Nonrelativistic Open String Model – Magnetic Monopole Mass and Lifetime Values

Candidate magnetic monopole string parameters are investigated using a nonrelativistic open string model with fixed endpoints. String parameters and lifetime values are derived as a function of the magnetic monopole mass. A wide variation in string parameter and lifetime values is predicted for the various monopole mass values utilized in this paper. The monopole mass lifetime values exceed 1022 y for the 10 to 1018 MeV/c2 mass range considered in this paper. 1.0 Introduction String theory is an elegant mathematical formulation1-7 that has yet to be experimentally verified. Specific particle parameter values and associated decay modes are uncertain and have been qualitatively discussed8-29. These uncertainties are exemplified by estimates of the monopole mass and lifetime values25. This paper applies the nonrelativistic open string model proposed in Refs. 28 and 29 to calculate a range of magnetic monopole string parameter and lifetime values as a function of assumed monopole mass values. Since the magnetic monopole mass values are uncertain25, a wide range of values, encompassing 10 to 1018 MeV/c2, is utilized in this paper25. This range of mass values is based on a variety of assumed monopole origins25. The magnitude of the monopole mass and associated lifetime values has implications for both particle physics and cosmology. Magnetic monopoles are predicted by a number of theories including string theory and various Grand Unification Theories. The discovery of magnetic monopoles would provide an importance benchmark for advancing various theories as well as the development or more comprehensive approaches including a better quantification of realistic Grand Unification Theories as well as the possible development of a Theory of Everything. In addition, monopole detection would open new research avenues in both particle physics and cosmology. Using Refs. 25 and 27-29 as a guide, this paper defines a model to calculate the monopole lifetime and associated string parameters as a function of magnetic monopole mass using the nonrelativistic open string model with fixed endpoints28,29. By constraining the model to reproduce a selected monopole mass, a set of parameters that provide an initial representation for the monopole string and associated lifetime Qeios, CC-BY 4.0 · Article, July 3, 2021 Qeios ID: ACMO7R · https://doi.org/10.32388/ACMO7R 1/18 are derived. Determination of these string parameters and lifetime values is fraught with obvious uncertainty. The present approach provides string parameters that establish an initial, but not definitive, set as the basis to explore in future work. As noted in Refs. 28 and 29, subsequent work will include a model string incorporating charge, electric and magnetic fields, multiple interacting strings including loops, various boundary conditions, interaction types, gauge theories, and symmetry conditions. The deviation in string parameters from the base case established in this paper will illuminate the dependence of the various parameters on specific string properties. 2.0 Nonrelativistic Open String Model Overview The model proposed in this paper assumes the production of cosmic strings following the big bang or during a big bang/crunch cycle of cosmic events. In this paper, it is assumed that particles result from the emission of the vibrational energy of the string. The fields associated with these particles can be derived from a number of symmetry classes. A simple example would be an Abelian-Higgs theory with a complex scalar field and a U(1) gauge field27-29. This class of fields is shown by Matsunami et al.27 to produce a string with a lifetime, defined in Section 6.0 that is proportional to the square of the string length. Following the Abelian-Higgs field theory with a U(1) gauge approach, the decay of strings into requisite particles occurs episodically with an associated energy loss. This energy loss is associated with the magnetic monopole mass In Ref. 28, a representative sample of string parameters for a set of baryons, leptons, and mesons was determined. This determination was based on specific mass and lifetime values for the set of selected particles that included the proton, neutron, and lambda baryons; electron, muon, and tau leptons; and charged pions and charged B mesons28. In Ref. 29, neutrino string parameters and lifetime values were determined in a similar manner. Since the magnetic monopole mass and lifetime values are uncertain25, these circumstances require a somewhat different approach than utilized in Ref. 28. The approach that is utilized is based on the approach of Ref. 29. Given these uncertainties, magnetic monopole masses are assumed to vary between 10 and 1018 MeV/c2 where this mass range is suggested in Ref. 25. For each assumed mass, string parameter and lifetime values are derived from the best three fits to the particle mass value. These parameter values and lifetimes are summarized in Table 1 – 5. 3.0 Model Parameter Specification The string model utilized in this paper is limited to nonrelativistic velocities. The energy of the string available for monopole decay is based on its total vibrational energy (kinetic plus potential energy). In this paper, assumed monopole mass values are utilized to calculate the associated lifetime and string parameter values. Key model parameters include the string density, which is related to the tension, and the length, amplitude, and velocity. Bounds on the string tension (S), derived from pulsar timing measurements 22-24, 27, are based on the gravitational wave background produced by decaying cosmic string loops. This bound, Qeios, CC-BY 4.0 · Article, July 3, 2021 Qeios ID: ACMO7R · https://doi.org/10.32388/ACMO7R 2/18 GS ≤ 10-11, is based on Newton’s gravitational constant (G) and is derived from simulations that ignore the field composition of the string. This would correspond to a string mass density of about 1.4x1017 kg/m. As a matter of comparison, a density of 1.4x1027 kg/m is derived from the Planck energy divided by the Planck length. Ref. 20 suggests that a string density of 1021 kg/m is an appropriate string density. These results imply that a range of density values are possible. Accordingly, the string density is permitted to vary over a range of values. Matsunami et al.27 suggest that particle radiation is associated with a string length that is < 10-19 m. Longer-lived particles that do not decay or that have extended lifetimes (e.g., protons and electrons) would be expected to have significantly longer string lengths. This assertion was also noted in Refs. 28 and 29. In addition, cosmological strings are expected to be mildly relativistic27. Ref. 27 utilizes values of 0.33 c and 0.6 c in their calculations. The model proposed in this paper28,29 uses a nonrelativistic approach and limits the string velocity to values less than used in Ref. 27 (i.e., β ≤ 0.05). These parameter values will be used as a guide and not a specific limitation in this paper. Reasonable variations will be considered in subsequent discussion. In particular, the density is permitted to vary between 107 and 1.4x1027 kg/m. The string length is permitted to vary within the 10-21 to 1046 m. As noted above, the string velocity is assumed to be nonrelativistic. Amplitude values are restricted to be less than the string length. 4.0 Base Case String Model Cosmic strings have extremely large masses that greatly exceed the values considered in this paper. The particle masses are assumed to be generated by the kinetic and potential energies of the vibrating string. The resulting particle mass does not depend on the total inclusive string mass. In this paper, the inherent string mass is treated as a renormalized vacuum or zero point energy with particles associated with the vibrational energy of the string. As a base case, a one-dimensional string of finite length and fixed endpoints is assumed. The model details are provided in Refs. 28 and 29 and only salient features will be addressed in this paper. 5.0 Magnetic Monopole Mass Assuming a uniform energy density over the string length, the energy (E) of a particle corresponding to the string vibrational energy density28,29 with total length L is

Specific particle parameter values and associated decay modes are uncertain and have been qualitatively discussed  . These uncertainties are exemplified by estimates of the monopole mass and lifetime values 25 . This paper applies the nonrelativistic open string model proposed in Refs. 28 and 29 to calculate a range of magnetic monopole string parameter and lifetime values as a function of assumed monopole mass values. Since the magnetic monopole mass values are uncertain 25 , a wide range of values, encompassing 10 to 10 18 MeV/c 2 , is utilized in this paper 25 . This range of mass values is based on a variety of assumed monopole origins 25 .
The magnitude of the monopole mass and associated lifetime values has implications for both particle physics and cosmology. Magnetic monopoles are predicted by a number of theories including string theory and various Grand Unification Theories. The discovery of magnetic monopoles would provide an importance benchmark for advancing various theories as well as the development or more comprehensive approaches including a better quantification of realistic Grand Unification Theories as well as the possible development of a Theory of Everything. In addition, monopole detection would open new research avenues in both particle physics and cosmology.
Using Refs. 25 and 27-29 as a guide, this paper defines a model to calculate the monopole lifetime and associated string parameters as a function of magnetic monopole mass using the nonrelativistic open string model with fixed endpoints 28,29 . By constraining the model to reproduce a selected monopole mass, a set of parameters that provide an initial representation for the monopole string and associated lifetime Qeios, CC-BY 4.0 · Article, July 3, 2021 Qeios ID: ACMO7R · https://doi.org/10.32388/ACMO7R 1/18 are derived.
Determination of these string parameters and lifetime values is fraught with obvious uncertainty. The present approach provides string parameters that establish an initial, but not definitive, set as the basis to explore in future work. As noted in Refs. 28 and 29, subsequent work will include a model string incorporating charge, electric and magnetic fields, multiple interacting strings including loops, various boundary conditions, interaction types, gauge theories, and symmetry conditions. The deviation in string parameters from the base case established in this paper will illuminate the dependence of the various parameters on specific string properties.

Nonrelativistic Open String Model Overview
The model proposed in this paper assumes the production of cosmic strings following the big bang or during a big bang/crunch cycle of cosmic events. In this paper, it is assumed that particles result from the emission of the vibrational energy of the string. The fields associated with these particles can be derived from a number of symmetry classes. A simple example would be an Abelian-Higgs theory with a complex scalar field and a U(1) gauge field [27][28][29] . This class of fields is shown by Matsunami et al. 27 to produce a string with a lifetime, defined in Section 6.0 that is proportional to the square of the string length.
Following the Abelian-Higgs field theory with a U(1) gauge approach, the decay of strings into requisite particles occurs episodically with an associated energy loss.

Model Parameter Specification
The string model utilized in this paper is limited to nonrelativistic velocities. The energy of the string available for monopole decay is based on its total vibrational energy (kinetic plus potential energy). In this paper, assumed monopole mass values are utilized to calculate the associated lifetime and string parameter values.
Key model parameters include the string density, which is related to the tension, and the length, amplitude, and velocity. Bounds on the string tension (S), derived from pulsar timing measurements [22][23][24]27 , are based on the gravitational wave background produced by decaying cosmic string loops. This bound, Matsunami et al. 27 suggest that particle radiation is associated with a string length that is < 10 -19 m.
Longer-lived particles that do not decay or that have extended lifetimes (e.g., protons and electrons) would be expected to have significantly longer string lengths. This assertion was also noted in Refs. 28 and 29. In addition, cosmological strings are expected to be mildly relativistic 27  These parameter values will be used as a guide and not a specific limitation in this paper. Reasonable variations will be considered in subsequent discussion. In particular, the density is permitted to vary between 10 7 and 1.4x10 27 kg/m. The string length is permitted to vary within the 10 -21 to 10 46 m. As noted above, the string velocity is assumed to be nonrelativistic. Amplitude values are restricted to be less than the string length.

Base Case String Model
Cosmic strings have extremely large masses that greatly exceed the values considered in this paper.
The particle masses are assumed to be generated by the kinetic and potential energies of the vibrating string. The resulting particle mass does not depend on the total inclusive string mass. In this paper, the inherent string mass is treated as a renormalized vacuum or zero point energy with particles associated with the vibrational energy of the string.
As a base case, a one-dimensional string of finite length and fixed endpoints is assumed. The model details are provided in Refs. 28 and 29 and only salient features will be addressed in this paper.

Magnetic Monopole Mass
Assuming a uniform energy density over the string length, the energy (E) of a particle corresponding to the string vibrational energy density 28,29 with total length L is where μ is the string mass per unit length, A is the amplitude, and ω is the angular frequency.
An application of Eq. 1 permits an estimate of the magnetic monopole's rest mass energy (ε). As noted in Refs. 28 and 29, Eq. 1 can be written as where λ = 2L based on a first harmonic assumption 28,29 , and v is the string velocity.

Magnetic Monopole Lifetime
Matsunami et al. 27 provide a relationship for the string lifetime (τ) where ξ is the number of episodes per period, and ε is the average energy lost per unit time which the model assumes to be the magnetic monopole rest mass energy. The string described in Section 4 is used as the basis for estimating the magnetic monopole lifetime.

Model Assumptions and Limitations
The magnetic monopole lifetime and associated string parameters are derived by assuming the following: 1. The model, defined in Sections 2 -4, specifies the string parameters that characterize the monopole.
2. One episode per period is assumed which is consistent with the fundamental mode assumption of Section 5.
3. The average energy lost per unit time (e.g., over a period) is the string kinetic plus potential energy.
Since the string is nonrelativistic, this is assumed to be the monopols's rest mass. The magnetic monopole lifetime is derived from the rest mass energy of the particle (ε) and is defined by Eqs. 2 and 3.
4 Only the string kinetic plus potential energy contributes to the monopole mass. The inherent string mass (ρL) is essentially a renormalizable constant (i.e., it is the vacuum or zero point energy), because the magnetic monopole energy is much smaller than this inherent mass.
5. The specific magnetic monopole decay modes and their associated decay products are not specified or considered.

Results and Discussion
The model results provide specific magnetic monopole string parameter and lifetime values as a function of mass. Model results suggest that long-lived magnetic monopole lifetime values are obtained for a wide range of string parameters. The string parameters (i.e., density, length, amplitude, and velocity) supporting these lifetime values are addressed, and their variation with monopole mass are discussed in subsequent commentary. Tables 1, 2, 3, 4, and 5 summarize, as a function of magnetic monopole mass, the monopole string density, length, amplitude, beta value, and lifetime values, respectively. The three best fits to the assumed monopole mass are provided in these tables.
Given the nature of the proposed calculations and associated uncertainties, a preliminary goal of fitting the particle masses and lifetimes to within 1% of their assumed values was set. This appears to be a reasonable criterion for the initial calculations.
In Tables 1 -5, the notation H (high), M (medium), and L (low) is used to label the columns of the three best parameter fits to the assumed magnetic monopole mass value. The parameter set yielding the largest

Magnetic Monopole Masses
The magnetic monopole masses summarized in Tables 1 -5 are limited to values between 10 and 10

String Density
As noted in Table 1, there is significant variation in the string density as a function of magnetic monopole mass for the L, M, and H Cases. In particular, the string density values reside within the range of 10 10 -10 27 kg/m. In view of this variation, definitive conclusions regarding the string density are not possible. Therefore, a more global analysis must be utilized. To facilitate a global analysis, an averaged logarithmic string parameter (ALSP) Ω(E) is defined by the relationship: where the averaged logarithmic string parameters are ALSμ for the string density, ALSL for the string length, ALSA for the string amplitude, and ALAτ for the string lifetime. The averaged string velocity (ASβ) is addressed in subsequent discussion.
The ALSμ for the string density is plotted as a function of magnetic monopole mass in Fig. 1. As expected, the ALSμ ( Fig. 1 dashed curve derived from the Table 1 data) still exhibits considerable variation, but it is less severe than the individual Case L, M, and H variations. The solid curve in Table 1  is also a long-lived particle with a lifetime range, noted in Table 5, which is similar to that of the proton and electron.

String Length
Following Ref. 27, the string length associated with the decay of unstable particles should be <10 -19 m.
As noted in previous discussion, this value provides an indication of an expected unstable particle string length, and the results of other open string nonrelativistic models may differ.
The monopole string length values summarized in Table 2 Table 2 further suggest a long-lived magnetic monopole.

String Amplitude
The monopole string amplitude summarized in Table 3 has a range between 10 -22 and 10 -4 m. As noted with the other string parameters, there is considerable variability in the amplitude values. This variability is reduced using the ALSA values.
Qeios, CC-BY 4.0 · Article, July 3, 2021 Qeios ID: ACMO7R · https://doi.org/10.32388/ACMO7R 9/18  The magnetic monopole amplitude is also larger in magnitude than the proton and electron values 28 . As noted in Reference 28, the proton and electron amplitude values are in the range of 10 -20 -10 -13 m and 10 -19 -10 -17 m, respectively. These results suggest that long-lived particles have larger amplitudes than short-lived particles 28 . This result continues to suggest that the magnetic monopole has a long lifetime.

String Velocity
The string velocity is restricted to β ≤ 0.05. In Reference 28, the baryon, lepton, and meson results suggest that there is no general velocity relationship between values of β and the particle mass or lifetime and associated string parameters. Similar results occur for the neutrino results 29 . There is also considerable scatter in the magnetic monopole string velocity values summarized in Table 4.   The Table 4 and Fig. 4 values are not clustered near the maximum β value (i.e., 0.05) that suggests that the model is favoring a nonrelativistic solution. This conclusion is model dependent and must be verified with a more refined approach including electromagnetic fields and other symmetry assumptions that were noted previously.

Particle Lifetime
Following Eq. 3 and the associated discussion, the particle lifetime values are strongly dependent on the string length, tension, and particle mass. The particle mass (Eq. 2) involves multiple parameters, but the lifetime (Eq. 3) only depends on a subset of these parameters.
The variation in lifetime values as a function of magnetic monopole mass is illustrated by an examination of Table 5. As summarized in Table 5, the monopole lifetime values vary significantly and range between about 10 22 and 10 66 y. In the spirit of the model assumptions and limitations, the results of  The ALSτ values are plotted in Fig. 5 (dashed curve) and exhibit considerable variation. In Fig. 5, the solid curve represents the linear fit to the ALSτ values τ(E) = alog 10 τ ALSτ (E) + b (10) where the parameters a = -0.90480678 y and b = 51.63738806 y. The linear fit provides a more stable set of lifetime values, but there is still a significant variation with mass.
Qeios, CC-BY 4.0 · Article, July 3, 2021 Qeios ID: ACMO7R · https://doi.org/10.32388/ACMO7R 14/18 The predicted magnetic monopole lifetime range is similar to that for the string model lifetimes for the proton and electron [28]. Nonrelativistic string model predictions for the proton (electron) lifetime are 10 37 -10 58 y (10 29 -10 59 y), respectively. The relative consistency of the string density, length, and amplitude values for the proton, electron, and neutrino further support a long-lived value for the magnetic monopole lifetime 28,29 .

Generalization to Closed String Models
Bagchi et al. 26 note that there is a natural emergence of an open string from a closed string given selected parameter limits. There is also a condensation of perturbative closed string modes to an open string. Reference 26 provides an important calculation that has the potential to generalize the open string model of this paper to closed string models.