The QCD static potential: perturbative calculations · The static potential QWG6. Nara - December 2...

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



■ Renormalon effects in the static potential

Outline of the talk




■ Comparison with lattice

Outline of the talk




■ Comparison with lattice

■ Conclusions

The static potential


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

+ · · ·

)






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)


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



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


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}



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



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


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


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



a3 = a(3)3 n3

f+a(2)3 n2

f+a(1)3 nf+a


(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



a3 = a(3)3 n3

f+a(2)3 n2

f+a(1)3 nf+a


(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


Consider the non-relativistic bound state scales

m (≫ ΛQCD) hard scale

p ∼ mv soft scale

E ∼ mv2 ultrasoft scale

v ≪ 1 (αs(mv) ∼ v)





p ∼ mv soft scale


v ≪ 1 (αs(mv) ∼ v)

The expansion is organized around the Coulombic state





p ∼ mv soft scale


v ≪ 1 (αs(mv) ∼ v)






p ∼ mv soft scale


v ≪ 1 (αs(mv) ∼ v)



All those bound state scales will get entangled in a typicaldiagram





hard



hard soft



hard soft ultrasoft



We can construct Effective Field Theories to disentangle theeffects from those scales




QCD




QCDm≫mv,mv2

−→ NRQCD




QCDm≫mv, mv2

−→ NRQCDm≫mv≫mv2

−→ pNRQCD




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




pNRQCD can be organized as an expansion in r (multipoleexpansion) and 1/m

Potentials appear as Wilson coefficients in the EFT


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)



E0(r) = limT→∞

i

Tln

*

P exp

(

−ig

I

r×Tdz

µAµ(z)

)+

= Vs(r; µ) − ig2

Nc

V2A

Z

∞

0dt e


Expectation value of Wilson loop operator



E0(r) = limT→∞

i

Tln

*

P exp

(

−ig

I

r×Tdz

µAµ(z)

)+

= Vs(r; µ) − ig2

Nc

V2A

Z

∞

0dt e


Expectation value of Wilson loop operator Matching coefficient



E0(r) = limT→∞

i

Tln

*

P exp

(

−ig

I

r×Tdz

µAµ(z)

)+

= Vs(r; µ)−ig2

Nc

V2A

Z

∞

0dt e


Expectation value of Wilson loop operator Matching coefficient Ultrasoft contribution (retardation effects)



E0(r) = limT→∞

i

Tln

*

P exp

(

−ig

I

r×Tdz

µAµ(z)

)+

= Vs(r; µ)−ig2

Nc

V2A

Z

∞

0dt e



■ Left hand side must be µ independent (static energy)



E0(r) = limT→∞

i

Tln

*

P exp

(

−ig

I

r×Tdz

µAµ(z)

)+

= Vs(r; µ)−ig2

Nc

V2A

Z

∞

0dt e



■ 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)


The ultrasoft logarithms can be resummed by solving the renormalizationgroup equations


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)


The solution of the renormalization group equation for the singlet and octetpotential is


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


■ 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)




■ 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





■ 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


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

)



In the weak coupling regime at the static limit, we account for them with theshift

Vs,o → Vs,o + Λs,o Λs,o ∼ ΛQCD



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


The renormalization group equations for Λs,o are given by


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



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



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)

)


We have to match those structures to the ambiguities in a proper definition ofthe Borel integral.


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


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

>

>

>

=

>

>

>

;


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)


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



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


(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


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.)





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





Vs + Λs = Vs (r, µ, ρ) + K1 + K2f(r, µ)


1

r≫

αs


α2s

r

Starting at three loop level we have the presence of two arbitraryconstants.





Vs + Λs = Vs (r, µ, ρ) + K1 + K2f(r, µ)


1

r≫

αs


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


0.1 0.2 0.3 0.4

r

r0

-1.5

-1.0

-0.5

r0HVsHrL+LsHrLL


(plus next-to-leading us log resummation)


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

H LL

Impact of the variation of αs (ΛMS

= 0.602(48)r−10 )



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

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


■ Calculation of the static potential at short distances

Conclusions



■ New ingredients since last QWG meeting

Conclusions




◆ Fermionic part of the three loop coefficient a3 is known

Conclusions




◆ Fermionic part of the three loop coefficient a3 is known◆ Inclusion of ultrasoft effects in the renormalon analysis

Conclusions





■ The use of Effective Field Theories allows us to calculate theultrasoft effects, and helps in dealing with the renormalonsingularity

Conclusions





■ 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

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The QCD static potential: perturbative calculations · The static potential QWG6. Nara - December 2...

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