002947299 001__ 2947299
002947299 005__ 20251031040831.0
002947299 0248_ $$aoai:cds.cern.ch:2947299$$pcerncds:FULLTEXT$$pcerncds:CERN:FULLTEXT$$pcerncds:CERN
002947299 037__ $$9arXiv:reportnumber$$aCERN-TH-2025-211
002947299 037__ $$9arXiv:reportnumber$$aImperial-TP-2025-KG-1
002947299 037__ $$9arXiv$$aarXiv:2510.21922$$chep-th
002947299 035__ $$9arXiv$$aoai:arXiv.org:2510.21922
002947299 035__ $$9Inspire$$aoai:inspirehep.net:3073735$$d2025-10-30T00:01:17Z$$h2025-10-31T03:00:15Z$$mmarcxml$$ttrue$$uhttps://inspirehep.net/api/oai2d
002947299 035__ $$9Inspire$$a3073735
002947299 041__ $$aeng
002947299 100__ $$aElder, Benjamin$$tROR:https://ror.org/041kmwe10$$uImperial Coll., London$$vAbdus Salam Centre for Theoretical Physics, Imperial College London, London SW7 2AZ, United Kingdom
002947299 245__ $$9arXiv$$aConstrained instantons in scalar field theories
002947299 269__ $$c2025-10-24
002947299 300__ $$a30 p
002947299 500__ $$9arXiv$$a30 pages, 16 figures
002947299 520__ $$9arXiv$$aInstantons, localised saddle points of the action, play an important role in describing non-perturbative aspects of quantum field theories, for example vacuum decay or violation of conservation laws associated with anomalous symmetries. However, there are theories in which no saddle point exists. In this paper, we revisit the idea of constrained instantons, proposed initially by Affleck in 1981, and develop it into a complete method for computing the vacuum decay rate in such cases. We apply this approach to the massive scalar field theory with a negative quartic self-interaction using two different constraints. We solve the field equations numerically and find a two-branch structure, with two distinct solutions for each value of the constraint. By counting the negative modes, we identify one branch of solutions as the constrained instantons and the other as the minima of the action subject to the constraint. We discuss their significance for the computation of the vacuum decay rate.
002947299 541__ $$aarXiv$$chepcrawl$$d2025-10-28T04:01:08.343324$$e10870916
002947299 540__ $$3preprint$$aarXiv nonexclusive-distrib 1.0$$uhttp://arxiv.org/licenses/nonexclusive-distrib/1.0/
002947299 595__ $$aCERN-TH
002947299 65017 $$2arXiv$$anlin.PS
002947299 65017 $$2SzGeCERN$$aNonlinear Systems
002947299 65017 $$2arXiv$$amath.MP
002947299 65017 $$2SzGeCERN$$aMathematical Physics and Mathematics
002947299 65017 $$2arXiv$$amath-ph
002947299 65017 $$2SzGeCERN$$aMathematical Physics and Mathematics
002947299 65017 $$2arXiv$$ahep-th
002947299 65017 $$2SzGeCERN$$aParticle Physics - Theory
002947299 690C_ $$aCERN
002947299 690C_ $$aPREPRINT
002947299 700__ $$aGawrych, Kinga$$mk.gawrych22@imperial.ac.uk$$tROR:https://ror.org/041kmwe10$$uImperial Coll., London$$vAbdus Salam Centre for Theoretical Physics, Imperial College London, London SW7 2AZ, United Kingdom$$vTheoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
002947299 700__ $$aRajantie, Arttu$$tROR:https://ror.org/01ggx4157$$tROR:https://ror.org/041kmwe10$$uCERN$$uImperial Coll., London$$vAbdus Salam Centre for Theoretical Physics, Imperial College London, London SW7 2AZ, United Kingdom$$vTheoretical Physics Department, CERN, 1211 Geneva 23, Switzerland
002947299 8564_ $$82791555$$s20786$$uhttps://cds.cern.ch/record/2947299/files/phi6_nu_k_font_linear.png$$y00018 The projection prefactor $\nu$ as a function of $K_6\equiv (m^2/\lambda^2)\kappa$ for the $\phi^6$ constraint. The red dashed line marks $\kappa=\kappa_{\rm crit}$.
002947299 8564_ $$82791556$$s25351$$uhttps://cds.cern.ch/record/2947299/files/phi6_delta_S_rmax_10-6_font.png$$y00026 $\phi^6$ constraint. The difference between the value of the action computed at $R_{\min}=10^{-7}$ and $R_{\max}=50$ for $K_6 = 10^{-6}$, and the action computed for different values of $R_{\min}$ (left) and $R_{\max}$ (right).
002947299 8564_ $$82791557$$s24726$$uhttps://cds.cern.ch/record/2947299/files/phi6_xi_kappa_font.png$$y00016 Action and constraint values for the $\phi^6$ constraint. Left: The action as a function of $K_6\equiv (m^2/\lambda^2)\kappa$. The red line denotes the value of the corresponding massless instanton action. The solid black line indicates the constrained instantons, that contribute to the vacuum decay rate, while the grey dashed line indicates those solutions that do not. Right: The constraint as a function of $K_6$.
002947299 8564_ $$82791558$$s34622$$uhttps://cds.cern.ch/record/2947299/files/phi6_fits_k10-8_font.png$$y00012 Solutions for different values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint, together with the short-range massless instanton fits (dashed red) and long-range exponential fits (dashed green) (eq.~\eqref{eq:small-large-fits}).
002947299 8564_ $$82791559$$s27564$$uhttps://cds.cern.ch/record/2947299/files/phi6_delta_S_rmin_10-6_font.png$$y00025 $\phi^6$ constraint. The difference between the value of the action computed at $R_{\min}=10^{-7}$ and $R_{\max}=50$ for $K_6 = 10^{-6}$, and the action computed for different values of $R_{\min}$ (left) and $R_{\max}$ (right).
002947299 8564_ $$82791560$$s34462$$uhttps://cds.cern.ch/record/2947299/files/phi6_fits_k10-11_font.png$$y00011 Solutions for different values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint, together with the short-range massless instanton fits (dashed red) and long-range exponential fits (dashed green) (eq.~\eqref{eq:small-large-fits}).
002947299 8564_ $$82791561$$s22936$$uhttps://cds.cern.ch/record/2947299/files/phi3_S_kappa_font_sign.png$$y00005 Left: The action $S$ of our solutions as a function of $K_3\equiv \kappa/(m\lambda^{1/2})$ for the $\phi^3$ constraint. The dashed red line denotes the value of the massless instanton action \eqref{eq:mlessact}. The solid black line represents the solutions that contribute to the tunnelling rate, while the grey dashed line shows the ones that do not. Right: The constraint $\xi$ as a function of $K_3$ for the same solutions.
002947299 8564_ $$82791562$$s36571$$uhttps://cds.cern.ch/record/2947299/files/phi6_CIs_font.png$$y00010 Solutions for several values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint.
002947299 8564_ $$82791563$$s31925$$uhttps://cds.cern.ch/record/2947299/files/phi6_fits_kcrit_font.png$$y00013 Solutions for different values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint, together with the short-range massless instanton fits (dashed red) and long-range exponential fits (dashed green) (eq.~\eqref{eq:small-large-fits}).
002947299 8564_ $$82791564$$s30175$$uhttps://cds.cern.ch/record/2947299/files/phi3_fits_k-1_font.png$$y00004 Constrained instantons for the $\phi^3$ constraint at different values of $K_3\equiv \kappa/(m\lambda^{1/2})$. The red and green dashed lines show the short and long distance fits given by eq.~\eqref{eq:small-large-fits}
002947299 8564_ $$82791565$$s29537$$uhttps://cds.cern.ch/record/2947299/files/phi6_legendre_font_1.png$$y00020 A comparison between the absolute value of the derivative of the unconstrained action with respect to the Lagrange multiplier (black) and the calculated values of the constraint (red dashed). Left: $\phi^3$ constraint. Right: $\phi^6$ constraint.
002947299 8564_ $$82791566$$s24399$$uhttps://cds.cern.ch/record/2947299/files/phi3_S_xi_font.png$$y00007 The action as a function of the constraint $\bar\xi$ for the $\phi^3$ constraint. The black solid line corresponds to the solutions that contribute to the tunnelling rate. The dashed red line shows the massless instanton action.
002947299 8564_ $$82791567$$s32008$$uhttps://cds.cern.ch/record/2947299/files/phi3_fits_kcrit_font.png$$y00003 Constrained instantons for the $\phi^3$ constraint at different values of $K_3\equiv \kappa/(m\lambda^{1/2})$. The red and green dashed lines show the short and long distance fits given by eq.~\eqref{eq:small-large-fits}
002947299 8564_ $$82791568$$s24815$$uhttps://cds.cern.ch/record/2947299/files/phi3_xi_kappa_font_sign.png$$y00006 Left: The action $S$ of our solutions as a function of $K_3\equiv \kappa/(m\lambda^{1/2})$ for the $\phi^3$ constraint. The dashed red line denotes the value of the massless instanton action \eqref{eq:mlessact}. The solid black line represents the solutions that contribute to the tunnelling rate, while the grey dashed line shows the ones that do not. Right: The constraint $\xi$ as a function of $K_3$ for the same solutions.
002947299 8564_ $$82791569$$s33156$$uhttps://cds.cern.ch/record/2947299/files/phi6_potential_font.png$$y00009 Variation in the shape of the modified potential with the $\phi^6$ constraint for different values of $K_6\equiv (m^2/\lambda^2)\kappa$.
002947299 8564_ $$82791570$$s26053$$uhttps://cds.cern.ch/record/2947299/files/phi3_delta_S_rmin_1_font.png$$y00023 $\phi^3$ constraint. The difference between the value of the action computed at $R_{\min}=10^{-7}$ and $R_{\max}=50$ for $K_3 = -1$, and the action computed for different values of $R_{\min}$ (left) and different values of $R_{\max}$ (right).
002947299 8564_ $$82791571$$s34814$$uhttps://cds.cern.ch/record/2947299/files/phi3_fits_k-10-3_font.png$$y00002 Constrained instantons for the $\phi^3$ constraint at different values of $K_3\equiv \kappa/(m\lambda^{1/2})$. The red and green dashed lines show the short and long distance fits given by eq.~\eqref{eq:small-large-fits}
002947299 8564_ $$82791572$$s27262$$uhttps://cds.cern.ch/record/2947299/files/phi6_S_kappa_font.png$$y00015 Action and constraint values for the $\phi^6$ constraint. Left: The action as a function of $K_6\equiv (m^2/\lambda^2)\kappa$. The red line denotes the value of the corresponding massless instanton action. The solid black line indicates the constrained instantons, that contribute to the vacuum decay rate, while the grey dashed line indicates those solutions that do not. Right: The constraint as a function of $K_6$.
002947299 8564_ $$82791573$$s27186$$uhttps://cds.cern.ch/record/2947299/files/phi3_delta_S_rmax_1_font.png$$y00024 $\phi^3$ constraint. The difference between the value of the action computed at $R_{\min}=10^{-7}$ and $R_{\max}=50$ for $K_3 = -1$, and the action computed for different values of $R_{\min}$ (left) and different values of $R_{\max}$ (right).
002947299 8564_ $$82791574$$s29901$$uhttps://cds.cern.ch/record/2947299/files/phi6_S_xi_font_1.png$$y00017 Action as a function of the constraint value for the $\phi^6$ constraint. The red line represents the corresponding value of the massless instanton action. The black solid line corresponds to the configurations that contribute to the tunnelling rate, while the grey dashed line denotes those that do not.
002947299 8564_ $$82791575$$s29755$$uhttps://cds.cern.ch/record/2947299/files/phi3_CIs_font.png$$y00000 Solutions for the $\phi^3$ constraint for several values of $K_3\equiv \kappa/(m\lambda^{1/2})$.
002947299 8564_ $$82791576$$s24828$$uhttps://cds.cern.ch/record/2947299/files/phi3_delta_S_wp_1_font.png$$y00027 The difference between the action computed at a working precision of 40 digits and the action computed for different values of the working precision. Left: $\phi^3$ constraint, $K_3 = -1$. Right: $\phi^6$ constraint, $K_6 = 10^{-6}$.
002947299 8564_ $$82791577$$s18835$$uhttps://cds.cern.ch/record/2947299/files/phi6_integral_checks_font_1.png$$y00022 $\Delta I_1$ (blue) and $\Delta I_2$ (red) of eq.~\eqref{eq:integralchecks} as a function of $\kappa$. Left: $\phi^3$ constraint. Right: $\phi^6$ constraint.
002947299 8564_ $$82791578$$s31718$$uhttps://cds.cern.ch/record/2947299/files/phi6_fits_k10-4_font.png$$y00014 Solutions for different values of $K_6\equiv (m^2/\lambda^2)\kappa$ with the $\phi^6$ constraint, together with the short-range massless instanton fits (dashed red) and long-range exponential fits (dashed green) (eq.~\eqref{eq:small-large-fits}).
002947299 8564_ $$82791579$$s30729$$uhttps://cds.cern.ch/record/2947299/files/phi3_legendre_fonts_1.png$$y00019 A comparison between the absolute value of the derivative of the unconstrained action with respect to the Lagrange multiplier (black) and the calculated values of the constraint (red dashed). Left: $\phi^3$ constraint. Right: $\phi^6$ constraint.
002947299 8564_ $$82791580$$s20294$$uhttps://cds.cern.ch/record/2947299/files/phi3_integral_checks_font_1.png$$y00021 $\Delta I_1$ (blue) and $\Delta I_2$ (red) of eq.~\eqref{eq:integralchecks} as a function of $\kappa$. Left: $\phi^3$ constraint. Right: $\phi^6$ constraint.
002947299 8564_ $$82791581$$s25746$$uhttps://cds.cern.ch/record/2947299/files/phi6_delta_S_wp_10-6_font.png$$y00028 The difference between the action computed at a working precision of 40 digits and the action computed for different values of the working precision. Left: $\phi^3$ constraint, $K_3 = -1$. Right: $\phi^6$ constraint, $K_6 = 10^{-6}$.
002947299 8564_ $$82791582$$s23602$$uhttps://cds.cern.ch/record/2947299/files/phi3_nu_k_font_sign.png$$y00008 The projection prefactor $\nu$ as a function of $K_3\equiv \kappa/(m\lambda^{1/2})$ for the $\phi^3$ constraint. The red dashed line marks $\kappa = \kappa_\mathrm{crit}$.
002947299 8564_ $$82791583$$s36223$$uhttps://cds.cern.ch/record/2947299/files/phi3_fits_k-10-5_font.png$$y00001 Constrained instantons for the $\phi^3$ constraint at different values of $K_3\equiv \kappa/(m\lambda^{1/2})$. The red and green dashed lines show the short and long distance fits given by eq.~\eqref{eq:small-large-fits}
002947299 8564_ $$82791584$$s1298514$$uhttps://cds.cern.ch/record/2947299/files/2510.21922.pdf$$yFulltext
002947299 960__ $$a11
002947299 980__ $$aPREPRINT