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Low energy chiral lagrangian parameters for scalar and pseudoscalar mesons
SUPPLEMENTS ELSEVIER
Nuclear Physics B (Proc. Suppl.) 119 (2003)
242-244
www.elsevier.com/locatclnpc
Low Energy Chiral Lagrangian Parameters for Scalar and Pseudoscalar Mesons W. Bardeena, E. Eichten*aand
H. Thackerb
aFermilab, PO Box 500, Batavia, IL 60510 bDept. of Physics, U niversity of Virginia,
Charlottesville,
VA 22901
We present results of a high-statistics study of scalar and pseudoscalar meson propagators in quenched QCD at two values of lattice spacing, p = 5.7 and 5.9, with clover-improved Wilson fermions. The study of the chiral limit is facilitated by the pole-shifting ansatz of the modified quenched approximation. Pseudoscalar masses and decay constants are determined as a function of quark mass and quenched chiral log effects are estimated. A study of the flavor singlet q’ hairpin diagram yields a precise determination of the 7’ mass insertion. The corresponding value of the quenched chiral log parameter 6 is compared with the observed QCL effects. Removal of QCL effects from the scalar propagator allows a determination of the mass of the lowest lying isovector scalar qtj meson.
1. Quenched
Chiral
Perturbation
Theory
The lowest order chiral Lagrangian mented by the chiral symmetry breaking L5 of O(p2mz) and La of O(mi) (which the mass-dependence of the chiral slope meson decay constants) is given by[l]:. L = $Tr(L$UtPU)
+ cTr(XtU
suppleterms model for the
M;
3 -+n~$(~Trln(Li’)
- iTrln(U))2
(1)
fP;ij=
JZfro(1
-
- LS)Xij[l
$L,axij[(
+
(3)
xi = 2romi, xij = 1/(2~i) - l/(2&,)). and jij = Jijfxij. are defined in [l]. constants:
+ S(fij + Iij)]
+ Iij) - (jii + Jjj)]}
(4)
While for the axial vector decay constants: f*;ij
= &f(l
+ 0*256(Iii + Ijj - 2Iij))
{ 1 + $XijL,[l masses and chiral
- Ls)xij[l
+ 0.25d(Iii + Ijj + 2Iij)){l
+ $(4&3
(2)
We use explicit formulas for the pseudoscalar masses, and pseudoscalar and axial vector decay constants, consistent through order p” (for details see [l]). The coefficients L5, Lg follow the notation of Gasser and Leutwyler [2]. 2. Pseudoscalar eters
+ $8(2La
where TOis a slope parameter, (xi + sj)/2 and rni q ln(1 + Here Iij = (Iiixi + Ijjxj)/xij The 100~ integrals Iij and Jij For the pseudoscalar decay
where Chairpin
+ SI,j){l
+ S(&j + Iij)] + ~8LsXij~~ij) l
+ utx)
+ Lgr(d,UWU(XW + UQ) + LsTr(XtUXtU + UtxUtx) + Lhairpin
= xij(l
+ d(PIij - ?ij)}
(5)
param-
squared up to first order in the hairpin mass, LS and Lg is:
We study 350 configurations on a 163 x 32 lattice with /3 = 5.9. Clover fermions 1.50) and the MQA proce(Csw = dure[3] was used. We consider six K values (.1397, .1394, .1391, .1388, .1385, .1382) with
*Presenter
IC, = .140143.
The
expression
0920-5632/03/$
- see front
for
matter
doi: 10.1016/S0920-5632(03)01514-7
the
pseudoscalar
0 2003 Published
mass
by Elsevier
Science
B.V.
The fits are shown
in Figures
l-2.
FV Burdeen
Diagonal
et ul. /Nuclear
Pion
Physics
B (Proc.
Suppl.)
119 (2003)
Chiral
Masses
243
242-244
Slope
Parameter
1.7 12( 39)
2.0
--I
0.20 x
0.15 -
*
N* E
CI 0.10 -
i) LI
0.05 -
I
0.00 * 0.00
0.01
0.02 quark
Diagonal
0.03 0.04 mass
Pseudoscalar
0.05
0.06
i.6 c 0
10
Decay
Pseudoscalar
Constant
0.22
1
Decay
30
20 scan
(*,,nJ
Constant
f-ps
= 0.1501(27)
0.22 L
-7
0.21 0.20 & ,’
0.19 0.16 -
I-
0.14
0.00
0.05
0.10 z
0.15
0.20
ala’
I
0
10
Diagonal
Axial
Vector
Decay
0.14
Axial
Constant
0.16
Vector
30
20
(np$
m,
Decay
scan
Constant
f-ax
= 0.0915(13)
I
1
:::Fr-1
0.12 2 2
0.10 -
x
f
I
5
0.12
x
x
x 0.11
P
f
xx
x 0.10
$
xx
I
xx xx
x
I
xx
I.
x
I XX
XXX
I f
$
I
1
0.06 -
006t . 0.00
0.05
0.10
m,
Figure
2
0.15
0.20
1. rnz, fps and fax versus diagonal masses.
o.oet
0
’ 10 (K,JC,)
I 20 scan
I 30
Figure 2. rni, fps and fax and ratios to chiral fits.
I+! Bardeen et al. /Nuclear Physics B (Proc. Suppl.) I19 (2003) 242-244
244
0.01 0
’ 5
’ 10
’ 15
II
-0.002’
0
.
’ 5
’ 10
’ 15
1
T
Figure 3. Hairpin propagattr for K.= .1394.
3. Hairpins
and Scalar
and double pole fit
Propagators
In the quenched theory the disconnected part of the eta prime correlator (hairpin term) has
Figure 4. Isovector scalar correlators for K~ = K~ = .1397 (0) and K~ = K@= .1382 (x).
1
Parameter
f = f7T
( p = 5.9 ( 0.091(2)
1 p = 5.7 1 0.100(2)
a double pole (~z~~\)2 whose coefficient determines me. The corr”elator for K = .1394 and a double pole fit(m, = 0.261 and me = 0.211) is shown in Figure 3. The S parameter is given by:
As the quenched chiral limit is approached the isovector scalar correlator shows negative norm behaviour. The correlators for the lightest and heaviest K values are shown in Figure 4. Properly accounting for the effects of the hairpin - pion bubble[4] allows a good fit of the isovector scalar correlator for all quark masses. One output is the (a,-,) mass.
and .613(17) for m, = .666,.612,.553,.492,.423 and .330 GeV respectively. Extrapolating to - 0 we obtain mo = .642(15). The corresniing value for p = 5.7 is mo = .548(25). Extracting an and al masses (at /cc) from scalar and axial vector propagators gives ma,, = 1.33(5) and 1.34(6) GeV and m,, = 1.27(4) and 1.12(8) Gev for p = 5.9 and 5.7 respectively. The value for m,o = 1.330(50) GeV suggests that the observed as (980) resonance is a KK “molecule” and not an ordinary qQ meson.
4. Results
REFERENCES
Using the MQA technique, meson properties (masses and decay constants) can be extracted with sufficient accuracy to allow a fit of higher order chiral parameters, L5 and LB. The physical results for p = 5.9 (l/alp = 1.619 GeV ZA = 0.865) are compared with our /3 = 5.7 (l/al, = 1.115 GeV ZA = 0.845) results[l]. The physical values of me (in GeV) extracted from the hairpin analysis (at /? = 5.9) are .526(14), .533(14), .554(14), .580(14), .606(014)
1. W. Bardeen, A. Duncan, E. Eichten, and H. Thacker, Phys. Rev. D62 (2000) 114505. 2. J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465. 3. W. Bardeen, A. Duncan, E. Eichten, and H. Thacker, Phys. Rev. D57 (1998) 1633; Phys. Rev. D59 (1999) 014507. 4. W. Bardeen, A. Duncan, E. Eichten, N. Isgur and H. Thacker, Phys. Rev. D 65 (2002) 014509.
(6)
Nuclear Physics B (Proc. Suppl.) 119 (2003)
242-244
www.elsevier.com/locatclnpc
Low Energy Chiral Lagrangian Parameters for Scalar and Pseudoscalar Mesons W. Bardeena, E. Eichten*aand
H. Thackerb
aFermilab, PO Box 500, Batavia, IL 60510 bDept. of Physics, U niversity of Virginia,
Charlottesville,
VA 22901
We present results of a high-statistics study of scalar and pseudoscalar meson propagators in quenched QCD at two values of lattice spacing, p = 5.7 and 5.9, with clover-improved Wilson fermions. The study of the chiral limit is facilitated by the pole-shifting ansatz of the modified quenched approximation. Pseudoscalar masses and decay constants are determined as a function of quark mass and quenched chiral log effects are estimated. A study of the flavor singlet q’ hairpin diagram yields a precise determination of the 7’ mass insertion. The corresponding value of the quenched chiral log parameter 6 is compared with the observed QCL effects. Removal of QCL effects from the scalar propagator allows a determination of the mass of the lowest lying isovector scalar qtj meson.
1. Quenched
Chiral
Perturbation
Theory
The lowest order chiral Lagrangian mented by the chiral symmetry breaking L5 of O(p2mz) and La of O(mi) (which the mass-dependence of the chiral slope meson decay constants) is given by[l]:. L = $Tr(L$UtPU)
+ cTr(XtU
suppleterms model for the
M;
3 -+n~$(~Trln(Li’)
- iTrln(U))2
(1)
fP;ij=
JZfro(1
-
- LS)Xij[l
$L,axij[(
+
(3)
xi = 2romi, xij = 1/(2~i) - l/(2&,)). and jij = Jijfxij. are defined in [l]. constants:
+ S(fij + Iij)]
+ Iij) - (jii + Jjj)]}
(4)
While for the axial vector decay constants: f*;ij
= &f(l
+ 0*256(Iii + Ijj - 2Iij))
{ 1 + $XijL,[l masses and chiral
- Ls)xij[l
+ 0.25d(Iii + Ijj + 2Iij)){l
+ $(4&3
(2)
We use explicit formulas for the pseudoscalar masses, and pseudoscalar and axial vector decay constants, consistent through order p” (for details see [l]). The coefficients L5, Lg follow the notation of Gasser and Leutwyler [2]. 2. Pseudoscalar eters
+ $8(2La
where TOis a slope parameter, (xi + sj)/2 and rni q ln(1 + Here Iij = (Iiixi + Ijjxj)/xij The 100~ integrals Iij and Jij For the pseudoscalar decay
where Chairpin
+ SI,j){l
+ S(&j + Iij)] + ~8LsXij~~ij) l
+ utx)
+ Lgr(d,UWU(XW + UQ) + LsTr(XtUXtU + UtxUtx) + Lhairpin
= xij(l
+ d(PIij - ?ij)}
(5)
param-
squared up to first order in the hairpin mass, LS and Lg is:
We study 350 configurations on a 163 x 32 lattice with /3 = 5.9. Clover fermions 1.50) and the MQA proce(Csw = dure[3] was used. We consider six K values (.1397, .1394, .1391, .1388, .1385, .1382) with
*Presenter
IC, = .140143.
The
expression
0920-5632/03/$
- see front
for
matter
doi: 10.1016/S0920-5632(03)01514-7
the
pseudoscalar
0 2003 Published
mass
by Elsevier
Science
B.V.
The fits are shown
in Figures
l-2.
FV Burdeen
Diagonal
et ul. /Nuclear
Pion
Physics
B (Proc.
Suppl.)
119 (2003)
Chiral
Masses
243
242-244
Slope
Parameter
1.7 12( 39)
2.0
--I
0.20 x
0.15 -
*
N* E
CI 0.10 -
i) LI
0.05 -
I
0.00 * 0.00
0.01
0.02 quark
Diagonal
0.03 0.04 mass
Pseudoscalar
0.05
0.06
i.6 c 0
10
Decay
Pseudoscalar
Constant
0.22
1
Decay
30
20 scan
(*,,nJ
Constant
f-ps
= 0.1501(27)
0.22 L
-7
0.21 0.20 & ,’
0.19 0.16 -
I-
0.14
0.00
0.05
0.10 z
0.15
0.20
ala’
I
0
10
Diagonal
Axial
Vector
Decay
0.14
Axial
Constant
0.16
Vector
30
20
(np$
m,
Decay
scan
Constant
f-ax
= 0.0915(13)
I
1
:::Fr-1
0.12 2 2
0.10 -
x
f
I
5
0.12
x
x
x 0.11
P
f
xx
x 0.10
$
xx
I
xx xx
x
I
xx
I.
x
I XX
XXX
I f
$
I
1
0.06 -
006t . 0.00
0.05
0.10
m,
Figure
2
0.15
0.20
1. rnz, fps and fax versus diagonal masses.
o.oet
0
’ 10 (K,JC,)
I 20 scan
I 30
Figure 2. rni, fps and fax and ratios to chiral fits.
I+! Bardeen et al. /Nuclear Physics B (Proc. Suppl.) I19 (2003) 242-244
244
0.01 0
’ 5
’ 10
’ 15
II
-0.002’
0
.
’ 5
’ 10
’ 15
1
T
Figure 3. Hairpin propagattr for K.= .1394.
3. Hairpins
and Scalar
and double pole fit
Propagators
In the quenched theory the disconnected part of the eta prime correlator (hairpin term) has
Figure 4. Isovector scalar correlators for K~ = K~ = .1397 (0) and K~ = K@= .1382 (x).
1
Parameter
f = f7T
( p = 5.9 ( 0.091(2)
1 p = 5.7 1 0.100(2)
a double pole (~z~~\)2 whose coefficient determines me. The corr”elator for K = .1394 and a double pole fit(m, = 0.261 and me = 0.211) is shown in Figure 3. The S parameter is given by:
As the quenched chiral limit is approached the isovector scalar correlator shows negative norm behaviour. The correlators for the lightest and heaviest K values are shown in Figure 4. Properly accounting for the effects of the hairpin - pion bubble[4] allows a good fit of the isovector scalar correlator for all quark masses. One output is the (a,-,) mass.
and .613(17) for m, = .666,.612,.553,.492,.423 and .330 GeV respectively. Extrapolating to - 0 we obtain mo = .642(15). The corresniing value for p = 5.7 is mo = .548(25). Extracting an and al masses (at /cc) from scalar and axial vector propagators gives ma,, = 1.33(5) and 1.34(6) GeV and m,, = 1.27(4) and 1.12(8) Gev for p = 5.9 and 5.7 respectively. The value for m,o = 1.330(50) GeV suggests that the observed as (980) resonance is a KK “molecule” and not an ordinary qQ meson.
4. Results
REFERENCES
Using the MQA technique, meson properties (masses and decay constants) can be extracted with sufficient accuracy to allow a fit of higher order chiral parameters, L5 and LB. The physical results for p = 5.9 (l/alp = 1.619 GeV ZA = 0.865) are compared with our /3 = 5.7 (l/al, = 1.115 GeV ZA = 0.845) results[l]. The physical values of me (in GeV) extracted from the hairpin analysis (at /? = 5.9) are .526(14), .533(14), .554(14), .580(14), .606(014)
1. W. Bardeen, A. Duncan, E. Eichten, and H. Thacker, Phys. Rev. D62 (2000) 114505. 2. J. Gasser and H. Leutwyler, Nucl. Phys. B250 (1985) 465. 3. W. Bardeen, A. Duncan, E. Eichten, and H. Thacker, Phys. Rev. D57 (1998) 1633; Phys. Rev. D59 (1999) 014507. 4. W. Bardeen, A. Duncan, E. Eichten, N. Isgur and H. Thacker, Phys. Rev. D 65 (2002) 014509.
(6)