Three-loop Strong Coupling Constant
αMS
Quentin Mason (Cambridge), Howard Trottier (Simon Fraser)
Christine Davies (Glasgow)
Kerryann Foley, Peter Lepage & Matthew Nobes (Cornell)
Alan Gray & Junko Shigemitsu (Ohio State)
13 – Gluon Action
Symanzik Improved Gluon Action
"
1 !
a
a
abc b c 2
SG = Tr ∂µ Aν − ∂ν Aµ − gf Aµ Aν + ψ("D + m)ψ
4
SG = βpl
#
x;µ<ν
(1 − Pµν ) + βrt
βpl
βrt = −
(1 + 0.4805αs )
20 u20
#
1
Pµν = Re Tr $ " Rµν
3
!
HPQCD
#
x;µ!=ν
(1 − Rµν ) + βpg
βpg
#
x;µ<ν<σ
βpl
= − 2 0.03325αs
u0
(QCD)
(1 − Cµνσ )
(Coefficients)
'!!!! #
&
# !! #
&
!!!
!!!
1
1
= Re Tr $ !!!
Cµνσ = Re Tr $! ! !!! ! ! ! ! ! ! ! ! ! "
"
! %
3
3
&
! !!
! ! !
&
Quentin Mason
13 – Gluon Action
Symanzik Improved Gluon Action
"
1 !
a
a
abc b c 2
SG = Tr ∂µ Aν − ∂ν Aµ − gf Aµ Aν + ψ("D + m)ψ
4
SG = βpl
#
x;µ<ν
(1 − Pµν ) + βrt
#
x;µ!=ν
(1 − Rµν ) + βpg
#
x;µ<ν<σ
βpl
βpl
βrt = −
(1 + 0.4805αs )
βpg = − 2 0.03325αs
20 u20
u0
−0.3879 Nf αs
−0.01541 Nf αs
#
1
Pµν = Re Tr $ " Rµν
3
!
HPQCD
(QCD)
(1 − Cµνσ )
(Coefficients)
'!!!! #
&
# !! #
&
!!!
!!!
1
1
= Re Tr $ !!!
Cµνσ = Re Tr $! ! !!! ! ! ! ! ! ! ! ! ! "
"
! %
3
3
&
! !!
! ! !
&
Quentin Mason
Program for SM constants
Fundamental Constants:
(3-loops)
αs
mu , md , ms
(1-loop)
mc , mb
(1-loop)
CKM matrix elements: Vcd , Vcs , Vcb , Vub
-
eg |Vub | = 3.27 (24)exp (52)theory × 10−3
Quentin Mason (Cambridge)
Strong coupling from the Lattice
Simulation
Lattice PT
2
3
!O" = c0 + c1 αlat + c2 αlat
+ c3 αlat
Lattice PT
αlat
α
−→ Continuum PT
=
(q) + f1 (q)
αV2 + f2 (q) αV3 + . . .
is α
inV “lattice”
regularisation
action dependent
most series are
poorly convergent
Continuum
PT + 4-loop running
prefer a physical coupling with O(1) coefficients
αV (d/a) −→ αMS (MZ )
Quentin Mason (Cambridge)
Strong coupling from the Lattice
Simulation
Lattice PT
2
3
!O" = c0 + c1 αlat + c2 αlat
+ c3 αlat
Lattice PT
αlat
−→ Continuum PT
= αV (q) + f1 (q) αV2 + f2 (q) αV3 + . . .
Continuum PT + 4-loop running
αV (d/a) −→ αMS (MZ )
Quentin Mason (Cambridge)
Background Field
gA −→ Bconst + gAQuantum
Consider the IR background field 2-pt function:
ΓMS (q)µν
"
(δµν q 2 − qµ qν ) !
=−
1 − νMS (q)
2
gMS
known analytically:
2
4
q 2 ν(q) ∼ gMS
(a + b ln q 2 ) + gMS
(A + B ln q 2 + C ln2 q 2 )
Γlat (q)µµ
2
gMS
2
glat
"
3q̂ 2 !
= − 2 1 − νlat (q) + O(a)
glat
calculate numerically for range of q
and fit to remove lattice artifacts
1 − νlat (q)
=
1 − νMS (q)
Quentin Mason (Cambridge)
Two-loop Diagrams
77
79
O(αlat ), δmadditive
O(αlat )
Figure 5.3: Two-loop Fermionic Background field Diagrams.
2
O(nf αlat
)
The numbering
scheme comes from [69]. After combining 7+11 and 9+17 everything
is IR finite. 15-18 are the 4 continuum diagrams, and the last row are
one- and two-loop lattice counter-term diagrams from (2.6).
15
0.2
0.5
0.3
0.2
26
0.1
QM Fit 0.1
HT Fit
Combined Fit
QM 29 points
HT 10 points 0
PSfrag replacements
0.4
26
`
´
Total one-loop Symanzik improved glue with up to p10 e + f ln(p) + g ln2 (p) corrections
rag replacements
g replacements
0.3
0.26
0.24
0.22
0.2
0.18
-0.1
0.16
0.14
0.12
-0.2
0.1
0.26
0.08
0.1
0.22
0
PSfrag replacements
-0.3
0.001
0.24
0.2
0.16
0.14
0.12
-0.2
0.1
0.08
PSfrag replacements
-0.3
0.001
0.01
0.1
0.1
1
1
p
(2)
pb2 νlat
0.1
1
10
p
P
(2)
FIG. 4: The final fit for the two-loop part of the constant in pb2 νlat = i c4,i + β1 ln p, after the known logarithm was subtracte
out, shown over the complete range of external momenta calculated for the Asqtad action. Inset shows that data for p ∼ 0.
controls the quality of the final fit, and the agreement between the two independent evaluations.
0.18
-0.1
0.01
1
P
0.003
10
Two-loop fermionic results for Wilson quarks in four different gauges
ξ=1
ξ = 3/2
ξ = 2.405
ξ=5
0.0025
FIG. 4: The final fit for the two-loop part of the constant in
= i c4,i + β1 ln p, after the known logarithm was subtracted
out, shown over the complete range of external momenta calculated for the Asqtad action. Inset shows that data for p ∼ 0.1
controls the quality of the final fit, and the agreement between the two independent evaluations. 0.002
0.0015
Two-loop fermionic results for Wilson quarks in four different gauges
0.003
0.001
0.002
ξ=1
ξ = 3/2 0.0005
ξ = 2.405
ξ=5
0
0.0015
-0.0005
0.001
-0.001
0.0005
-0.0015
0
-0.002
0.0025
-0.0005
-0.0025
0.1
-0.001
-0.0015
1
p
P
(2)
FIG. 5: Fits to the fermionic two-loop part of pb2 νlat = i c4,i +β1 ln p, in four different gauges. The coefficient of the logarithm
are set to the known analytic values. The fits agree at 1σ after comparing to the identically gauge dependent MS calculation
Philosophy for Lattice PT
Uµ (x) = eigT
a
Aa (x+ µ̂
2)
Automate generation of Feynman Rules
Automate diagram generation
Numerical evaluation
sacrifice precision for flexibility, reusability and robustness
√
errors decrease ∼ CPU time
Only tricky part is IR divergences
twisted BC, subtractions
Quentin Mason (Cambridge)
Operators
Wilson loops
3rd Order
1x1, 1x2, 1x3, 1x4, 2x2, 2x3
Mean-link
Asqtad
Naive
Wilson
Clover
3rd Order
! Tr "U #Landau
lim
T →∞
←
1
ln
T
Method:
!
mixture of “direct” !φ" =
and “susceptibility”
T
→
!
R
!
Static Potential
2nd Order !
!
DU φ e−S[U ]
! "
#$
$
1 d
−S[U ]−φJ $
!φ" = −
ln DU e
$
V dJ
J=0
Quentin Mason (Cambridge)
Vacuum Bubble Diagrams
96
98
2
O(nf αlat
)
2
O(αlat
)
Figure 5.8: The three-loop pure glue Feynman diagrams for the expectation of a
Figure 5.9: The three-loop fermionic Feynman diagrams for the expectation of
Wilson loop, evaluated from derivatives of vacuum bubbles using the
susceptibilty method (5.45). The magenta vertex is the amputated
one-loop gluon propagator of figure 5.7. The diagram numbering is consistent with [72]. The lattice counter-terms are shown separately. The
square vertex g 4 A4 comes from the measure, and the cross, [double-
a Wilson loop, evaluated from derivatives of vacuum bubbles using
the susceptibilty method (5.45). The diagram numbering is consistent with [69] except this D917=D7+D10+D15+D19+D25 and this
D97=D17+D20+D21+D24+D26. The notation F2 DX indicates that
diagram X is propotional to n2f rather than nf , and CT indicates lat-
Static Potential @ small R
!
"
αV (q)
αV (q ∗ (R))
β02 2
V (q) ≡ −4πCF
=⇒ V (R) = −CF
1+
α + ...
q2
R
48 V
Compute ∆V12 ≡ V (R2 ) − V (R1 ) in lattice & continuum PT then:
∆V12 correction = ∆V12 continuum − ∆V12 lattice
−Cf
!
1
1
−
R2
R1
"
#
$
2
d1 αV + (d2 + dnf nf )αV
, 2nd Order, high q*
2.31
2.00
0.1747(2)
0.1402(4)
0.02(7)
-0.04(6)
-0.033(2)
-0.031(2)
0.84
1.59
0.0524(3)
0.0714(3)
-0.l2(6)
-0.10(7)
-0.018(2)
-0.022(2)
1.73
0.0750(4)
-0.09(7)
-0.022(1)
2.31
0.0596(4)
0.00(8)
-0.020(2)
Static Potential
1.2
1
0.8
0.6
0.4
0.2
1
2
3
4
R in GeV
Figure 2. ∆V (R) on the fine (squares) and coarse
2
Static Potential
0.4
0.3
0.2
∆V (R/a) = V (R/a) − V (1/a)
0.5
Perturbative Potential
Potential from Simulation
1.5
2
R/a
2.5
3
Figure 1. ∆V (R) from the static potential on the
F
formula for Y reproduces the nonperturbative value from the
simulation.
We computed cn for n ≤ 3 using Feynman diagrams. We
estimated higher-order coefficients by simultaneously fitting
results from different lattice spacings to the same perturbative formula. This is possible because the coupling
10 αV (d/a)
!
2
3 a. We paramechanges
with different lattice
spacings
!O"
= cvalue
cn αVn
1 αV (d/a) + c2 αV + c3 αV +
terized the running coupling in our fit by a single scale paramn=4
eter, ΛV , according to the standard third-order formula
!
$
"
#
2
2
4π
2β1 ln ln(q /ΛV )
αV (q) =
1− 2
+ ···
β0 ln(q 2 /Λ2V )
β0
ln(q 2 /Λ2V )
(2)
where β0 , β1 . . . are constants (see [10, 11] for the full formulas).
We used a constrained fitting procedure, based upon
Bayesian methods, to do our fits [12]. Given simulation results Yi ± σYi for three different lattice spacings ai ± σai , we
minimized an augmented χ2 function,
Fit Details
χ2
&
3
10
2
%
%
(Yi − n cn αnV (d/ai ))
(cn − cn )2
≡
+
2
2
σ
σ
c
Y
n
i
n=1
i=1
'
(2
3
The fits reveale
loop expansions a
find
log W11 = −
log W12
+
= −
+
The large 5α3V co
to agree with sim
The coupling αV
our lattice spacing
Wilson loop we e
These large coe
sults. There are
ficients. One is
2(n+m)
u0
where u
Creutz ratios of th
Each procedure s
cients we obtain w
spacings: for exam
)
*
formula for Y reproduces the nonperturbative value from the
simulation.
We computed cn for n ≤ 3 using Feynman diagrams. We
estimated higher-order coefficients by simultaneously fitting
results from different lattice spacings to the same perturbative formula. This is possible because the coupling
10 αV (d/a)
!
2
3 a. We paramechanges
with different lattice
spacings
!O"
= cvalue
cn αVn
1 αV (d/a) + c2 αV + c3 αV +
terized the running coupling in our fit by a single scale paramn=4
eter, ΛV , according to the standard third-order formula
!
$
"
#
2
2
4π
2β1 ln ln(q /ΛV )
αV (q) =
1− 2
+ ···
β0 ln(q 2 /Λ2V )
β0
ln(q 2 /Λ2V )
(2)
where β0 , β1 . . . are constants
(see [10, 11] for the full formu- "
!
las).
log W11
= −3.068 αV (3.33/a) 1−1.068 αV +1.69(4) αV2 −5(2) αV3 −1(7) αV4 +. . .
We used a constrained fitting procedure, based upon
Bayesian methods, to! do our fits [12]. Given simulation re- "
W12
4
−0(2)
α
+. . .
log 6sults
= −0.949
αV (1.82/a)
1+0.160
αV −0.54(8)
αV2 −2(1) a
αV3 ±
V
Y
±
σ
for
three
different
lattice
spacings
σ
,
i
Yi
i
ai we
u0
minimized an augmented χ2 function,
Fit Details
The fits reveale
loop expansions a
find
log W11 = −
log W12
+
= −
+
The large 5α3V co
to agree with sim
The coupling αV
our lattice spacing
Wilson loop we e
These large coe
sults. There are
ficients. One is
2(n+m)
u0
where u
" Creutz ratios of th
!
W13
4
αV3 −0(2) α
+. . . Each procedure s
log
= 1.323 α3V (1.21/a) &
1−0.39(1)
αV −0.3(2)
αV2 +2(1)
10
V
2
n
2
% (Yi −
%
W22
(cn − cn )
n cn αV (d/ai ))
cients we obtain w
χ2 ≡
+
2
2
σYi
σcn
spacings: for exam
n=1
i=1
'
(2
)
*
3
(3)
αV (d/a)
0.8
0.6
0.4
0.2
2
4
6
d/a (GeV)
8
(3)
αV (d/a)
0.8
0.6
0.4
0.2
2
4
6
d/a (GeV)
8
(3)
αV (d/a)
0.8
0.6
0.4
0.2
2
4
6
d/a (GeV)
8
(3)
αV (d/a)
0.8
0.6
0.4
0.2
2
4
6
d/a (GeV)
8
(3)
αV (d/a)
0.8
0.6
0.4
0.2
2
4
6
d/a (GeV)
8
(3)
αV (d/a)
0.8
0.6
0.4
0.2
2
4
6
d/a (GeV)
8
(3)
αV (d/a)
0.8
0.6
0.4
0.2
2
4
6
d/a (GeV)
8
(3)
αV (d/a)
0.8
0.6
0.4
0.2
2
4
6
d/a (GeV)
8
Systematic Errors
a−1
c1 . . . c3
cn for n ≥ 4
V → MS → MZ
condensate
mu , md , ms
mc
mb
simulation errors
total uncertainty
log W11 log W13 /W22
0.0007
0.0008
0.0001
0.0004
0.0008
0.0005
0.0005
0.0006
0.0002
0.0001
0.0003
0.0001
0.0002
0.0002
0.0001
0.0001
0.0000
0.0000
0.0012
0.0012
√
V ( 2a) − V (a)
0.0008
0.0006
0.0006
0.0006
0.0001
0.0001
0.0002
0.0001
0.0001
0.0014
TABLE I: Sources of the uncertainties in our final determinations
(5)
of the coupling αMS (MZ ). These errors should be compared
in the loop
in u, d and s
larger mass
also negligi
the correctio
Our coup
simulation t
contribution
tions correc
both the he
been accura
third-order
estimates of
from 28 dif
an order of
(3)
ΛV
(MeV)
400 425 450
log W11
log W12
log W13
log W14
log W22
log W23
log WBR
log WCC
log W12 /u60
(3)
ΛV (MeV)
400 425 450
log W11
log W12
log W13
log W14
log W22
log W23
log WBR
log WCC
log W12 /u60
log W13 /u80
log W14 /u10
0
log W22 /u80
log W23 /u10
0
log WBR /u60
log WCC /u60
log W /W
(3)
ΛV (MeV)
400 425 450
log W11
log W12
log W13
log W14
log W22
log W23
log WBR
log WCC
log W12 /u60
log W13 /u80
log W14 /u10
0
log W22 /u80
log W23 /u10
0
log WBR /u60
log WCC /u60
log W13 /W22
2
log W11 W22 /W12
3
log WCC WBR /W11
log WCC /WBR
log W14 /W23
log W11 W23 /W12 W13
√
V (√2a) − V (a)
(3)
ΛV (MeV)
400 425 450
log W11
log W12
log W13
log W14
log W22
log W23
log WBR
log WCC
log W12 /u60
log W13 /u80
log W14 /u10
0
log W22 /u80
log W23 /u10
0
log WBR /u60
log WCC /u60
log W13 /W22
2
log W11 W22 /W12
3
log WCC WBR /W11
log WCC /WBR
log W14 /W23
log W11 W23 /W12 W13
√
V (√2a) − V (a)
V ( 3a) − V (a)
V (2a)
√ − V (a)
V (√5a) − V (a)
V ( 6a) − V (a)
V (3a) − V (a)
αlat /W11
0.116 0.118 0.120
(5)
αMS (MZ )
PDG 2004 results
Average
Hadronic Jets
+ -
e e rates
Photo-production
Fragmentation
Z width
ep event shapes
Polarized DIS
Deep Inelastic Scattering (DIS)
! decays
Spectroscopy (Lattice)
" decay
0.1
0.12
#s(MZ)
0.14
2004
World
Average:
0.1187(20)
PDG 2004 results
Average
Hadronic Jets
+ -
e e rates
Photo-production
Fragmentation
Z width
2004
World
Average:
0.1187(20)
ep event shapes
Polarized DIS
Deep Inelastic Scattering (DIS)
! decays
Spectroscopy (Lattice)
" decay
0.1
0.12
#s(MZ)
My old
2-loop
result
0.14
3-loop
result:
0.1177(13)
Summary & Outlook
First 2+1 α determination; no extrapolation in nf
10 hadronic quantities agree on the lattice spacing
Three loop accuracy, three lattice spacings, and twentyeight operators
Lattice agrees with PDG and is more accurate
1% accuracy is limit without 4th order PT or super-fine
Hope to show convergence of nf=2 results between
Asqtad, Naive, Wilson and Clover
Sub-1% using super-fine Asqtad ?
Quentin Mason (Cambridge)