renormalon analysis
■ The use of Effective Field Theories allows us to calculate theultrasoft effects, and helps in dealing with the renormalonsingularity
Conclusions
QWG6. Nara - December 2 2008 – 19 / 19
■ Calculation of the static potential at short distances
■ New ingredients since last QWG meeting
◆ Fermionic part of the three loop coefficient a3 is known◆ Inclusion of ultrasoft effects in the renormalon analysis
■ The use of Effective Field Theories allows us to calculate theultrasoft effects, and helps in dealing with the renormalonsingularity
■ Comparison with lattice data is very good