Solutions of the Renormalisation Group Equation in Minimal Supersymmetric Standard Model

Renormalisation Group Equation(RGE) for color and top couplings sector of MSSM has been solved. The mass of the top comes out to be 180.363$\pm$ 10.876 GeV and $\beta_{top}$=$\frac{\pi}{2}$. It is conjectured that the masses of the other 11 fermions and the CKM phase angle $\phi$ can be theoretically estimated. The results confirm the fact that the quarks and leptons have been created having equal mass $\sim$ 115 GeV at the MSSM GUT scale $\sim ~2.2\times 10^{16}$ GeV.

The Standard Model for Electroweak and Strong interaction has been proposed with many variations like Standard Model(SM), two Higgs Doublet Model(2HDM), Minimal Supersymmetric Standard Model(MSSM) and such others [1]. Most of them have been phenomenologically successful but this success has brought in relatively large number of free parameters [2]. Besides the gauge couplings of SU C (3), SU L (2) and U Y (1), in the matter sector there are Higgs and Yukawa couplings [2,3]. The aim of a good model building is not only to reduce the number of free parameters but should perturbatively tenable and obtain result in the agreement with experiment. To have an idea of the parameters, the vertex function of the model satisfy the renormalisation group equation [4], The notations are usually taken from reference [4]. To give an example of what we imply by reliable calculation is to note that the gauge coupling coefficients β gi satisfy the following equations in MSSM which we shall consider in this letter, where µ is the mass scale of the theory. c 1 =6.6, and c 2 =1. This means that the SU L (2) and U(1) Y groups are not asymptotically free but c 3 =−3 means that the color SU C (3) is asymptotically free. Therefore one can make a perturbative expansion in powers of g 3 . It is not certain that the other β-coefficients obtained from product groups are such that the theory is asymptotically free and one can find reliable perturbative results. There is sufficient evidence to show that the top quark mass coupling decreases as energy increases like g 3 . So from the entire parameter space of SM, we isolate a small region containing the top coupling and g 3 of the color group. The strategy is to see that if the experimentally acceptable results are obtained in this small region; we can then continue this region analytically to gradually encroach and cover the entire parameter space so that the perturbative result will be meaningful everywhere. This was also the basis of reduction of couplings technique of Zimmermann et al [4]. They developed a technique of reduction of coupling with one Higgs doublet and n generations and have shown that the cancellation of divergeces in the Higgs propagator is compatible with renormalisation group invariance and reduction of couplings. For three generations, they have predicted m top =81 GeV and m H =61 GeV with estimated error about 10-15%.
Pendleton and Ross [5] have exclusively extended the Kube, Siebold and Zimmermann's [6] work to include the other gauge coupling parameters g 1 , g 2 and g 3 in the same type of standard model. Their result, taking g 3 coupling alone, is also m top = 2 9 g 3 v=81 GeV. Theoretical prediction of the mass of the top, before it was discovered, have been made by several authors [7]. Faraggi has obtained m top ≃ 175 − 180 GeV in Superstring derived standard like model, but, in getting this result, he has taken m bottom mtop = 1 8 at the unification scale. We attempt here to extend earlier works to include the successful MSS Model into Yukawa coupling parameters calculations.
It will be helpful to state the gauge sector values [8] for later use. We shall take t X =log(M X /M Z )=33.0, 4π/g 2 U =24.6 and M S = M Z . The values of g 2 1 , g 2 2 , g 2 3 at m Z obtained from the R.G. equation above, are 4π/g 2 1 = 59.24, 4π/g 2 2 = 29.85 and 4π/g 2 3 =8.85. These are reasonable and consistent with the experimental results. In this letter, we are primarily concerned in the top mass within the top-color sector. In the renormalisation group equation for the top in MSSM, there is an added complication of having two Higgs. Therefore we write for the top [9] where v=174 GeV. Letting g 1 = g 2 = 0, m lepton = 0, m quark =0, except the top, the MSSM, RGE is and The top Yukawa and color sector equations, which have to be solved self consistently, become Following reference [4], we introduce ρ top (g 3 ) = . From equation (9), this satisfies the equation To simplify further, we set ρ top = Γ top v 2 top and using equation (10) and (11), we get Equating equation (11) to equation (14), we obtain The equation, exhibiting the pole, is On integration, C top is independent of g 3 . This is similar to or the general solution of Kubo et al [4] for g 1 = g 2 =0. C top may depend on the other gauge couplings g 1 ,g 2 at higher energies without invalidating equation (17). We solve the equation for the v top by noting that On simplification This can be deduced to be the same as the corresponding equation for m top of Faraggi [7] translated to our notation.
Integrating equation (19), 3 ) (20) Using v 2 top in m 2 top = M 2 top v 2 top , one obtains the solutions of (9) and (10) This can be confirmed by direct substitution. Observationally, one intends to know the value of m top (t) for a given mass m 1 , such that t 1 = log( m1 MZ ). The function in the square bracket, optimally stable like the equation (17), with respect to the variation of g 3 at this mass M 1 (t 1 ) or t=t 1 , should be These are the results from the explicitly known asymptotically free region. Experimentally, the top quark has a mass of 174.3± 5.1 [10] GeV and in excellent agreement with our result. Eventhough the extrapolation is too drastic, the decrease in the value of running tanβ as the mass changes fron M Z to M X , confirms the result which was first reported by Parida and Purkayastha [9]. The value of tangent hovers around one. So, to get a overall picture of the general nature of solutions of the MSSM R.G. Equations, it is fairly adequate to solve the equations for the fermion couplings M F like M top of equation (3).
We continue to expand the parameter space, retaining the top and bottom Yukawa couplings(M top ,M bottom ) and color gauge coupling g 3 . The equations are [7] We eliminate M bottom from r.h.s, by noting hence This is a very good result in view of the shrunk region of parameter space. As a step further, we expand the region of validity of perturbation to include SU(2) and U(1) i.e. g 2 = 0 and g 1 = 0. Then the equation for top becomes This equation (40) has been exactly solved by Deo and Maharana [8] and the result is K U 1 = 13/15, K U 2 = 3 and K U 3 = 16/3 are the coefficients of the coupling constant as given in references [8,11]. We found that the top mass originated from the unification mass m U ∼ 114 GeV.
This also confirms the perturbative 'stability' of this approach. Taking g 1 =0 and g 2 =0, and extending to large t-values, the value of m U of equation (31), changes by a few GeV only.
In reference [8], it has been shown that all the 12 fermions (quarks and leptons) at the GUT scale had the same mass of about 115 GeV. Encouraged by this, we now enlarge the region of applicability of the analysis for the whole parameter space. Essentially, there are 13 parameters, 12 fermion masses and precisely one CKM phase angle φ for the three generations. We now write the full renormalization equation for all the fermions, A F is a group theoretic factor whose value is '6' for quarks, i.e. for F =1,2,· · · ,6 and '4' for the leptons i.e. for F =7,8,· · · , 12. The positive values indicate that the field theory containing Yukawa couplings only, may not be asymptotically free. Y F is the mixing term which can be put in matrix form In MSSM, the matrix A F H is specified by the 144 elements given below [11,12], Then one can write an exact solution for M F (M Z ), as given in reference [12] The first term of equation (45) is A F M 3 F which is perturbatively very small except for the top. Therefore we neglect the 2nd term in (48). For fermions other than top, i.e. F varying from 2 to 12, where But, the main problem of finding general solutions, is to calculate I F . This integral is Next, we consider the equality, The integrand should have the poles like the ones in equation (16). The most solutions have M lepton =0 as the gauge factor G lepton does not contain the color gauge coupling g 3 . So,we shall let 'H' to be summed over from 1 to 6 in we have where n F in an integer. We have neglected 1 in the numerator and want to make the factor multiplying n F , independent of F by taking the average, using 1 12 G =1. Retracing back, we replace M F (t) = m U e iθF (t)m 2 U and find that so that I F = n F t X 1 12 This is an excellent result in spite of the approximate estimates. Starting from the unification mass m U =115 GeV and using the equation (59), we now calculate masses of all 11 fermions for different values of n F and identify them in the Table-I