Abstract
Solitons in classical field theories correspond to states in quantum field theories. If the spatial dimension is infinite, then momentum eigenstates are not normalizable. This leads to infrared divergences, which are generally regularized via wave packets or by compactification. However, in some applications both possibilities are undesirable. In the present note, we introduce a finite inner product on translation-invariant kink states that allows us to compute probabilities involving these nonnormalizable states. Essentially, it is the quotient of the usual inner product by the translation group. We present a surprisingly simple formula for the reduced inner product, which requires no knowledge of the zero-mode dependence of the states but includes a correction which accounts for the mixing between zero modes and normal modes as the kink moves. As an application, we show that initial and final state corrections to meson multiplication vanish. However, we find that the pole of the subleading term in the initial state requires an infinitesimal imaginary shift.
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References
J.-L. Gervais, A. Jevicki and B. Sakita, Collective Coordinate Method for Quantization of Extended Systems, Phys. Rept. 23 (1976) 281 [INSPIRE].
R.F. Dashen, B. Hasslacher and A. Neveu, Nonperturbative Methods and Extended Hadron Models in Field Theory 2. Two-Dimensional Models and Extended Hadrons, Phys. Rev. D 10 (1974) 4130 [INSPIRE].
N. Graham and H. Weigel, Quantum corrections to soliton energies, Int. J. Mod. Phys. A 37 (2022) 2241004 [ar**v:2201.12131] [INSPIRE].
J.-L. Gervais and A. Jevicki, Point Canonical Transformations in Path Integral, Nucl. Phys. B 110 (1976) 93 [INSPIRE].
H.J. de Vega, Two-Loop Quantum Corrections to the Soliton Mass in Two-Dimensional Scalar Field Theories, Nucl. Phys. B 115 (1976) 411 [INSPIRE].
J. Verwaest, Higher Order Correction to the Sine-Gordon Soliton Mass, Nucl. Phys. B 123 (1977) 100 [INSPIRE].
M.A. Shifman, A.I. Vainshtein and M.B. Voloshin, Anomaly and quantum corrections to solitons in two-dimensional theories with minimal supersymmetry, Phys. Rev. D 59 (1999) 045016 [hep-th/9810068] [INSPIRE].
A. Hayashi, S. Saito and M. Uehara, Pion-nucleon scattering in the Skyrme model and the P wave Born amplitudes, Phys. Rev. D 43 (1991) 1520 [INSPIRE].
A. Hayashi, S. Saito and M. Uehara, Pion-nucleon scattering in the soliton model, Prog. Theor. Phys. Suppl. 109 (1992) 45 [INSPIRE].
I.V. Melnikov, C. Papageorgakis and A.B. Royston, Accelerating solitons, Phys. Rev. D 102 (2020) 125002 [ar**v:2007.11028] [INSPIRE].
I.V. Melnikov, C. Papageorgakis and A.B. Royston, Forced Soliton Equation and Semiclassical Soliton Form Factors, Phys. Rev. Lett. 125 (2020) 231601 [ar**v:2010.10381] [INSPIRE].
J.F. Wheater and P.D. Xavier, The Size of a Soliton, ar**v:2207.01274 [INSPIRE].
J. Evslin, Manifestly Finite Derivation of the Quantum Kink Mass, JHEP 11 (2019) 161 [ar**v:1908.06710] [INSPIRE].
J. Evslin and H. Guo, Two-Loop Scalar Kinks, Phys. Rev. D 103 (2021) 125011 [ar**v:2012.04912] [INSPIRE].
J. Evslin and S.B. Gudnason, Dwarf Galaxy Sized Monopoles as Dark Matter?, ar**v:1202.0560 [INSPIRE].
H.-Y. Schive, T. Chiueh and T. Broadhurst, Cosmic Structure as the Quantum Interference of a Coherent Dark Wave, Nature Phys. 10 (2014) 496 [ar**v:1406.6586] [INSPIRE].
C. Adam et al., Solvable self-dual impurity models, JHEP 07 (2019) 150 [ar**v:1905.06080] [INSPIRE].
J. Evslin, C. Halcrow, T. Romanczukiewicz and A. Wereszczynski, Spectral walls at one loop, Phys. Rev. D 105 (2022) 125002 [ar**v:2202.08249] [INSPIRE].
B. Schwesinger, H. Weigel, G. Holzwarth and A. Hayashi, The Skyrme Soliton in Pion, Vector and Scalar Meson Fields: πN Scattering and Photoproduction, Phys. Rept. 173 (1989) 173 [INSPIRE].
T.H.R. Skyrme, A Nonlinear field theory, Proc. Roy. Soc. Lond. A 260 (1961) 127 [ar**v:1961.0018] [INSPIRE].
S.B. Gudnason and C. Halcrow, A Smörgåsbord of Skyrmions, JHEP 08 (2022) 117 [ar**v:2202.01792] [INSPIRE].
M.A.A. Martin, R. Schlesier and J. Zahn, Semiclassical energy density of kinks and solitons, Phys. Rev. D 107 (2023) 065002 [ar**v:2204.08785] [INSPIRE].
J. Evslin, ϕ4 kink mass at two loops, Phys. Rev. D 104 (2021) 085013 [ar**v:2104.07991] [INSPIRE].
J. Evslin and H. Guo, Excited Kinks as Quantum States, Eur. Phys. J. C 81 (2021) 936 [ar**v:2104.03612] [INSPIRE].
H. Guo, Leading quantum correction to the Φ4 kink form factor, Phys. Rev. D 106 (2022) 096001 [ar**v:2209.03650] [INSPIRE].
J. Evslin and A. García Martín-Caro, Spontaneous emission from excited quantum kinks, JHEP 12 (2022) 111 [ar**v:2210.13791] [INSPIRE].
H. Liu, J. Evslin and B. Zhang, Meson production from kink-meson scattering, Phys. Rev. D 107 (2023) 025012 [ar**v:2211.01794] [INSPIRE].
H. Weigel, Quantum Instabilities of Solitons, AIP Conf. Proc. 2116 (2019) 170002 [ar**v:1907.10942] [INSPIRE].
K.E. Cahill, A. Comtet and R.J. Glauber, Mass Formulas for Static Solitons, Phys. Lett. B 64 (1976) 283 [INSPIRE].
J. Evslin, Normal ordering normal modes, Eur. Phys. J. C 81 (2021) 92 [ar**v:2007.05741] [INSPIRE].
A. Alonso-Izquierdo, D. Miguélez-Caballero, L.M. Nieto and J. Queiroga-Nunes, Wobbling kinks in a two-component scalar field theory: Interaction between shape modes, Physica D 443 (2023) 133590 [ar**v:2207.10989] [INSPIRE].
H. Weigel and N. Graham, Vacuum polarization energy of the Shifman-Voloshin soliton, Phys. Lett. B 783 (2018) 434 [ar**v:1806.07584] [INSPIRE].
I. Takyi, M.K. Matfunjwa and H. Weigel, Quantum corrections to solitons in the Φ8 model, Phys. Rev. D 102 (2020) 116004 [ar**v:2010.07182] [INSPIRE].
I. Takyi, B. Barnes and J. Ackora-Prah, Vacuum Polarization Energy of the Kinks in the Sinh-Deformed Models, Turk. J. Phys. 45 (2021) 194 [ar**v:2012.12343] [INSPIRE].
Y. Zhong, Normal modes for two-dimensional gravitating kinks, Phys. Lett. B 827 (2022) 136947 [ar**v:2112.08683] [INSPIRE].
Y. Zhong, Singular Pöschl-Teller II potentials and gravitating kinks, JHEP 09 (2022) 165 [ar**v:2207.12681] [INSPIRE].
M.P. Hertzberg, Quantum Radiation of Oscillons, Phys. Rev. D 82 (2010) 045022 [ar**v:1003.3459] [INSPIRE].
A. Kovtun, Analytical computation of quantum corrections to a nontopological soliton within the saddle-point approximation, Phys. Rev. D 105 (2022) 036011 [ar**v:2110.05222] [INSPIRE].
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Evslin, J., Liu, H. A reduced inner product for kink states. J. High Energ. Phys. 2023, 70 (2023). https://doi.org/10.1007/JHEP03(2023)070
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DOI: https://doi.org/10.1007/JHEP03(2023)070