Remarks on the dimension-four chiral Lagrangian

Hai-Yang Cheng

doi:10.1103/PhysRevD.34.166

Access by Xinjiang University

Remarks on the dimension-four chiral Lagrangian

Hai-Yang Cheng

Physics Department, Indiana University, Bloomington, Indiana 47405

PDF

Phys. Rev. D 34, 166 – Published 1 July, 1986

DOI: https://doi.org/10.1103/PhysRevD.34.166

Abstract

The profound fact that the dimension-four operators in the effective chiral Lagrangian, including quartic-derivative terms, are uniquely determined by the integration of spurious (nontopological) chiral anomalies has rich applications to low-energy hadron physics. The form factor $f_{-}$ and the Callan-Treiman relation in $K_{l 3}$ decays, the axial-vector $K_{l 4}$ form factors, and the structure-dependent form factor ξ in π→eν $e^{+}$ $e^{-}$ are derived from the effective Lagrangian. A discussion of the parameter γ in weak radiative π and K decays is also given.

References (46)

K. C. Chou, H. Y. Guo, K. Wu and X. C. Song, Phys. Lett. 134B, 67 (1984); G. C. Rossi, M. Testa and K. Yoshida, ibid. 134B, 78 (1984); H. Kawai and S.-H.H. Tye, ibid. 140B, 403 (1984); A. Dhar and Spenta R. Wadia, Phys. Rev. Lett. 52, 959 (1984); J. L. Petersen, Acta Phys. Polon. B16, 271 (1985); OÜ. Kaymakcalan, S. Rajeev and J. Schechter, Phys. Rev. D 30, 594 (1984).
N. K. Pak and P. Rossi, Nucl. Phys. B250, 279 (1985).
R. I. Nepomechie, Ann. Phys. (N.Y.) 158, 67 (1984).
K. Seo, Gifu Report No. GWJC-1, 1984 (unpublished).
J. Balog, Phys. Lett. 149B, 197 (1984); Nucl. Phys. B258, 361 (1985).
A. A. Andrianov, Phys. Lett. 157B, 425 (1985).
J. Wess and B. Zumino, Phys. Lett. 37B, 95 (1971).
R. MacKenzie, F. Wilczek and A. Zee, Phys. Rev. Lett. 53, 2203 (1984); I. J. R. Aitchison and C. M. Fraser, Phys. Lett. 146B, 63 (1984); G. Bhattacharya and S. Rajeev, in Proceedings of the Sixth Annual Montreal-Rochester-Syracuse-Toronto Meeting, Syracuse, 1984 (Syracuse University, Syracuse, New York, 1984); P. Simić, Phys. Rev. Lett. 55, 40 (1985); A. Zaks, Nucl. Phys. B260, 241 (1985).
L.-H. Chan, Phys. Rev. Lett. 55, 21 (1985).
A. Manohar and H. Georgi, Nucl. Phys. B234, 189 (1984); J. F. Donoghue, E. Golowich and B. R. Holstein, Phys. Rev. D 30, 587 (1984).
J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984); Nucl. Phys. B250, 465 (1985).
T. H. R. Skyrme, Proc. R. Soc. London A260, 127 (1961).
G. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B228, 552 (1983).
J. F. Donoghue, E. Golowich and B. R. Holstein, Phys. Rev. Lett. 53, 747 (1984); T. N. Pham and T. N. Truong, Phys. Rev. D 31, 3027 (1985). Loop corrections to the D-wave π pi scattering length, which can be of order ln $m_{π}^{2}$ (rather than $m_{π}$ ${}^{2}\ln$ $m_{π}^{2}$ ), are not considered in both references in the determination of gamma.
J. A. Cronin, Phys. Rev. 161, 327 (1969).
J. S. R. Chisholm, Nucl. Phys. 26, 469 (1961).
S. Weinberg, Physica 96A, 327 (1979).
This is analogous to an example in QED. The value of LAMBDA in G( $α_{μ}$ )= 1/2 ln ( μ / Λ $)^{2}$ (G being a function of $α_{mu}$ ) is determined once the renormalized coupling constant $α_{mu}$ in a given regularization scheme is derived in terms of the physical fine-structure constant alpha; see R. Coquereaux, Ann. Phys. (N.Y.) 125, 401 (1980).
E. Witten, Nucl. Phys. B160, 57 (1979).
There are two examples which show the importance of one-loop corrections for some physical quantities: one is the decay constant $f_{pi}$ (or $f_{K}$ , $f_{eta}$ ) and the other is the D-wave π pi scattering length $a_{2}^{2}$ . At the renormalization scale μ = $m_{eta}$ , the one-loop contribution to $f_{pi}$ is 6%, while the correction from the dimension-four Lagrangian is only 2%. The one-loop correction is, however, negligible in $f_{K}$ / $f_{π}$ -1; see Eqs. (10.7) and (10.12) of the second reference of Ref. 11. The contribution to $a_{2}^{2}$ from the quartic-derivative Lagrangian is proportional to γ - case 1 over case 4 (Ref. 14), which vanishes at γ = case 1 over case 4; see Eq. (10).
As discussed in Ref. 20, the decay contant $f_{pi}$ receives large loop corrections. Since we do not consider the one-loop contribution (see the discussion at the end), we use f= $f_{pi}$ throughout in this paper.
C. Callan and S. Treiman, Phys. Rev. Lett. 16, 153 (1966).
T. N. Pham, Phys. Rev. D 30, 234 (1984).
H. Leutwyler and M. Roos, Z. Phys. C 25, 91 (1984).
Particle Data Group, Rev. Mod. Phys. 56, S1 (1984).
M. Milosević, D. Tadić and J. Trampetić, Nucl. Phys. B187, 514 (1981).
G. Kramer, W. F. Palmer and S. S. Pinsky, Phys. Rev. D 30, 89 (1984).
L. Rosselet et al., Phys. Rev. D 15, 574 (1977).
S.-C. Chao, G. Kramer, W. F. Palmer and S. S. Pinsky, Phys. Rev. D 30, 1916 (1984).
D. A. Bryman, P. Depommier and C. Leroy, Phys. Rep. 88, 151 (1982).
C.-Y. Lee, Phys. Rev. D 32, 658 (1985).
N. Paver and M. D. Scadron, Nuovo Cimento 78A, 159 (1983).
S. B. Gerasimov, Yad. Fiz. 29, 513 (1979) [Sov. J. Nucl. Phys. 29, 259 (1979)].
R. Montemayor and M. Moreno, Phys. Rev. D 20, 250 (1979).
V. N. Pervushin and M. K. Volkov, Phys. Lett. 58B, 74 (1975).
P. Pascual and R. Tarrach, Nucl. Phys. B146, 509 (1978).
M. K. Volkov and V. N. Pervushin, Phys. Lett. 51B, 356 (1974).
E. B. Dally et al., Phys. Rev. Lett. 48, 375 (1982).
The $g_{9}$ term of ${scrL}_{2}$ [Eq. (6)] yields < $r^{2}$ > ${}_{π}$ =3/2 $π^{2}$ $f_{π}^{2}$ . The inclusion of the loop correction, which is of order ln $m_{π}^{2}$ , improves the agreement with experiment.
L.-H. Chan, Louisiana Report No. DOE/ER 05490-67, 1985 (unpublished).
Current algebra leads to a sum rule for the axial-vector form factor $f_{A}$ (Ref. 42). However, no prediction for $f_{A}$ can be made unless both the spectrum function and the charge radius of a pion are calculated ``theoretically'' in some models.
T. Das, V. S. Mathur and S. Okubo, Phys. Rev. Lett. 19, 859 (1967).
A. Stetz et al., Nucl. Phys. B138, 285 (1978).
J. Heintze et al., Nucl. Phys. B149, 365 (1979).
Y. Akiba et al., Phys. Rev. D 32, 2911 (1985).
The agreement between the coupling constants $g_{i}$ ( μ $)_{expt}$ determined in Ref. 11 and the theoretical values of $g_{i}$ given in Eq. (7) at the renormalization scale μ = $m_{eta}$ implies that at least at μ = $m_{eta}$ the loop contributions are generally small for most physial quantities.

[c1] K. C. Chou, H. Y. Guo, K. Wu and X. C. Song, Phys. Lett. 134B, 67 (1984); G. C. Rossi, M. Testa and K. Yoshida, ibid. 134B, 78 (1984); H. Kawai and S.-H.H. Tye, ibid. 140B, 403 (1984); A. Dhar and Spenta R. Wadia, Phys. Rev. Lett. 52, 959 (1984); J. L. Petersen, Acta Phys. Polon. B16, 271 (1985); OÜ. Kaymakcalan, S. Rajeev and J. Schechter, Phys. Rev. D 30, 594 (1984).

[c2] N. K. Pak and P. Rossi, Nucl. Phys. B250, 279 (1985).

[c3] R. I. Nepomechie, Ann. Phys. (N.Y.) 158, 67 (1984).

[c4] K. Seo, Gifu Report No. GWJC-1, 1984 (unpublished).

[c5] J. Balog, Phys. Lett. 149B, 197 (1984); Nucl. Phys. B258, 361 (1985).

[c6] A. A. Andrianov, Phys. Lett. 157B, 425 (1985).

[c7] J. Wess and B. Zumino, Phys. Lett. 37B, 95 (1971).

[c8] R. MacKenzie, F. Wilczek and A. Zee, Phys. Rev. Lett. 53, 2203 (1984); I. J. R. Aitchison and C. M. Fraser, Phys. Lett. 146B, 63 (1984); G. Bhattacharya and S. Rajeev, in Proceedings of the Sixth Annual Montreal-Rochester-Syracuse-Toronto Meeting, Syracuse, 1984 (Syracuse University, Syracuse, New York, 1984); P. Simić, Phys. Rev. Lett. 55, 40 (1985); A. Zaks, Nucl. Phys. B260, 241 (1985).

[c9] L.-H. Chan, Phys. Rev. Lett. 55, 21 (1985).

[c10] A. Manohar and H. Georgi, Nucl. Phys. B234, 189 (1984); J. F. Donoghue, E. Golowich and B. R. Holstein, Phys. Rev. D 30, 587 (1984).

[c11] J. Gasser and H. Leutwyler, Ann. Phys. (N.Y.) 158, 142 (1984); Nucl. Phys. B250, 465 (1985).

[c12] T. H. R. Skyrme, Proc. R. Soc. London A260, 127 (1961).

[c13] G. Adkins, C. R. Nappi and E. Witten, Nucl. Phys. B228, 552 (1983).

[c14] J. F. Donoghue, E. Golowich and B. R. Holstein, Phys. Rev. Lett. 53, 747 (1984); T. N. Pham and T. N. Truong, Phys. Rev. D 31, 3027 (1985). Loop corrections to the D-wave π pi scattering length, which can be of order ln $m_{π}^{2}$ (rather than $m_{π}$ ${}^{2}\ln$ $m_{π}^{2}$ ), are not considered in both references in the determination of gamma.

[c15] J. A. Cronin, Phys. Rev. 161, 327 (1969).

[c16] J. S. R. Chisholm, Nucl. Phys. 26, 469 (1961).

[c17] S. Weinberg, Physica 96A, 327 (1979).

[c18] This is analogous to an example in QED. The value of LAMBDA in G( $α_{μ}$ )= 1/2 ln ( μ / Λ $)^{2}$ (G being a function of $α_{mu}$ ) is determined once the renormalized coupling constant $α_{mu}$ in a given regularization scheme is derived in terms of the physical fine-structure constant alpha; see R. Coquereaux, Ann. Phys. (N.Y.) 125, 401 (1980).

[c19] E. Witten, Nucl. Phys. B160, 57 (1979).

[c20] There are two examples which show the importance of one-loop corrections for some physical quantities: one is the decay constant $f_{pi}$ (or $f_{K}$ , $f_{eta}$ ) and the other is the D-wave π pi scattering length $a_{2}^{2}$ . At the renormalization scale μ = $m_{eta}$ , the one-loop contribution to $f_{pi}$ is 6%, while the correction from the dimension-four Lagrangian is only 2%. The one-loop correction is, however, negligible in $f_{K}$ / $f_{π}$ -1; see Eqs. (10.7) and (10.12) of the second reference of Ref. 11. The contribution to $a_{2}^{2}$ from the quartic-derivative Lagrangian is proportional to γ - case 1 over case 4 (Ref. 14), which vanishes at γ = case 1 over case 4; see Eq. (10).

[c21] As discussed in Ref. 20, the decay contant $f_{pi}$ receives large loop corrections. Since we do not consider the one-loop contribution (see the discussion at the end), we use f= $f_{pi}$ throughout in this paper.

[c22] C. Callan and S. Treiman, Phys. Rev. Lett. 16, 153 (1966).

[c23] T. N. Pham, Phys. Rev. D 30, 234 (1984).

[c24] H. Leutwyler and M. Roos, Z. Phys. C 25, 91 (1984).

[c25] Particle Data Group, Rev. Mod. Phys. 56, S1 (1984).

[c26] M. Milosević, D. Tadić and J. Trampetić, Nucl. Phys. B187, 514 (1981).

[c27] G. Kramer, W. F. Palmer and S. S. Pinsky, Phys. Rev. D 30, 89 (1984).

[c28] L. Rosselet et al., Phys. Rev. D 15, 574 (1977).

[c29] S.-C. Chao, G. Kramer, W. F. Palmer and S. S. Pinsky, Phys. Rev. D 30, 1916 (1984).

[c30] D. A. Bryman, P. Depommier and C. Leroy, Phys. Rep. 88, 151 (1982).

[c31] C.-Y. Lee, Phys. Rev. D 32, 658 (1985).

[c32] N. Paver and M. D. Scadron, Nuovo Cimento 78A, 159 (1983).

[c33] S. B. Gerasimov, Yad. Fiz. 29, 513 (1979) [Sov. J. Nucl. Phys. 29, 259 (1979)].

[c34] R. Montemayor and M. Moreno, Phys. Rev. D 20, 250 (1979).

[c35] V. N. Pervushin and M. K. Volkov, Phys. Lett. 58B, 74 (1975).

[c36] P. Pascual and R. Tarrach, Nucl. Phys. B146, 509 (1978).

[c37] M. K. Volkov and V. N. Pervushin, Phys. Lett. 51B, 356 (1974).

[c38] E. B. Dally et al., Phys. Rev. Lett. 48, 375 (1982).

[c39] The $g_{9}$ term of ${scrL}_{2}$ [Eq. (6)] yields < $r^{2}$ > ${}_{π}$ =3/2 $π^{2}$ $f_{π}^{2}$ . The inclusion of the loop correction, which is of order ln $m_{π}^{2}$ , improves the agreement with experiment.

[c40] L.-H. Chan, Louisiana Report No. DOE/ER 05490-67, 1985 (unpublished).

[c41] Current algebra leads to a sum rule for the axial-vector form factor $f_{A}$ (Ref. 42). However, no prediction for $f_{A}$ can be made unless both the spectrum function and the charge radius of a pion are calculated ``theoretically'' in some models.

[c42] T. Das, V. S. Mathur and S. Okubo, Phys. Rev. Lett. 19, 859 (1967).

[c43] A. Stetz et al., Nucl. Phys. B138, 285 (1978).

[c44] J. Heintze et al., Nucl. Phys. B149, 365 (1979).

[c45] Y. Akiba et al., Phys. Rev. D 32, 2911 (1985).

[c46] The agreement between the coupling constants $g_{i}$ ( μ $)_{expt}$ determined in Ref. 11 and the theoretical values of $g_{i}$ given in Eq. (7) at the renormalization scale μ = $m_{eta}$ implies that at least at μ = $m_{eta}$ the loop contributions are generally small for most physial quantities.

Physical Review D

Export citation

Export citation

Remarks on the dimension-four chiral Lagrangian

Abstract

References (46)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation