Higgs Boson Mass and Other Constants Estimated by Polymer Statistics Methods

The Higgs boson mass, the fine structure constant, the Weinberg angle, the critical exponents are estimated by polymer statistics methods. Friction forces have electroweak nature. New phase transitions by changing the even number of components of an ordering field, n, near the α relaxation transition point in melt-crystallized polymers is revealed. The transition from n = 0 to n = 2 is found. In frameworks of scaling theory of phase transitions and critical phenomena the results obtained are in a good agreement with experimental and theoretical data.


Introduction
About the α−relaxation transition the draw ratio at break, λ br , has a maximum [1,2] in copolymers of low pressure ethylene -acrylic acids with the small number of the latter comonomer (~0.1-0.3mol% [1,3]) as well as in linear polyethylene (PE) (see [4] and refs. in it). At this point the expression λ n 2 /λ br 2 = P(N) (obtained for room temperature) should be changed as formulated in [2]. Here, λ n is the neck draw ratio, P(N) = l a /l a cr is the probability of collision of chain ends, l a is the mean thickness of amorphous layers in isotropic material, l a cr is the same value at N = N cr , N = M w /µ is the polymerization degree, µ = 14kg/kmol is the molecular mass of repeated CH 2 group, lnN cr ≈ 15.89. Figure 1 shows the dependence of λ br on N at room [5] and elevated [4] temperatures. The theoretical curves λ br = (N/N cr ) (1−νd)/2 are also presented in Figure 1, ν is the critical exponent of correlation radius, d is the space dimension. It has been recently shown [6] that ν = 0.5(1 + t -1/t) where t = 1 + 2π (n + 2)/f, f = 137, n is the number of components of ordering field; it should be underlined that n may be only by the even number, otherwise, the gauge symmetry breaks (see Appendix). The theoretical curves in Figure 1 seem to confirm our assumption that near the α -relaxation transition the phase transition from n = 0 to n = 2 occurs. Here, we have used precise value for lnN cr ≈ 15.69 in comparison with [2].

Two Phase Model of Polymer Structure; An Estimation of Density Ratio
Let a polymer chain with the polymerization degree, N, walk over two lattices having a total boundary, both being the simple cubic. In that case z is the number of neighbour monomers surrounding a site of the crystalline lattice (z = 2d = 6 for three-dimensional space). If we will regard z* as the same parameter for the second amorphous lattice and introduce the probability p=z*/z< 1 then we can write the partition function, Z: where p can be defined as the probability discovering a monomer of the chain in the first phase while 1 -p is the same probability for the second phase. Assuming Z = z* we find From the solution of the equation (1) we obtain and Z = 3 + 3 1/2 ≈ 4.732: The other solution p -= 1/Z = 1 -p has no interest for subsequent considerations. Taking the logarithm of the expression (1) we obtain lnZ = lnp + lnz.
Multiplying the equation (2) by c/lnz where c is the mean concentration of chain monomers in the first phase we find clnZ /lnz = c lnp /lnz + c = C where C is a mean concentration. We suppose that C is the concentration of monomers of the chain in the second phase  Experimental data are obtained at room temperature [5] and at 75C [4], they are marked by circles and by squares, respectively; the theoretical curves λbr = exp(-0.5(νd -1)ln(N/Ncr)) correspond to Ncr≈exp(15.69) and ν from Table 2 for n = 0 (dash line) and n = 2 (solid line)  Thus, we can draw an important conclusion: for the two-phase model of polymer structure the density ratio of the most dense phase to the second phase is the constant which is equal to lnz/lnZ≈ 1.153 for three-dimensional space. It should be underlined that this relation is correct within the error which is less than 4% for a number of semicrystalline polymers. In Table 1 the experimental data [7] are presented for 12 polymers. The list may be expanded to a great extent.
The value of Z ≈ 4.732 is very close to that of 4.68 which is a characteristic of the polymer statistics for the excluded volume problem [8]. This enables us to solve our problems in frameworks of the scaling theory of critical phenomena. This is the second important conclusion obtained by help of the present model.

3D-exponents
In Table 2 the theoretical values of critical exponentsν, γ, β are presented. If n = 1 or n = 3 then such systems can be regarded as examples of materials when states with n = 0 and n = 2 (or n = 2 and n = 4) must have the same probability, while n = 6 seems to be realized in some percolation systems such as gels [8]. The experimental value of the exponent η is not determined exactly. The theoretical estimations [9,10] show its weak dependence on n. In this connection, we may attribute any reasonable meaning to η. Then for β and γ the following evaluations can be obtained by standard ways (γ + β = δβ) [9] if to suppose η= δ/f and to use the identity δ = -1 + 2d/(d -2 + η). Thus, η ≈ 0.0350, γ = νd(δ -1)/(δ + 1), β = νd/(δ + 1). These values are presented in Table 2. They are in a good accordance with recent theoretical data [10]. However, more accurate experimental checking could be made in future. If 1/f = 0 and d = 4 then we obtain the Landau exponents.
Thus, we can see that friction forces have electroweak nature. Firstly, λ n is expressed by help of the β-exponent and connected with the energy dissipation during drawing [2,3].
Secondly, the exponent η is dealing with a dynamic exponent [2,3,8] and characterized the viscosity of swelling polymer coil [2,3]. where m e ≈ 0.511MeV is the electron mass. The rest exponents are expressed in terms of the similarity laws [8][9][10]. Friction forces have electroweak nature.

Appendix
Following Edwards [11] a behaviour of polymer chain with the excluded volume can be described by help of function, ψ(r), which is the solution of the diffusion with the boundary condition ψ(r, r`) = δ (r -r`) a 3 at N = 0, a is the monomer diameter (the lattice constant); ϕ (r) is the potential connected with the excluded volume parameter, V, as follows Let ψ(r) be an eigen function of the operator Ĥ(r). Let us assume also that there exists an uniform field H with a vector potential A(r). While the symmetry of the operator Ĥ(r) can be broken due to the vector potential is not the periodic function of coordinates, a physical translation symmetry of the polymer system does not change, as the lattice over that the macromolecule makes random walks is periodic. Further we will use the well-known solution about the symmetry of electron states in the lattice with a magnetic field [12]. Let us choose the gauge for the vector potential of the uniform field: If the field h will be where f is a simple number, a 3 is one of the lattice periods, and v is the volume of the elementary unit (v = a 3 for the primitive cubic lattice which we will use for the sake of simplicity), then for the translations where i, j, k are the unit vectors of the orthogonal basis, here, m, n, and l are the whole numbers, g(L, L`) ≡ 1 in (A.2) and Ŝ L Ŝ L` = Ŝ L` Ŝ L . Returning to estimations of the critical exponents we have found [13] that the wave vector q gives the main contribution to the integral for receiving the pair correlation function of concentration fluctuation. The wave vector q has a very important meaning for polymer statistics, since it is connected with the fluctuation interactions of the chain ends, and, consequently, with the field h corresponding to the chain ends [14,8].
Let h designate h·k in (A.3) and let us choose N = exp [1/(ha 2 )]. Then N<N cr at h>h cr where N cr is the value of the polymerization degree at which the transition from a swelling coil to ideal (h → h cr > 0) can be revealed [13]. Where n is the number of components of an ordering field, f=137 to obtain the best agreement with the estimation ν ≈ 1 -6 -1/2 at n = 0 [3,13,15], t = 1 + 2π (n + 2)/f. We have used the expression ν − 0.5 ~ n + 2 by the Wilson ε-expansion [9,16]. Thus, n may be the even number only, otherwise, g(L, L`) can have the pure imaginary value in (A.2).
The formula (A.8) can be rewritten in other way supposing 2π (n + 2)/f by a small parameter