- Email: [email protected]
Vacuum polarisation and the muon g − 2 in lattice QCD
SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 293-295
Vacuum polarisation
and the muon g - 2 in lattice &CD*
M. Gijckeler”>b, R. Horsley’,
W. Kiirzinger d+, D. Pleiterd, P.E.L. Rakowf and G. Schierholzd,g
“Institut
fiir Theoretische Physik, Universitgt
Leipzig, D-04109 Leipzig, Germany
bInstitut
fiir Theoretische Physik, Universitgt
Regensburg, D-93040 Regensburg, Germany
“School of Physics, University dJohn von Neumann-Institut Vnstitut
fiir Theoretische
fDepartment
of Edinburgh, fir Computing
Edinburgh
NIC, DESY Zeuthen, D-15735 Zeuthen, Germany
Physik, Freie Universitst
of Mathematical
Sciences, University
gDeutsches Elektronen-Synchrotron
EH9 352, UK
Berlin, D-14195 Berlin, Germany of Liverpool,
DESY, D-22603 Hamburg,
Liverpool
L69 3BX, UK
Germany
We measure the hadronic contribution to the vacuum polarisation tensor, and use it to estimate the hadronic contribution to (g - 2),, the muon anomalous magnetic moment. 1. Introduction The vacuum polarisation
J&(q)
= i 5
II
is defined by
There is an important connecting R and II
set of dispersion relations
d4x e’““(OlTJ,(z)J,(O))O)
s (q/lqv- q2gpv)WQ”>.
Jp is the electromagnetic current, Q2 = -4 . q. The l-loop diagram is logarithmically divergent, II is additively renormalised. The value of II can be shifted up and down by a constant depending on scheme and scale without any physical effects, but the Q2 dependence of II(Q2) is physically meaningful, and it must be independent of scheme or regularisation. The vacuum polarisation tensor enters physics in several important ways. It is responsible for the running of o?, which must be known very precisely for high-precision electro-magnetic calculations (for example of g - 2). Secondly, the optical theorem implies that the cross-section for e+e- -+ hadrons is proportional to the imaginary part of H at timelike q. This cross-section is usually given in terms of the ratio R, %z+e-+hadro&) R(s) = ge+e-+p+p(S) . *Talk presented by P. &kow 0920-5632/$ - see front matter 0 2004 Elsevier B.V doi:l0.1016/j.nuclphysbps.2003.12.137
Z-Z
(1)
(2)
Q2 h,
2. The lattice
R(s) d”(s + Q2),+l
*
calculation
Our present calculation is carried out in the quenched approximation, using clover fermions and an O(a) improved electromagnetic current. We have used three ,8 values (6.0, 6.2 and 6.4) with several quark masses in each case. Full details of the calculation can be found in [l]. II can be split into two parts, a fermion line connected contribution Cn, and a disconnected contribution An, Fig. 1. -127r21T(Q2) = c
e; CII(Q”, mf>
(4) fJ’
We only calculate Cn, but the term An (which violates Zweig’s rule) is expected to be very small. In Fig. 2 we compare our lattice data with continuum perturbation theory. Firstly, we see very
All rights reserved
294
M. Giickeler et al. /Nuclear Physics B (Pmt. Suppl.) 129&130 (2004) 293-295
persion relations eq.(3). These relations connecting R(s) to II(Q2) are very stable, so even a crude model of R is likely to give a good estimate of II. The simplest model of R consists of &function contributions from the vector mesons (p,w, 4) and a flat continuum beginning at a threshold so, R(s) = c
e2f (A6(s - m$) + BO(s - SO)) .
(5)
f Using the dispersion relation gives Figure 1. Examples of fermion-line-connected and disconnected contributions to the vacuum polarisation. good agreement between the two lattice sizes, showing that finite size effects are not a major problem. Perturbation theory works very well except at the largest Q2 values, where discretisation errors start to show up, and at low Q2 (below N 2 GeV2). Both lattice sizes agree, so this is not a finite size effect. Can we understand this in terms of non-perturbative physics?
CII(&~) = Bln[a2(Q2 + so)] -
A
Q2
+ m$ + K-(6)
We have already measured mv and fv (the decay constant) so the weight and position of the Sfunction are already known. B and se are found by fitting. As seen in Fig. 3, eq.(6) gives an excellent fit, and can be used to extrapolate our data to lower Q2. -20 % -25
-20 CII -25
-40
Figure 3. Eq. (6) compared with the lattice data. -45
10-l
1
10’
Q'[GeV']
lo2
3. The muon anomalous Figure 2. A check forjinite size eflects, comparing a 164 lattice (white) and a 324 lattice (black) at ,8 = 6.0, n = 0.1345. The curves show continuum perturbation theory. The dashed curve is at zero quark mass, the solid curve includes mass eflects. One approach,
introduced
in [2], uses the dis-
magnetic
moment
The anomalous magnetic moment of the muon can be calculated to very high order in QED (5 loop), and measured very precisely. (g - 2), is more sensitive to high-energy physics than (g - 2),, by a factor rnE/rnz, so it is a more promising place to look for signs of new physics, but to identify new physics we need to know the conventional contributions very accurately. QED
M. Giickeler et al. /Nuclear Physics B (Proc. Suppl.) 129&I30
perturbative calculations take good account of muon and electron loops, but at the two-loop level quarks can be produced, which in turn will produce gluons. The dominant contribution comes from photons with virtualities N rni, which is a region where QCD perturbation theory will not work well. The traditional route for estimating these hadronic contributions is by using a dispersion relation involving the cross-section R(s).
a.045
By distorting the contour of integration be rewritten as had
=p
,
0.035
1 L
0.03
0.025
1 f
0 02 1 t 0
0.8
0.4 m;,
0.6
0.8
-
_I ‘I 1 11 I,1 1
[Gef]
Figure 4. The results for the integral If, extrapolations to the continuum limit.
%Tn
and our
c
f
4. Conclusions
where lf(mf)
,
this can
2
al,
295
If
0.015
(7)
(2004) 293-295
=lw$$F
($)
[G(Q2)
- Cfl(O)](9)
The kernel in this relation is
For each data set we calculate the integral If. The integral is dominated by Q2 N 3mi, so some Q2 extrapolation is needed. In Fig. 4 we extrapolate the integral 1f to the continuum limit and to the physical quark masses (m$ = rng, 2m& - ma) using the ansatz If = (Al + A2a2) + (I31 + &a”)m&.
(11)
We have seen that perturbation theory works above Q2 N 2 GeV2. We can describe the entire Q2 region very well with a dispersion relation using a simple model of R(s) [2]. From the low Q2 region of the vacuum polarisation we can extract a lattice value for upd, the hadronic contribution to the muon’s anomalous magnetic moment, which is of the right order of magnitude. Our estimate could be improved by using a larger lattice size, enabling us to reach lower Q2 which would reduce uncertainties from extrapolation. Naturally, dynamical calculations would be very interesting. Acknowledgements This work was supported by the European Community’s Human Potential Program and by the DFG and BMBF.
Our values are I” = Id = 0.0389(21);
1’ = 0.0287(g) .
(12)
Adding these values, we get our final answer ahad - a2em 41”+Id+IS = 446(23) x lo-l’(13) P 37r2 9 This value agrees with the pioneering lattice calculation of [3], apd = 460(78) x lo-lo. However both lattice values lie lower than the experimental value 683.6(8.6) x 10-r’ [4]. Possibly this shortfall is due to the absence of two-pion states in the quenched calculation.
REFERENCES Ph.D. Thesis (in Ger1. W. Ktizinger, Freie Universitgt Berlin (2001); man), http://www.diss.fu-berlin.de/2001/220/ 2. M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 448. 3. T. Blum, Phys. Rev. Lett. 91 (2003) 052001 4. F. Jegerlehner, J. Phys. G 29 (2003) 101; K. Hagiwara et al., Phys. Lett. B 557 (2003) 69.
Nuclear Physics B (Proc. Suppl.) 129&130 (2004) 293-295
Vacuum polarisation
and the muon g - 2 in lattice &CD*
M. Gijckeler”>b, R. Horsley’,
W. Kiirzinger d+, D. Pleiterd, P.E.L. Rakowf and G. Schierholzd,g
“Institut
fiir Theoretische Physik, Universitgt
Leipzig, D-04109 Leipzig, Germany
bInstitut
fiir Theoretische Physik, Universitgt
Regensburg, D-93040 Regensburg, Germany
“School of Physics, University dJohn von Neumann-Institut Vnstitut
fiir Theoretische
fDepartment
of Edinburgh, fir Computing
Edinburgh
NIC, DESY Zeuthen, D-15735 Zeuthen, Germany
Physik, Freie Universitst
of Mathematical
Sciences, University
gDeutsches Elektronen-Synchrotron
EH9 352, UK
Berlin, D-14195 Berlin, Germany of Liverpool,
DESY, D-22603 Hamburg,
Liverpool
L69 3BX, UK
Germany
We measure the hadronic contribution to the vacuum polarisation tensor, and use it to estimate the hadronic contribution to (g - 2),, the muon anomalous magnetic moment. 1. Introduction The vacuum polarisation
J&(q)
= i 5
II
is defined by
There is an important connecting R and II
set of dispersion relations
d4x e’““(OlTJ,(z)J,(O))O)
s (q/lqv- q2gpv)WQ”>.
Jp is the electromagnetic current, Q2 = -4 . q. The l-loop diagram is logarithmically divergent, II is additively renormalised. The value of II can be shifted up and down by a constant depending on scheme and scale without any physical effects, but the Q2 dependence of II(Q2) is physically meaningful, and it must be independent of scheme or regularisation. The vacuum polarisation tensor enters physics in several important ways. It is responsible for the running of o?, which must be known very precisely for high-precision electro-magnetic calculations (for example of g - 2). Secondly, the optical theorem implies that the cross-section for e+e- -+ hadrons is proportional to the imaginary part of H at timelike q. This cross-section is usually given in terms of the ratio R, %z+e-+hadro&) R(s) = ge+e-+p+p(S) . *Talk presented by P. &kow 0920-5632/$ - see front matter 0 2004 Elsevier B.V doi:l0.1016/j.nuclphysbps.2003.12.137
Z-Z
(1)
(2)
Q2 h,
2. The lattice
R(s) d”(s + Q2),+l
*
calculation
Our present calculation is carried out in the quenched approximation, using clover fermions and an O(a) improved electromagnetic current. We have used three ,8 values (6.0, 6.2 and 6.4) with several quark masses in each case. Full details of the calculation can be found in [l]. II can be split into two parts, a fermion line connected contribution Cn, and a disconnected contribution An, Fig. 1. -127r21T(Q2) = c
e; CII(Q”, mf>
(4) fJ’
We only calculate Cn, but the term An (which violates Zweig’s rule) is expected to be very small. In Fig. 2 we compare our lattice data with continuum perturbation theory. Firstly, we see very
All rights reserved
294
M. Giickeler et al. /Nuclear Physics B (Pmt. Suppl.) 129&130 (2004) 293-295
persion relations eq.(3). These relations connecting R(s) to II(Q2) are very stable, so even a crude model of R is likely to give a good estimate of II. The simplest model of R consists of &function contributions from the vector mesons (p,w, 4) and a flat continuum beginning at a threshold so, R(s) = c
e2f (A6(s - m$) + BO(s - SO)) .
(5)
f Using the dispersion relation gives Figure 1. Examples of fermion-line-connected and disconnected contributions to the vacuum polarisation. good agreement between the two lattice sizes, showing that finite size effects are not a major problem. Perturbation theory works very well except at the largest Q2 values, where discretisation errors start to show up, and at low Q2 (below N 2 GeV2). Both lattice sizes agree, so this is not a finite size effect. Can we understand this in terms of non-perturbative physics?
CII(&~) = Bln[a2(Q2 + so)] -
A
Q2
+ m$ + K-(6)
We have already measured mv and fv (the decay constant) so the weight and position of the Sfunction are already known. B and se are found by fitting. As seen in Fig. 3, eq.(6) gives an excellent fit, and can be used to extrapolate our data to lower Q2. -20 % -25
-20 CII -25
-40
Figure 3. Eq. (6) compared with the lattice data. -45
10-l
1
10’
Q'[GeV']
lo2
3. The muon anomalous Figure 2. A check forjinite size eflects, comparing a 164 lattice (white) and a 324 lattice (black) at ,8 = 6.0, n = 0.1345. The curves show continuum perturbation theory. The dashed curve is at zero quark mass, the solid curve includes mass eflects. One approach,
introduced
in [2], uses the dis-
magnetic
moment
The anomalous magnetic moment of the muon can be calculated to very high order in QED (5 loop), and measured very precisely. (g - 2), is more sensitive to high-energy physics than (g - 2),, by a factor rnE/rnz, so it is a more promising place to look for signs of new physics, but to identify new physics we need to know the conventional contributions very accurately. QED
M. Giickeler et al. /Nuclear Physics B (Proc. Suppl.) 129&I30
perturbative calculations take good account of muon and electron loops, but at the two-loop level quarks can be produced, which in turn will produce gluons. The dominant contribution comes from photons with virtualities N rni, which is a region where QCD perturbation theory will not work well. The traditional route for estimating these hadronic contributions is by using a dispersion relation involving the cross-section R(s).
a.045
By distorting the contour of integration be rewritten as had
=p
,
0.035
1 L
0.03
0.025
1 f
0 02 1 t 0
0.8
0.4 m;,
0.6
0.8
-
_I ‘I 1 11 I,1 1
[Gef]
Figure 4. The results for the integral If, extrapolations to the continuum limit.
%Tn
and our
c
f
4. Conclusions
where lf(mf)
,
this can
2
al,
295
If
0.015
(7)
(2004) 293-295
=lw$$F
($)
[G(Q2)
- Cfl(O)](9)
The kernel in this relation is
For each data set we calculate the integral If. The integral is dominated by Q2 N 3mi, so some Q2 extrapolation is needed. In Fig. 4 we extrapolate the integral 1f to the continuum limit and to the physical quark masses (m$ = rng, 2m& - ma) using the ansatz If = (Al + A2a2) + (I31 + &a”)m&.
(11)
We have seen that perturbation theory works above Q2 N 2 GeV2. We can describe the entire Q2 region very well with a dispersion relation using a simple model of R(s) [2]. From the low Q2 region of the vacuum polarisation we can extract a lattice value for upd, the hadronic contribution to the muon’s anomalous magnetic moment, which is of the right order of magnitude. Our estimate could be improved by using a larger lattice size, enabling us to reach lower Q2 which would reduce uncertainties from extrapolation. Naturally, dynamical calculations would be very interesting. Acknowledgements This work was supported by the European Community’s Human Potential Program and by the DFG and BMBF.
Our values are I” = Id = 0.0389(21);
1’ = 0.0287(g) .
(12)
Adding these values, we get our final answer ahad - a2em 41”+Id+IS = 446(23) x lo-l’(13) P 37r2 9 This value agrees with the pioneering lattice calculation of [3], apd = 460(78) x lo-lo. However both lattice values lie lower than the experimental value 683.6(8.6) x 10-r’ [4]. Possibly this shortfall is due to the absence of two-pion states in the quenched calculation.
REFERENCES Ph.D. Thesis (in Ger1. W. Ktizinger, Freie Universitgt Berlin (2001); man), http://www.diss.fu-berlin.de/2001/220/ 2. M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 448. 3. T. Blum, Phys. Rev. Lett. 91 (2003) 052001 4. F. Jegerlehner, J. Phys. G 29 (2003) 101; K. Hagiwara et al., Phys. Lett. B 557 (2003) 69.