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Kink fluctuation asymptotics and zero modes

  • Regular Article - Theoretical Physics
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Abstract

In this paper we propose a refinement of the heat-kernel/zeta function treatment of kink quantum fluctuations in scalar field theory, further analyzing the existence and implications of a zero-energy fluctuation mode. Improved understanding of the interplay between zero modes and the kink heat-kernel expansion delivers asymptotic estimations of one-loop kink mass shifts with remarkably higher precision than previously obtained by means of the standard Gilkey–DeWitt heat-kernel expansion.

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Notes

  1. We pass to use QFT terminology: the linear waves in the quantized theory are the light mesons, whereas the minima of U are the vacua of the system.

  2. The kinks are also static solutions of the PDE equation (3).

  3. We refer to this formula as the first DHN formula, to be distinguished from the “second” DHN formula applicable to the quantum sine-Gordon breathers.

  4. After all, it is an asymptotic series such that there is an optimum truncation order approximation to the exact value.

  5. In [19] it is explained that these models are well defined only for |ϕ|≤1.

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Acknowledgements

We warmly thank our collaborators in previous research of this topic, W. Garcia Fuertes, M. Gonzalez Leon and M. de la Torre Mayado, for illuminating conversations about different aspects of this subject.

We also gratefully acknowledge that this work has been partially financed by the Spanish Ministerio de Educacion y Ciencia (DGICYT) under grant: FIS2009-10546.

Author information

Authors and Affiliations

  1. Departamento de Matematica Aplicada and IUFFyM, Universidad de Salamanca, Salamanca, Spain

    A. Alonso Izquierdo

  2. Departamento de Fisica Fundamental and IUFFyM, Universidad de Salamanca, Salamanca, Spain

    J. Mateos Guilarte

Authors
  1. A. Alonso Izquierdo
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  2. J. Mateos Guilarte
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Corresponding author

Correspondence to A. Alonso Izquierdo.

Appendix

Appendix

In this appendix we display a Mathematica code which automatizes the computation of the quantum correction to the kink mass in one-component scalar field theory by applying formula (42), derived from the modified asymptotic series approach. The algorithm is divided into three subroutines: the identification of the density coefficients c n (x,x)=0 C n (x) by means of (34); the computation of the Seeley coefficients by integrating the density coefficients, finally evaluating the formula (42) to obtain an estimation of the kink mass quantum correction.

  • Calculation of the c n (x,x) densities.

    The following Mathematica code:

    densitycoefficients[potential_, vacuum1_, vacuum2_, nmax_] := Module[{var1, var2, var3, tomax, d1, v, v0, oper, f0, f6, x7, coef, k, coa, j, co}, (var1[ph1_] = potential /. {y -> ph1}; var2[ph1_] = Simplify[PowerExpand[Sqrt[2 var1[ph1]]]]; var3[ph1_] = Sign[var2[(vacuum1 + vacuum2)/2]] var2[ph1]; coef = {}; v[x_] = Simplify[(D[var1[ph1], {ph1, 2}]) /. {ph1 -> ph1[x]}]; v0 = Sqrt[Simplify[(D[var1[ph1], {ph1, 2}]) /. {ph1 -> vacuum1}]]; f0[x_ ] = (var3[ph1]/(Sqrt[Integrate[var3[ph1], {ph1, vacuum1, vacuum2}]])) /. {ph1 -> ph1[x]}; d1[fun_] := Simplify[(D[fun, x]) /. {ph1’[x] -> var3[ph1[x]]}]; oper[fu8_, n1_] := Simplify[Nest[f6, x7, n1] /. {f6 -> d1, x7 -> fu8}]; tomax = 2 nmax; For[k = 0, k < tomax + 0.5, coa[0, k] = 0; k++]; coa[0, 0] = 1; co[n_, k_] := Simplify[(1/(n + k)) (coa[n - 1, k + 2] - Sum[Binomial[k, r5] oper[v[x] - v0^2, r5] coa[n - 1, k - r5], {r5, 0, k}] - 2 v0 f0[x] oper[f0[x], k] KroneckerDelta[0, n - 1] - f0[x] oper[f0[x], k] (1 + 2 k) (2^n (v0)^(2 n - 1))/((2 n - 1)!!))]; For[j = 1, j < nmax + 0.5, tomax = tomax - 2; For[k = 0, k < tomax + 0.5, coa[j, k] = co[j, k]; If[k == 0, coef = Append[coef, coa[j, 0]]]; k++]; j++]; Return[coef])];

    defines the module densitycoefficients[potential_, vacuum1_, vacuum2_, nmax_], which performs the work of calculating the coefficients c n (x,x). The arguments of this computational function are potential, the U(y) potential written by prescription as a function of the y variable, vacuum1 and vacuum2, the two vacua connected by the kink solution in increasing order, and nmax, the N-order truncation chosen in the computation of ΔE(ϕ K ;N).

  • Calculation of the Seeley coefficients.

    The Mathematica module seeleycoefficients[potential_, vacuum1_, vacuum2_, nmax_] depends on the same arguments as the previous one:

    seeleycoefficients[potential_, vacuum1_, vacuum2_, nmax_] := Module[{coef, densi, f, f1, f2, a = {},j}, (coef = densitycoefficients[potential, vacuum1, vacuum2, nmax]; f1[y_] = Simplify[PowerExpand[Sqrt[2 potential]]]; f2[y_] = Sign[f1[(vacuum1 + vacuum2)/2]] f1[y]; For[j = 1, j < nmax + 0.5, f[y_] = Simplify[(coef[[j]] /. {ph1[x] -> y})/f2[y]]; a = Append[a, Integrate[f[y], {y, vacuum1, vacuum2}]]; j++]; Return[a])];

    and provides us with the value of the Seeley coefficients. This subroutine calls the previous function in order to accomplishes its task.

  • Estimation of the quantum correction.

    The subroutine quantumcorrection[potential_, vacuum1_, vacuum2_, nmax_],

    quantumcorrection[potential_, vacuum1_, vacuum2_, nmax_] := Module[{v0, corr, a}, (v0 = Sqrt[Simplify[(D[potential, {y, 2}]) /. {y -> vacuum1}]]; a = Chop[seeleycoefficients[potential, vacuum1, vacuum2, nmax]]; corr = -(v0/Pi) - (1/(8. Pi)) (Sum[(a[[n]] (v0^(-2 n + 2)) Gamma[n - 1]), {n, 2, nmax}]); Return[corr])];

    completes the work by providing us with the one-loop kink mass quantum correction in the (1+1) dimensional scalar field theory model characterized by the potential term U(y). This function calls the two previous ones.

The KinkMassQuantumCorrection_Modified.nb file containing this Mathematica code can be downloaded at the web page http://campus.usal.es/~mpg/General/Mathematicatools, which includes examples and demos. We recommend this option in order to avoid transcription errors in the code.

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Izquierdo, A.A., Guilarte, J.M. Kink fluctuation asymptotics and zero modes. Eur. Phys. J. C 72, 2170 (2012). https://doi.org/10.1140/epjc/s10052-012-2170-3

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