- Email: [email protected]
2K→ππ with physical final state
Nuclear Physics B (Proc. Suppl.) 140 (2005) 381–383 www.elsevierphysics.com
∆I = 3/2 K → ππ with physical final state Changhoan Kima a
Department of Physics, Columbia University, New York, NY, 10027
We present an on-shell ∆I = 3/2 K → ππ matrix element calculation. We achieve a ππ excited state with non-zero relative momentum by imposing anti-periodic boundary conditions on the π. By calculating the matrix element with a varying strange quark mass, we can interpolate to the on-shell matrix element.
1. Introduction It has been a challenging problem to calculate K → ππ matrix elements because of the strong interaction effects. In order to account for these effects, we employ lattice gauge theory. There were important lattice calculations of these matrix elements[1,2], in which they calculate K → π and K → |0 and then deduce K → ππ matrix elements using chiral perturbation theory. In this paper, we present a direct ∆I = 3/2, K → ππ matrix element calculation, using an interacting ππ final state with non-zero relative momentum. Although there are promising methods which have been successfully applied[3,4], it is much more difficult to extract excited states in lattice calculations than the ground state, which appears as a leading exponential in correlation functions. An idea to extract ππ states with non-zero relative momentum as a leading exponential is to impose anti-periodic boundary conditions on the π. To do this, we proposed applying the G-parity or H-parity operation on the boundary in [5]. We briefly review H-parity boundary conditions used this calculation. We define the H-parity operation as: H (u, d) = (−u, d)
(1)
a non-zero relative momentum. Since the conventional weak operators have ∆Iz = 1/2, the Kaon cannot decay into Iz = ±2 ππ. In order to circumvent this problem, we calculate matrix elements of different operators with ∆Iz = 3/2, Iz =3/2 , and rely on the Wigner-Eckart theorem OI=3/2 to deduce the conventional matrix elements, 3/2
K|O1/2 |ππ = I =3/2
z Those OI=3/2 can be deduced easily by applying lowering operators to the conventional quanties. For operators in the (27,1) representation,
(27,1) I =3/2
z (O(m) )I=3/2
±
H|π = −|π while
0
0
H|π = |π .
≡
(¯ sd)V −A (¯ ud)V −A ,
(3)
and for operators in the (8,8) representation, (8,8) I =3/2
z (O(m) )I=3/2
≡
(¯ sd)V −A (¯ ud)V +A ,
(4)
where the m subscript is attached for color mixed operators. We use the same notation for the corresponding matrix elements. To generate the ππ states we use an interpolating operator, p ¯ 1 )A† (x1 )]γ5 [A(x2 )u(x2 )]eip·x1 = [d(x Oππ x1 ,x2
Then, pions are eigenstates of H-parity, ±
2, 32 ; 1, 12 |2, 32 ; 12 , 12 3/2 K|O3/2 |ππ 2, 32 ; 2, 32 |2, 32 ; 12 , 12
(2)
·
¯ 3 )A† (x3 )]γ5 [A(x4 )u(x4 )]e−ip·x3 . (5) [d(x
x3 ,x4
2. States and operators Under H-parity boundary condition only π ± satisfy anti-periodic boundary conditions. Thus, only the Iz = ±2 ππ states will be forced to have 0920-5632/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2004.11.146
where A(x) is the Coulomb gauge fixing matrix. We vary the number of hypersurfaces, n to which the H-parity operation is applied in order to vary the relative momentum. Depending on n, we
382
C. Kim / Nuclear Physics B (Proc. Suppl.) 140 (2005) 381–383
Mass 5.568e-1(25) 7.034e-1(33) 8.005e-1(73) 8.90e-1(12)
Free ptl energy 0.54 0.67 0.78 0.87
Table 1 Measured ππ energies for various momenta.
(6)
separately, and average them. Thus, we expect relatively smaller fluctuations, for example, in the n = 1 case than the n = 3 case. For the Kaon, we use the operator OK = [¯ u(x)A† (x)]γ5 [A(y)s(y)]. (7) x,y
Note, we use a wall sink to calculate the needed matrix element normalizations. The correlation functions for the matrix elements calculation, p (Tππ )O(t)OK (TK ), Gππ O K (t) ≡ Oππ
(8)
behaves in the asymptotic region Tπ t TK Gππ O K (t) → e−mK (t−tK ) e−Eππ (tπ −t)
Lattice Calculation From Phenomolgical parametrization -30
0
100
200
300
400
500
Relative Momentum
Figure 1. Calculated values for the ππ phase shift with the relative momentum chosen for x-axis. lattice spacing of 1.3GeV. The ensemble is composed of 129 configurations and each configuration is separated by 200 sweeps. A sweep includes one heatbath update and 4 over relaxation updates. Forward and backward time propagators were constructed from linear combinations of propagators computed with periodic and antiperiodic boundary conditions in the time direction. This is equivalent to using an unphysical, doubled lattice in the time direction with periodic boundary conditions. The bare quark mass for the light quarks is 0.015 and the resulting pion mass is 352Mev. We have used a series of s-quark masses and the resulting Kaon masses range from 712MeV to 1.29GeV. 4. I = 2 ππ state and matrix elements
p ·0|Oππ |π + π + π + π + |O|K + K + |OK |0.
In order to extract on-shell quantities, we calculate matrix elements for each final state with various Kaon masses and then interpolate to the onshell point. The magnitude of 0|Oππ |π + π + and K + |OK |0 can be deduced from the correlation functions of ππ and Kaon states: p Gππ ππ (t) → |0|Oππ |π + π + |2 e−Eππ (tπ −t) GK K (t) → |K + |PK |0|2 e−mK (t−tK ) .
-10
-20
can √ vary the resulting relative momentum around nπ/16. We also choose p in Eq. 5 according to n. For n = 0, we set p = (0, 0, 0), for n = 1, π π π , 0, 0), (0, L , 0), (0, 0, L , 0, 0), for n = 2, p = (L π π π π π p = (L , L , 0) and for n = 3, p = ( L , L , L ). For n with multiple p’s, we calculate the correlation functions p p† (0)Oππ (t) Gππ,ππ (t) ≡ Oππ
0
Phase Shift
Momentum 0 π/L √ √2π/L 3π/L
(9) (10)
3. Simulation parameters In this calculation, we have used an ensemble of quenched lattice configurations that were generated with the DBW2 action, a 163 × 32 volume and a coupling β = 0.87 giving an inverse
Table 1 presents the resulting ππ energies, showing measurable shifts from those of the noninteracting ππ states. We then calculate phase shift values, plotted in Fig. 1 together with a fit to experimental values. We see significant disagreement for higher momenta, possibly the result of quenching errors. To extract the lattice matrix elements, we fit the Kaon, ππ and matrix element correlation functions simultaneously, thereby obtaining the most information and making the fits more stable. Typical data, for the operator O(27,1) , is shown in Fig. 2. The resulting, lattice-normalized matrix elements are shown in Tab. 2 and the interpolated, on-shell lattice matrix elements in Tab. 3. (27,1) because this Note, we omit the data for Om
383
C. Kim / Nuclear Physics B (Proc. Suppl.) 140 (2005) 381–383
O mK 0.46 0.54 0.62 0.69 0.81 0.91 1.00
× 100 0 -1.18(2) -1.22(2) -1.26(2) -1.28(3) -1.30(4) -1.29(4) -1.24(5)
O(8,8) × 100 mK 0 0.46 -8.9(1) 0.54 -8.3(1) 0.62 -7.9(1) 0.69 -7.5(1) 0.81 -7.0(1) 0.91 -6.5(1) 1.00 -6.0(1)
π/16 -1.99(3) -2.04(4) -2.08(5) -2.11(5) -2.13(7) -2.09(9) -2.0(1)
π/16 -8.6(2) -7.9(2) -7.4(2) -7.0(2) -6.3(2) -5.7(2) -5.1(2)
√ 2π/16 -5.4(2) -5.0(2) -4.6(3) -4.3(3) -3.9(4) -3.5(4) -3.1(5)
√ 3π/16 -1.9(1) -1.9(1) -1.9(1) -2.0(1) -2.1(2) -2.3(3) -2.5(4) √ 3π/16 -3.8(4) -3.6(4) -3.4(5) -3.1(6) -2.5(8) -2(1) -7e-1(1)
Oi × 100 Oi 0 (27,1) O -1.23(2) O(8,8) -8.2(1) (8,8) Om -29.9(5)
π/16 -2.11(6) -6.9(2) -28.2(8)
√
2π/16 -2.2(1) -3.9(4) -18(1)
π/16 -33.4(7) -31.3(7) -29.7(8) -28.5(8) -26.4(9) -24.4(9) -22.5(9)
√ 2π/16 -23.2(8) -21.7(9) -21(1) -20(1) -18(1) -17(1) -16(1)
√ 3π/16 -19(1) -18(1) -17(1) -17(2) -16(2) -14(3) -11(3)
Table 2 Lattice matrix elements of the three ∆I = 3/2 operators.
√ 3π/16 -2.3(3) -2(1) -14(3)
Table 3 On-shell matrix elements, interpolated to the point mK = Eππ
1
0.5
0
s-mass 0.08 s-mass 0.12 s-mass 0.16 s-mass 0.2 s-mass 0.28 s-mass 0.36 s-mass 0.44
-0.5
-1
(8,8)
Om × 100 mK 0 0.46 -32.1(6) 0.54 -30.2(5) 0.62 -28.8(6) 0.69 -27.8(6) 0.81 -26.0(6) 0.91 -24.4(6) 1.00 -22.7(6)
√ 2π/16 -1.88(5) -1.94(6) -2.01(8) -2.07(9) -2.2(1) -2.2(1) -2.2(2)
Effective Mass
(27,1)
0
5
10
15
20
25
Time
Figure 2. Effective mass difference plot for the O(27,1) operator for various strange quark masses. Here the source is at t = 0 and the sink at t = 22. elements is still underway, the lattice matrix elements from 129 gauge configurations are quite accurate with statistical errors less than 5% for the ππ final states with lower relative momentum. Thus, calculations with more realistic kinematics appear promising. REFERENCES
operator is identical to O(27,1) through a Fierz transfromation. 5. Conclusions By imposing H-parity boundary condition in various directions we can have ππ states with different relative momentum in same volume. We calculate the phase shift for each relative momentum. However, the phase shift for ππ states with higher relative momentum differs from experimental data and requires more study. Although the final analysis to deduce the physical matrix
1. T. Blum et al. [RBC Collaboration], Phys. Rev. D 68, 114506 (2003) [arXiv:heplat/0110075]. 2. J. I. Noaki et al. [CP-PACS Collaboration], Phys. Rev. D 68, 014501 (2003) [arXiv:heplat/0108013]. 3. M. Luscher and U. Wolff, Nucl. Phys. B 339, 222 (1990). 4. T. Yamazaki et al. [CP-PACS Collaboration], arXiv:hep-lat/0402025. 5. C. h. Kim and N. H. Christ, Nucl. Phys. Proc. Suppl. 119, 365 (2003) [arXiv:heplat/0210003].
∆I = 3/2 K → ππ with physical final state Changhoan Kima a
Department of Physics, Columbia University, New York, NY, 10027
We present an on-shell ∆I = 3/2 K → ππ matrix element calculation. We achieve a ππ excited state with non-zero relative momentum by imposing anti-periodic boundary conditions on the π. By calculating the matrix element with a varying strange quark mass, we can interpolate to the on-shell matrix element.
1. Introduction It has been a challenging problem to calculate K → ππ matrix elements because of the strong interaction effects. In order to account for these effects, we employ lattice gauge theory. There were important lattice calculations of these matrix elements[1,2], in which they calculate K → π and K → |0 and then deduce K → ππ matrix elements using chiral perturbation theory. In this paper, we present a direct ∆I = 3/2, K → ππ matrix element calculation, using an interacting ππ final state with non-zero relative momentum. Although there are promising methods which have been successfully applied[3,4], it is much more difficult to extract excited states in lattice calculations than the ground state, which appears as a leading exponential in correlation functions. An idea to extract ππ states with non-zero relative momentum as a leading exponential is to impose anti-periodic boundary conditions on the π. To do this, we proposed applying the G-parity or H-parity operation on the boundary in [5]. We briefly review H-parity boundary conditions used this calculation. We define the H-parity operation as: H (u, d) = (−u, d)
(1)
a non-zero relative momentum. Since the conventional weak operators have ∆Iz = 1/2, the Kaon cannot decay into Iz = ±2 ππ. In order to circumvent this problem, we calculate matrix elements of different operators with ∆Iz = 3/2, Iz =3/2 , and rely on the Wigner-Eckart theorem OI=3/2 to deduce the conventional matrix elements, 3/2
K|O1/2 |ππ = I =3/2
z Those OI=3/2 can be deduced easily by applying lowering operators to the conventional quanties. For operators in the (27,1) representation,
(27,1) I =3/2
z (O(m) )I=3/2
±
H|π = −|π while
0
0
H|π = |π .
≡
(¯ sd)V −A (¯ ud)V −A ,
(3)
and for operators in the (8,8) representation, (8,8) I =3/2
z (O(m) )I=3/2
≡
(¯ sd)V −A (¯ ud)V +A ,
(4)
where the m subscript is attached for color mixed operators. We use the same notation for the corresponding matrix elements. To generate the ππ states we use an interpolating operator, p ¯ 1 )A† (x1 )]γ5 [A(x2 )u(x2 )]eip·x1 = [d(x Oππ x1 ,x2
Then, pions are eigenstates of H-parity, ±
2, 32 ; 1, 12 |2, 32 ; 12 , 12 3/2 K|O3/2 |ππ 2, 32 ; 2, 32 |2, 32 ; 12 , 12
(2)
·
¯ 3 )A† (x3 )]γ5 [A(x4 )u(x4 )]e−ip·x3 . (5) [d(x
x3 ,x4
2. States and operators Under H-parity boundary condition only π ± satisfy anti-periodic boundary conditions. Thus, only the Iz = ±2 ππ states will be forced to have 0920-5632/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysbps.2004.11.146
where A(x) is the Coulomb gauge fixing matrix. We vary the number of hypersurfaces, n to which the H-parity operation is applied in order to vary the relative momentum. Depending on n, we
382
C. Kim / Nuclear Physics B (Proc. Suppl.) 140 (2005) 381–383
Mass 5.568e-1(25) 7.034e-1(33) 8.005e-1(73) 8.90e-1(12)
Free ptl energy 0.54 0.67 0.78 0.87
Table 1 Measured ππ energies for various momenta.
(6)
separately, and average them. Thus, we expect relatively smaller fluctuations, for example, in the n = 1 case than the n = 3 case. For the Kaon, we use the operator OK = [¯ u(x)A† (x)]γ5 [A(y)s(y)]. (7) x,y
Note, we use a wall sink to calculate the needed matrix element normalizations. The correlation functions for the matrix elements calculation, p (Tππ )O(t)OK (TK ), Gππ O K (t) ≡ Oππ
(8)
behaves in the asymptotic region Tπ t TK Gππ O K (t) → e−mK (t−tK ) e−Eππ (tπ −t)
Lattice Calculation From Phenomolgical parametrization -30
0
100
200
300
400
500
Relative Momentum
Figure 1. Calculated values for the ππ phase shift with the relative momentum chosen for x-axis. lattice spacing of 1.3GeV. The ensemble is composed of 129 configurations and each configuration is separated by 200 sweeps. A sweep includes one heatbath update and 4 over relaxation updates. Forward and backward time propagators were constructed from linear combinations of propagators computed with periodic and antiperiodic boundary conditions in the time direction. This is equivalent to using an unphysical, doubled lattice in the time direction with periodic boundary conditions. The bare quark mass for the light quarks is 0.015 and the resulting pion mass is 352Mev. We have used a series of s-quark masses and the resulting Kaon masses range from 712MeV to 1.29GeV. 4. I = 2 ππ state and matrix elements
p ·0|Oππ |π + π + π + π + |O|K + K + |OK |0.
In order to extract on-shell quantities, we calculate matrix elements for each final state with various Kaon masses and then interpolate to the onshell point. The magnitude of 0|Oππ |π + π + and K + |OK |0 can be deduced from the correlation functions of ππ and Kaon states: p Gππ ππ (t) → |0|Oππ |π + π + |2 e−Eππ (tπ −t) GK K (t) → |K + |PK |0|2 e−mK (t−tK ) .
-10
-20
can √ vary the resulting relative momentum around nπ/16. We also choose p in Eq. 5 according to n. For n = 0, we set p = (0, 0, 0), for n = 1, π π π , 0, 0), (0, L , 0), (0, 0, L , 0, 0), for n = 2, p = (L π π π π π p = (L , L , 0) and for n = 3, p = ( L , L , L ). For n with multiple p’s, we calculate the correlation functions p p† (0)Oππ (t) Gππ,ππ (t) ≡ Oππ
0
Phase Shift
Momentum 0 π/L √ √2π/L 3π/L
(9) (10)
3. Simulation parameters In this calculation, we have used an ensemble of quenched lattice configurations that were generated with the DBW2 action, a 163 × 32 volume and a coupling β = 0.87 giving an inverse
Table 1 presents the resulting ππ energies, showing measurable shifts from those of the noninteracting ππ states. We then calculate phase shift values, plotted in Fig. 1 together with a fit to experimental values. We see significant disagreement for higher momenta, possibly the result of quenching errors. To extract the lattice matrix elements, we fit the Kaon, ππ and matrix element correlation functions simultaneously, thereby obtaining the most information and making the fits more stable. Typical data, for the operator O(27,1) , is shown in Fig. 2. The resulting, lattice-normalized matrix elements are shown in Tab. 2 and the interpolated, on-shell lattice matrix elements in Tab. 3. (27,1) because this Note, we omit the data for Om
383
C. Kim / Nuclear Physics B (Proc. Suppl.) 140 (2005) 381–383
O mK 0.46 0.54 0.62 0.69 0.81 0.91 1.00
× 100 0 -1.18(2) -1.22(2) -1.26(2) -1.28(3) -1.30(4) -1.29(4) -1.24(5)
O(8,8) × 100 mK 0 0.46 -8.9(1) 0.54 -8.3(1) 0.62 -7.9(1) 0.69 -7.5(1) 0.81 -7.0(1) 0.91 -6.5(1) 1.00 -6.0(1)
π/16 -1.99(3) -2.04(4) -2.08(5) -2.11(5) -2.13(7) -2.09(9) -2.0(1)
π/16 -8.6(2) -7.9(2) -7.4(2) -7.0(2) -6.3(2) -5.7(2) -5.1(2)
√ 2π/16 -5.4(2) -5.0(2) -4.6(3) -4.3(3) -3.9(4) -3.5(4) -3.1(5)
√ 3π/16 -1.9(1) -1.9(1) -1.9(1) -2.0(1) -2.1(2) -2.3(3) -2.5(4) √ 3π/16 -3.8(4) -3.6(4) -3.4(5) -3.1(6) -2.5(8) -2(1) -7e-1(1)
Oi × 100 Oi 0 (27,1) O -1.23(2) O(8,8) -8.2(1) (8,8) Om -29.9(5)
π/16 -2.11(6) -6.9(2) -28.2(8)
√
2π/16 -2.2(1) -3.9(4) -18(1)
π/16 -33.4(7) -31.3(7) -29.7(8) -28.5(8) -26.4(9) -24.4(9) -22.5(9)
√ 2π/16 -23.2(8) -21.7(9) -21(1) -20(1) -18(1) -17(1) -16(1)
√ 3π/16 -19(1) -18(1) -17(1) -17(2) -16(2) -14(3) -11(3)
Table 2 Lattice matrix elements of the three ∆I = 3/2 operators.
√ 3π/16 -2.3(3) -2(1) -14(3)
Table 3 On-shell matrix elements, interpolated to the point mK = Eππ
1
0.5
0
s-mass 0.08 s-mass 0.12 s-mass 0.16 s-mass 0.2 s-mass 0.28 s-mass 0.36 s-mass 0.44
-0.5
-1
(8,8)
Om × 100 mK 0 0.46 -32.1(6) 0.54 -30.2(5) 0.62 -28.8(6) 0.69 -27.8(6) 0.81 -26.0(6) 0.91 -24.4(6) 1.00 -22.7(6)
√ 2π/16 -1.88(5) -1.94(6) -2.01(8) -2.07(9) -2.2(1) -2.2(1) -2.2(2)
Effective Mass
(27,1)
0
5
10
15
20
25
Time
Figure 2. Effective mass difference plot for the O(27,1) operator for various strange quark masses. Here the source is at t = 0 and the sink at t = 22. elements is still underway, the lattice matrix elements from 129 gauge configurations are quite accurate with statistical errors less than 5% for the ππ final states with lower relative momentum. Thus, calculations with more realistic kinematics appear promising. REFERENCES
operator is identical to O(27,1) through a Fierz transfromation. 5. Conclusions By imposing H-parity boundary condition in various directions we can have ππ states with different relative momentum in same volume. We calculate the phase shift for each relative momentum. However, the phase shift for ππ states with higher relative momentum differs from experimental data and requires more study. Although the final analysis to deduce the physical matrix
1. T. Blum et al. [RBC Collaboration], Phys. Rev. D 68, 114506 (2003) [arXiv:heplat/0110075]. 2. J. I. Noaki et al. [CP-PACS Collaboration], Phys. Rev. D 68, 014501 (2003) [arXiv:heplat/0108013]. 3. M. Luscher and U. Wolff, Nucl. Phys. B 339, 222 (1990). 4. T. Yamazaki et al. [CP-PACS Collaboration], arXiv:hep-lat/0402025. 5. C. h. Kim and N. H. Christ, Nucl. Phys. Proc. Suppl. 119, 365 (2003) [arXiv:heplat/0210003].