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Chiral Perturbation Theory for Mesons

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A Primer for Chiral Perturbation Theory

Part of the book series: Lecture Notes in Physics ((LNP,volume 830))

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Abstract

Chiral perturbation theory provides a systematic method for discussing the consequences of the global flavor symmetries of QCD at low energies by means of an effective field theory. The effective Lagrangian is expressed in terms of those hadronic degrees of freedom which, at low energies, appear as observable asymptotic states. At very low energies these are just the members of the pseudoscalar octet \((\pi,K,\eta)\) which are regarded as the Goldstone bosons of the spontaneous breaking of the chiral \(\hbox{SU}(3)_L\times\hbox{SU}(3)_R\) symmetry down to \(\hbox{SU}(3)_V.\) The nonvanishing masses of the light pseudoscalars in the “real” world are related to the explicit symmetry breaking in QCD by the light-quark masses

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Notes

  1. 1.

    The toy model serves pedagogical purposes only. As a (quantum) field theory it is not consistent because the energy is not bounded from below [14].

  2. 2.

    The existence of mass-degenerate states of opposite parity is referred to as parity doubling .

  3. 3.

    The subscript \(V\) (for vector) indicates that the generators result from integrals of the zeroth component of vector-current operators and thus transform with a positive sign under parity.

  4. 4.

    Recall that each quark is assigned a baryon number 1/3.

  5. 5.

    In this section, we explicitly write out sums over flavor indices, because a summation over repeated indices is \(not\) implied in the final results of Eqs. 3.27 and 3.28.

  6. 6.

    The commutation relations also remain valid for \(equal\) times if the symmetry is explicitly broken.

  7. 7.

    Accordingly, the right coset of \(g\) is defined as \(Hg=\{hg|h\in H\}.\) An \(invariant\) subgroup has the additional property that the left and right cosets coincide for each \(g\) which allows for a definition of the factor group \(G/H\) in terms of the complex product. However, here we do not need this property.

  8. 8.

    There is a subtlety here, because \(F_0\) is traditionally reserved for the three-flavor chiral limit , whereas the two-flavor chiral limit (at fixed \(m_s\)) is denoted by \(F.\) In this section, we will use \(F_0\) for both cases.

  9. 9.

    Since the Goldstone bosons are pseudoscalars, a true parity transformation is given by \(\phi_a(t,\vec{x})\mapsto -\phi_a(t,-\vec{x})\) or, equivalently, \(U(t,\vec{x})\mapsto U^\dagger(t,-\vec{x}).\)

  10. 10.

    In view of the coupling to the external fields \(s+ip\) and \(s-ip\) (see Eq. 1.161) to be discussed in Sect. 3.4.3, we distinguish between \({{\fancyscript{M}}}\) and \({{\fancyscript{M}}}^\dagger\) even though for a real, diagonal matrix they are the same.

  11. 11.

    Including all of the infinite number of effective functionals \(Z^{(2n)}_{\rm eff}[v,a,s,p]\) will generate a result which is equivalent to that obtained from \(Z_{\rm QCD}[v,a,s,p].\)

  12. 12.

    In principle, we could also “gauge” the U(1)\(_V\) symmetry. However, this is primarily of relevance to the two-flavor sector in order to fully incorporate the coupling to the electromagnetic four-vector potential (see Eq. 1.165). Since in the three-flavor sector the quark-charge matrix is traceless, this important case is included in our considerations.

  13. 13.

    Under certain circumstances it is advantageous to introduce for each object with a well-defined transformation behavior a separate covariant derivative. One may then use a product rule similar to the one of ordinary differentiation.

  14. 14.

    Throughout this monograph we will reserve the notation \({{\fancyscript{O}}}(q^n)\) for power counting in chiral perturbation theory, whereas \(O(x^n)\) denotes terms of order \(x^n\) in the usual mathematical sense.

  15. 15.

    There is a certain freedom in the choice of the elementary building blocks. For example, by a suitable multiplication with \(U\) or \(U^\dagger\) any building block can be made to transform as \(V_R \ldots V_R^\dagger\) without changing its chiral order. The present approach most naturally leads to the Lagrangian of Gasser and Leutwyler [53].

  16. 16.

    At \({{\fancyscript{O}}}(q^2)\) invariance under \(C\) does not provide any additional constraints.

  17. 17.

    Recall that the entries \(V_{ud}\) and \(V_{us}\) of the Cabibbo-Kobayashi-Maskawa matrix are real.

  18. 18.

    Of course, in the chiral limit, the pion is massless and, in such a world, the massive leptons would decay into Goldstone bosons, e.g., \(e^-\to\pi^-\nu_e.\) However, at \({{\fancyscript{O}}}(q^2),\) the symmetry-breaking term of Eq. 3.55 gives rise to Goldstone-boson masses, whereas the decay constant is not modified at \({{\fancyscript{O}}}(q^2).\)

  19. 19.

    We will refer to this parameterization as the square-root parameterization because of the square root multiplying the unit matrix.

  20. 20.

    Recall that \(\Upgamma(z)\) is single-valued and analytic over the entire complex plane, save for the points \(z=-n,\,n=0,1,2,\ldots,\) where it possesses simple poles with residue \((-1)^n/n!.\)

  21. 21.

    Note that the convention \(\varepsilon=2-{\frac{n}{2}}\) is also commonly used in the literature.

  22. 22.

    More generally, renormalization is simply the process of expressing the parameters of the Lagrangian in terms of physical observables, independent of the presence of divergences [51].

  23. 23.

    Note that Ref. [43] uses a slightly different convention which corresponds to the replacement \((\delta Z_\phi M^2 +Z_\phi\delta M^2)\to\delta M^2;\) analogously for \(\delta m^2.\)

  24. 24.

    Note that the number of independent momenta is \(not\) the number of faces or closed circuits that may be drawn on the internal lines of a diagram. This may, for example, be seen using a diagram with the topology of a tetrahedron which has four faces but \(N_L=6-(4-1)=3\) (see, e.g., Chap. 6-2 of Ref. [63]).

  25. 25.

    In distinction to the \(\overline{\hbox{MS}}\) scheme commonly used in Standard Model calculations, the \(\widetilde{\hbox{MS}}\) scheme contains an additional finite subtraction term. To be specific, in \(\widetilde{\hbox{MS}}\) one uses multiples of \( 2/(n-4)- \left[ \ln (4\pi )+\Upgamma '(1)+1\right]\) instead of \(2/(n-4)-\left[ \ln (4\pi )+\Upgamma '(1)\right]\) in \(\overline{\hbox{MS}}.\)

  26. 26.

    For pedagogical reasons, we make use of the physical fields. From a technical point of view, it is often advantageous to work with the Cartesian fields and, at the end of the calculation, express physical processes in terms of the Cartesian components.

  27. 27.

    Note that we work in the three-flavor sector and thus with the exponential parameterization of \(U.\)

  28. 28.

    When deriving the Feynman rule of Exercise 3.14, we took account of 24 distinct combinations of contracting four field operators with four external lines. However, the Feynman diagram of Eq. 3.134 involves only 12 possibilities to contract two fields with each other and the remaining two fields with two external lines.

  29. 29.

    In order to conform with our previous convention of Eq. 3.35, we need to replace \(U\) of Ref. [103] by \(U^\dagger.\) Furthermore, \(F_\pi\) of Ref. [103] corresponds to \(2F_0.\) Finally, \(\partial^2 U U^\dagger- U\partial^2 U^\dagger=2\partial_\mu(\partial^\mu U U^\dagger).\)

  30. 30.

    The subscript ano refers to anomalous.

References

  1. Adeva, B., et al., DIRAC Collaboration: Phys. Lett. B 619, 50 (2005)

    Google Scholar 

  2. Adler, S.L.: Phys. Rev. 177, 2426 (1969)

    Article  ADS  Google Scholar 

  3. Adler, S.L., Bardeen, W.A.: Phys. Rev. 182, 1517 (1969)

    Article  ADS  Google Scholar 

  4. Adler, S.L.: In: Deser, S., Grisaru, M., Pendleton, H. (eds.) Lectures on Elementary Particles and Quantum Field Theory, 1970 Brandeis University Summer Institute in Theoretical Physics, vol. 1. M.I.T. Press, Cambridge (1970)

    Google Scholar 

  5. Ahrens, J., et al.: Eur. Phys. J. A 23, 113 (2005)

    Article  ADS  Google Scholar 

  6. Akhoury, R., Alfakih, A.: Ann. Phys. 210, 81 (1991)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  7. Ananthanarayan, B., Colangelo, G., Gasser, J., Leutwyler, H.: Phys. Rep. 353, 207 (2001)

    Article  ADS  MATH  Google Scholar 

  8. Antipov, Y.M., et al.: Phys. Lett. B 121, 445 (1983)

    Article  ADS  Google Scholar 

  9. Balachandran, A.P., Trahern, C.G.: Lectures on Group Theory for Physicists. Bibliopolis, Naples (1984)

    Google Scholar 

  10. Balachandran, A.P., Marmo, G., Skagerstam, B.S., Stern, A.: Classical Topology and Quantum States (Chap. 12.2). World Scientific, Singapore (1991)

    Google Scholar 

  11. Bardeen, W.A.: Phys. Rev. 184, 1848 (1969)

    Article  ADS  Google Scholar 

  12. Bär, O., Wiese, U.J.: Nucl. Phys. B 609, 225 (2001)

    Article  ADS  Google Scholar 

  13. Batley, J.R., et al., NA48/2 Collaboration: Eur. Phys. J. C 54, 411 (2008)

    Google Scholar 

  14. Baym, G.: Phys. Rev. 117, 886 (1960)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Bell, J.S., Jackiw, R.: Nuovo Cim. A 60, 47 (1969)

    Article  ADS  Google Scholar 

  16. Bijnens, J., Cornet, F.: Nucl. Phys. B 296, 557 (1988)

    Article  ADS  Google Scholar 

  17. Bijnens, J., Bramon, A., Cornet, F.: Z. Phys. C 46, 599 (1990)

    Article  ADS  Google Scholar 

  18. Bijnens, J.: Int. J. Mod. Phys. A 8, 3045 (1993)

    Article  ADS  Google Scholar 

  19. Bijnens, J., Ecker, G., Gasser, J.: In: Maiani, L., Pancheri, G., Paver, N. (eds.) The Second DA\(\Phi\)NE Physics Handbook, vol. 2, pp. 125–143. SIS, Frascati (1995)

    Google Scholar 

  20. Bijnens, J., Colangelo, G., Ecker, G., Gasser, J., Sainio, M.E.: Phys. Lett. B 374, 210 (1996)

    Article  ADS  Google Scholar 

  21. Bijnens, J., Colangelo, G., Ecker, G., Gasser, J., Sainio, M.E.: Nucl. Phys. B 508, 263 (1997)

    Article  ADS  Google Scholar 

  22. Bijnens, J., Colangelo, G., Ecker, G.: JHEP 9902, 020 (1999)

    Article  ADS  Google Scholar 

  23. Bijnens, J., Girlanda, L., Talavera, P.: Eur. Phys. J. C 23, 539 (2002)

    Article  ADS  Google Scholar 

  24. Bijnens, J.: Prog. Part. Nucl. Phys. 58, 521 (2007)

    Article  ADS  Google Scholar 

  25. Bijnens, J., Jemos, I.: In: PoS (CD09), 087 (2009)

    Google Scholar 

  26. Bissegger, M., Fuhrer, A., Gasser, J., Kubis, B., Rusetsky, A.: Phys. Lett. B 659, 576 (2008)

    Article  ADS  Google Scholar 

  27. Bissegger, M., Fuhrer, A., Gasser, J., Kubis, B., Rusetsky, A.: Nucl. Phys. B 806, 178 (2009)

    Article  ADS  MATH  Google Scholar 

  28. Borasoy, B., Lipartia, E.: Phys. Rev. D 71, 014027 (2005)

    Article  ADS  Google Scholar 

  29. Burgess, C.P.: Phys. Rep. 330, 193 (2000)

    Article  ADS  Google Scholar 

  30. Bürgi, U.: Nucl. Phys. B 479, 392 (1996)

    Article  ADS  Google Scholar 

  31. Cabibbo, N.: Phys. Rev. Lett. 93, 121801 (2004)

    Article  ADS  Google Scholar 

  32. Cabibbo, N., Isidori, G.: JHEP 0503, 021 (2005)

    Article  ADS  Google Scholar 

  33. Callan, C.G., Coleman, S.R., Wess, J., Zumino, B.: Phys. Rev. 177, 2247 (1969)

    Article  ADS  Google Scholar 

  34. Chisholm, J.: Nucl. Phys. 26, 469 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  35. Colangelo, G., Gasser, J., Leutwyler, H.: Phys. Lett. B 488, 261 (2000)

    Article  ADS  Google Scholar 

  36. Colangelo, G., Gasser, J., Leutwyler, H.: Phys. Rev. Lett. 86, 5008 (2001)

    Article  ADS  Google Scholar 

  37. Colangelo, G., Gasser, J., Leutwyler, H.: Nucl. Phys. B 603, 125 (2001)

    Article  ADS  Google Scholar 

  38. Colangelo, G., Gasser, J., Kubis, B., Rusetsky, A.: Phys. Lett. B 638, 187 (2006)

    Article  ADS  Google Scholar 

  39. Colangelo, G., Gasser, J., Rusetsky, A.: Eur. Phys. J. C 59, 777 (2009)

    Article  ADS  Google Scholar 

  40. Colangelo, G., et al.: arXiv:1011.4408 [hep-lat]

    Google Scholar 

  41. Coleman, S.: J. Math. Phys. 7, 787 (1966)

    Article  ADS  Google Scholar 

  42. Coleman, S.R., Wess, J., Zumino, B.: Phys. Rev. 177, 2239 (1969)

    Article  ADS  Google Scholar 

  43. Collins, J.C.: Renormalization. Cambridge University Press, Cambridge (1984)

    Book  MATH  Google Scholar 

  44. Donoghue, J.F., Wyler, D.: Nucl. Phys. B 316, 289 (1989)

    Article  ADS  Google Scholar 

  45. Donoghue, J.F.: In: PoS (EFT09), 001 (2009)

    Google Scholar 

  46. Ebertshäuser, T., Fearing, H.W., Scherer, S.: Phys. Rev. D 65, 054033 (2002)

    Article  ADS  Google Scholar 

  47. Ecker, G., Gasser, J., Pich, A., de Rafael, E.: Nucl. Phys. B 321, 311 (1989)

    Article  ADS  Google Scholar 

  48. Ecker, G.: arXiv:hep-ph/0507056

    Google Scholar 

  49. Fearing, H.W., Scherer, S.: Phys. Rev. D 53, 315 (1996)

    Article  ADS  Google Scholar 

  50. Frlež, E., et al.: Phys. Rev. Lett. 93, 181804 (2004)

    Article  ADS  Google Scholar 

  51. Gasser, J., Leutwyler, H.: Phys. Rep. 87, 77 (1982)

    Article  ADS  Google Scholar 

  52. Gasser, J., Leutwyler, H.: Ann. Phys. 158, 142 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  53. Gasser, J., Leutwyler, H.: Nucl. Phys. B 250, 465 (1985)

    Article  ADS  Google Scholar 

  54. Gasser, J., Sainio, M.E., Švarc, A.: Nucl. Phys. B 307, 779 (1988)

    Article  ADS  Google Scholar 

  55. Gasser, J., Ivanov, M.A., Sainio, M.E.: Nucl. Phys. B 745, 84 (2006)

    Article  ADS  Google Scholar 

  56. Gell-Mann, M., Oakes, R.J., Renner, B.: Phys. Rev. 175, 2195 (1968)

    Article  ADS  Google Scholar 

  57. Georgi, H.: Weak Interactions and Modern Particle Theory. Benjamin/Cummings, Menlo Park (1984)

    Google Scholar 

  58. Georgi, H.: Ann. Rev. Nucl. Part. Sci. 43, 207 (1993)

    Article  ADS  Google Scholar 

  59. Gerber, P., Leutwyler, H.: Nucl. Phys. B 321, 387 (1989)

    Article  ADS  Google Scholar 

  60. Goldstone, J., Salam, A., Weinberg, S.: Phys. Rev. 127, 965 (1962)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  61. Guskov, A., On behalf of the COMPASS collaboration: J. Phys. Conf. Ser. 110, 022016 (2008)

    Google Scholar 

  62. Holstein, B.R.: Comm. Nucl. Part. Phys. A 19, 221 (1990)

    Google Scholar 

  63. Itzykson, C., Zuber, J.B.: Quantum Field Theory. McGraw-Hill, New York (1980)

    Google Scholar 

  64. Jones, H.F.: Groups, Representations and Physics. Hilger, Bristol (1990)

    Book  MATH  Google Scholar 

  65. Kamefuchi, S., O’Raifeartaigh, L., Salam, A.: Nucl. Phys. 28, 529 (1961)

    MathSciNet  Google Scholar 

  66. Kaplan, D.B.: arXiv:nucl-th/9506035

    Google Scholar 

  67. Kermani, M., et al., CHAOS Collaboration: Phys. Rev. C 58, 3431 (1998)

    Google Scholar 

  68. Knecht, M., Moussallam, B., Stern, J., Fuchs, N.H.: Nucl. Phys. B 457, 513 (1995)

    Article  ADS  Google Scholar 

  69. Leibbrandt, G.: Rev. Mod. Phys. 47, 849 (1975)

    Article  MathSciNet  ADS  Google Scholar 

  70. Leutwyler, H.: In: Ellis, R.K., Hill, C.T., Lykken, J.D. (eds.) Perspectives in the Standard Model. Proceedings of the 1991 Advanced Theoretical Study Institute in Elementary Particle Physics, Boulder, CO, 2–28 June 1991. World Scientific, Singapore (1992)

    Google Scholar 

  71. Leutwyler, H.: Ann. Phys. 235, 165 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  72. Li, L.F., Pagels, H.: Phys. Rev. Lett. 26, 1204 (1971)

    Article  ADS  Google Scholar 

  73. Manohar, A.V., Georgi, H.: Nucl. Phys. B 234, 189 (1984)

    Article  ADS  Google Scholar 

  74. Manohar, A.V.: arXiv:hep-ph/9606222

    Google Scholar 

  75. Nakamura, K., et al., Particle Data Group: J. Phys. G 37, 075021 (2010)

    Google Scholar 

  76. Necco, S.: In: PoS (Confinement8), 024 (2008)

    Google Scholar 

  77. O’Raifeartaigh, L.: Group Structure of Gauge Theories. Cambridge University Press, Cambridge (1986)

    Book  MATH  Google Scholar 

  78. Pich, A.: arXiv:hep-ph/9806303

    Google Scholar 

  79. Pich, A.: In: PoS (Confinement8), 026 (2008)

    Google Scholar 

  80. Pislak, S., et al., BNL-E865 Collaboration: Phys. Rev. Lett. 87, 221801 (2001)

    Google Scholar 

  81. Pislak, S., et al.: Phys. Rev. D 67, 072004 (2003) [Erratum, ibid. D 81, 119903 (2010)]

    Article  ADS  Google Scholar 

  82. Polchinski, J.: arXiv:hep-th/9210046

    Google Scholar 

  83. Preston, M.A.: Physics of the Nucleus. Addison-Wesley, Reading (1962)

    MATH  Google Scholar 

  84. Rosselet, L., et al.: Phys. Rev. D 15, 574 (1977)

    Article  ADS  Google Scholar 

  85. Roy, S.M.: Phys. Lett. B 36, 353 (1971)

    Article  ADS  Google Scholar 

  86. Scherer, S., Fearing, H.W.: Phys. Rev. D 52, 6445 (1995)

    Article  ADS  Google Scholar 

  87. Scherer, S.: Adv. Nucl. Phys. 27, 277 (2003)

    Article  Google Scholar 

  88. Schwinger, J.S.: Phys. Lett. B 24, 473 (1967)

    Article  ADS  Google Scholar 

  89. Terent’ev, M.V.: Yad. Fiz. 16, 162 (1972)

    Google Scholar 

  90. Terent’ev, M.V.: Sov. J. Nucl. Phys. 16, 87 (1973)

    MathSciNet  Google Scholar 

  91. ’t Hooft, G., Veltman, M.J.: Nucl. Phys. B 44, 189 (1972)

    Article  MathSciNet  ADS  Google Scholar 

  92. ’t Hooft, G., Veltman, M.J.: Nucl. Phys. B 153, 365 (1979)

    Article  MathSciNet  ADS  Google Scholar 

  93. Unkmeir, C., Scherer, S., L’vov, A.I., Drechsel, D.: Phys. Rev. D 61, 034002 (2000)

    Article  ADS  Google Scholar 

  94. Vafa, C., Witten, E.: Nucl. Phys. B 234, 173 (1984)

    Article  MathSciNet  ADS  Google Scholar 

  95. Veltman, M.J.: Diagrammatica. The Path to Feynman Rules. Cambridge University Press, Cambridge (1994)

    Book  Google Scholar 

  96. Watson, K.M.: Phys. Rev. 95, 228 (1954)

    Article  ADS  MATH  Google Scholar 

  97. Weinberg, S.: Phys. Rev. Lett. 17, 616 (1966)

    Article  ADS  Google Scholar 

  98. Weinberg, S.: Phys. Rev. Lett. 18, 188 (1967)

    Article  ADS  Google Scholar 

  99. Weinberg, S.: Phys. Rev. 166, 1568 (1968)

    Article  ADS  Google Scholar 

  100. Weinberg, S.: Physica A 96, 327 (1979)

    Article  ADS  Google Scholar 

  101. Weinberg, S.: The Quantum Theory of Fields. Foundations, vol. I (Chap. 12). Cambridge University Press, Cambridge (1995)

    Google Scholar 

  102. Wess, J., Zumino, B.: Phys. Lett. B 37, 95 (1971)

    Article  MathSciNet  ADS  Google Scholar 

  103. Witten, E.: Nucl. Phys. B 223, 422 (1983)

    Article  MathSciNet  ADS  Google Scholar 

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Authors and Affiliations

  1. Johannes Gutenberg-Universität Mainz, Institut für Kernphysik, Johann-Joachim-Becher-Weg 45, 55099, Mainz, Germany

    Dr. Stefan Scherer

  2. Department of Physics and Astronomy, University of South Carolina, 712 Main Street, Columbia, SC, 29208, USA

    Dr. Matthias R. Schindler

  3. Department of Physics, The George Washington University, Washington, DC, USA

    Dr. Matthias R. Schindler

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  1. Dr. Stefan Scherer
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Scherer, S., Schindler, M.R. (2011). Chiral Perturbation Theory for Mesons. In: A Primer for Chiral Perturbation Theory. Lecture Notes in Physics, vol 830. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19254-8_3

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