the qcd static potential at nnnllbibliography (1) n. brambilla, x. garcia i tormo, j. soto and a....
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The QCD static potential at NNNLLAntonio Vairo
University of Milano and INFN
In collaboration with N. Brambilla, X. Garcia i Tormo and J. Soto
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Summary
1. Potential: definition
1.1 pNRQCD
2. Potential: calculation in PT
2.1 Static potential
3. Conclusions
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Bibliography
(1) N. Brambilla, X. Garcia i Tormo, J. Soto and A. Vairoin preparation
(2) N. Brambilla, X. Garcia i Tormo, J. Soto and A. VairoThe logarithmic contribution to the QCD static energy at N4LO
Phys. Lett. B 647 (2007) 185 arXiv:hep-ph/0610143 .
(3) A. Pineda and J. SotoThe Renormalization group improvement of the QCD static potentials
Phys. Lett. B 495 (2000) 323. arXiv:hep-ph/0007197 .
(4) N. Brambilla, A. Pineda, J. Soto and A. VairoPotential NRQCD: an effective theory for heavy quarkonium
Nucl. Phys. B 566 (2000) 275 arXiv:hep-ph/9907240 .
(5) N. Brambilla, A. Pineda, J. Soto and A. VairoThe infrared behaviour of the static potential in perturbative QCD
Phys. Rev. D 60 (1999) 091502 arXiv:hep-ph/9903355 .
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1. Definition
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The potential is what to write in a Schrödinger equation
E φ =
„p2
m+ V (r)
«
φ
In a full theory, V must come from a double expansion:
• a non-relativistic expansion ∼ p/m, rm: V → V (0) + V (1)/m + · · · ;
• an expansion in E r, since V is a function of r (or p at h.o. in the non-relativisticexpansion): V → V + energy-dependent effects (e.g. Lamb-shift).
A potential V describes the interaction of a non-relativistic bound state, p ∼ mv, E ∼ mv2, v ≪ 1,
once the expansions in mv/m and mv2/mv have been exploited.
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Non-relativistic scales in QCD
Near threshold:
E ≈ 2m +p2
m+ . . . with v =
p
m≪ 1
• The perturbative expansion breaks down when αs ∼ v:
+ + ... ≈1
E −“
p2
m+ V
”
αs
“
1 +αs
v+ . . .
”
• The system is non-relativistic : p ∼ mv and E =p2
m+ V ∼ mv2.
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Non-relativistic scales in QCD
Scales get entangled.
... ... ...
∼ mv
p ∼ m
E ∼ mv2
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EFTs for systems made of two heavy quarks
mv2
µ
perturbative matching(short−range quarkonium)
nonperturbative matching(long−range quarkonium)
mv
m
µ
perturbative matching perturbative matching
QCD
NRQCD
pNRQCD
... ... ...
��������
��������
...
+ ...+
∼ mv
p ∼ m
E ∼ mv2
• They exploit the expansion in v/ factorization of low and high energy contributions.• They are renormalizable order by order in v.• In perturbation theory (PT), RG techniques provide resummation of large logs.
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pNRQCD
pNRQCD is obtained by integrating out modes associated with the scale1
r∼ mv
+ + ...... ... ...
+ ...++ ...
NRQCD pNRQCD
1
E − p2/m − V (r)
• The Lagrangian is organized as an expansion in 1/m , r, and αs(m):
LpNRQCD =X
k
X
n
1
mk× ck(αs(m/µ)) × V (rµ′, rµ) × On(µ′, λ) rn
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pNRQCD formαs ≫ ΛQCD
L = −1
4F a
µνF µν a + Tr
S†
„
i∂0 −p2
m− Vs
«
S
+ O†
„
iD0 −p2
m− Vo
«
O
ff
LO in r
θ(T ) e−iTHs θ(T ) e−iTHo
“
e−iR
dt Aadj”
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pNRQCD formαs ≫ ΛQCD
L = −1
4F a
µνF µν a + Tr
S†
„
i∂0 −p2
m− Vs
«
S
+ O†
„
iD0 −p2
m− Vo
«
O
ff
+VATrn
O†r · gES + S†r · gEOo
+VB
2Tr
n
O†r · gEO + O†Or · gEo
+ · · ·
LO in r
NLO in r
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pNRQCD formαs ≫ ΛQCD
O†r · gES O†{r · gE, O}
+VATrn
O†r · gES + S†r · gEOo
+VB
2Tr
n
O†r · gEO + O†Or · gEo
NLO in r
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The potential is a Wilson coefficient of an EFT. In general, it undergoes renormalization,
develops scale dependence and satisfies renormalization group equations, which allow
to resum large logarithms.
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2. Calculation
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The Static Potential
=
NRQCD
+
pNRQCD
+ ...eig
HdzµAµ
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The Static Potential
=
NRQCD
+
pNRQCD
+ ...eig
HdzµAµ
limT→∞
i
Tln = Vs(r, µ) − i
g2
Nc
V 2A
Z ∞
0dt e−it(Vo−Vs) 〈Tr(r · E(t) r · E(0))〉(µ) + . . .
ultrasoft contribution
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The Static Potential
=
NRQCD
+
pNRQCD
+ ...eig
HdzµAµ
limT→∞
i
Tln = Vs(r, µ) − i
g2
Nc
V 2A
Z ∞
0dt e−it(Vo−Vs) 〈Tr(r · E(t) r · E(0))〉(µ) + . . .
ultrasoft contribution
* The µ dependence cancels between the two terms in the right-hand side:
Vs ∼ ln rµ, ln2 rµ, ...
ultrasoft contribution ∼ ln(Vo − Vs)/µ, ln2(Vo − Vs)/µ, ... ln rµ, ln2 rµ, ...
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Static Wilson loop
limT→∞
i
Tln = −CF
αs(1/r)
r
"
1 + a1αs(1/r)
4π+ a2
„αs(1/r)
4π
«2
+ . . .
#
is known at two loops:
a1 =31
9CA −
10
9nf + 2γEβ0,
Billoire 80
a2 =
„4343
162+ 4π2 −
π4
4+
22
3ζ(3)
«
C2A −
„899
81+
28
3ζ(3)
«
CAnf
−
„55
6− 8ζ(3)
«
CF nf +100
81n2
f + 4γEβ0a1 +
„π2
3− 4γ2
E
«
β20 + 2γEβ1
Schr oder 99, Peter 97
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Appelquist–Dine–Muzinich diagrams
...... ... = −CF C3
A
12
αs
r
α3s
πln
»CAαs
2r× r
–
| {z }
∼ exp(−i(Vo − Vs) T )
Appelquist Dine Muzinich 78, Brambilla Pineda Soto Vairo 99
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Static octet potential
limT→∞
i
Tln
����
����
〈φadjab
〉
=1
2Nc
αs(1/r)
r
"
1 + b1αs(1/r)
4π+ b2
„αs(1/r)
4π
«2
+ . . .
#
Is known at two loops.
b1 = a1
b2 = a2 + C2A(π4 − 12π2)
Kniehl et al 04
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VA
The first contributing diagrams are of the type:
�Therefore
VA(r, µ) = 1 + O(α2s )
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Chromoelectric field correlator:〈E(t)E(0)〉
Is known at NLO.
�× ×
LO
�× × �× ×
(a) (b)
�× × �× ×
(c) (d)
�× × �× ×
(e) (f)
�× × �× ×
(g) (h)
NLOEidem uller Jamin 97
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Static singlet potential at N4LO
Vs(r, µ) = −CFαs(1/r)
r
"
1 + a1αs(1/r)
4π+ a2
„αs(1/r)
4π
«2
+
„16 π2
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#
aL24 = −
16π2
3C3
A β0
aL4 = 16π2C3
A
»
a1 + 2γEβ0 + nf
„
−20
27+
4
9ln 2
«
+CA
„149
27−
22
9ln 2 +
4
9π2
«–
Brambilla et al 99, 06
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Static singlet potential at N4LO
Vs(r, µ) = −CFαs(1/r)
r
"
1 + a1αs(1/r)
4π+ a2
„αs(1/r)
4π
«2
+
„16 π2
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#
• The logarithmic contribution at N3LO may be extracted from the one-loopcalculation of the ultrasoft contribution;
• the single logarithmic contribution at N4LO may be extracted from the two-loopcalculation of the ultrasoft contribution.
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Static energy at N4LO
E0(r) = −CF αs(1/r)
r
(
1 +αs(1/r)
4π[a1 + 2γEβ0]
+
„αs(1/r)
4π
«2 »
a2 +
„π2
3+ 4γ2
E
«
β20 + γE (4a1β0 + 2β1)
–
+
„αs(1/r)
4π
«3 »16π2
3C3
A lnCAαs(1/r)
2+ a3
–
+
„αs(1/r)
4π
«4 »
aL24 ln2 CAαs(1/r)
2+ aL
4 lnCAαs(1/r)
2+ a4
–
+ · · ·
)
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Renormalization group equations
8>>>>>>>>>>>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>>>>>>>>>>>:
µd
dµαVs
=2
3
αs
πV 2
A
»αVo
6+
4
3αVs
–3 “
1 +αs
πc”
µd
dµαVo
=2
3
αs
πV 2
A
»αVo
6+
4
3αVs
–3 “
1 +αs
πc”
µd
dµαs = αsβ(αs)
µd
dµVA = 0
µd
dµVB = 0
Vs = −CFαVs
r, Vo =
1
2Nc
αVo
r, c =
−5nf + CA(6π2 + 47)
108.
Pineda Soto 00, Brambilla Garcia Soto Vairo 07
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Static singlet potential at NNNLL
αVs(µ) = αVs
(1/r) +C3
A
6β0α3
s (1/r)
„
1 +3
4
αs(1/r)
πa1
«
lnαs(1/r)
αs(µ)„
β1
4β0− 6c
« »αs(µ)
π−
αs(1/r)
π
–ff
Vs = −CFαVs
r, a1 =
31
9CA −
10
9nf + 2γEβ0, c =
−5nf + CA(6π2 + 47)
108.
Brambilla Garcia Soto Vairo 07
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Static singlet potential at NNNLL
1.5 2 2.5 3 3.5 4 4.5 5
0.4
0.6
0.8
1
1.5 2 2.5 3 3.5 4 4.5 5
0.4
0.6
0.8
1
αVs
r−1 in GeV
(µ = αs(r)/r; nf = 4)
−−− 2-loop−−− NNLL−−− NNNLL [a3 = (4.32 ± 25%) 104]
[preliminary]
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Static singlet potential at NNNLL
1.5 2 2.5 3 3.5 4 4.5 5
0.4
0.6
0.8
1
αVs
r−1 in GeV
(µ = αs(r)/r; nf = 4)
−−− N4LO [a3 = 4.32 104]
−−− NNNLL [a3 = 4.32 104]
[preliminary]
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Conclusions
Non-relativistic EFTs provide a rigorous definition of the potential between two heavyquarks.
• In the perturbative regime, we have calculated the NNNLL expression of the staticpotential (up to the three-loop non-logarithmic piece) .
• This may become a key ingredient for the precision calculation of several thresholdobservables, e.g. quarkonium masses and top-quark pair production cross sectionnear threshold.