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Left: The logarithm of the ratio of ¯ M − (t) 2 to ¯ m 2 (t) 2 is evaluated in the regions of β ∈ (0, π 2 ) and t ∈ (−6, 0). Right: The logarithm of the ratio of ¯ M + (t) 2 to ¯ m 2 (t) 2 is calculated in the regions of β ∈ (0, π
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We study the improvement of an effective potential by a renormalization group (RG) equation in a two real scalar system. We clarify the logarithmic structure of the effective potential in this model. Based on the analysis of the logarithmic structure of it, we find that the RG improved effective potential up to |$L$|th-to-leading log order can be c...
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Many different quantities can be described as 'dimensionless' and have the same SI unit "one", so their different natures cannot be determined from the unit name alone. In scientific unit systems like the SI, a set of base quantities and associated units can lead to such many-to-one relationships between quantities, dimensions, and units. So, it is...
Citations
... Thus, we compute the Coleman-Weinberg potential at the renormalization scale associated with the heavy field χ. By setting a field-dependent renormalization scale µ = µ H (|φ|,χ) we are able to study quantum corrections even in the limit of small W χχ [31,50]. The renormalized Coleman-Weinberg potential with the 1-loop contribution of field χ q is ...
We propose a general form of the UV-completed Friedberg-Lee-Sirlin (FLS) model. As can be seen from the mechanical interpretation, UV-completion allows thin-wall approximation for non-topological solitons. The 1-loop renormalized effective potential for the UV-completed FLS model is constructed under the assumption of mass hierarchy. By special choice of parameters, we studied how the Coleman-Weinberg mechanism induces a new mass scale $\omega_{-}$ in the classical FLS model. It can be expounded in terms of a stable condensate. As a consequence, the asymptotic of a non-topological soliton's energy at large charge is changed from $\sim Q^{3/4}$ to the linear law.
... This is because higher-order corrections make the minimum of the effective potential dependent on the flat direction of spontaneous symmetry breaking at tree level, leading to a preferred direction for the true minimum. It is worth noting that our approach to improving the effective potential is based on the summation of leading logarithms, which should not be confused with the RG improvement presented in Ref. [25]. In that work, the author considers μ 2 ¼ μ 2 0 e 2t and expresses the renormalized coupling constants as functions of t. ...
... Our approach, on the other hand, involves using the RG equation to derive a recurrence relation that enables us to calculate the leading higher-loop contributions to the effective potential, as originally described in Ref. [2]. This has led us to obtain a new result for the preferred flat direction compared to Ref. [25]. A promising avenue for future research would be to combine our method with the one used in [25] to achieve a better approximation of the effective potential. ...
... This has led us to obtain a new result for the preferred flat direction compared to Ref. [25]. A promising avenue for future research would be to combine our method with the one used in [25] to achieve a better approximation of the effective potential. ...
The Gildener-Weinberg models are of particular interest in the context of extensions to the Standard Model of particle physics. These extensions may encompass a variety of theories, including double Higgs models, grand unification theories, and proposals for dark matter, among others. In order to rigorously test these models experimentally, obtaining precise results is of crucial importance. In this study, we employ the renormalization group equation and its one-loop functions to obtain a deeper understanding of the higher-loop effective potential. Our findings reveal that the radiatively generated mass of the light particle in the Gildener-Weinberg approach experiences a substantial correction. Furthermore, our results suggest that not all flat directions are equivalent and some may be preferred by nature.
... This is because higher-order corrections make the minimum of the effective potential dependent on the flat direction of spontaneous symmetry breaking at the tree level, leading to a preferred direction for the true minimum. It is worth noting that our approach to improving the effective potential is based on the summation of leading logarithms, which should not be confused with the RG improvement presented in Ref. [31]. In that work, the author considers µ 2 = µ 2 0 e 2t and expresses the renormalized coupling constants as functions of t. ...
... Our approach, on the other hand, involves using the RG equation to derive a recurrence relation that enables us to calculate the leading higher-loop contributions to the effective potential, as originally described in Ref. [2]. This has led us to obtain a new result for the preferred flat direction compared to Ref. [31]. A promising avenue for future research would be to combine our method with the one used in [31] to achieve a better approximation of the effective potential. ...
... This has led us to obtain a new result for the preferred flat direction compared to Ref. [31]. A promising avenue for future research would be to combine our method with the one used in [31] to achieve a better approximation of the effective potential. ...
The Gildener-Weinberg models are of particular interest in the context of extensions to the Standard Model of particle physics. These extensions may encompass a variety of theories, including double Higgs models, Grand Unification Theories, and proposals for Dark Matter, among others. In order to rigorously test these models experimentally, obtaining precise results is of crucial importance. In this study, we employ the renormalization group equation and its one-loop functions to obtain a deeper understanding of the higher-loop effective potential. Our findings reveal that the radiatively generated mass of the light particle in the Gildener-Weinberg approach experiences a substantial correction. Furthermore, our results suggest that not all flat directions are equivalent and some may be preferred by nature.
... We show that the counter terms of S[ρ 1 , 0, 0] in Eq.(63) andΓ tree eff in Eq.(65) are determined so that the divergences are canceled. We replace the bare mass terms and the bare coupling constants with the renormalized ones using the relations from Eqs. (2)(3)(4)(5)(6)(7)(8). We also use the fact that there is no wave function renormalization for scalars from their one-loop diagrams. ...
The low energy effective potential for the model with a light scalar and a heavy scalar is derived. We perform the path integration for both heavy and light scalars and derive the low energy effective potential in terms of only the light scalar. The effective potential is independent of the renormalization scale approximately. By setting the renormalization scale equal to the mass of the heavy scalar, one finds the corrections with the logarithm of the ratio of the two scalar masses. The large logarithm is summed with the renormalization group (RG) and the RG improved effective potential is derived. The improved effective potential includes the one-loop correction of the heavy scalar and the leading logarithmic corrections due to the light scalar. We study the correction to the vacuum expectation value of the light scalar and the dependence on the mass of the heavy scalar.
... We checked that, when letting S c have a varying field value, we recover the full set of RGEs; see, e.g., Ref. [45]. ...
In many realizations of beyond the Standard Model theories, new massive particles are introduced, leading to a multiscale system with widely separated energy scales. In this setting the Coleman-Weinberg effective potential, which describes the vacuum of the theory at the quantum level, has to be supplemented with a prescription to handle the hierarchy in mass scales. In any quantum field theory involving scalar fields and multiple, highly differing mass scales, it is, in general, not possible to choose a single renormalization scale that will remove all the large logarithms in the effective potential. In this paper, we focus on the so-called decoupling method, which freezes the effects of heavy particles on the renormalization group running of the light degrees of freedom at low energies. We study this for a simple two-scalar theory and find that, while the decoupling method leads to an acceptable and convergent effective potential, the method does not solve the fine-tuning problem that is inherent to the hierarchy problem of multiscale theories. We also consider an alternative implementation of the decoupling approach, which gives different results for the shape of the potential, but still leads to similar conclusions on the amount of fine-tuning in the model. We suggest a way to avoid running into this fine-tuning problem by adopting a prescription on how to fix parameters in such decoupling approaches.
... We show that the counter terms of S[ρ 1 , 0, 0] in Eq.(63) andΓ tree eff in Eq.(65) are determined so that the divergences are canceled. We replace the bare mass terms and the bare coupling constants with the renormalized ones using the relations from Eqs. (2)(3)(4)(5)(6)(7)(8). We also use the fact that there is no wave function renormalization for scalars from their one-loop diagrams. ...
The low energy effective potential for the model with a light scalar and a heavy scalar is derived. We perform the path integration for both heavy and light scalars and derive the low energy effective potential in terms of only the light scalar. The effective potential is independent of the renormalization scale approximately. By setting the renormalization scale equal to the mass of the heavy scalar, one finds the corrections with the logarithm of the ratio of the two scalar masses. The large logarithm is summed with the renormalization group (RG) and the RG improved effective potential is derived. The improved effective potential includes the one-loop correction of the heavy scalar and the leading logarithmic corrections due to the light scalar. We discuss the implication of the corrections to the parameters of the mass squared dimension as well as the cosmological constants.
... This method was generalized in ref. [11], which applied it to the two-scalar case model discussed in section 4, and to the Higgs-Yukawa model. A recent attempt at applying the methods of ref. [9] to the two-scalar model is given in ref. [12]. In section 7.4 we compare our method in various examples to these earlier results. ...
A bstract
We apply effective field theory (EFT) methods to compute the renormalization group improved effective potential for theories with a large mass hierarchy. Our method allows one to compute the effective potential in a systematic expansion in powers of the mass ratio, as well as to sum large logarithms of mass ratios using renormalization group evolution. The effective potential is the sum of one-particle irreducible diagrams (1PI) but information about which diagrams are 1PI is lost after matching to the EFT, since heavy lines get shrunk to a point. We therefore introduce a tadpole condition in place of the 1PI condition, and use the renormalization group improved value of the tadpole in computing the effective potential. We explain why the effective potential computed using an EFT is not the same as the effective potential of the EFT. We illustrate our method using the O ( N ) model, a theory of two scalars in the unbroken and broken phases, and the Higgs-Yukawa model. Our leading-log result, obtained by integrating the one-loop β -functions, correctly reproduces the log-squared term in explicit two-loop calculations. Our method does not have a Goldstone boson infrared divergence problem.
... We checked that, when letting Sc have a varying field value, we recover the full set of RGEs, see e.g. ref.[45]. ...
In many realizations of beyond the Standard Model theories, new massive particles are introduced, leading to a multi-scale system with widely separated energy scales. In this setting the effective potential, which takes into account quantum corrections for the scalar sector, has to be supplemented with a prescription to handle the hierarchy in mass scales. In this paper, we focus on the so-called decoupling method, which freezes the effects of heavy particles on the RG running of the light degrees of freedom at low energies. For a two-scalar theory, we disentangle the effects of the high-energy degrees of freedom on the shape of the potential and on the fine-tuning of the model parameters. We find that, while the decoupling method leads to an acceptable and convergent effective potential, the method does not solve the fine-tuning problem that is inherent to the hierarchy problem of multi-scale theories. We also consider two alternative implementations of the decoupling, which give different results for the shape of the potential, but still lead to similar conclusions on the amount of fine-tuning in the model.
... This method was generalized in Ref. [11], which applied it to the two-scalar case model discussed in Sec. 4, and to the Higgs-Yukawa model. A recent attempt at applying the methods of Ref. [9] to the two-scalar model is given in Ref. [12]. In Sec. 7 we compare our method in various examples to these earlier results. ...
We apply effective field theory (EFT) methods to compute the renormalization group improved effective potential for theories with a large mass hierarchy. Our method allows one to compute the answer in a systematic expansion in powers of the mass ratio, as well as to sum the logarithms using renormalization group evolution. The effective potential is the sum of one-particle irreducible diagrams (1PI) but information about which diagrams are 1PI is lost after matching to the EFT, since heavy lines get shrunk to a point. We therefore introduce a tadpole condition in place of the 1PI condition. We also explain why the effective potential computed using an EFT is not the same as the effective potential of the EFT. We illustrate our method using the $O(N)$ model, a theory of two scalars in the unbroken and broken phases, and the Higgs-Yukawa model. Our leading-log result obtained by integrating the one-loop $\beta$-functions correctly reproduces the log-squared term in explicit two-loop calculations.
... where β ρ , β ρ , β κ are the beta functions, which are given in the one-loop level by [36]: ...
In the standard model, the weak scale is the only parameter with mass dimensions. This means that the standard model itself cannot explain the origin of the weak scale. On the other hand, from the results of recent accelerator experiments, except for some small corrections, the standard model has increased the possibility of being an effective theory up to the Planck scale. From these facts, it is naturally inferred that the weak scale is determined by some dynamics from the Planck scale. In order to answer this question, we rely on the multiple point criticality principle as a clue and consider the classically conformal $\mathbb{Z}_2\times \mathbb{Z}_2$ invariant two-scalar model as a minimal model in which the weak scale is generated dynamically from the Planck scale. This model contains only two real scalar fields and does not contain any fermions or gauge fields. In this model, due to a Coleman–Weinberg-like mechanism, the one-scalar field spontaneously breaks the $ \mathbb{Z}_2$ symmetry with a vacuum expectation value connected with the cutoff momentum. We investigate this using the one-loop effective potential, renormalization group and large-$N$ limit. We also investigate whether it is possible to reproduce the mass term and vacuum expectation value of the Higgs field by coupling this model with the standard model in the Higgs portal framework. In this case, the one-scalar field that does not break $\mathbb{Z}_2$ can be a candidate for dark matter and have a mass of about several TeV in appropriate parameters. On the other hand, the other scalar field breaks $\mathbb{Z}_2$ and has a mass of several tens of GeV. These results will be verifiable in near-future experiments.
